Handbook of Quantitative Finance and Risk Management (v. 1-3)

Handbook of Quantitative Finance and Risk Management

Editors Cheng-Few Lee, Rutgers University, USA Alice C. Lee, State Street Corp., USA John Lee, Center for PBBEF Research, USA

Advisory Board Ivan Brick, Rutgers University, USA Stephen Brown, New York University, USA Charles Q. Cao, Penn State University, USA Chun-Yen Chang, National Chiao Tung University, Taiwan Wayne Ferson, Boston College, USA Lawrence R. Glosten, Columbia University, USA Martin J. Gruber, New York University, USA Hyley Huang, Wintek Corporation, Taiwan Richard E. Kihlstrom, University of Pennsylvania, USA E. H. Kim, University of Michigan, USA Robert McDonald, Northwestern University, USA Ehud I. Ronn, University of Texas at Austin, USA



Disclaimer: Any views or opinions presented in this publication are solely those of the authors and do not necessarily represent those of State Street Corporation. State Street Corporation is not associated in any way with this publication and accepts no liability for the contents of this publication.

Cheng-Few Lee  Alice C. Lee  John Lee Editors

Handbook of Quantitative Finance and Risk Management

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Editors Cheng-Few Lee Rutgers University Department of Finance and Economics 94 Rockafeller Road New Brunswick, NJ 08854-8054, Janice H. Levin Bldg. USA [email protected]

John Lee Center for PBBEF Research North Brunswick, NJ USA [email protected]

Alice C. Lee State Street Corp. Boston, MA USA [email protected]

ISBN 978-0-387-77116-8 e-ISBN 978-0-387-77117-5 DOI 10.1007/978-0-387-77117-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2010921816 c Springer Science+Business Media, LLC 2010  All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Quantitative finance and risk management is a combination of economics, accounting, statistics, econometrics, mathematics, stochastic process, and computer science and technology. This handbook is the most comprehensive handbook in quantitative finance and risk management, which integrates theory, methodology, and application. Due to the importance of quantitative finance and risk management in the finance industry, it has become one of the most popular subjects in business schools and departments of mathematics, operation research, and statistics. In addition, the finance industry has many job opportunities for people with good training in quantitative finance and risk management. Thus, a handbook should have a broad audience and be of interest to academics, educators, students, and practitioners. Based on our years of experience in industry, teaching, research, textbook writing, and journal editing on the subject of quantitative finance and risk management, this handbook will review, discuss, and integrate theoretical, methodological and practical issues of quantitative finance and risk management. This handbook is organized into five parts as follows: Part I. Overview of Quantitative Finance and Risk Management Research Part II. Portfolio Theory and Investment Analysis Part III. Options and Option Pricing Theory Part IV. Risk Management Part V. Theory, Methodology, and Applications Part I of this handbook covers three chapters: they are “Chapter 1. Theoretical Framework of Finance,” “Chapter 2. Investment, Dividend, Financing, and Production Policies,” and “Chapter 3. Research Methods of Quantitative Finance and Risk Management.” Part II of this handbook covers 18 chapters of portfolio theory and investment analysis. Part III of this handbook includes 21 chapters of options and option pricing theory. Part IV of this handbook includes 23 chapters of theory and practice in risk management. Finally, Part V of this handbook covers 44 chapters of theory, methodology, and applications in quantitative finance and risk management. In the preparation of this handbook, first, we would like to thank the members of advisory board and contributors of this handbook. In addition, we note and appreciate the extensive help from our Editor, Ms. Judith Pforr, our research assistants Hong-Yi Chen, Wei-Kang Shih and Shin-Ying Mai, and our secretary Ms. Miranda Mei-Lan Luo. Finally, we would like to thank the Wintek Corporation and the Polaris Financial Group for the financial support that allowed us to write this book. There are undoubtedly some errors in the finished product, both typographical and conceptual. We invite readers to send suggestions, comments, criticisms, and corrections to the author Professor Cheng-Few Lee at the Department of Finance and Economics, Rutgers University at Janice H. Levin Building Room 141, Rockefeller Road, Piscataway, NJ 08854-8054. New Brunswick, NJ Boston, MA North Brunswick, NJ

Cheng-Few Lee Alice C. Lee John Lee v

About the Editors

Cheng-Few Lee is Distinguished Professor of Finance at Rutgers Business School, Rutgers University and was chairperson of the Department of Finance from 1988 to 1995. He has also served on the faculty of the University of Illinois (IBE Professor of Finance) and the University of Georgia. He has maintained academic and consulting ties in Taiwan, Hong Kong, China, and the United States for the past three decades. He has been a consultant to many prominent groups, including the American Insurance Group, the World Bank, the United Nations, The Marmon Group Inc., Wintek Corporation, and Polaris Financial Group. Professor Lee founded the Review of Quantitative Finance and Accounting (RQFA) in 1990 and the Review of Pacific Basin Financial Markets and Policies (RPBFMP) in 1998, and serves as managing editor for both journals. He was also a co-editor of the Financial Review (1985– 1991) and the Quarterly Review of Economics and Business (1987–1989). In the past 36 years, Dr. Lee has written numerous textbooks ranging in subject matters from financial management to corporate finance, security analysis and portfolio management to financial analysis, planning and forecasting, and business statistics. In addition, he edited a popular book entitled Encyclopedia of Finance (with Alice C. Lee). Dr. Lee has also published more than 170 articles in more than 20 different journals in finance, accounting, economics, statistics, and management. Professor Lee was ranked the most published finance professor worldwide during the period 1953–2008. Professor Lee was the intellectual force behind the creation of the new Masters of Quantitative Finance program at Rutgers University. This program began in 2001 and has been ranked as one of the top ten quantitative finance programs in the United States. These top ten programs are located at Carnegie Mellon University, Columbia University, Cornell University, New York University, Princeton University, Rutgers University, Stanford University, University of California at Berkley, University of Chicago, and University of Michigan. Alice C. Lee is currently a Director in the Model Validation Group, Enterprise Risk Management, at State Street Corporation. Most recently, she was an Assistant Professor of Finance at San Francisco State University. She has more than 20 years of experience and has a diverse background, which includes academia, engineering, sales, and management consulting. Her primary areas of teaching and research are corporate finance and financial institutions. She is coauthor of Statistics for Business and Financial Economics, 2e (with Cheng F. Lee and John C. Lee) and Financial Analysis, Planning and Forecasting, 2e (with Cheng F. Lee and John C. Lee). In addition, she has co-edited other annual publications including Advances in Investment Analysis and Portfolio Management (with Cheng F. Lee). John C. Lee is a Microsoft Certified Professional in Microsoft Visual Basic and Microsoft Excel VBA. He has a bachelor and masters degree in accounting from the University of Illinois at Urbana-Champaign. John has more than 20 years’ experience in both the business and technical fields as an accountant, auditor, systems analyst, and as a business software developer. He has authored a book on how to use MINITAB and Microsoft Excel to do statistical analysis; this book is vii

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a companion text to Statistics of Business and Financial Economics, of which he is one of the co-authors. John has been a senior technology officer at the Chase Manhattan Bank and assistant vice president at Merrill Lynch. He is currently Director of the Center for PBBEF Research.

About the Editors

Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I

v

Overview of Quantitative Finance and Risk Management Research

1

Theoretical Framework of Finance : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Discounted Cash-Flow Valuation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 M and M Valuation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Markowitz Portfolio Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Arbitrage Pricing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Option Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Futures Valuation and Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 3 6 10 10 12 14 15 22 22

2

Investment, Dividend, Financing, and Production Policies: Theory and Implications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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2.1 2.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Investment and Dividend Interactions: The Internal Versus External Financing Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Interactions Between Dividend and Financing Policies . . . . . . . . . . . . . . . . . . 2.4 Interactions Between Financing and Investment Decisions . . . . . . . . . . . . . . . 2.5 Implications of Financing and Investment Interactions for Capital Budgeting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Implications of Different Policies on the Beta Coefficient . . . . . . . . . . . . . . . . 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2A Stochastic Dominance and its Applications to Capital-Structure Analysis with Default Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2A.2 Concepts and Theorems of Stochastic Dominance . . . . . . . . . . . . . . . 2A.3 Stochastic-Dominance Approach to Investigating the Capital-Structure Problem with Default Risk . . . . . . . . . . . . . . . . . . . 2A.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 25 28 30 34 36 36 38 38 38 39 40

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Contents

Research Methods in Quantitative Finance and Risk Management : : : : : : : : : : 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Econometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Other Disciplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part II

41 41 41 43 46 48 49 50

Portfolio Theory and Investment Analysis

4

Foundation of Portfolio Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5: 3 Cheng-Few Lee, Alice C. Lee, and John Lee 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Risk Classification and Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 Portfolio Analysis and Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.4 The Efficient Portfolio and Risk Diversification . . . . . . . . . . . . . . . . . . . . . . . . 60 4.5 Determination of Commercial Lending Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.6 The Market Rate of Return and Market Risk Premium . . . . . . . . . . . . . . . . . . . 66 4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5

Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng-Few Lee, Joseph E. Finnerty, and Hong-Yi Chen 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Measurement of Return and Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Utility Theory, Utility Functions, and Indifference Curves . . . . . . . . . . . . . . . 5.4 Efficient Portfolios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Capital Asset Pricing Model and Beta Forecasting : : : : : : : : : : : : : : : : : : : : : : : : Cheng-Few Lee, Joseph E. Finnerty, and Donald H. Wort 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 A Graphical Approach to the Derivation of the Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Mathematical Approach to the Derivation of the Capital Asset Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Market Model and Risk Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Growth Rates, Accounting Betas, and Variance in EBIT . . . . . . . . . . . . . . . . . 6.6 Some Applications and Implications of the Capital Asset Pricing Model . . . 6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 6A Empirical Evidence for the Risk-Return Relationship . . . . . . . . . . . . . Appendix 6B Anomalies in the Semi-strong Efficient-Market Hypothesis . . . . . . . . Index Models for Portfolio Selection : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng-Few Lee, Joseph E. Finnerty, and Donald H. Wort 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Single-Index Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Multiple Indexes and the Multiple-Index Model . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 69 69 71 77 91 91 93 93 93 96 97 100 104 105 105 106 109 111 111 111 118 121 122

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Appendix 7A A Linear-Programming Approach to Portfolio-Analysis Models . . . . 122 Appendix 7B Expected Return, Variance, and Covariance for a Multi-index Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Performance-Measure Approaches for Selecting Optimum Portfolios : : : : : : : : Cheng-Few Lee, Hong-Yi Chen, and Jessica Shin-Ying Mai 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Sharpe Performance-Measure Approach with Short Sales Allowed . . . . . . . . 8.3 Treynor-Measure Approach with Short Sales Allowed . . . . . . . . . . . . . . . . . . . 8.4 Treynor-Measure Approach with Short Sales Not Allowed . . . . . . . . . . . . . . . 8.5 Impact of Short Sales on Optimal-Weight Determination . . . . . . . . . . . . . . . . 8.6 Economic Rationale of the Treynor Performance-Measure Method . . . . . . . . 8.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 8A Derivation of Equation (8.6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 8B Derivation of Equation (8.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 8C Derivation of Equation (8.15) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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The Creation and Control of Speculative Bubbles in a Laboratory Setting : : : : James S. Ang, Dean Diavatopoulos, and Thomas V. Schwarz 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Bubbles in the Asset Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Portfolio Optimization Models and Mean–Variance Spanning Tests : : : : : : : : : : Wei-Peng Chen, Huimin Chung, Keng-Yu Ho, and Tsui-Ling Hsu 10.1 Introduction of Markowitz Portfolio-Selection Model . . . . . . . . . . . . . . . . . . . 10.2 Measurement of Return and Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Efficient Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Mean–Variance Spanning Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Alternative Computer Program to Calculate Efficient Frontier . . . . . . . . . . . . 10.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 Combining Fundamental Measures for Stock Selection : : : : : : : : : : : : : : : : : : : : Kenton K. Yee 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Bayesian Triangulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Triangulation in Forensic Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Bayesian Triangulation in Asset Pricing Settings . . . . . . . . . . . . . . . . . . . . . . . 11.5 The Data Snooping Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6 Using Guidance from Theory to Mitigate Data Snooping . . . . . . . . . . . . . . . . 11.7 Avoiding Data-Snooping Pitfalls in Financial Statement Analysis . . . . . . . . . 11.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 11A Proof of Theorem 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11A.1 Generalization of Theorem 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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125 125 128 130 132 132 133 133 133 134 135

137 139 140 145 161 163

165 166 166 172 175 182 184

185 187 189 190 194 195 197 199 200 201 201

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12 On Estimation Risk and Power Utility Portfolio Selection : : : : : : : : : : : : : : : : : : Robert R. Grauer and Frederick C. Shen 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3 The Multiperiod Investment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5 Alternative Ways of Estimating the Joint Return Distribution . . . . . . . . . . . . . 12.6 Alternate Ways of Evaluating Investment Performance . . . . . . . . . . . . . . . . . . 12.7 The Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.9 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13 International Portfolio Management: Theory and Method : : : : : : : : : : : : : : : : : : Wan-Jiun Paul Chiou and Cheng-Few Lee 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Overview of International Portfolio Management . . . . . . . . . . . . . . . . . . . . . . . 13.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 Forming the Optimal Global Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 The Benefits of International Diversification Around the World . . . . . . . . . . . 13.6 The Optimal Portfolio Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221

14 The Le Chatelier Principle in the Markowitz Quadratic Programming Investment Model: A Case of World Equity Fund Market : : : : : : : : : : : : : : : : : : Chin W. Yang, Ken Hung, and Jing Cui 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 The Le Châtelier Principle in the Markowitz Investment Model . . . . . . . . . . . 14.4 An Application of the Le Châtelier Principle in the World Equity Market . . . 14.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

203 203 205 206 206 208 210 216 217 218

221 222 226 226 227 229 232 233 235 235 236 236 237 245 245

15 Risk-Averse Portfolio Optimization via Stochastic Dominance Constraints : : : : Darinka Dentcheva and Andrzej Ruszczy´nski 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 The Portfolio Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Stochastic Dominance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 The Dominance-Constrained Portfolio Problem . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Optimality and Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

247

16 Portfolio Analysis : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Jack Clark Francis 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Inputs for Portfolio Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3 The Security Analyst’s Job . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4 Four Assumptions Underlying Portfolio Analysis . . . . . . . . . . . . . . . . . . . . . . . 16.5 Different Approaches to Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 A Portfolio’s Expected Return Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 The Quadratic Risk Formula for a Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8 The Covariance Between Returns from Two Assets . . . . . . . . . . . . . . . . . . . . .

259

247 248 249 252 254 256 257 257

259 259 259 260 260 261 261 262

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16.9 Portfolio Analysis of a Two-Asset Portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.10 Mathematical Portfolio Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.11 Calculus Minimization of Risk: A Three-Security Portfolio . . . . . . . . . . . . . . 16.12 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

262 265 265 266 266

17 Portfolio Theory, CAPM and Performance Measures : : : : : : : : : : : : : : : : : : : : : : Luis Ferruz, Fernando Gómez-Bezares, and María Vargas 17.1 Portfolio Theory and CAPM: Foundations and Current Application . . . . . . . 17.2 Performance Measures Related to Portfolio Theory and the CAPM: Classic Indices, Derivative Indices, and New Approaches . . . . . . . . . . . . . . . . . . . . . . . 17.3 Empirical Analysis: Performance Rankings and Performance Persistence . . . 17.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267

18 Intertemporal Equilibrium Models, Portfolio Theory and the Capital Asset Pricing Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Stephen J. Brown 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Intertemporal Equilibrium Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Relationship to Observed Security Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Intertemporal Equilibrium and the Capital Asset Pricing Model . . . . . . . . . . . 18.5 Hansen Jagannathan Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.6 Are Stochastic Discount Factors Positive? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

267 274 277 280 280 283 283 283 284 285 285 286 286 287

19 Persistence, Predictability, and Portfolio Planning : : : : : : : : : : : : : : : : : : : : : : : : Michael J. Brennan and Yihong Xia 19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Detecting and Exploiting Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.3 Stock Price Variation and Variation in the Expected Returns . . . . . . . . . . . . . . 19.4 Economic Significance of Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Forecasts of Equity Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 19A The Optimal Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 19B The Unconditional Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 19C The Myopic Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 19D The Optimal Buy-and-Hold Strategy . . . . . . . . . . . . . . . . . . . . . . . . . .

289

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies : : : : : Lan-chih Ho, John Cadle, and Michael Theobald 20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Theory of Alternative Portfolio Insurance Strategies . . . . . . . . . . . . . . . . . . . . 20.3 Empirical Comparison of Alternative Portfolio Insurance Strategies . . . . . . . 20.4 Recent Market Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.5 Implications for Financial Market Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

319

289 290 296 298 303 314 314 315 316 317 317

319 319 324 329 331 332 332

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Contents

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Reto Francioni, Sonali Hazarika, Martin Reck, and Robert A. Schwartz 21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Microstructure’s Challenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The Perfectly Liquid Environment of CAPM . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 What Microstructure Analysis Has to Offer: Personal Reflections . . . . . . . . . 21.5 From Theory to Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.6 Deutsche Börse: The Emergence of a Modern, Electronic Market . . . . . . . . . 21.7 Conclusion: The Roadmap and the Road . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 21A Risk Aversion and Risk Premium Measures . . . . . . . . . . . . . . . . . . . . 21A.1 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21A.2 Risk Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 21B Designing Xetra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21B.1 Continuous Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21B.2 Call Auction Trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21B.3 Electronic Trading for Less Liquid Stocks . . . . . . . . . . . . . . . . . . . . . . 21B.4 Xetra’s Implementation and the Migration of Liquidity to Xetra Since 1997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part III

333 333 334 335 339 344 345 347 347 349 349 349 350 350 351 351 352

Options and Option Pricing Theory

22 Options Strategies and Their Applications : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 355 :: Cheng Few Lee, John Lee, and Wei-Kang Shih 22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 22.2 The Option Market and Related Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 22.3 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 22.4 Risk-Return Characteristics of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 22.5 Examples of Alternative Option Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 22.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 23 Option Pricing Theory and Firm Valuation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng Few Lee, Joseph E. Finnerty, and Wei-Kang Shih 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Basic Concepts of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.3 Factors Affecting Option Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.4 Determining the Value of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.5 Option Pricing Theory and Capital Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.6 Warrants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Applications of the Binomial Distribution to Evaluate Call Options : : : : : : : : : : Alice C. Lee, John Lee, and Jessica Shin-Ying Mai 24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2 What Is an Option? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.3 The Simple Binomial Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4 The Generalized Binomial Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . 24.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

377 377 377 380 384 387 390 391 392 393 393 393 393 395 397 397

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25 Multinomial Option Pricing Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng Few Lee and Jack C. Lee 25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 Multinomial Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3 A Lattice Framework for Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 25A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Two Alternative Binomial Option Pricing Model Approaches to Derive Black-Scholes Option Pricing Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng-Few Lee and Carl Shu-Ming Lin 26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 The Two-State Option Pricing Model of Rendleman and Bartter . . . . . . . . . . 26.3 The Binomial Option Pricing Model of Cox, Ross, and Rubinstein . . . . . . . . 26.4 Comparison of the Two Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 26A The Binomial Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Normal, Lognormal Distribution and Option Pricing Model : : : : : : : : : : : : : : : : Cheng Few Lee, Jack C. Lee, and Alice C. Lee 27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.3 The Lognormal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.4 The Lognormal Distribution and Its Relationship to the Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.5 Multivariate Normal and Lognormal Distributions . . . . . . . . . . . . . . . . . . . . . . 27.6 The Normal Distribution as an Application to the Binomial and Poisson Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.7 Applications of the Lognormal Distribution in Option Pricing . . . . . . . . . . . . 27.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Bivariate Option Pricing Models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng Few Lee, Alice C. Lee, and John Lee 28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.2 The Bivariate Normal Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.3 American Call Option and the Bivariate Normal CDF . . . . . . . . . . . . . . . . . . . 28.4 Valuating American Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.5 Non-Dividend-Paying Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.6 Dividend-Paying Stocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Displaced Log Normal and Lognormal American Option Pricing: A Comparison : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Ren-Raw Chen and Cheng-Few Lee 29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.2 The American Option Pricing Model Under the Lognormal Process . . . . . . . 29.3 The Geske-Roll-Whaley Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 29A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

399 399 399 402 406 406 406 409 409 409 415 417 418 418 419 421 421 421 422 422 423 425 426 428 428 429 429 429 430 431 433 433 438 438 439 439 439 440 442 442 443

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30 Itô’s Calculus and the Derivation of the Black–Scholes Option-Pricing Model : George Chalamandaris and A.G. Malliaris 30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.2 The ITÔ Process and Financial Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.3 ITÔ’S Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.4 Stochastic Differential-Equation Approach to Stock-price Behavior . . . . . . . 30.5 The Pricing of an Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.6 A Reexamination of Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.7 Extending the Risk-Neutral Argument: The Martingale Approach . . . . . . . . . 30.8 Remarks on Option Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 30A An Alternative Method To Derive the Black–Scholes Option-Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.1 Assumptions and the Present Value of the Expected Terminal Option Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.2 Present Value of the Partial Expectation of the Terminal Stock Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A.3 Present Value of the Exercise Price under Uncertainty . . . . . . . . . . . . 31 Constant Elasticity of Variance Option Pricing Model: Integration and Detailed Derivation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Y.L. Hsu, T.I. Lin, and C.F. Lee 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The CEV Diffusion and Its Transition Probability Density Function . . . . . . . 31.3 Review of Noncentral Chi-Square Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 The Noncentral Chi-square Approach to Option Pricing Model . . . . . . . . . . . 31.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 31A Proof of Feller’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

447 447 447 451 452 454 455 458 463 465 465 466 466 467 469 471 471 471 473 474 478 478 478

32 Stochastic Volatility Option Pricing Models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Cheng Few Lee and Jack C. Lee 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Nonclosed-Form Type of Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Review of Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.4 Closed-Form Type of Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 32A The Market Price of the Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

481

33 Derivations and Applications of Greek Letters: Review and Integration : : : : : : Hong-Yi Chen, Cheng-Few Lee, and Weikang Shih 33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Delta () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.3 Theta .‚/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.4 Gamma ./ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.5 Vega ./ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.6 Rho ./ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.7 Derivation of Sensitivity for Stock Options Respective with Exercise Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.8 Relationship Between Delta, Theta, and Gamma . . . . . . . . . . . . . . . . . . . . . . . 33.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

491

481 481 485 485 489 489 489

491 491 494 496 498 500 501 502 503 503

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34 A Further Analysis of the Convergence Rates and Patterns of the Binomial Models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : San-Lin Chung and Pai-Ta Shih 34.1 Brief Review of the Binomial Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.2 The Importance of Node Positioning for Monotonic Convergence . . . . . . . . . 34.3 The Flexibility of GCRR Model for Node Positioning . . . . . . . . . . . . . . . . . . . 34.4 Numerical Results of Various GCRR Models . . . . . . . . . . . . . . . . . . . . . . . . . . 34.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 34A Extrapolation Formulas for Various GCRR Models . . . . . . . . . . . . . . 35 Estimating Implied Probabilities from Option Prices and the Underlying : : : : : Bruce Mizrach 35.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.2 Black Scholes Baseline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.3 Empirical Departures from Black Scholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.4 Beyond Black Scholes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.5 Histogram Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.6 Tree Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.7 Local Volatility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.8 PDF Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.9 Inferences from the Mixture Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.10 Jump Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35.11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Are Tails Fat Enough to Explain Smile : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Ren-Raw Chen, Oded Palmon, and John Wald 36.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.3 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.4 Data and Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 36A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.1 The Derivation of the Lognormal Model Under No Rebalancing . . . 36A.2 Continuous Rebalancing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.3 Smoothing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36A.4 Results of Sub-Sample Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Option Pricing and Hedging Performance Under Stochastic Volatility and Stochastic Interest Rates : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Gurdip Bakshi, Charles Cao, and Zhiwu Chen 37.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.2 The Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.3 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.4 Empirical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 37A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

505 505 506 507 507 510 513 513 515 515 516 517 518 518 520 522 522 524 526 528 528 531 531 532 533 537 541 541 542 542 543 543 544

547 547 549 556 557 571 571 572

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38 Application of the Characteristic Function in Financial Research : : : : : : : : : : : H.W. Chuang, Y.L. Hsu, and C.F. Lee 38.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.2 The Characteristic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.3 CEV Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.4 Options with Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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39 Asian Options : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Itzhak Venezia 39.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.2 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

583

40 Numerical Valuation of Asian Options with Higher Moments in the Underlying Distribution : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Kehluh Wang and Ming-Feng Hsu 40.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.2 Definitions and the Basic Binomial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.3 Edgeworth Binomial Model for Asian Option Valuation . . . . . . . . . . . . . . . . . 40.4 Upper Bound and Lower Bound for European Asian Options . . . . . . . . . . . . . 40.5 Upper Bound and Lower Bound for American Asian Options . . . . . . . . . . . . . 40.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 The Valuation of Uncertain Income Streams and the Pricing of Options : : : : : : Mark Rubinstein 41.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Uncertain Income Streams: General Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Uncertain Income Streams: Special Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 41A The Bivariate Normal Density Function . . . . . . . . . . . . . . . . . . . . . . . . 42 Binomial OPM, Black-Scholes OPM and Their Relationship: Decision Tree and Microsoft Excel Approach : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : John Lee 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Call and Put Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 One Period Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Two-Period Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Using Microsoft Excel to Create the Binomial Option Trees . . . . . . . . . . . . . . 42.6 Black-Scholes Option Pricing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.7 Relationship Between the Binomial OPM and the Black-Scholes OPM . . . . 42.8 Decision Tree Black-Scholes Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 42A Excel VBA Code: Binomial Option Pricing Model . . . . . . . . . . . . . .

575 575 576 577 581 581

583 584 586 586

587 587 588 589 591 593 594 602 602 605 605 606 608 611 613 613 614

617 617 617 618 621 622 624 625 626 626 627 627

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Part IV

Risk Management

43 Combinatorial Methods for Constructing Credit Risk Ratings : : : : : : : : : : : : : : : 639 :: Alexander Kogan and Miguel A. Lejeune 43.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 43.2 Logical Analysis of Data: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 641 43.3 Absolute Creditworthiness: Credit Risk Ratings of Financial Institutions . . . 643 43.4 Relative Creditworthiness: Country Risk Ratings . . . . . . . . . . . . . . . . . . . . . . . 648 43.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660 Appendix 43A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662 44 The Structural Approach to Modeling Credit Risk : : : : : : : : : : : : : : : : : : : : : : : : Jing-zhi Huang 44.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.2 Structural Credit Risk Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.3 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 An Empirical Investigation of the Rationales for Integrated Risk-Management Behavior : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Michael S. Pagano 45.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45.2 Theories of Risk-Management, Previous Research, and Testable Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45.3 Data, Sample Selection, and Empirical Methodology . . . . . . . . . . . . . . . . . . . . 45.4 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

665 665 665 668 671 671 675 675 677 685 689 694 694

46 Copula, Correlated Defaults, and Credit VaR : : : : : : : : : : : : : : : : : : : : : : : : : : : : Jow-Ran Chang and An-Chi Chen 46.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.3 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

697

47 Unspanned Stochastic Volatilities and Interest Rate Derivatives Pricing : : : : : : Feng Zhao 47.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47.2 Term Structure Models with Spanned Stochastic Volatility . . . . . . . . . . . . . . . 47.3 LIBOR Market Models with Stochastic Volatility and Jumps: Theory and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47.4 Nonparametric Estimation of the Forward Density . . . . . . . . . . . . . . . . . . . . . . 47.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 47A The Derivation for QTSMs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 47B The Implementation of the Kalman Filter . . . . . . . . . . . . . . . . . . . . . . Appendix 47C Derivation of the Characteristic Function . . . . . . . . . . . . . . . . . . . . . . .

713

697 698 703 710 711

713 716 723 734 746 746 748 750 751

xx

48 Catastrophic Losses and Alternative Risk Transfer Instruments : : : : : : : : : : : : : Jin-Ping Lee and Min-Teh Yu 48.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.2 Catastrophe Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.3 Catastrophe Equity Puts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.4 Catastrophe Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.5 Reinsurance with CAT-Linked Securities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 A Real Option Approach to the Comprehensive Analysis of Bank Consolidation Values : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Chuang-Chang Chang, Pei-Fang Hsieh, and Hung-Neng Lai 49.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.3 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 49A The Correlations Between the Standard Wiener Process Generated from a Bank’s Net Interest Income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 49B The Risk-Adjusted Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 49C The Discrete Version of the Risk-Adjusted Process . . . . . . . . . . . . . . 50 Dynamic Econometric Loss Model: A Default Study of US Subprime Markets C.H. Ted Hong 50.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.2 Model Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.3 Default Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.4 Prepayment Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.5 Delinquency Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 50A Default and Prepayment Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 50B General Model Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 50C Default Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 50D Prepayment Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 The Effect of Default Risk on Equity Liquidity: Evidence Based on the Panel Threshold Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Huimin Chung, Wei-Peng Chen, and Yu-Dan Chen 51.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 51A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Put Option Approach to Determine Bank Risk Premium : : : : : : : : : : : : : : : : : : : Dar Yeh Hwang, Fu-Shuen Shie, and Wei-Hsiung Wu 52.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Evaluating Insurer’s Liability by Option Pricing Model: Merton (1977) . . . . 52.3 Extensions of Merton (1977) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Applications for Merton (1977) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

753 753 753 757 760 763 764 766 767 767 768 771 775 777 777 778 778 778 779 779 780 782 792 797 800 802 802 803 803 805 807 807 808 812 815 815 816 819 819 820 820 823

Contents

xxi

52.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 52A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 52B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Keiretsu Style Main Bank Relationships, R&D Investment, Leverage, and Firm Value: Quantile Regression Approach : : : : : : : : : : : : : : : : : : : : : : : : : : Hai-Chin Yu, Chih-Sean Chen, and Der-Tzon Hsieh 53.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Data and Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.4 Empirical Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.5 Conclusions and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

825 826 826 827 829 829 831 831 836 840 841

54 On the Feasibility of Laddering : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Joshua Ronen and Bharat Sarath 54.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

843

55 Stock Returns, Extreme Values, and Conditional Skewed Distribution : : : : : : : Thomas C. Chiang and Jiandong Li 55.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.2 The AGARCH Model Based on the EGB2 Distribution . . . . . . . . . . . . . . . . . . 55.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.4 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.5 Distributional Fit Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.6 The Implication of the EGB2 Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

853

56 Capital Structure in Asia and CEO Entrenchment : : : : : : : : : : : : : : : : : : : : : : : : Kin Wai Lee and Gillian Hian Heng Yeo 56.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.2 Prior Research and Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.3 Data and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 56A Variables Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

863

57 A Generalized Model for Optimum Futures Hedge Ratio : : : : : : : : : : : : : : : : : : Cheng-Few Lee, Jang-Yi Lee, Kehluh Wang, and Yuan-Chung Sheu 57.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57.2 GIG and GH Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57.3 Futures Hedge Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57.4 Estimation and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 57A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

873

843 845 849 851 851

853 854 855 856 859 859 861 862

863 864 865 867 871 871 872

873 876 877 879 880 880 881

xxii

58 The Sensitivity of Corporate Bond Volatility to Macroeconomic Announcements : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Nikolay Kosturov and Duane Stock 58.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.2 Theory and Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.3 Data and Return Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.4 Descriptive Statistics of Daily Excess Returns . . . . . . . . . . . . . . . . . . . . . . . . . 58.5 OLS Regressions of Volatility and Excess Returns . . . . . . . . . . . . . . . . . . . . . . 58.6 Conditional Variance Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.7 Alternative GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 58A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Raw Material Convenience Yields and Business Cycle : : : : : : : : : : : : : : : : : : : : : Chang-Wen Duan and William T. Lin 59.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59.2 Characteristics of Study Commodities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59.5 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Alternative Methods to Determine Optimal Capital Structure: Theory and Application : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Sheng-Syan Chen, Cheng-Few Lee, and Han-Hsing Lee 60.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60.2 The Traditional Theory of Optimal Capital Structure . . . . . . . . . . . . . . . . . . . . 60.3 Optimal Capital Structure in the Contingent Claims Framework . . . . . . . . . . 60.4 Recent Development of Capital Structure Models . . . . . . . . . . . . . . . . . . . . . . 60.5 Application and Empirical Evidence of Capital Structure Models . . . . . . . . . 60.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Actuarial Mathematics and Its Applications in Quantitative Finance : : : : : : : : : Cho-Jieh Chen 61.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Actuarial Discount and Accumulation Functions . . . . . . . . . . . . . . . . . . . . . . . 61.3 Actuarial Mathematics of Insurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Actuarial Mathematics of Annuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Actuarial Premiums and Actuarial Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.6 Applications in Quantitative Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 The Prediction of Default with Outliers: Robust Logistic Regression : : : : : : : : : Chung-Hua Shen, Yi-Kai Chen, and Bor-Yi Huang 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Literature Review of Outliers in Conventional and in Logit Regression . . . . . 62.3 Five Validation Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Source of Data and Empirical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

883 883 884 886 886 897 899 903 910 912 913 915 915 917 919 921 922 930 931 933 933 934 936 941 948 950 950 953 953 953 955 958 959 961 963 963 965 965 966 967 969 969 973 976

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63 Term Structure of Default-Free and Defaultable Securities: Theory and Empirical Evidence : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 979 Hai Lin and Chunchi Wu 63.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979 63.2 Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 980 63.3 Bond Pricing in Dynamic Term Structure Model Framework . . . . . . . . . . . . . 980 63.4 Dynamic Term Structure Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 981 63.5 Models of Defaultable Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 988 63.6 Interest Rate and Credit Default Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996 63.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1001 64 Liquidity Risk and Arbitrage Pricing Theory : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1007 Umut Çetin, Robert A. Jarrow, and Philip Protter 64.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1007 64.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1009 64.3 The Extended First Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 64.4 The Extended Second Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 1012 64.5 Example (Extended Black–Scholes Economy) . . . . . . . . . . . . . . . . . . . . . . . . . 1015 64.6 Discontinuous Supply Curve Evolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016 64.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017 Appendix 64A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018 65 An Integrated Model of Debt Issuance, Refunding, and Maturity : : : : : : : : : : : : 1025 Manak C. Gupta and Alice C. Lee 65.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025 65.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026 65.3 Operationalizing the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029 65.4 Numerical Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032 65.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1036 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037 Part V

Theory, Methodology, and Applications

66 Business Models: Applications to Capital Budgeting, Equity Value, and Return Attribution : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1041 ::: Thomas S. Y. Ho and Sang Bin Lee 66.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1041 66.2 The Model Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1042 66.3 Simulation Results of the Capital Budgeting Decisions . . . . . . . . . . . . . . . . . . 1045 66.4 Relative Valuation of Equity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1048 66.5 Equity Return Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1050 66.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1051 Appendix 66A Derivation of the Risk Neutral Probability . . . . . . . . . . . . . . . . . . . . . . 1052 Appendix 66B The Model for the Fixed Operating Cost at Time T . . . . . . . . . . . . . . 1052 Appendix 66C The Valuation Model Using the Recombining Lattice . . . . . . . . . . . . 1053 Appendix 66D Input Data of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054

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67 Dividends Versus Reinvestments in Continuous Time: A More General Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1055 Ren-Raw Chen, Ben Logan, Oded Palmon, and Larry Shepp 67.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 67.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055 67.3 The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1057 67.4 Expected Bankruptcy Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1058 67.5 Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059 67.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1059 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1060 68 Segmenting Financial Services Market: An Empirical Study of Statistical and Non-parametric Methods : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1061 Kenneth Lawrence, Dinesh Pai, Ronald Klimberg, Stephen Kudbya, and Sheila Lawrence 68.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 68.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1062 68.3 Evaluating the Classification Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064 68.4 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 68.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 68.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066 69 Spurious Regression and Data Mining in Conditional Asset Pricing Models : : : 1067 Wayne Ferson, Sergei Sarkissian, and Timothy Simin 69.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067 69.2 Spurious Regression and Data Mining in Predictive Regressions . . . . . . . . . . 1068 69.3 Spurious Regression, Data Mining, and Conditional Asset Pricing . . . . . . . . . 1069 69.4 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 69.5 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1071 69.6 Results for Predictive Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073 69.7 Results for Conditional Asset Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . 1080 69.8 Solutions to the Problems of Spurious Regression and Data Mining . . . . . . . 1086 69.9 Robustness of the Asset Pricing Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1087 69.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1088 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1089 70 Issues Related to the Errors-in-Variables Problems in Asset Pricing Tests : : : : : 1091 Dongcheol Kim 70.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091 70.2 The Errors-in-Variables Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092 70.3 A Correction for the Errors-in-Variables Bias . . . . . . . . . . . . . . . . . . . . . . . . . . 1094 70.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1099 70.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1108 71 McMC Estimation of Multiscale Stochastic Volatility Models : : : : : : : : : : : : : : : 1109 German Molina, Chuan-Hsiang Han, and Jean-Pierre Fouque 71.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1109 71.2 Multiscale Modeling and McMC Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1110 71.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113 71.4 Empirical Application: FX Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113 71.5 Implication on Derivatives Pricing and Hedging . . . . . . . . . . . . . . . . . . . . . . . . 1118

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71.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119 Appendix 71A Proof of Independent Factor Equivalence . . . . . . . . . . . . . . . . . . . . . . 1119 Appendix 71B Full Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1120 72 Regime Shifts and the Term Structure of Interest Rates : : : : : : : : : : : : : : : : : : : : 1121 Chien-Chung Nieh, Shu Wu, and Yong Zeng 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1121 72.2 Regime-Switching and Short-Term Interest Rate . . . . . . . . . . . . . . . . . . . . . . . 1122 72.3 Regime-Switching Term Structure Models in Discreet Time . . . . . . . . . . . . . . 1126 72.4 Regime-Switching Term Structure Models in Continuous Time . . . . . . . . . . . 1128 72.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133 73 ARM Processes and Their Modeling and Forecasting Methodology : : : : : : : : : : 1135 Benjamin Melamed 73.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135 73.2 Overview of ARM Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136 73.3 The ARM Modeling Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1139 73.4 The ARM Forecasting Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1140 73.5 Example: ARM Modeling of an S&P 500 Time Series . . . . . . . . . . . . . . . . . . 1145 73.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1148 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1149 74 Alternative Econometric Methods for Information-based Equity-selling Mechanisms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1151 Lee Cheng-Few and Yi Lin Wu 74.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1151 74.2 The Information Contents of Equity-Selling Mechanisms . . . . . . . . . . . . . . . . 1152 74.3 Alternative Econometric Methods for Information-Based Equity-Selling Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153 74.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1161 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1162 75 Implementation Problems and Solutions in Stochastic Volatility Models of the Heston Type : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1165 Jia-Hau Guo and Mao-Wei Hung 75.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165 75.2 The Transform-Based Solution for Heston’s Stochastic Volatility Model . . . 1165 75.3 Solutions to the Discontinuity Problem of Heston’s Formula . . . . . . . . . . . . . 1168 75.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1170 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1171 76 Revisiting Volume vs. GARCH Effects Using Univariate and Bivariate GARCH Models: Evidence from U.S. Stock Markets : : : : : : : : : : : : : : : : : : : : : : 1173 Zhuo Qiao and Wing-Keung Wong 76.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173 76.2 The Mixture of Distribution Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175 76.3 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175 76.4 Empirical Findings in NYSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176 76.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1178 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1179 Appendix 76A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1180

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77 Application of Fuzzy Set Theory to Finance Research: Method and Application : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1183 Shin-Yun Wang and Cheng Few Lee 77.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1183 77.2 Fuzzy Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184 77.3 Applications of Fuzzy Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1190 77.4 A Example of Fuzzy Binomial OPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194 77.5 An Example of Real Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196 77.6 Fuzzy Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1197 77.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1198 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1199 78 Hedonic Regression Analysis in Real Estate Markets: A Primer : : : : : : : : : : : : : 1201 Ben J. Sopranzetti 78.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1201 78.2 The Theoretical Foundation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1201 78.3 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202 78.4 The Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1202 78.5 Empirical Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203 78.6 The Semi-Log Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204 78.7 The Box-Cox Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205 78.8 Problems with Hedonic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205 78.9 Recent Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1206 78.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207 79 Numerical Solutions of Financial Partial Differential Equations : : : : : : : : : : : : : 1209 Gang Nathan Dong 79.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209 79.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1209 79.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210 79.4 Finite Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1210 79.5 Finite Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217 79.6 Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1218 79.7 Empirical Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1219 79.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1220 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1220 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1221 80 A Primer on the Implicit Financing Assumptions of Traditional Capital Budgeting Approaches : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1223 Ivan E. Brick and Daniel G. Weaver 80.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223 80.2 Textbook Approaches to NPV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224 80.3 Theoretical Valuation of Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226 80.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228 80.5 Personal Tax and Miller Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229 80.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1231 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232 81 Determinants of Flows into U.S.-Based International Mutual Funds : : : : : : : : : 1235 Dilip K. Patro 81.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235 81.2 Motivation and Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236

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81.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237 81.4 Methodology and Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1238 81.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253 Appendix 81A Econometric Analysis of Panel Data . . . . . . . . . . . . . . . . . . . . . . . . . . 1253 82 Predicting Bond Yields Using Defensive Forecasting : : : : : : : : : : : : : : : : : : : : : : : 1257 Glenn Shafer and Samuel Ring 82.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1257 82.2 Game-Theoretic Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1260 82.3 Defensive Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265 82.4 Predicting Bond Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1269 82.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1271 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1271 83 Range Volatility Models and Their Applications in Finance : : : : : : : : : : : : : : : : : 1273 Ray Yeutien Chou, Hengchih Chou, and Nathan Liu 83.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1273 83.2 The Price Range Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274 83.3 The Range-Based Volatility Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1276 83.4 The Realized Range Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278 83.5 The Financial Applications and Limitations of the Range Volatility . . . . . . . . 1279 83.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1279 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1280 84 Examining the Impact of the U.S. IT Stock Market on Other IT Stock Markets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1283 Zhuo Qiao, Venus Khim-Sen Liew, and Wing-Keung Wong 84.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283 84.2 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284 84.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285 84.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289 Appendix 84A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1290 85 Application of Alternative ODE in Finance and Economics Research : : : : : : : : 1293 Cheng-Few Lee and Junmin Shi 85.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293 85.2 Ordinary Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294 85.3 Applications of ODE in Deterministic System . . . . . . . . . . . . . . . . . . . . . . . . . 1295 85.4 Applications of ODE in Stochastic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297 85.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1300 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1300 86 Application of Simultaneous Equation in Finance Research : : : : : : : : : : : : : : : : 1301 Carl R. Chen and Cheng Few Lee 86.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301 86.2 Two-Stage and Three-Stage Least Squares Method . . . . . . . . . . . . . . . . . . . . . 1302 86.3 Application of Simultaneous Equation in Finance Research . . . . . . . . . . . . . . 1305 86.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306

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87 The Fuzzy Set and Data Mining Applications in Accounting and Finance : : : : : 1307 Wikil Kwak, Yong Shi, and Cheng-Few Lee 87.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1307 87.2 A Fuzzy Approach to International Transfer Pricing . . . . . . . . . . . . . . . . . . . . 1307 87.3 A Fuzzy Set Approach to Human Resource Allocation of a CPA Firm . . . . . 1312 87.4 A Fuzzy Set Approach to Accounting Information System Selection . . . . . . . 1316 87.5 Fuzzy Set Formulation to Capital Budgeting . . . . . . . . . . . . . . . . . . . . . . . . . . . 1319 87.6 A Data Mining Approach to Firm Bankruptcy Predictions . . . . . . . . . . . . . . . 1324 87.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329 88 Forecasting S&P 100 Volatility: The Incremental Information Content of Implied Volatilities and High-Frequency Index Returns : : : : : : : : : : : : : : : : : 1333 Bevan J. Blair, Ser-Huang Poon, and Stephen J. Taylor 88.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333 88.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334 88.3 Methodology for Forecasting Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336 88.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1338 88.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1343 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344 89 Detecting Structural Instability in Financial Time Series : : : : : : : : : : : : : : : : : : : 1345 Derann Hsu 89.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345 89.2 Genesis of the Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1345 89.3 Problems of Multiple Change Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347 89.4 Here Came the GARCH and Its Brethrens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1348 89.5 Examples of Structural Shift Analysis in Financial Time Series . . . . . . . . . . . 1349 89.6 Implications of Structural Instability to Financial Theories and Practice . . . . 1352 89.7 Direction of Future Research and Developments . . . . . . . . . . . . . . . . . . . . . . . 1353 89.8 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354 90 The Instrument Variable Approach to Correct for Endogeneity in Finance : : : 1357 Chia-Jane Wang 90.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357 90.2 Endogeneity: The Statistical Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1358 90.3 Instrumental Variables Approach to Endogeneity . . . . . . . . . . . . . . . . . . . . . . . 1358 90.4 Validity of Instrumental Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1361 90.5 Identification and Inferences with Weak Instruments . . . . . . . . . . . . . . . . . . . . 1364 90.6 Empirical Applications in Corporate Finance . . . . . . . . . . . . . . . . . . . . . . . . . . 1366 90.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368 91 Bayesian Inference of Financial Models Using MCMC Algorithms : : : : : : : : : : 1371 Xianghua Liu, Liuling Li, and Hiroki Tsurumi 91.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1371 91.2 Bayesian Inference and MCMC Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 1371 91.3 CKLS Model with ARMA-GARCH Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374 91.4 Copula Model for FTSE100 and S&P500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1376 91.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1379 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1380

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92 On Capital Structure and Entry Deterrence : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1381 Fathali Firoozi and Donald Lien 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1381 92.2 The Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1382 92.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1384 92.4 Capital Structure and Entry Deterrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386 92.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1388 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1389 93 VAR Models: Estimation, Inferences, and Applications : : : : : : : : : : : : : : : : : : : : 1391 Yangru Wu and Xing Zhou 93.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 93.2 A Brief Discussion of VAR Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1391 93.3 Applications of VARs in Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1393 93.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397 94 Signaling Models and Product Market Games in Finance: Do We Know What We Know? : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1399 Kose John and Anant K. Sundaram 94.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1399 94.2 Supermodularity: Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1400 94.3 Supermodularity in Signaling Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1400 94.4 Supermodularity in Product Market Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403 94.5 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406 94.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407 95 Estimation of Short- and Long-Term VaR for Long-Memory Stochastic Volatility Models : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1409 Hwai-Chung Ho and Fang-I Liu 95.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409 95.2 Long Memory in Stochastic Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1410 95.3 VaR Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1411 95.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414 96 Time Series Modeling and Forecasting of the Volatilities of Asset Returns : : : : 1417 Tze Leung Lai and Haipeng Xing 96.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417 96.2 Conditional Heteroskedasticity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1417 96.3 Regime-Switching, Change-Point and Spline-GARCH Models of Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1421 96.4 Multivariate Volatility Models and Applications to Mean–Variance Portfolio Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1424 96.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425 97 Listing Effects and the Private Company Discount in Bank Acquisitions : : : : : 1427 Atul Gupta and Lalatendu Misra 97.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427 97.2 Why Acquiring Firms May Pay Less for Unlisted Targets . . . . . . . . . . . . . . . . 1428 97.3 Sample Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1430 97.4 Event Study Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1431 97.5 Findings Based on Multiples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433

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97.6 Cross-Sectional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1439 97.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1442 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443 98 An ODE Approach for the Expected Discounted Penalty at Ruin in Jump Diffusion Model (Reprint) : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1445 Yu-Ting Chen, Cheng-Few Lee, and Yuan-Chung Sheu 98.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445 98.2 Integro-Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1446 98.3 Explicit Formula for ˆ – ODE Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1448 98.4 The Constant Vector Q: Second Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1453 98.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458 Appendix 98A Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458 Appendix 98B Toolbox for Phase-Type Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 1462 Appendix 98C First Order Derivative of ˆ at Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1462 99 Alternative Models for Estimating the Cost of Equity Capital for Property/Casualty Insurers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1465 Alice C. Lee and J. David Cummins 99.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465 99.2 Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1466 99.3 Model-Specification and Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467 99.4 Data Description and Cost of Equity Capital Estimates . . . . . . . . . . . . . . . . . . 1470 99.5 Evaluations of Simulations and Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476 99.6 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1480 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1481 100 Implementing a Multifactor Term Structure Model : : : : : : : : : : : : : : : : : : : : : : : 1483 Ren-Raw Chen and Louis O. Scott 100.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1483 100.2 A Multifactor Term Structure Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1483 100.3 Pricing Options in the Multifactor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1485 100.4 Calibrating a Multifactor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487 100.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1488 101 Taking Positive Interest Rates Seriously : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1489 Enlin Pan and Liuren Wu 101.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1489 101.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1490 101.3 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1491 101.4 The Hump-Shaped Forward Rate Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1494 101.5 Fitting the US Treasury Yields and US Dollar Swap Rates . . . . . . . . . . . . . . . 1495 101.6 Extensions: Jumps in Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1498 101.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1500 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1500 Appendix 101A Factor Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1501 Appendix 101B Extended Kalman Filter and Quasilikelihood . . . . . . . . . . . . . . . . . . 1502

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102 Positive Interest Rates and Yields: Additional Serious Considerations : : : : : : : : 1503 Jonathan Ingersoll 102.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503 102.2 A Non-Zero Bound for Interest Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1503 102.3 The Cox–Ingersoll–Ross and Pan–Wu Term Structure Models . . . . . . . . . . . . 1504 102.4 Bubble-Free Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1506 102.5 Multivariate Affine Term-Structure Models with Zero Bounds on Yields . . . 1511 102.6 Non-Affine Term Structures with Yields Bounded at Zero . . . . . . . . . . . . . . . . 1514 102.7 Non-Zero Bounds for Yields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516 102.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 Appendix 102A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 102A.1 Derivation of the Probability and State price for rT D 0 for the PW Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517 102A.2 Bond Price When rt D 0 Is Accessible for Only the Risk-Neutral Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1519 102A.3 Properties of the Affine Exponentially Smoothed Average Model . . 1520 102A.4 Properties of the Three-Halves Power Interest Rate Process . . . . . . . 1521 103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1523 Cheng-Few Lee, Dilip K. Patro, and Bo Liu 103.1 Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1523 103.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1524 103.3 Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1526 103.4 Data and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1527 103.5 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534 103.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1545 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553 104 A Semimartingale BSDE Related to the Minimal Entropy Martingale Measure : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1555 Michael Mania, Marina Santacroce, and Revaz Tevzadze 104.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555 104.2 Some Basic Definitions, Conditions, and Auxiliary Facts . . . . . . . . . . . . . . . . 1556 104.3 Backward Semimartingale Equation for the Value Process . . . . . . . . . . . . . . . 1558 104.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1564 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565 105 The Density Process of the Minimal Entropy Martingale Measure in a Stochastic Volatility Model with Jumps (Reprint) : : : : : : : : : : : : : : : : : : : : : 1567 Fred Espen Benth and Thilo Meyer-Brandis 105.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1567 105.2 The Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1568 105.3 The Minimal Entropy Martingale Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1569 105.4 The Density Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1571 105.5 The Entropy Price of Derivatives and Integro-Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1573 105.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1574 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1575

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106 Arbitrage Detection from Stock Data: An Empirical Study : : : : : : : : : : : : : : : : : 1577 Cheng-Der Fuh and Szu-Yu Pai 106.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577 106.2 Arbitrage Detection: Volatility Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579 106.3 Arbitrage Detection: Mean Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583 106.4 Empirical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1586 106.5 Conclusions and Further Researches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1590 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1591 107 Detecting Corporate Failure : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1593 Yanzhi Wang, Lin Lin, Hsien-Chang Kuo, and Jenifer Piesse 107.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593 107.2 The Possible Causes of Bankruptcy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594 107.3 The Methods of Bankruptcy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1594 107.4 Prediction Model for Corporate Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1596 107.5 The Selection of Optimal Cutoff Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1603 107.6 Recent Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604 107.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604 108 Genetic Programming for Option Pricing : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1607 N.K. Chidambaran 108.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1607 108.2 Genetic Program Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608 108.3 Black–Scholes Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1611 108.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613 108.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1614 109 A Constant Elasticity of Variance (CEV) Family of Stock Price Distributions in Option Pricing, Review, and Integration : : : : : : : : : : : : : : 1615 Ren-Raw Chen and Cheng-Few Lee 109.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1615 109.2 The CEV Diffusion and Its Transition Density . . . . . . . . . . . . . . . . . . . . . . . . . 1616 109.3 The CEV Option Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1619 109.4 Computing the Non-Central Chi-Square Probabilities . . . . . . . . . . . . . . . . . . . 1622 109.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1623 Appendix 109A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1623 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1625 References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1627 Author Index : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1685 Subject Index : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1709

Contents

List of Contributors

James S. Ang, Florida State University, Tallahassee, FL, USA Gurdip Bakshi, University of Maryland, College Park, MD, USA Hamid Beladi, University of Texas at San Antonio, San Antonio, TX, USA Fred Espen Benth, University of Oslo and Agder University College, Kristiansand, Norway Bevan J. Blair, Ingenious Asset Management, London, UK Michael J. Brennan, University of California at Los Angeles, Los Angeles, CA, USA Ivan Brick, Rutgers University, Newark, NJ, USA Stephen J. Brown, New York University, New York, NY, USA John Cadle, University of Birmingham, Birmingham, UK Charles Cao, Department of Finance, Smeal College of Business, Pennsylvania State University, University Park, PA, USA Umut Çetin, Technische Universität Wien, Vienna, Austria George Chalamandaris, Athens University of Economics and Business, Athens, Greece Chuang-Chang Chang, National Central University, Taipei, Taiwan, ROC Jow-Ran Chang, National Tsing Hua University, Hsinchu, Taiwan, ROC An-Chi Chen, KGI Securities Co. Ltd., Taipei, Taiwan, ROC Carl R. Chen, University of Dayton, Dayton, OH, USA Chih-Sean Chen, Chung Yuan University, Taoyuan County, Taiwan, ROC Cho-Jieh Chen, University of Alberta, Edmonton, AB, Canada Hong-Yi Chen, Rutgers University, Newark, NJ, USA Ren-Raw Chen, Fordham University, New York, NY, USA Sheng-Syan Chen, National Taiwan University, Taipei, Taiwan, ROC Wei-Peng Chen, Shih Hsin University, Taipei, Taiwan, ROC Yi-Kai Chen, National University of Kaohsiung, Kaohsiung, Taiwan, ROC Yu-Dan Chen, National Chiao Tung University, Hsinchu, Taiwan, ROC Yu-Ting Chen, National Chao Tung University, Hsinchu, Taiwan, ROC Zhiwu Chen, Yale University, New Haven, CT, USA

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Thomas C. Chiang, Drexel University, Philadelphia, PA, USA N. K. Chidambaran, Fordham University, New York, NY, USA Wan-Jiun Paul Chiou, Shippensburg University, Shippensburg, PA, USA Heng-chih Chou, Ming Chuan University, Taipei, Taiwan, ROC Ray Y. Chou, Academia Sinica, Taipei, Taiwan, ROC H.W. Chiang, National Taiwan University, Taipei, Taiwan, ROC Huimin Cheng, National Chiao Tung University, Hsinchu, Taiwan, ROC San-Lin Cheng, National Taiwan University, Taipei, Taiwan, ROC Jing Cui, Clarion University of Pennsylvania, Clarion, PA, USA J. D. Cumming, Temple University, Philadelphia, PA, USA Darinka Dentcheva, Stevens Institute of Technology, Hoboken, NJ, USA Dean Diavatopoulos, Villanova University, Philadelphia, PA, USA Gang Nathan Dong, Rutgers University, Newark, NJ, USA Chang-Wen Duan, Tamkang University, Taipei, Taiwan, ROC Luis Ferruz, University of Zaragoza, Zaragoza, Spain Wayne Fresón, University of Southern California, Los Angeles, CA, USA Joseph E. Finnerty, University of Illinois at Urbana-Champaign, Champaign, IL, USA Fathali Firoozi, University of Texas at San Antonio, San Antonio, TX, USA Jean-Pierre Fouque, University of California, Santa Barbara, CA, USA Reto Francioni, Deutsche Börse, Frankfurt, Germany Jack Clark Francis, Baruch College, New York, NY, USA Cheng-Der Fuh, National Central University and Academia Sinica, Taipei, Taiwan, ROC Fernando Gómez-Bezares, University of Deusto, Bilbao, Spain Robert R. Grauer, Simon Fraser University, Burnaby, BC, Canada Jia-Hau Guo, Soochow University, Taipei, Taiwan, ROC Atul Gupta, Bentley University, Waltham, MA, USA Manak C. Gupta, Temple University, Philadelphia, PA, USA Chuan-Hsiang Han, National Tsing Hua University, Hsinchu, Taiwan, ROC Sonali Hazarika, Baruch College, New York, NY, USA Hwai-Chung Ho, Academia Sinica and National Taiwan University, Taipei, Taiwan, ROC Keng-Yu Ho, National Taiwan University, Taipei, Taiwan, ROC Lan-chih Ho, Central Bank of the Republic of China, Taipei, Taiwan, ROC Thomas S. Y. Ho, Thomas Ho Company, Ltd, New York, NY, USA C.H. Ted Hong, Beyondbond Inc., New York, NY, USA Tsui-Ling Hseu, National Chiao Tung University, Hsinchu, Taiwan, ROC Der-Tzon Hsieh, National Taiwan University, Taipei, Taiwan, ROC

List of Contributors

List of Contributors

xxxv

Pei-Fang Hsieh, Department of Finance, National Central University, Chung Li City, Taiwan, ROC Derann Hsu, University of Wisconsin–Milwaukee, Milwaukee, WI, USA Ming-Feng Hsu, Tatung University, Taipei, Taiwan, ROC Ying Lin Hsu, National Chung Hsing University, Taichung, Taiwan, ROC Bor-Yi Huang, Department of Business Management, Shih Chien University, Taipei, Taiwan, ROC Dar-Yeh Huang, National Taiwan University, Taipei, Taiwan, ROC Jingzhi Huang, Pennsylvania State University, University Park, PA, USA Ken Hung, Texas A&M International University, Laredo, TX, USA Mao-Wei Hung, National Taiwan University, Taipei, Taiwan, ROC Jonathan E. Ingersoll, Jr., Yale School of Management, New Haven, CT, USA Robert A. Jarrow, Cornell University, Ithaca, NY, USA Kose John, New York University, New York, NY, USA Dongcheol Kim, Korea University Business School, Seoul, Korea Ronald Klimberg, St. Joseph’s University, Philadelphia, PA, USA Alexander Kogan, Rutgers University, Newark, NJ, USA Nikolay Kosturov, University of Oklahoma, Norman, OK, USA Stephen Kudbya, New Jersey Institute of Technology, Newark, NJ, USA Hsien-chang Kuo, National Chi-Nan University and Takming University of Science and Technology, Nantou Hsien, Taiwan, ROC Wikil Kwak, University of Nebraska at Omaha, Omaha, NE, USA Hung-Neng Lai, Department of Finance, National Central University, Chung Li City, Taiwan, ROC Tze Leung Lai, Stanford University, Stanford, CA, USA Kenneth Lawrence, New Jersey Institute of Technology, Newark, NJ, USA Sheila Lawrence, Rutgers University, Newark, NJ, USA Alice C. Lee, State Street Corp., Boston, MA, USA Cheng-Few Lee, Rutgers University, New Brunswick, NJ, USA and National Chiao Tung University, Hsinchu, Taiwan, ROC Han-Hsing Lee, National Chiao Tung University, Hsinchu, Taiwan, ROC Jack C. Lee, National Chiao Tung University, Hsinchu, Taiwan, ROC Jang-Yi Lee, Tunghai University, Taichung, Taiwan, ROC Jin-Ping Lee, Feng Chia University, Taichung, Taiwan, ROC John Lee, Center for PBBEF Research, Hackensack, NJ, USA Kin Wai Lee, Nanyang Technological University, Singapore, Singapore Sang Bin Lee, Hanyang University, Seoul, Korea Miguel A. Lejeune, George Washington University, Washington, DC, USA

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Jiandong Li, Central University of Finance and Economics, P.R. China Liuling Li, Rutgers University, New Brunswick, NJ, USA Donald Lien, University of Texas at San Antonio, San Antonio, TX, USA Venus Khim-Sen Liew, Universiti Malaysia Sabah, Sabah, Malaysia Carle Shu Ming Lin, Rutgers University, New Brunswick, NJ, USA Hai Lin, Xiamen University, Xiamen, Fujian, China Lin Lin, Department of Banking and Finance, National Chi-Nan University, 1 University Rd., Puli, Nantou Hsien, Taiwan 545, ROC T. I. Lin, National Chung Hsing University, Taichung, Taiwan, ROC William T. Lin, Tamkang University, Taipei, Taiwan, ROC Bo Liu, Citigroup Global Market Inc., New York, NY, USA Fang-I Liu, National Taiwan University, Taipei, Taiwan, ROC Nathan Liu, National Chiao Tung University, Hsinchu, Taiwan, ROC Xianghua Liu, Rutgers University, Piscataway, NJ, USA Ben Logan, Bell Labs, USA Jessica Mai, Rutgers University, Newark, NJ, USA A.G. Malliaris, Loyola University Chicago, Chicago, IL, USA Michael Mania, A. Razmadze Mathematical Institute, Georgia and Georgian-American University, Tbilisi, Georgia Benjamin Melamed, Rutgers Business School, Newark and New Brunswick, NJ, USA Thilo Meyer-Brandis, University of Oslo, Oslo, Norway Lalatendu Misra, University of Texas at San Antonio, San Antonio, TX, USA Bruce Mizrach, Rutgers University, New Brunswick, NJ, USA German Molina, Statistical and Applied Mathematical Sciences Institute, NC, USA Chien-Chung Nieh, Tamkang University, Taipei, Taiwan, ROC Michael S. Pagano, Villanova University, Philadelphia, PA, USA Dinesh Pai, Rutgers University, Newark, NJ, USA Szu-Yu Pai, National Taiwan University, Taipei, Taiwan, ROC Oded Palmon, Rutgers University, New Brunswick, NJ, USA Enlin Pan, Chicago Partners, Chicago, IL USA Dilip K. Patro, Office of the Comptroller of the Currency, Washington, DC, USA Jenifer Piesse, University of London, London, UK Ser-Huang Poon, University of Manchester, Manchester, UK Philip Protter, Cornell University, Ithaca, NY, USA Zhuo Qiao, University of Macau, Macau, China Martin Reck, Deutsche Börse, Frankfurt, Germany Samuel Ring, Rutgers University, Newark, NJ, USA

List of Contributors

List of Contributors

xxxvii

Joshua Ronen, New York University, New York, NY, USA Mark Rubinstein, University of California, Berkley, CA, USA Andrzej Ruszczynski, Rutgers University, Newark, NJ, USA Marina Santacroce, Politecnico di Torino, Department of Mathematics, C.so Duca degli Abruzzi 24, 10129 Torino, Italy Bharat Sarath, Baruch College, New York, NY, USA Sergei Sarkissian, McGill University, Montreal, QC, Canada Robert A. Schwartz, Baruch College, New York, NY, USA Thomas V. Schwarz, Grand Valley State University, Allendale, MI, USA Louis O. Scott, Morgan Stanley, New York, NY, USA Glenn Shafer, Rutgers University, Newark, NJ USA Chung-Hua Shen, National Taiwan University, Taipei, Taiwan, ROC Frederick C. Shen, Coventree Inc, Toronto, ON, Canada Larry Shepp, Rutgers University, Piscataway, NJ, USA Yuan-Chung Sheu, National Chao Tung University, Hsinchu, Taiwan, ROC Junmin Shi, Rutgers University, Newark, NJ, USA Yong Shi, University of Nebraska at Omaha, Omaha, NE, USA and Chinese Academy of Sciences, Beijing, China Fu-Shuen Shie, National Taiwan University, Taipei, Taiwan, ROC Pai-Ta Shih, Department of Finance, National Taiwan University, Taipei 106, Taiwan, ROC Wei-Kang Shih, Rutgers University, Newark, NJ, USA Timothy Simin, Pennsylvania State University, University Park, PA, USA Ben J. Sopranzetti, Rutgers University, Newark, NJ, USA Duane Stock, University of Oklahoma, Norman, OK, USA Anant Sunderam, Tuck School, Hanover, NH, USA Stephen J. Taylor, Lancaster University, Lancaster, UK Revaz Tevzadze, Institute of Cybernetics, Georgia and Georgian-American University, Tbilisi, Georgia Michael Theobald, Accounting and Finance Subject Group, University of Birmingham, Birmingham, UK Hiroki Tsurumi, Rutgers University, New Brunswick, NJ, USA María Vargas, University of Zaragoza, Zaragoza, Aragon, Spain Itzhak Venezia, Hebrew University, Jerusalem, Israel John Wald, Pennsylvania State University, University Park, PA, USA Chia-Jane Wang, Manhattan College, New York, NY, USA Kehluh Wang, National Chiao Tung University, Hsinchu, Taiwan, ROC

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Shin-Yun Wang, National Dong Hwa University, Hualien, Taiwán, ROC Yanzhi Wang, Yuan Ze University, Taoyuan, Taiwán, ROC Daniel Weaver, Rutgers University, Piscataway, NJ, USA Wing-Keung Wong, Hong Kong Baptist University, Hong Kong, Kowloon Tong, Hong Kong Donald H. Wort, California State University East Bay, Hayward, CA, USA ChunChi Wu, University of Missouri, Columbia, MO, USA Liuren Wu, Baruch College, New York, NY, USA Shu Wu, The University of Kansas, Lawrence, KS, USA Wei-Hsiung Wu, National Taiwan University, Taipei, Taiwan, ROC Yangru Wu, Rutgers Business School, Newark and New Brunswick, NJ, USA Yi Lin Wu, National Tsing Hua University, Hsinchu, Taiwan, ROC Yihong Xia, Wharton School, Pennsylvania, PA, USA Haipeng Xing, SUNY at Stony Brook, Stony Brook, NY, USA Chin W. Yang, Clarion University of Pennsylvania, Clarion, PA, USA Kenton K. Yee, Columbia Business School, New York, NY, USA Gillian Hian Heng Yeo, Nanyang Technological University, Singapore, Singapore Hai-Chin Yu, Chung Yuan University, Taoyuan, Taiwan, ROC Min-Teh Yu, Providence University, Taichung, Taiwan, ROC Yong Zeng, The University of Missouri at Kansas City, Kansas City, MO, USA Feng Zhao, Rutgers University, Newark, NJ, USA Xing Zhou, Rutgers Business School, Newark and New Brunswick, NJ, USA

List of Contributors

Chapter 1

Theoretical Framework of Finance

Abstract The main purpose of this chapter is to explore important finance theories. First, we discuss discounted cash-flow valuation theory (classical financial theory). Second, we discuss the Modigliani and Miller (M and M) valuation theory. Third, we examine Markowitz portfolio theory. We then move on to the capital asset pricing model (CAPM), followed by the arbitrage pricing theory. Finally, we will look at the option pricing theory and futures valuation and hedging. Keywords Discounted cash-flow valuation r M and M valuation theory r Markowitz portfolio theory r Capital asset pricing model r Arbitrage pricing theory r Option pricing model r Futures valuation and hedging

1.1 Introduction Value determination of financial instruments is important in security analysis and portfolio management. Valuation theories are the basic tools for determining the intrinsic value of alternative financial instruments. This chapter provides a general review of the financial theory that most students of finance would have already received in basic corporate finance and investment classes. Synthesis and integration of the valuation theories are necessary for the student of investments in order to have a proper perspective of security analysis and portfolio management. The basic policy areas involved in the management of a company are (1) investment policy, (2) financial policy, (3) dividend policy, and (4) production policy. Since the determination of the market value of a firm is affected by the way management sets and implements these policies, they are of critical importance to the security analyst. The security analyst must evaluate management decisions in each of these areas and convert information about company policy into price estimates of the firm’s securities. This chapter examines these policies within a financial theory framework, dealing with valuation models.

There are six alternative but interrelated valuation models of financial theory that might be useful for the analysis of securities and the management of portfolios: 1. Discounted cash-flow valuation theory (classical financial theory) 2. M and M valuation theory 3. Capital asset pricing model (CAPM) 4. Arbitrage Pricing Theory (APT) 5. Option-pricing theory (OPT) 6. Futures Valuation and Hedging The discounted cash-flow valuation and M and M theories are discussed in the typical required corporate-finance survey course for both bachelor’s and master’s programs in business. The main purpose of this chapter is to review these theories and discuss their interrelationships. The discounted cash-flow model is first reviewed by some of the basic valuation concepts in Sect. 1.2. In the second section, the four alternative evaluation methods developed by M and M in their 1961 article are discussed. Their three propositions and their revision with taxes are explored, including possible applications of their theories in security analysis. Miller’s inclusion of personal taxes is discussed in Sect. 1.3. Section 1.4 discusses the Markowitz portfolio theory. Section 1.5 includes a brief overview of CAPM concepts. Section 1.6 introduces the Arbitrage Pricing Theory (APT). Sections 1.6 and 1.7 discuss the option-pricing theory and the futures valuation and hedging. Conclusion is presented in Sect. 1.8.

1.2 Discounted Cash-Flow Valuation Theory Discounted cash-flow valuation theory is the basic tool for determining the theoretical price of a corporate security. The price of a corporate security is equal to the present value of future benefits of ownership. For example, for common stock, these benefits include dividends received while the stock is owned plus capital gains earned during the ownership period. If we assume a one-period investment and a world of certain cash flows, the price paid for a share of

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_1, 

3

4

1

stock, P0 , will equal the sum of the present value of a certain dividend per share, d1 (assumed to be paid as a single flow at year end), and the selling price per share P1 : P0 D

d1 C P1 1Ck

(1.1)

in which k is the rate of discount assuming certainty. P1 can be similarly expressed in terms of d2 and P2 : P1 D

d2 C P2 1Ck

(1.2)

If P1 in Equation (1.1) is substituted into Equation (1.2), a two-period expression is derived: P0 D

d2 d1 P2 C C .1 C k/ .1 C k/2 .1 C k/2

(1.3)

It can be seen, then, that an infinite time-horizon model can be expressed as the P0 D

1 X t D1

dt .1 C k/t

MV 0 D

1 X t D1

Dt .1 C k/t

where: PV D present value of the bond; n D the number of periods to maturity; CFt D the cash flow (interest and principal) received in period t; kb D the required rate of return of the bondholders (equal to risk-free rate i plus a risk premium).

1.2.1.1 Perpetuity The first (and most extreme) case of bond valuation involves a perpetuity, a bond with no maturity date and perpetual interest payments. Such bonds do exist. In 1814, the English government floated a large bond issue to consolidate the various small issues it had used to pay for the Napoleonic Wars. Such bonds are called consols, and the owners are entitled to a fixed amount of interest income annually in perpetuity. In this case, Equation (1.6) collapses into the following: PV D

(1.4)

Since the total market value of the firms’ equity is equal to the market price per share multiplied by the number of shares outstanding, Equation (1.4) may be re-expressed in terms of total market value MV 0 : (1.5)

Theoretical Framework of Finance

CF kb

(1.7)

Thus, the valuation depends directly on the periodic interest payment and the required rate of return for the bond. It can be seen that required rates of return, necessitated by a higher rate of inflation or an increase in the perceived risk of the bond, lower the present value, decreasing the bond’s market value. For example, if the stated annual interest payment on the perpetuity bond is $50 and the required rate of return in the market is 10%, the price of the security is stated: PV D $50=0:10 D $500

in which Dt D total dollars of dividends paid during year t. Using this basic valuation approach as a means of expressing the appropriate objective of the firm’s management, the valuation of a firm’s securities can be analyzed in a world of certainty.

If its issuing price had been $1,000, it can be seen that the required rate of return would have been only 5% (kb D CF=PV D $50=$1; 000 D 0:05, or 5%).

1.2.1 Bond Valuation

1.2.1.2 Term Bonds

Bond valuation is a relatively easy process, as the income stream the bondholder will receive is known with a high degree of certainty. Barring a firm’s default, the income stream consists of the periodic coupon payments and the repayments of the principal at maturity. These cash flows must be discounted to the present using the required rate of return for the bond. The basic principles of bond valuation are represented in the equation: n X CFt (1.6) PV D .1 C kb /t t D1

Most bonds are term bonds, which mature at some definite point in time. Thus, Equation (1.6) should be respecified to take this fact into account: PV D

n X t D1

It Pn C .1 C kb /t .1 C kb /n

(1.8)

where: It D the annual coupon interest payment; Pn D the principal amount (face value) of the bond; and n D the number of periods to maturity.

1 Theoretical Framework of Finance

Again, it should be noted that the market price, PV, of a bond is affected by changes in the rate of inflation. If inflation increases, the discount rate must also increase to compensate the investor for the resultant decrease in the value of the debt repayment. The present value of each period’s interest payment thus decreases, and the price of the bond falls. The bondholder is always exposed to interest-rate risk, the variance of bond prices resulting from fluctuations in the level of interest rates. Interest-rate risk, or price volatility of a bond caused by changes in interest-rate levels, is directly related to the term to maturity. There are two types of risk premiums associated with interest-rate risk as it applies to corporate bonds. The bond maturity premium refers to the net return from investing in long-term government bonds rather than the short-term bills. Since corporate bonds generally possess default risk, another of the components of corporate bond rates of return is default premium. The bond default premium is the net increase in return from investing in long-term corporate bonds rather than in long-term government bonds. Additional features of a bond can affect its valuation. Convertible bonds, those with a provision for conversion into shares of common stock, are generally more valuable than a firm’s straight bonds for several reasons. First, the investor receives the potential of positive gains from conversion, should the market price of a firm’s common stock rise above the conversion price. If the stock price is greater than the conversion price, the convertible bond generally sells at or above its conversion value. Second, the bondholder also receives the protection of fixed income payment, regardless of the current price of the stock – assuring the investor that the price of the bond will be at least equal to that of a straight bond, should stock prices fail to increase sufficiently. Third, for any given firm the coupon rate of return from its bonds is generally greater than the dividend rate of return (dividend yield) from its common stock – thus causing a measure of superiority for a convertible bond over its conversion into common stock until stock dividends rise above the bond’s coupon rate. Even then, the convertible bond may be preferred by investors because of the higher degree of certainty of interest payments versus dividends that would decline if earnings fall. A sinking fund provision may also increase the value of a bond, at least at its time of issue. A sinking-fund agreement specifies a schedule by which the sinking-fund will retire the bond issue gradually over its life. By providing cash to the sinking-fund for use in redeeming the bonds, this provision ensures the investor some potential demand for the bond, thus increasing slightly the liquidity of the investment. Finally, the possibility that the bond may be called will generally lower the value relative to a noncallable bond. A call provision stipulates that the bond may be retired by the issuer at a certain price, usually above par or face value. Therefore, in periods of large downward interest movements,

5

a company may be able to retire a high coupon bond and issue new bonds with a lower interest payment requirement. A call feature increases the risk to investors in that their expected high interest payments may be called away from them, if overall interest rate levels decline.

1.2.2 Common-Stock Valuation Common-stock valuation is complicated by an uncertainty of cash flows to the investor, necessarily greater than that for bond valuation.1 Not only might the dividends voted to shareholders each period change in response to management’s assessment concerning the current level of earnings stability, future earnings prospects, or other factors, but the price of the stock may also either rise or fall – resulting in either capital gains or losses, if the shares are sold. Thus, the valuation process requires the forecasting of both capital gains and the stream of expected dividends. Both must also be discounted at the required rate of return of the common stockholders. P0 D

d2 d1 Pn C CC 1Ck .1 C k/2 .1 C k/n

(1.9)

where: P0 D the present value, or price, of the common stock per share; d D the dividend payment per share; k D the required rate of return of the common stockholders; and Pn D the price of the stock in period n when sold. However, Pn can also be expressed as the sum of all discounted dividends to be received from period n forward into the future. Thus, the value at the present time can be expressed as an infinite series of discounted dividend payments: 1 X dt (1.4) P0 D .1 C k/t t D1 in which dt is the dividend payment in period t. Several possibilities exist regarding the growth of dividend payments over time. First, dividends may be assumed to be a constant amount, and the formula for the stock’s valuation is simple Equation (1.7), where CF is the constant dividend and k is the required rate of return of the common stockholder. Second, dividends may be expected to grow at some constant rate, g. In such a case, a dividend at time t is simply the compound value of the present dividend (i.e., 1

This is true because foregoing interest puts the firm into default, while missing dividend payments does not.

6

1

Pt D .1 C g/t d0 ). Under this assumption, if g < k, the valuation equation can be simplified to the following: P0 D

d1 .k  g/

(1.10)

This equation represents the Gordon growth model. Note that a critical condition for this model is that the constant growth of dividends must be less than the constant required rate of return. The zero growth situation is a special case of this model, in which: d1 (1.11) P0 D k Finally, dividends can exhibit a period of supernormal growth (i.e., g is greater than k) before declining to the normal growth situation assumed in the Gordon model (g is less than k). Supernormal growth often occurs during the “takeoff” phase in a firm’s life cycle. That is, a firm may experience a life cycle analogous to that of a product: first, a low-profit introductory phase, then a takeoff phase of high growth and high profits, leveling off at a plateau during its mature stage, perhaps followed by a period of declining earnings. Computer and electronics manufacturers experienced a period of supernormal growth during the 1960s, as did semiconductor firms during the 1970s. Bioengineering firms appear to be the super growth firms of the 1980s. The valuation of a supernormal growth stock requires some estimate of the length of the supernormal growth period. The current price of the stock will then consist of two components: (1) the present value of the stock during the supernormal growth period, and (2) the present value of the stock price at the end of the supernormal growth period: P0 D

n X d0 .1 C gs /t t D1

.1 C k/t

C

dnC1 kgn

.1 C k/n

exclusion of portfolio concepts and the interrelationship with the overall market indexes. Most of the classical models are also static in nature, overlooking the concept of dynamic growth. Nevertheless, a fundamental approach to security valuation – the stream of dividends approach – has evolved from this theory.

1.3 M and M Valuation Theory Modigliani and Miller (M and M 1961) have proposed four alternative valuation methods to determine the theoretical value of common stocks. This section discusses these valuation approaches in some detail. M and M’s four more or less distinct approaches to the valuation of common stock are as follows: 1. The discounted cash-flow approach; 2. the current earnings plus future investment opportunities approach; 3. the stream of dividends approach; and 4. the stream of earnings approach Working from a valuation expression referred to by M and M as the “fundamental principle of valuation”: 1 .d1 C P1 / (1.13) 1Ck M and M further developed a valuation formula to serve as a point of reference and comparison among the four valuation approaches: P0 D

V0 D

1 X t D0

(1.12)

where: gs D supernormal growth rate; n D the number of periods before the growth drops from supernormal to normal; k D the required rate of return of the stockholders; and gn D the normal growth rate of dividends (assumed to be constant thereafter).

1 .Xt  It / .1 C k/t C1

(1.14)

where: V0 D the current market value of the firm; Xt D net operating earnings in period t; and It D new investment during period t. In this context, the discounted cash-flow approach can be expressed: V0 D

1 X t D0

As we can see from our development of the discounted cashflow financial theory, the primary determinant of value for securities is the cash flow received by the investors. Anything that affects the cash flow, such as the dividend policy, investment policy, financing policy, and production policy of the firm, needs to be evaluated in order to determine a market price. Some shortcomings of this approach include the overemphasis on the evaluation of the individual firm to the

Theoretical Framework of Finance

1 .Rt  Ot / .1 C k/t C1

(1.15)

in which Rt is the stream of cash receipts by the firm and Ot is the stream of cash outlays by the firm. This fundamental principle is based on the assumption of “perfect markets,” “rational behavior,” and “perfect certainty” as defined by M and M. Since Xt differs from Rt and It differs from Ot only by the cost of goods sold and depreciation expense, if .Rt  Ot / equals .Xt  It /, then (1.15) is equivalent to (1.14) and the discounted cash-flow approach is an extension of

1 Theoretical Framework of Finance

7

Equation (1.13), the fundamental valuation principle. Hence, the security analyst must be well versed in generally accepted accounting principles in order to evaluate the worth of accounting earnings of .Xt  It /. The investment-opportunities approach seems in some ways the most natural approach from the standpoint of an investor. This approach takes into account the ability of the firm’s management to issue securities at “normal” market rates of return and invest in the opportunities, providing a rate higher than the normal rate of return. From this framework, M and M developed the following expression, which they show can also be derived from Equation (1.14):

in the pre-M and M period. Assuming an infinite time horizon, this approach defines the current market price of a share of common stock as equal to the discounted present value of all future dividends: P0 D

t D0

X0 X It .kt  k/ C k .1 C k/t C1 t D0 1

(1.16)

in which X0 is the perpetual net operation earning and kt is the “higher than normal” rate of return on new investment It . From the expression it can be seen that if a firm cannot generate a rate of return of its new investments higher than the normal rate, k, the price/earnings ratio applied to the firm’s earnings will be equal to 1/k, thus implying simple expansion rather than growth over time. An important variable for security analysis is a firm’s P/E ratio (or earnings multiple), defined as:

Conceptually the P/E ratio is determined by three factors: (1) the investor’s required rate of return .K/; (2) the retention ratio of the firm’s earning, b, where b is equal to 1 minus the dividend payout ratio; and (3) the firm’s expected return on investment .r/. Using the constant-growth model Equation (1.10):

V0 D

1 X

1 .Dt / .1 C k/t C1

(1.19)

With no outside financing, it can be seen that Dt D Xt  It and: 1 X 1 V0 D .Xt  It / .1 C k/t C1 t D0 which is Equation (1.14). With outside financing through the issuance of shares of new common stock, it can be shown that: V0 D

1 X t D0

1 .Dt C Vt C1  mt C1  Pt C1 / .1 C k/t C1

(1.20)

Vt C1  .mt C1 /.Pt C1 / D It  .Xt  Dt / Thus, Equation (1.20) can also be written in the form of Equation (1.14): V0 D

1 X t D0

d1 kg

E1 .1  b/ P0 D k  .br/ P0 1b D E1 k  .br/

(1.18)

in which mt C1 is the number of new shares issued at price Pt C1 . For the infinite horizon, the value of the firm is equal to the investments it makes and the new capital it raises, or:

Market price P=E ratio D Earnings per share

P0 D

1 .dt / .1 C k/t C1

Restating in terms of total market value:

t D0

V0 D

1 X

(1.17)

in which b is the retention rate and E1 is the next period’s expected profit. The P0 =E ratio is theoretically equal to the payout ratio of a firm divided by the difference between the investor’s required return and the firm’s growth rates. In the above equation a direct relationship has been identified between price/earnings ratio and discount cash-flow valuation model. The stream-of-dividends approach has been by far the most popular in the literature of valuation; it was developed

1 .Xt  It / .1 C k/t C1

(1.14)

Given the M and M ideal assumptions, the above result implies the irrelevance of dividends because the market value of the dividends provided to new stockholders must always be precisely the same as the increase in current dividends. This is in direct disagreement with the findings of the discounted cash-flow model, where dividends are a major determinant of value. In this case, dividends have no impact on value, and the firm’s investment policy is the most important determinant of value. Security analysis should concern itself with the future investment opportunities of the firm and forget about dividends. M and M also developed the stream-earnings approach, which takes account of the fact that additional capital must be acquired at some cost in order to maintain the stream of future earnings at its current level. The capital to be raised

8

1

is It and its cost is K percent per period thereafter; thus, the current value of the firm under this approach can be stated:

V0 D

1 X t D0

1 .Xt  It / .1 C k/t C1

which, again, is Equation (1.14). Under none of these four theoretical approaches does the term Dt remain in the final valuation expression and because Xt ; It , and k are assumed to be independent of Dt , M and M conclude that the current value of a firm is independent of its current and future dividend decisions. The amount gained by stockholders is offset exactly by the decline in the market value of their stock. In the short run, this effect if observed when a stock goes ex-dividend – that is, if the market price of the stock falls by the amount of the dividend on the last day the old shareholders are entitled to receive a dividend payment. The stock’s value depends only on the expected future earnings stream of the firm. Security analysts spend much time and effort forecasting a firm’s expected earnings. While the above analysis ignores the case in which external financing is obtained through the issuance of debt, in such a situation M and M’s position then rests upon their indifference proposition with respect to leverage (M and M 1958), discussed elsewhere in this chapter. Since that analysis shows that under a set of assumptions consistent with their “fundamental principal of valuation” the real cost of debt in a world of no taxation is equal to the real cost of equity financing, M and M conclude that the means of external financing used to offset the payment of dividends does not affect their hypothesis that dividends are irrelevant. Prior to Miller and Modigliani’s (1961) article, the classical view held that dividend policy was a major determinant of the value of the corporation and that firms should seek their “optimal payout ratios” to maximize their value. M and M’s conclusions about the irrelevance of dividend policy given investment policy, collided head-on with the existing classical view. The view that the value of the firm is independent of dividend policy also extends into a world with corporate taxes but without personal taxes.

1.3.1 Review and Extension of M and M Proposition I

Theoretical Framework of Finance

cast doubt upon the existence of such an optimal structure. The specific assumptions that they made, consistent with the dividend irrelevance analysis previously outlined, include the following: 1. Capital markets are perfect (frictionless). 2. Both individuals and firms can borrow and lend at the risk-free rate. 3. Firms use risk-free debt and risky equity. 4. There are only corporate taxes (that is, there are no wealth taxes or personal income taxes). 5. All cash flow streams are perpetuities (that is, no growth). Developing the additional concepts of risk class and homemade leverage, M and M derived their well-known Proposition I, both with and without corporate taxes.2 If all firms are in the same risk class, then their expected risky future net operating cash flow .XP / varies only by a scale factor. Under this circumstance, the correlation between two firms’ net operating income (NOI) within a risk class should be equal to 1.0. This implies that the rates of return will be equal for all firms in the same risk class, that is: Rit D

XP it  XP i t 1 Xi t 1

and because XPit D CXP jt where C is the scale factor: Rjt D

CXP jt  CXPjt D Rit CXP jt 1

(1.22)

in which Rit and Rjt are rates of return for the i th and j th firms, respectively. Therefore, if two streams of cash flow differ by only a scale factor, they will have the same distributions of returns and the same risk, and they will require the same expected return. The concept of homemade leverage is used to refer to the leverage created by individual investors who sell their own debt, while corporate leverage is used to refer to the debt floated by the corporation. Using the assumption that the cost of homemade leverage is equal to the cost of corporate leverage, M and M (1958) derived their Proposition I both with and without taxes. However, the Proposition I with taxes was not correct, and they subsequently corrected this result in their 1963 paper. Mathematically, M and M’s Proposition I can be defined: Vj D .Sj C Bj / D Xj =k

The existence of optimal capital structure has become one of the important issues for academicians and practitioners in finance. While classical finance theorists argue that there is an optimal capital structure for a firm, the new classical financial theory developed y M and M (1958, 1963) has

(1.21)

2

(1.23)

In 1985 Franco Modigliani won the Nobel Prize for his work on the life cycle of savings and his contribution to what has become known as the M and M theory, discussed in this section.

1 Theoretical Framework of Finance

9

and Proposition I with taxes can be defined as VjL D

.1  j /Xj Ij D VjU C Bj C  k r

(1.24)

In Equation (1.23), Bj ; Sj , and Vj are, respectively, the market value of debt, common shares, and the firm. Xj is the expected profit before deduction of interest, k the required rate of return or the cost of capital in risk class k. In Equation (1.24), k is the required rate of return used to capitalize the expected returns net of tax for the unlevered firm with long-run average earnings before tax and interest of .Xj / in risk class k. j is the corporate tax rate for the j th firm, Ij is the total interest expense for the j th firm, and r is the market interest rate used to capitalize the certain cash inflows generated by risk-free debt. Bj is total risk-free debt floated by the j th firm, and V L and V U are the market values of the leveraged and unleveraged firms, respectively. By comparing these two equations, we find that the advantages of a firm with leverage will increase that firm’s value by j Bj – that is, the corporate tax rate times the total debt floated by that firm. One of the important implications of this proposition is that there is no optimal capital structure for the firm unless there are bankruptcy costs associated with its debt flotation. If there are bankruptcy costs, then a firm will issue debt until its tax benefit is equal to the bankruptcy cost, thus providing, in such a case, an optimal capital structure for the firm. In addition to the bankruptcy costs, information signaling (see Leland and Pyle 1977, and Ross 1977a) and differential expectations between shareholders and bondholders can be used to justify the possible existence of an optimal structure of a firm. The existence of optimal capital structure is an important issue for security analysts to investigate because it affects the value of the firm and the value of the firm’s securities. Is a firm with a high level of debt more valuable than a similar firm with very little debt? M and M say it doesn’t matter or that the highly leveraged firm is more valuable. The important assumptions used to prove the M and M Proposition I with taxes are that (1) there are no transaction costs, (2) homemade leverage is equal to corporate leverage, (3) corporate debt is riskless, and (4) there is no bankruptcy cost. Overall, M and M’s Proposition I implies that there is no optimal capital structure. If there is a tax structure that systematically provides a lower after-tax real cost of debt relative to the after-tax real cost of equity, the corporation will maximize the proportion of debt in its capital structure and will issue as much debt as possible to maximize the tax shield associated with the deductibility of interest. Stiglitz (1969) extends M and M’s proposition using a general equilibrium state preference framework. He is able to show that M and M’s results do not depend on risk classes, competitive capital markets, or agreement by

investors. The only two fundamental assumptions are that there is no bankruptcy and individuals can borrow at the same rate as firms. Stiglitz (1974) develops the argument that there may be a determinate debt-equity ratio for the economy as a whole, but not for the individual firm.

1.3.2 Miller’s Proposition on Debt and Taxes Miller (1977) argues that although there is no optimal capital structure for an individual firm, in the aggregate case there may be an optimal structure. In balancing bankruptcy cost against tax shelter, an optimal capital structure is derived, just as the classical view has always maintained. The Tax Reform Act of 1986 taxes dividends and longterm capital gains at the same top rate of 28%. This is a major change from the old 50% rate on dividends and 20% rate on long-term capital gains. The new tax bill has also shifted the major tax burden to corporations and away from individuals. Even though the maximum corporate tax rate will decrease to 34% from the current top rate of 46%, corporations will be paying more taxes because of the loss of the Investment Tax Credits and the Accelerated Cost Recovery System depreciation allowances. These changes in the tax code will shift the emphasis of corporate management from retaining earnings in order to generate price appreciation and capital gains to the payout of corporate funds in the form of dividends. In his presidential address at the Annual Meeting of the American Finance Association, Merton Miller (1977) incorporates personal taxes into the Modigliani and Miller (1958, 1963) argument for the relationship between the firm’s leverage and cost of capital. M and M’s Proposition I shows that the value of the leveraged firm equals the value of the unleveraged firm plus the tax shield associated with interest payments, as shown by Equation (1.24): V L D V U C tc B

(1.25)

where: V L D the value of the leveraged firm; V U D the value of the unleveraged firm; tc D the corporate tax rate; and B D the value of the firm’s debt. Miller generalizes the M and M relationship shown in Equation (1.25) to include personal taxes on dividends and capital gains as well as taxes on interest income, to yield:   .1  tc /.1  tps / B VL D VU C 1 .1  tpB /

(1.26)

10

1

Theoretical Framework of Finance

in which tps is the personal tax rate on income from stock and tpB is the personal tax rate on income from bonds.

In the second approach, the maximization problem in Equation (1.30) can be rewritten as

1.4 Markowitz Portfolio Theory

82 9 31=2 ˆ > n n X < X =   5 4 W i R i C 1 Cov Ri ; Rj  p Max L D ˆ > : j D1 i D1 ; i D1 n X

Professors Markowitz, Miller, and Sharpe earned their Nobel Prize in applied economics in 1989. Section 1.3 briefly discussed the M and M theory. In the next section, we will discuss CAPM. In this section we will discuss Prof. Markowitz portfolio theory. In the paper entitled “Markowitz, Miller, and Sharpe: The First Nobel Laureates in Finance,” Lee (1991) has discussed this historical event in details. Markowitz suggests two constrained maximization approaches to obtain the optimal portfolio weight. The first approach is to minimize the risk or variance of the portfolio, subject to the portfolio’s attaining some target expected rate of return, and also subject to the portfolio weight summing to one. The problem can be stated mathematically: Min p2 D

n n X X

Wi Wj ij

(1.27)

Wi E .Ri / D E 

(1.28)

i D1 j D1

Subject to

n X

1:

C2

n X

! Wi  1

(1.31)

i D1

where Ri is the average rate of rates of return of the portfolio given targeted standard deviation of the portfolio P . Essentially Equation (1.31) maximizes the expected rates of return of the portfolio given the targeted standard deviation of the portfolio. In this book, a large portion is dedicated to the portfolio theory and its application. Chapter 10 discusses portfolio optimization models and mean-variance spanning tests. Chapter 12 discusses the estimation risk and power utility portfolio selection. Chapter 13 discusses theory and methods in the international portfolio management. Chapter 17 provides discussion on portfolio theory, CAPM, and the performance measures. Chapter 18 discusses the intertemporal equilibrium models, portfolio theory, and the capital asset pricing model.

1.5 Capital Asset Pricing Model

i D1 

where E is the target expected return and 2:

n X

Wi D 1:0

(1.29)

i D1

The first constraint simply says that the expected return on the portfolio should equal the target return determined by the portfolio manager. The second constraint says that the weights of the securities invested in the portfolio must sum to one. The Lagrangian objective function can be written: Min L D

n n X X

W i W j C 1

i D1 j D1

C2

n X

n X 

Wi E .Ri /  E 



i D1

! Wi  1

(1.30)

i D1

Taking the partial derivatives of this equation with respect to each of the variables, W1 ; W2 ; W3 ; 1 ; 2 and setting the resulting five equations equal to zero yields the minimization of risk subject to the Lagrangian constraints. This system of five equations and five unknowns can be solved by the use of matrix algebra. Equations (1.30) minimizes the portfolio variance given the portfolio’s targeted expected rate of return.

At about the same time as M and M were developing their work, developments in portfolio theory were leading to a model describing the formation of capital asset prices in world of uncertainty: the capital asset pricing model (CAPM). The CAPM is a generalized version of M and M theory in which M and M theory is provided with a link to the market: E.Rj / D Rf C ˇj ŒE.Rm /  Rf

(1.32)

where: Rj D the rate of return for security j ; ˇj D a volatility measure relating the rate of return on security j with that of the market over time; Rm D the rate of return for the overall market (typically measured by the rate of return reflected by a market index, such as the S&P 500); and Rf D the risk-free rate available in the market (usually the rate of return on U.S. Treasury bills is used as a proxy). In the CAPM framework, the valuation of a company’s securities is dependent not only on its cash flows but also on those of other securities available for investment. It is assumed that much of the total risk, as measured by standard

1 Theoretical Framework of Finance

11

deviation of return, can be diversified away by combining the stock of a firm being analyzed with those of other companies. Unless the cash flows from these securities are perfectly positively correlated, smoothing or diversification will take place. Thus, the security return can be divided into two components: a systematic component that is perfectly correlated with the overall market return and an unsystematic component that is independent of the market return: Security return D Systematic return C Unsystematic return (1.33) Since the security return is perfectly correlated with the market return, it can be expressed as a constant, beta, multiplied by the market return .Rm /. The beta is a volatility index measuring the sensitivity of the security return to changes in the market return. The unsystematic return is residual of the relationship of Rj with Rm . As has been previously noted, the standard deviation of the probability distribution of a security’s rate of return is considered to be an appropriate measure of the total risk of that security. This total risk can be broken down into systematic and unsystematic components, just as noted above for security return: Total security risk D Systematic risk C Unsystematic risk (1.34) Diversification is achieved only when securities that are not perfectly correlated with one other are combined. The unsystematic risk components tend to cancel each other as they are all residuals from the relationship of security returns with the overall market return. In the process, the portfolio risk measure declines without any corresponding lowering of portfolio return (see Fig. 1.1). It is assumed in this illustration that the selection of additional securities as the portfolio size is increased is performed in some random manner, although any selection process other than intentionally choosing perfectly correlated securities will suffice. Unsystematic risk is shown to be gradually eliminated until the remaining portfolio risk is completely market related. While for an actual portfolio the systematic risk will not remain constant as securities are added, the intent is to show that the unsystematic-risk portion can be diversified away, leaving the market related systematic portion as the only relevant measure of risk. Empirical studies have shown that a portfolio of about 20 securities not highly correlated with one another will provide a high degree of diversification. Although capital-market theory assumes that all investors will hold the market portfolio, it is neither necessary nor realistic to assume that all investors will be satisfied with the market level of risk. There are basically two ways that investors can adjust their risk level within the CAPM theoretical framework. First, funds for investment can

Fig. 1.1 Diversification process

be divided between the market portfolio and risk-free securities. The capital-market line (CML) is derived assuming such a tradeoff function. This is illustrated in Fig. 1.2, in which point M is the market portfolio and points on the CML below and above M imply lending and borrowing at the risk-free rate. The second way of adjusting the portfolio risk level is by investing in a fully diversified portfolio of securities (that is, the correlation coefficient of the portfolio with the market, rpm is equal to 1.0) that has a weighted average beta equal to the systematic-risk level desired: ˇp D

n X

Wj Bj

(1.35)

j D1

in which Wj is the proportion of total funds invested in security j . In the CAPM, systematic risk as measured by beta is the only risk that need be undertaken; therefore, it follows that no risk premium should be expected for the bearing of unsystematic risk. With that in mind, the relationship between expected return and risk can be better defined through the illustration of the security-market line (SML) in Fig. 1.3 in which Rm and ˇm are the expected return and risk level of the market portfolio. In equilibrium, all securities and combinations of securities are expected to lie along this line. In contrast, only fully diversified portfolios would be expected to fall along the CML, because only with full diversification is total risk equal to systematic risk alone. In addition to the static CAPM developed by Sharpe (1964) and others Merton has discussed intertemporal CAPM. In Chap. 6 we will discuss the static CAPM and beta forecasting and in Chap. 18 we will focus on intertemporal CAPM.

12

1

Theoretical Framework of Finance

bij D a coefficient called a factor loading that quantifies the sensitivity of asset i ’s returns to the movements in the common factor ıQj (and is analogous to the beta in the CAPM); and 2Q i D an error term, or unsystematic risk component, idiosyncratic to the i th asset, with mean zero and variance equal to Q 22 . Moreover, it is assumed that the 2Q i reflects the random influence of information that is unrelated to other assets. Thus, the following condition is assumed to hold:

Fig. 1.2 Capital market line

 ˚ E 2Q i j2Q j D 0

(1.37)

as well as 2i and 2j independence for all i ¤ j . Also, for any two securities i and j :  ˚ E 2Q i ; 2Q j D 0

Fig. 1.3 Security market line

1.6 Arbitrage Pricing Theory 1.6.1 Ross’s Arbitrage Model Specification This section focuses on two related forms of the arbitrage pricing model (APM). The first is the model as originally proposed by Ross (1976). The initial and probably the most prominent assumption made by APM concerns the return-generating process for assets. Specifically, individuals are assumed to believe (homogeneously) that the random returns on the set of assets being considered are governed by a k-factor generating model of the form: rQi D Ei C bi1 ıQ1 C : : : C bi k ıQk C 2Q i .i D 1; : : : ; n/

(1.36)

where: rQi D random return on the i th asset; Ei D expected return on the i th asset; ıQj D j th factor common to the returns of all assets under consideration with a mean of zero, common factors that in essence capture the systematic component of risk in the model;

(1.38)

for all i and j , where i ¤ j . If this last condition did not hold – that is, if there was too strong a dependence between 2Q i and 2Q j – it would be equivalent to simply saying that the k-hypothesized common factors existed. Finally, it is assumed that for the set of n assets under consideration that n is much greater than the number of factors k. Before developing Ross’s riskless arbitrage argument, it is essential to examine Equation (1.36) more closely and draw some implications from its structure. First, consider the effect of omitting the unsystematic risk terms eQi . Equation (1.36) would then imply that each asset i has returns rQ that are an exact linear combination of the returns on a riskless asset (with constant return) and the returns on k other factors or assets (or column vectors) ıQi ; : : : ; ıQk . Moreover, the riskless return and each of the k factors can be expressed as a linear combination of k C 1 other return – for example, r, through rkC1 – in this type of setting. Taking this logic one step further, since any other asset return is a linear combination of the factors, it must also be a linear combination of the returns of the first k C 1 assets. Hence, portfolios composed from the first k C 1 assets must be perfect substitutes for all other assets in the market. Consequently, there must be restrictions on the individual returns generated by the model, as perfect substitutes must be priced equivalently. This sequence of mathematical logic is the core of APT. That is, only a few systematic components of risk exist in the economy, and consequently many portfolios will be close substitutes, thereby demanding the same value. To initiate Ross’s arbitrage argument about APT, it is best to start with the assumption of Equation (1.36). Next, presume an investor who is contemplating an alteration of the currently held portfolio, the difference between any new portfolio and the old portfolio will be quantified by changes

1 Theoretical Framework of Finance

13

in the investment proportions xi .i D 1; : : : ; n/. The xi represents the dollar amount purchased or sold of asset i as a fraction of total invested wealth. The investor’s portfolio investment is constrained to hold to the following condition: n X

xi D 0

(1.39)

i D1

In words, Equation (1.39) says that additional purchases of assets must be financed by sales of others. Portfolios that require no net investment such as x  .xj ; : : : ; xn / are called arbitrage portfolios. Now, consider an arbitrage portfolio chosen in the following manner. First, the portfolio must be chosen to be well diversified by keeping each element, x, of order 1=n in size. Second, the x of the portfolio must be selected in such a way as to eliminate all systematic risk (for each h): xbh

X

xi bih D 0 .h D 1; : : : ; k/

(1.40)

i D1

The returns on any such arbitrage portfolios can be described: Q x rQ D .xE/ C .xb/ıQ1 C : : : C .xbk /ıQk C .x 2/  xE C .xb1 /ıQ1 C : : : C .xbk /ıQk D xE where x rQ D

n P

xi rQi and xE D

i D1

n P

xi Ei . Note that the term

i D1

Q is (approximately) eliminated by the effect of holding a .x 2/ well-diversified portfolio of n assets where n is large. Using the law of large numbers, if  2 denotes the average variance of the eQi terms, and assuming for simplicity that each x, approximately equals 1=n and that the 2i are mutually independent: 1X Q D Var Var.x 2/ 2i n i

! D

2 Var.2Q i / D n2 n2

(1.41)

Thus if n is large the variance of x 2Q will be negligible. Reconsidering the steps up to this point, note that a portfolio has been created that has no systematic or unsystematic risk and using no wealth. Under conditions of equilibrium, it can be stated unequivocally that all portfolios of these n assets that satisfy the conditions of using no wealth and having no risk must also earn no return on average. In other words, there are no free lunches in an efficient market, at least not for any extended period of time. Therefore the expected return on the arbitrage portfolio can be expressed:

x rQ D xE D

n X

xi Ei D 0

(1.42)

i D1

Another way to state the preceding statements and results is through linear algebra. In general, any vector x with elements on the order of 1=n that is orthogonal to the constant vector and to each of the coefficient vectors bh .h D 1; : : : ; k/ must also be orthogonal to the vector of expected returns. A further algebraic consequence of this statement is that the expected return vector E must be a linear combination of the constant vector and the b vectors. Using algebraic terminology, there exist k C 1 weights .0 ; 1 ; : : : ; k / such that: E i D 0 C 1 b i l C : : : C k b i k ;

for all i

(1.43)

In addition, if there exists a riskless asset with return E0 , which can be said to be the common return on all zero-beta assets – that is, bih D 0 (for all h) – then: E 0 D 0 Utilizing this definition and rearranging: E i  E 0 D 1 b i l C : : : C k b i k

(1.44)

The pricing relationship depicted in Equation (1.44) is the central conclusion of the APT. Before exploring the consequences of this pricing model through a simple numerical example, it is best to first give some interpretation to the h , the factor risk premium. If portfolios are formed with a systematic risk of 1 relative to factor h and no risk on other factors, then each h can be interpreted as: h D E h  E 0

(1.45)

In words, each h can be thought of as the excess return or market risk premium on portfolios with only systematic factor h risk. Hence, Equation (1.44) can be rewritten: Ei  E0 D .E 1  E0 /bi1 C : : : C .E k  E0 /bi k

(1.46)

The implications that arise from the arguments concerning APT that have been constructed thus far can be summarized in the following statement: APT yields a statement of relative pricing on subsets of the universe of assets. Moreover, note that the arbitrage pricing model of Equations (1.44) or (1.46) can be tested by examining only subsets of the set of all return. Consequently, the market portfolio plays no special role in APT, since any well-diversified portfolio could serve the same purpose. Hence, it can be empirically tested on any set of data, and the results should be generalizable to the entire market.

14

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Theoretical Framework of Finance

Even though the APT is very general and based on few assumptions, it provides little guidance concerning the identification of the priced factors. Hence empirical research must achieve two goals. 1. Identify the number of factors. 2. Identify the various economics underlying each factor. Chapter 64 will discuss the relationship between the liquidity risk and the arbitrage pricing theory. Fig. 1.4 Theoretical and actual values of a call option

1.7 Option Valuation Option contracts give their holders the right to buy and sell a specific asset at some specified price on or before a specified date. Since these contracts can be valued in relation to common stock, the basic concepts involved have a number of applications to financial theory and to the valuation of other financial instruments. While there are a variety of option contracts – for example, call options, put options, combinations of calls and puts, convertible securities, and warrants – this chapter’s discussion is limited to call options. A call option gives the holder the right to buy a share of stock at a specified price, known as the exercise price, and the basic American option can be exercised at any time through the expiration date. The value of the option at expiration is the difference between the market price of the underlying stock and its exercise price (with a minimum value of zero, of course). While several factors affect the value of an option, the most important factor is the price volatility of the stock – the greater the volatility, the greater the value of the option, other things remaining the same. We will also note that the longer the time left before expiration and the higher the level of interest rates in the market, the greater the option value, all other things held the same. The theoretical value of a call option at expiration is the difference between the market price of the underlying common stock, ps and the exercise price of the option, E, or zero, whichever is greater: C D Max.Ps  E; 0/

(1.47)

When the price of the stock is greater than the exercise price, the option has a positive theoretical value that will increase dollar for dollar with the price of the stock. When the market price of the stock is equal to or less than the exercise price, the option has a theoretical value of zero, as shown in Fig. 1.4. Nevertheless, as long as some time remains before expiration, the actual market price of the option (referred to as the option premium at the time of issue) is likely to be greater than its theoretical value. This increment above the

theoretical value is called the time value or speculative value of the option, and its size will depend primarily on the perceived likelihood of a profitable move on the price of the stock before expiration of the option. The full range of possible values for the market price of the option is from the theoretical value on the low side to the market price of the stock itself on the high side. For the option price to be equal to the stock price, an infinite time to expiration would be implied. For the option price to be equal to the theoretical value only, imminent expiration would be implied. For virtually all options for which the value would be determined, however, the option price would fall somewhere between these two extremes. Because an option costs less than its underlying stock, the percentage change in option price is greater than the percentage change in stock price, given some increase in the market price of the stock. Thus, a leveraged rate of return can be earned by investment in the option rather than the stock. As stock price continues to increase, the difference between the percentage change in option price and the percentage change in stock price will tend to converge. Thus far it has been shown that the value of an option will be a function of the underlying stock price, the exercise price of the option, and the time to maturity. Yet there is still another factor that is probably the single most important variable affecting the speculative value of the option. That is the price volatility of the underlying stock. The greater the probability of significant change in the price of the stock, the more likely it is that the option can be exercised at a profit before expiration. There is another factor affecting the speculative premium for options. This is the level of interest rates in the market – specifically for option analysis, the call money rate charged by brokers for the use of margin in common-stock accounts. As this concept is discussed later, it is sufficient here to point out that the leverage achieved through option investment is similar to that achieved through direct margin purchase of the underlying common stock, but without the explicit interest cost involved in the latter. Thus, the higher the call money rate, the greater the savings from the use of options and the greater the speculative value of the option.

1 Theoretical Framework of Finance

15

To summarize, there are five variables necessary to determine the value of an American call option (ignoring dividends on the common stock): 1. and 2. Stock price–Exercise price: The relationship between these two prices determines whether the option has a positive theoretical value. 3. Time to maturity: The longer the time to maturity, the greater the speculative value of the option because the chances for a profitable movement in the price of the stock are increased. 4. Volatility of stock price: There is a positive relationship between the volatility of the underlying stock price and the speculative value of the option because with greater volatility, there is greater potential for gain on the upside and greater benefit from the downside protection involved with the option. 5. Interest rate: The higher the call money rate for direct margin purchase of common stock, the greater the relative value of being able to achieve equal amounts of leverage through the alternative of option purchase. The factors that affect the value of an option can be written in a functional form: C D f .S, X;  2 ; T,rf /

(1.48)

where: C D value of the option; S D stock price; X D exercise price;  2 D variance of the stock; T D time to expiration; and rf D risk-free rate. The value of the option increases as a function of the value of the stock for a given exercise price and maturity date. The lower the exercise price, the greater the value of the option. The longer the time to maturity, the higher the value of the option. The holder of an option will prefer more variance in the price of the stock to less. The greater the variance (price volatility) the greater the probability that the stock price will exceed the exercise price and thus benefit the holder. Considering two related financial securities – common stock and the option on the common stock – it is possible to illustrate how a risk-free hedged position can be developed. In this way, unprofitable price movements in one of the securities will be offset by profitable price movements in the other. The hedge ratio determines the portion of stock held long in relation to the options in the short position (or vice versa). With a complete hedge, the value of the hedged position can be the same regardless of the stockprice outcome. In efficient financial markets, the rate of return earned on perfectly hedged positions will be the risk-free rate. Consequently, it is then possible to determine

the appropriate option price at the beginning of the period. If the actual market price is above or below this value, arbitrage would then drive the option price toward its correct level. This process and the development of the Black-Scholes (1973) continuous type of option-pricing model will be discussed in Chaps. 23, 24, and 27. Cox et al. (1979) discrete type of binomial option-pricing model will be analyzed in Chaps. 25, 26, and 28.

1.8 Futures Valuation and Hedging A basic assumption of finance theory is that investors are risk averse. If we equate risk with uncertainty, can we question the validity of this assumption? What evidence is there? As living, functional proof of the appropriateness of the risk aversion assumption, there exist entire markets whose sole underlying purpose is to allow investors to display their uncertainties about the future. These particular markets, which primary focus on the future, are called just that, futures markets. These markets allow for the transfer of risk from hedgers (risk-averse individuals) to speculators (risk-seeking individuals). A key element necessary for the existence of futures markets is the balance between the number of hedgers and speculators who are willing to transfer and accept risk. A future contract is a standardized legal agreement between a buyer and a seller, who promise now to exchange a specified amount of money for goods or services at a future time. Of course, there is nothing really unusual about a contract made in advance of delivery. For instance, whenever something is ordered rather than purchased on the spot, a futures (or forward) contract is involved. Although the price is determined at the time of the order, the actual exchange of cash for the merchandise takes place later. For some items the lag is a few days, while for others (such as a car) it may be months. Moreover, a futures contract imparts a legal obligation to both parties of the contract to fulfill the specifications. To guarantee fulfillment of this obligation, a “goodfaith” deposit, also called margin, may be required from the buyer (and the seller, if he or she does not already own the product). To ensure consistency in the contracts and to help develop liquidity, futures exchanges have been established. These exchanges provide a central location and a standardized set of rules to enhance the credibility of these markets and thus generate an orderly, liquid arena for the price determination of individual commodities at distinct points in the future. A substantial increase in the number of types of futures contracts offered by the exchanges has been occurring over the last decade. At the same time, the growth in futures trading volume has been phenomenal. Two explanations can be offered for this increase in futures activity. These increases

16

can be intuitively correlated with the growing levels of uncertainty in many facts of the economic environment – for example, inflation and interest rates. A second view is based on the argument that even though the world has not become any more uncertain, the increased integration of financial and real markets has increased the risk exposure of any given individual. The tremendous growth in the home-mortgage and consumer-debt financial markets has allowed the purchase of more expensive real assets. This increase in the rise of individual financial leverage has increased individual exposure to interest-rate fluctuations, thereby increasing the requirements for risk-sharing across markets or between individuals with varied portfolios. Futures markets have the potential to help people manage or transfer the uncertainties that plague the world today. This section examines the basic types of futures contracts offered and the functioning of futures markets. In addition the uses of financial and index futures are illustrated, and the theoretical pricing concepts related to these financial instruments are discussed. The important terms associated with futures contracts and futures markets are defined and an analysis of futures market follows. A theory of valuation is introduced, and the section closes with a discussion of various hedging strategies and concepts.

1.8.1 Futures Markets: Overview In the most general sense, the term commodity futures is taken to embrace all existing futures contracts. Nevertheless, for purposes of clarity and classification its meaning here is restricted to a limited segment of the total futures markets. Accordingly, futures contracts can be classified into three main types. 1. Commodity futures 2. Financial futures 3. Index futures Within this classification commodity futures include all agriculturally related futures contracts with underlying assets, such as corn, wheat, rye, barley, rice, oats, sugar, coffee, soybeans, frozen orange juice, pork bellies, live cattle, hogs, and lumber. Also within the commodity-futures framework are futures contracts written on precious metals, such as gold, silver, copper, platinum, and palladium, and contracts written on petroleum products, including gasoline, crude oil, and heating oil. Many of the futures contracts on metals and petroleum products have been introduced as recently as the early 1980s. Producers, refineries, and distributors, to name only a few potential users, employ futures contracts to assure a particu-

1

Theoretical Framework of Finance

lar price or supply – or both – for the underlying commodity at a future date. Futures-market participants are divided into two broad classes: hedgers and speculators. Hedging refers to a futuresmarket transaction made as a temporary substitute for a cashmarket transaction to be made at a later date. The purpose of hedging is to take advantage of current prices by using futures transactions. For example, banks and corporations can be hedgers when they use futures to fix future borrowing and lending rates. Futures market speculation involves taking a short or long futures position solely to profit from price changes. If you think that interest rates will rise because of an increase in inflation, you can sell T-bill futures and make a profit if interest rates do rise and the value of T-bills falls. Financial futures are a trading medium initiated with the introduction of contracts on foreign currencies at the International Monetary Market (IMM) in 1972. In addition to futures on foreign currencies, financial futures include contracts based on Treasury bonds (T-bonds), Treasury bills (T-bills), Treasury notes (T-notes), bank certificates of deposit, Eurodollars, and GNMA mortgage securities. These latter types of financial futures contracts are also referred to as interest-rate futures as their underlying asset is an interest-bearing security. While foreign-currency futures arose with the abolition of the Bretton Woods fixed exchange-rate system during the early 1970s, interest-rate futures surged in popularity and number following the change in U.S. monetary policy in October 1979. The effect of the Federal Open Market Committee’s decision to deemphasize the traditional practice of “pegging” interest rates was to greatly increase the volatility of market interest rates. Thus, interest-rate changes have become a highly prominent risk to corporations, investors, and financial institutions. Index futures represent the newest and boldest innovation in the futures market to date. An index-futures contract is in for which the underlying asset is actually a portfolio of assets – for example, the Major Market Index (MMI) includes 20 stocks traded on the NYSE and the S&P index includes 500 stocks. Contracts on more diverse types of indexes include a high-quality bond index, an interest-rate index composed of interest-bearing market securities, and the consumer price index. The S&P 500 index, requiring delivery of the 500 stocks constituting the S&P 500 stock index, would certainly have dampened enthusiasm for this and similar index contracts. Because of this, an index-futures contract is settled on the basis of its cash value when the contract matures. The cash value of the contract is equal to the closing index value on its last trading day multiplied by a dollar amount of $500. Many portfolio managers are taking advantage of index futures to alter their portfolios risk-return distributions.

1 Theoretical Framework of Finance

17

1.8.2 The Valuation of Futures Contracts The discussions of each of the three classifications of futures contracts have pointed out pricing idiosyncrasies and have examined specific pricing models for particular types of contracts. Nevertheless, the underlying tenets of any particular pricing model have their roots in a more general theoretical framework of valuation. Consequently, the focus is now on the traditional concepts of futures contracts valuation.

Ft;T D St .1 C Rf;T t /

1.8.2.1 The Arbitrage Argument An instant before the futures contract matures, its price must be equal to the spot (cash) price of the underlying commodity, or: (1.49) Ft;T D St where: Ft;T D the price of the futures contract at time t, which matures at time T , where T > t and T  t is a very small interval of time; and St D the spot price of the underlying commodity at time t. If Equation (1.49) did not hold arbitrage condition would prevail. More specifically, when t D T at the maturity of the contract, all trading on the contract ceases and the futures price equals the spot price. If an instant before maturity Ft;T < St , one could realize a sure profit (an arbitrage profit) by simultaneously buying the futures contract (which is undervalued) and selling the spot commodity (which is overvalued). The arbitrage profit would equal: St  Ft;T

carry out the arbitrage process, the trader would incur certain costs. For instance, if the spot commodity were purchased because it is undervalued relative to the futures, the trader or arbitrageur would incur an opportunity or interest cost. Any funds he or she tied up in the purchase of the commodity could alternatively be earning some risk-free interest rate Rf through investment in an interest-bearing risk-free security. Therefore, the futures price should account for the interest cost of holding the spot commodity over time, and consequently Equation (1.49) can be modified to:

(1.50)

However, if Ft;T > St is the market condition an instant before maturity, smart traders would recognize this arbitrage condition and sell futures contracts and buy the spot commodity until t D T and Ft;T D St . In fact, the effect of selling the futures and buying the spot would bid their prices down and up, respectively. Thus, the arbitrage process would alleviate any such pricing disequilibrium between the futures contract and its underlying spot commodity.

1.8.2.2 Interest Costs The previous simplified argument demonstrated that the futures and spot prices must be equal an instant before the contract’s maturity. This development assumes no costs in holding the spot commodity or carrying it (storing it) across time. If such a market condition held, Equation (1.49) could be extended to apply to any point of time where t < T . However, by having to buy or sell the spot commodity to

(1.51)

where Rf;T t is the risk-free opportunity cost or interest income that is lost by tying up funds in the spot commodity over the interval T  t.

1.8.2.3 Carrying Costs Since theories on the pricing of futures contracts were developed long before the introduction of financial or index futures, the costs of storing and insuring the spot commodity were considered relevant factors in the price of a futures contract. That is, someone who purchased the spot commodity to hold from time t to a later period T incurs the costs of actually housing the commodity and insuring it in case of fire or theft. In the case of livestock such as cattle or hogs, the majority of this cost would be in feeding. The holder of a futures contract avoids these costs borne by the spot holder, making the value of the contract relative to the spot commodity increase by the amount of these carrying costs. Therefore, Equation (1.51) can be extended: Ft;T D St .1 C Rf;T t / C CT t

(1.52)

where CT t is the carrying costs associated with the spot commodity for the interval T  t.

1.8.2.4 Supply and Demand Effects As for other financial instruments or commodities, the price of a futures contract is affected by expectations of future supply and demand conditions. The effects of supply and demand for the current spot commodity (as well as for the future spot commodity) have not yet been considered in this analysis. If the probability exists that future supplies of the spot commodity might significantly differ from current supplies, then this will affect the futures price. The discussion up to this point has assumed that the aggregate supply of the commodity was fixed over time and that demand remained

18

1

constant; however, for agricultural, financial, and index futures this is a very unrealistic assumption. For instance, if it is expected that the future available supply of wheat for time T will decline because of poor weather, and demand is unchanged, one would then expect the future spot price of wheat to be higher than the current spot price. Furthermore, a futures contract on wheat that matures at time T can also be considered to represent the expected spot price at time T and consequently should reflect the expected change in supply conditions. In a more extreme fashion, if there is no current supply of wheat, then the futures price would reflect only future supply conditions and the expected future spot price at time T . This can be expressed as: ST / Ft;T D Et .e

(1.53)

S T / is the spot price at a future point T exwhere Et .e pected at time t, where t < T . The tilde above ST indicates that the future spot price is a random variable because future factors such as supply cannot presently be known with certainty. Equation (1.53) is called the unbiased-expectations hypothesis because it postulates that the current price of a futures contract maturing at time T represents the market’s expectation of the future spot price at time T . Which of these expressions for the price of a futures contract at time t will hold in the market – the arbitrage pricing relationship in Equation (1.52) or the unbiased-expectations hypothesis in Equation (1.53)? As the markets are assumed to be efficient the answer is that, the market price of the futures contract will take on the minimum value of either of these two pricing relationship, or: S T /; St .1 C Rf;T t / C CT t Ft;T D MinŒEt .e

(1.54)

For any storable commodity on a given day t, the futures price Ft;T will be higher than the spot price St on day tI Ft;T > St . The amount by which the futures price exceeds the spot price .Ft;T St / is called the premium. In most cases this premium is equal to the sum of financial costs St Rf;T t and carrying costs CT t . The condition of Ft;T > St is associated with a commodity market called a normal carryingchange market. In general, the difference between the futures price Ft;T and spot price St is called the basis. Basis D Ft;T  St

(1.55)

1.8.2.5 The Effect of Hedging Demand John Maynard Keynes (1930), who studied the futures markets as a hobby, proposed that for some commodities there was a strong tendency for hedgers to be concentrated on the

Theoretical Framework of Finance

short side of the futures market. That is, to protect themselves against the risk of a price decline in the spot commodity, the spot holder or producer (such as a farmer) would hedge the risk by selling futures contracts on his or her particular commodity. This demand for hedging, producing an abundant supply of futures contracts, would force the market price below that of the expected spot price at maturity (time T ). Moreover, the hedgers would be transferring their price S T / and Ft;T risk to speculators. This difference between Et .e when Ft;T < Et .e S T /, can be thought of as a risk premium paid to the speculators for holding the long futures position and bearing the price risk of the hedger. This risk premium can be formulated as: S T /  Ft;T Et .RP / D Et .e

(1.56)

where Et .RP / is the expected risk premium paid to the speculator for bearing the hedger’s price risk. Keynes described this pricing phenomenon as normal backwardation. When the opposite conditions exist – hedgers are concentrated on the long side of the market and bid up the futures spot pricing Ft;T over the expected future spot S T / – the pricing relationship is called contango (that is, Et .e S T //. To reflect the effect of normal Et .RP / D Ft;T  Et .e backwardation or contango on the current futures price, the S T / term in Equation (1.56) must be adjusted for the efEt .e fects of hedging demand: S T / C E.RP /; St .1 C Rf;T t / C CT t Ft;T D Min ŒEt .e (1.57) Equation (1.57) expresses a broad pricing framework for the value of a futures contract. Over the life of the futures contract the futures price must move toward the cash price, because at the maturity of the futures contract the futures price will be equal to the current cash price. If hedgers are in a net short position, then futures prices must lie below the expected future spot price, and futures prices would be expected to rise over the life of the contract. However, if hedgers are net long, then the futures price must lie above the expected futures spot price and the price of the futures would be expected to fall. Either a falling futures price (normal backwardation) or a rising futures price (contango) determines the boundaries within which the actual futures price will be located. However, numerous other factors can alter and distort the relationship shown by Equation (1.57). For instance, the analysis implicitly assumes that interest rates remain constant from time t to the contract’s maturity date at time T . However, since market interest rates fluctuate, an increasing or decreasing term structure of interest rates would bias the price of the futures contract higher or lower. In fact, the more accurate one’s forecast of future interest rates, the more accurate the current valuation of the futures contract.

1 Theoretical Framework of Finance

Empirical research casts rather strong doubt on the size of the expected risk-premium component of futures prices, particularly for financial and index futures. In fact, the expectation of speculators, along with actual futures contract supply and demand conditions in the pit, can combine to reverse the effect expected by Keynes. This results in part from the makeup of futures’ users, a clear majority of whom are not hedgers as suggested by Keynes. Additionally, a futures contract for which an illiquid level of trading volume exists would put the bid and offer prices for the contract further apart. A seller of such a futures contract would require more than the theoretical fair price as compensation for the risk undertaken. The risk is of prices starting to rise in an illiquid market in which the position cannot be immediately closed out. It costs money to maintain the position; therefore, a premium is required to cover this cost.

19

Table 1.1 The components of basis risk Type of risk Components Expiration-date risk

Futures contracts are not usually available for every month. If a hedger needed a futures contract for July and the only contracts that were available were for March, June, September, and December, the hedger would have to select either the June or September contract. Either of these contracts would have a different price series than a July contract (if one existed). Hence, the hedger cannot form a perfect hedge and is faced with the chance that the basis may change

Location risk

The hedger requires delivery of the futures contract in location Y, but the only futures contracts available are for delivery in location X. Hence, the hedger cannot form a perfect hedge because of the transportation costs from X to Y; this may cause the basis to change

Quality risk

The exact standard or grade of the commodity required by the hedger is not covered by the futures contract. Therefore, the price movement of commodity grade A may be different from the price movement of commodity grade B, which will cause the basis to change and prevent the hedger from forming a perfect hedge

Quantity risk

The exact amount of the commodity needed by the hedger is not available by a single futures contract or any integer multiple thereof. Hence, the amount of the commodity is not hedged exactly; this prevents the hedger from forming a perfect hedge, and the underhedged or overhedged amount is subject to risk

1.8.3 Hedging Concepts and Strategies The underlying motivation for the development of futures markets is to aid the holders of the spot commodity in hedging their price risk; consequently, the discussion now focuses on such an application of futures market. Four methodologies based on various risk-return criteria are examined; moreover, to fully clarify the hedger’s situation some of the common problems and risks that arise in the hedging process are analyzed.

1.8.3.1 Hedging Risks and Costs As mentioned previously, hedging refers to a process designed to alleviate the uncertainty of future price changes for the spot commodity. Typically this is accomplished by taking an opposite position in a futures contract on the same commodity that is held. If an investor owns the spot commodity, as is usually the case (a long position), the appropriate action in the futures market would be to sell a contract (a short position). However, disregarding for the moment the correct number of futures contracts to enter into, a problem arises if the prices of the spot commodity and the futures contract on this commodity do not move in a perfectly correlated manner. This nonsynchronicity of spot and futures prices is related to the basis and is called the basis risk. The basis has been defined as the difference between the futures and spot prices. Basis risk is the chance that this difference will not remain constant over time. Four types of risk contribute to basis risk; these are defined in Table 1.1. These four types of risks prevent the hedger from forming a perfect hedge (which would have zero risk). Even though the hedger

is reducing the amount of risk, it has not been reduced to zero. It is often said that hedging replaces price risk with basis risk. The potential causes of basis risk are not necessarily limited to those identified in Table 1.1. Hence, basis risk is the prominent source of uncertainty in the hedging process. Other potential causes are (1) supply-demand conditions and (2) cross-hedging consequences. Even if the futures contract is written on the exact commodity that the hedger holds, differing supply and demand conditions in the spot market and futures market could cause the basis to vary over time. Occasionally, speculators in the futures market will bid the futures price above or below its equilibrium position, due perhaps to the excitement induced by an unexpected news release. Of course, the market forces of arbitrage will eventually bring the spot and futures prices back in line. The limiting case is at the expiration of the futures contract, when its price must converge to the spot price.

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Consequently, the disequilibrating influence on the basis stemming from supply-demand forces can be alleviated by entering a futures contract that matures on the exact day that the hedger intends to sell the spot commodity. But although most futures contracts are quite flexible, it is unlikely that any contract would correlate so precisely with the hedger’s needs. In some cases (such as for agricultural commodities, where futures contracts are offered that mature each month) the basis risk due to nonsimultaneous maturities is not so great. However, for other commodities, particularly financial instruments, futures contracts maturing 3 months apart are more typically offered. Thus, at the time the hedger needs to sell the spot commodity in the market, any protection in price risk over the hedging period could conceivably be wiped out by a temporary adverse change in the basis. Cross-hedging refers to hedging with a futures contract written on a nonidentical commodity (relative to the spot commodity). Although not often necessary with agricultural futures, cross-hedging is frequently the best that can be done with financial and index futures. Changes in the basis risk induced by the cross-hedge are caused by less-than-perfect correlation of price movements between the spot and futures prices – even at maturity. That is, because the spot commodity and futures contract commodity are different, their respective prices will tend to be affected (even though minutely at times) by differing market forces. While the futures price must equal the price of its underlying spot commodity at the contract’s maturity date, this condition does not necessarily hold when the hedger’s commodity is not the “true” underlying asset. Therefore, even when the liquidation of the spot commodity coincides with the maturity of the futures contract, there is no guarantee of obtaining the original price of the commodity that held at the initiation of the hedge.

1

Theoretical Framework of Finance

where:

St D change in the spot price at time t;

Ft D change in the futures price at time t; a D constant; H D hedged ratio; and et D residual term at time t. Furthermore, the hedge ratio measure can be better understood by defining it in terms of its components: Xf Xs

D

 S; F DH 2  F

(1.59)

where: Xf and Xs D the dollar amount invested in futures and spot;  S; F D the covariance of spot and futures price changes; and 2 D the variance of futures price changes.  F Thus H , the minimum-variance hedge ratio computed in variability, is also a measure of the relative dollar amount to be invested in futures per dollar of spot holdings. In a sense it is a localized beta coefficient similar in concept to the beta of a stock à la capital asset pricing theory. As a measure of hedging effectiveness, Johnson utilizes the squared simple-correlation coefficient between spot and futures price changes, 2 . More formally, Johnson’s hedgingeffectiveness measure can be ascertained by first establishing the following expression: HE D 1 

VH Vu

(1.60)

where: 1.8.3.2 The Johnson Minimum-Variance Hedge Strategy Developed within the framework of modern portfolio theory, the Johnson hedge model (1960) retains the traditional objective of risk minimization but defines risk as the variance of return on a two-asset hedge portfolio. As in the two-parameter world of Markowitz (1959), the hedger is assumed to be infinitely risk averse (that is, the investor desires zero variance). Moreover, with the risk-minimization objective defined as the variance of return of the combined spot and futures position, the Johnson hedge ratio is expressed in terms of expectations of variances and covariances for price changes in the spot and futures markets. The Johnson hedge model can be expressed in regression form as: (1.58)

St D a C H Ft C et

2 Vu D variance of the unhedged spot position D Xs2  S 2  S D variance of spot price changes; and VH D the variance of return for the hedged portfolio D 2 .1  2 /. Xs2  S

By substituting the minimum-variance hedge position in the futures, Xf :  2 XS2  S .1  2 / D 2 HE D 1  2 XS2  S 

(1.61)

In simpler terms then, the Johnson measure of hedging effectiveness is the R2 of a regression of spot-price changes on futures-price changes. To utilize this hedging method, it is necessary to regress historical data of spot-price changes on futures-price changes. The resulting beta coefficient from the regression would be the localized Johnson hedge ratio, and the regression R2 would represent the expected degree of

1 Theoretical Framework of Finance

variance minimization using this hedge ratio over the hedging horizon. “Localized” and “expected” must be emphasized because, first of all, although the Johnson hedge ratio can be re-estimated, it nonetheless is a static measure based on historical data. What held for the past may not hold precisely for the future. Moreover, large price moves may distort this hedge ratio considerably. Hence, R2 is what can be expected based on the past in terms of variance reduction for the total hedge position. It should not be expected to hold exactly.

1.8.3.3 The Howard-D’Antonio Optimal Risk-Return Hedge Strategy The classic one-to-one hedge is a naïve strategy based upon a broadly defined objective of risk minimization. The strategy is naïve in the sense that a hedging coefficient of one is used regardless of past or expected correlations of spotand futures-price changes. Working’s strategy brings out the speculative aspects of hedging by analyzing changes in the basis and, accordingly, exercising discrete judgment about when to hedge and when not to hedge. The underlying objective of Working’s decision rule for hedgers is one of profit maximization. Finally, Johnson (1960), in applying the mean-variance criteria of modern portfolio theory, emphasizes the risk-minimization objective but defines risk in terms of the variance of the hedged position. Although Johnson’s method improves on the naïve strategy of a one-to-one hedge, it however, essentially disregards the return component associated with a particular level of risk. Rutledge (1972) uses both mean and variance information to derive hedge ratio. In a recent paper by Howard and D’Antonio (1984), a hedge ratio and measure of hedging effectiveness are derived in which the hedger’s risk and return are both explicitly taken into account. Moreover, some of the variable relationships derived from their analysis help explain some of the idiosyncrasies of hedging that occur in practice. Using a mean-variance framework, the HowardD’Antonio strategy begins by assuming that the “agent” is out to maximize the expected return for a given level of portfolio risk. With a choice of putting money into three assets – a spot position, a futures contract, and a risk-free asset – the agent’s optimal portfolio will depend on the relative risk-return characteristics of each asset. For a hedger, the optimal portfolio may contain a short futures position, a long futures position, or no futures position at all. In general, the precise futures position to be entered into will be determined by (1) the risk-free rate, (2) the expected returns and the standard deviations for the spot and futures positions, and (3) the correlation between the return on the spot position and the return on the futures.

21

Howard and D’Antonio arrive at the following expressions for the hedge ratio and the measure of hedging effectiveness: Hedge ratio H D

.œ  ¡/ ” .1  œ¡/

(1.62)

and s Hedging effectiveness HE D

1  2œ¡ C œ2 1  ¡2

(1.63)

where:   D f =s D relative variability of futures and spot returns; ’ D r f =.r s  i / D relative excess return on futures to that of spot; ” D Pf =Ps D current price ratio of futures to spot; œ D ˛= D .r f =f /=Œ.r s  i /=s D risk-to-excessreturn relative of futures versus the spot position; Ps ; Pf D the current price per unit for the spot and futures respectively; ¡ D simple correlation coefficient between the spot and futures returns; s D standard deviation of spot returns; f D standard deviation of futures returns; r s D mean return on the spot over some recent past interval; r f D mean return on the futures over some recent past interval; and i D risk-free rate. By analyzing the properties of œ these authors discern some important insights for the coordinated use of futures in a hedge portfolio. Numerically, œ expresses the relative attractiveness of investing in futures versus the spot position. When œ < 1; œ D 1, and œ > 1, the futures contract offers less, the same, and more excess return per unit of risk than the spot position, respectively. Since this analysis is being undertaken from a hedger’s point of view, it is assumed œ < 1. An assumption that œ > 1 would inappropriately imply that theoretically it is possible to hedge the futures position with the spot asset. From a practitioner’s perspective it is also important to note that even when œ ¤ ¡, a hedged position using the futures may not provide a real net improvement in the riskreturn performance of a portfolio. Unless HE is significantly greater than 1, other factors such as transaction costs, taxes, the potential for margin calls, and liquidity may negate the overall benefit of hedging with futures. This point, along with the previous results about hedging, helps explain why certain futures contracts highly correlated with their underlying

22

assets are not used extensively as hedging vehicles as might be expected. This section has focused on the basis concepts of futures markets. Important terms were defined and basic models to evaluate futures contracts were discussed. Finally, hedging concepts and strategies were analyzed and alternative hedging ratios were investigated in detail. These concepts and valuation models can be used in security analysis and portfolio management related to futures and forward contracts. Please see Chap. 57 for a generalized model for optimum futures hedge ratio.

1.9 Conclusion This chapter has reviewed and summarized alternative valuation theories – discounted cash flow, M and M, CAPM, APT, OPT, and futures valuation – and the Markowitz portfolio theory that are basic to introductory courses in financial management or investments. These theories can directly and indirectly become guidelines for further study of security analysis and portfolio management. Derivations and applications of these valuation models to security analysis and portfolio management are studied in detail in later parts of this handbook.

References Barnea, A., R. A. Haugen, and L. W. Senbet. 1981 “Market imperfections, agency problems, and capital structure: a review.” Financial Management 10, 7–22. Beranek, W. 1981. “Research directions in finance.” Quarterly Journal of Economics and Business 21, 6–24. Black, F. and M. Scholes. “The pricing of options and corporate liabilities.” Journal of Political Economy (May–June, 1973): 673–654. Brealey, R. and S. Mers. 1988. Principles of corporate finance, McGraw-Hill, New York. Brigham, E. F. 1988. Financial management: theory and practice, 4th ed., Dryden Press, Hinsdale. Copeland, T. E. and J. F. Weston. 1988. Financial theory and corporate policy, 3rd ed., Addison-Wesley, New York. Cox, J., S. A. Ross, and M. Rubinstein. 1979. “Option Pricing: A Simplified Approach.” Journal of Financial Economics, 7, 229–263. Cox, J. C. and M. Rubinstein. 1985. Option markets, Prentice-Hall, Englewood Cliffs, NJ. DeAngelo, H. and R. W. Masulis. 1980. “Optimal capital structure under corporate and personal taxation.” Journal of Financial Economics 8, 3–29. Fama, E. F. and M. H. Miller. 1972. Theory of finance, Holt, Rinehart and Winston, New york. Galai, D. and R. W. Masulis. 1976. “The option pricing model and the risk factor of stock.” Journal of Financial Economics 3, 53–81. Haley, C. W. and L. D. Schall. 1979. Theory of financial decision, 2nd ed., McGraw-Hill, New York. Howard, C. T. and L. J. D’Antonio. 1984. “A risk-return measure of hedging effectiveness.” Journal of Financial and Quantitative Analysis 19, 101–112.

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Theoretical Framework of Finance

Hsia, C. C. 1981. “Coherence of the modern theories of finance.” The Financial Review, 16, 27–42. Jensen, M.C. and W.H. Meckling. 1978. “Can the corporation survive?” Financial Analysts Journal, 34, 31–37. Johnson, L. L. 1960. “The theory of hedging and speculation in commodity futures.” Review of Economic Studies 27, 139–151. Keynes, J. M. 1930. A treatise on money, Vol 2. Macmillan & Co., London. Lee, C. F. 1983. Financial analysis and planning: theory and application. a book of readings. Addison-Wesley, MA. Lee, C. F. 1985. Financial analysis and planning: theory and applications. Addison-Wesley, MA. Lee, C. F. 1991. “Markowitz, Miller, and Sharpe: the first Nobel laureates in finance,” Review of Quantitative Finance and Accounting 1 209–228. Lee, C. F. and J. E. Finnerty. 1990. Corporate finance: theory, method, and applications. Harcourt Brace Jovanovich, Orlando, FL. Lee, C. F. and J. C. Junkus. 1983. “Financial analysis and planning: an overview.” Journal of Economics and Business 34, 257–283. Leland, H. and D. Pyle. 1977. “Informational asymmetries, financial structure, and financial intermediation.” Journal of Finance 32, 371–387. Mao, H. C. F. 1969. Quantitative analysis of financial decisions, The Macmillan Company, New York. Markowitz, H. 1959. Portfolio selection, Wiley, New York. Miller, M. H. 1977. “Debt and taxes.” Journal of Finance 32, 101–175. Miller, M. H. 1988. “The Modigliani–Miller proposition after 30 years.” Journal of Economic Perspectives 2, 99–120. Miller, M. H. and F. Modigliani. 1961. “Dividend policy growth and the valuation of share.” Journal of Business 34, 411–433. Modigliani, F. and M. Miller. 1958. “The cost of capital, corporation finance and the theory of investment.” American Economic Review 48, 261–297. Modigliani, F. and M. Miller. 1963. “Corporate income taxes and the cost of capital: a correction.” American Economic Review 53, 433–443. Pogue, G. A. and K. Lull. 1974. “Corporate finance: an overview.” Sloan Management Review, 15, 19–38. Reilly, F. K. 1985. Investment analysis and portfolio management, 2nd ed., Dryden Press, Hinsdale. Ross, S. 1976. “The arbitrage theory of capital asset pricing.” Journal of Economic Theory, 13, 341–360. Ross, S. A. 1977a. “The determination of financial structure: the incentive signalling approach.” Bell Journal of Economics 8, 23–40. Ross, S. 1977b. “Return, risk and arbitrage.” in Risk and return in finance, I Friend and J. L. Bicksler. (Eds.). vol. 1, Ballinger, Cambridge, MA. Ross, S. 1978. “Mutual fund separation in financial theory – the separating distributions.” Journal of Economic Theory 15, 254–286. Rutledge, D. J. S. 1972. “Hedgers’ demand for futures contracts: a theoretical framework with applications to the United States soybean complex.” Food Research Institute Studies 11, 237–256. Stiglitz, J. E. 1969. “A re-examination of the Modigliani-Miller theorem.” The American Economic Review 54, 784–793. Stiglitz, J. E. 1974. “On the irrelevance of corporate financial policy.” The American Economic Review 54, 851–866. Taggart, R. A Jr. 1980. “Taxes and corporate capital structure in an incomplete market.” Journal of Finance 35, 645–659. Van Horne, J. C. 1985. Financial management and policy, 6th ed., Prentice-Hall, Englewood Cliffs, NJ. Weston, J. F. 1981. “Developments in finance theory.” Financial Management 10 (Tenth Anniversary Issue), 5–22. Weston, J. F. and T. E. Copeland. 1986. Managerial finance, 8th ed., Dryden Press, Hinsdale. Working, H. 1953. “Hedging reconsidered.” Journal of Farm Economics 35, 544–561.

Chapter 2

Investment, Dividend, Financing, and Production Policies: Theory and Implications

Abstract The purpose of this chapter is to discuss the interaction between investment, financing, and dividends policy of the firm. A brief introduction of the policy framework of finance is provided in Sect. 2.1. Section 2.2 discusses the interaction between investment and dividends policy. Section 2.3 discusses the interaction between dividends and financing policy. Section 2.4 discusses the interaction between investment and financing policy. Section 2.5 discusses the implications of financing and investment interactions for capital budgeting. Section 2.6 discusses the implications of different policies on the beta coefficients. The conclusion is presented in Sect. 2.7.

investment and financing policy is discussed; the role of the financing mix in analyzing investment opportunities will receive detailed treatment. Again, the recognition of risky debt will be allowed, so as to lend further practicality to the analysis. The recognition of financing and investment effects are covered in Sect. 2.5, where capital-budgeting techniques are reviewed and analyzed with regard to their treatment of the financing mix. In this section several numerical comparisons are offered, to emphasize that the differences in the techniques are nontrivial. Section 2.6 will discuss implication of different policies on systematic risk determination. Summary and concluding remarks are offered in Sect. 2.7.

Keywords Investment policy r Financing policy r Dividend policy r Capital structure r Financial analysis r Financial planning, Default risk of debt r Capital budgeting r Systematic risk

2.2 Investment and Dividend Interactions: The Internal Versus External Financing Decision

2.1 Introduction This chapter discusses the three-way interaction between investment, financing, and dividend decisions. As shown in financial policy literature, there exists a set of ideal conditions under which there will be no interaction effects between the areas of concern. However, the ideal conditions imposed by academicians to analyze the effects dividend and financing policy have on the investment decision and the value of the firm are not realistic, when one considers financial management in the real world. Thus, an overview of the interactions of financing, investment, and dividend decisions is important for those concerned with financial analysis and planning. This chapter sequentially addresses the three interaction effects. In Sect. 2.2 the relation between investment and dividend policy is explored through a discussion of internal vs. external financing, with the emphasis on the use of retained earnings as a substitute for new equity, or vice versa. The interaction between corporate financing and dividend policies will be covered in Sect. 2.3, where default risk on debt is recognized. Then in Sect. 2.4, the interaction between

Internal financing consists primarily of retained earnings and depreciation expense, while external financing is comprised of new equity and new debt, both long and short term. Decisions on the appropriate mix of these two sources for a firm are likely to affect both the payout ratio and the capital structure of the firm, and this in turn will generally affect its market value. In this section an overview of internal and external financing is provided, with the discussion culminating in a summary of the impacts that earnings retention or earnings payout (with or without supplemental financing from external sources) can have on a firm’s value and on planning and forecasting.

Internal Financing Changes in equity accounts between balance-sheet dates are generally reported in the statement of retained earnings. Retained earnings are most often the major internal source of funds made available for investment by a firm. The cost of these retained earnings is generally less than the cost associated with raising capital through new common-stock issues.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_2, 

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24

Table 2.1 Payout ratio – composite for 500 firms

2

Investment, Dividend, Financing, and Production Policies: Theory and Implications

1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973

0.58 0:567 0:549 0:524 0:517 0:548 0:533 0:547 0:612 0:539 0:491 0:414

It follows that retained earnings, rather than new equity, should be used to finance further investment if equity is to be used and the dividend policy (dividends paid from retained earnings) is not seen to matter. The availability of retained earnings is then determined by the firm’s profitability and the payout ratio, the latter being indicative of dividend policy. Thus we find that the decision to raise funds externally may be dependent on dividend policy, which in turn may affect investment decisions. The payout ratios indicated in Table 2.1 show that, on average, firms in the S&P 500 retained more than 50% of their earnings after 1971 rather than pay them out in dividends. This is done because of the investment opportunities available for a firm, and indicates that retained earnings are a major source of funds for a firm.

External Financing External financing usually takes one of two forms, debt financing or equity financing. We leave the debt vs. equity question to subsequent sections, and here directly confront only the decision whether to utilize retained earnings or new common stock to finance the firm. We have previously shown that the market value of the firm is unaffected by such factors if dividend policy and capital structure are irrelevant. It should also be clear that dividend policy and capital structure can affect market values. Therefore, the consideration of an optimal internal-external financing combination is important in the field of financial management. This optimal combination is a function of the payout ratio, the debt-to-equity ratio, and the interaction between these two decision variables. From this it can be shown (and this will be the topic of discussion in the following paragraphs) that different combinations of internal and external

1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985

0.405 0:462 0:409 0:429 0:411 0:38 0:416 0:435 0:422 0:399 0:626 0:525

1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997

0.343 0:559 0:887 0:538 0:439 0:418 0:137 0:852 0:34 0:346 0:327 0:774

1998 1999 2000 2001 2002 2003 2004 2005 2006

0.234 0.435 0.299 0.383 0.04 0.259 0.311 0.282 0.301

financing will have different effects on the growth rate of earnings per share, of dividends per share, and, presumably, of the share price itself. Higgins (1977) considered the amount of growth a firm could maintain if it was subject to an external equity constraint and sought to maintain certain debt and payout ratios. He was able to show the sustainable growth rate in sales, S , to be: S D

p.1  d /.1 C L/ .rr/.ROE/ D t  p.1  d /.1 C L/ 1  .rr/.ROE/

(2.1)

where p, profit margin on sales; d , dividend payout ratio; L, debt-to-equity ratio; t, total asset-to-sales ratio; rr, retention rate; ROE, return on equity. Higgins (1977) used 1974 U.S. manufacturing firms’ composite financial statement data as an example to calculate S  . Using the figures p D 5:5%; d D 33%; L D 88% and t D 73%, he obtained: S D

.0:055/.1  0:33/.1 C 0:88/ D 10:5% 0:73  .0:055/.1  0:33/.1 C 0:88/

Using this method we can calculate sustainable growth rate in sales for financial analysts and planning models. From this we can see that a firm with many valuable investment opportunities may be forced to forego some of these opportunities due to capital constraints, and the value of the firm will not be maximized as it could have been. Dividend and capital-structure decisions relate directly to investment decisions. While it may not be reasonable to assume that the firm could not issue new equity, the question is at what cost it can be raised under such a constraint. In an even more practical vein, Higgins also incorporated inflation into his model, acknowledging the fact that most depreciation methods that serve to make depreciation a source of funds are founded on historical costs, and not the

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

replacement values the firm must pay to sustain operations at their current level. Introducing the new variables defined below, it is possible to define sustainable growth in real and nominal terms, the former being the item of interest here. Let c, nominal current assets to nominal sales; f, nominal fixed assets to real sales; j, inflation rate. So real sustainable growth Sr is Sr D

.1 C j /p.1  d /.1 C L/  jc : .1 C j /c C f  .1 C j /p.1  d /.1 C L/ (2.2)

Using figures from the manufacturing sector and an inflation rate of 10%, which at the time of writing was approximately the actual inflation rate then prevailing, it was found that real sustainable growth was only a third of that of the nominal figure; this serves to further emphasize the importance of the interaction between dividend policy and investment decisions, since the former acts as a constraint on the latter. Higgins (1977) paper has recently been generalized by Johnson (1981) and Higgins (1981). The usual caveat associated with the dividend-irrelevance proposition is that investment policy is unaffected; that is, new equity is issued to replace those retained earnings that are paid out. Here we emphasize new equity with the intent of avoiding financing-policy questions. Knowing that the flotation costs involved with new issues could be avoided by employing retained earnings, the effect such a strategy has on firm value is largely an empirical question. Any such tests directed toward this issue must also recognize that there may be a preference for dividends that may dominate or mitigate the flotation-cost effects. With these factors in mind, Van Horne and McDonald (1971) ran cross-sectional regressions on samples of utility stocks and electronics manufacturers to see whether dividend payouts and the rates of new issues of common stocks had significant effects on priceearnings ratios. The utility industry results indicate that, at the lower end of the new-equity issue spectrum, the dividendpreference effect overshadowed the flotation-cost effect, a result that appeared to shift in direction when higher levels of new equity financing were considered. Further analysis into the dividend-preference question has been performed by Litzenberger and Ramaswamy (1979). Using a generalized capital-asset-pricing model, they examined this trade-off between dividend preference and tax effects, all in an effort to justify the contention of an optimal dividend policy. The results presented by Van Horne and McDonald were admittedly tentative (especially since the electronics industry sample yielded few corroborating results), and most of the t-statistics associated with the utility sample new-issue coefficients were quite low. The strongest statement that could be made following this analysis is that there appears to be little detrimental effect on firm valuation from following a strategy of main-

25

taining a high payout rate and financing further investment with new-equity issues.

2.3 Interactions Between Dividend and Financing Policies If we were able to hold the capital structure of firms constant, then we would be able to determine the advantage or disadvantage of internal financing relative to external financing. Van Horne and McDonald, as briefly mentioned before, were able to empirically test for the effects created by either policy, providing the substance for a major part of the following discussion. If we were to go into more depth and specifically consider firms that issue risky debt while maintaining shareholder’s limited-liability status, and still keeping to the chosen internal or external equity plan, then the interactions between financing and investment decisions can affect the relative positions of stock and bondholders. Black (1976) argued that a possible strategy a firm could follow to transfer economic resources from bondholders to stockholders is to pay as generous dividends as possible. In this way the internal-external financing plan is predetermined, as paying large dividends jeopardizes the bondholders’ position as assets are siphoned away from the firm, all to the gain of the shareholders. In this section the interactions between dividend and financing policy will be analyzed in terms of (1) cost of equity capital and (2) the default-risk viewpoint of debt.

Cost of Equity Capital and Dividend Policy1 Van Horne and McDonald (1971) chose to develop a cross-sectional model to test for the significance of dividend payouts in the valuation process in the electric utility industry. The use of the electric utility industry proved to be operationally less difficult than other industries because it was desired to hold interfirm differences constant, and independent variable selection to this end was relatively straightforward. Since the authors were interested in capitalization rates, year-end P/E ratios were selected to be the dependent variables.1 This value was thought to be a function of three variables: growth in assets, dividend payout, and financial risk. The latter variable is considered sufficient for this sample due to the homogeneity of the other aspects of the included firm’s risks. The model can be written out and more carefully defined by the following; P =E D a0 C a1 g C a2 p C a3 R C u;

(2.3)

26

2

Investment, Dividend, Financing, and Production Policies: Theory and Implications

where g, compound growth of assets over eight previous years; p, dividend payout ratio on an annual basis; Lev, interest charges/[operating revenues – operating expenses]; u, error term. While the growth and risk factors do not correspond exactly to what most capital-market theory tells us is relevant, research with respect to this particular industry lends credence to these variables as defined here (see Malkiel 1970). Upon regressing the data for the 86 companies included in the sample, all three independent variables are found to be statistically significant. Of the 86 firms, all of which paid dividends, 37 firms also had new equity issues. Testing to see whether the 37 firms that issued new equity came from the same population as those that did not raise new equity, the researchers found no essential difference. Since we know there are nontrivial flotation costs associated with the issuance of new equity, the finding of no difference between the sub samples implies one of two things: either the costs associated with new issues are relatively too small compared to the total costs of the firm to be detected (despite their large absolute size, or size relative to the new issue by itself), or a net preference for dividends by holders of electric utility stocks exists that acts to offset the aforementioned expenses. The main thrust of this paper was to assess the impact of new equity flotation costs on firm value. With the number of firms issuing new equity, Van Horne and McDonald were able to calculate new-issue ratios – that is, new issues/total shares outstanding at year-end – and separate these firms into four groups, leaving those firms issuing new equity in a separate classification, as indicated in the upper portion of Table 2.2. Adding dummy variables to Equation (2.3), the effect of new issue rates could be analyzed: P =E D a0 C a1 .g/ C a2 p C a3 .Lev/ C a4 .F1 / Ca5 .F2 / C a6 .F3 / C a7 .F4 /:

(2.4)

where g, compound growth rate; Lev, financial risk measured by times interest earned, and F1 ; F2 ; F3 ; F4 , dummy variables representing levels of new equity financing. We would expect negative coefficients on the dummy variables if the flotation costs were to be relevant, but in fact all coefficients were positive, though only one was significant (possibly due to small sample size and small relative differ-

Table 2.2 New-issue ratios of electric utility firms

ences). Empirical results are indicated in the lower portion of Table 2.2. However, by replacing the dummy variable of Group E for any one of the four dummy variables discussed above, Van Horne and McDonald found that the estimate of its coefficient is negative. One question pertaining to the figures presented here stems from the interpretation of the significance of the dummy variables. As it turns out, the class B dummy-variable coefficient is greater than the class A coefficient, supposedly because of a preference for dividend payout, while actually that payout was lower in class B. The question is whether the dummy that is intended to substantiate the dividend-preference claim through the new-issue ratio actually tells us that investors instead attach a higher value to higher earnings retention. The finding of higher P/E ratios resulting from earnings retention would be consistent with the Litzenberger-Ramaswamy framework cited earlier, and the new-issue effect is somewhat confounded with the dividend effect. From the data we can see that the coefficients did decrease as the new-issue ratios increased, though they were not significant, and, not surprisingly, new-issue ratios were rather highly negatively correlated with dividend payout ratios. From this we can say, although with some hesitation, that external equity appears to be a more costly alternative compared to internal financing when pushed to rather extreme limits. Over more moderate ranges, or those more closely aligned with the industry averages, this claim cannot be made with the same degree of certainty, since we cannot be certain that the payout ratio does not have a positive influence on share price, and therefore an inverse relationship to the cost of equity capital.

Default Risk and Dividend Policy With the development of the Option Pricing Model, Black and Scholes (1973) have made available a new method of valuing corporate liabilities or claims on the firm. Chen (1978) has reviewed some recent developments in the theory of risky debt, and examines in a more systematic fashion the determinants of the cost of debt capital. Both Reseck (1970) and Hellwig (1981) have shown that the Modigliani

F dummy variable grouping A

B

C

D

New-issue ratio interval 0 0.001–0.05 0.05–0.1 0.1–0.15 Number of firms in interval 49 16 11 6 Mean dividend payout ratio 0:681 0:679 0:678 0:703 Dummy variable coefficient 1:86 3:23 1:26 0:89 Dummy variable t -statistic 1:33 2:25 0:84 0:51 From Van Horne and McDonald (1971), Reprinted by permission

E 0.15 and up 4 0.728 N.A. (N.A.)

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

and Miller (1958 and 1963) arbitrage process that renders capital structure irrelevant is generally invalidated if the debt under consideration is risky and the shareholders enjoy limited liability. From this and the 1963 M&M article, we can show that there do exist optimal structures for firms under the more realistic conditions mentioned above. In the option-pricing framework the two claimants of the firm, the debt holders and the equity holders, are easily seen to have conflicting interests; thus one group’s claim can only be put forth at the expense of the other party. The assumption that the total firm value is unchanged is utilized to highlight the wealth-transfer effects, but, as we see in the following section, that need not be the case. It is now necessary to find a way to value corporate debt when default risk is introduced. If the total value of the firm is known or given (this value should be reasonably well known or approximated prior to debt valuation as the former is an upper bound on the latter), then once the debt value is established the total equity value falls out as the residual. It is also required that we know how to perform the transfer of wealth (since the stated goal of corporate finance is to maximize the shareholders’ wealth), and learn of any possible consequences that could result from such attempts. Merton (1974) sought to find the value of risky corporate debt, assuming that the term structure of riskless interest rates was flat and known with certainty. This was an indirect goal, since the true emphasis lay in finding a way to value the equity of the firm as a continuous option. The further assumption of consol-type debt was also employed in an attempt to avoid the transactions-cost arguments involved with rolling over debt at maturity. Invoking Modigliani and Miller’s Proposition I (M&M 1958) and allowing for risky debt, we find the value of the stock to be the difference between the value of the firm as a whole and the value of the total debt financing employed to support that whole. Explicitly, S DV 

C .1  L/; r

(2.5)

where S , total value of the firm’s stock; V , total firm value; C , constant coupon payment on the perpetual bonds; r, riskless rate of interest; L, a complicated risk factor associated with possible default on the required coupon payment. The last term of Equation (2.5) is more often represented by B, the value of the total debt claims outstanding against the firm, stated in a certainty-equivalent form. It should be apparent from this equation that by introducing a greater probability of default on the debt while maintaining firm value, the equity holders gain at the expense of the bondholders. Rendleman (1978) took Merton’s model and adapted it to allow for the tax deductibility of interest charges. As in the Merton article, debt is assumed to be of perpetual type, consols – a justifiable assumption as most firms roll over

27

their debt obligations at maturity. The tax benefit introduced is assumed to be always available to the firm. In instances where the interest expense is greater than earnings before taxes, the carryback provision of the tax code is used to obtain a refund for the amount of the previously unused tax shield and, in the event the three-year carryback provision is not sufficient to make use of the tax shield, Rendleman suggests that a firm could sell that advantage to another firm by means of a merger or an acquisition, when the other firm involved could use the tax benefit. Thus the coupon (net) to be paid each time period is given by C.1  T /. At first glance it is apparent that the equity holders gain from this revision, as they are alleged to in the 1963 taxcorrected model of M&M. But something else is at work as well: Since the firm is subject to a lower debt-service requirement, it can build a larger asset base, which serves to act as insurance in the case of default probabilities owing to the small net coupon payment. In short, the risk premium associated with debt is reduced and the value of the stock, given in Equation (2.5) can be rewritten as shown below: S DV 

C .1  T / .1  L /; r

(2.6)

where all the variables are as defined before and L is less than the L given before because of the lessened default risk. It has been argued above that the bondholders benefit when interest is tax-deductible, excluding the possibility here that an overzealous attempt to lever the firm is quickly undertaken when the tax-deductibility feature is introduced, and Equations (2.5) and (2.6) seem to indicate that the value of the shareholders’ claim also increases, although we have not yet provided any theoretical justification for such a statement. Rendleman’s analysis gave us the L-value, which is the clue to this seeming problem. Management can act in a way to jeopardize the bondholders’ claims by issuing more debt and thus making all debt more risky, or by undertaking projects that are riskier than the average project undertaken by the firm. This is the subject of the next section. Another possibility is to pay large dividends so as to deplete the firm of its resources, thus paying off the shareholders but hurting the bondholders. Black (1976) goes so far as to suggest that the firm could liquidate itself, pay out the total as a dividend, and leave the bondholders holding the proverbial bag. Bond covenants, of course, prevent this sort of action (hence, agency theory), and if a firm could not sell its growth opportunities not yet exploited in the market, this may not maximize the shareholder’s wealth, either, but within bounds it does seem a reasonable possibility. In their review article, Barnea, Haugen, and Senbet (BHS, 1981) discuss the issues related to market imperfections, agency problems, and the implications for the consideration of optimal capital structures.2 The authors make use of M&M’s valuation theory and option-pricing theory to

28

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Investment, Dividend, Financing, and Production Policies: Theory and Implications

reconcile the differences between academicians and practitioners about the relevance of financing and dividend policies. They arrive at the conclusion that, without frictionless capital markets, agency problems can give rise to potential costs. These costs can be minimized through complex contractual arrangements between the conflicting parties. Potential agency costs may help to explain the evolution of certain complexities in capital structure, such as conversion privileges of corporate debt and call provisions. If these agency costs are real, then financial contracts that vary in their ability to reduce these costs may very well sell at different equilibrium prices or yields, even if the financial marketplaces are efficient. An optimal capital structure can be obtained when, for each class of contract, the costs associated with each agency problem are exactly balanced by the yield differentials and tax exposures. Overall, BHS show that optimal capital structures can exist and that this is still consistent with the mainstream of classical finance theory.

2.4 Interactions Between Financing and Investment Decisions Myers (1974) has analyzed in detail the interactions of corporate financing and investment decisions and the implications therein for capital budgeting. He argues that the existence of these interaction effects may be attributable to the recognition of transaction costs, corporate taxes, or other market imperfections. Ignored in his analysis is the probability of default on debt obligations, or, as could otherwise be interpreted, changes in this default risk. As alluded to in the previous section, if we consider possible effects of default risk on the firm’s investment and financing decisions, then further analysis must be performed to determine how these interactions can affect the wealth positions of shareholders and bondholders. In this section we will analyze this issue by considering both the risk-free debt case and the risky debt case. Chapter 92 and Chap. 60 discuss the theoretical and empirical issues of capital structure.

2.4.1 Risk-Free Debt Case Following Myers (1974), the basic optimization framework is presented in accordance with well-accepted mathematical programming techniques as discussed in the literature of capital-rationing. It is presented as a general formulation; it should be considered one approach to analyzing interactions and not the final word per se. Specific results derived by Chambers et al. (1982), will be used later to demonstrate the importance of considering alternative financing mixes when evaluating investment opportunities.

We identify a firm Q, which faces several investment opportunities of varying characteristics. The objective is to identify those projects that are in the stockholders’ interest (i.e., they maximize the change in the firm’s market value ex-dividend at the end of the successive time periods t D 0; 1; : : : ; T) and undertake them in order of their relative values. We specify relative values so that project divisibility remains possible, as was assumed by Myers. Required is a financing plan that specifies the desired mix of earnings retained, debt outstanding, and proceeds from the issuance of new equity shares. Let dV be a general function of four factors in a direct sense: (i) the proportion of each project j accepted, xj ; (ii) the stock of debt outstanding in period t; yt ; (iii) the total cash dividends paid in period t, Dt ; and (iv) the net proceeds from equity issued in period t; Et . For future use let Z be denoted as the debt capacity of the firm in period t, this being defined as a limit on yt imposed internally by management or by the capital markets, and Ct is the expected net after-tax cash flow to the firm in time period t. The problem can now be written as: Maximize dV .xj ; yt ; Dt ; Et / D W; subject to the constraints; .a/ Uj D xj  1  0 .j D 1; 2; : : : ; J /I .b/ UtF D yt  Zt  0 .t D 1; 2; : : : ; T /I

(2.7)

.c/ UtC D Ct Œyt yt 1 .1C.1  /r/ CDt Et  0; where £ and r are the tax rates and borrowing rates, respectively. Both are assumed constant for simplicity, but in actuality both could be defined as functions of other variables. Constraints (a) and (b) specify the percentage of the project undertaken cannot exceed 100%, and the debt outstanding cannot exceed the debt capacity limit. Constraint (c) is the accounting identity, indicating that outflows of funds equal inflows, and could be interpreted as the restriction that the firm maintain no excess funds. Constraint (b) can be used to investigate the interaction between the firm’s financing and investment decisions by examining the necessary conditions for optimization. If we assign the symbols Lj ; Lft , and Lct to be the shadow prices on constraints (a)–(c), respectively, and let Aj D dW=dxj , Ft D dW=dyt ; Zjt D dZ t =dxj , and Cjt D dCt =dxj , we can rewrite Equation (2.7) in Lagrangian form as the following: Max W 0 D W  Lj .xj  1/  Lf t .yt  Zt / Lct fCt  Œyt  yt 1 .1 C .1  t/r/ g C.Dt  Et /: (2.70 )

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

The necessary first-order conditions for the optimum are shown as Equations (2.8) through (2.11), with accompanying explanations. For each project: Aj C

T X

ΠLf t Zjt C Lct Cjt  Lj  0

(2.8)

t D0

This can be interpreted as follows: The percentage of a project undertaken should be increased until its incremental cost exceeds the sum of the incremental value of the project, the latter consisting of the added debt capacity and the value of the cash flows generated by that project. The incremental increase in value of the firm obtained by increasing xj to this maximum point is termed the Adjusted Present Value, APV (“adjusted” because of the consideration of interaction effects). We can examine each of these effects in turn. For the debt constraint in each period: Ft  Lf t C Lct  Lc;t 1 Œ1 C .1  /r  0:

(2.9)

For the constraint implied on dividends: dW  Lct  0I dDt

(2.10)

29

tax purposes. This procedure will be further investigated in the following section when we compare it with other wellaccepted capital-budgeting techniques. When dividend policy is not irrelevant, the standard argument arises as to whether the preference for dividends, given their general tax status, outweighs the transaction costs incurred by the firm when dividends are paid and new financing must be raised. In this framework, dividends are viewed as another cash flow and, as such, the issue centers on whether the cash flows from the project plus any increases in the debt capacity of the firm are adequate to cover the cost of financing, through whatever means. Although we are unable to discern the effect of dividend policy on the value of the firm, it is not outside the solution technique to incorporate such efforts by including the related expenses as inputs to the numerical-solution procedure. It is generally assumed that these interaction effects exist, so if we are to consider disregarding them, as some would insist, we must know what conditions are necessary for lack of dependence of financing and investment decisions. As Myers points out, only in a world of perfect markets with no taxes is this the case. Otherwise, the tax deductibility of the interest feature of debt suggests that the APV method gives a more accurate assessment of project viability than does the standard NPV method.

and for the new equity constraint: dW C Lct  0: dEt

(2.11)

While Equations (2.8)–(2.11) are all of interest, the focus is on Equation (2.8), which tells us that the Net Present Value (NVP) rule commonly put forth should be replaced by the APV rule, which accounts for interaction effects. Specifically considering the financing constraint, Equation (2.9), we would like to be able to find the value of this constraint, which is most easily done by assuming for the moment that dividend policy is irrelevant. Combined with Equations (2.8) and (2.9) and the definition of APV we obtain3; APV j D Aj C

T X

Zjt Ft

(2.12)

t D0

which, in the spirit of the M&M with-tax firm-valuation model, tells us that the value of a project is given by the increase in value that would occur in an unlevered firm, plus the value of the debt the project is capable of supporting. This follows from the firms ability to deduct interest expenses for

Risky Debt Case Rendleman (1978) not only examined the risk premiums associated with risky debt, but also considered the impact that debt financing could have on equity values, with taxes and without. The argument is to some extent based on the validity (or lack thereof) of the perfect-market assumption often invoked, which, interestingly enough, turns out to be a double-edged sword. Without taxes, the original M&M article claims, the investment decision of a firm should be made independent of the financing decision. But the financing base of the firm supports all the firm’s investment projects, not some specific project. From this we infer that the future investments of a firm and the risk premiums embodied in the financing costs must be considered when the firm takes on new projects. If, for example, the firm chooses to take on projects of higherthan-average risk, then this may have an adverse effect on the value of the outstanding debt (to the gain of the shareholders), and the converse holds true as well. It follows that the management of a firm should pursue more risky projects to transfer some of the firm’s risk from the shareholders to

30

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Investment, Dividend, Financing, and Production Policies: Theory and Implications

the bondholders, who do not receive commensurate return for that risk. If the bondholders anticipate this action on the part of management, then it is all the more imperative that management takes the action because the bondholders are requiring and receiving a risk premium, for which they are not incurring the “standard” or market-consensus level of risk. Myers (1977) presents an argument in which a firm should issue no risky debt. The rationale for this strategy is that a firm possesses certain real options, or investment opportunities, that unfold over time. With risky debt some of these investment projects available only to the firm of interest may not be undertaken if it is in the interest of the shareholders to default on a near-term scheduled debt payment. In this way risky debt induces a suboptimal investment policy and firm value is not maximized as it would be if the firm issued no risky debt. One problem here is that we do not have strict equivalence between equity-value maximization and firm-value maximizations, so investment policy cannot be thought of entirely in terms of firm-value maximization. This is a quirk associated with the option-pricing framework when applied to firm valuation, and it clouds the determination of what is suboptimal in the finance area.4 Although we concede that we aren’t entirely sure as to how we should treat the interaction effects of financing and investment policy, we can state with some degree of certainty that, even in the absence of the tax deductibility of interest, these two finance-related decisions are interdependent and, as a result, the financial manager should remain wary of those who subscribe to the idea that financing decisions do not matter. Allowing the tax deductibility of interest, it is ironic to note that the conclusions are not nearly as clear-cut as before. If the firm undertakes further, more risky projects, the value of debt may actually increase if the firm does not issue further debt. The shareholders gain from the new project only on its own merits. If no additional debt financing is raised, it is impossible to obtain a larger tax shield and the debt may actually become more secure as a result of the larger asset base. If, however, the firm does issue more debt and in that way acts to jeopardize the currently outstanding debt, the number of considerations multiply, and analysis becomes exceedingly difficult because the value of the project by itself, plus the value of the added tax shield, needs to be considered in light of the possible shifting of wealth due to transfers of risk among claimants of the firm. Thus it seems that, when we allow the real world to influence the model, as it well should, the only thing we can say for certain about financing and investment decisions is that each case must be considered separately.

2.5 Implications of Financing and Investment Interactions for Capital Budgeting This section is intended to briefly review capital-budgeting techniques, explain how each in turn neglects accounting for financing influences in investment-opportunity analysis, and discuss the method by which they do incorporate this aspect of financial management into the decision process. We will draw heavily on the work of Chambers et al. (1982), in making particular distinctions between methods most often covered in corporate-finance textbooks and presumably used in practice, presenting numerical comparisons derived from varying sets of circumstances. Chambers, Harris, and Pringle (CHP) examined four standard, discounted cash-flow models and considered the implications of using each as opposed to the other three. By way of simulation, they were able to deal with differences in financing projects as well as with possible differences in the risk underlying the operating cash flows generated by each project. The problem inherent in this and any project evaluation (not withstanding other difficulties) is in concentrating on the specification of the amount of debt used to finance the investment project. Project debt is therefore defined as the additional debt capacity afforded the firm as a result of accepting the project, and can alternatively be described as the difference between the firm’s optimal debt level with the project and without it. Conceptually this is a fairly concrete construct, but it still leaves some vague areas when we are actually performing the computations. It is essential to arrive at some value estimate of the estimated cash flows of a project. The CHP analysis considered the following four methods: (i) the Equity Residual method; (ii) the “after-tax Weighted Average Cost-ofCapital” method; (iii) the “Arditti-Levy” weighted cost-ofcapital method; and (iv) the Myers “Adjusted Present Value” method. For simplicity, Table 2.3 contains the definitions of the symbols used in the following discussion, after which we briefly discuss the formulation of each method, and then elaborate on the way in which each method incorporates interacting effects of financing and investment decisions.

Equity-Residual Method The equity-residual method is formulated in Equation (2.13) in a manner that emphasizes the goal of financial management, the pursuance of the interests of shareholders:

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

Table 2.3 Definitions of variables

31

Rt D Pretax operating cash revenues of the project during period t; Ct D Pretax operating cash expenses of the project during period t; dept D Additional depreciation expense attributable to the project in period t; £c D Applicable corporate tax rate; I D Initial net cash investment outlay; Dt D Project debt outstanding during period t; NP D Net proceeds of issuing project debt at time zero; rt D Interest rate of debt in period t; ke D Cost of the equity financing of the project; kw D After-tax weighted-average cost of capital (i.e., debt cost is after-tax); kAL D Weighted average cost of capital – debt cost considered before taxes; ¡ D Required rate-of-return applicable to unlevered cash-flow series, given the risk class of the project r, ke , and ¡ are all assumed to be constant over time

NPV.ER/ D

N X Œ. Rt  Ct  dept  rD t /.1  c / C Dt  . Dt  Dt C1 /

.1 C ke /t

t D1

The formula presented above can be interpreted as stating that the benefit of the project to the shareholders is the present value of the cash flows not going to pay operating expenses or to service or repay debt obligations. With these flows identified as those going to shareholders, it is appropriate, and rather easy, to discount these flows at the cost of equity. The only difficulty involved is identifying this cost of equity, a problem embodied in all capital-budgeting methods.

 ŒI  NP :

(2.13)

2.5.1 Arditti and Levy Method The Arditti-Levy method is most similar to the after-tax weighted-average cost-of-capital method. This formulation can be written as: NPV.AL/ D

N X Œ. Rt  Ct  dept  rD t /.1  c / C dept C rD t I .1 C kAL /t tD1

(2.15)

After-Tax, Weighted-Average, Cost-of-Capital Method The after-tax, weighted-average, cost-of-capital method, depicted in Equation (2.14), has two noticeable differences from the formulation of the equity-residual method: NPV D

N X . Rt  Ct  dept /.1  c / C dept t D1

.1 C kw /t

I:

(2.14)

First, no flows associated with the debt financing appear in the numerator, or as relevant cash flows. Second, the cost of capital is adjusted downward, with kw being a weighted average of debt and equity costs, the debt expense accounted for on an after-tax basis. In that way debt financing is reckoned with in an indirect manner. With the assumption that r and ke are constant over time, kw can be affected only by the debtto-equity ratio, a problem most often avoided by assuming a fixed debt-to-equity ratio.

It was necessary to restate the after-tax, weighted-average, cost-of-capital formula in this manner because the tax payment to the government has an influence on the net cash flows, and for that reason the cash-flow figures would be misleading. To rectify this problem the discount rate must now be adjusted as the interest tax shield is reflected in the cash flows and double counting would be involved. While the after-tax, weighted-average cost of capital recognized the cost of debt in the discount rate, it was akin to the equity-residual method in considering only returns to equity. The Arditti and Levy formulas imply that a weighted average discount rate including the lower cost of debt could only (or best) be used if all flows to all sources of financing were included; hence the term rDt is found at the end of the first term. This can be rationalized if one considers the case of a firm where there is one owner. The total cash flow to the owner is the relevant figure, and the discount rate applicable is simply a weighted average of the two individual required rates of return.

32

2

Table 2.4 Application of four capital budgeting techniques

Investment, Dividend, Financing, and Production Policies: Theory and Implications

Inputs: (1) ke D 0:112 (2) r D 0:041 (3) c D 0:46 (4)  D 0:0802 (5) w D 0:6 Method NPV results Discount rates 1. Equity-residual 2. After-tax WACC 3. Arditti-Levy WACC 4. Myers APV

$230:55 270:32 261:67 228:05

ke D 0:112 kw D 0:058 kAL D 0:069 r D 0:041 and  D 0:0802

From Chambers et al. (1982). Reprinted by permission

2.5.2 Myers Adjusted-Present-Value Method This method, derived in Sect. 2.4, is closely related to the Arditti-Levy method except for the exclusion of the interestexpense flows to the bondholders. In treating the financing mix, Myers implicitly assumes that the tax shield, created by the interest payments and afforded to the equity holders, has the same risk for the equity holders as the coupon or interest payments have for the bondholders. Instead of aggregating all factors and attempting to arrive at a suitable weightedaverage discount rate, Myers found it less difficult to leave operating- and financing-related flows separated, and to discount each at an appropriate rate. This formulation can be written as: APV D

N X . Rt  Ct  dept /.1  c / C dept t D1

I C

.1 C  /t N X c rD t : .1 C r /t t D1

(2.16)

This formulation appears to be closely related to the Modigliani and Miller with-tax firm-valuation model, and well it should, given Myers’ motivation for this work. We choose not to discuss it here, but the reader should be aware that Myers’ emphasis was not solely on the tax advantage of debt, as the last term in the above equation tends to imply. The four methods are obviously comparable, but usually give different figures for the net present value. The equityresidual, after-tax, weighted-average cost-of-capital and the Arditti-Levy weighted-average, cost-of-capital formulations are comparable if the value of the debt outstanding remains a constant proportion of the remaining cash flows; this is often taken to mean a constant debt ratio. This will not guarantee that the Myers APV method will yield the same results, although the Myers method is equivalent to the other three only if the project life can be described as one period, or as infinite with constant perpetual flows in all periods. By way of numerical examples we now address the task of determining the factors that create the differences in the net-presentvalue figures generated by each method. The first example involves the simplest case, where a $1,000 investment generates operating flows of $300 per

year for all 5 years of the project’s life. Debt financing comprises $600 of the project’s total financing and the principal is repaid over the five time periods in equal amounts. The discount and tax rates employed are included below in Table 2.4 with the results for each of the four methods. Because the Arditti-Levy method recognizes the acquisition of the debt capital and uses a lower discount rate, as does the after-tax WACC, these two methods give the highest netpresent-value figures. The equity-residual method recognizes only the $400 outflow at time zero, but the higher discount suppresses the net-present-value figure. The Myers APV method, though discounting financing-related flows at the cost of debt, also attains a low net-present-value figure because the majority of the flows are discounted at a higher unlevered cost-of-equity rate. Basically we can speak of three differences in these methods that create the large discrepancies in the bottom-line figures. The risk factor, reflected in discount rates that may vary over time, is a major element, as was evidenced in the example shown above. The pattern of debt and debt payments will also be an important factor, particularly so when the debt repayment is not of the annuity form assumed earlier. Finally, the recognition and valuation of the debt tax shields will play an important role in net-present-value determination, especially when the constant-debt-repayment assumption is dropped and the interest expenses and associated tax shields grow larger. In the CHP study the valuation models described earlier were employed in a simulation procedure that would allow the assessment of the investment proposal. The inputs to the capital-budgeting procedures were varied across simulations, and the effects of the changes in each input were scrutinized from a sensitivity-analysis viewpoint. They considered, but did not dwell upon, the effects of changing discount rates over time or by some scaling factor on the bottom-line valuation figures; further discussion was deferred primarily because of the multitude of possible combinations that would be of interest. Compounding the problem, one would also be interested in different debt-equity combinations and the effects these would have with changing discount rates, as the weighting scheme of the appropriate discount rates plays an integral part in the analysis. In avoiding this aspect of the analysis, the projects evaluated in the forthcoming discussion will be viewed as being of equivalent risk, or as coming

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

Table 2.5 Inputs for simulation

Project

Net cash inflows per year

33

Project life

1 2 3 4

$300 per year 5 years $253.77 per year 5 years $124.95 per year 20 years $200 per year, years 1–4 5 years $792.58 in year 5 For each project the initial outlay is $1,000 at time t D 0, with all subsequent outlays being captured in the yearly flows Debt schedule Market value of debt outstanding remains a constant proportion of the project’s market value Equal principal repayments in each year Level debt, total principal repaid at termination of project r D 0:041 Inputs: ke D 0:112 .M &M / D 0:0986 kw D 0:085 c D 0:46 M D 0:085 W D 0:3 From Chambers et al. (1982). Reprinted by permission

from the same risk class. Lest the reader feel shortchanged, it is suggested that one examine each method and verify for him or herself what changes would be produced in the netpresent-value figures with varying debt schedules. More in tune with the basic theme of this chapter, we go on to consider the effects that changing financing mixes, debt-payment patterns, and project lives have on the figures attained for each of the four methods considered, the first (financing mix) being the major issue involved. In confronting mix effects, we are required to select a model for valuing debt in the Myers APV method because ¡ (the unlevered cost of equity capital) is unobservable. In this case we must choose between numerous alternatives, the most notable being those of Modigliani and Miller (1963), where debt provides an interest-tax shield, and that of Miller (1977), where the inclusion of personal taxes on investor interest income has the effect of perfectly offsetting the tax shield the firm receives, rendering the debt tax advantage moot. In the simulation results of CHP presented in the following paragraphs, the Myers cost of unlevered equity capital is computed using each method, with subscripts denoting the particular method used. Four projects of varied cash-flow patterns and lives are to be presented and valued, each of which will be simulated with three different debt schedules. Brief descriptions of the projects and debt schedules can be found in Table 2.5, along with the fixed inputs to be used in the actual computations. The projects’ cash flows, as listed in the table, were actually manipulated in a predetermined way, so that the more interesting cases would be presented. Project 1 is simply a base figure, while project 2 has a cash-flow pattern that makes the net present value 0, when using the after-tax, weightedaverage cost of capital. Project 3 has a longer life, and thus will serve as a method of determining the effects of increasing project life. Finally, Project 4 has four level payments

with a larger final payment in year 5, intended to simulate an ongoing project with terminal value in year 5. The results of the simulations are presented in Table 2.6. For purposes of comparison, the net-present-value figures of the after-tax, weighted-average cost of capital are reported first, for two reasons. The after-tax weighted-average cost of capital is probably the best-known technique of capital budgeting, and its net present value figures are insensitive to the debt schedule. The initial weights of the debt and equity used to support the project are all that are used in calculating the weighted discount rate, rendering the repayment pattern of debt irrelevant. As indicated earlier, the after-tax weighted-average cost of capital, Arditti-Levy weighted-average cost of capital, and the equity-residual methods are all equivalent if the debt ratio is held constant. It is also of interest here that the Myer’s APV figures, using the Miller method for determining the cost of capital, are constant over debt schedules; they are, for all intents and purposes, the same as the after-tax, weightedaverage cost-of-capital figures, a finding that is reasonable if one believes that the cost of debt capital is the same as that of unlevered equity. Of all the methods cited above, the equityresidual method is the most sensitive to the debt schedule employed, with both principal and interest payments included in the cash flows, a feature compounded by the higher discount rates included in the computations. The two methods that include the interest tax shield, the Arditti-Levy method and the Myers APV (M&M) method are also sensitive to the amount of debt outstanding, the interest tax shield being of greater value the longer the debt is outstanding. In all cases the latter method gives the lowest net-present-value figures due to the treatment of the interest tax shield. In further simulations CHP showed (and it should be no surprise) that as higher levels of debt are employed and higher tax rates are encountered, the magnitude of the

34

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Investment, Dividend, Financing, and Production Policies: Theory and Implications

Table 2.6 Simulation results

Net-present-value under alternative debit schedule Project

Capital budgeting

Constant debt ratio

Equal principal

Level debt

1

After-tax WACC Arditti-levy WACC Equity-residual Myers APV (M&M) Myers APV (M) After-tax WACC Arditti-Levy WACC Equity-Residual Myers APV (M&M) Myers APV (M) After-tax WACC Arditti-Levy WACC Equity Residual Myers APV (M&M) Myers APV (M) After-tax WACC Arditti-Levy WACC Equity Residual Myers APV (M&M) Myers APV (M)

182 182 182 160 182 0 0 0 18 0 182 182 182 138 182 182 182 182 155 182

182 179 167 157 182 0 1 3 19 0 182 169 128 119 182 182 174 147 146 182

182 187 202 166 182 0 7 32 10 0 182 186 194 150 182 182 182 183 156 182

2

3

4

From Chambers et al. (1982). Reprinted by permission

differences is amplified, and the method employed in the capital-budgeting decision takes on greater and greater importance. Even so it was argued that changes associated with changes in the inputs of the longest-lived project, where the changes in the net-present-value figures were the most pronounced, were not that great when compared with the outcomes associated with changing the estimates of the cash flows by as little as 5%. The importance of this finding is that projects that are of short duration and are financed with relatively little debt are not as sensitive to the capitalbudgeting technique employed. But, in the case of longerlived projects, the method selected can have serious implications for the acceptance or rejection of a project, particularly when higher levels of debt financing are employed. Analysts undoubtedly possess their own views as to which method is stronger conceptually and, in the event of capitalbudgeting procedures, should be aware of the debt policy to be pursued. Even in light of these views, it may be prudent to use the Myers APV method with the M&M unlevered-equity cost-determination method as the first screening device for a project or set of projects. Since this method yields the most conservative figures, any project that appears profitable following this analysis should be undertaken, and any project failing this screening can be analyzed using the methods chosen by the financial manager, if further analysis is thought to be warranted.

In this book, Chap. 56 discusses the capital structure and CEO entrenchment in Asia. Moreover, Chap. 60 discusses the theory and application of alternative methods to determine optimal capital structure. Finally, Chap. 92 provides discussion on the capital structure and entry deterrence.

2.6 Implications of Different Policies on the Beta Coefficient Investment, financing, dividend, and production policy are four important policies in financial management decision. In previous sections, we have discussed investment, financing and dividend policies. In this section, we will discuss the impacts of financing, production and the dividend policy on beta coefficient determination.

Impact of Financing Policy on Beta Coefficient Determination Suppose that the security is a share in the common stock of a corporation. Let us assume that that this corporation increases the proportion of debt in its capital structure, all other relevant factors remaining unchanged. How would you

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

expect this change to affect the firm’s beta? We have seen that an increase in the level of debt leads to an increase in the riskiness of stockholders’ future earnings. To compensate for this additional risk, stockholders will demand a higher expected rate of return. Therefore, from Equation (2.17), beta must rise for this company’s stock. We see that, all other things equal, the higher the proportion of debt in a firm’s capital structure, the higher the beta of shares of its common stock.

B.1  c / (2.17) ˇL D ˇU 1 C S where ˇL is the leveraged bet; ˇU is the unlevered operating beta; B is the amount of debt; S is the amount of equity; and c is the corporate tax rate. When the market model is used to estimate a firm’s beta, the resulting estimate of the beta is the market assessment of both operating and financial risk. This is called leveraged beta. Hamada (1972) and Rubinstein (1973) suggest that Equation (2.17) can be modified to calculate the unleveraged beta. This beta is an estimate of the firm’s operating or business, risk.

Impact of Production Policy on Beta Coefficient Determination In this section, we will first discuss production policy. Then we will discuss implications of different policies on the beta coefficient determination. Production policy refers to how a company uses different input mix to produce its products. The company’s production process can be classified into either capital intensive or labor intensive, which depend on whether capital labor ratio (K=L Ratio) is larger or smaller than one. We can use either Cobb-Douglas production function or Variable Elasticity of Substitution (VES) production function to show how capital and labor affect the change of the beta coefficient. The Cobb-Douglas production function in two factors, capital .K/ and labor .L/ can be defined as follows. (2.18) Q D K a Lb where Q is firm’s output, a and b are positive parameters. Lee et al. (1990) have derived the theoretical relationship between beta coefficient and the capital labor ratio in terms of the Cobb-Douglas production functions as follows in Equation (2.18). ˇD

.1 C r/Cov.e; Q RQ m / Q Var.Rm / f Œ1  .1  E/b g

(2.19)

35

where r, the risk-free rate; RQ m , return on the market portfolio; e, Q random price disturbances with zero mean; E D .@P =@Q/.Q=P /, an elasticity constant; b, contribution of labor to total output; D 1 cov.Qv; RQ m /, and , the market price of systematic risk. In addition, the VES production function in two factors capital .K/ and labor .L/ can be defined as follows in Equation (2.19). Q D K ˛.1s/ ŒL C .  1/K ˛s

(2.20)

where Q is firm’s output and ˛; s, and  are parameters with the following constraints: ˛ > 0; 0<s .1  /=.1  s/3 : Lee et al. (1995) have derived the theoretical relationship between beta coefficient and the capital labor ratio in terms of the VES production functions as follows. ˚ .1 C r/ C ov.e; Q RQ m /  Œˆ.1  E/˛s 1 w.pQ/1 .1  /K C ov.Qv; RQ m / ˇD Var.RQ m / fˆ  Œˆ.1  E/˛s  w .pQ/1 .1  /K (2.21) where p, expected price of output; , reciprocal of the price elasticity of demand, 0    1I w, expected wage rate; vQ , random shock in the wage rate with zero mean; ˆ D 1  cov.e; Q RQ m /, and r; RQ m ; e; Q E; ;  are as defined in Equation (2.19).

Impact of Dividend Policy on Beta Coefficient Determination Impact of payout ratio on beta coefficient is one of the important issues in finance research. By using dividend signaling theory, Lee et al. (1993) have derived the relationship between the beta coefficient and pay out ratio as follows: ˇi D ˇpi Œ1 C F .di /

(2.22)

ˇi , the firm’s systematic risk when the market is informationally imperfect and the information asymmetry can be resolved by dividends; ˇpi , the firm’s systematic risk when market is informationally perfect;  , a signaling cost incurred

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Investment, Dividend, Financing, and Production Policies: Theory and Implications

if firm’s net after-tax operating cash flow X falls below the promised dividend DI di , firm’s dividend payout ratio; F .di /, cumulative normal density function in term of payout ratio. Equation (2.21) implies that beta coefficient is a function of payout ratio. In sum, this section has shown that how financing, production, and dividend policy can affect the beta coefficient. This information discussed in this section is important for calculating the cost of capital equity.

2.7 Conclusion In this chapter we have attacked many of the irrelevant propositions of the neoclassical school of finance theory, and in so doing have created a good news-bad news sort of situation. The good news is that by claiming that financial policies are important, we have justified the existence of academicians and a great many practicing financial managers. The bad news is that we have made their lot a great deal more difficult as numerous tradeoffs were investigated, the more general of these comprising the title of the chapter. In the determination of dividend policy, we examined the relevance of the internal-external equity decision in the presence of nontrivial transaction costs. While the empirical evidence was found to be inconclusive because of the many variables that could not be controlled, there should be no doubt mind that flotation costs (incurred when issuing new equity to replace retained earnings paid out) by themselves have a negative impact on firm value. But if the retained earnings paid out are replaced whole or in part by debt, the equity holders may stand to benefit because the risk is transferred to the existing bondholders – risk they do not receive commensurate return for taking. Thus, if the firm pursues a more generous dividend-payout policy while not changing the investment policy, the change in the value of the firm depends on the way in which the future investment is financed. The effect that debt financing has on the value of the firm was analyzed in terms of the interest tax shield it provides and the extent to which the firm can utilize that tax shield. In Myers’ analysis we also saw a that a limit on borrowing could be incorporated so that factors such as risk and the probability of insolvency would be recognized when making each capital-budgeting decision. When compared to other methods widely used in capital budgeting, Myers’ APV formulation was found to yield more conservative benefit estimates. While we do not wish to discard the equityresidual, after-tax weighted cost-of-capital method or the Arditti-Levy weighted cost-of-capital method, we set forth Myers’ method as the most appropriate starting point when a firm is first considering a project, reasoning that if the project

was acceptable following Myers’ method, it would be acceptable using the other methods – to an even greater degree. If the project was not acceptable following the APV criteria, it could be reanalyzed with one of the other methods. The biases of each method we hopefully made clear with the introduction of debt financing. In Sect. 2.6 we have discussed how different policies can affect the determination of beta coefficient. In essence, this chapter points out the vagaries and difficulties of financial management in practice. Virtually no decision concerning the finance function can be made independent of the other variables under management’s control. Profitable areas of future research in this area are abundant; some have already begun to appear in the literature under the heading “simultaneous-equation planning models.” Any practitioner would be well advised to stay abreast of developments in this area.

Notes 1. Alderson (1984) has applied this kind of concept to the problem of corporate pension funding. He finds grounds for the integration of pension funding with capital structure decisions. In doing so, he argues that unfunded vested liabilities are an imperfect substitute for debt. 2. BHS (1981) discusses the relationship between debt capacity and the existence of optimal capital structure. 3. If the dividend policy is irrelevant, then Lct D Lc;t1 D 0; if dW=dyt D Ft is positive, then the constraints U will always be binding, Equation (2.7b) will be strict equalities, and Lft for all t. Substituting Lct D 0 and Lft D Ft into Equation (2.8), we obtain Equation (2.12). 4. This statement can be explained by the agency theory, which was developed by Jensen and Meckling (1976) and was extended by Barnea et al. (1981). According to the agency theory and the findings of Stiglitz (1972), the manager might not act out of the equityholders’ best interest. For his own benefit, the manager might shift the wealth of a firm from the equityholders to bondholders.

References Alderson, M. J. 1984. Unfunded pension liabilities and capital structure decisions: a theoretical and empirical investigation, PhD dissertation, The University of Illinois, Urbana-Champaign Arditti, F. D. 1980. “A survey of valuation and the cost of capital.” Research in Finance 2, 1–56 Arditti, F. D. and Y. C. Peles. 1977. “Leverage, bankruptcy, and the cost of capital.” in Natural resources, uncertainty and general equilibrium systems: essays in memory of Rafael Lusky, A. Blinder and P. Friedman (Eds.). Academic Press, New York

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications Arditti, F. D. and H. Levy. 1977. “The weighted average cost of capital as a cutoff rate: a critical analysis of the classical textbook weighted average.” Financial Management 6(3), 24–34 Ang, J. S. and J. H. Chua. 1982. “Mutual funds: different strokes for different folks.” The Journal of Portfolio Management 8, 43–47 Barnea, A., R. A. Haugen and L. W. Senbet. 1981. “Market imperfections, agency problems, and capital structure: a review.” Financial Management 10, 7–22 Baron, D. P. 1975. “Firm valuation, corporate taxes, and default risk.” Journal of Finance 30, 1251–1264 Baumol, W. and B. G. Malkiel. 1967. “The firm’s optimum debtequity combination and the cost of capital.” Quarterly Journal of Economics 81(4), 546–578 Black, F. 1976. “The dividend puzzle.” Journal of Portfolio Management, 5–8 Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 81(3) 637–654 Chambers, D. R., R. S. Harris, and J. J. Pringle. 1982. “Treatment of financing mix in analyzing investment opportunities.” Financial Management 11, 24–41 Chen, A. H. 1978. “Recent development in cost of debt capital.” Journal of Finance 33, 863–877 Conine, T. E. Jr. 1980. “Debt capacity and the capital budgeting decision: a comment.” Financial Management, 20–22 DeAngelo, H. and L. DeAngelo. 2006. “The irrelevance of the MM dividend irrelevance theorem.” Journal of Financial Economics 79, 293–315 Fama, E. F. and M. H. Miller. 1972. The theory of finance, Holt, Rinehart and Winston, New York Gahlon, J. M. and R. D. Stover. 1979. “Debt capacity and the capital budgeting decision: a caveat.” Financial Management 8, 55–59 Gupta, M. C. and A. C. Lee. 2006. “An integrated model of debt issuance, refunding, and maturity.” Review of Quantitative Finance and Accounting 26(2), 177–199 Gupta, M. C., A. C. Lee, and C. F. Lee. 2007. Effectiveness of dividend policy under the capital asset pricing model: a dynamic analysis. Working Paper, Rutgers University Hamada, R. S. 1972. “The effect of the firm’s capital structure on the systematic risk of common stocks.” Journal of Finance 27(2), 435–452 Hanoch, G. and H. Levy. 1969. “The efficiency analysis of choices involving risk.” Review of Economic Studies 36, 335–346 Hellwig, M. F. 1981. “Bankruptcy, limited liability, and the ModiglianiMiller theorem.” The American Economic Review 71, 155–170 Higgins, R. C. 1977. “How much growth can a firm afford?” Financial Management 6(3), 7–16 Higgins, R. C. 1981. “Sustainable growth under inflation.” Financial Management, 36–40 Hong, H. and A. Rappaport. 1978. “Debt capacity and the capitalbudgeting decision: a caveat.” Financial Management, 7–11 Jean, W. 1975. “Comparison of moment and stochastic-dominance ranking methods.” Journal of Financial and Quantitative Analysis 10, 151–162 Jensen, M. and W. Meckling. 1976. “Theory of the firm: managerial behavior, agency costs, and ownership structure.” Journal of Financial Economics 3, 305–360 Johnson, D. J. 1981. “The behavior of financial structure and sustainable growth in an inflationary environment.” Financial Management 10, 30–35 Kim, E. H. 1978. “A mean-variance theory of optimal capital structure and corporate debt capacity.” Journal of Finance 33, 45–64 Kim, E. H. 1982. “Miller’s equilibrium, shareholder leverage clienteles, and optimal capital structure.” Journal of Finance 37, 301–319 Kim, E. H., J. J. McConnell, and P. R. Greenwood. 1977. “Capitalstructure rearrangements and me-first roles in an efficient capital market.” Journal of Finance 32, 811–821

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Lee, C. F. and A. C. Lee. 2006. Encyclopedia of finance, Springer, New York Lee, C. F., W. Chunchi, and D. Hang. 1993. “Dividend policy under conditions of capital markets and signaling equilibrium.” Review of Quantitative Finance and Accounting 3, 47–59 Lee, C. F., S. Rahman, and K. Thomas Liaw. 1990. “The impacts of market power and capital-labor ratio on systematic risk: a Cobb-Douglas approach.” Journal of Economics and Business 42(3), 237–241 Lee, C. F., K. C. Chen, and K. Thomas Liaw. 1995. “Systematic risk, wage rates, and factor substitution.” Journal of Economics and Business 47(3), 267–279 Levy, H. and M. Sarnat. 1972. Investment and portfolio analysis, Wiley, New York Litzenberger, R. H. and K. Ramaswamy. 1979. “The effect of personal taxes and dividends on capital-asset prices: theory and empirical evidence.” Journal of Financial Economics 7, 163–195 Litzenberger, R. H., and H. B. Sosin. 1979. “A comparison of capitalstructure decision of regulated and nonregulated firms.” Financial Management 8, 17–21 Malkiel, B. G. 1970. “The valuation of public-utility equity.” Bell Journal of Economics and Management Science 1, 143–160 Martin, J. D. and D. F. Scott. 1976. “Debt capacity and the capital budgeting decision.” Financial Management 5(2), 7–14 Martin, J. D. and D. F. Scott Jr. 1980. “Debt capacity and the capital budgeting decision: a revisitation.” Financial Management 9(1), 23–26 Merton, R. C. 1974. “On pricing of corporate debt: the risk structure of interest rates.” Journal of Finance 29, 449–470 Miller, M. H. 1977. “Debt and taxes.” Journal of Finance 32, 261–275 Modigliani, F. and M. H. Miller. 1958. “The cost of capital, corporate finance, and theory of investment.” American Economic Review 48(3), 261–297 Modigliani, F. and M. H. Miller. 1963. “Corporate income taxes and the cost of capital: a correction.” American Economic Review 53(3), 433–443 Myers, S. C. 1974. “Interaction of corporate financing and investment decision.” Journal of Finance 29, 1–25 Myers, S. C. 1977. “Determinants of corporate borrowing.” Journal of Financial Economics 5, 147–175 Porter, R. B. and J. E. Gaumwitz. 1972. “Stochastic dominance vs. mean-variance portfolio analysis: an empirical evaluation.” American Economic Review 57, 438–446 Rendleman, R. J. Jr. 1978. “The effects of default risk on firm’s investment and financing decisions.” Financial Management 7(1), 45–53 Reseck, R. W. 1970. “Multidimensional risk and the Modigliani-Miller hypothesis.” Journal of Finance 25, 47–51 Rhee, S. G. and F. L. McCarthy. 1982. “Corporate debt capacity and capital-budgeting analysis.” Financial Management 11, 42–50 Rubinstein, M. E. 1973. “A mean-variance synthesis of corporate financial theory.” Journal of Finance 28, 167–187 Stiglitz, J. 1972. “Some aspects of pure theory of corporate finance: Bankruptcies and take-overs.” Bell Journal of Economics and Management Science 3, 458–482 Tuttle, L., and R. H. Litzenberger. 1968. “Leverage, diversification, and capital-market effects on a risk-adjusted capital-budgeting framework.” Journal of Finance 23, 427–443 Van Horne, J. C. and J. D. McDonald. 1971. “Dividend policy and new equity financing.” Journal of Finance 26, 507–519

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Appendix 2A Stochastic Dominance and its Applications to Capital-Structure Analysis with Default Risk

EF U.X / > EG U.X /

where EF U.X / and EG U.X / are expected utilities. Mathematically, they can be defined as: Zx

2A.1 Introduction EF U.X / D Mean-variance approaches were extensively used in the literature to derive alternative finance theories and perform related empirical studies. Besides mean-variance approaches, there is a more general approach, stochasticdominance analysis, which can be used to choose a portfolio, to evaluate mutual-fund performance, and to analyze the optimal capital-structure problem. Levy and Sarnat (1972), Porter and Gaumwitz (1972), Jean (1975), and Ang and Chua (1982) have discussed the stochastic-dominance approach to portfolio choice and mutual-fund performance evaluation in detail. Baron (1975), Arditti and Peles (1977), and Arditti (1980) have used the theory of stochastic dominance to investigate the optimal capital-structure question. In this appendix we will discuss only how stochastic-dominance theory can be used to analyze the issue of optimal capital structure with default risk.

2A.2 Concepts and Theorems of Stochastic Dominance

(2A.1)

U.X /f .X /dx

(2A.2a)

U.X /g.X /dx

(2A.2b)

x

Zx EF U.X / D x

where U.X / D the utility function, X D the investment dollar-return variable, f .X / and g.X / are probability distributions of X . It should be noted that the above-mentioned first-order stochastic dominance does not depend upon the shape of the utility function with positive marginal utility. Hence, the investor can either be risk-seeking, risk neutral, or risk-averse. If the risk-aversion criterion is imposed, the utility function is either strictly concave or nondecreasing. If utility functions are nondecreasing and strictly concave, then the second-order stochastic-dominance theorem can be defined. Mathematically, the second-order dominance can be defined as: Zx ŒG.T /  F .t/ dt > 0; where G.t/ ¤ F .t/ for some t: x

The expected utility rule can be used to introduce the economics of choice under uncertainty. However, this decision rule has been based upon the principle of utility maximization, where either the investor’s utility function is assumed to be a second-degree polynomial with a positive first derivative and a negative second derivative, or the probability function is assumed to be normal. A stochastic-dominance theory is an alternative approach of preference orderings that does not rely upon these restrictive assumptions. The stochastic-dominance technique assumes only that individuals prefer more wealth to less. An asset is said to be stochastically dominant over another if an individual receives greater wealth from it in every (ordered) state of nature. This definition is known as first-order stochastic dominance. Mathematically, it can be described by the relationship between two cumulativeprobability distributions. If X symbolizes the investment dollar return variable and F .X / and G.X / are two cumulativeprobability distributions, then F .X / will be preferred to G.X / by every person who is a utility maximizer and whose utility is an increasing function of wealth if F .X /  G.X / for all possible X , and F .X / < G.X / for some X . For the family of all monotonically nondecreasing utility functions, Levy and Sarnat (1972) show that first-order stochastic dominance implies that:

(2A.3) Equation (2A.3) specifies a necessary and sufficient condition for an asset F to be preferred over a second asset G by all risk-averters. Conceptually, Equation (2A.3) means that in order for asset F to dominate asset G for all risk-aversion investors, the accumulated area under the cumulative-probability distribution G must be greater than the accumulated area for F for any given level of wealth. This implies that, unlike first-order stochastic dominance, the cumulative density functions can cross. Stochastic dominance is an extremely important and powerful tool. It is properly founded on the basis of expected utility maximization, and even more important, it applies to any probability distribution. This is because it takes into account every point in the probability distribution. Furthermore, we can be sure that, if an asset demonstrates second-order stochastic dominance, it will be preferred by all risk-aversion investors, regardless of the specific shape of their utility functions. Assume that the density functions of earnings per share (EPS) for both firm A and firm B are f .X / and g.X /, respectively. Both f .X / and g.X / have normal distributions and the shape of these two distributions are described in Fig. 2A.1. Obviously the EPS of firm A will dominate the EPS of firm B if an investor is risk-averse because they both

2 Investment, Dividend, Financing, and Production Policies: Theory and Implications

39

Fig. 2A.1 Probability distributions of asset F and asset G

offer the same expected level of wealth .f D g / and because the variance of g.X / is larger than that of f .X /. Based on both the first-order and the second-order stochastic-dominance theorems mentioned earlier, a new theorem needed for investigating capital structure with default risk can be defined as: Theorem 2A.1. Let F , G be two distributions with mean values 1 and 2 respectively, such that, for some X0 < 1, F  G for X  X0 (and F < G for some X1 < X0 ) and F G for some X X0 , then F dominates G (for concave utility functions) if and only if 1 2 . The proof of this theorem can be formed in either Hanoch and Levy (1969) or Levy and Sarnat (1972). Conceptually, this theorem states that if two cumulative distributions, F and G, intersect only once, then if F is below G to the left of the intersection point and has a higher mean than does G, the investment with cumulative distribution F dominates that with cumulative return distribution G on a second-degree stochastic-dominance basis.

earnings X, before taxes and financial charges, such that, in any state of nature that occurs, both firms have the same earnings. In addition, it is assumed that this random variable X is associated with a cumulative-distribution function F .X /. Firm A is assumed to be financed solely by equity, while firm B is financed by both debt and equity. The market value V1 of firm A equals E1 , the value of its equity, while the market value V2 of firm B equals the market value of its equity .E2 / plus the value of its debt .D2 /. Debt is assumed to sell at its par value and to carry a gross coupon rate of r .r > 0/. The firm can generally use the coupon payments as a tax shield and therefore the after-tax earnings for firms A and B can be defined as X.1T / and .X rD2 /.1T /, respectively (where T is the corporate-profit tax rate.) If an investor purchases a fraction of firm A, this investment results in dollar returns of:  0; if X  0; Y1 D (2A.4) ˛X.1  T / if X > 0; with cumulative probability function: 

2A.3 Stochastic-Dominance Approach to Investigating the Capital-Structure Problem with Default Risk The existence of default risk is one of the justification of why an optimal capital structure might exist for a firm. To analyze this problem. Baron (1975), Arditti and Peles (1977), and Arditti (1980) have used the stochastic-dominance theorem described in the previous sections to indicate the effects of debt-financing bonds on relative values of levered and unlevered firms. Baron analyzed the bonds in terms of default risk, tax rates, and debt levels. Consider two firms, or the same firm before and after debt financing, with identical probability distribution of gross

G1 .Y / D

F .0/ F .Y =.1  T /

if Y D 0; if Y > 0:

(2A.5)

If the investor purchased a fraction ’ of the equity and ’ of the debt of firm B, this dollar return would be: 8 if X  0; ˆ E.B/ and Var.A/ D Var.B/ or E.A/ D E.B/ and Var.A/ < Var.B/ where: E.A/ and E.B/ D the expected returns of A and B: and Var (A) and Var (B) D their respective variances or risk. The mean returns and variance of every investment opportunity can be calculated and plotted as a single point on a mean-standard deviation diagram, as shown in Fig. 4.3. All points below curve EF represent portfolio combinations that are possible. Point D represents a portfolio of investments with a return RD and risk D . All points above EF are combinations of risk and returns that do not exist. Point B would therefore represent risk and return that cannot be obtained with any combination of investments. The EF curve is also called the efficient frontier because all points below the curve are dominated by a point found on the curve. For instance, suppose a firm is willing to assume a maximum level of risk D . It can obtain a return of RD with portfolio D or move to point C on the frontier and receive a higher return RC with that portfolio. Therefore, C dominates D because it would be preferred to D. For the same level of risk, it has a higher return. A similar argument could be made in terms of risk. If the firm wants to achieve a return of RA , it will select portfolio

4.4 The Efficient Portfolio and Risk Diversification Utilizing the definitions of standard deviation of expected return of a portfolio discussed previously, this section discussed the concepts of the efficient portfolio and risk diversification. Fig. 4.3 The efficient frontier in portfolio analysis

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A over D, because A represents the same return at a smaller level of risk or standard deviation: A < B . Therefore, point D is not efficient but points A and C are. A decision maker could, therefore, select any point on the frontier and be secure in knowing that a better portfolio is not available. Example 4.4 further illustrates this concept. Example 4.4. To show how the portfolio concepts and methods discussed in this section can be used to do practical analysis, monthly rates of return for January 1980 to December 1984 for Pennzoil and Coca Cola are used as examples. The basic statistical estimates for these two firms are average monthly rates of return and the variance-covariance matrix. The average monthly rates of return for Pennzoil (PZ) and Coca Cola (CK) are 0.0093 and 0.0306, respectively. The variances and covariances are listed in the following table. Variance-covariance matrix PZ CK Pennzoil Coke

0.0107391 0.000290

0.000290 0.0495952

From Equation (4.7), we have: 0:0495952 C 0:000290 0:0107391 C 0:0495952 C 0:00058 0:0498852 D 0:069143 D 0:8189

W1 D

W2 D 1:0  0:8189 D 0:1811 Using the weight estimates and Equations (4.2) and (4.3): E.RN P / D .0:8189/.0:0093/ C .0:1811/.0:0306/ D 0:01316 P2 D .0:8189/2.0:0107391/ C .0:1811/2.0:0495952/

4.4.2 Corporate Application of Diversification The effect of diversification is not necessarily limited to securities but may have wider applications at the corporate level. Frequently, managers will justify undertaking many product lines because of the effects of diversification. Instead of “putting all of the eggs in one basket,” the investment risks are spread out among many lines of services or products in hope of reducing the overall risks involved and maximizing returns. To what degree this diversification takes place in other types of corporate decisions depends on the decision maker’s preference for risk and return. The overall goal is to reduce business risk fluctuations of net income. However, it should be noted that investors can do their own homemade diversification, which generally reduces the need of corporate diversification. This type of corporate diversification can be taken to the multinational level. For example, General Motors has overseas divisions throughout the world. Although these divisions all produce the same product, autos and auto parts, GM’s status as a multinational corporation allows it to take advantage of the diversifying effects of different exchange rates and political and economic climates.

4.4.3 The Dominance Principle The dominance principle has been developed as a means of conceptually understanding the risk/return tradeoff. As with the efficient-frontier analysis, we must assume an investor prefers returns and dislikes risks. For example, as depicted in Fig. 4.4, if an individual is prepared to experience risk associated with A , he or she can obtain a higher expected return with portfolio A.RN A / than with portfolio B.RN B /. Thus A dominates B and would be preferred. Similarly, if an individual were satisfied with a return of RN B , he or she would select portfolio B over C because the risks associated with B

C2.0:8189/.0:1811/.0:000290/ D 0:0088 P D 0:0940 when 12 is less than 1.00 it indicates that the combination of the two securities will result in a total risk less than their added respective risks. This is the diversification effect. If 12 were equal to 1.00, this would mean that the combination of the two securities has no diversification effect at all. The correlation coefficient of 12 D 0:0125659 indicates that a portfolio combining Pennzoil and Coca Cola would show a diversification effect and a reduction in risk.

Fig. 4.4 The dominance principle in portfolio analysis

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are less .B < C /. Therefore using the dominance principle reinforces the choice of an efficient portfolio and is also a method in determining it. Figure 4.4 makes clear that points A and B are directly comparable because they have a common standard deviation, A . Points B and C are directly comparable because of a common return, RN B . Now consider portfolio D. How does its risk versus return compare with the other portfolios shown in Fig. 4.4? It is difficult to say because the risk and return are not directly comparable using the dominance principle. This is the basic limitation of the dominance principle – that portfolios without a common risk or return factor are not directly comparable.

4.4.4 Three Performance Measures For such a situation it becomes necessary to use some other performance measure. There are basically three important portfolio performance measures taught in investment courses: (1) the Sharpe, (2) the Treynor, and (3) the Jensen measures. The Sharpe measure (SP) (Sharpe 1966) is of immediate concern. Given two of the portfolios depicted in Fig. 4.4, portfolios Band D, their relative risk-return performance can be compared using the equations: SPD D

RN D  Rf RN B  Rf and SPB D D B

where: SPD ; SPB RN D ; RN B Rf D ; B

D Sharpe performance measuresI D the average return of each portfolioI D risk-free rateI and D the respective standard deviation on risk of each portfolio:

Because the numerator is the average return reduced by the risk-free rate, it represents the average risk premium of each portfolio. Dividing the risk premium by the total risk per portfolio results in a measure of the return (premium) per unit of risk for each portfolio. The Sharpe performance measure equation will therefore allow a direct comparison of any portfolio, given its risk and returns. Consider Fig. 4.5; portfolio A is being compared to portfolio B. If a riskless rate exists, then all investors would prefer A to B because combinations of A and the riskless asset give higher returns for the same level of risk than combinations of the riskless asset and B. The preferred portfolio lies on the ray passing through Rf that is furthest in the counterclockwise direction (the ray that has the greatest slope). Example 4.5 further illustrates this concept.

Fig. 4.5 Combinations of portfolio and the risk-free investment

Example 4.5. An insurance firm is trying to decide between two investment funds. From past performance it was able to calculate the average returns and standard deviations for these funds. The current T-bill rate is 9.5% and the firm will use this as a measure of the risk-free rate.

Average return R (percent) Standard deviation  (percent)

Smyth fund

Jones fund

18 20

16 15

Risk-free rate D Rf (percent) D 9:5 Using the Sharpe performance measure, the risk-return measurements for these two firms are: 0:18  0:095 D 0:425 0:20 0:16  0:095 D 0:433 D 0:15

SPSmyth D SPJones

It is clear that the Jones fund has a slightly better performance and would be the better alternative of the two. The Sharpe measure looks at the risk-return decision from the point of view of an investor choosing a portfolio to represent the majority of his or her investment. An investor choosing a portfolio to represent a large part of his or her wealth would likely be concerned with the full risk of the portfolio, and the standard deviation is a measure of that risk. On the other hand, if the risk level of the portfolio is already determined by the investor and what is important is to evaluate the performance of the portfolio over and above the total market performance, perhaps the proper risk measure would be the relationship between the return on the portfolio and the return on the market, or beta. All combinations of a riskless asset and a risky portfolio lie on a straight line con-

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necting them. The slope of the line connecting the risky asset A and the risk-free rate is .RN A  Rf /=A . Here, as in the Sharpe measure, an investor would prefer the portfolio on the most counterclockwise ray emanating from the riskless asset. This measure, called the Treynor measure (TP), developed by Treynor in 1965, examines differential return when beta is the risk measure. Example 4.6 provides further illustration.

SPA D 0:84

The Treynor performance measure uses the beta coefficient (systematic risk) instead of total risk for the j th portfolio .j / as a risk measure. Applications of the TP are similar to the SP as discussed previously.1 Jensen (1968, 1969) has proposed a measure referred to as the Jensen differential performance index (Jensen’s measure). The differential return can be viewed as the difference in return earned by the portfolio compared to the return that the capital asset pricing line implies should be earned. Consider the line connecting the riskless rate and the market portfolio. A manager could obtain any point along this line by investing in the market portfolio and mixing this with the riskless asset to obtain the desired risk level. If the constructed portfolio is actively managed, then one measure of performance is the difference in return earned by actively managing the portfolio, versus what would have been earned if the portfolio had been passively constructed of the market portfolio and the riskless asset to achieve the same risk level. The slope of the line connecting the riskless asset and the market portfolio is .RN M  Rf /=ˇM , and the intercept must be the riskless rate. The beta on the market portfolio is one; therefore, the CAPM equation results in:

SPB D 0:73

RN P D Rf C .RN M  Rf /ˇP :

Example 4.6. Rank the portfolios shown in the table based on the Sharpe measure. Assume Rf D 8%. If Rf D 5%, how does the order change? Portfolio

Return (percent)

Risk (percent)

A B C D E

50 19 12 9 8:5

50 15 9 5 1

Solution Sharpe measure SPM D

RN M  Rf 

SPC D 0:44 SPD D 0:20 SPE D 0:50 Ranked by the Sharpe measure, A > B > E > C > D. The Sharpe measure indicates that portfolio A is the most desirable: it has the highest return per unit of risk. For Rf D 5 percent SPA D 0:90 SPB D 0:933 SPC D 0:77 SPD D 0:80 SPE D 0:35

Jensen’s measure (JM) is the differential return of the managed portfolio’s actual return less the return on the portfolio of identical beta that lies on the line connecting the riskless asset and the market portfolio. Algebraically, Jensen’s measure is expressed: JM D RN P  ŒRf C .RN M  Rf /ˇp Figure 4.6 depicts portfolio rankings for the Jensen measure, and Example 4.7 provides further illustration. Example 4.7. Rank the portfolio in the table according to Jensen’s measure:

The order changes to E > B > A > D > C. E is now the best portfolio as it has the highest return per unit of risk. The Treynor measure can be expressed by the following: TP D

RN j  Rf ˇj

where: RN j D average return of j th portfolioI Rf D ris  free rateI and ˇj D beta coefficient for j th portfolio:

Fig. 4.6 Jensen’s measure for portfolio rankings

1

Discussion of the Treynor measure adapted from Treynor (1965). Adapted by permission.

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1. 2. 3. 4.

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Assuming RM D 10 percent and Rf D 8 percent Assuming RM D 12 percent and Rf D 8 percent Assuming RM D Rf D 8 percent Assuming RM D 12 percent and Rf D 4 percent

Portfolio

Ri (percent)

(percent)

ˇi

A B C D E

50 19 12 9 8:5

50 15 9 5 1

2.5 2.0 1.5 1.0 0.25

Solution

2.

3.

4.

D

ŒRN P  Rf ŒRN M  Rf  P m

D SPP  SPm .commom constant/l If the Jensen measure (JM) is divided by ˇP , it is equivalent TM to the Treynor measure plus some constant common to all portfolios: ŒRN P  Rf ŒRN M  Rf ˇP JM D  ˇP ˇP ˇP D TMP  ŒRN M  Rf

Ri D Rf Cˇi .RM Rf /

1.

ŒRN P  Rf ŒRN M  Rf .pm / JM D  p P P m m

JM D .Ri Rf /ˇi .RM Rf /

JMA D 37 percent JMB D 7 percent JMC D 1 percent JMD D 2 percent JME D 0 percent Ranked by the Jensen measure A > B > C > E > D. JMA D 32 percent JMB D 3 percent JMC D  2 percent JMD D 3 percent JME D 0:5 percent The rank change to A > B > E > C > D. JMA D 42 percent JMB D 11 percent JMC D 4 percent JMD D 1 percent JME D 0:5 percent The rank is A > B > C > D > E. JMA D 26 percent JMB D 1 percent JMC D 4 percent JMD D 3 percent JME D 2:5 percent The rank now is A > E > B > D > C.

Interrelationship among Three Performance Measure. It should be noted that all three performance measures are interrelated. For instance, if pm D pm =p m D 1, then the Jensen measure divided by p becomes equivalent to the Sharpe measure. Since ˇp D pm =m2 and pm D pm =p m the Jensen measure (JM) must be multiplied by 1=p to derive the equivalent Sharpe measure:

D TMP  commom constant Example 4.8 provides further illustration.

Example 4.8. Continuing with the example used for the Sharpe performance measure in Example 4.5, assume that in addition to the information already provided, the market return is 10%, the beta of the Smyth Fund is 0.8, and the Jones Fund beta is 1.1. Then, according to the capital asset pricing line, the implied return earned should be: RN Smyth D 0:095 C .0:10  0:095/.0:8/ D 0:099 RN Jones D 0:095 C .0:10  0:095/.1:1/ D 0:1005 Using the Jensen measure, the risk-return measurements for these two firms are: JMSmyth D 0:18  0:099/ D 0:081 JMJones D 0:16  0:1005 D 0:0595 From these calculations, it is clear that the Smyth Fund has a better performance and would be the better alternative of the two. Note that this is the opposite of the results determined from the Sharpe performance measure in Example 4.5. Computing the Treynor measure would reinforce the Jensen results. More analysis of these performance measures will be undertaken in detail in next chapters.

4.5 Determination of Commercial Lending Rate This section concerns a process for estimating the lending rate a financial institution would extend to a firm or the borrowing rate a firm would think is reasonable based on economic, industry, and firm-specific factors.

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As shown previously, part of the rate of return is based on the risk-free rate. The risk-free rate Rf must first be forecasted for three types of economic conditions – boom, normal, and poor. The second component of the lending rate is the risk premium .RP /. This can be calculated individually for each firm by examining the change in earnings before interest and taxes (EBIT) under the three types of economic conditions. The EBIT is used by the lender as an indicator of the ability of the potential borrower to repay borrowed funds. Table 4.2 has been constructed based on the methods discussed previously. In total there are nine possible lending rates under the three different economic conditions. The construction of these lending rates is shown in Table 4.3. This table shows that during a boom the risk-free rate is set at 12%, but the risk premium can taken on different values. These is a 40% chance that it will be 3.0%, a 30% chance it will be 5.0%, and a 30% chance it will be 8.0%. The products of the RP probabilities and the Rf probability are the joint probabilities of occurrence for the lending rates computed from these parameters. Therefore, there is a 10% chance that a firm will be faced with a 15% lending rate during a boom, a 7.5% chance of a 17% rate, and a 7.5% chance of an 18%

Table 4.2 Possible lending rates Economic conditions Rf (percent)

Probability

Boom

12.0

0.25

Normal

10.0

0.50

Poor

8.0

0.25

rate. This process applies for the other conditions, normal and poor, as well. Based upon the mean and variance Equations (4.1) and (4.2) it is possible to calculate the expected lending rate and its variance. Using the information provided in Table 4.3, the weighted average can be calculated: R D.0:100/.15%/ C .0:075/.17%/ C .0:075/.20%/ C .0:200/.13%/ C .0:150/.15%/ C .0:150/.18%/ C .0:100/.11%/ C .0:075/.13%/ C .0:075/.16%/ D15:1% With a standard deviation of: h  D .0:100/.15  15:1/2 C .0:075/.17  15:1/2 C .0:075/.20  15:1/2 C .0:200/.13  15:1/2 C .0:150/.5  15:1/2

EBIT ($ millions)

Probability

Rp (percent)

2.5 1.5 0.5 2.5 1.5 0.5 2.5 1.5 0.5

0.40 0.30 0.30 0.40 0.30 0.30 0.40 0.30 0.30

3 5 8 3 5 8 3 5 8

Table 4.3 Construction of actual lending rates Economic conditions

(A) Rf (percent)

(B) Probability

(C) Rp (percent)

(D) Probability

.B  D/ Joint probability of occurrence

.A C C/ Lending rate (percent)

Boom

12

0.25

Normal

10

0.50

Poor

8

0.25

3.0 5.0 8.0 3.0 5.0 8.0 3.0 5.0 8.0

0.40 0.30 0.30 0.40 0.30 0.30 0.40 0.30 0.30

0.100 0.075 0.075 0.200 0.150 0.150 0.100 0.075 0.075

15 17 20 13 15 18 11 13 16

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Fig. 4.7 Probability of Xi in the intervals ˙1; ˙2; ˙3

C .0:150/.18  15:1/2 C .0:100/.11  15:1/2 C .0:075/.13  15:1/2 i1=2 C .0:075/.16  15:1/2 D.0:001 C 0:271 C 1:801 C 0:882 C 0:0015 C 1:2615 C 1:681

away. The remaining risk is systematic risk that which influences all risky assets. The market rate of return can be calculated using one of several types of market indicator series, such as the DowJones Industrial Average, the Standard and Poor (S&P) 500, or the New York Stock Exchange Index, using the following equation: It  It 1 D Rmt It 1

C 0:331 C 0:061/ D2:51% If this distribution is indeed approximately normal, the mean and standard deviation can be employed to make some statistical inferences. Figure 4.7 makes it clear that 68.3% of the observations of a standard normal distribution are within one standard deviation of the mean, 95.4% are within three. Since the average lending rate is assumed to be normally distributed with a mean of 15.1% and a standard deviation of 2.51%, it is clear that almost all (99.7%) of the lending rates will lie in the range of 7.57–22.63%, because 7.57% is three standard deviations below 15.1 and 22.63% is three standard deviations above the mean. It is also clear that 68.3% of the rates will lie in the range of 12.59–17.61%.

4.6 The Market Rate of Return and Market Risk Premium The market rate of return is the return that can be expected from the market portfolio. This portfolio is of all risky assets – that is, stocks, bonds, real estate, coins, and so on. Because all risky assets are included, the market portfolio is a completely diversified portfolio. All unsystematic risks related to each individual asset would, therefore, be diversified

(4.9)

where: Rmt D market rate of return at time tI It D market index at tI and It 1 D market index at t  1: This equation calculates the percent change in the market index during period t and the previous period t  1. This change is the rate of return an investor would expected to receive in t had he or she invested in t  1. A risk-free investment is one in which the investor is sure about the timing and amount of income streams arising from that investment. However, for most types of investments, investors are uncertain about the timing and amount of income of their investments. The types of risks involved in investments can be quite broad, from the relatively riskless T-bills to highly risky speculative stocks. The reasonable investor dislikes risks and uncertainty and would, therefore, require an additional return on his investment to compensate for this uncertainty. This return, called the risk premium, is added to the nominal risk-free rate. The risk premium is derived from several major sources of uncertainty or risk, as was discussed at the beginning of this chapter. Table 4.4 illustrates this concept. In this table the market rate of return using the S&P 500 was calculated using

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Table 4.4 Market returns and T-bill by quarters Year

Month

S&P 500

2005 2006

Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

1; 248:29 1; 280:08 1; 280:66 1; 294:87 1; 310:61 1; 270:09 1; 270:20 1; 276:66 1; 303:82 1; 335:85 1; 377:94 1; 400:63 1; 418:30 1; 438:24 1; 406:82 1; 420:86 1; 482:37 1; 530:62 1; 503:35 1; 455:27 1; 473:99 1; 526:75 1; 549:38 1; 481:14 1; 468:36 1; 378:55 1; 330:63 1; 322:70 1; 385:59 1; 400:38 1; 280:00 1; 267:38 1; 282:83 1; 164:74 968:75 896:24 887:88

2007

2008

Equation (4.9) to devise average monthly returns. Monthly T-bill rates are listed in the column B. The T-bill rate was deducted from the market return rate .Rm  Rf / to devise the risk premium. In the last 2 months of 2007 and for most months of 2007 the market was in decline, with low returns resulting in each month. This allowed the T-bill investors to obtain a higher than the market return and resulted in negative risk premiums. The second half of 2006 demonstrated an increasing market level and higher market returns. In October 1982 the return was 3.15%, the highest in the past 10 months. This al-

(A) Market return (percent) 2:55 0:05 1:11 1:22 3:09 0:01 0:51 2:13 2:46 3:15 1:65 1:26 1:41 2:18 1:00 4:33 3:25 1:78 3:20 1:29 3:58 1:48 4:40 0:86 6:12 3:48 0:60 4:75 1:07 8:60 0:99 1:22 9:21 16:83 7:48 0:93

(B) T-bill rate (percent)

(A–B) Risk premium (percent)

0.34 0.36 0.37 0.38 0.39 0.39 0.41 0.42 0.41 0.40 0.41 0.41 0.41 0.42 0.42 0.41 0.40 0.39 0.40 0.40 0.36 0.32 0.31 0.25 0.27 0.17 0.14 0.12 0.12 0.15 0.15 0.14 0.14 0.07 0.04 0.01

2:21 0:32 0:74 0:84 3:48 0:38 0:10 1:71 2:05 2:75 1:23 0:85 0:99 2:60 0:58 3:92 2:86 2:17 3:60 0:89 3:22 1:16 4:71 1:11 6:38 3:65 0:74 4:64 0:95 8:75 1:14 1:08 9:35 16:90 7:53 0:94

lowed market rates to leap beyond the riskless T-bill rate, and the result was a positive risk premium. During the period of fluctuation in the level of stock market prices, the first half of 2007 revealed a fluctuated risk premium. Theoretically it is not possible for a risk premium required by investors to be negative. Taking on risk involves some positive cost. Nevertheless, using short-run estimators as in Table 4.4 may result in negative figures because they reflect the fluctuations of the market. The basic problem with using actual market data to assess risk premiums is the difference between expected returns (which are always positive)

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and realized returns (which may be positive or negative). It becomes evident that investors’ expectations will not always be realized.

4.7 Conclusion This chapter has defined the basic concepts of risk and risk measurement. The efficient-portfolio concept and its implementation was demonstrated using the relationships of risk and return. The dominance principle and performance measures were also discussed and illustrated. Finally, the interest rate and market rate of return were used as measurements to show how the commercial lending rate and the market risk premium can be calculated. Overall, this chapter has introduced uncertainty analysis assuming previous exposure to certainty concepts. Further application of the concepts discussed in this chapter as related to security analysis and portfolio management are explored in later chapters.

References Ben-Horin, M. and H. Levy. 1980. “Total risk, diversifiable risk and non-diversifiable risk: a pedagogic note.” Journal of Financial and Quantitative Analysis 15, 289–295. Bodie, Z., A. Kane, and A. Marcus. 2006. Investments, 7th Edition, McGraw-Hill, New York. Bowman, R. G. 1979. “The theoretical relationship between systematic risk and financial (accounting) variables.” Journal of Finance 34, 617–630. Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2006. Modern portfolio theory and investment analysis, 7th Edition, Wiley, New York. Evans, J. L. and S. H. Archer. 1968. “Diversification and the reduction of dispersion: an empirical analysis.” Journal of Finance 23, 761–767.

C.-F. Lee et al. Francis, J. C. and S. H. Archer. 1979. Portfolio analysis, Prentice-Hall, Englewood Cliffs, NJ. Ibbotson, R. G. and R. A. Sinquefield. 1976. “Stocks, bonds, bills, and inflation: simulations of the future (1976–2000).” Journal of Business 49, 313–338. Jensen, M. C. 1968. “The performance of mutual funds in the period 1945–1964.” Journal of Finance 23, 389–416. Jensen, M. C. 1969. “Risk, the pricing of capital assets, and the evaluation of investment portfolios.” Journal of Business 42, 167–185. Lee, C. F. and S. N. Chen. 1981. “The sampling relationship between Sharpe’s performance measure and its risk proxy: sample size, investment horizon and market conditions.” Management Science 27(6), 607–618. Lee, C. F. and S. N. Chen. 1984. “On the measurement errors and ranking of composite performance measures.” Quarterly Review of Economics and Business 24, 6–17. Lee, C. F. and S. N. Chen. 1986. “The effects of the sample size, the investment horizon and market conditions on the validity of composite performance measures: a generalization,” Management Science 32(11), 1410–1421. Lee, C. F. and A. C. Lee. 2006. Encyclopedia of finance, Springer, New York. Lee, C. F., J. C. Lee, and A. C. Lee. 2000. Statistics for business and financial economics, World Scientific Publishing Co, Singapore. Maginn, J. L., D. L. Tuttle, J. E. Pinto, and D. W. McLeavey. 2007. Managing investment portfolios: a dynamic process, CFA Institute Investment Series, 3rd Edition, Wiley, New Jersey. Markowitz, H. M. 1959. Portfolio selection: efficient diversification of investments, Wiley, New York, NY. Modigliani, F. and G. A. Pogue. 1974. “An introduction to risk and return.” Financial Analysis Journal 30, 69–86. Robicheck, A. A. and R. A. Cohn. 1974. “The economic determinants of systematic risk.” Journal of Finance 29, 439–447. Schall, L. D. 1972. “Asset valuation, firm investment, and firm diversification.” Journal of Business 45, 11–28. Sharpe, W. F. 1966. “Mutual fund performance.” Journal of Business 39, 119–138. Thompson, D. J. 1976. “Sources of systematic risk in common stocks.” Journal of Business 49, 173–188. Tobin, J. 1958. “Liquidity preference as behavior toward risk.” Review of Economic Studies 25, 65–86. Treynor, J. 1965. “How to rate management of investment funds.” Harvard Business Review 43, 63–75. Wackerly, D., W. Mendenhall, and R. L. Scheaffer. 2007. Mathematical statistics with applications, 7th Edition, Duxbury Press, California.

Chapter 5

Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model Cheng-Few Lee, Joseph E. Finnerty, and Hong-Yi Chen

Abstract In this chapter, we first introduce utility function and indifference curve. Based on utility theory, we derive the Markowitz’s model and the efficient frontier through the creation of efficient portfolios of varying risk and return. We also include methods of solving for the efficient frontier both graphically and mathematically, with and without explicitly incorporating short selling. Keywords Markowitz model r Utility theory r Utility functions r Indifference curve r Risk averse r Short selling r Dyl model r Iso-return line r Iso-variance ellipse r Critical line r Lagrange multipliers

5.1 Introduction In this chapter, we address basic portfolio analysis concepts and techniques discussed in the Markowitz portfolioselection model and other related issues in portfolio analysis. Before Harry Markowitz (1952, 1959) developed his portfolio-selection technique into what is now modern portfolio theory (MPT), security-selection models focused primarily on the returns generated by investment opportunities. The Markowitz theory retained the emphasis on return, but it elevated risk to a coequal level of importance, and the concept of portfolio risk was born. Whereas risk had been considered an important factor and variance an accepted way of measuring risk, Markowitz was the first to clearly and rigorously show how the variance of a portfolio can be reduced through the impact of diversification. He demonstrated that by combining securities that are not perfectly positively correlated into a portfolio, the portfolio variance can be reduced. The Markowitz model is based on several assumptions concerning the behavior of investors:

1. A probability distribution of possible returns over some holding period can be estimated by investors. 2. Investors have single-period utility functions in which they maximize utility within the framework of diminishing marginal utility of wealth. 3. Variability about the possible values of return is used by investors to measure risk. 4. Investors use only expected return and risk to make investment decisions. 5. Expected return and risk as used by investors are measured by the first two moments of the probability distribution of returns-expected value and variance. 6. Return is desirable; risk is to be avoided. It follows, then, that a security or portfolio is considered efficient if there is no other investment opportunity with a higher level of return at a given level of risk and no other opportunity with a lower level of risk at a given level of return.

5.2 Measurement of Return and Risk This section focuses on the return and risk measurements utilized in applying the Markowitz model to efficient portfolio selection.

5.2.1 Return Using the probability distribution of expected returns for a portfolio, investors are assumed to measure the level of return by computing the expected value of the distribution. E.RP / D

H.-Y. Chen () and C.-F. Lee Rutgers University, New Brunswick, NJ, USA e-mail: [email protected]

n X

Wi E .Ri /

(5.1)

i D1

where:

J.E. Finnerty University of Illinois, Urbana-Champaign, IL, USA e-mail: [email protected] C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_5, 

n X

Wi D 1:0I

i D1

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n D the number of securities; Wi D the proportion of the funds invested in security i ; Ri , RP D the return on i th security and portfolio p; and E . / D the expectation of the variable in the parentheses.

Since rAA and rBB D 1:0 by definition, terms can be simplified and rearranged, yielding:

Thus, the return computation is nothing more than finding the weighted average return of the securities included in the portfolio. The risk measurement to be discussed in the next section is not quite so simple, however. For only in the case of perfect positive correlation among all its components is the standard deviation of the portfolio equal to the weighted average standard deviation of its component securities.

For a three-security portfolio, the variance of portfolio can be defined:

5.2.2 Risk Risk is assumed to be measurable by the variability around the expected value of the probability distribution of returns. The most accepted measures of this variability are the variance and standard deviation. The variance of a single security is the expected value of the sum of the squared deviations from the mean, and the standard deviation is the square root of the variance. The variance of a portfolio combination of securities is equal to the weighted average covariance of the returns on its individual securities: n n X     X Wi Wj Cov Ri ; Rj Var Rp D

(5.2)

i D1 j D1

Covariance can also be expressed in terms of the correlation coefficient as follows:   Cov Ri ; Rj D rij i j D ij

(5.3)

Where rij D correlation coefficient between the rates of return on security i; Ri , and the rates of return on security j; Rj , and i , and j represent standard deviations of Ri and Rj respectively. Therefore: n n X   X Wi Wj rij i j Var Rp D

(5.4)

i D1 j D1

For a portfolio with two securities, A and B, the following expression can be developed: B B X   X Var Rp D Wi Wj rij i j i DA j DA

D WA WA rAA A A C WA WB rAB A B CWB WA rBA B A C WB WB rBB B B

  Var Rp D WA2 A2 C WB2 B2 C 2WA WB rAB A B

(5.5)

C P C   P Var Rp D Wi Wj rij i j i DA j DA

D WA WA rAA A A C WA WB rAB A B CWA WC rAC A C C WB WA rAB B A CWB WB rBB B B C WB WC rBC B C CWC WA rCA C A C WC WB rCB C B CWC WC rC C C C Again simplifying and rearranging yields:   Var Rp D WA2 A2 C WB2 B2 C WC2 C2 C 2.WA WB rAB A B CWA WC rAC A C C WB WC rBC B C / (5.6) Thus, as the number of securities increases from three to two, there is one more variance term and two more covariance terms. The general formula for determining the number of terms that must be computed (NTC) to determine the variance of a portfolio with N securities is NTC D N variancesC

N2  N covariances 2

For the two-security example two variances were involved  ı A2 and B2 , and 22  2 2 D .4  2/=2 D 1 covariances, AB , or three computations. ı For the three-security case, three  variances and 32  3 2 D .9  3/=2 D 3 covariances were needed, AB ; AC , and BC , or six computations. ıWith  four securities, the total number would be 4 C 42  4 2 D 4 C .16  4/=2 D 4 C 6 D 10 computations. As shown from these examples, the number of covariance terms increases by N  1, and the number of variance terms increases by one, as N increases by 1 unit. If rAB D rAC D rBC D 1, then the securities are perfectly positively correlated with each other, and Equation (5.6) reduces to:   Var Rp D .WA A C WB B C WC C /2

(5:60 )

This implies that the standard deviation of the portfolio is equal to the weighted average standard deviation of its component securities. In other words: Standard deviation of Rp D WA A C WB B C WC C

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

Table 5.1 Portfolio size and variance computations Number of securities Number of Var and Cov terms 2 3 4 5 10 15 20 25 50 75 100 250 500

3 6 10 15 55 120 210 325 1; 275 2; 850 5; 050 31; 375 125; 250

Table 5.1 clearly illustrates the tremendous estimation and computational load that exists using the Markowitz diversification approach. Later chapters will illustrate how this problem can be alleviated.

5.3 Utility Theory, Utility Functions, and Indifference Curves In this section, utility theory and functions, which are needed for portfolio analysis and capital asset models, will be discussed in detail. Utility theory is the foundation for the theory of choice under uncertainty. Following Henderson and Quandt (1980), cardinal and ordinal theories are the two major alternatives used by economists to determine how people and societies choose to allocate scarce resources and to distribute wealth among one another over time.1

5.3.1 Utility Functions Economists define the relationships between psychological satisfaction and wealth as “utility.” An upward-sloping relationship, as shown in Fig. 5.1, identifies the phenomena of increasing wealth and increasing satisfaction as being directly related. These relationships can be classified into linear, concave, and convex utility functions. 1

A cardinal utility implies that a consumer is capable of assigning to every commodity or combination of commodities a number representing the amount or degree of utility associated with it. An ordinary utility implies that a consumer needs not be able to assign numbers that represent (in arbitrary unit) the degree or amount of utility associated with commodity or combination of commodity. The consumer can only rank and order the amount or degree of utility associated with commodity.

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In Fig. 5.1a, for each unit change in wealth, there is a linear utility function and equal increase in satisfaction or utility. A doubling of wealth will double satisfaction, and so on. If an investor’s utility function is a linear utility function, we call this kind of investor a risk-neutral investor. This is probably not very realistic: a dollar increase in wealth from $2 to $1 is probably more important than an increase from $2 million to $1 million, because the marginal utility diminishes with increased wealth. In Fig. 5.1b the concave utility function shows the relationship of an increase in wealth and a less-than-proportional increase in utility. In other words, the marginal utility of wealth decreases as wealth increases. As mentioned above, the $1 increase from $2 to $1 of wealth is more important to the individual than the increase from $1,000,001 to $1 million. Each successive increase in wealth adds less satisfaction as the level of wealth rises. We call an investor with a concave utility function a risk-averse investor. Finally, Fig. 5.1c is a convex utility function, which denotes a more than proportional increase in satisfaction for each increase in wealth. Behaviorally, the richer you are the more satisfaction you receive in getting an additional dollar of wealth. Investors with convex utility functions are called risk-seeking investors. The utility theory primarily used in finance is that developed by Von Neumann and Morgenstern (VNM 1947). VNM define investor utility as a function of rates of return or wealth. Mao (1969) points out that the VNM utility theory is really somewhere between the cardinal and ordinal utility theories. The function associated with the VNM’s utility theory in terms of wealth can be defined: U D f .w ; w / where w indicates expected future wealth and w represents the predicted standard deviation of the possible divergence of actual future wealth from w . Investors are expected to prefer a higher expected future wealth to a lower value. Moreover, they are generally risk averse as well. That is, they prefer a lower value of w to a higher value, given the level of w .2 These assumptions imply that the indifference curves relating w and w will be upward sloping, as indicated in Fig. 5.2. In Fig. 5.2, each indifference curve is an expected utility isoquant showing all the various combinations of risk and return that provide an equal amount of expected utility for the investor. In explaining how investment decisions or portfolio choices are made, utility theory is used here not to imply that individuals actually make decisions using a utility curve, but rather as an expository vehicle that helps explain how 2

Technically, these conditions can be represented mathematically by @U =@w > 0 and @U =@w < 0.

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Fig. 5.1 Utility functions

utilities of the various possible returns. The weights are the probabilities of occurrence associated with each of the possible returns. It is calculated by the following formula: E .U / D

n X

U .wi /Pi

(5.7)

i

where: E .U / D expected utility; U .wi / D the utility of the i th outcome wi ; and Pi D the Probability of the i th outcome. Example 5.1 provides further illustration. Example 5.1. Given investments A and B as shown in the table, determine the utilities of A and B for the given utility functions. Fig. 5.2 Indifference curves of utility functions

A Outcome wi 10 5 1

Probability 2/5 2/5 1/5

B Outcome wi 9 3

Probability investors presumably act. In general, humans behave as if 2/3 more is better than less (the utility curve is upward slop1/3 ing) and marginal utility is decreasing (the utility curve is concave). In an uncertain environment it becomes necessary to ascertain how different individuals will react to risky situ- 1. U.w/ D wi ations. The risk is defined as the probability of success or 2. U.w/ D w2i failure. Alternatively, risk could be described as variability 3. U.w/ D w2i  wi of outcomes, payoffs, or returns. This implies that there is a distribution of outcomes associated with each investment Solution decision. What is needed is a linkage between utility or expected utility and risk. Expected utility has been defined as 1. For U.w/ D wi the numerical value assigned to the probability distribution associated with a particular portfolio’s return. This numeriUtility A D 25 .10/ C 25 .5/ C 15 .1/ D 6 51 cal value is calculated by taking a weighted average of the Utility B D 23 .9/ C 13 .3/ D 7

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

73

Fig. 5.3 Risk-neutral investors and fair games

2. For U.w/ D w2i Utility A D 25 .100/ C 25 .25/ C 15 .1/ D 50 51 Utility B D 23 .81/ C 13 .9/ D 57 3. U.w/ D w2i  wi (use results from 1 and 2) Utility A D50 51  6 15 D 44 Utility B D57  7 D 50 In all three cases, B has the higher degree of utility because it has a higher expected value as well as a smaller dispersion than A. Linear utility function and risk. It is useful now to consider how the shape of an individual’s utility function affects his or her reaction to risk. Assume that an individual who has $5,000 and whose behavior is a linear utility function (Fig. 5.1a) is offered a chance to gain $10,000 with a probability of half or to lose $10,000 with a probability of half. What should he or she pay for such an opportunity? The answer is nothing, for as shown in Fig. 5.3, this individual would be no better or worse off accepting or rejecting this opportunity. If he rejected the offer, his wealth would be $5,000 with utility U1 ; if he paid nothing for the opportunity, his wealth would remain as $5,000 with Utility U1 . Any payment for this chance would reduce his wealth and therefore be undesirable. This is so because the expected value of the fair game is zero: 1 2

.10; 000/ C 12 .10; 000/ D 0

Figure 5.3 illustrates this linear utility function concept. In the following section we will analyze the implication of concave utility function. Concave utility function and risk. Now consider an individual whose behavior is a concave utility function. If this individual participates and wins, her utility is shown by point UW in Fig. 5.4. But if she loses, her position is shown by UL . The expected value of this fair game, having a 50% chance of winning and a 50% chance of losing, is shown by point A.3 The utility of the fair game is UF . A comparison of UF with her initial position, Ui , shows that the investor should not accept this fair game. As shown in Fig. 5.4 the utility of winning .UW  Ui / is less than the utility of losing .Ui  UL /. Therefore, the utility of doing nothing is greater than the expected utility of accepting the fair game. In fact, the individual should be willing to pay up to the difference between the utility of winning and the utility of losing .Ui  UL /  .UW  Ui / to avoid being involved in this situation. Alexander and Francis (1986) theoretically analyze this issue in more detail. Hence investors with concave utility functions are said to be risk averse. That is, they would reject a fair game because the utility derived from winning is less than the utility lost should they lose the game. In other words, the expected utility of participating in a fair game is negative. Example 5.2 will provide illustration of linear, concave, and convex utility function cases for different investment decisions. 3

Following Fama (1970) and Alexander and Francis (1986:177), a fair game means that the expected returns, given information set  , equal the expected returns without the information set. Note that this does not mean the expected returns are zero or positive – they could be negative.

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For a risk-averse investor QUi > E (UF) Þ E(UF)-Ui < 0 \Reject a fair game

Uw -Ui =

Ui -UL =

E(UF)

E(Ul) Concave Utility Function Linear Utility Function

Fig. 5.4 Risk-averse investors and fair games

E.UF / D 12 .Uw C UL / is the expected utility of the fair game under a concave utility function. E.U` / is the expected utility of the fair game under a linear utility function.

This implies the investor is risk averse and would reject a fair gamble. 3.

u.w/ D e 2w u .w/ D 2e 2w 00 u .w/ D 4e 2w < 0 0

Example 5.2. Given the following utility functions for four investors, what can you conclude about their reaction towards a fair game? 1. 2. 3. 4.

u.w/ D w C 4 u.w/ D w  12 w2 (quadratic utility function) u.w/ D e 2w (negative exponential utility function), and u.w/ D w2  4w

Evaluate the second derivative of the utility functions according to the following rules. 00

u .w/ < 0 implies risk averse 00 u .w/ D 0 implies risk neutral 00 u .w/ > 0 implies risk seeker Solution 1.

u.w/ D w C 4 u0 .w/ D 1 00 u .w/ D 0 This implies risk neutrality; it is indifference for the investor to accept or reject a fair gamble.

2.

u.w/ D w  12 w2 u0 .w/ D w2 00 u .w/ D 2w < 0 .We assume wealth is nonnegative:/

This implies risk adversity; the investor would reject a fair gamble. 4.

u.w/ D w2  4w u .w/ D 2w  4 00 u .w/ D 2 > 0 0

This implies risk preference; the investor would seek a fair gamble. The convex utility function is not realistic in real-world decisions; therefore, it is not further explored at this point. The following section discusses the implications of alternative utility functions in terms of indifference curves.

5.3.2 Risk Aversion and Utility Values But when risk increases along with return, the most attractive portfolio is not obvious. How can investors quantify the rate at which they are willing to trade off return against risk? By assuming that each investor can assign a welfare, or utility, score to competing investment portfolios based on the expected return and risk of those portfolios. Higher utility values are assigned to portfolios with more attractive risk-return

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

profiles. Portfolios receive higher utility scores for higher expected returns and lower scores for higher volatility. Assigning a portfolio with expected return and variance of returns, the following utility score: U D E.r/ 

1

A s2 2

Where U D utility E.r/ D expected return on the asset or portfolio A D coefficient of risk aversion s 2 D variance of returns Example: Consider three investors with different degrees of risk aversion: A1 D 2:0; A2 D 3:0, and A3 D 4:0, all of whom are evaluating the three portfolios in Table 5.2. Because the risk-free rate is assumed to be 5%, utility score implies that all three investors would assign a utility score of 0.05 to the risk-free alternative. Table 5.2 presents the utility scores that would be assigned by each investor to each portfolio. The portfolio with the highest utility score for each investor appears in bold. Notice that the high-risk portfolio, H, would be chosen only by the investor with the lowest degree of risk aversion, A1 D 2:0, while the low-risk portfolio, L. would be passed over even by the most risk-averse of our three investors. All three portfolios beat the risk-free alternative for the investors with levels of risk aversion given in Table 5.2.

5.3.3 Capital Allocation Across Risky and Risk-Free Portfolios From the previous discussion we know that the riskier investments offer higher average returns, while less risky investments offers lower average returns. Therefore, a rational investor makes optimal portfolio choice between risky and risk-free securities instead of making all-or-nothing choices from these investment classes. They can and do construct their portfolios using securities from all asset classes. For example, some of the portfolios may be in risk-free Treasury bills, some in high-risk stocks.

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The most straightforward way to control the risk of the portfolio is through the fraction of the portfolio invested in Treasury bills and other safe money market securities versus risky assets. The capital allocation decision is an example of an asset allocation choice – a choice among broad investment classes, rather than among the specific securities within each asset class. Most investment professionals consider asset allocation the most important part of portfolio construction. Example 5.3. Assume that the total market value of a private fund is $500,000, of which $100,000 is invested in a risk-free asset for practical purposes. The remaining $400,000 is invested in risky securities, where $240,000 in equities (E) and $160,000 in long-term bonds (B). Under such assumption, the risky portfolio consists 60% of E and 40% of B, and the weight of the risky portfolio in the mutual fund is 80%. Suppose that the fund manager wishes to decrease risk by reducing the allocation to the risky portfolio from 80 to 70% and not change the proportion of each asset in the risky portfolio. The risky portfolio would then total only 0:7 $500; 000 D $350; 000, requiring the sale of $50,000 of the original $400,000 of risky holdings, with the proceeds used to purchase more shares in risk-free asset. Total holdings in the risk-free asset will increase $500; 000 .10:7/ D $150; 000, the original holdings ($100,000) plus the new contribution ($50,000). To leave the proportions of each asset in the risky portfolio unchanged. Because the weights of E and B in the risky portfolio are 60 and 40%, respectively, the fund manager should sell 0:6 $50; 000 D $30; 000 of E and 0:4 $50; 000 D $20; 000 of B. After the sale, the proportions of each asset in the risky portfolio are in fact unchanged: E W WE D

240; 000  30; 000 D 0:6 400; 000  50; 000

B W WB D

160; 000  20; 000 D 0:4 400; 000  50; 000

5.3.4 Indifference Curves Indifference (utility-function) curves are abstract theoretical concepts. They cannot as a practical matter be used to actually measure how individuals make investment deci-

Table 5.2 Utility scores of alternative portfolios for investors with varying degrees of risk aversion Investor Utility score portfolio L Utility score portfolio L Utility score portfolio L risk aversion (A) E.r/ D :07I  D :05 E.r/ D :09I  D :1 E.r/ D :13I  D :2 2.0

:07 

3.0

:07 

4.0

:07 

1 2 1 2 1 2

 2  :052 D :0675  3  :052 D :0663 2

 4  :05 D :0650

1  2  :12 2 :09  12  3  12 :09  12  4  :12

:09 

D :0800

:13 

D :0750

:13 

D :0700

:13 

1 2 1 2 1 2

 2  :22 D :0900  3  :22 D :0700  4  :22 D :0500

76 Fig. 5.5 Indifference curves for various types of individuals

C.-F. Lee et al.

a

b

Risk-Averse investor d

c

Risk-Seeking Investor sions – or any other decisions for that matter. They are, however, useful tools for building models that illustrate the relationship between risk and return. An investor’s utility function can be utilized conceptually to derive an indifference curve, which shows individual preference for risk and return.4 An indifference curve can be plotted in the risk-return space such that the investor’s utility is equal all along its length. The investor is indifferent to various combinations of risk and return, hence the name indifference curve. Various types of investor’s indifference curves are shown in Fig. 5.5. In the same level of satisfaction, the risk-averse investor requires more return for an extra unit of risk than the return he asks for previous one unit increase of risk. Therefore, the indifference curve is convex. Figure 5.5a presents two indifference curves that are risk-averse and U1 > U2 (higher return and lower risk). For a risk-neutral investor, the indifference curve is a straight line. In the same level of satisfaction, the investor requires the same return for an extra unit of risk as the return he asks for previous one unit increase of risk. Figure 5.5b presents two indifference curves that are

4

Risk-Neutral Investor

By definition, an indifference curve shows all combinations of products (investments) A and B that will yield same level of satisfaction or utility to consume. This kind of analysis is based upon ordinal rather than cardinal utility theory.

Level of Risk-Aversion

risk-neutral, and U1 > U2 . For a risk-seeking investor, the indifference curve is a concave function. In the same level of satisfaction, the investor requires less return for an extra unit of risk than the return he asks for previous one unit increase of risk. Figure 5.5c presents two indifference curves that are risk-seeking and U1 > U2 . The more risk-averse individuals will claim more premiums when they face uncertainty (risk). In Fig. 5.5d, when facing 0 , investor 1 and investor 2 have the same expected return, E .R0 /. However, when the risk adds to 1 , the expected return E .R1 / investor 1 asks is higher than the expected return E .R2 / investor 2 asks. Thus, investor 1 is more risk-averse than investor 2. Therefore, in return-risk plane, the larger slope of indifference curve that the investor has, the higher risk-averse level the investor is. Later in this chapter, different levels of risk and return are evaluated for securities and portfolios when a decision must be made concerning which security or portfolio is better than another. It is at that point in the analysis that indifference curves of hypothetical individuals are employed to help determine which securities or portfolios are desirable and which ones are not. Basically, values for return and risk will be plotted for a number of portfolios as well as indifference curves for different types of investors. The investor’s optimal portfolio will then be the one identified with the highest level of utility for the various indifference curves.

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

Investors generally hold more than one type of investment asset in their portfolio. Besides securities an investor may hold real estate, gold, art, and so on. Thus, given the measures of risk and return for individual securities developed, the measures of risk and return may be used for portfolios of risky assets. Risk-averse investors hold portfolios rather than individual securities as a means of eliminating unsystematic risk; hence the examination of risk and return will continue in terms of portfolios rather than individual securities. Indifference curves can be used to indicate investors’ willingness to trade risk for return; now investors’ ability to trade risk for return needs to be represented in terms of indifference curves and efficient portfolios, as discussed in the next section.

5.4 Efficient Portfolios Efficient portfolios may contain any number of asset combinations. Two examples are shown, a two-asset combination and a three-asset portfolio; both are studied from graphical and mathematical solution perspectives. The degree to which a two-security portfolio reduces variance of returns depends on the degree of correlation between the returns of the securities. This can be best illustrated by expressing the variability in terms of standard deviation (the square root of the variance): p D

q

WA2 A2 C WB2 B2 C 2WA WB rAB A B

(5.8)

First, assume that rAB D 1:0, which would mean that securities A and B are perfectly positively correlated. Then: p D

q

WA2 A2 C WB2 B2 C 2WA WB A B

or:

q p D

.WA A C WB B /2

so: p D WA A C WB B

(5.9)

With the securities perfectly positively correlated, the standard deviation of the portfolio combination is equal to the weighted average of the standard deviation of the component securities. Since the correlation coefficient cannot be greater than 1.0, the weighted average represents the highest possible values of the portfolio standard deviation as discussed previously in this chapter. In this case, there is no diversification taking place. With any correlation coefficient less than 1.0, there will be a diversification effect, and this effect will be larger the lower the value of the correlation coefficient. The ultimate diversification impact occurs if rAB D 1:0, perfect negative correlation.

p D

77

q

WA2 A2 C WB2 B2  2WA WB A B

This can be reduced to: q p D .WA A  WB B /2 D jWA A  WB B j

(5.10)

Example 5.3 provides further illustration. Example 5.4. Two-security portfolio diversification. Assume that RA D 10 percent; RB D 8 percent; A D 4 percent, and B D 3 percent. Since the values for both RA and A are greater than those for RB and B , respectively, there is no dominant choice between the two securities. Higher return is associated with higher risk. Additionally, assume that rAB D 1:0 and that the securities are equally weighted in the portfolio. If follows then that: q .0:5/2 .4/2 C .0:5/2 .3/2 C .2/ .0:5/ .0:5/ .1:0/ .4/ .3/ p D .0:25/ .16/ C .0:25/ .9/ C .2/ .0:25/ .12/ p D 4 C 2:25 C 6 p D 12:25

p D

D 3:5

Notice that this is the same result that would have been achieved using Equation (5.9), p D .0:5/ .4/ C .0:5/ .3/ D 2 C 1:5 D 3:5 percent. If rAB D 0:5 there still is positive correlation, but as it is not perfect positive correlation, there is a diversification effect. q .0:5/2 .4/2 C .0:5/2 .3/2 C .2/ .0:5/ .0:5/ .0:5/ .4/ .3/ p D 4 C 2:25 C 3 p D 9:25 D 3:04

p D

This reduction of p from 3.50 to 3.04 is the effect of diversification. Notice that the first two terms under the square root radical were unaffected and that the third term was only half the size it was with no diversification. The diversification impact from a lower correlation coefficient occurs in the covariance term only. If security A and B were independent of one another, that is, rAB D 0, it is clear that the covariance term would equal zero. p p D p 4 C 2:25 C .2/ .0:5/ .0:5/ .0/ .4/ .3/ D p4 C 2:25 C 0 D 6:25 D 2:5 and the diversification impact reduces p to 2.50. If security A and B were perfectly and negatively correlated, that is, AB D 1, substituting the numbers from the example, first into Equation (5.8), gives

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p p D p 4 C 2:25 C .2/ .0:5/ .0:5/ .1:0/ .4/ .3/ D p6:25  6 D 0:25 D 0:5 and into Equation (5.10): p D .0:5/ .4/  .0:5/ .3/ D 2  1:5 D 0:5 with the appropriate weighting factors, the p may be reduced to zero if there is negative correlation. For example, if WA D 3=7 and WB D 4=7 in this problem: p D .3=7/ .4/  .4=7/ .3/ D0

Fig. 5.6 The relationship between correlation and portfolio risk

5.4.1 Portfolio Combinations Assume that the actual correlation coefficient is 0.50. By varying the weights of the two securities and plotting the combinations in the risk-return space, it is possible to derive a set of portfolios (see Table 5.3) that form an elliptical curve (see Fig. 5.6). The amount of curvature in the ellipse varies with the degree of correlation between the two securities. If there were perfect positive correlation, all of the combinations would lie on a straight line between points A and B. If the combination were plotted with the assumption of perfect negative correlation, the curvature would be more pronounced and one of the points .WA D 0:43; WB D 0:57/ would actually be on the vertical axis (a risk of zero). This process could be repeated for all combination of investments; the result is graphed in Fig. 5.7. In Fig. 5.7 the area within curve XVYZ is the feasible opportunity set representing all possible portfolio combinations. The curve YV represents all possible efficient portfolios and is the efficient frontier. The line segment VX is on the feasible opportunity set, but not on the efficient frontier; all points on VX represent inefficient portfolios. The portfolios and securities that lie on the frontier VX in Fig. 5.7 would not be likely candidates for investors to hold. This is so because they do not meet the criteria of maximizing expected return for a given level of risk or minimizing risk Table 5.3 Possible risk-return combination .RA D 0:10; RB D 0:08; A D 0:04; B D 0:03; AB D 0:5/ Portfolio WA WB RP (percent) P (percent) 1 2 3 4 5

1:00 0:75 0:50 0:25 0:00

0:00 0:25 0:50 0:75 1:00

10:0 9:5 9:0 8:5 8:0

4:00 3:44 3:04 2:88 3:00

Fig. 5.7 The minimum-variance set

for a given level of return. This is easily seen by comparing the portfolio represented by points X and X 0 . As investors always prefer more expected return than less for a given level of risk, X 0 is always better than X . Using similar reasoning, investors would always prefer V to X because it has both a higher return and a lower level of risk. In fact, the portfolio at point V is identified as the minimum-variance portfolio, because no other portfolio exists that has a lower variance. For each of the combinations of individual securities and inefficient portfolios in Fig. 5.7 there is a corresponding portfolio along the efficient frontier that either has a higher return given the same risk or a lower risk given the same return. However, points on the efficient frontier do not dominate one another. While point Y has considerably higher return than point V , it also has considerably higher risk. The opti-

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

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number of securities making up the portfolios will result in a function similar to that illustrated in Fig. 5.9. As shown, the additional reduction in portfolio variance rapidly levels off as the number of securities is increased beyond five. An earlier study by Evans and Archer (1968) shows that the percentage of diversification that can be achieved with randomly selected, equally weighted portfolios level off rapidly beyond a portfolio size of about 15.

5.4.2 Short Selling

Fig. 5.8 Indifference curves and the minimum-variance set

mal portfolio along the efficient frontier is not unique with this model and depends upon the risk/return tradeoff utility function of each investor. Portfolio selection, then, is determined by plotting investors’ utility functions together with the efficient-frontier set of available investment opportunities. No two investors will select the same portfolio except by chance or if their utility curves are identical. In Fig. 5.8, two sets of indifference curves labeled U and U 0 are shown together with the efficient frontier. The U curves have a higher slope, indicating a greater level of risk aversion. The investor is indifferent to any combination of Rp and p along a given curve, for example, U1 ; U2 , or U3 . The U 0 curve would be appropriate for a less risk-averse investor – that is, one who would be willing to accept relatively higher risk to obtain higher levels of return. The optimal portfolio would be the one that provides the highest utility – a point in the northwest direction (higher return and lower risk). This point will be at the tangent of a utility curve and the efficient frontier. The tangency point investor in Fig. 5.8 is point X 0 ; for the risk-averse it is point Y . Each investor is logically selecting the optimal portfolio given his or her risk-return preference, and neither is more correct than the other. Individual investors sometimes find it necessary to restrict their portfolios to include a relatively small number of securities. Mutual-fund portfolios, on the other hand, often contain securities from more than 500 different companies. To give some idea of the number of securities that is necessary to achieve a high level of diversification, Levy and Sarnat (1971) considered a naïve strategy of equally weighted portfolios – that is, of dividing the total investment into equal proportions among component securities. Plotting the variance of the portfolios developed in this manner against the

Short selling (or “going short”) is a very regulated type of market transaction. It involves selling shares of a stock that are borrowed in expectation of a decline in the security’s price. When and if the price declines, the investor buys an equivalent number of shares of the same stock at the new lower price and returns to the lender the stock that was borrowed. The Federal Reserve Board requires short selling customers to deposit 50% of the net proceeds of such short sales with the brokerage firm carrying out the transaction. Another key requirement of a short sale, set by the Securities and Exchange Act of 1934, is that the short sale must occur at a price higher than the preceding sale – or at the same price as the preceding sale, if that took place at a higher price than a preceding price. This is the so-called uptick or zero-tick rule. It prevents the price of a security from successively falling because of continued short selling. Relaxing the assumption of no short selling in this development of the efficient frontier involves a modification of the analysis of the previous section. The efficient frontier analyzed in the previous sections was bounded on both ends by Y and the minimum variance portfolio V , respectively, as shown in Fig. 5.7. Point Y is called the maximum-return portfolio, as there is no other portfolio with a higher return. This point is normally an efficient security or portfolio with the greatest level of risk and return. It could also be a portfolio of securities, all having the same highest levels of risk and return. Point Z is normally a single security with the lowest level of return, although it could be a portfolio of securities, all having the same low level of return. The Black (1972) model is identical to the Markowitz model except that it allows for short selling.5 That is, the nonnegativity constraint on the amount that can be invested in each security is relaxed, WA ? 0. A negative value for the weight invested in a security is allowed, tantamount to allowing a short sale of the security. The new efficient frontier that can be derived with short selling is shown in Fig. 5.10a.

5

Most texts do not identify the Markowitz model with restrictions on short sale. Markowitz (1952), in fact, excluded short sales.

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Fig. 5.9 Native diversification reduces risk to the systematic level in a randomly selected portfolio

The major difference between the frontier in Fig. 5.10a (short selling) and Fig. 5.10b (no short selling) is the disappearance of the end points Y and Z. An investor could sell the lowest-return security .Y /. If the number of short sales is unrestricted, then by a continuous short selling of X and reinvesting in Y the investor could generate an infinite expected return. Hence the upper bound of the highest-return portfolio would no longer be Y but infinity (shown by the arrow on the top of the efficient frontier). Likewise the investor could short sell the highest-return security U and reinvest the proceeds into the lowest-yield security X , thereby generating a return less than the return on the lowest-return security. Given no restriction on the amount of short selling, an infinitely negative return can be achieved, thereby removing the lower bound of X on the efficient frontier. But rational investors will not short sell a high-return stock and buy a low-return stock and buy a low-return stock. The portfolios on VZ 0 always dominate those of VX, as shown in Fig. 5.10a. Whether an investor engages in any of this short-selling activity depends on the investor’s own unique set of indifference curves. Hence, short selling generally will increase the range of alternative investments from the minimum-variance portfolio to plus or minus infinity. However, in the Black model with short selling, no provision was made for the SEC margin requirement. Dyl (1975) imposed the margin requirement on short selling and added it to the Markowitz development of the efficient frontier. The Dyl model. Dyl introduced short selling with margin requirements by creating a new set of risky securities, the ones sold short, which are negatively correlated with the existing set of risky securities. These new securities greatly enhance the diversification effect when they are placed in portfolios. The Dyl model affects the efficient frontier in two ways: (1) If the investor were to combine in equal weight any long position in a security or portfolio with a short position in a security or a portfolio, the resulting portfolio would yield zero return and zero variance and (2) any combination of unequal weighted long or short positions would yield

Fig. 5.10 (a) The efficient frontier with short selling. (b) Efficient frontiers with and without short selling and margin requirements

portfolios with higher returns and lower risk levels. Overall, these two effects will yield an efficient frontier that dominates the Markowitz efficient frontier. Figure 5.10b compares the Dyl and Markowitz efficient frontiers.

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

Even though the inclusion of short selling changes the location and boundaries of the efficient frontier, the concavity of the curve is still intact. This is important in that it preserves the efficient frontier as the locus of optimal portfolios for risk-average investors. As long as the efficient frontier remains concave, by using indifference curves it will be possible to locate the optimal portfolio for each investor. Techniques for calculating the efficient frontier with short selling. Since there are thousands of securities from which the investor can choose for portfolio formation, it can be very costly and time consuming to calculate the efficient frontier set. One way of determining the optimal investment proportions in a portfolio is to hold the return constant and solve for the weighting factors (W1 : : : ; Wn for n securities) that minimize the variance, given the constraint that all weights sum to one and that the constant return equals the expected return developed from the portfolio. The optimal weights can then be obtained by minimizing the Lagrange function C for portfolio variance. C D



n P Wi Wj rij i j C 1 1  Wi i D1 j  D1 i D1  n P C2 E   Wi E .Ri / n P n P

(5.11)

i D1

in which 1 and 2 are the Lagrange multipliers, E  and E .Ri / are targeted rate of return and expected rate of return for security i , respectively, ij is the correlation coefficient between Ri and Rj , and other variables are as previously defined. Function C has n C 2 unknowns: W1 ; : : : ; Wn ; 1 , and 2 . By differentiating C with respect to Wi ; 1 and 2 and equating the first derivatives to zero, nC2 equations can be derived to solve for n C 2 unknowns. As shown in the empirical example later in this chapter, matrix algebra is best suited to this solution process. By using this approach the minimum variance can be computed for any given level of expected portfolio return (subject to the other constraint that the weights sum to one). In practice it is best to use a computer because of the explosive increase in the number of calculations as the number of securities considered grows. The efficient set that is generated by the aforementioned approach [Equation (5.11)] is sometimes called the minimum-variance set because of the minimizing nature of the Lagrangian solution. Thus far no specific distribution for measuring returns has been assumed. In most cases specific distribution of returns will be needed when applying the portfolio model. The normal and log normal are two of the most commonly used distribution. The normal distribution. As you undoubtedly are aware, the normal (probability) distribution is a bell-shaped curve centered on the mean of a given population or sample distri-

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bution. The area under the curve is an accumulation of probabilities that sum to one. With half of the area lying to the left and half to the right of the mean, probability statements may be made about the underlying sample that makes up the distribution. Within the scope of the Markowitz model, the normal distribution may be applied because of the use of the meanvariance assumptions. The formation of probability statements from the underlying sample is a result of the standard deviation (square root of the variance) quantifying the spread of the distribution. If the sample is normally distributed, approximately 68% of the observations will lie within one standard deviation on either side of the mean, approximately 95% will lie within two standard deviations, and approximately 99% will lie within three standard deviations. This ability to ascertain intervals of confidence for the observations allows the utilization of the normal distribution to predict ranges within which the portfolio returns will lie. Utilizing the standard normal distribution (mean of zero, standard deviation of one), any set of mean-variance data can be standardized to develop probability statements about returns on a given portfolio. Standardization is an algebraic operation in which the mean is subtracted from a given return and this difference is divided by the standard deviation. The resultant metric is then standard normal and can be compared with tabulated values to calculate probabilistic quantities of occurrence. Within this framework, any hypothesized level of return can be assigned a probability of occurrence given a probabilistic value of occurrence that is based upon the nature of the sample. For expository purposes, suppose the mean return on a particular investment is 10% for a given period, and that historically these returns have a variance of 0.16%. It is of interest to know what the probabilities are of obtaining a 15% or greater return, or a return less than or equal to 8%. These probabilities may be evaluated by using the normal distribution, as stated in notation below: ˇ

ˇ P X 0:15 ˇRp D 0:1; p2 D 0:0016 ˇ

ˇ P X  0:18 ˇRp D 0:1; p2 D 0:0016 by standardizing the probability statements to z values:

0:15  0:1 D 1:25 P z 0:04

0:08  0:1 P z D 0:5 0:04

(5.12)

These standardized values can then be compared with the tabulated z values found in tables of the standardized normal distribution. For z 1:25 the probability is 10.56%

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(50% – 39.44%), and for z D 0:05 the probability is 0.3085. Therefore this investment has an 11.51% chance of obtaining a 15% or more return, and a 30.85% chance of obtaining 8% or less. It is extremely difficult to make probabilistic statements when using the normal distribution. Given that the return data utilized are ex post, the predictive ability of the normal distribution based on historical data is limited. The past does not always predict the future; hence, this problem is somewhat alleviated by an assumption that the security under consideration is in a static state. Nevertheless, as shown by research and common sense, this assumption is rather bold, and the use of the normal distribution should be limited to comparisons of various past portfolio returns. Another difficulty with the normal distribution is that if the distribution of the sample is skewed in any way, the standard deviation will not properly reflect the equivalent areas under the curve on either side of the mean. This problem will be discussed and solved in the next section through the application of the log normal distribution. A variable is log normally distributed if a logarithm of this variable is normally distributed. The log normal distribution. The log normal distribution discussed in the next chapter will explore further of this section. The approach outlined in the last section for delineating the efficient frontier assumes that the security-return data are normally distributed. In reality, most financial researchers would agree that security-return data tend to be positively skewed. This skewness can be a serious problem in accurately developing the efficient frontier with the Markowitz model because of the assumption that only the first two moments of the return distribution, mean and variance, are important. One reason for returns being positively skewed is the inability of the investor to lose more than 100% of his or her investment, effectively creating a lower bound to portfolio returns. This is called the limited-liability constraint. But since capital gains and dividends could conceivably be infinite, the upper tail of the distribution of returns has no upper bounds. The range of probable returns is spread towards the positive side and therefore contains a potential bias when it is utilized in developing statistical estimates. If the return distributions are significantly skewed, the efficient frontier can be more accurately determined with the use of logarithmically transformed holding-period returns. That is, each holding-period return for security i with holding period T .1 C RiT / is transformed by computing its natural logarithm, ln.1 C RiT /. The logarithmic transformation converts a data set of discretely compounded returns into continuously compounded returns. The distribution of discrete time returns will be more positively skewed the larger the differencing interval used to measure the returns; that is,

C.-F. Lee et al.

if skewness exists in the return distribution, annual data will be more positively skewed than monthly data, which will be more skewed than weekly data. The continuous compounding implied in the logarithmically transformed data will virtually eliminate any positive skewness existing in the raw return data when rates of return are log normally distributed. In practice, it is possible to directly derive the efficient frontier using means and variances of logarithmically transformed data. Under this circumstance, assume that utility functions use continuous returns and variances, and then use the ln.1 C Rp / transformation on discrete data. To deal with the untransformed log normally distributed data, Elton et al. (1976) use the mean and variance of log normal distributed as defined in Equations (5.13) and (5.14) to derive the efficient frontier. In other words, they first delineate the efficient frontier by using the means and variances of the untransformed data and then use Equations (5.13) and (5.14) to determine the subset of the ŒE.1 C Rp /; .1 C Rp / efficient frontier in terms of log-transformed data, ln.1 C Rp /. E.1 C Rp / D eE .rp /C1=2

2

(5.13)

where rp D ln.1 C Rp / and  is the standard deviation of rp .

2

2  2 .1 C Rp / D e2E .rp /C e  1

(5.14)

  where rp D ln.1 C Rp / and  2 D Var rp . Use of the logarithmic transformation can also be shown to eliminate less desirable portfolios in the lowest-return segment of the efficient frontier computer using the raw data.6 Before moving on to an example, however, it will be useful to briefly summarize the portfolio-selection process. Again, the Markowitz model of portfolio selection is a mathematical approach for deriving optimal portfolios; that is, portfolios that satisfy the following conditions. 1. The least risk for a given level of expected return (minimum-variance portfolios). 2. The greatest expected return for a given level of risk (efficient portfolios). How does a portfolio manager apply these techniques in the real world? The process would normally begin with a universe of securities available to the fund manager. These securities would be determined by the goals and objectives of the mutual fund. For example, a portfolio manager who runs a mutual fund specializing in health-care stocks would be required to select securities from the universe of health-care stocks. This would greatly reduce the analysis of the fund manager by limiting the number of securities available. 6

See Baumol (1963).

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

The next step in the process would be to determine the proportions of each security to be included in the portfolio. To do this, the fund manager would begin by setting a target rate of return for the portfolio. After determining the target rate of return, the fund manager can determine the different proportions of each security that will allow the portfolio to reach this target rate of return. The final step in the process would be for the fund manager to find the portfolio with the lowest variance given the target rate of return. The next section uses a graphical approach to derive the optimal portfolios for three securities.

5.4.3 Three-Security Empirical Solution To facilitate a realistic example, actual data have been taken from a set of monthly returns generated by the Dow-Jones 30 Industrials. This example focuses on the returns and risk of the first three industrial companies – AXP (American Express), XOM (Exxon Mobil), and JNJ (Johnson & Johnson) – for the period January 2001–September 2007. The data used are tabulated in Table 5.4. Both graphical and mathematical analyses are employed to obtain an empirical solution. Graphical analysis. To begin to develop the efficient frontier graphically, it is necessary to move from the three dimensions necessitated by the three-security portfolio to a twodimensional problem by transforming the third security into an implicit solution from the other two.7 To do this it must be noted that since the summation of the weights of the three securities is equal to unity, then implicitly: W3 D 1  W1  W2

(5.15)

Additionally, the above relation may be substituted into Equation (5.1):   E Rp D W1 E .R1 / C W2 E .R2 / C W3 E .R3 / D W1 E .R1 / C W2 E .R2 / C E .R3 /

Table 5.4 Data for three securities Company E .ri / i2 JNJ AXP XOM 7

0:0053 0:0055 0:0126

0:0455 0:0614 0:0525

D ŒE .R1 /  E .R3 / W1 C ŒE .R2 / E .R3 / W2 C E .R3 / Finally, inserting the values for the first and second securities yields:   E Rp D .0:0053  0:0126/ W1 C .0:0055  0:0126/ W2 C 0:0126

(5.17)

D 0:0073W1  0:0071W2C0:0126 As can be seen, Equation (5.17) is a linear function in two variables and as such is readily graphable. Since given a certain level of portfolio return the function will solve jointly for the weights of securities 1 and 2, it solves indirectly for the weight of security 3. The variance formula shown in Equation (5.2) is converted in a similar manner by substituting in Equation (5.15) as follows: p2 D

3 3 X X

    Wi Wj Cov Ri ; Rj D Var Rp

i D1 j D1

D W12 11 C W22 22 C W32 33 C 2W1 W2 12 C2W1 W3 13 C 2W2 W3 23 D W12 11 C W22 22 C .1  W1  W2 /2 33 12 C2W1 W2 C 2W1 .1  W1  W2 / 13 C2W2 .1  W1  W2 / 23 D .11 C 33  213 / W12 C .233 C 212 213  223 / W1 W2 C .22 C 33  223 / W22 C .233 C 213 / W1 C .233 C 213 / W2 C 33

(5.18)

Inserting the covariances and variances of the three securities from Table 5.4:

D W1 E .R1 / CW2 E .R2 / C .1  W1  W2 / E .R3 / W1 E .R3 /  W2 E .R3 /

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(5.16)   Cov Ri ; Rj 12 D 0:0009 23 D 0:0010 13 D 0:0004

The process of finding the efficient frontier graphically described in this section was originally developed by Markowitz (1952). Francis and Archer (1979) have discussed this subject in detail.

p2 D Œ0:0455 C 0:0525  2 .0:0004/ W12 C Œ2 .0:0525/ C2 .0:0009/  2 .0:0004/  2 .0:0010/ W1 W2 CŒ0:0614 C 0:0525  2 .0:0010/ W22 CŒ2 .0:0525/ C 2 .0:0004/ W1 CŒ2 .0:0525/ C 2 .0:0010/ W2 C 0:0525 D 0:0972W12 C 0:104W1 W2 C 0:1119W22 0:1042W1  0:103W2 C 0:0525

(5.19)

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Minimum-risk portfolio. Part of the graphical solution is the determination of the minimum-risk portfolio. Standard partial derivatives are taken of Equation (5.18) with respect to the directly solved weight factors as follows: @p2 @W1

D 2 .11 C 33  213 / W1 C .233 C 212 213  223 / W2 C .233 C 213 / D 0

@p2 @W2

D .233 C 212  213  223 / W1

(5.20)

C2 .22 C 33  223 / W2 C .223  233 / D 0 When these two partial derivatives are set equal to zero and the unknown weight factors are solved for, the minimum risk portfolio is derived. Using the numeric values from Table 5.4:

Table 5.5 Iso-return lines Target return W2 1:0 0:5 0:0 0:5 1:0

0:0082 W1 1:5753 1:0890 0:6027 0:1164 0:3699

0:01008 W1 1:3178 0:8315 0:3452 0:1411 0:6274

0:01134 W1 1:1452 0:6589 0:1726 0:3137 0:8000

other to solve for W2 . For example, if W2 D 0 and   values E Rp D 0:0082, then W1 can be solved as follows (see Table 5.5 for final results). 0:0082 D 0:0073W1  0:0071 .0/ C 0:0126 0:0073W1 D 0:0126  0:0082 D 0:01154 W1 D 0:6027

In a similar fashion, other points are calculated and are listed in Table 5.5. A line may be drawn between the points to de@p2 D 2Œ0:0455 C 0:0525  2 .0:0004/ W1 C Œ2 .0:0525/ velop the various iso-expected return function lines. @W1 There are a multitude of possible return functions, but C2 .0:0009/  2 .0:0004/  2 .0:0010/ only these three lines .IR1 ; IR2 and IR3 ) in terms of the data listed in Table 5.5 are shown in Fig. 5.11. W2 C Œ2 .0:0525/ C 2 .0:0004/ In Fig. 5.11 vertical axis and horizontal axis represent W1 D 0:1944W1 C 0:104W2  0:1042 D 0 and W2 , respectively. Each point on the iso-expected return line of Fig. 5.11 rep@p2 D Œ2 .0:0525/ C 2 .0:0009/  2 .0:0004/  2 .0:0010/ resents a different combination of weights placed in the three @W2 securities. The issue now is to determine the portfolio that W1 C 2Œ.0:0614/ C 0:0525  2 .0:0010/ lies on the iso-expected return line with the lowest variance. W2 C Œ2 .0:0010/  2 .0:0525/ (5.21) Table 5.6 shows the portfolio variances as we move along the iso-expected return line. For example, moving along the D 0:104W1 C 0:2238W2  0:103 D 0 iso-expected return line associated with an expected return of 0.82%, note that the minimum variance portfolio is asBy solving these two equations simultaneously the weights sociated with weights of 0.2577 in security 2 and 0.3521 in of the minimum-risk portfolio are derived. This variance rep- security 1. resents the lowest possible portfolio-variance level achievable, given variance and covariance data for these stocks. Iso-variance ellipses. As shown in Table 5.6 the minimumThis can be represented by the point V of Fig. 5.8. This variance portfolio can be found by moving along the isosolution is an algebraic exercise that yields W1 D 0:3857 expected return line until the minimum-variance portfolio is and W2 D 0:2810 and, therefore, through Equation (5.15), reached. To better visualize this, examine the variance of a portfolio as the proportions in security 1 and security 2 W3 D 0:3333. are changed. A plot of these points will allow the family of The iso-expected return line. The variance and return iso-variance ellipses to be traced. The iso-variance ellipses equations have been derived, and it is now time to com- are the ellipses that have the same variance on every point plete the graphing procedure. To begin the graphing, the iso- of this ellipse curve. It should be noted that an ellipse is expected return function lines must be delineated given var- an egg-shaped circle of points with a common center and ious levels of expected return. The iso-expected return line orientation. The minimum-risk portfolio variance is the cenis a line that has the same expected return on every point ter of all possible ellipses as it has the least risk. It will be of the line. Utilizing Equation (5.17), three arbitrary returns desirable, then, to find this minimum risk value, as no other are specified: 0.0082, 0.01008, 0.01134 (these correspond weighting scheme of these three securities will develop a to 70, 80, and 90% annual returns of Exxon Mobil). These lesser risk. The solutions of Equation (5.21) have determined three monthly returns are then set equal to Equation (5.17), that the weights associated with the minimum-risk portfolio and graphing is now possible by setting W1 equal to zero or are W1 D 0:3857; W2 D 0:2810, and W3 D 0:3333. Substi-

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

85

Fig. 5.11 Iso-return lines

Table 5.6 Portfolio variance along the iso-return line

0.0082 w2 1 0:75 0:5 0:25 0 0:0795 0:151 0:25 0:2577 0:5 0:75 1

w1 1:5753 1:3322 1:0890 0:8459 0:6027 0:5254 0:4559 0:3596 0:3521 0:1164 0:1267 0:3699

0.01008 Var 0.1806 0.1225 0.0771 0.0447 0.0250 0.0214 0.0194 0.0182 0.0182 0.0242 0.0431 0.0748

w1 1:3178 1:0747 0:8315 0:5884 0:3452 0:2679 0:1983 0:1021 0:0946 0:1411 0:3842 0:6274

0.01134 Var 0.1618 0.1091 0.0693 0.0423 0.0281 0.0263 0.0258 0.0268 0.0269 0.0383 0.0626 0.0998

w1 1:1452 0:9021 0:6589 0:4158 0:1726 0:0953 0:0257 0:0705 0:0780 0:3137 0:5568 0:8000

Var 0.1564 0.1074 0.0713 0.0479 0.0374 0.0368 0.0373 0.0397 0.0400 0.0549 0.0829 0.1238

Note: Underlined variances indicate minimum variance portfolios

tuting this information listed in Table 5.4 into Equation (5.2), the variance of the minimum-risk portfolio is: n n X     X Wi Wj Cov Ri Rj Var Rp D i D1 j D1

be completed by taking Equation (5.18) andholding one of  , constant. portfolio variance Var R the weights, say W p  2 Bring the Var Rp to the right-hand side of the equation and notice that the equation is of a quadratic form and can be solved using the quadratic formula:

D W12 11 C W22 22 C W32 33 C 2W1 W2 12

W1 D

C2W2 W3 23 C 2W1 W3 13 D .0:3857/2 .0:0455/ C .0:2810/2 .0:0614/ C .0:3333/2 .0:0525/ C2 .0:3857/ .0:2810/ .0:0009/ C2.0:2810/.0:3333/.0:0010/ C 2.0:3857/ .0:3333/.0:0004/ D 0:017934 Note that the variance of the minimum-risk portfolio can be used as a base for graphing the iso-variance ellipses. It can

b ˙

p b 2  4ac 2a

(5.22)

where: a D all coefficients of W12 ; b D all coefficients of W1 ; and c D all coefficients that are not multiplied by W1 , or W12 : or a D 11 C 33 C 13 ; b D .233 C 212  213  223 / W2  233 C 213 ; and c D .22 C33  223 / W22 C .233 C 223 / W2 C 33  Var Rp .

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Substituting the numbers from the data of Table 5.4 into Equation (5.18) yields: Var.Rp / D 0:0972W21 C 0:104W1 W2 C 0:1119W22 0:1042W1  0:103W2 C 0:0525 0 D 0:0972W21 C 0:104W1 W2 C 0:1119W22 0:1042W1  0:103W2 C 0:0525Var.Rp / where: a D 0:0972I b D 0:104W2  0:1042I and c D 0:1119W22  0:103W2 C 0:0525  Var.Rp /: When these expressions are plugged into the quadratic formula: p  .0:104W2  0:1042/ ˙ b 2  4ac W1 D (5.23) 2 .0:0972/ where b 2 D .0:104W2  0:1042/2 4ac D 4f.0:0972/Œ0:1119W22  0:103W2 C0:0525  Var.Rp / g This is a solution for the two points of W1 (W11 and W12 ) for a given portfolio variance and weight of the second security .W2 /. When selecting the level of Var .Rp / it is best to choose a value that is slightly larger than the minimumrisk value. This assures the calculation of a possible portfolio, since no portfolio of these three securities may have less risk. Additionally, an initial selection of a value for W2 close to the minimum-risk portfolio W2 will be desirable. In sum, Equation (5.23) can be used to construct the isovariance ellipse for given Var .Rp / in terms of arbitrary W2 . By jointly considering iso-expected return lines and isovariance ellipses, efficient portfolios can be identified that are defined as the portfolios with the minimum variance given the expected rate of return. This task can be done by computer, as mentioned in the last section. To obtain the weights of an efficient portfolio, the following set of instructions must be entered into the computer. 1. Find the portfolio weights that minimize portfolio variance, subject to the target expected rate-of-return constraint. For this case, target rates of return are 0.82, 1.008, or 1.1134%. The sum of the portfolio weights for all stocks in the portfolio must be equal to one. Mathematically, this instruction is defined in Equation (5.15).

2. The expected return and variance for a three-security portfolio is defined in Equations (5.16) and (5.18), respectively. 3. The estimated expected return, variance, and covariance as defined in Table 5.4 should be entered into the computer for estimation. Using E.Rp / D 0:82 percent as an example, W1 D 0 and W2 D 0:6197. From the relationship W3 D 1  W1  W2 D 1  0  0:6197 D 0:3803. In other words, the point H represents three portfolio weights for the three stocks. The computer substitutes weights, variances, and covariances (listed in Table 5.4) into Equation (5.18) to obtain portfolio variance. The portfolio variance consistent with the weights of point H is computed. The computer now moves by some predetermined distance either northeast or southwest along the 0.82% iso-expected return line. From this kind of search, the minimum variance associated with this 0.82% iso-expected return line is identified to be 0.0182, as indicated in Table 5.6. By a similar procedure, the minimum variances associated with 1.008 and 1.134% iso-expected return lines are identified to be 0.0258 and 0.0368, respectively. When minimum variances associated with predefined iso-expected return are identified, the optimal weights associated with expected returns equal to 1.008 and 1.134% in terms of data indicated in Table 5.4 are also calculated. These results are indicated in Table 5.6. Using the minimum variances associated with isoexpected returns (0.82, 1.008, and 1.134%), three isovariance ellipses can be drawn as IV 1 ; IV 2 , and IV 3 . When Var.Rp / D 0:0182 and W2 D 0:3, the quadratic formula from Equation (5.23) above yields two roots. p .0:0730/C

W11 D D 0:4247

.0:0730/2 .0:0052/ 2.0:0972/

p .0:0730/2 W12 D .0:0730/ 2.0:0972/ .0:0052/ D 0:3263 To solve for W11 and W12 we need b 2 > 4ac. Solving for the variance: .0:104W2  0:1042/2  4f.0:0972/ > Œ0:1119W22 0:103W2 C 0:0525  Var.Rp / g If W2 and Var .Rp / are not selected so that b 2 > 4ac, Equation (5.23) will require taking the square root of a negative number, and mathematically the solution will be an imaginary number. Solving W2 in terms of an imaginary number is not meaningful; therefore we only consider the case when b 2 > 4ac.

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

The variance chosen was the minimum variance of the portfolio associated with E.Rp / D 0:0082. Actually, any variance might have realistically been chosen as long as it exceeded the minimum-risk portfolio variance. Table 5.7 lists various W11 and W12 values for given levels of W2 and Var .Rp /. In addition, Table 5.7 presents the value of b 2 and 4ac to check whether or not the root is a real number. It should be noted that all possible variances are higher than the minimum-risk portfolio variance. Data from Table 5.7 are used to draw three iso-variance ellipses, as indicated in Figs. 5.12 and 5.13. The critical line and efficient frontier. After the iso-expected return functions and iso-variance ellipses have been plotted, it is an easy task to delineate the efficient frontier. By definition, the efficient portfolio is the portfolio with the highest return for any given risk. In Fig. 5.13, the efficient portfolios are those where a given iso-expected return line is just tangent to the variance ellipse. MRPABC is denoted as Table 5.7 Various W1 given W2 and Var.Rp / W2 Var.Rp / b b2 0:28 0:29 0:30 0:31 0:32 0:28 0:29 0:30 0:31 0:32 0:28 0:29 0:30 0:31 0:32

Fig. 5.12 Iso-variance ellipses

0:0182 0:0182 0:0182 0:0182 0:0182 0:0258 0:0258 0:0258 0:0258 0:0258 0:0368 0:0368 0:0368 0:0368 0:0368

0:0751 0:0740 0:0730 0:0720 0:0709 0:0751 0:0740 0:0730 0:0720 0:0709 0:0751 0:0740 0:0730 0:0720 0:0709

0:0056 0:0055 0:0053 0:0052 0:0050 0:0056 0:0055 0:0053 0:0052 0:0050 0:0056 0:0055 0:0053 0:0052 0:0050

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the critical line; all portfolios that lie between points MRP and C are said to be efficient, and the weights of these portfolios may be read directly from the graph. From the graph, portfolio weights for the portfolios that minimize variances, given a 0.82, 1.008, or 1.134% expected rate of return are listed in Table 5.6. It is possible, given these various weights, to calculate the E.Rp / and the variances of these portfolios, as indicated in Table 5.8. The efficient frontier is then developed by plotting each risk-return combination, as shown in Fig. 5.14. In this section a step-wise graphical approach has been employed to obtain optimal weights for a three-security portfolio. In the next section a mathematical optimization approach is used to calculate the optimal weight for a threesecurity portfolio. Mathematical analysis. The same efficient frontier can be obtained mathematically through the use of the Lagrangian multipliers. The Lagrangian method allows the minimization

C 0:0142 0:0138 0:0135 0:0131 0:0128 0:0066 0:0062 0:0059 0:0055 0:0052 0:0044 0:0048 0:0051 0:0055 0:0058

4 ac 0:0055 0:0054 0:0052 0:0051 0:0050 0:0026 0:0024 0:0023 0:0021 0:0020 0:0017 0:0019 0:0020 0:0021 0:0023

W11 0:4385 0:4325 0:4247 0:4149 0:4025 0:6707 0:6652 0:6594 0:6534 0:6470 0:8268 0:8213 0:8157 0:8099 0:8039

W12 0:3339 0:3293 0:3263 0:3254 0:3272 0:1017 0:0965 0:0916 0:0870 0:0827 0:0543 0:0596 0:0647 0:0696 0:0742

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Fig. 5.13 Iso-variance ellipses and iso-return lines

Table 5.8 Weights, Rp , and p for efficient points Portfolio W1 W2 W3 A B C

0:3521 0:1983 0:3902

0:2577 0:1510 0:6507

0:3902 0:0795 0:8252

E.Rp /

Var.Rp /

0:0082 0:01008 0:01134

0:0182 0:0258 0:0368

 .Rp / 0:1349 0:1606 0:1918

Portfolios A, B, and C represent expected returns of 0.82, 1.008, and 1.134%, respectively Fig. 5.14 The efficient frontier for the data of Table 5.8

or maximization of an objective function when the objective function is subject to some constraints. One of the goals of portfolio analysis is minimizing the risk or variance of the portfolio, subject to the portfolio attaining some target expected rate of return, and also subject to the portfolio weights’ summing to one. The problem can be stated mathematically: Min p2 D

n n X X i D1 j D1

Wi Wj ij

Subject to

(a)

n P

Wi E.Ri / D E 

i D1

where E  is the target expected return and

(b)

n P i D1

Wi D 1:0

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

The first constraint simply says that the expected return on the portfolio should equal the target return determined by the portfolio manager. The second constraint says that the weights of the securities invested in the portfolio must sum to one. The Lagrangian objective function can be written: C D

n n X X





Wi Wj Cov Ri Rj C 1 1 

i D1 j D1

"

C2 E  

n X

# Wi E .Ri /

n X

! Wi

i D1

(5.11)

i D1

Taking the partial derivatives of this equation with respect to each of the variables, W1 ; W2 ; W3 ; 1 ; 2 and setting the resulting five equations equal to zero yields the minimization of risk subject to the Lagrangian constraints. This system of five equations and five unknowns can be solved by the use of matrix algebra. Briefly, the Jacobian matrix of these equations is 2

211 6 221 6 6 231 6 4 1 E .R1 /

212 213 222 223 232 233 1 1 E .R2 / E .R3 /

3 2 3 2 3 W1 0 1 E .R1 / 6W2 7 6 0 7 1 E .R2 / 7 7 6 7 6 7 6 7 6 7 1 E .R3 / 7 7 6W3 7 D 6 0 7 4 1 5 4 1 5 5 0 0 E 0 0 2

89

W A1 K 3 2 3 W1 0:9442 6 W2 7 6 0:6546 7 6 7 6 7 6 W3 7 D 6 0:5988 7 6 7 6 7 4 1 5 4 0:1937 5 2 20:1953 2

(5.26)

Using the matrix operation on the other two arbitrary returns yields the results shown in Table 5.8. As shown in the composition of Portfolios A, B, and C, the weights of security 3 are negative, implying that this security should be sold short in order to generate an efficient portfolio. It should be noted that the results of Table 5.8 are similar to those of Table 5.6. Therefore, both graphical and mathematical methods can be used to calculate the optimal weights. With the knowledge of the efficient-portfolio weights, given that E.Rp / is equal to 0.00106, 0.00212, and 0.00318, the variances of the efficient portfolios may be derived from plugging the numbers into Equation (5.2). Taking the square root of the variances to derive the standard deviation, the various risk-return combinations can be plotted and the efficient frontier graphed, as shown in Fig. 5.13 in the previous section. As can be seen from comparing this figure to the graphical derivation of the efficient frontier derived graphically, the results are almost the same.

(5.24) Therefore, it is possible to premultiply both sides of the matrix Equation (5.24), AW D K, by the inverse of A (denoted A1 ) and solve for the W column. This is possible because all values in the A and K matrices are known or arbitrarily set. The first problem is the inversion of the matrix of coefficients. First, derive the determinant of matrix A. Next, after developing the signed, inverted matrix of cofactors for A, divide the cofactors by the determinant, resulting in the inversion of the original matrix. Finally, premultiply the column vector for the investment weights. Plugging the data listed in Table 5.4 and E  D 0:00106 into the matrix above yields 2

0:0910 6 0:0036 6 6 6 0:0008 6 4 1 0:0053

0:0018 0:1228 0:0020 1 0:0055

0:0008 0:0020 0:1050 1 0:0126

3 2 3 2 3 1 0:0053 W1 0 6 7 6 7 1 0:0055 7 7 6 W2 7 6 0 7 7 6 7 6 7 1 0:0126 7 6W3 7 D 6 0 7 7 6 7 6 7 5 4 1 5 4 1 5 0 0 2 0 0 0:00106 (5.25)

When matrix A is properly inverted and postmultiplied by K, the solution vector A1 K is derived:

5.4.4 Portfolio Determination with Specific Adjustment for Short Selling By using a definition of short sales developed by Lintner (1965) the computation procedure for the efficient frontier can be modified. Lintner defines short selling as putting up an amount of money equal to the value of the security sold short. Thus the short sale is a use rather than a source of funds to the short seller. The total funds the investor invests short, plus the funds invested long, must add up to the original investment. The proportion of funds invested short is jXi j, since Xi < 0. The constraint in the minimization problem concerning the weights of the individual securities needs to be modified to incorporate this fact. Additionally, the final portfolio weight (output of the matrix inversion) must be rescaled so the sum of the absolute value of the weights equals one. By defining short sales in this manner, the efficient frontier does not extend to infinity (as shown in Fig. 5.10a) but resembles the efficient frontier in Fig. 5.10b. Monthly rates of return for Johnson & Johnson, American Express, and Exxon Mobile are used to perform the analysis in this section. The sample period is from January

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2001 to September 2007. The Markowitz model determines optimal asset allocation by minimizing portfolio variance using a constrained optimization procedure: 



Min Var Rp D

3 3 X X

Wi Wj ij

(5.2)

Security JNJ AXP XOM

Monthly E.Ri /

i2

12

13

23

0.0053 0.0055 0.0126

0.0455 0.0614 0.0525

0.0009

0.0004

0.0010

i D1 j D1

Subject to: (a) (b)

3 P i D1 3 P

Substituting these values into the matrix: 2 3 2 0:0910 6 0:0018 6 6 6 0:0008 6 4 0:0053 1

Wi E .Ri / D E (5.1) 

jWi j D 1:0 (5.1)

i D1

Where the E  is the investors’ desired rate of return, and where the absolute value of the weights jWi j allows for a given W to be negative (sold short) but maintains the requirement that all funds are invested or their sum equals one. The Lagrangian function is Min L D

n n X X

W i W j C 1

i D1 j D1

C2

n X

!

n X 

Wi E .Ri /  E



0:0018 0:0614 0:0020 0:0055 1

E  D 0:013

By solving using the identity-matrix technique the weights for the three securities can be obtained: Johnson & Johnson D 0:0406 American Express D 0:0146 Exxon Mobile D 1:0552



i D1

By using the following relationship to rescale these weights so that the second constraint for the sum of the absolute values of the weights to equal one is satisfied:

Wi  1

i D1

Again, derivatives with respect to Wi ’s and ’s are found. Setting these equations equal to zero leaves the following system of equations in matrix form: C  

X K 2 3 3 2 W1 0 211 212 21n E .R1 / 1 6 6 221 7 7 6 222 22n E .R2 / 1 7 7 6 W2 7 6 0 7 6 6 6 7 6 : 7 :: :: :: :: 7 6 :: 7 6 :: 7 6 : 7 : : : :7 6 : 7D 6 : 7 6 : 7 6 6 7 7 6 6 2n1 2n2    2nn E .Rn / 1 7 6 Wn 7 6 0 7 7 6 6 7 6 7 4 E .R1 / E .R2 /    E .Rn / 0 0 5 4 1 5 4 E 5 1 1  1 0 0 2 1 2

3 3 2 0:0008 0:0053 1 0 W1 7 7 6 6 0:0020 0:0055 1 7 7 6 W2 7 6 0 7 7 7 6 7 6 0:0525 0:0126 1 7 6 W3 7 D 6 0 7 7 6 7 7 6 0:0126 0 0 5 4 1 5 4 E 5 2 1 1 0 0

3

To solve this matrix:

WjAj D

jAj jAj C jBj C jC j

(5.27)

The resealed absolute weights are: 0:0406 D 0:0366 0:0406 C 0:0146 C 1:0552 0:0146 WAXP D D 0:0131 0:0406 C 0:0146 C 1:0552 1:0552 WXOM D D 0:9503 0:0406 C 0:0146 C 1:0552 WJNJ D

The return on this portfolio is: CX C1 CX IX X

DK D C1 K D C1 K D C1 K

Rp D .0:0366/ .0:0053/ C .0:0131/ .0:0055/ C .0:9503/ .0:0126/ D 1:22%

where C1 is the inverse of matrix C and I is the identity matrix. The solution to the above formula will give the weights in terms of E  . By using Johnson & Johnson, American Express, and Exxon Mobile a three-security portfolio will be formed and optimal weights will be solved for.

The variance: p2 D .0:0366/2 .0:0455/ C .0:0131/2 .0:0614/ C .0:9503/2 .0:0525/ C 2 .0:0366/ .0:0131/ .0:0009/ C 2 .0:0366/ .0:9503/ .0:0004/ C 2 .0:0131/ .0:9503/ .0:0010/ D 0:0475

5 Risk-Aversion, Capital Asset Allocation, and Markowitz Portfolio-Selection Model

5.4.5 Portfolio Determination Without Short Selling The minimization problem under study can be modified to include the restriction of no short selling by adding a third constraint: Wi 0; i D 1; : : : ; N The addition of this nonnegativity constraint precludes negative values for the weights (that is, no short selling). The problem now is a quadratic programming problem similar to the ones solved so far, except that the optimal portfolio may fall in an unfeasible region. In this circumstance the next best optimal portfolio is elected that meets all of the constraints. An example of this situation is shown in Fig. 5.15.8 In Fig. 5.15a the optimal portfolio is located in the region of positive values for weights Wi , and thereby satisfies the

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constraint Wi 0. In Fig. 5.15b the optimal portfolio shown by Wi falls in a region where W1 are negative, and so the constraint is not satisfied. The next best optimal portfolio is shown by B, and this is the solution of the problem that satisfies the nonnegativity constraint. Additional discussion of the problems is presented in later chapters. Most recently, Lewis (1988) has developed a simple algorithm for the portfolioselection problem. His method is based on an interactive use of the Markowitz critical-line method for solving quadratic programs. In ı Fig. 5.15a, b the vertical axis is defined as .RN p  Rf / p , this is a Sharpe performance mean, as defined in Chap. 4, and can be regarded as the objective function for a portfolio selection.

5.5 Conclusion This chapter has focused on the foundations of Markowitz’s model and on derivation of efficient frontier through the creation of efficient portfolios of varying risk and return. It has been shown that an investor can increase expected utility through portfolio diversification as long as there is no perfect positive correlation among the component securities. The extent of the benefit increases as the correlation is lower, and also increases with the number of securities included. The Markowitz model can be applied to develop the efficient frontier that delineates the optimal portfolios that match the greatest return with a given amount of risk. Also, this frontier shows the dominant portfolios as having the lowest risk given a stated return. This chapter has included methods of solving for the efficient frontier both graphically and through a combination of calculus and matrix algebra, with and without explicitly incorporating short selling. The next chapter will illustrate how the crushing computational load involved in implementing the Markowitz model can be alleviated through the use of the index portfolio, and how the tenets of the Markowitz efficient frontier are still met.

References Alexander, G. J. and J. C. Francis. 1986. Portfolio analysis, PrenticeHall, Englewood Cliffs, NJ. Baumol, W. J. 1963. “An expected gain-confidence limit criterion for portfolio selection.” Management Science 10, 171–182. Fig. 5.15 Optimal portfolio in feasible and unfeasible regions ı In Fig. 5.15a and b the vertical axis is defined as .Rp  Rf / p , which is the ı Sharpe performance measure as defined in Chap. 4. .Rp  Rf / p is the objective function for portfolio optimization; it will be explored in Chap. 8.

8

Bertsekas, D. 1974. “Necessary and sufficient conditions for existence of an optimal portfolio.” Journal of Economic Theory 8, 235–247. Black, F. 1972. “Capital market equilibrium with restricted borrowing.” Journal of Business 45, 444–455. Blume, M. 1970. “Portfolio theory: a step toward its practical application.” Journal of Business 43, 152–173.

92 Bodie, Z., A. Kane, and A. Marcus. 2006. Investments, 7th Edition, McGraw-Hill, New York. Brealey, R. A. and S. D. Hodges. 1975. “Playing with portfolios.” Journal of Finance 30, 125–134. Breen, W. and R. Jackson. 1971. “An efficient algorithm for solving large-scale portfolio problems.” Journal of Financial and Quantitative Analysis 6, 627–637. Brennan, M. J. 1975. “The optimal number of securities in a risky asset portfolio where there are fixed costs of transaction: theory and some empirical results.” Journal of Financial and Quantitative Analysis 10, 483–496. Cohen, K. and J. Pogue. 1967. “An empirical evaluation of alternative portfolio-selection models.” Journal of Business 46, 166–193. Dyl, E. A. 1975. “Negative betas: the attractions of selling short.” Journal of Portfolio Management 1, 74–76. Elton, E. J. and M. Gruber. 1974. “Portfolio theory when investment relatives are log normally distributed.” Journal of Finance 29, 1265–1273. Elton, E. J. and M. Gruber. 1978. “Simple criteria for optimal portfolio selection: tracing out the efficient frontier.” Journal of Finance 13, 296–302. Elton, E. J., M. Gruber, and M. E. Padberg. 1976. “Simple criteria for optimal portfolio selection.” Journal of Finance 11, 1341–1357. Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2006. Modern portfolio theory and investment analysis, 7th Edition, Wiley, New York. Evans, J. and S. Archer. 1968. “Diversification and the reduction of dispersion: an empirical analysis.” Journal of Finance 3, 761–767. Fama, E. F. 1970. “Efficient capital markets: a review of theory and empirical work.” Journal of Finance 25, 383–417. Feller, W. 1968. An introduction to probability theory and its application, Vol. 1, Wiley, New York. Francis, J. C. and S. H. Archer. 1979. Portfolio analysis, Prentice-Hall, Englewood Cliffs, NJ. Gressis, N., G. Philiippatos, and J. Hayya. 1976. “Multiperiod portfolio analysis and the inefficiencies of the market portfolio.” Journal of Finance 31, 1115–1126. Henderson, J. and R. Quandt. 1980. Microeconomic theory: a mathematical approach, 3d Edition, McGraw-Hill, New York. Lee, C. F. and A. C. Lee 2006. Encyclopedia of finance, Springer, New York.

C.-F. Lee et al. Lee, C. F., J. C. Lee, and A. C. Lee. 2000. Statistics for business and financial economics, World Scientific Publishing Co, Singapore. Levy, H. and M. Sarnat. 1971. “A note on portfolio selection and investors’ wealth.” Journal of Financial and Quantitative Analysis 6, 639–642. Lewis, A. L. 1988. “A simple algorithm for the portfolio selection problem.” Journal of Finance 43, 71–82. Lintner, J. 1965. “The valuation of risk assets and the selection of risky investments in stock portfolio and capital budgets.” Review of Economics and Statistics 47, 13–27. Mao, J. C. F. 1969. Quantitative analysis of financial decisions, Macmillan, New York. Maginn, J. L., D. L. Tuttle, J. E. Pinto, and D. W. McLeavey. 2007. Managing investment portfolios: a dynamic process, CFA Institute Investment Series, 3rd ed., Wiley, New Jersey. Markowitz, H. M. 1952. “Portfolio selection.” Journal of Finance 1, 77–91. Markowitz, H. M. 1959. Portfolio selection. Cowles Foundation Monograph 16, Wiley, New York. Markowitz, H. M.1976. “Markowitz revisited.” Financial Analysts Journal 32, 47–52. Markowitz, H. M. 1987. Mean-variance analysis in portfolio choice and capital markets. Blackwell, Oxford. Martin, A. D., Jr. 1955. “Mathematical programming of portfolio selections.” Management Science 1, 152–166. Merton, R. 1972. “An analytical derivation of efficient portfolio frontier.” Journal of Financial and Quantitative Analysis 7, 1851–1872. Mossin, J. 1968. “Optimal multiperiod portfolio policies.” Journal of Business 41, 215–229. Ross, S. A. 1970. “On the general validity of the mean-variance approach in large markets,” in Financial economics: essays in honor of Paul Cootner, W. F. Sharpe and C. M. Cootner (Eds.). Prentice Hall, Englewood Cliffs, NJ, pp. 52–84. Sharpe, W. F. 1970. Portfolio theory and capital markets, McGraw-Hill, New York. Von Neumann, J. and O. Morgenstern. 1947. Theory of games and economic behavior, 2nd Edition, Princeton University Press, Princeton, NJ. Wackerly, D., W. Mendenhall, and R. L. Scheaffer. 2007. Mathematical statistics with applications, 7th Edition, Duxbury Press, California.

Chapter 6

Capital Asset Pricing Model and Beta Forecasting Cheng-Few Lee, Joseph E. Finnerty, and Donald H. Wort

Abstract In this chapter, using the concepts of portfolio analysis and the dominance principle, we derive the capital asset pricing model (CAPM). Then we show how total risk can be decomposed into systematic risk and unsystematic risk. Finally, we discuss the determination of beta and introduce different methods for forecast beta coefficient. Keywords Capital asset pricing model r Beta r Capital market line r Market risk premium r Security market line r Market model r Capital labor ratio r Efficient market hypothesis r Market beta r Accounting beta r Systematic risk r Unsystematic risk

6.1 Introduction One of the important financial theories is the capital asset pricing model, commonly referred to as CAPM, which constitutes the major topic of this chapter; in addition, alternative methods for forecasting beta (systematic risk) are discussed. Using the concepts of basic portfolio analysis and the dominance principle discussed in the last two chapters, two alternative methods are employed to derive the CAPM. The discussion concerns how the market model can be used to decompose (that is, divide) risk into two components, systematic and unsystematic risk. In addition, applications of the beta coefficient and procedures for forecasting a beta coefficient. A graphical approach is first utilized to derive the CAPM, after which a mathematical approach to the derivation is developed that illustrates how the market model can be used to decompose total risk into two components. This is followed J.E. Finnerty () University of Illinois at Urbana-Champaign, Champaign, IL, USA e-mail: [email protected] C.-F. Lee Rutgers University, New Brunswick, NJ, USA D.H. Wort University of California, Hayward, CA, USA

by a discussion of the importance of beta in security analysis and further exploration of the determination and forecasting of beta. The discussion closes with the applications and implications of the CAPM. The appendix offers empirical evidence of the risk-return relationship.

6.2 A Graphical Approach to the Derivation of the Capital Asset Pricing Model Following the risk-return tradeoff principle and the portfolio diversification process, Sharpe (1964), Lintner (1965), and Mossin (1966) have developed an asset pricing model that can determine both the market price of a portfolio and the price of an individual security. They focus on the pricing determination of those parts of security risk that can be eliminated through diversification as well as those that cannot. Systematic risk is that part of total risk that results from the common variability of stock prices and the subsequent tendency of stock prices to move together with the general market. The other portion of total risk is unsystematic risk, the result of variables peculiar to the firm or industry; for example, a labor strike or resource shortage. Systematic risk, also referred to as market risk, reflects the swings of the general market. Some stocks and portfolios can be very sensitive to movements in the market, while others show more independence and stability. The universally accepted notion for the measure of a stock’s or a portfolio’s relative sensitivity to the market based upon its past record is the Greek letter beta .“/. The estimation, application, and forecasting of beta are discussed in the following sections.

6.2.1 The Lending, Borrowing, and Market Portfolios The next chapters will assume that the efficient frontier was constructed with risky assets only. If there are both risk-free

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_6, 

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94

Fig. 6.1 The capital market line

and risky assets, investors would then have the choice of investing in one or the other, or in some combination of the two. The concept of the risk-free asset is generally proxied by a government security, such as the Treasury bill (T-bill). T-bills are backed by the Federal government and are default free, hence considered riskless. An investor’s portfolio can be composed of different combinations of riskless and risky assets. Figure 6.1 is a graph of different sets of portfolio opportunities; it includes the risk-free asset with a return of Rf . Since the riskless asset has zero risk, it is represented by a point on the vertical axis. With the additional alternative to invest in risk-free assets that yield a return of Rf , the investor is able to create new combinations of portfolios that combine risk-free assets with the risky assets. The investor is thus able to achieve any combination of risk and return that lies along the line connecting Rf and a tangent point Mp the market portfolio. All portfolios along the line Rf Mp C are preferred to the risky portfolio opportunities on the curve AM p B. Therefore, the points on the line Rf Mp C represent the best attainable combinations of risk and return. At point Rf the investor has all available funds invested in the riskless asset and expects to receive the return of Rf . The portfolios along the line Rf Mp are lending portfolios and contain combinations of investments in the risk-free asset and investments in a portfolio of risky assets Mp . These are called lending portfolios because investors are, in effect, lending money to the government at the risk-free rate, when they invest in the risk-free asset. At point Mp the investor wants only risky assets and has put his wealth into the risky-asset portfolio, which is called the market portfolio. The investor at this point is neither lending nor borrowing. An alternative view is that at this point the investor may be lending and borrowing equal amounts that just offset each other. At Mp investors receive a rate of return Rm and undertake risk m .

C.-F. Lee et al.

If it is assumed that the investor can borrow money at the risk-free rate Rf and invest this money in the risky portfolio Mp , he will be able to derive portfolios with higher rates of return but with higher risks along the line extending beyond Mp C . The portfolios along line Mp C are borrowing portfolios. They are so called because they contain a negative amount of the risk-free asset. The negative amount invested in the risk-free asset can be viewed as borrowing funds at the risk-free rate and investing in risky assets. The borrowing for investment is called margin and is controlled by the government. Therefore, the new efficient frontier becomes Rf Mp C and is referred to as the capital market line (CML). The capital market line describes the relationship between expected return and total risk.

6.2.2 The Capital Market Line An illustration and explanation of the capital market line have already been provided; the equation for the CML is    p  E Rp D Rf C E .Rm /  Rf m

(6.1)

where: Rf D the risk-free rate; Rm D return on market portfolio Mp ; Rp D return on the portfolio consisting of the risk-free asset and portfolio Mp ; and p m D the standard deviations of the portfolio and the market, respectively. The implementation of Equation (6.1) can be explained graphically. An investor has three choices in terms of investments. He may invest in the riskless asset Rf in the market portfolio Mp or in any other efficient portfolio along the capital market line, such as portfolio P of Fig. 6.1. If the investor puts his money into the riskless asset he can receive a certain return of Rf , if his investments are put into the market portfolio he can expect an average return of Rm and risk of m , if he invests in portfolio P he can expect an average return  of Rp with  risk of p . The difference between Rm and Rf Rm  Rf is called the market risk premium. The investor of portfolio P only needs to take on risk of p  so his risk premium is Rp  Rf . This is less than the market risk premium because the investor is taking on a smaller amount of risk, p < m . By geometric theory, triangles Rf PF and Rf Mp D are similar. Consequently, they are directly proportional. Thus:  p    E Rp  Rf D E .Rm /  Rf m

6 Capital Asset Pricing Model and Beta Forecasting

95

At equilibrium all investors will want to hold a combination   of the risk-free asset and the tangency portfolio Mp . Since the market is cleared at equilibrium – that is, prices are such that the demand for all marketable assets equals their supply – the tangency portfolio Mp must represent the market portfolio. An individual security’s proportional makeup .Xi / in the market portfolio will be the ratio of its market value (its equilibrium price times the total number of shares outstanding) to the total market value of all securities in the market, or: Xi D

Market value of individual asset Market value of all assets

(6.2)

Thus, at equilibrium, prices are such that supply is equated to demand, and all securities in the market will be represented in the market portfolio according to their market value. Given the capital market line (CML), assuming that investors are interested only in mean and variance of return, the CML will dominate all other attainable portfolios and securities because of the dominance principle, as discussed in Chap. 4. This implies that they will attempt to put some portion of their wealth into the market portfolio of risky assets, depending on their individual risk preference. Since it has been established that the market portfolio is the only relevant portfolio of risky assets, the relevant risk measure of any individual security is its contribution to the risk of the market portfolio. The beta coefficient relates the covariance between security and market to the market’s total variance, and it is the relevant market-risk measure. This will be explored in the next two sections. Notationally, beta is ˇi D

im Cov .Ri ; Rm / D m2 Var .Rm /

Example 6.1 provides further illustration about risk/return tradeoffs.

at too large a premium. C is an asset with expected negative return – for example, a lottery ticket. The price of the lottery ticket is too high to be justified by the expected value of winning the jackpot. Even though the jackpot may be large, the probability of winning it is very small; hence, its expected value is small. Therefore, the cost of the ticket is greater than the expected value of winning, which yields a negative return.

6.2.3 The Security Market Line: The Capital Asset Pricing Model An asset’s systematic risk with the market, ˇ, is the only relevant risk measure of capital asset pricing for the individual asset and the portfolio. Consequently, a derivation of the relationship between systematic risk and return can be made where the expected linear relationship between these two variables is referred to as the security market line (SML), illustrated by Fig. 6.2. To derive the graphical picture of the SML it is necessary to list a few assumptions concerning the investors and the securities market. 1. Investors are risk averse. 2. The CAPM is a one-period model because it is assumed that investors maximize the utility of their end-of-period wealth. 3. All investors have the same efficient frontier – that is, they have homogeneous expectations concerning asset returns and risk. 4. Portfolios can be characterized by their means and variances.

Example 6.1. Describe the kinds of assets that could characterize the risk/return tradeoffs depicted by A, B, and C in the figure.

Solution A is a correctly priced asset, perhaps a stock or a portfolio. B is an overpriced asset, perhaps a bond selling

Fig. 6.2 The capital asset pricing model showing the security market line

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5. There exists a risk-free asset with a return Rf , the rate at which all investors borrow or lend. The borrowing rate is equal to the lending rate. 6. All assets are marketable and perfectly divisible, and their supplies are fixed. 7. There are no transaction costs. 8. Investors have all information available to them at no cost. 9. There are no taxes or regulations associated with trading. All individual assets or portfolios will fall along the SML. The position of an asset will depend on its systematic risk or beta. The required risk premium for the market is Rm  Rf , and ˇm is equal to one. Given the SML, the return on a risky asset is equal to:   E .Ri / D Rf C ˇi E .Rm /  Rf

(6.3)

E.Ri / D the expected rate of return for asset i ; Rf D the expected risk-free rate; ˇi D the measure of normalized systematic risk (beta) of asset i , and E.Rm / D the expected return on the market portfolio. (Note: This security market line is also generally called the capital asset pricing model (CAPM). The mathematical derivation of this model will be shown in the next section.) The relationship between the CML and the SML can be seen by rearranging the definition of the beta coefficient: ˇi D

im Cov .Ri ; Rm / i;m i m i;m i D 2 D D 2 Var .Rm / m m m

(6.4)

where: i D standard deviation of a security’s rate of return; m D standard deviation of the market rate of return; i;m D the correlation coefficient of Ri and Rm ; and im im D : i m If im D 1, then Equation (6.3) reduces to:  i  E .Rm /  Rf E .Ri / D Rf C m

(6:30 )

If i;m D 1, this implies that this portfolio is an efficient portfolio. If i is an individual security, it implies that the returns and risks associated with the asset are perfectly correlated with the market as a whole. There are several implications: 1. Equation (6.3) is a generalized case of Equation (6.30 ). 2. The SML instead of the CML should be used to price an individual security or an inefficient portfolio. 3. The CML prices the risk premium in terms of total risk, and the SML prices the risk premium in terms of systematic risk.

Example 6.2 provides further illustration. Example 6.2. Suppose the expected return on the market portfolio is 10% and that Rf D 6 percent. Further, if that you were confronted with an investment opportunity to buy a security with return expected to be 12% and with ˇ D 1:2. Should you undertake this investment? Solution Ri D a C bˇi for the market portfolio with ˇ D 1 we have 0:10 D a C .b 1/ D a C b and for the risk-free rate wit ˇ D 0, we have 0:06 D a C .b 0/ D a Solving these equations for a and b yields: a D 0:06 b D 0:04 For the security with ˇ D 1:2 the expected return given the SML is Ri D 0:06 C 0:04.1:2/ D 0:108 Since 12% > 10:8%, the security is undervalued; that is, the security should be purchased since its return is above the equilibrium return of 10.8% for that level of the risk.

6.3 Mathematical Approach to the Derivation of the Capital Asset Pricing Model Using the assumptions about efficient and perfect markets stated earlier in the chapter, we can show how Sharpe derived the capital asset pricing model (CAPM). Sharpe (1964) used a general risky asset that did not lie along the CML and dubbed it i . Risk and return for the possible combinations of security i with the market portfolio M are shown in Fig. 6.3. The average return and standard deviation for any I-M combination can be approached in the Markowitz fashion for a two-asset case:   .i/ E Rp Dwi E .Ri / C .1  wi /E.Rm /

1=2   .ii/  Rp D w2i i2 C .1  wi /2 m2 C 2 .1  wi / wi im (6.5)

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along the ı security market line. It should be noted that the term im m2 represents the beta coefficient for the regression of Ri vs. Rm so that Equation (6.9) can be rewritten as   E .Ri / D Rf C E .Rm /  Rf ˇi

(6.10)

which is the formula for the CAPM.

6.4 The Market Model and Risk Decomposition

Fig. 6.3 The opportunity set provide by combinations of risky asset i and the market portfolio M

in which wi represents excess demand for i or demand greater than its equilibrium weight in portfolio M. The changes in mean and standard deviation as the proportion wi changes are represented by:   @E Rp D E .Ri /  E .Rm / @wi i1=2 h   1=2 w2i i2 C .1  wi /2 m2 C 2wi .1  wi / im @ Rp D h i @wi  2wi i2  2m2 C 2wi m2 C 2im  4wi im (6.6)

when wi D 0, the i th security is held in proportion to its total market value, and there is no excess demand for security i . This is the key insight to Sharpe’s argument, for when wi D 0, it is possible to equate the slope of the curve iMi0 with the capital market line, thus obtaining an expression for the return on any risky security i . At equilibrium when wi D 0, the slope along the iMi0 curve will be equal to   @E Rp   D @ Rp

@E .Rp / @wi @ .Rp / @wi

D

E .Ri /  E .Rm / 2 im m m

(6.7)

To use the capital asset pricing model the market model must be employed to estimate the beta (systematic risk). In addition, the market model can be used to do risk decomposition. Both the market model and risk decomposition are discussed in this section.

6.4.1 The Market Model As noted previously, the total risk of a portfolio or security can be conceived as the sum of its systematic and unsystematic risks. The equation that expresses these concepts states that the return on any asset at time t can be expressed as a linear function of the market return at time t plus a random error component. Thus, the market model is expressed as: Ri;t D ˛i C ˇi Rm;t C ei;t

(6.11)

where Ri;t D the return of the i th security in time t; ˛i D the intercept of the regression; ˇi D the slope; Rm;t D the market return at time t; and ei;t D random error term.

The slope of the capital market line at point M is E .Rm /  Rf m

6.4.2 Risk Decomposition (6.8)

Setting Equation (6.7) equal to Equation (6.8) and rearranging the terms to solve for E .Ri / gives the equation for the security market line or CAPM:  im  E .Ri / D Rf C E .Rm /  Rf m2

(6.9)

This represents the return on any risky asset i . At equilibrium, every risky asset will be priced so that it lies

The regression model of Equation (6.11), which describes the risky asset’s characteristics relative to the market portfolio, is also called the characteristic line. Since beta is the slope coefficient for this regression (market model) it demonstrates how responsive returns for the individual earning assets are to the market portfolio. By using the market model, the total variance for security i i2 can be represented by and decomposed into: (6.12) i2 D ˇi2 m2 C ei2

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in which ˇi2 m2 is the systematic-risk component of total risk and ei2 is the unsystematic component. The CAPM as developed here is expressed in terms of expected values. Since expected values are not directly measured, we must transform the CAPM into an expression that uses observable variables. Assuming that on average expected returns E .Ri;t / for a security equal realized returns, returns can be expressed as: Ri;t D E .Ri;t / C ˇi ŒRm;t  E .Rm;t / C ei;t

(6.13)

in which ei;t is a random error term. Assuming that the expected value of the error term is 0, that it is uncorrelated with the term ŒRm;t  E .Rm;t / and that Cov.ei;t ; ei;t 1 / D 0, the expression for E .Ri;t / can be substituted from the CAPM Equation (6.10) into Equation (6.13):        Ri;t D Rf;t C E Rm;t  Rf;t ˇi C Rm;t  E Rm;t ˇi C ei;t

Simplifying this equation:   Ri;t  Rf;t D ˛i C ˇi Rm;t  Rf;t C ei;t

(6.14)

where ˛i is the intercept. All variables of Equation (6.14) can be estimated from observed data. Equation (6.14) is called the risk-premium version of the market model. It is similar to the market model indicated in Equation (6.11), except that instead of using the total returns Ri and Rm , it uses the riskpremium portion of the returns, or Ri  Rf and Rm  Rf . Example 6.3 provides further illustration. Example 6.3. To show how Equations (6.11) and (6.12) can be used, we will use monthly return data from Exxon Mobil and American Express. The time period covers January 2000 to November 2007. Their average return, beta coefficient, total variance, and residual variance are as indicated in the table below. We can see that American Express with ˇ D 1:26176 is more sensitive to the fluctuations of the market than Exxon Mobil with ˇ D 0:64647.

Exxon Mobil American Express

Return

Beta

Residual variance

Total variance

0.0126

0.64647

0.00214

0.00275

0.0055

1.26176

0.00138

0.00377

Both the magnitude of total variance and beta of American Express were larger than those of Exxon Mobil, although the average rates of return for Exxon Mobil were higher than American Express. Hence, Exxon Mobil was a more desirable security, if that was the only choice the investor had.

6.4.3 Why Beta Is Important for Security Analysis Implications and applications of beta coefficients in security analysis will be discussed in this section. Beta is used as a measurement of risk: it gauges the sensitivity of a stock or portfolio relative to the market, as indicated in Equation (6.11). A beta of 1.0 is characteristic of a broad market index such as the NYSE index or the S&P 500. A beta of 2.0 indicates that the stock will swing twice as far in either direction than a fall or rise in the market average. If the market gains 15%, a security of beta D 2 is expected to gain 30%. Conversely, should the market fall 15%, the stock is expected to fall 30%. A beta of 0.50 indicates that the stock is more stable than the market and will move only half as much as the market. For example, if the market loses 20%, a stock with beta D 0:50 will lose only 10% of the market, and if the market should gain 20%, the stock will gain only 10%. Highbeta stocks have been classified as aggressive while low-beta stocks are referred to as defensive. From the table in Example 6.3 it is clear that Exxon Mobil is an offensive stock while American Express is a defensive stock. The CAPM has illustrated the concept that risks are associated with portfolios where the relevant (systematic) risk of an individual security is dependent upon the security’s effect on portfolio risk. Therefore, the CAPM equation E.Ri / D Rf C ˇi ŒE.Rm /  Rf represents a description of how rates of return are established in the marketplace assuming investors behave according to the assumptions of the model. Thus a stock’s beta measures its contributions to the risk of the portfolio and is therefore a measure of the stock’s riskiness relative to the market. In the previous section it was found that the rates of return of stock i are presumed to bear a linear relationship with the market rate of return as defined in Equation (6.11). Equation (6.11) is a fixed-coefficient market model. Following Fabozzi and Francis (1978), Sunder (1980), and Lee and Chen (1980), a random-coefficient market model can be defined as Equation (6.15a) or Equation (6.15b): Rit D ˛i C ˇit Rmt C eit   Rit  Rft D ˛i0 C ˇit0 Rmt  Rft C eit

(6.15a) (6.15b)

in which ˇit D ˇi C it :it represents the random fluctuation associated with the beta coefficient. Using the randomcoefficient market model, the total risk can be decomposed into three components, as defined in Equation (6.16): i2 D ˇj2 m2 C "i2 C 2 m2

(6.16)

in which 2 m2 represents an interaction risk between the market and the random fluctuation of the ˇ. Equation (6.16)

6 Capital Asset Pricing Model and Beta Forecasting

is a generalized case of Equation (6.12). It contains systematic risk, unsystematic risk, and a risk term that reflects any interaction between the systematic and unsystematic risk. Both fixed-coefficient and random-coefficient market models can be used to do security analysis and portfolio selection, discussed later in this text. The relationship between total risk, market risk, and firmspecific risk can be shown as: Total risk D Market risk C Firm-specific risk Since firm-specific risk can be eliminated by diversification: Relevant risk D Market risk Beta is important for the investment manager because it can be used to (1) select individual stocks for investment, (2) construct portfolios of financial assets with desired levels of risk and return, and (3) evaluate the performance of portfolio managers. Details of the various uses of beta and the CAPM are provided later in this text. Example 6.4 provides further illustration. Example 6.4. Given the SML Ri D 0:06 C 0:08ˇi . What should the expected return of a security be if it has a ˇ twice as great as a similar security returning 18%? Solution Ri D 0:06 C 0:08ˇ D 0:18 Solving for ˇ yields ˇ D 1:5 Therefore, the security’s ˇ is 2 1:5 D 3:0 and the required return is: Ri D 0:06 C 0:08 .3:0/ D 0:30 A 30% return is required.

6.4.4 Determination of Systematic Risk As mentioned above, systematic risk is nondiversifiable because it runs across industries and companies and affects all securities. Previous chapters discussed many factors that determine risk. At this point we are interested only in those management decisions that can be related to the degree of systematic risk a firm exhibits; namely, the operating decisions and the financing decisions of management.

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There are two dimensions of risk that affect a firm’s systematic risk. The first is financial risk, the additional risk placed on the firm and its stock holders due to the firm’s decision to be leveraged – that is, to take on additional debt. The second, business risk, is the riskiness involved with a firm’s operations, if it takes on no debt. A particular firm’s capital structure affects the riskiness inherent in the company’s common stock and thus affects its required rate of return and the price of the stock. A company’s capital-structure policy requires choosing between risk and return. Taking on increasing levels of debt increases the riskiness of the firm’s earning stream, but it usually also results in a higher expected rate of return. High levels of risk tend to lower a stock’s price, but a high level of expected rates of return tends to raise it. Therefore, striking a balance with the optimal capital structure maximizes the price of the stock. Business risk is the risk inherent in a firm’s operations. It can also be defined as the uncertainty inherent in projection of future operating income or earnings before interest and taxes (EBIT). Fluctuations in EBIT can result from a number of factors. On the national level these can be economic factors such as inflationary or recessionary times. At the industry level some factors may be the level of competition between similar industries, natural or man-made catastrophes, labor strikes, price controls, and so on. There are a host of possibilities that affect EBIT by raising or lowering its level. Uncertainty regarding future income flows is a function of the company’s business risk. It may fluctuate among industries, among firms, and cross time. The extent of business risk is dependent on the firm and the Industry. Cyclical industries such as steel production or automobile manufacture have especially high business risks because they are dependent on the strength of the economy. The retail food industry is considered to be quite stable because food is a necessary good that will be purchased regardless of the state of the economy. Business risk is dependent on several factors, the more important of which are listed here: Demand variability : Stability in the levels of demand for the firm’s product results in a reduction of business risk. 1. Sales price variability: Highly volatile prices result in high-risk, volatile markets; therefore, stability of prices results in reduction of business risk. 2. Suppliers’ price variability: Firms whose input prices are highly variable are exposed to higher levels of risk. 3. Output price flexibility relative to input prices: As a result of inflation, a firm that is able to raise its output prices with increasing input costs minimizes its business risk. When a firm uses debt or financial leverage, business risk and financial risk are concentrated on the stockholders. For

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example, if a firm is capitalized only with common equity, then the investors all share the business risk in proportion to their ownership of stock. If, however, a firm is 50% levered (50% of the corporation is financed by debt, the other half by common equity), the investors who put up the equity will then have to bear all business risk and some financial risk. The effect of leverage on return on assets (ROA) and return on equity (ROE) and its effect on the stockholders can be generalized as follows: 1. The use of leverage or debt generally increases ROE. 2. The standard deviation of ROA.ROA / is a measure of business risk while the standard deviation of ROE.ROE / is a measure of the risk borne by stockholders. ROA D ROE if the firm is not levered; otherwise with the use of debt ROE > ROA an indication that business risk is being borne by stockholders. 3. The difference between ROE and ROA is the actual risk stockholders face and a measure of the increased risk resulting from financial leverage. Thus 2 2  ROA : Risk of financial leverage D ROE

It becomes necessary, therefore, to define a company’s sustainable growth rate: P .1  D/ .1 C L/

S D g D S T  P .1  D/ .1 C L/

(6.17)

where: P D the profit margin on all sales; D D the target dividend payout ratio; L D the target debt to equity ratio; T D the ratio of total assets to sales; S D annual sales; and

S D the increase in sales during the year. How is Equation (6.17) derived? Assuming a company is not raising new equity, the cash to finance growth must come from retained profits and new borrowings: Retained profits D Profits  Dividends D Profit margin Total sales  Dividends

6.5 Growth Rates, Accounting Betas, and Variance in EBIT Besides leverage, other financial variables associated with the firm can affect the beta coefficient. These are the growth rate, accounting beta, and variance in EBIT.

D P .S C S/ .1  D/ Further, because the company wants to maintain a target debt-to-equity ratio equal to L, each dollar added to the owners’ equity enables it to increase its indebtedness by $L. Since the owners’ equity will rise by an amount equal to retained profits: New borrowings D Retained profit

6.5.1 Growth Rates

Target debt-to-equity ratio D P .S C S /.1  D/L

The growth rate can be measured in terms of the growth in total assets or the growth in sales. It is determined by the percentage change between two periods. salest  salest 1

100% or salest 1 total assetst  total assetst 1

100% total assetst 1 Another method to measure growth rates is presented by Higgins (1984). Growth and its management present special problems in financial planning. From a financial perspective growth is not always a blessing. Rapid growth can put considerable strain on a company’s resources, and unless management is aware of this effect and takes active steps to control it, rapid growth can lead to bankruptcy.

The use of cash represented by the increase in assets must equal the two sources of cash (retained profits and new borrowings): Uses of cash D Sources of cash Increases in assets D Retained profits C New borrowings

ST D P .S C S /.1  D/ C P .S C S /.1  D/L

S T D P .1  D/.1 C L/S C P .1  D/.1 C L/ S

6 Capital Asset Pricing Model and Beta Forecasting

S ŒT  P .1  D/.1 C L/ D P .1  D/.1 C L/S P .1  D/.1 C L/

S D S T  P .1  D/.1 C L/ In Equation (6.17) the S =S or g  is the firm’s sustainable growth rate assuming no infusion of new equity. Therefore, a company’s growth rate in sales must equal the indicated combination of four ratios, P, D, L, and T . In addition, if the company’s growth rate differs from g  , one or more of the ratios must change. For example, if a company grows at a rate in excess of g  , then it must either use its assets more efficiently or it must alter its financial policies. Efficiency is represented by the profit margin and asset-to-sales ratio. It therefore would need to increase its profit margin .P / or decrease its asset-to-sales ratio .T / to increase efficiency. Financial policies are represented by payout or leverage ratios. In this case, a decrease in its payout ratio .D/ or an increase in its leverage .L/ would be necessary to alter its financial policies to accommodate a different growth rate. It should be noted that increasing efficiency is not always possible and altering financial policies not always wise (For other available methods to alter growth the reader is referred to Chap. 8 of Lee (1985).)

6.5.2 Accounting Beta The accounting beta can be calculated from earnings-pershare data. Using EPS as an example, beta can be computed as follows: EPSi;t D ˛i C ˇi EPSm;t C ei;t where: EPSi;t D earnings per share of firm i at time t; EPSm;t D earnings per share of market average at time t; and ei;t D error term. The estimate of ˇi is the EPS type of accounting beta.

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The total variance of EBIT can be used to measure the overall fluctuation of accounting earnings for a firm.

6.5.4 Capital-Labor Ratio The capital-labor ratio has an impact on the magnitude of the beta coefficient. In order to examine this impact it is necessary to examine the capital-labor ratio. A production function is a function that can be seen as a function of labor and capital: Q D f .K; L/

(6.18)

where K D capital and L D labor: K=L (the capital-labor ratio) is generally used to measure a firm’s degree of capital intensity. Corporations often choose between increasing their capital intensity through installation of computers, use of robotics in place of labor, or increase in labor inputs. Small industries that specialize in hand-crafted or -tooled goods will have to increase their capital ratio in to increase production. However, it has been the trend in recent years for many growth-oriented firms, whether manufacturers or other members of the business sector, to increase their efficiencies through increased investments in capital. Auto manufacturers are finding that robots are able to assemble cars of high and consistent quality at only a fraction of the cost of workers. Capital intensity results in increased total risks and generally results in an increase in beta. Large investments are often needed to fully automate a plant or to computerize a bank completely. Taking on debt or issuing securities is normally how these capitalization increases are financed. If the capital-labor ratio is greater than one – that is, if K is greater than L – a firm is capital intensive. If the capital-labor ratio is less than one – that is, if K is less than L – then there is a reduction in capital intensity and a shift toward humanresource investment.

6.5.5 Fixed Costs and Variable Costs 6.5.3 Variance in EBIT The variance in EBIT .X / can be defined as: n  P

Xt  X

2

t D1

n1 in which Xt D earnings before interest and taxes in period t, and X D average EBIT.

Business risk is dependent upon the extent a firm builds fixed costs into its operations. If a large percentage of a firm’s costs are fixed, costs cannot decline proportionally when demand falls off. If fixed costs are high, then a slight drop in sales can lead to large declines in EBIT. Therefore, the higher a firm’s fixed costs are, the greater its business risk and, generally, the higher its beta. A firm with a large amount of fixed costs is said to have a large degree of operating leverage. An example of these costs may be the highly skilled workers of an engineering

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firm. The firm cannot hire and fire experienced and highly skilled workers easily; therefore, the workers must be retained and paid during a period of slack demand. Similarly, a firm that is highly leveraged will be characterized by the scenario that small changes in sales will result in large changes in operating income. Variable costs have the opposite effect, because they are adjustable to the firm’s needs. Should a drop in sales occur, variable costs can be lowered to meet the lowered output. The extent to which firms can control their operating leverage is dependent upon their technological needs. Companies that require large investments in fixed asset – such as steel mills, auto manufacturers, and airlines – will have large fixed costs and operating leverages. Therefore, how much a company is willing to undertake operating leverages must come into play during capital-budgeting decisions. If the company is risk averse, it may opt for alternatives with smaller investments and fixed costs. A company having a larger percentage of fixed costs generally implies that it uses more capital-intensive types of technology in production. For example, an auto manufacturer such as General Motors has a higher percentage of fixed costs than a food manufacturer such as General Foods. Therefore, GM’s capital-labor ratio can be expected to be higher than that of General Foods. An implication of this phenomenon is that General Foods has a lower beta than General Motors.

C.-F. Lee et al.

the estimation of beta. They conclude that using both price and accounting information shows promise for better beta estimates. Rosenberg and Marathe (1975) use 54 factors in six categories to estimate betas. Their factors include historical price information, price/earning ratios, financial ratios, and statistical measures associated with the market model. They support the notion that betas are determined by fundamental factors related to firms in addition to market pricing data. Given that the most useful beta is one that can be forecasted correctly, and that the beta is related to the financial and business risk of the firm as well as the degree to which a firm’s businesses co-vary with the total economy, it becomes necessary to determine the best way to forecast beta. The next section addresses this issue.

6.5.7 Market-Based versus Accounting-Based Beta Forecasting Market-based beta forecasts are based on market information alone. Historical betas of firms are used as a proxy for their future betas. This implies that the unadjusted sample beta, ˇOt , is equal to the population value of future beta: ˇt C1 D ˇOt

6.5.6 Beta Forecasting Analysts and investment managers who use the CAPM and beta are interested in whether the beta coefficient and the standard-deviation statistics for different securities and portfolios are stable through time or whether they change in a predictable fashion. If the beta coefficient and standard deviation are stable, then using a beta derived from current and historical price data is fine, because the beta today is the same as the beta in the future. However, if the beta coefficient and standard deviation are unstable or vary through time, the analyst or manager must forecast a beta’s future value before employing it. The available evidence on the stability of beta indicates that the beta on an individual security is generally not stable, while portfolios have stable betas. Hence we are faced with the problem of forecasting future betas in order to use the CAPM for individual securities. Beta forecasting refers to using the historical beta estimates or other historical financial information to forecast future betas. Beaver et al. (1970) argue that if it is possible to find the underlying determinants of beta, such knowledge can be used to help forecast future betas. Beaver et al. use accounting information from the financial statements of firms to help in

(6.19a)

Alternatively, there may be a systematic relationship between the estimated betas for the first period and those of the second period, as shown by Blume (1971): ˇOi;t C1 D a0 C a1 ˇOi;t

(6.19b)

in which ˇOi;t C1 and ˇOi;t estimated beta for the i th firm in period t C 1 and t respectively. Example 6.5 provides further illustration. Example 6.5. If aO 0 D 0:35; aO 1 D 0:80, and, ˇOi;t D 0:12, then the future beta can be either ˇt C1 D 1:2 or ˇOt C1 D 0:35 C .0:80/.1:2/ D 1:31 It is worthwhile to note that Value Line uses Equation (6.19b) to estimate the future beta. Accounting-based beta forecasts rely on the relationships of accounting information such as the growth rate of the firm, EBIT, leverage, and the accounting beta as a basis for forecasting beta. To use accounting information in beta forecasts, the historical beta estimates are first cross-sectionally related to accounting information such as growth rate, variance of EBIT, leverage, accounting beta, and so on:

6 Capital Asset Pricing Model and Beta Forecasting

ˇi D a0 C a1 X1i C a2 X2i C aj Xj i C : : : C am Xmi

103

(6.20)

where Xji is the j th accounting variables for i th firm, and aj is the regression coefficient. Some researchers have found that the historically based beta is the best forecast, while others have found that the accounting-based beta is best. Lee et al. (1986) use composite concepts to show that both accounting and market information are useful for beta forecasting. They find that neither beta forecasts based on market information nor those based on accounting information are conditionally more efficient with respect to each other. It can be inferred, then, that each set of forecasts contains useful information for the prediction of systematic risk. The statistical procedure used by Lee et al. (1986) is the ordinary least-squares method. Ordinary least squares is a statistical procedure for finding the best fitting straight line for a set of points; it seems in many respects a formalization of the procedure employed when fitting a line by eye. For instance, when visually fitting a line to a set of data, the ruler is moved until it appears that the deviations of the points from the prospective line have been minimized. If the predicted value of yi (dependent variable) obtained from the fitted line is denoted as yOi the prediction equation becomes: yOi D ˇO0 C ˇO1 xi where ˇO0 and ˇO1 represent estimates of the true ˇ0 and ˇ1 I xi is an independent variable. Graphically, the vertical lines drawn from the prediction line to each point represent the deviations of the points from the predicted value of y. Thus the deviation of the i th point is yi  yOi , where: SSE D

n X

.yi  yOi /

In order to find the best fit it is necessary to minimize the deviations of the points. A criterion of best fit that is often employed is known as the principle of least squares. It may be stated as follows: Choose the best fitting line as the one that minimizes the sum of squares of the errors (SSE) of the observed values of Y from those predicted. Further, the mean sum of squares of the error (MSSE) of the observed values of Y could be minimized from the predicted mean. Expressed mathematically:

MSSE D

n P

.yi  yOi /2

i D1

n

.yi  yOi /2

i D1

n

Mincer and Zarnowitz (1969) suggest a decomposition of the mean squared error term into three components representing

2    D .yi  yOi /2 C Sy  Sy C 1  2 Sy2

where y, and Sy represent the mean, standard deviation, and correlation coefficient of y, respectively. Using the ordinary least-squares method, Lee et al. (1986) show that it is possible to achieve better forecasts by combining two types of betas. Using an ordinary least-squares estimate of ˇOLS and a Bayesian adjustment procedure of Vasicek (1973) as defined in Equation (6.21):

ˇV D

ˇN C V ˇO V

ˇOOLS

ˇOOLS (6.21)

1 1

C V ˇO V ˇOOLS

where: ˇOLS D the least-squares estimate of a first-period individual beta; V ˇOOLS D variance estimate of ˇOLS ; ˇN D the

cross-sectional mean value of estimated ˇOLS ; O V ˇ D the cross-sectional variance of estimated ˇOLS ; and ˇV D the Bayesian adjusted beta. Equation (6.21) indicates that the Vasicek type of Bayesian adjustment beta is a weighted average of ˇN and ˇOLS . The weights are

2

i D1

n P

bias, inefficiency, and random error. Mathematically, this is represented:

w1 D

1 V ˇO 1 1 C

V ˇO V ˇOOLS



and w2 D

1

V ˇOOLS

1 1 C

V ˇO V ˇOOLS

The authors first use first-period regressions projected forward to obtain forecasts of the second-period betas. The summary of the stepwise regression for the first-period data is given in Table 6.1. Table 6.2 summarizes the overall mean squared errors together with the mean squared error decomposition. Lee et al. (1986) note that for forecasts developed without the Bayesian adjustment, the accounting-based forecast is better, in terms of overall mean squared error. This advantage results entirely from inefficiency in the Mincer-Zarnowitz sense of the market-based forecast. By contrast, the Bayesian-adjusted market-based forecasts suffer far less from this problem and, as a result, the mean squared prediction error for Bayesian-adjusted market-based

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Table 6.1 Summary of stepwise regression results for first-period data

Dependent variable adjusted R2 Independent variables Intercept Financial leverage Dividend payout Sales Operating income Assets

“O OLS 0.245

“O V 0.236

Coefficients (t -values) 0.911 1.037 (11.92) (12.71) 0.704 0.599 (5.31) (5.81) 0:175 0:108 .3:50/ .2:85/ 0.030 0.018 (3.02) (2.28) 0.011 0.026 (2.18) (2.28) – 0:026 – .2:41/

Source: Lee et al. (1986) Table 6.2 Mean squared error decompositions for forecasts of second-period beta Market-based forecasts Accounting-based forecasts Without Bayesian adjustment Bias Inefficiency Random error Total mean squared error

0.0003 0.0309 0.0581 0.0893

With Bayesian adjustment 0.0000 0.0090 0.0577 0.0667

Without Bayesian adjustmentl 0.0003 0.0015 0.0659 0.0677

With Bayesian adjustment 0.0000 0.0001 0.0638 0.0640

Source: Lee et al. (1986)

forecasts are only marginally larger than the accountingbased forecasts 0:0667 compared to 0.0640. Lee et al. (1986) tested a composite predictor for beta consisting of both a market beta and an accounting beta. Table 6.3 shows the results of using this composite method to forecast beta for both the Bayesian-adjusted and nonadjusted model. In Table 6.3, ˇOLS (2) represents the OLS estimated A and ˇOVA represent accountingbeta in the second period. ˇOLS based beta without and with Bayesian adjustment, respectively. By comparing both MSE (market) of 0.893 and 0.0667 and MSE(accounting) of 0.0677 and 0.0640 with MSE(composite) of 0.0545 and 0.0530, respectively, it can be concluded that on the basis of mean squared error, the composite predictor outperforms either the market- or accounting-information predictor. Of the models used for the composite predictor, it appears that the Bayesian adjustment case is superior. From research it appears, then, that accounting-based and market-based forecasts can be combined to produce a superior composite forecast of beta.

6.6 Some Applications and Implications of the Capital Asset Pricing Model Since its development, the uses of the capital asset pricing model (CAPM) have extended into all areas of corporate finance and investments. The CAPM can be applied to two

aspects of the capital-budgeting problem: (1) determining the cost of capital, and (2) assessing the riskiness of a project under consideration. The CAPM can also be useful in a realestate problem, deciding to lease or buy. The use of the CAPM can be extended into valuation of the entire firm. Because of its impact upon firm valuation, the CAPM has been of great use in the merger-analysis area of financial analysis. The CAPM has also been used to test various financial theories. By including a dividend term and considering its effects, the CAPM can be used to test the effects of the firm’s dividend policy. An area that has received a great deal of attention is the use of the CAPM in testing the efficient-market hypothesis. An application of the CAPM to the capital-budgeting process concerns the valuation of risky projects. If accurate estimates can be made about the systematic risk of a project, then the CAPM can be used to determine the return necessary to compensate the firm for the project’s risk. If the sum of the estimated cash flows discounted by the CAPM-calculated required rate of return is positive, then the firm should undertake the project. Rubinstein (1973) demonstrates how the CAPM can be used to value securities and to calculate their risk-adjusted equilibrium price. First, the CAPM must be converted to using price variables instead of expected return. It may be rewritten: P1  P0 E.Ri / D P0

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Table 6.3 Results for the estimation of market- and accounting-based composite with and without Bayesian adjustment O Without Bayesian adjustment “O OLS .2/ D a C b1 “O A OLS C b2 “OLS C w “O A OLS 0.399 0.102 MSE D 0:0545 O With Bayesian adjustment “O OLS .2/ D a C b1 “O A V C b2 “V C w Intercept 0:251 0:104

Coefficients Standard errors

Coefficients Standard errors

Intercept

“O A V

0:096 0:134

0.606 0.133 MSE D 0:0530

“O OLS 0:370 0:052

“O V 0:475 0:067

Source: Lee et al. (1986)

where: Ri D the expected returns for the i th firms; P1 D the price of stock in time l; and P0 D the price of stock in the previous period. Thus, the CAPM is redefined:  im  E .P1 /  P0 D Rf:t C E .Rm /  Rf P0 m2

(6.22)

or, rearranging Equation (6.22): P0 D

E .P1 /   1 C Rf C E .Rm /  Rf im2 m

Thus the rate of return used to discount the expected endof-period price contains a risk premium dependent upon the security’s systematic risk. CAPM has also been applied in the analysis of mergers. It has been shown that the risks of a portfolio can be substantially reduced with the inclusion of securities that are not perfectly correlated in terms of returns. This principle also applies with respect to the mergers between firms. The merging of two firms with different product lines, called a conglomerate merger, creates diversification, considered of great benefit. Suppose that one firm sells a product that is recession resistant, then a decrease in earnings of one division of the conglomerate will be offset by the steady earnings of another division. The overall result will be a relatively stable income stream despite shifting trends in the economy.

6.7 Conclusion This chapter has discussed the basic concepts of risk and diversification and how they pertain to the CAPM. The procedures for deriving the CAPM itself were presented, and the CAPM was shown to be an extension of the capital market

line theory. The possible uses of the CAPM in financial management were also indicated. The concept of beta and its importance to the financial manager was introduced. Beta represents the systematic risk of a firm and is a comparison measure between a particular firm’s security or portfolio risk and the market average. Systematic risk was further discussed through an investigation of the beta coefficient and the impact of other important financial variables on the magnitude of the beta coefficient. The statistical method of least squares and its application were introduced, and beta forecasts based on the leastsquares method were compared with those based on market information, accounting information, and a compositepredictor beta forecast composed of both accounting and market information. The composite predictor appears to yield a better forecast than either the market-information or accounting-information forecasts separately.

References Banz, R. W. 1981. “The relationship between return and market value of common stocks.” Journal of Financial Economics 9, 3–18. Basu, S. 1977. “Investment performance of common stocks in relation to their price-earnings ratios: a test of the efficient markets hypothesis.” Journal of Finance 32, 663–682. Beaver, W., P. Kettler, and M. Scholes. 1970. “The association between market determined and accounting determined risk measures.” Accounting Review 45, 654–682. Black, F., M. C. Jensen, and M. Scholes. 1972. “The capital asset pricing model: some empirical tests,” in Studies in the theory of capital markets, M. C. Jensen (Ed.). Praeger, New York, pp. 20–46. Blume, M. E. 1971. “On the assessment of risk.” Journal of Finance 26, 1–10. Blume, M. E. and I. Friend. 1973. “A new look at the capital asset pricing model.” Journal of Finance 28, 19–34. Blume, M. E. and F. Husick. 1973. “Price, beta, and exchange listing.” Journal of Finance 28, 19–34. Bodie, Z., A. Kane, and A. Marcus. 2006. Investments, 7th Edition, McGraw-Hill, New York. Brennan, M. J. 1971. “Capital market equilibrium with divergent borrowing and lending rate.” Journal of Financial and Quantitative Analysis 7, 1197–1205.

106 Douglas, G. W. 1969. “Risk in the equity markets: an empirical appraisal of market efficiency.” Yale Economic Essays 9, 3–45. Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2006. Modern portfolio theory and investment analysis, 7th Edition, Wiley, New York. Fabozzi, F. J. and J. C. Francis. 1978. “Beta as a random coefficient.” Journal of Financial and Quantitative Analysis 13, 101–116. Fama, E. F. 1968. “Risk, return and equilibrium: some clarifying comments.” Journal of Finance 23, 29–40. Fama, E. F, and J. MacBeth. 1973. “Risk, return and equilibrium: empirical tests.” Journal of Political Economy 31, 607–636. Francis, J. C. 1986. Investments: analysis and management, 4th Edition, McGraw-Hill, New York. Higgins, R. C. 1974. “Growth, dividend policy and cost of capital in the electric utility industry.” Journal of Finance 29, 1189–1201. Higgins, R. C. 1984. Analysis for financial management, Richard D. Irwin, Homewood, IL. Jensen, M. C. 1969. “Risk, the pricing of capital assets, and the evaluation of investment portfolio.” Journal of Business 42, 607–636. Jensen, M. C. 1972. “Capital markets: theory and evidence.” The Bell Journal of Economic and Management Science 3, 357–398. Keim, D. B. 1983. “Size-related anomalies and stock return seasonality: further empirical evidence.” Journal of Financial Economics 11, 13–32. Lee, C. F. 1985. Financial analysis and planning: theory and application, Addison-Wesley, Reading, MA. Lee, C. F. and S. N. Chen. 1980. “A random coefficient model for reexamining risk decomposition method and risk-return relationship test.” Quarterly Review of Economics and Business 20, 58–69. Lee, C. F. and A. C. Lee. 2006. Encyclopedia of finance, Springer, New York. Lee, C. F, P. Newbold, J. E. Finnerty, and C. C. Chu. 1986. “On accounting-based, market-based and composite-based beta predictions: methods and implications.” The Financial Review 21, 51–68. Lee, C. F., J. C. Lee, and A. C. Lee. 2000. Statistics for business and financial economics, World Scientific Publishing Co, Singapore. Lintner, J. 1965. “The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets.” Review of Economics and Statistics 47, 13–37. Maginn, J. L., D. L. Tuttle, J. E. Pinto, and D. W. McLeavey. 2007. Managing investment portfolios: a dynamic process, CFA Institute Investment Series, 3rd Edition, Wiley, Hoboken, NJ. Merton, R. C. 1980. “On estimating the expected return on the market, an exploratory investigation.” Journal of Financial Economics 8, 323–361. Miller, M. and M. Scholes. 1972. “Rates of return in relation to risk: a reexamination of some recent findings,” in Studies in theory of capital markets, M. C. Jensen (Ed.). Praeger, New York, pp. 47–78. Mincer, J. and V. Zarnowitz. 1969. “The evaluation of economic forecasts,” in Economic forecasts and expectations, J. Mincer (Ed.). National Bureau of Economic Research, New York. Mossin, J. 1966. “Equilibrium in a capital asset market.” Econometria 34, 768–873. Reinganum, M. R. 1981. “Misspecification of capital asset pricing: empirical anomalies based on earnings yields and market values.” Journal of Financial Economics 8, 19–46. Roll, R. 1977. “A critique of the asset pricing theory’s tests-part i: on past and potential testability of the theory.” Journal of Financial Economics 4, 129–176. Roll, R. 1978. “Ambiguity when performance is measured by the securities market line.” Journal of Finance 33, 1051–1069. Rosenberg, B. and V. Marathe. 1975. Tests of the capital asset pricing hypothesis, Working Paper No. 32 of the Research Program in France, Graduate School of Business and Public Administration, University of California, Berkeley.

C.-F. Lee et al. Rosenberg, B. and W. McKibben. 1973. “The prediction of systematic and specific risk in common stocks.” Journal of Finance and Quantitative Analysis 8, 317–333. Rubinstein, M. E. 1973. “A mean-variance synthesis of corporate financial theory.” Journal of Finance 28, 167–168. Sharpe, W. 1964. “Capital asset prices: a theory of market equilibrium under conditions of risk.” Journal of Finance 19, 425–442. Sharpe, W. 1966. “Mutual fund performance.” Journal of Business 39, 119–138. Sunder, S. 1980. “Stationarity of market risk: random coefficients tests for individual stocks.” Journal of Finance 35, 883–896. Vasicek, O. A. 1973. “A note on using cross-sectional information in Bayesian estimation of security betas.” Journal of Finance 28, 1233–1239. Wackerly, D., W. Mendenhall, and R. L. Scheaffer. 2007. Mathematical statistics with applications, 7th Edition, Duxbury Press, California.

Appendix 6A Empirical Evidence for the Risk-Return Relationship The validity of the CAPM can be borne out partly through observations of actual portfolios held in the marketplace. As discussed in previous chapters, there are two primary relationships between risk and return. First, the rates of return of efficient portfolios are linear functions of their riskiness as measured by their standard deviation. This is illustrated by the capital market line (CML). Second, the rate of return of an individual asset is determined by its contribution of risk to the portfolio, and this is measured by beta, where beta has a linear relationship with the security’s expected rate of return. This is illustrated by the security market line (SML). The performance of mutual funds can be employed to test the explanatory powers of the linear relationship between risk and return of the CML. Mutual funds are professionally managed and therefore the most visible type of portfolio, easily used for comparison testing. One study was performed by Sharpe (1966) to test the performance of a fund and the relationship between its rate of return and risk over time. Sharpe computed average annual returns and the standard deviations of these returns for 34 mutual funds from 1954 to 1963. His model implies that portfolios with higher risks will receive higher returns. This Sharpe found to be true for all 34 funds. He calculated the correlation between average returns and their standard deviations to be 0.836, indicating that more than 80% of the difference in returns was due to differences in risk. Sharpe also found that there was a linear relationship between returns and risks, except in the region of very high risks. Sharpe’s study provides basic support to the contention that the CML explains the relationship between risk and return, both in portfolio theory and in the marketplace. Another study was performed by Jensen (1969). He studied the correlation of beta coefficients (market sensitivity) and the expected return of mutual funds. On the basis

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of analysis of 115 mutual funds over a 9-year period he was able to conclude that high returns were associated with high volatility or high systematic risks. He also found evidence that beta coefficients are a valid and accurate measure of risk. Both the Sharpe and Jensen studies on the risk and return of mutual funds show that an empirical risk-return relationship does exist among mutual funds (However, Sharpe used the capital market line to perform his empirical tests while Jensen used the security market line, derived by Sharpe 1964.) The second implication of the risk-return relationship is that the risk premium on individual assets depends on the contribution each makes to the riskiness of the entire portfolio. The CAPM is a simple linear model expressed in terms of expected returns and expected risk. In its ex-ante form:   E .Ri / D Rf C E .Rm /  Rf ˇi

(6A.1)

Although many of the aforementioned extensions of the model support this simple linear form, others suggest that it may not be linear, that factors other than beta are needed to explain E .Ri /, or that the Rf is not the appropriate riskless rate. The first step necessary to empirically test the theoretical CAPM is to transform it from expectations (ex-ante) form into a form that uses observed data. On average, the expected rate of return on an asset is equal to the realized rate of return. This can be written: Rit D E .Rit / C ˇi ımt C eit where: ımt D Rmt  E.Rmt /I E.ımt / D 0I eit D a random error termI Cov .eit ; ımt / D 0I Cov .eit ; ei t 1 / D 0I and ˇit D Cov.Rit Rmt /Var.Rmt /: When CAPM is empirically tested it is usually written in the following form: 0 D 0 C 1 ˇp C ept Rpt

where:

1 D Rmt  Rf t 0 Rpt D Rpt  Rf t

(6A.2)

These relationships can be stated as follows. 1. The intercept term 0 should not be significantly different from zero. 2. Beta should be the only factor that explains the rate of return on a risky asset. If other terms, such as residual variance, dividend yields, price/earnings ratios, firm size, or beta squared are included in an attempt to explain return, they should have no explanatory power. 3. The relationship should be linear in beta. 4. The coefficient of beta, “II” should be equal to Rmt  Rft . 5. When the equation is estimated over very long periods of time, the rate of return on the market portfolio should be greater than the risk-free rate. The work of Jensen (1972) provides a comprehensive and unifying review of the theoretical developments and the empirical work done in the field until that year. In his paper he points out that the main result of the original papers in this area is the demonstration that one can derive the individual’s demand function for assets, aggregate these demands to obtain equilibrium prices (or expected returns) solely as a function of potentially measurable market parameters. Thus, the model becomes testable. Let us now summarize the empirical work of Douglas (1969), Black et al. (1972), and Fama and MacBeth (1973). The first published test of the CAPM was by Douglas (1969), who regressed the returns of a large cross-sectional sample of common stocks on their own variances and on their beta coefficients ˇi obtained by market models. His results are in variance with the Sharpe-Lintner-Mossin model, for he found that the return was positively related to the variance of the security but not the covariance with the index of returns. Douglas also summarizes some of the work of Lintner (1965), who estimates beta from a typical market model. Douglas then adds a term for the standard deviation of error (proxy for unsystematic risk). He finds that the coefficient for the unsystematic risk is both positive and significant, the intercept term is higher than the appropriate risk-free rate, and the coefficient for the market risk premium is too low. Black, Jensen, and Scholes observe that cross-sectional tests may not provide direct validation of the CAPM, and they proceed to construct a time-series test, which they consider more powerful. Their results lead them to assert that the usual form of the CAPM does not provide an accurate description of the structure of security returns. Their results indicate that ˇs are non-zero and are directly related to the risk level. Low-beta securities earn significantly more on average than predicted by the model, and high-risk securities earn significantly less on average than predicted by the model. They go on to argue for a two-factor model: Rit D .1  ˇi /Rzt C ˇi .Rmt / C eit

(6A.3)

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If E.Rz / D 0 then the Sharpe-Lintner-Mossin CAPM would be consistent with this model. However, the cross-sectional term for the intercept is a constant and not equal to zero. They then proceed to look for a rationale of this finding in Black’s zero-beta model. Fama and MacBeth (1973) test (1) a linear relationship between return on the portfolio and the portfolio’s beta, and (2) whether unsystematic risk has an effect between portfolio return and a risk measure in addition to beta. Their basic estimation equation is: 2 RQ it D Q0t C Q1t ˇ i C Q2t ˇ i C Q3t  i2 ./ C eQit

(6A.4)

in which Q0t is the intercept term, ˇ i is the average of the ˇi for all individual securities in portfolio i , and  i ./ is the average of the residual standard deviations from all securities in portfolio j . Although they find that there are variables in addition to the portfolio beta that systematically affect period-byperiod returns (which are apparently related to the average squared beta of the portfolio and the risk factor other than beta), they dismiss the latter as “almost surely proxies,” since “there is no rationale for their presence in our stochastic riskreturn model.” Their results seem to suggest that the SharpeLintner-Mossin model does not hold. The intercept factor, Q0t is generally greater than RF , and Q1t is substantially less than Rm  Rf . This seems to indicate that the zero-beta model is more consistent with the data. Blume and Husick (1973) find empirical evidence to indicate that historical rates of return may sometimes foreshadow changes in future betas, and that stocks with higher transaction costs should yield somewhat higher gross expected returns. Their data indicate that beta is not stationary over time, and that it does change over time as a function of price. Transaction-cost effects appear less important than the informational effects of price in explaining future returns or future betas. Therefore, the return-generating process may be more complex than what has been assumed. Blume and Friend (1973) examine both theoretically and empirically the reasons why the CAPM does not adequately explain differential returns on financial assets. Empirically the risk-return tradeoffs implied by stocks on the New York Stock Exchange for three different periods after World War II cast doubt on the validity of the CAPM either in its S-L-M form or zero-beta form. However, they do confirm the linearity of the relationship for NYSE stocks. Rosenberg and McKibben (1973) observe that predictions of the riskiness of returns on common stocks can be based on fundamental accounting data for the firm and also on the previous history of stock prices. This paper tries to combine both sources of information to provide efficient predictions. A stochastic model of the parameters is built and the mean squared error is used as a criterion for the evaluation of the

forecasting performance of estimators. They conclude that the results “strongly confirm the usefulness of the specific risk predictions based on the accounting descriptors.” Merton (1980) is concerned with the estimation of the expected return on the market. He notes that the current practice for estimating the expected market return adds the historical average realized excess market returns to the current observed interest rate. However, while this model explicitly reflects the dependence of the market return on the interest rate, it fails to account for the effects of changes in the level of market risk. Three models of equilibrium expected market returns are elaborated, and estimation procedures that incorporate the prior restriction that equilibrium expected excess returns on the market must be positive are derived and applied to return data for the period 1926–1978. The following are the principal conclusions of the study. 1. The nonnegativity restriction above should be explicitly included as part of the specifications. 2. Estimators that use realized returns should be adjusted for heteroskedasticity. Roll (1977) directs an attack on the empirical tests of the CAPM. While recognizing that the theory is testable in principle, he asserts that “no correct and unambiguous test to the theory (has) appeared in the literature and there is practically no possibility that such a test can be accomplished in the future.” This conclusion is derived from the mathematical equivalence between the individual return beta linearity and the market portfolio’s mean variance efficiency. Therefore, any valid test presupposes complete knowledge of the market portfolio’s composition. The major results reported by Roll from his theoretical inquiry include the following: 1. The only testable hypothesis is that the market portfolio is mean-variance efficient. 2. All other so-called implications of the CAPM are not independently testable. 3. In any sample there will always be an infinite number of ex-post mean-variance efficient portfolios; betas calculated will satisfy the linearity relation exactly, whether or not the true market portfolio is mean-variance efficient. 4. The theory is not testable unless the exact composition of the true market portfolio is known and used in the tests. 5. Using a proxy for the market portfolio does not solve the problem, for the proxy itself might be mean-variance efficient even when the true market portfolio is not, and conversely. 6. Empirical tests that reject the S-L-M model have results fully compatible with the S-L-M model and a specification error in the measured market portfolio. 7. If the selected index is mean-variance efficient, then the betas of all assets are related to their mean returns by the same linear function (all assets and portfolios fall exactly on the SML).

6 Capital Asset Pricing Model and Beta Forecasting

8. For every ranking of performances obtained with a mean-variance inefficient index, there exists another nonefficient index that reverses the ranking. Roll’s critique is a broad indictment of most of the accepted empirical evidence concerning the CAPM theory.

Appendix 6B Anomalies in the Semi-strong Efficient-Market Hypothesis Three anomalies in the semi-strong efficient-market hypothesis are noteworthy. Four authors – Basu, Banz, Reinganum, and Keirn – deal with these three anomalies: (1) P/E ratios; (2) size effects; and (3) the January effect. Basu (1977) empirically notes that a firm with a low P/E ratio, when adjusted for risk, has an excess return over firms that have a high P/E ratio. If this is true, then there are implications for the market’s efficiency, the validity of the CAPM, or both. However, Basu found that the excess returns, when adjusted for transaction costs, taxes, and so forth, were so much smaller as to be insignificant. Therefore, CAPM and market efficiency were supported. In conjecturing why a difference in P/E ratios could affect the returns of the firm, we believe that the relationship may involve the firm’s ability to raise debt. A low P/E ratio may indicate more difficulty in raising capital than a high P/E ratio. This difficulty in raising capital could result in different lending and borrowing rates for different firms. Therefore, a Brennan version of the standard CAPM pricing model may be more applicable. Reinganum (1981) also empirically tests the P/E effect and finds the same results as Basu. In addition, Reinganum was concerned with the efficiency of the market. In order to see if the market was informationally efficient, Reinganum also looked at returns of firms with neither high nor low P/E ratios. He found these firms to be correctly priced. From these results he conjectures that the market was informationally efficient, but CAPM did not allow for the P/E effect on returns. Therefore an APT model with the P/E effect as one of its factors would be preferable to the CAPM model. Banz (1981) empirically tests the effect of firm size, finding that small-company stock returns were higher than

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large-company stock returns. Banz argues that the P/E ratio serves as a proxy for the size of a firm and not vice versa. His conjecture about why this anomaly exists centers around informational distribution. A small firm’s information distribution is somewhat limited, which causes investors to be wary of buying the stock and depresses the price. Banz suggests the APT valuation model may be more robust than the CAPM in that the APT would be able to capture the size effect by using a P/E ratio as a proxy for the size effect as a factor in the model. Since the distribution of information affects a firm’s ability to raise capital (less information on a firm may cause a firm to pay a premium for capital) the premium would indicate that the lending and borrowing rates of different firms are not the same. A valuation model capturing different lending and borrowing rates was provided by Brennan (1971). The utilization of this model could be implemented where rates differed for large and small firms. If small firms do not borrow in the capital market, then Black’s noborrowing CAPM would be more suitable than the standard CAPM. Keim (1983) empirically tests one of the most baffling anomalies, the January effect. He found that for stocks with excess returns, over 50% of these excess returns were realized in January. In addition, 50% of the January excess return occurs in the first week of January. This phenomenon of excess returns occurring in the month following the taxyear-end has been found empirically in Great Britain also. Although Keim offers no rationale for this phenomenon, others have tried to find tax reasons for this anomaly. The selling of assets in January rather than December to postpone capital gains taxes to the next taxable year has been one suggested rationale. Unlike the P/E and size effects, this anomaly does not have a clear proxy that could be utilized as a factor in an APT model. CAPM does not capture the effect well, and without any theoretical or economical rationale for this effect, any valuation model would be hard pressed to account for it. Many of the authors seem to conclude that the standard CAPM is not working well, and that alternative valuation models should be considered to capture these anomalies. This might imply that security analysis and portfolio management techniques can be used to beat the market.

Chapter 7

Index Models for Portfolio Selection Cheng-Few Lee, Joseph E. Finnerty, and Donald H. Wort

Abstract In this chapter, we discuss both the single-index model and multiple-index portfolio selection model. We use constrained maximization instead of minimization procedure to calculate the portfolio weights. We find that both single-index and multi-index models can be used to simplify the Markowitz model for portfolio section. Keywords Single-index model r Multi-index model r Market model r Multiple indexes r Linear programming approach r Lagrange multipliers

7.1 Introduction Previously, we have presented and discussed the Markowitz model for delineating the efficient frontier. Numerous examples were shown that indicated the potentially crushing number of computations resulting from the calculations for even a three-security portfolio. This chapter offers some simplifying assumptions that reduce the overall number of calculations through the use of the Sharpe single-index and multiple-index models. The essential difference between the single- and multipleindex models is the assumption that the single-index model explains the return of a security or a portfolio with only the market. The multiple-index model describes portfolio returns through the use of more than one index. The investor may quantify the return on a portfolio by seeking an index that is representative of the market together with indexes that are representative of the industries of which the component securities are members or exhibit some other

D.H. Wort () University of California, Hayward, CA, USA e-mail: [email protected] C.-F. Lee Rutgers University, New Brunswick, NJ, USA J.E. Finnerty University of Illinois at Urbana-Champaign, Champaign, IL, USA

common factor. More is said about the multiple-index model later in the chapter; for now the single-index model is the focus of discussion.

7.2 The Single-Index Model The major simplifying assumption that yields the index model from Markowitz’s portfolio theory is that covariances between individual securities contained in the portfolio are zero. This assumption greatly reduces the number of calculations needed to find the set of efficient portfolios. The use of the index model necessitates additional statistical estimates for the parameters of the index; nevertheless, these additions are minor in comparison to the reduction in the calculation load as a result of ignoring the covariance terms between securities. Suggested by Markowitz (1959), the single-index model was fully developed by Sharpe (1970), who assumed that the covariances could be overlooked. The return of an individual security was tied to two factors – a random effect and the performance of some underlying market index. Notationally: Rit D ai C bi RI t C eit

(7.1)

where: ai and bi D regression parameters for the i th firm; RIt D the tth return of some underlying market index; Rit D the tth return on security i ; and eit D the tth random effect for the i th security. Equation (7.1) is the market model as discussed in the last chapter. This regression makes several assumptions about the random effect term. 1. The expected value of the tth random effect for security i is zero. More explicitly, E .eit / D 0. 2. The variance of the error terms is constant. This amounts to the assumption that the errors are homoscedastic. 3. There is no relationship between the errors and the return on the market: Cov .eif ; RIt / D 0.

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4. The random effects are not serially correlated: E.eit ; eitCn / D 0. 5. The i th security’s random effect is unrelated to any other random effects of any other security: E.eit ; ejt / D 0. The fifth assumption guarantees that the regression coefficients ai and bi are the best unbiased linear estimators of the true parameters. An investigation of some of the results of the previous assumptions is in order. The expected value of the return on security i is equal to the sum of the intercept, the adjusted return on the index, and some random effect. This can be expressed:

returns of two different securities are not interrelated but only connected to the market index, the covariance between the two securities can be derived from the twice-adjusted variance of the market index (once by the b coefficient of the first security with the market and again by the b coefficient of the second security with the market). The investigation starts with a statement about the covariance of the two securities:    ij D E .Rit  ri / Rjt  rj D EŒ..ai C bi RIt C eit /.ai C bi rI //

..aj C bj RIt C ejt /.aj C bj rI //    D E .bi .RIt  rI / C eit / bj .RIt  rI / C ejt   D bi bj E .RIt  rI /2 C bi E ejt .RIt  rI /

E.Ri / D E.ai C bi RIt C eit / D E.ai / C E.bi RIt / C E.eit /

(7.2)

Because ai and bi are constants, and E .eit / is equal to zero: ri D E.Ri / D ai C bi rI

(7.3)

where rI is equal to the mean of the returns on the market index. If the mean of the returns on security i equals ri , then the variance of security i is equal to the expected value of squared deviations from rit . This translates to: i2 D E.Rit  ri /2 D EŒ.ai C bi RIt C eit /  .ai C bi rI / 2 D EŒbi .RIt  rI / C eit 2 D bi2 E .RIt  rI /2 C 2bi EŒeit .RIt  rI / C E.eit /2 (7.4) By the third assumption the covariance of the random effect and the deviation of the index return from its mean are zero; also, the expected value of the squared errors is equal to the variance of the random effect. Thus: i2 D bi2 E.RIt  rI /2 C E.eit /2 D bi2 I2 C ei2

(7.5)

This is equivalent to saying that the variance of the returns on a security is made up of some adjusted quantity of the variance of the market (usually referred to as systematic risk) plus the variance of the random effects exclusive to that particular security (unsystematic risk). The last result to be investigated from these assumptions involves a minor step into abstraction in which the possibility of interaction between two securities and the market index is considered. It is suggested that because variations in the

C bj E.eit .RIt  rI // C E.eit ejt /

(7.6)

Since according to the third and fourth assumptions the last three terms of the last summation are equal to zero: ij D bi bj I2

(7.7)

The number of calculations necessary to utilize the ı Markowitz model is N C .N 2  N / 2, as shown in the last chapter. For a portfolio with a hundred securities, this translates to 5,050 calculations. With the Sharpe singleindex model, only 100 estimates are needed for the various security-regression coefficients and only one variance calculation, the variance of the returns on the market. In addition, 100 estimates of the unsystematic risk ei2 are also needed for the single-index model. Hence, the single-index model has dramatically reduced the input information needed.1

7.2.1 Deriving the Single-Index Model So far only the Sharpe single-index model has been utilized to study the returns of a single security i as determined by its relation to the returns on a market index. Expected return of a portfolio. Now consider the return on a portfolio of n securities. The return of a portfolio of n securities is the weighted summation of the individual returns of the component securities. Notationally: n   X E Rpt D xi E .Ri / i D1

1

Discussion of the single-index model adapted in part from Sharpe (1970). Adapted by permission.

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Where Rpt is the rate of return for a portfolio in period t and Xi is the weight associated with the i th security. Rit D ai C bi RIt C eit Rpt D D

n X i D1 n X

xi .ai C bi RIt C eit / xi .ai C eit / C

i D1

D

n X

n X

xi .bi RIt /

i D1

xi ai C

i D1

n X

.xi bi / .RIt / C

i D1

n X

xi eit

(7.8a)

i D1

It was shown previously that when the number of component securities in a portfolio approaches 15, the unique risk of the component securities is reduced through diversification. In Equation (7.10) the last term, the weighted sum of the random effect variances, approach zero as n increases. So, again, as the number of securities increases, the unsystematic risk is reduced and the remaining risk of the portfolio is the adjusted variance of the market index. Example 7.1 further illustrates this concept. Example 7.1. Given the following information, what should the ˇ of the portfolio .bp / be?

Thus, E.Rpt / D

n X

xi ai C

i D1

n X

.xi bi /E.RIt /

(7.8b)

Variance of a portfolio. To derive the variance of the portfolio p2 , consider first that the mean return of the portfolio is equal to the expected value of the return on the portfolio. Then, following the definition of p2 in previous chapters and Equations (7.5) and (7.7), we can derive P2 as follows. The variance of the portfolio is equal to P2 D EŒ.Rpt  E.Rpt //

(7.9a)

Substituting Equations (7.8a) and (7.8b) into Equation (7.9A), we obtain h i2  Xn Xn 2 P D E .RIT  E.RIT // .xi bi / C xi ei i D1

D D D

Xn

i D1

hXn

i D1

i D1

2 Xn .xi bi / I2 C

i D1

Xn

i D1

j D1

xi bi

.xi xj bi bj /I2 C

i hXn j D1

i

2 xi ei Xn

xj bj I2 C

i D1

Xn i D1

i D1

i D1

xj bj D bp . Hence, the last equation reduces to: n X p2 D bp2 I2 C xi2 ei2 i D1

Solution (7.10):

Substituting related information into Equation

bp D D

I2 P2 0:082 0:041

D 2:0 Therefore, bp D 1:414. Equation (7.8b) implies that the portfolio can be viewed as an investment in n basic securities and a weighted adjusted return in the market, or n X

.xi bi / E.RIt t/

i D1

The return on the market can be decomposed as a combination of the expected return plus some random effect. When this random effect is positive, the atmosphere is bullish and when it is negative, the atmosphere is bearish. Notationally:

D anC1 C enC1;t

xi2 ei2

Because the weighted sum of the bi coefficients is equal n P xi bi D bp , similarly to the coefficient of the portfolio

j D1

xi2 ei2 D 0

RIt D E.RIt / C enC1;t

xi2 ei2

(7.9b)

n P

n X

i D1

This equation indicates that the return of a portfolio may be decomposed into the summation of the weighted returns peculiar to the individual securities and the summation of the weighted adjusted return on the market index. Thus, the portfolio may be viewed as a combination of n basic securities and a weighted adjusted return from an investment in the market index.

Xn

p2 D 0:082 I2 D 0:041

(7.10)

(7.11)

This bit of algebraic maneuvering enables the weighted adjusted investment in the market to be viewed as an investment in an artificial security, the .n C 1/th of an n-security portfolio. The weight for this .n C 1/th security is the sum of the n weights multiplied by their respective related coefficients to the market index. Thus: xnC1 D

n X i D1

xi bi

(7.12)

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The reason for this divergence in notation resulting in the definition of the .n C 1/th security’s return and weight is that Equation (7.8) can be simplified to yield a working model for portfolio analysis. Substituting the last results for XnC1 and RIt into Equation (7.8) yields: Rpt D

n X

D

t D1

relationship n X

xi .ai C eit / C xnC1 .anC1 C enC1;t /

RIt D n˛i C

t D1

i D1 nC1 X

Example 7.2. Given RIt ; Rit and ˛i as indicated in the table n P shown below and the fact that ˇi RIt D 56, using the

n X

ˇi RIt C

t D1

find the values for ˛i ; ˇi and ei . xi .ai C eit /

(7.13)

Solution n X

Because the expected value of the random-effect terms is zero, the summation that results after the application of the expectations operator can be expressed: E.Rpt / D

n X

n P

RIt D 28; so ˇi D

t D1

D

nC1 X

xi ai C xnC1 E.RIt /

D

(7.14)

This yields a formula for the return of a portfolio that is easily applied to portfolio analysis. Before proceeding, however, it is necessary to simplify the variance formula so that it may be used as well. Remembering that the variance of the portfolio is the expected value of the squared deviations from the expected market return, the last results concerning Rpt and E.Rpt / may be applied:

Substitute the values

iD1

n X

n P

ˇi RIt D 56; n D 4

t D1

RIt D n˛i C

n X

ˇi RIt C

t D1

n X

eit

t D1

and solve for ˛i . RIt

Rit

˛i

ˇi RIt

ei

4 6 10 8 28

12 14 20 18 64

2 2 2 2 8

8 12 16 16 56

12  2  8 D 2 14  2  12 D 0 20  2  20 D 2 18  2  16 D 0 0

iD1

iD1

The ˇi RIt column in the table is filled by simply multiplying ˇ.D 2/ by the RIt column, The ei ’s are the amounts such that Rit D ˛ C ˇi Rmt C eit is an equality so Rit D ˛i C ˇi RIt C eit is satisfied, From the last column of the table we know that n P eit D 0. Therefore

iD1

12 xi eit A

iD1 nC1 X

RIt D 64;

t D1

iD1

92 82 3 nC1 = < nC1 X X DE 4 xi .ai C eit /5  xi E .ai / ; :

D Var @

56 28

into the regression line

92 82 3 nC1 = < nC1 X X DE 4 xi .ai C eit /5  xi ŒE .ai / C E .eit / ; :

0

n P t D1

82 3 2 392 nC1 = < nC1 X X   5 4 4 Var Rpt D E xi .ai C eit /  E xi .ai C eit /5 ; :

nC1 X

RIt

D2 xi ai

i D1

iD1

ˇi RIt

t D1 n P t D1

i D1

DE@

eit

t D1

i D1

0

n X

1 xi eit A

i D1

(7.15)

iD1

This result follows from the assumption that the covariances are equal to zero. Additionally, each variance term is only a weighted sum of the errors around the market return. The direct usefulness of the last two conclusions will become apparent when solving for the security weights. Example 7.2 provides further illustration.

˛i D

64  56 C 0 D2 4

In this section, the expected return E.Rpt / and the variance of a portfolio Var.Rpt / in terms of the single-index model has been derived. In the following section, the optimal portfolio selection procedures discussed in previous chapters are used to explore the single-index optimum-portfolio selection model.

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7.2.2 Portfolio Analysis and the Single-Index Model Before beginning the portfolio analysis using the single-index model, it is necessary to explain a derivation of security weights through the use of the Lagrangian calculus maximization discussed in previous chapters. The maximization procedure maximizes a linear combination of the following two equations:2 E.Rp / D

nC1 X

E.Ri /

or

i D1

Max nC1 X   Var Rp D Var xi eit

! (7.16a)

i D1

subject to nC1 P

xi D 1 and

i D1

xnC1 D

n P

(7.16b) xi bi

i D1

The first constraint is equivalent to requiring that the sum of the weights of the component securities is equal to one. The second constraint requires that the weight of the market index within the portfolio returns is equal to the summation of the weighted adjustment factors of the component securities. This requirement is as described in Equation (7.16b). Combining the above two objective functions with the two constraints yields a Lagrangian function: ! n X     xi  1 P D ˆE Rp  Var Rp C 1

C2

n X

!

i D1

xi bi  xnC1

i D1

Dˆ

nC1 X

xi ai 

i D1

C2

nC1 X

Var .eit / C 1

i D1 n X

xi bi  xnC1

!

n X

! xi  1

i D1

(7.16c)

i D1

The only difference between this maximization function and the minimization shown in previous chapters, beyond intent,

2 The maximization of the negative of the variance is equivalent to the minimization of the variance itself. This can be viewed as originating from the negative spectrum of the number line and maximizing towards zero.

is that instead of fixing some arbitrary value of return needed .E  /, an attempt can be made to quantify the level of risk aversion that the investors of the portfolio require, thereby placing the portfolio on the efficient frontier not by desired return but by level of utility, as discussed in previous chapters. This indication of risk aversion is denoted by the Greek letter ˆ (Phi). When low values are exhibited, risk aversion is pronounced; when high values are in evidence, substantial risk taking is allowed. This notion of the level of risk aversion is best pictured in Fig. 7.1. In Fig. 7.1, A represents an investor’s objective function to minimize the risk only – therefore, the aggressiveness to return is zero. C represents an investor’s objective function to maximize returns only – therefore, the aggressiveness to return is infinite. At B an investor’s attitude toward return and risk is between A and C. Figure 7.2 provides further illustration of the approach being utilized. What is depicted is the variation of the objective function P as the parameter denoting risk is varied. Again, notice that the risk-return relation for a low ˆ is lower than that for the more risk-taking, high value of ˆ. Points A and C are as discussed for Fig. 7.1. Lines BD and B 0 D 0 represent the objective function when the risk-aversion parameter is equal to one. Note that BD instead of B 0 D 0 represents the maximization of the objective function. In sum, different values of ˆ generate different optimal objective functions. Consider now the three-security portfolio. In this framework, the preceding objective function expands to: P D ˆx1 a1 C ˆx2 a2 C ˆx3 a3 Cˆx4 a4  x12 Var .e1t / x22 Var .e2t /  x32 Var .e3t /  x42 Var .e4t / C1 x1 C 1 x2 C1 x3  1 C 2 x1 b1 C 2 x2 b2 C 2 x3 b3  2 x4 (7.17) Take note of all terms in the preceding equation that contain a4 and x4 . By utilizing the relations of the individual securities to the market, it has been possible to delete most of the calculations necessitated by the full Markowitz variancecovariance model. The return that will be generated from an optimal portfolio derived through this maximization rests on the estimation of the future expected return and variance of the index. This will lead to the availability of ranging analysis for these estimations, which will produce robust estimates for the security weights highlighted at the end of this section. To proceed with the Lagrangian maximization it is necessary to notice that the above equation has six unknowns – the four weights and the two Lagrangian coefficients. All other values are expected to be known or estimated. To maximize, the partial derivative of the objective function is taken with respect to each of the six variables:

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Fig. 7.1 Level of risk aversion and investors’ investment attitude

Fig. 7.2 Level of risk aversion and objective function

32 D Var .e3t / I and

@P D ˆa1  2x1 12 C 1 C 2 b1 D 0 @x1

42 D Var .e4t / :

@P D ˆa2  2x2 22 C 1 C 2 b2 D 0 @x2

Equation (7.18) is a set of six equations in six unknowns when set equal to zero. These six equations can be rewritten in matrix format:

@P D ˆa3  2x3 32 C 1 C 2 b3 D 0 @x3 @P D ˆa4  2x4 42  2 D 0 @x4 @P D x1 C x2 C x3  1 D 0 1 @P D x1 b1 C x2 b2 C x3 b3  x4 D 0 2 Where: 12 D Var .e1t / I 22 D Var .e2t / I

(7.18)

A 2 212 0 0 0 6 0 222 0 0 6 6 0 0 232 0 6 6 0 0 0 242 6 4 1 1 1 0 b2 b3 1 b1

1 1 1 0 0 0

3 b1 b2 7 7 b3 7 7 17 7 05 0

x D k 2 3 2 3 x1 ˆa1 6x2 7 6ˆa2 7 6 7 6 7 6x3 7 6 7 6 7 D 6ˆa3 7 6x4 7 6ˆa4 7 6 7 6 7 41 5 4 1 5 2 0

(7.19a) Equation (7.19a) has related the individual securities to the index and has discarded the use of numerous covariance terms. The majority of the elements of the matrix indicated in the first matrix of Equation (7.19a) are zero.

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As in previous chapters, to solve for the x column vector of variables the P matrix is inverted and each side of the equation is pre multiplied by A1 . This yields the x column vector of variables on the left-hand side and A1 multiplied by k, the solution column vector, on the right-hand side. Ax D k A1 Ax D A1 k Ix D A1 k

(7.19b)

where I is the identity matrix. The procedure of solving Equations (7.19a) and (7.19b) is discussed in later chapters. A linear-programming approach to solve this kind of model is discussed in the first appendix of this chapter. In order to estimate a single-index type of optimal portfolio the estimates of a market model are needed; a discussion of the market model and beta estimates is therefore needed as well.

7.2.3 The Market Model and Beta Equation (7.1) defines the market model: Rit D ai C bi RIt C eit

(7.1)

From this market model, ai ; bi ; Var .eit / and Var .RIt / can be estimated: all are required for the single-index type of optimal portfolio. So far the regression coefficient has been referred to as bi when, in fact, it is a risk relationship of a security with the market. Quantified as the covariance of security i with the market index divided by the variance of the return of the market, beta is a relational coefficient of the returns of security i as they vary with the returns of the market. Notationally: bi D

i:I D I2

n X

Œ.Rit  rit / .RIt  rIt /

t D1

.RIt  rIt /2

(7.20)

Of course, due to the continuous nature of security returns, the result of the above calculation is an estimate and therefore subject to error. This error can be quantified by the use of standard error estimates, which provide the ability to make interval estimations for future predictions. The following discussion is related to the beta estimate and its forecasting as discussed in previous chapters. Because of the linkage between security returns and the firm’s underlying fundamental nature, beta estimates will vary over time. It is the job of the security analyst to decide how to modify not only the beta estimate but also the period

of time from which the sample of returns is to be drawn. Some popular security-evaluation techniques use a tiered growth model to correspond to the product life-cycle theory. Within the scope of this theory, beta estimates for the relation of that security to the market will vary with respect to the time period chosen for the sample of returns. It is obvious that the returns generated by a firm in its infancy have very little correlation with returns during the growth or maturity phase. It is up to the analyst to judge which phase a company may be in and to adjust the beta estimates accordingly. Assume for expository purposes that all stocks move in perfect alignment with the market, and therefore the security returns all produce beta estimates of 1. It could then be said that any estimation of betas above 1 would indicate positive sampling error, and all estimates below 1 negative sampling error. Future periods would show returns that caused movement from either side of the estimate deviation back towards 1. Blume (1975) shows that the adjustment of a beta in one period could accurately predict the adjustment in subsequent periods. Utilizing regression analysis, Blume demonstrates that a historical beta could be adjusted successfully to predict future levels. Large brokerage houses have taken this theory a little further by applying weighted averages of historic and average betas in adjusting beta estimates. This has the advantage of being simple to calculate, but it has the ability nevertheless to adjust beta back towards an average level (This kind of beta-adjustment process was discussed in the last chapter; the Value Line beta estimate is essentially based on this process, as explored in previous chapters.) Historic beta estimates give the analyst a feeling for how the firm is generating returns to an investor but give very little guidance about the firm’s underlying fundamental nature. It is held that a firm’s balance-sheet ratios over time can indicate how risky the firm is relative to the market. Using seven financial quantities of a firm, Beaver et al. (1970) show that betas can be estimated by multiple regression relating the financial levels to the riskiness of the firm. Using 1. 2. 3. 4. 5. 6. 7.

Dividend payout Asset growth Leverage Liquidity Asset size Earnings volatility, and Accounting beta (relation of earnings of the firm with the earnings of the economy as a whole)

BKS show not only that each of the variables carries a logical relation to the market but also that the multiple regression is significant. Other studies, most notably Rosenberg and Guy (1976a, b), use many more fundamental factors in the regression for estimating betas. Historic betas are of consequence because of their relation with market returns; nevertheless, the sampling period

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tends to crowd out the information contained in returns due to recent developments for the firm. Fundamental betas tend to recognize significant changes in the makeup of a firm – for example, a change in debt structure, or liquidity. The disadvantage related to fundamental betas is that they treat intercompany responsiveness as constant. It is patently obvious that a small firm taking on a large debt will be at much more risk than a Ford or GM doing the same. Because of the advantages and disadvantages apparent in both types of beta estimates, it has been shown that a combination of the estimates is more suitable to risk classification. Rosenberg and McKibben (1973) found that intercompany responsiveness is substantial, so an analysis was undertaken to introduce these variations into the regression. Utilizing a set of dummy variables to capture the advantages of both types of betas, Rosenberg and Marathe (1974, Berkeley Working Paper Series) have described a multiple regression for the beta estimate of a firm that include information such as variability, level of success, relative size, and growth potential. By using a very large and complex model they found that the forecasting ability of the regression analysis is substantial, but they were inconclusive as to whether the benefit of the analysis outweighs the computational cost. Forecasting future beta levels holds its roots in the difficulty of forecasting the fundamental nature of the firm itself. This section has briefly looked over some of the available adjustment processes for beta estimation. While reasonable in its scope, this coverage of the adjustment process is by no means complete. Additional research material is cited at the end of this chapter for use in further study. It should be noted during future readings that the more sophisticated an adjustment process is, the more likely it is that the computational costs will outweigh the improvement of the beta forecast.

7.3 Multiple Indexes and the Multiple-Index Model The previous section reviewed the possibilities of adjusting the beta estimate in the single-index model to capture some of the information concerning a firm not contained in the historic returns. The multiple-index model (MIM) pursues the same problem as beta adjustment, but approaches the problem from a different angle. The multiple-index model tackles the problem of relation of a security not only to the market by including a market index, but also to other indexes that quantify other movements. For example, U.S. Steel has returns on its securities that are related to the market, but due to the declining nature of the American steel industry there is also some relation to the steel industry itself. If an index

C.-F. Lee et al.

could be developed that represented the movement of the U.S. steel industry, it could be utilized in the return analysis, and a more accurate estimate of possible security returns for U.S. Steel could be derived. The assumption underlying the single-index model is that the returns of a security vary only with the market index. The expansion provided by the multiple-index model includes factors that affect a security’s return beyond the effects of the market as a whole. Realistically any index might be used, but well-known, published indexes are usually incorporated. These may include general business indicators, industry-specific indicators, or even self-constructed indexes concerning the structure of the firm itself. The model to be addressed contains L indexes; nevertheless, a few examples of possible indexes of interest are offered. The multi-index model is related to the arbitrage pricing model developed by Ross (1976, 1977), which will be explored in detail in the next chapter. The covariance of a security’s return with other market influences can be added directly to the index model by quantifying the effects through the use of additional indexes. If the single-index model were expanded to take into account interest rates, factory orders, and several industry-related indexes, the model would change to:     I1 C bi2 I2 C    C biM IM C ci Ri D ai C bi1

(7.21)

In this depiction of the multiple-index model, Ij is the actual level of index j , while bij is the actual responsiveness of security i to index j . If the component of the security return is not related to any of the indexes then this index model can be divided into two parts: (1) ai , the expected value of the unique return, and (2) ci which represents the distribution of the random effect. ci has a mean effect of zero and a variance of c2i . This model can be utilized with multiple regression techniques, but if the indexes are unrelated to each other the calculations would be made much simpler. This assumption reduces the number of calculations, as compared with the Markowitz full variance-covariance model, but is obviously more complex than the single-index model. To assure that the indexes are unrelated, the index variables can be orthogonalized (made uncorrelated) by completing inter-regressions on the indexes themselves. Assume there is a hypothetical model that deals with two indexes:   I1 C bi2 I2 C ci Ri D ai C bi1

(7.22)

Suppose the indexes are the market index and an index of wholesale prices. If these two indexes are correlated, the correlation may be removed from either index.

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To remove the relation between I1 and I2 , the coefficients of the following equation can be derived by regression analysis: I2 D e0 C e1 I1 C di where:

 2 DE bi1 .I1t C I 1 /2 C 2bi1 bi 2 .I1t C I 1 /.I2t C I 2 / C bi22 .I2t C I 2 /2 C 2bi1 .I1t C I 1 /eit  C2bi 2 .I2t C I 2 /eit C eit2   2 D bi1 E.I1t C I 1 /2 C bi1 bi 2 E .I1t C I 1 /.I2t C I 2 /

e0 and e1 D the regression coefficients; and di D the random error term. By the assumptions of regression analysis, di is uncorrelated with I1 . Therefore:

C bi22 E.I2t C I 2 / C bi1 EŒ.I1t C I 1 /ei   C bi 2 EŒ.I2t C I 2 /ei C E ei2 But by assumption:

  dOi D I2  eO0 C eO1 I1

EŒ.I1t C I 1 /.I2t C I 2 / D 0 EŒ.I1t C I 1 /ei D 0

which is an index of the performance of the sector index without the effect of I1 (the market removed). Defining: I2 D dOi D I2  eO0  eO1 I1

EŒ.I2t C I 2 /ei D 0 and E.I1t C I 1 /2 D 12

an index is obtained that is uncorrelated with the market. By solving for I2 and substituting into Equation (7.22):

E.I2t C I 2 /2 D 22   E ei2 D ei2 I

  I1 C bi2 I2  bi2 eO0  bi2 eO1 I1 C ci Ri D ai C bi1

therefore,

Rearranging:       bi2 eO1 I1 C bi2 I2 C ci Ri D ai  bi2 eO0 C bi1 If the first set of terms in the brackets are defined as ai and the second set of terms are defined as bi1 ; bi2 D bi 2 ; I1 D I1 and ci D ei , the equation can be expressed: Ri D ai C bi1 I1 C bi 2 I2 C ei

(7.23)

in which I1 and I2 are totally uncorrelated: the goal has been achieved. As will be seen later, these simplifying calculations will make the job of determining variance and covariance much simpler. The expected return can be expressed:

D E.ai / C E.bi1 I1t / C E.bi 2 I2t / C E.eit / (7.24)

Since ai ; bi1 and bi 2 are constants, E .ei / D 0 by assumption, where I 1 D E.I1t / and I 2 D E.I2t /. Variance can be expressed:  2 DE .Rit  Ri /2

(7.25)

Covariance between security i and security j can be expressed ij DEŒ.Rit  Ri /.Rjt  Rj / Œwhere Ri DE.Rit / and Rj D E.Rjt /  ˚ DE .ai Cbi1 I1 Cbi 2 I2t C e1t /  .ai C bi1 I 1 C bi 2 I 2 / 

.aj C bj1 I1t C bj 2 I2t C ejt /  .aj C bj1 I 1 C bj 2 I 2 / DEfŒbi1 .I1t  I 1 / C bi 2 .I2t  I 2 / C ei Œbj1 .I1t  I 1 / C .bj 2 .I2t  I 2 / C ejt g D EŒbi1 bj1 .I1t  I 1 /2 C bi 2 bj 2 .I2t  I 2 /2

E.Rit / D E.ai C bi1 I1t C bi 2 I2t C eit /

D ai C bi1 I 1 C bi 2 I 2

2 2 i2 D bi1 1 C bi22 22 C ei2

where Ri D E .Rit /

DEŒ.ai C bi1 I1 C bi 2 I2 C eit /  .ai C bi1 I 1 C bi 2 I 2 / 2 DEŒ.a1 C ai / C bi1 .I1t C I 1 / C bi 2 .I2t C I 2 / C eit 2

D bi1 bj1 12 C bi 2 bj 2 22

(7.26)

since all remaining expected values of the cross-product terms equal zero. The extended results of expected return variance and covariance for a multi-index and more than two indexes can be found in the second appendix of this chapter. One simplifying way of applying the multiple-index model is to start with the basic market model and add indexes to reflect industry-related effects. If the firm has 100% of its operations in one industry, Equation (7.23) can be used to

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represent a two-index model with market index and industry index. In general this approach reduces the number of data inputs to 4N C 2I C 2. These data inputs are (1) the expected return and variance for each stock and market index; (2) the covariance between the individual security and the market index and the industry index; and (3) the mean and variance of each industry index. Although this is a larger number of data inputs than for the simple market model, the accuracy of the estimation of security return increases. So the tradeoff is one of more information (higher cost to use) versus greater accuracy of the forecasted security return. Care must be taken in applying the multiple-index model. It is often the case that the additional information resulting from the application of a higher-complexity model is outweighed by the computational cost increase. In an attempt at making the model as parsimonious as possible, it is necessary to judge the increased information gained by utilizing the more complex model. This can be accomplished by examining the mean square error for the forecast of the actual historic values. Although not within the scope of this text, the ability to judge the accuracy of a forecast is essential and can be acquired from any good statistical forecasting text. Example 7.3 provides further illustration of the single-index model.

Going back to the calculus maximization derivation of optimal portfolio weights, recall that a linear combination of the following two factors is being maximized: nC1   X E .Ri / or E Rp D i D1 nC1 X   Var Rp D Var xi eit

! (7.27a)

i D1

subject to nC1 P

xi D 1 and

i D1 n P

(7.16B)

xi bi D xnC1

i D1









P D ˆE Rp  Var Rp C 1 n

P xi bi  xnC1 C2 i D1

n P

xi  1

i D1

nC1 P 2 xi ai  xi Var .Ri / C 1 i D1 i D1 n

P xi bi  xnC1 C2

Dˆ

nC1 P





n P

xi  1

i D1

i D1

(7.16C) Example 7.3. During the discussion of the single-index model, a method was presented for determining optimal portfolio weights given different levels of risk aversion. In this section a three-security portfolio is examined in which the returns of the securities are related to a market index.

Ri Alpha Beta Var (residual)

JNJ

AXP

XOM

S&P 500

Risk-free

Artificial security

0.0053 0.0046 0.29 0.0019

0.0055 0.0022 1.21 0.0016

0.0126 0.0109 0.65 0.0022

0.0025 – – –

0.0023 – – –

– – – 0.0015

The only difference between this maximization function and the minimization shown in previous chapters, beyond intent, is that instead of fixing some arbitrary value of return needed .E  /, an attempt is made to quantify the level of risk aversion that the holders of the portfolio require, thereby placing the portfolio on the efficient frontier, not by desired return, but by level of utility. Consider again a three-security portfolio. In this framework, the preceding objective function expands to: P D ˆx1 a1 C ˆx2 a2 C ˆx3 a3 C ˆx4 a4  x12 Var .R1 /  x22 Var .R2 /  x32 Var .R3 /  x42 Var .R4 / C 1 x1 C 1 x2 C 1 x3  1 C 2 x1 b1 C 2 x2 b2 C 2 x3 b3  2 x4

Varaiance-covariance matrix

JNJ AXP XOM S&P 500

JNJ

AXP

XOM

S&P 500

0.0020 0.0009 0.0004 0.0004

0.0009 0.0038 0.0010 0.0018

0.0004 0.0010 0.0027 0.0010

0.0004 0.0018 0.0010 0.0015

The securities and the index have the following observed parameter estimates taken from actual monthly returns during the period January 2001 to September 2008 (see table). The table includes the calculation of the beta estimate for each security. Check these figures, remembering that beta is equal to the covariance of security i with the market divided by the variance of the market returns.

(7.17)

Take note of all terms in the preceding equation that contain a4 and x4 . By utilizing the individual securities’ relations to the market, it has been possible to delete most of the calculations necessitated by the Markowitz full variancecovariance model. The return that will be generated from an optimal portfolio derived through this maximization rests on the estimation of the future expected return and variance of the index. By partially differentiating the above objective function with respect to each of the weights and the ’s, it is possible to develop the following six equations in six unknowns:

7 Index Models for Portfolio Selection

121 n n X   X xi xj ij Var Rp D

@P D ˆa1  2x1 12 C 1 C 2 b1 D 0 @x1

i D1 j D1

@P D ˆa2  2x2 22 C 1 C 2 b2 D 0 @x2

Taking the covariance expressions:

@P D ˆa3  2x3 32 C 1 C 2 b3 D 0 @x3

  Var Rp D Œ..0:0652/2 0:0020/C..0:4341/2 0:0038/ C ..1:4994/2 0:0027/ C 2 ..0:0652/

@P D ˆa4  2x4 42 C 1 C 2 b4 D 0 @x4

.0:4341/ 0:0009/

@P D x1 C x2 C x3  1 D 0 @1

C 2 ..0:0652/ .1:4994/ 0:0004/ (7.18)

As shown in Equation (7.18), these last six equations can be transformed into Jacobian Matrix notation. A matrix can be developed by utilizing the set of data in the first table of this problem and a ˆ of 1.0 (denoting moderate risk aversion): A x D k 2 3 2 3 3 0:013 0 0 0 1 1:15 0:0084 x1 0:015 0 0 1 0:907 6x2 7 6 0 60:01277 6 6 7 6 7 7 0 0:006 0 1 0:557 6x3 7 6 0 60:01877 6 7 6 7 D 6 7 0 0 0:004 0 0 7 6x4 7 6 0 60:01257 4 1:0 5 4 5 4 1 1:0 1:0 0 0 0 1:0 5 2 1:15 0:90 0:55 1:0 0 0 0 2

When the P matrix is invented and premultiplies each side of the equation: x D A1 k 3 2 3 2 0:06523 x1 6x2 7 60:434147 7 6 7 6 6x3 7 6 1:49937 7 7 6 7D6 6x4 7 6 0:42381 7 7 6 7 6 41 5 40:005215 0:00129 2 This solution vector shows that investment should short 6:52% in Johnson & Johnson, short 43:41% in American Express, and long 149.94% in Exxon Mobil. Additionally, the weight of x4 is the sum of the weighted adjustments as indicated in Equation (7.16b). The return on this portfolio is the weighted sum of the individual returns: E.Rp / D

n X

xi E.Ri /

D 0:0055 A portfolio has been developed that is efficient within the realm of this model. In the table below are the portfolios derived by varying the utility factor ˆ from 0 (totally risk averse) to 2 (more aggressive risk posture). This last table offers the ability to develop an efficient frontier under the Sharpe single-index model. The figure shows that the portfolios start to trace out an efficient boundary. For a fuller graph, the analyst would continue the calculations with ever higher values of ˆ stretching the efficient frontier.   Var Rp

Portfolio

ˆ

1

0.0

0.5233

0.1531

0.3236

0.0077

0.0013

0.0361

2

0.5

0.2290

0.1405

0.9115

0.0119

0.2223

0.4715

X1

X2

Rp

X3

p

3

1.0

0.0652

0.4341

1.4994

0.0162

0.0055

0.0742

4

1.5

0.3595

0.7278

2.0873

0.0204

0.0109

0.1044

5

2.0

0.6538

1.0214

2.6751

0.0246

0.0185

0.1360

0.03 0.025 Expected Retrun E(R)

@P D x1 b1 C x2 b2 C x3 b3  x4 D 0 @2

C 2 ..0:4341/ .1:4994/ 0:0010/

0.02 0.015 0.01 0.005

i D1

D Œ.0:0652 0:0053/ C .0:4341 0:0055/

0 0.0000

0.0200

0.0400

0.0600

0.0800

0.1000

0.1200

0.1400

0.1600

Risk

C .1:4994 0:0126/ D 0:0162 The variance of the portfolio is the sum of the weighted variance and covariance terms:

7.4 Conclusion This chapter has discussed the essentials of single- and multiple-index models. The theoretical underpinnings of the

122

theories have been explored and numerical examples have been provided. An efficient boundary has been derived under the guidelines of the model and the quantitative analysis of the related parameters has been studied. It has been shown that both single-index and multi-index models can be used to simplify the Markowitz model for portfolio section; remember, however, that the multi-index model is much more complicated than the single-index model.

References Beaver, W., P. Kettler, and M. Scholes. 1970. “The association between market determined and accounting determined risk measures.” The Accounting Review 45, 654–682. Blume, M. 1975. “Betas and their regression tendencies.” Journal of Finance 20, 785–795. Bodie, Z., A. Kane, and A. Marcus. 2006. Investments, 7th Edition, McGraw-Hill, New York. Brenner, M. 1974. “On the stability of the distribution of the market component in stock price changes.” Journal of Financial and Quantitative Analysis 9, 945–961. Elton, E. J., M. J. Gruber, and T. Urich. 1978. “Are betas best?” Journal of Finance 23, 1375–1384. Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2006. Modern portfolio theory and investment analysis, 7th Edition, Wiley, New York. Fouse, W., W. Jahnke, and B. Rosenberg. 1974. “Is beta phlogiston?” Financial Analysts Journal 30, 70–80. Frankfurter, G. and H. Phillips. 1977. “Alpha-beta theory: a word of caution.” Journal of Financial Management 3, 35–40. Frankfurter, G. and J. Seagle. 1976. “Performance of the sharpe portfolio selection model: a comparison.” Journal of Financial and Quantitative Analysis 11, 195–204. Gibbons, M. R. 1982. “Multivariate tests of financial models, a new approach.” Journal of Financial Economics 10, 3–27. Haugen, R. and D. Wichern. 1975. “The intricate relationship between financial leverage and the stability of stock prices.” Journal of Finance 20, 1283–1292. Jacob, N. 1974. “A limited-diversification portfolio selection model for the small investor.” Journal of Finance 19, 847–856. King, B. 1966. “Market and industry factors in stock price behavior.” Journal of Business 39, 139–140. Latane, H., D. Tuttle, and A. Young. 1971. “How to choose a market index.” Financial Analysts Journal 27, 75–85. Lee, C. F. and A. C. Lee. 2006. Encyclopedia of finance, Springer, New York. Lee, C. F., J. C. Lee, and A. C. Lee. 2000. Statistics for business and financial economics, World Scientific Publishing Co, Singapore. Levy, R. 1974. “Beta coefficients as predictors of return.” Financial Analysts Journal 30, 61–69. Maginn, J. L., D. L. Tuttle, J. E. Pinto, and D. W. McLeavey. 2007. Managing investment portfolios: a dynamic process, CFA Institute Investment Series, 3rd Edition, Wiley, Hoboken, NJ. Markowitz, H. 1959. Portfolio selection. Cowles Foundation Monograph 16, Wiley, New York. Morgan, I. G. 1977. “Grouping procedures for portfolio formation.” Journal of Finance 21, 1759–1765. Roll, R. 1969. “Bias in fitting the Sharpe model to time series data.” Journal of Financial and Quantitative Analysis 4, 271–289. Rosenberg, B. and J. Guy. 1976a. “Prediction of beta from investment fundamentals.” Financial Analysts Journal 32, 60–72.

C.-F. Lee et al. Rosenberg, B. and J. Guy. 1976b. “Prediction of beta from investment fundamentals, part II.” Financial Analysts Journal 32, 62–70. Rosenberg, B. and W. McKibben. 1973. “The prediction of systematic and specific risk in common stocks.” Journal of Financial and Quantitative Analysis 8, 317–333. Rosenberg, B. and V. Marathe. 1974. The prediction of investment risk: systematic and residual risk, Berkeley Working Paper Series. Ross, S. A. 1976. “Arbitrage theory of capital-asset pricing.” Journal of Economic Theory 8, 341–360. Ross, S. A. 1977. “Return, risk and arbitrage,” in Risk and return in finance, Vol. 1, I. Friend and J. L. Bicksler (Eds.). Ballinger, Cambridge, pp. 187–208. Sharpe, W. F. 1967. “A linear programming algorithm for mutual fund portfolio selection.” Management Science 13, 499–510. Sharpe, W. F. 1970. Portfolio theory and capital markets, McGraw-Hill, New York. Smith, K. V. 1969. “Stock price and economic indexes for generating efficient portfolios.” Journal of Business 42, 326–335. Stone, B. 1973. “A linear programming formulation of the general portfolio selection problem.” Journal of Financial and Quantitative Analysis 8, 621–636. Wackerly, D., W. Mendenhall, and R. L. Scheaffer. 2007. Mathematical statistics with applications, 7th ed., Duxbury Press, California.

Appendix 7A A Linear-Programming Approach to Portfolio-Analysis Models Sharpe (1967) developed a simplified portfolio-analysis model designed to be formulated as a linear-programming problem. Jacob (1974) developed a linear-programming model for small investors that delineates efficient portfolios composed of only a few securities. Both of these approaches have as their objectives: (1) a reduction in the amount of data required and (2) a reduction in the amount of computer capability required to solve the portfolio-selection problem. Sharpe approaches the problem of capturing the essence of mean-variance portfolio selection in a linear-programming formulation by the following: 1. Making a diagonal transformation of the variables that will convert the problem into a diagonal form, and 2. Using a piecewise linear approximation for each of the terms for variance. The LP that results from the use of market responsiveness as the risk measure and the imposition of an upper limit on investment in each security is Max P D 

" n X i D1

# xi E.Ri /  .1  /

"

n X

# xi ˇi

i D1

(7A.1)

7 Index Models for Portfolio Selection

subject to:

n P

123

a reasonable linear approximation to the first term in the objective function is provided by:

xi D 1

i D1

!2 " n # n X 1 1 X xi ˇi  2 .Rm / D  2 .Rm / xi ˇi K i D1 K i D1

0  xi  U

where: xi D the fraction of the portfolio invested in security i ; E .Ri / D the expected returns of security i ; ˇi D the beta coefficient of security i ; U D the maximum fraction of the portfolio that may be held in any one security; and  D a parameter reflecting the degree of risk aversion. The  is used in generating the efficient frontier. The value of  corresponds to the rate of substitution of return for risk measured by the individual security ˇ, while the U maximum percentage to be invested in any security greatly reduces the number of securities that will be needed for diversification. Building on this framework, Jacob derives a LP model that allows the small investor explicit control over the number of securities held. Jacob’s LP model incorporates the effects of unsystematic risk as well as systematic risk, the beta in the Sharpe LP. Jacob’s model is to minimize Systematic risk   1  2 Rp D K

Unsystematic risk n X

!2 xi ˇi

 2 .Rm / C

i D1

1 K

2 X n

xi2  2 .ei /

i D1

After division by K and rearrangement of terms, the objective function can be restated:   1 2 2 ZD xi ˇi  .Rm / C  .ei / K i D1 n X

Letting Zi D ˇi  2 .Rm / C .1=K/ 2 .ei /, the problem can be cast as: Minimize Z D 

n X

Xi Zi  .1  /

i D1

n X

Xi E.Ri /

i D1

subject to: n X

Xi D 1:0

0  Xi  1=K

i D 1; 2; : : : ; N

i D1

in which  is varied from zero to one. This problem can be solved by the linear-programming approach suggested by Sharpe (1967). An example of solving this problem can be found in Jacob (1974).

(7A.2) subject to: 1 K 1 K

n P i D1 n P

  xi E .Ri /  E  Rp xi D K

xi D 0 or 1

i D1

where: xi D the investment in security i – all, or nothing; E .Ri / D the expected return on security i equal to ˛i C ˇi Rm K D the desired upper bound on the number of securities the investor is willing to consider, usually in the range of 15–20; ˇi D the measure of systematic risk;  2 .e  i/ D  the measure of unsystematic risk; and  E Rp D the lowest acceptable rate of return the investor is willing to earn on his or her portfolio. An additional simplification is to turn the objective function of Equation (7A.2) into a linear relationship. Since the decision variables .xi / are binary valued, the unsystematicrisk term of the objective

is already linear. Given n function P xi ˇi =K is very close to unity, that the portfolio beta i D1

Appendix 7B Expected Return, Variance, and Covariance for a Multi-index Model Using the orthogonalization technique discussed in the text, the multi-index model (MIM) is transformed into Ri D ai C bi1 I1 C bi 2 I2 C    C biL IL C ci

(7B.1)

in which all Ij are uncorrelated with each other. To interpret the transformed indexes, notice that I2 is now the difference between the actual level of the index and the level it would be, given the level of the other indexes. Also, bi 2 is now the sensitivity of security i ’s return to a change in I2 , given that all other indexes are held constant. It is also convenient, beyond making the indexes uncorrelated, to assume that the covariance of the residuals with the indexes is equal to zero. With this final assumption, the MIM can be recapped as follows. Generalized equation: Ri D ai C bi1 I1 C bi 2 I2 C    C biL IL C ci

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for all securities i D 1 to N , and:

Remembering that:

E.ci / D 0 Var .ci / D ci2 Var .Ij / D Ij2 Cov .Ij ; Ik / D EŒ.Ij  I j /.Ik  I k / D 0; I j D E.Ij /; I ci D E.Ici / 5. Cov.ci ; Ij / D EŒci .Ij  I j / D 0 6. Cov.ci ; cj / D E.ci ; cj / D 0 for all i, j, and k.

1. 2. 3. 4.

The last statement is equivalent to the residuals being unrelated, which is to say that the only reason for common comovement of returns is due to the movement of the indexes. Although this is like ignoring some of a model’s shortcomings, it facilitates a usable model to obtain a quantified idea of future return patterns. The expected value of the model with the multiple indexes is

E



Ii  I i

     Ij  I j D 0 and E Ii  I i ci D 0

the only nonzero term involving index one is expressed: 2 2 2 E.I1  I 1 /2 D bi1 I1 bi1

Because all terms involving ci are zero and E.ci /2 D ci2 : 2 2 2 2 I1 C bi22 I22 C    C biL IL C ci2 i2 D bi1

(7B.3)

The covariance of the returns between security i and j utilizing the multiple-index model can be expressed:    ij D E .Ri  ri / Rj  rj Again, substituting for Ri ; ri ; Rj , and rj :

E.Ri / D E.ai C bi1 I1 C bi 2 I2 C    C biL IL C ci / D E.ai / C E.bi1 I1 / C E.bi 2 I2 / C   

ij DEŒŒ.ai C bi1 I1 C bi 2 I2 C    C biL IL C ci /  .ai C bi1 I 1 C bi 2 I 2 C    C biL I L / 

C E.biL IL / C E.ci /

Œ.aj C bj1 I1 C bj 2 I2 C    C bjL IL C cj / because a and the b’s are constants, and the expected value of the residuals is equal to zero ri D ai C bi1 I 1 C bi 2 I 2 C    C biL I L

(7B.2)

where I j is the expected value of index j . The variance of the returns using multiple indexes is:

 .aj C bj1 I 1 C bj 2 I 2 C    C bjL I L / Again, noting that the ai cancel and combining the terms involving the same b’s: ij D EfŒbi1 .I1  I 1 / C bi 2 .I2  I 2 / C    C biL .IL  I L / C ci

i2 D E.Ri  ri /2 where ri is the expected value of the returns of security i . Substituting Ri and ri from above: i2

DEŒ.ai C bi1 I1 C bi 2 I2 C    C biL IL C ci /    ai C bi1 I 1 C bi 2 I 2 C    C biL I L 2

Rearranging, and noticing that the ai cancel, yields:       i2 D EŒbi1 I1  I 1 C bi2 I2  I 2 C    C biL IL  I L C ci 2

Next the terms in the brackets are squared. To proceed with this, concentrate on the first index, and the rest of the terms involving the other indexes follow directly. The first index times itself and all other terms yields  2    2 I1  I 1 C bi1 bi 2 I1  I 1 I2  I 2 C    EŒbi1      Cbi1 biL I1  I 1 IL  I L C bi1 I1  I 1 .ci /

Œbj1 .I1  I 1 / C bj 2 .I2  I 2 / C    C bjL .IL  I L / C cj g Next multiply out terms, again concentrating on the terms involving bi1 : EŒbi1 bj1 .I1  I 1 /2 C bi1 bi 2 .I1  I 1 /.I2  I 2 / C    C bi1 bjL .I1  I 1 /.IL  I L / C bi1 .I1  I 1 /cj Because the covariance between two indexes is zero, and the covariance between any residual and an index is zero: 2 bi1 bj1 E.I1  I 1 /2 D bi1 bj1 I1

To conclude, remember that the covariance of the residuals is equal to zero and therefore: 2 2 ij D bi1 bj1 I1 C bi 2 bj 2 I22 C    C biL bjL IL

(7B.4)

Chapter 8

Performance-Measure Approaches for Selecting Optimum Portfolios Cheng-Few Lee, Hong-Yi Chen, and Jessica Shin-Ying Mai

Abstract In this chapter, following Elton et al. (Journal of Finance 31:1341–57, 1976; Modern portfolio theory and investment analysis, 7th edn. Wiley, New York, 2006), we introduce the performance-measure approaches to determine optimal portfolios. We find that the performance-measure approaches for optimal portfolio selection are complementary to the Markowitz full variance-covariance method and the Sharpe index-model method. The economic rationale of the Treynor method is also discussed in detail. Keywords Sharpe performance measure approach r Lintner’s method of short sales r Treynor measure r Short sales r Standard method of short sales

8.1 Introduction Previously, we have discussed Markowitz’s (1952, 1959) full variance-covariance approach to determine optimal weights of a portfolio. Moreover, we also utilized Sharpe’s indexmodel approach to simplify Markowitz’s optimal portfolioselection process. This chapter assumes the existence of a risk-free borrowing and lending rate and advances one step further to simplify the calculation of the optimal weights of a portfolio and the efficient frontier. First discussed are Lintner’s (1965) and Elton et al. (1976) Sharpe performancemeasure approaches for determining the efficient frontier with short sales allowed. This is followed by a discussion of the Treynor performance-measure approach for determining the efficient frontier with short sales allowed. The Treynor measure approach is then analyzed for determining the efficient frontier with short sales not allowed. And finally, Dow Jones 30 data from January 2003 through December 2007 are employed to demonstrate how the Treynor method can be applied in the real world. Overall, this chapter

C.-F. Lee, H.-Y. Chen, and J.S.-Y. Mai () Rutgers University, New Brunswick, NJ, USA e-mail: [email protected]

relates the performance-measure concepts and methods to the portfolio-selection models, and make more accessible the insights of optimal portfolio selection.

8.2 Sharpe Performance-Measure Approach with Short Sales Allowed In deriving the capital asset pricing model (CAPM), Lintner (1965) suggests a performance-measure approach for determining the efficient frontier discussed in previous chapters. Lintner arrived at this approach through a sequence of logical steps. Following previous chapters, the objective function for portfolio selection can be expressed: Max L D

n X

W i R i C 1

i D1

82 9 31=2 ˆ > n n X < X =   4 Cov Ri ; Rj 5  p ˆ > : j D1 i D1 ; C2

n X

! Wi  1

(8.1)

i D1

where: Ri D average rates of return for security i ; Wi (or  the optimal weight for i th (or j th) security;  Wj / D Cov Ri ; Rj D the covariance between Ri and Rj ; p D the standard deviation of a portfolio; and 1 ; 2 D Lagrangian multipliers. Markowitz’s portfolio selection model minimizes the variance given the targeted expected rate of return. Equation (8.1) maximizes the expected rates of return given targeted standard deviation. If a constant risk-free borrowing and lending rate Rf is subtracted from Equation (8.1):

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_8, 

125

126

C.-F. Lee et al.

Max L0 D

n X 

 Wi Ri  Rf C 1

i D1

82 9 31=2 ˆ > n n X < X =   5 4 Wi Wj Cov Ri ; Rj  p ˆ > : i D1 j D1 ; C2

n X

! Wi  1

(8.2)

i D1

Equations (8.1) and (8.2), both formulated as a constrained maximization problem, can be used to obtain optimum portfolio weights Wi .i D 1; 2;    ; n/. Since Rf is a constant, the optimum weights obtained from Equation (8.1) will be equal to those obtained for Equation (8.2). Previous chapters used the methodology of Lagrangian multipliers; it can be shown that Equation (8.2) can be replaced by a nonconstrained maximization method as follows. Incorporating n P Wi  1 into the objective function by the constant i D1

substituting n X

Rf D .1/ Rf D

! Wi Rf D

i D1

n X

Wi Rf

i D1

into Equation (8.2): Max L0 D

n X

  Wi Ri  Rf C 1

i D1

20 11=2 3 n n   6@X X 7 Wi Wj Cov Ri ; Rj A p 5 (8.2a) 4 i D1 j D1

A two-Lagrangian multiplier problem has been reduced to a one-Lagrangian problem as indicated in Equation (8.2a). By using a special property of the relationship between !1=2 n  n P n    P P Ri  Rf and Wi Wj Cov Ri ; Rj the i D1

i D1 j D1

constrained optimization of Equation (8.2a) can be reduced to an unconstrained optimization problem, as indicated in Equation (8.3).1 n P



Wi Ri  Rf



i D1

Max L D n P i D1

Wi2 i2 C

n P n P

Fig. 8.1 Linear efficient frontier

  where ij D Cov Ri ; Rj . Alternatively, the objective function of Equation (8.3) can be developed as follows. This ratio L is equal to excess average rates of return for the i th portfolio divided by the standard deviation of the i th portfolio. This is a Sharpe performance measure. Following Sharpe (1964) and Lintner if there is   (1965), a risk-free lending and borrowing rate Rf and short sales are allowed, then the efficient frontier (efficient set) will be linear,   as discussed in previous chapters. In terms of return Rp standard-deviation p space, this linear efficient frontier is indicated as line Rf E in Fig. 8.1. AEC represents a feasible investment opportunity in terms of existing securities to be included in the portfolio when there is no risk-free lending and borrowing rate. If there is a risk-free lending and borrowing rate, then the efficient frontier becomes Rf E. An infinite number of linear lines represent the combination of a riskless asset and risky portfolio, such as Rf A; Rf B and Rf E. It is obvious that line Rf E has the highest slope, as represented by Rp  Rf (8.4) ‚D p in which Rp D

Wi Ri ; Rf , and p are defined as in

i D1

Equation (8.2). Thus the efficient set is obtained by maximizn P Wi D 1, Equation (8.4) ing ‚. By imposing the constraint i D1

is expressed:

! n X Rp  Rf C Wi  1 ‚ D p i D1 0

!1=2

.i ¤ j /

Wi Wj ij

i D1 j D1

Since the ratio of Equation (8.3) is homogeneous of degree zero with respect to Wi . In other words, the ratio L is unchanged by any proportionate change in the weight of Wi .

(8.5)

By using the procedure of deriving Equations (8.2a), (8.5) becomes n P

(8.3) 1

n P

‚0 D

  Wi Ri  Rf

i D1

n P

i D1

Wi2 i2

C

n P n P i D1 i D1

.i ¤ j / ij

(8.5a)

8 Performance-Measure Approaches for Selecting Optimum Portfolios

This equation is equivalent to Equation (8.3). This approach is used by Elton and Gruber (1987) to derive their objective function for optimal portfolio selection.2 Following the maximization procedure discussed earlier in previous chapters, it is clear that there are n unknowns to be solved in either Equation (8.3) or Equation (8.5a). Therefore, calculus must be employed to compute n firstorder conditions to formulate a system of n simultaneous equations: 1

dL D0 dW 1

2

dL D0 dW 2

127

Example 8.1. Let R1 D 15% R2 D 12% R3 D 20% 3 D 9% 1 D 8% 2 D 7% r13 D 0:4 r23 D 0:2 r12 D 0:5 Rf D 8% Substituting this information into Equation (8.6a): 15  8 D 64H1 C .0:5/ .8/ .7/ H2 C .0:4/ .8/ .9/ H3 12  8 D .0:5/ .8/ .7/ H1 C 49H2 C .0:2/ .7/ .9/ H3 20  8 D .0:4/ .8/ .9/ H1 C .0:2/ .7/ .9/ H2 C 81H3 Simplifying:

:: :

:: :

7 D 64H1 C 28H2 C 28:8H3

n

dL D0 dW n

4 D 28H1 C 49H2 C 12:6H3 12 D 28:8H1 C 12:6H2 C 81H3

From Appendix 8A, the n simultaneous equations used to solve Hi are R1  Rf D C H2 12 C H3 13 C    C Hn 1n R2  Rf D H1 12 C H2 22 C H3 23 C    C Hn 2n :: :

Using Cramer’s rule, H1 ; H2 , and H3 can be solved for as follows:

H1 12

Rm  Rf D H1 1n C H2 2n C H3 3n C    C Hn n2 (8.6) where:

H1 D

Rp Rf p2

.

The HS are proportional to the optimum portfolio weight Wi .i D 1; 2; : : : ; n/ by a constant factor K. To determine the optimum weight Wi ; Hi is first solved from the set of equations indicated in Equation (8.6). Having done so the Hi must be called to calculate Wi , as indicated in Equation (8.7). Wi D

Hi n P

ˇ ˇ ˇ 7 ˇ ˇ ˇ 4 ˇ ˇ ˇ ˇ 12

R1  Rf D H1 12 C H2 12 C H3 13 R3  Rf D H1 13 C H2 23 C H3 32 Example 8.1 provides further illustration. 2

Elton et al. (2006).

ˇ

49 12:6 ˇˇ

12:6 81

ˇ ˇ ˇ

D

33648:1  27:1177 6350:4 D D 3:97% 160030 160030 ˇ ˇ ˇ 64 7 28:8 ˇ ˇ ˇ ˇ ˇ ˇ 28 4 12:6 ˇ ˇ ˇ ˇ 28:8 12 81 ˇ

H2 D ˇ ˇ ˇ 64 28 28:8 ˇ ˇ ˇ ˇ ˇ ˇ 28 49 12:6 ˇ ˇ ˇ ˇ 28:8 12:6 81 ˇ D

.2540:2 C 9676:6 C 20736/  .9676:8 C 15876 C 3317:8/ 160030

D

32952:8  28870:6 D 2:55% 160030

H3 D

R2  Rf D H1 12 C H2 22 C H3 23

ˇ ˇ

28 28:8 ˇˇ

.4233:6 C 1451:5 C 27783/  .1111:3 C 9072 C 16934:4/ .10160:6 C 10160:6 C 254016/  .10160:6 C 63504 C 40642:6/

i D1

If there are only three securities, then Equation (8.6) reduces to:

ˇ

49 12:6 ˇˇ ˇ 12:6 81 ˇˇ

ˇ ˇ ˇ 64 ˇ ˇ ˇ 28 ˇ ˇ ˇ ˇ 28:8

(8.7)

Hi

ˇ ˇ

28 28:8 ˇˇ

D

Hi D kW i .i D 1; 2; : : : ; n/; and kD

(8.6b)

(8.6a)

ˇ ˇ ˇ 64 ˇ ˇ ˇ 28 ˇ ˇ ˇ ˇ 28:8 ˇ ˇ ˇ ˇ 64 ˇ ˇ 28 ˇ ˇ ˇ ˇ 28:8

ˇ ˇ

28 28:8 ˇˇ ˇ 49 12:6 ˇˇ ˇ ˇ ˇ ˇ ˇ 28:8 ˇˇ ˇ 12:6 ˇˇ ˇ 81 ˇˇ

12:6 81 28 49 12:6

.3225:6 C 2469:6 C 37632/  .3225:6 C 9480 C 9878:4/ 160030 20815:2 D D 13:01% 160030

D

128

C.-F. Lee et al.

Using Equation (8.7), W1 ; W2 , and W3 are obtained: W1 D

H1 3 P

D

Hi

3:97 3:97 D 3:97 C 2:55 C 13:01 18:53

The efficient frontier for this example is shown in the figure. A represents an efficient portfolio with Rp D 17:94 percent and p2 D 50:90 percent.

i D1

D 20:33% H2 2:55 W2 D 3 D P 19:53 Hi

8.3 Treynor-Measure Approach with Short Sales Allowed

i D1

Using the single-index market model discussed in previous chapter, Elton et al. (1976) define:

D 13:06% H3 13:01 W3 D 3 D P 19:53 Hi

p D

i D1

n P iD1

Wi2 ˇi2 m2

C

D 66:61%

n P n P iD1 j D1

Wi Wj ˇi ˇj m2

C

n P iD1

!1=2 2 Wi2 2i

j ¤i

Rp and p2 can be calculated by employing these weights:

Substituting of this value of p into Equation (8.3):

Rp D .15/ .0:2033/ C .12/ .0:1306/ C .20/ .0:0061/ D 3:049 C 1:5672 C 13:322

n P iD1

LD

D 17:9382%

  Wi Ri  Rf

p D

n P iD1

Wi2 ˇi2 m2

C

n n P P iD1 j D1

Wi Wj ˇi ˇj m2

C

n P iD1

!1=2 2 Wi2 2i

j ¤i

p2

D

3 X

Wi2 i2

i D1

C

3 3 X X

(8.8)

Wi Wj rij i j

i ¤j

i D1 j D1

D .0:2033/2 .64/ C .0:1306/2 .49/ C .0:6661/2 .81/

In order to find the set of Wi s that maximizes L, take the derivative of the above equation with respect to each Wi . Let

C2 .0:2033/ .0:1306/ .0:5/ .8/ .7/

Hi D

C2 .0:2033/ .0:6661/ .0:4/ .8/ .9/ C2 .0:2/ .0:1306/ .0:6661/ .7/ .9/ D 2:645 C 0:836 C 35:939 C 1:487 C 7:8 C 2:192 D 50:899%

Rp  Rf p2

! Wi

  since Rp  Rf =p2 is a constant factor for each security to be included in the portfolio. Hence, it can be cancelled by using a standard scaling method: Wi D

Efficient frontier for Example 8.1

Hi n P Hi

(8.9)

i D1

From Appendix 8B Hi can be obtained as follows: 1 n P .Rj Rf / 2  2 B m j D1 2i C ˇ 2

C B i B C i ˇ2 A  2 @ P j 2i 1 C m2 2 0

Hi D

  Ri  Rf 2 2i

j D1

(8.10)

ej

Equation (8.10) can be modified to: ˇi Hi D 2 2i

Ri  Rf  C ˇi

! (8.11)

8 Performance-Measure Approaches for Selecting Optimum Portfolios

 ˇ in which Ri  Rf ˇ ˇi is the Treynor performance measure. C  can be defined as m2 C D

i P .Ri Rf /ˇi j D1

1C

m2

2 2j

i P

ˇj2

j D1

2 2j

(8.12)

The Hi s must be calculated for all of the stocks in the portfolio. If Hi is a positive value, this indicates the stock will be held long, whereas a negative value indicates that the stock should be sold short. This method is called the Treynor measure approach. The argument will be clearer when the case of portfolio selection with short sales not allowed is discussed. To determine the optimum portfolio from the Hi s (such that 100% of funds are invested) the weights must be scaled. One method follows the standard definition of short sales, which presumes that a short sale of stock is a source of funds to the investor; it is called the standard method of short sales. This standard scaling method is indicated in Equation (8.9). In Equation (8.9), Hi can be positive or negative. This scaling factor includes a definition of short sales and the constraint: n X

jWi j D 1

i D1

(1) Security number

(2) Ri

(3) Ri  Rf

(4) “i

(5) 2 2i

1 2 3 4 5

15 13 10 9 7

10 8 5 4 2

1 2 1.43 1.33 1

30 50 20 10 30

From the information in the table, using Equations (8.10) and (8.11). Hi .i D 1; 2; : : : ; 5/ can be calculated: H1 D

1 30



 3:067 D 0:2311

13  5  3:067 D 0:0373 2



10  5 1:43 H3 D  3:067 D 0:0307 20 1:43



95 1:33 H4 D  3:067 D 0:0079 10 1:33

75 1 H5 D  3:067 D 0:0356 30 1

H2 D

2 50



15  5 1

According to Lintner’s method:

A second method (Lintner’s (1965) method of short sales) assumes that the proceeds of short sales are not available to the investor and that the investor must put up an amount of funds equal to the proceeds of the short sale. The additional amount of funds serves as collateral to protect against adverse price movements. Under these assumptions, the constraints on the Wi s can be expressed: n X

129

5 X

Now to scale the Hi values into an optimum portfolio, apply Equation (8.13): W1 D

jWi j D 1

W2 D

i D1

And the scaling factor is expressed as: Wi D

Hi n P jHi j

W3 D (8.13)

W4 D

i D1

W5 D

Example 8.2 provides further illustration. Example 8.2. The following example shows the differences in security weights in the optimal portfolio due to the differing short-sale assumptions. Data associated with regressions of the single-index model are presented in the table. The mean return, R, the excess return Ri Rf , the beta coef2 are presented ficient ˇi , and the variance of the error term 2i in columns 2–5.

jHi j D 0:3426

i D1

0:2311 D 0:6745 0:3426 0:0373 D 0:1089 0:3426 0:0307 D 0:0896 0:3426 0:0079 D 0:0231 0:3426 0:0356 D 0:1039 0:3426

Thus, the Lintner model states that an investor should invest 67.45% in security 1, 10.89% in security 2, and 8.96% in security 3. The investor should then sell short 2.31 and 10.39% of securities 4 and 5, respectively. If this same example using the standard defini 5 is scaled

P tion of short sales Hi , which provides funds to the i D1

investor:

130

C.-F. Lee et al. 5 X

where:

Hi D 0:2556

m2

i D1

W1 D W2 D W3 D W4 D W5 D

Ci

0:2311 D 0:9041 0:2556 0:0373 D 0:1459 0:2556 0:0307 D 0:1201 0:2556 0:0079 D 0:0309 0:2556 0:0356 D 0:1393 0:2556

i P

ˇj2

j D1

2 2j

(8.15)

Example 8.3 provides further illustration.

8.4 Treynor-Measure Approach with Short Sales Not Allowed Elton et al. (1976) also derive a Treynor-measure approach with short sales not allowed. From Appendix 8C, Equation (8.11) should be modified to: !

Ri  Rf  Ci C i ˇi

ˇj

If all securities have positive ˇi s, the following three-step procedure from Elton et al. can be used to choose securities to be included in the optimum portfolio.3 ı  1. Use the Treynor performance measure Ri  Rf ˇi to rank the securities in descending order. 2. Use Equation (8.15) to calculate Ci for all securities. 3. Include in ı the portfolio all securities for which  Ri  Rf ˇi is larger than Ci .

Example 8.3. The tape showing Center for Research in Security Prices was the source of 5 years of monthly return data, from January 2003 to December 2007, for the 30 stocks in the Dow Jones Industrial Averages. The value-weighted average of the NYSE index was used as the market while 3-month Treasury-bill rates were used as the risk-free rate. The single-index model was used with an ordinary leastsquares regression procedure to determine each stock’s beta. The following data were compiled for each stock. 1. 2. 3. 4. 5.

The mean monthly return Ri The mean excess return Ri  Rf The beta coefficient The variance of the residual errors  ı The Treynor performance measure Ri  Rf ˇi

All data are listed in the worksheet on next page, which lists the companies in descending order of the Treynor performance measure.4 To calculate the Ci as defined in Equation (8.15), cali h  . 2  . 2 i  P , culate Rj  Rf ˇj 2j ; Rj  Rf ˇj 2j

(8.14)

j D1

.

and

2 ˇj2 2j :

presented

ˇi 2i

j D1

Rj Rf 2 2j

1 C m2

Using this definition of short sales provides that the investor should invest 90.41% of his or her money in security 1, and so on. If all Wi s are added together, they equal 100%. This is true because the definition says that the funds received from selling short a security should be used to purchase more of the other securities. The difference between Lintner’s method and the standard method are due to the different definitions of short selling discussed earlier. The standard method assumes that the investor has the proceeds of the short sale, while Lintner’s method assumes that the short seller does not receive the proceeds and must provide funds as collateral. The method discussed in this section does not require the inputs of covariance among individual securities. Hence, it is a simpler method than that of the Sharpe performancemeasure method discussed in the previous section. The relative advantage of the Sharpe performance method over the Treynor performance method is exactly identical to the relative advantage of the Markowitz model over the Sharpe single-index model. In sum, the Treynor performance method for portfolio selection requires the information of both the risk-free rate and the market rates of return.

Hi D

D

i P

0:00207;

i P

. i P 2 ˇj2 2j

is

calculated

and

j D1

in the worksheet. Substituting m2 D h .

i  . 2 i P 2 Rj  Rf ˇj 2j , and ˇj2 2j

j D1

j D1

into Equation (8.15) produces Ci for every firm as listed in the last column in the worksheet. Using company HPQ as an example: 3 If the beta coefficient ˇi for i th security is positive, then the size of Hi depends on the sign of the term ı in parentheses. Therefore, if a security with a particular Ri  Rf ˇi is included in the optimum portfolio, all ı securities with a positive beta that have higher values of  R i  Rf ˇi must be included in the optimum portfolio. 4 This set of data has been analyzed in later chapters. The names of these 30 firms can be found in Table 19.1.

XOM MCD JNJ BA T CAT HPO CVX GE DIS KO JPM INTC AA UTX IBM AXP HD MRK VZ DD KFT GM MSFT C PG MMM WMT PFE BAC Rm Var.Rm / Rf

Ticker

Mean Ri 0:0185 0:0258 0:0043 0:0193 0:0115 0:0128 0:0202 0:0101 0:0088 0:0117 0:0080 0:0120 0:0121 0:0132 0:0077 0:0068 0:0076 0:0063 0:0038 0:0035 0:0038 0:0016 0:0004 0:0000 0:0010 0:0013 0:0017 0:0009 0:0035 0:0054 0:00887 0:002433333 0:000615571

0.0067 0.0073 0.0045 0.0080 0.0083 0.0113 0.0084 0.0102 0.0052 0.0067 0.0056 0.0070 0.0104 0.0098 0.0106 0.0066 0.0061 0.0083 0.0100 0.0066 0.0066 0.0064 0.0144 0.0114 0.0073 0.0099 0.0111 0.0059 0.0070 0.0095

Standard deviation i

Worksheet for Dow Jones industrial averages

R i  Rf 0:0160 0:0234 0:0018 0:0168 0:0091 0:0104 0:0178 0:0077 0:0064 0:0093 0:0056 0:0096 0:0096 0:0107 0:0053 0:0043 0:0051 0:0038 0:0014 0:0011 0:0014 0:0008 0:0028 0:0025 0:0035 0:0011 0:0042 0:0015 0:0059 0:0079

(1)

0.7905 1.3600 0.1178 1.2084 0.7139 0.8210 1.4747 0.7931 0.7268 1.0829 0.7845 1.3579 1.6463 1.9542 1.1547 1.1726 1.4069 1.2612 0.6459 0.8905 1.1670 0.7377 1.6717 1.0800 1.2265 0.3752 0.9803 0.2021 0.7079 0.7140

ˇi

(2)

0.00228 0.00204 0.00118 0.00299 0.00370 0.00720 0.00288 0.00587 0.00130 0.00197 0.00152 0.00183 0.00482 0.00341 0.00591 0.00177 0.00103 0.00319 0.00572 0.00217 0.00176 0.00213 0.01076 0.00706 0.00226 0.00574 0.00682 0.00206 0.00264 0.00506

2 2i

(3)

(5) .Ri Rf /ˇi ei2

5:5603 15:5812 0:1837 6:8054 1:7527 1:1; 863 9:0985 1:0377 3:5539 5:0882 2:8695 7:1293 3:2879 6:1519 1:0314 2:8744 6:9941 1:5228 0:1537 0:4538 0:9278 0:2825 0:4414 0:3761 1:8779 0:0729 0:6010 0:1498 1:5809 1:1104

(4) .Ri Rf / ˇi

0:0203 0:0172 0:0157 0:0139 0:0127 0:0127 0:0121 0:0097 0:0087 0:0086 0:0071 0:0071 0:0058 0:0055 0:0046 0:0037 0:0037 0:0031 0:0021 0:0012 0:0012 0:0011 0:0017 0:0023 0:0028 0:0030 0:0043 0:0076 0:0083 0:0110

5:5603 21:1416 21:3253 28:1307 29:8834 31:0697 40:1682 41:2059 44:7598 49:8480 52:7175 59:8467 63:1346 69:2865 70:3180 73:1924 80:1865 81:7093 81:8631 82:3169 83:2447 82:9622 82:5208 82:1447 80:2668 80:1939 79:5929 79:4430 77:8622 76:7518

(6) Cumulative sum of column (5) 273:92 905:36 11:73 488:69 137:56 93:62 754:92 107:18 406:76 593:91 403:88 1; 008:07 562:52 1; 120:50 225:46 776:21 1; 913:36 499:04 72:94 366:18 775:66 255:97 259:71 165:15 664:60 24:53 140:89 19:79 189:74 100:70

ˇi2 ei2

(7)

0.002929 0.007540 0.007574 0.008514 0.008683 0.008788 0.009362 0.009370 0.009317 0.009234 0.009086 0.008788 0.008563 0.008158 0.008065 0.007709 0.007029 0.006862 0.006833 0.006667 0.006344 0.006201 0.006051 0.005951 0.005548 0.005534 0.005439 0.005422 0.005246 0.005136 0.009370

Max C1

(9) C1

273:92 1; 179:29 1; 191:02 1; 679:71 1; 817:28 1; 910:90 2; 665:82 2; 773:00 3; 179:75 3; 773:66 4; 177:53 5; 185:61 5; 748:12 6; 868:63 7; 094:09 7; 870:31 9; 783:66 10; 282:71 1; 0355:65 1; 0721:83 1; 1497:49 1; 1753:45 1; 2013:16 1; 2178:30 1; 2842:91 1; 2867:43 1; 3008:32 13; 028:81 13; 217:85 13; 318:55

(8) Cumulative sum of column (7)

8 Performance-Measure Approaches for Selecting Optimum Portfolios 131

132

C.-F. Lee et al. Positive optimum weight for eight securities

 CHPQ D

Ticker

Hi

XOM MCD JNJ BA T CAT HPQ CVX Total C D

6.0189 6.4370 0.8050 2.1887 0.7820 0.4428 1.3769 0.0421 18.0935 0.009370

ˇi ="i2

.R i  Rf / ˇi

(A) optimum percentage

(B) mean

.A/  .B/

346.5158 665.6941 99.5753 404.4066 192.6984 114.0371 511.8963 135.1436

0.0203 0.0172 0.0157 0.0139 0.0127 0.0127 0.0121 0.0097

0.3327 0.3558 0.0445 0.1210 0.0432 0.0245 0.0761 0.0023 1.0000

0.0185 0.0258 0.0043 0.0193 0.0115 0.0128 0.0202 0.0101 Rp

0.00615 0.00919 0.00019 0.00233 0.00050 0.00031 0.00154 0.00002 0.02023

.0:0006155/ .40:6817114/ D 0:009362497 1 C .0:0006155/ .265:818464/

From Ci of the worksheet it is clear that there are ten securities included in the portfolio. The estimated ı 2thatshould beı ; Ri  Rf ˇi and Ci of these ten securities are ˇi 2i listed in the table at the top of this page. Substituting this information into Equation (8.14) produces Hi for all ten securities. Using security 5 as an example: HXOM D .346:5158/.0:0202988  0:009370285/ D 6:018948255 Using Equation (8.12) the optimum weights can be estimated for all ten securities, as indicated in the table. In other words, 33.27% of our fund should be invested in security XOM, 35.58% in security MCD, 4.45% in security JNJ, 12.10% in security BA, 4.32% in security T, 2.45% in security CAT, 7.61% in security HPQ, 0.23% in security CVX, based on the optimal weights, the average rate for the portfolio Rp is calculated as 2.02%, as presented in the last column of the table above.

8.5 Impact of Short Sales on Optimal-Weight Determination The Markowitz model of portfolio analysis. The Markowitz model requires a large number of inputs, as it is necessary to estimate the covariance between each pair of securities. Previously, the analysis was simplified by the assumption that security returns were related through a common response to some market index. This model, known as Sharpe’s

single-index model, greatly reduces the number of inputs necessary to analyze the risk and return characteristics of portfolios. In both the Markowitz and Sharpe models the analysis is facilitated by the presence of short selling. This chapter discusses a method proposed by Elton and Gruber for the selection of optimal portfolios. Their method involves ranking securities based on their excess return to beta ratio, and choosing all securities with a ratio greater than some particular cutoff level C  . It is interesting to note that while the presence of short selling facilitated the selection of the optimum portfolio in both the Markowitz and Sharpe models, it complicates the analysis when we use the Elton and Gruber approach. From Example 8.3, using the Dow Jones Industrial Average (DJIA), the absence of the short selling allowed formation of the optimal portfolio using only eight stocks.

8.6 Economic Rationale of the Treynor Performance-Measure Method Cheung and Kwan (1988) have derived an alternative simple rate of optimal portfolio selection in terms of the singleindex model. First, Cheung and Kwan relate Ci as defined in Equation (8.12) to the correlation coefficient between the portfolio rates of return with i securities Ri and market rates of return Rm .i /: Ci i D (8.16) m ‚i where: i D i m =i m I i m D covariance between Ri and Rm I ı  ‚i D Ri  Rf i ; the Sharpe performance

8 Performance-Measure Approaches for Selecting Optimum Portfolios

measure associatedwith the i th portfolioI and i and m D standard deviation for ith portfolio and market portfolio, respectively. Cheung and Kwan show that Pi and Ci display the same functional behavior for optimal portfolio selection. In other words, if portfolios are formed by adding securities successively from the highest rank to the lowest rank, the optimal portfolio is reached when the correlation of the portfolio returns and the index is at its maximum. Since the expected return on the index must be positive if the investor is to invest in stocks, an objective of the investor using the single-index model to establish the risk-return tradeoff is to pick securities that benefit the most from a market upswing. Hence, the role of index in the selection of securities for an optimum portfolio is demonstrated explicitly. Based on the single-index model and the risk decomposition discussed in previous chapters, the following relationships can be defined: 1: 2: 3:

i D

i m i m

i m D ˇi m2 2 i2 D ˇi2 m2 C 2i

(8.17)

From Equation (8.17), Cheung and Kwan define i in terms of ˇi2 ; m2 , and i . i D

q

ı

2 ˇi2 m2 i2 C 2i D

s 1

2 2i i2

2 in which 2i is the nonsystematic risk for the i th portfolio. They use both i and ‚i to select securities for an optimum portfolio, and they conclude that i can be used to replace ‚i in selecting securities for an optimum portfolio. Nevertheless, ‚i information is still needed to calculate the weights for each security. Hence, Cheung and Kwan’s i criteria is good only for understanding Elton and Gruber’s performance-measure method for portfolio selection.

8.7 Conclusion Following Elton et al. (1976) and Elton and Gruber (1987) we have discussed the performance-measure approaches to selecting optimal portfolios. We have shown that the performance-measure approaches for optimal portfolio selection are complementary to the Markowitz full variancecovariance method and the Sharpe index-model method.

133

These performance-measure approaches are thus worthwhile for students of finance to study following an investigation of the Markowitz variance-covariance method and Sharpe’s index approach.

References Alexander, G. J. and B. J. Resnick. 1985. “More on estimation risk and single rule for optimal portfolio selection.” Journal of Finance 40, 125–134. Bodie, Z., A. Kane, and A. Marcus. 2006. Investments, 7th Edition, McGraw-Hill, New York. Chen, S. N. and S. J. Brown. 1983. “Estimation risk and simple rules for optimal portfolio selection.” Journal of Finance 38, 1087–1093. Cheung, C. S. and C. C. Y. Kwan. 1988. “A note on simple criteria for optimal portfolio selection.” Journal of Finance 43, 241–245. Elton, E. J., M. J. Gruber, and M. W. Padberg. 1976. “Simple criteria for optimal portfolio selection.” Journal of Finance 31, 1341–1357. Elton, E. J., M. J. Gruber, S. J. Brown, and W. N. Goetzmann. 2006. Modern portfolio theory and investment analysis, 7th Edition, Wiley, New York. Kwan, C. C. Y. 1984. “Portfolio analysis using single index, multiindex, and constant correlation models: a unified treatment.” Journal of Finance 39, 1469–1483. Lee, C. F. and A. C. Lee. 2006. Encyclopedia of finance. Springer, New York. Lee, C. F., J. C. Lee, and A. C. Lee. 2000. Statistics for business and financial economics, World Scientific Publishing Co, Singapore. Lintner, J. 1965. “The valuation of risk asset on the selection of risky investments in stock portfolio and capital budgets.” The Review of Economics and Statistics 57, 13–37. Maginn, J. L., D. L. Tuttle, J. E. Pinto, and D. W. McLeavey. 2007. Managing investment portfolios: a dynamic process, CFA Institute Investment Series, 3rd Edition, Wiley, Hoboken, NJ. Markowitz, H. 1952. “Portfolio selection.” Journal of Finance 7, 77–91. Markowitz, H. 1959. Portfolio selection: efficient diversification of investments, Wiley, New York. Sharpe, W. F. 1963. “A simplified model for portfolio analysis.” Management Science 9, 277–293. Sharpe, W. F. 1964. “Capital asset prices: a theory of market equilibrium under conditions of risk.” Journal of Finance 19, 425–442. Wackerly, D., W. Mendenhall, and R. L. Scheaffer. 2007. Mathematical statistics with applications, 7th Edition, Duxbury Press, California.

Appendix 8A Derivation of Equation (8.6) The objective function L as defined in Equation (8.3) can be rewritten: # " n X   Max L D Wi Ri  Rf i D1

2 31=2 n n n X X X 4 Wi2 i2 C Wi Wj ij 5 i ¤j i D1

i D1 j D1

134

C.-F. Lee et al.

0

Then, following the product and chain rule, we have: d dL D dW i d Wi 2 4

n X

"

n X

  Wi Ri  Rf

Wi i2 C

n n X X

D4

n X

Wj ij 5

n n X X

i D1

n P

i ¤j kD

31=2

4

n X

i D1 j D1

Wi i2 C

n n X X

i D1

2 D4

n X

k

i D1

Wj ij 5 31=2

Wi2 i2 C

i D1

n n X X

42Wi i2 C 2

n X

i D1 j D1

Wi i2

C

n X

! Wi ij

  C Ri  Rf D 0

j ¤i

There is one equation like this for each value of i .

33=2

R1  Rf D H1 12 C H2 12 C    C Hn 1n

Wi Wj ij 5 3

R2  Rf D H1 21 C H2 22 C    C Hn 2n :: :

Wj ij 5

Rn  Rf D H1 n1 C H2 n2 C    C Hn n2

i D1 j D1

2

i ¤j Wi Wj ij

Ri  Rf D Hi 1i C H2 2i C    C Hi i2 C    C Hn ni

i D1 n X

n P n P

Define Hi D kW i , where the Wi are the fractions to invest in each security and the Hi are proportional to this fraction. Substituting Hi for kW i :

Wi Wj ij 5

Ri  Rf # " n X   1 Wi Ri  Rf 2

4

Wi2 i2 C

dL D .kW1 1i C kW2 2i C    C kWi i2 C    dW i CkWn ni / C .Ri  Rf / D0



2

i D1

Therefore:

i D1 j D1



  Wi Ri  Rf

i D1

i D1 j D1

Wi2 i2 C

(8A.2)

yields

31=2

n n X X

n P i D1

Wi Wj ij 5

" n # X   d

Wi Ri  Rf dW i i D1 " n # X   d C Wi Ri  Rf dW i i D1 2

Wj ij A

Defining

31=2

i D1 j D1

Wi2 i2 C

1

j D1

i D1

i D1

2

@Wi i2 C

#

n X

j D1

D0

.i D 1; 2; : : : ; n/

(8A.1) 2

Multiplying n P

Equation

by

4

n P

i D1

#1=2 Wi Wj ij

(8A.1)

Wi2 i2 C

n P

Appendix 8B Derivation of Equation (8.10)

i D1

Following the optimization procedure Equation (8A.2) in Appendix 8A:

; i ¤ j , and rearranging yields:

j D1

0 n B   P B Wi Ri  Rf B B i D1 Ri  Rf D B n n P n P BP 2 2 B Wi i C Wi Wj ij @ i D1 j D1 j D1 i ¤j

1 C C C C C C C A

n P

  dL D Ri  Rf  dW i 0

  Wi Ri  Rf p2

n X j D1

j ¤i

deriving

i D1

@Wi ˇi2 m2 C ˇi D0

for

1

2 A Wj ˇj m2 C Wi 2i

8 Performance-Measure Approaches for Selecting Optimum Portfolios

Let Hi D

h

Rp  Rf

.

i p2 Wi and, solving for any Hi ,

Ri  Rf Hi D  2 2i

ˇi m2

n P

Hj ˇj

j D1

(8B.1)

2 2i

135

Hi D

Hi ˇi D



Ri  Rf ˇi 2 2i

ˇi m2 

n P j D1 2 2i

d P d X

Hj ˇj2

j D1

Hj ˇj D

(8B.2)

Hj ˇj D

2 1 C 2i

2 2i

n P

ˇj2

j D1

2 2i

n X

Hj ˇj D

j D1

(8B.3) and let:

By substituting Equation (8B.3) into Equation (8B.1), Equation (8.10) is obtained.

Appendix 8C Derivation of Equation (8.15) This appendix discusses the use of performance measure to examine the optimal portfolio with short sales not allowed. Therefore, Equation (8B.1) from Appendix 8.B must be modified: n Ri  Rf ˇi m2 X  2 ˇj Hj C i Hi D 2 2i 2i j D1



Rj Rf 2 2j

ˇj (8C.3)

d P j D1

ˇj 2 2j

Notice since the set d contains all stocks with positive Hi :

n P .Rj Rf /ˇj j D1

j D1



1 C m2

j D1

Adding together the n equation of this form yields: n X

(8C.2)

Multiplying both sides by ˇj , summing over all stocks in d , and rearranging yields:

Multiplying both sides of the equation by ˇi : 

d Ri  Rf ˇi m2 X  Hj ˇj and i D 0 2 2 2i 2i j D1

(8C.1)

where Hi 0; i 0, and i Hi D 0 for all i . The justification of this equation can be found in Elton et al. (1976). Assuming all stocks that would be in an optimal portfolio (called d ) can be found, and then arranging these stocks as i D 1; 2; : : : ; d , for the subpopulation of stocks that make up the optimal portfolio:

Hj ˇj

j D1

d P

C D

d X

j D1 m2

Rj Rf 2 2j

ˇj

n P

ˇj2

j D1

2 2j

1 C m2

(8C.4)

Using Equation (8C.4), the following equation for Hi is obtained after substitution and rearranging from Equation (8C.1): ˇi Hi D 2 2i

Ri  Rf C ˇi

! C i

(8C.5)

Since i 0, the inclusion of i can only increase the value of Hi . Therefore, if Hi is positive with i D 0 can never make it zero and the security should be included. If Hi < 0 when i D 0, positive values of i can increase Hi . However, because the product of i and Hi must equal zero, as indicated in Equation (8C.1), positive values of i imply Hi D 0. Therefore, any security Hi < 0 when i D 0 must be rejected. Therefore, Equation (8.15) in the text can be used to estimate the optimal weight of a portfolio.

Chapter 9

The Creation and Control of Speculative Bubbles in a Laboratory Setting James S. Ang, Dean Diavatopoulos, and Thomas V. Schwarz

Abstract Persistent divergence of an asset price from its fundamental value has been a subject of much theoretical and empirical discussion. This paper takes an alternative approach of inquiry – that of using laboratory experiments – to study the creation and control of speculative bubbles. The following three factors are chosen for analysis: the compensation scheme of portfolio managers, wealth and supply constraints, and the relative risk aversion of traders. Under a short investment horizon induced by a tournament compensation scheme, speculative bubbles are observed in markets of speculative traders and in mixed markets of conservative and speculative traders. These results maintain with super-experienced traders who are aware of the presence of a bubble. A binding wealth constraint dampens the bubbles as does an increased supply of securities. These results are unchanged when traders risk their own money in lieu of initial endowments provided by the experimenter. Keywords Speculative bubbles r Experimental asset markets r Fundamental asset values r Tournament r Market efficiency r Behavioral finance

Speculative bubbles are induced in this study under a pictoria laboratory setting, where a New York Stock Exchange type of double oral auction market (without a specialist) involving many traders is modeled. Speculative bubbles occur when buyers are willing to bid higher and higher prices for an asset, which, in retrospect, is far in excess of its worth based on fundamentals. The bubbles ultimately burst and prices drop to a much lower level.1 The stock market crash in the U.S. in 1987, in Japan in 1991–1992, the dot com bubble in 2000, and the recent housing bubble in the U.S. are examples.2 In recent years, academicians and practitioners are slowly but grudgingly coming to the realization that the extant theories of stock market behavior (e.g., efficient market hypothesis and capital asset pricing theory), fail to explain the magnitude of fluctuations in the stock market. Not only have stock prices been found to fluctuate too much relative to fundamentals, but also there have been occurrences of speculative bubbles that could not be explained by arrival of new information. Several plausible explanations for bubbles are offered such as rational bubbles (Shiller 1988; West 1988), irrational bubbles (Ackert et al. 2002; Lei et al. 2001), judgment error (Ackert et al. 2006) and herding behavior (Froot et al. 1992).3

9.1 Introduction The purpose of this study is to investigate the formation of speculative bubbles in asset prices under a laboratory setting. Specifically, we investigate how to create, control, and dismantle bubbles, as well as the conditions in which bubbles may or may not arise. J.S. Ang () Department of Finance, Florida State University, Tallahassee, FL 32306, USA e-mail: [email protected] D. Diavatopoulos Villanova University, Villanova, PA 19085, USA e-mail: [email protected] T.V. Schwarz Grand Valley State University, Grand Rapids, MI 49504, USA e-mail: [email protected]

1 Stiglitz (1990), in his overview of a symposium on bubbles, defines the existence of bubbles to be: “if the reason that the price is high today is only because investors believe that the selling price will be high tomorrow – when ‘fundamental’ factors do not seem to justify such a price.” Similarly, he defines the breaking of a bubble as marked price declines that occur without any apparent new information. 2 Other notable example of bubbles include the Dutch tulip mania in the seventeenth century, the South Sea Islands Company bubbles (Voth and Temin 2003), John Low’s Mississippi Company Scheme Bubbles of the eighteenth century, the South Sea Islands Company bubbles, John Low’s Mississippi Company scheme bubbles of the eighteenth century, the U.S. stock market boom of the late 1920s, the Florida land price bubbles of the 1920s, the great bull market of the 1950s and 1960s, and the high-tech stock boom of the early 1980s and the boom and bust of the California and Massachusetts housing markets in recent years. However, due to the difficulties in specifying the fundamentals, there is still disagreements as to whether these cases could be explained by the fundamental example of Garber (1990) versus White (1990). 3 Outstanding surveys of this literature are provided by Porter and Smith (2003), Camerer (1989) and Sunder (1992).

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_9, 

137

138

While some work has been done to show that bubbles can be abated with experience (Dufwenberg et al. 2005), an understanding of the formation of speculative bubbles is still important to researchers for several reasons. First, bubbles could cause significant disruptions in the asset market, not only by creating a large redistribution of wealth among investors, but also by adversely affecting the supply of funds to the market as well as resource allocation among and within firms. Second, the identification of factors affecting the formation of bubbles is crucial in aiding regulators in designing policies to reduce the occurrence or magnitude of bubbles. In particular, if bubbles can be replicated in a laboratory setting, then various proposals to dampen bubbles could also be tested and compared for their effectiveness. Roll (1989) summarizes the difficulty with examining recent empirical results of the 1987 Crash in this regard. Third, an understanding of the dynamic process of bubble formation would contribute to our knowledge of how to model the behavior of asset prices (DeLong et al. 1989; Cutler et al. 1991). In spite of some interesting recent theoretical developments, empirical research on the existence of bubbles tends to be inconclusive and with low power; for example, Gurkaynak (2005), West (1988), Flood and Hodrick (1990). A major problem is the difficulty of specifying the fundamental value of an asset, as bubbles are defined as the price in excess of the fundamental values (Bierman 1995; Robin and Ruffieux 2001). Without being able to calculate the time series of the asset’s fundamental value, price movement could simply be caused by factors affecting the fundamental valuation of the asset; for example, change in risk aversion, arrival of new information, and so forth. And if the fundamental value could only be measured imperfectly using proxies such as past dividends, the imprecise estimates would, of course, reduce the power of any test. The experimental approach reduces this problem (Cason and Noussair 2001). By design, the value of the fundamentals can be specified in advance; hence there is no measurement problem. Any gross and persistent divergence of the asset price from the prespecified fundamental value can now be attributed to bubbles (Siegel 2003). In addition to reducing the identification/measurement problem, performing laboratory experiments to study asset bubbles has two other advantages. First, it allows different characteristics of the market institutions and participants to be introduced in a controlled manner. That is, relevant factors may be manipulated to create or discourage the formation of bubbles. This is an important feature because some of these factors may not be isolated in the real world for detailed study while other factors are simply proposals in the design of market institutions of the future. Second, by controlling the information available to market participants, we can control the role played by unrelated, or exogenous events (e.g., sunspots). Thus, the laboratory experiment approach to study asset market behavior

J.S. Ang et al.

complements the theory/model building process. The three types of variables chosen for analysis in this study are the following: 1. The compensation scheme of a portfolio manager. Allen and Gorton (1988) have argued that compensation schemes for portfolio managers may induce bubbles even in a finite horizon. Also, recent literature on tournaments (see James and Issac 2000; Ehrenberg and Bognanno 1990 and others) has shown that the level and structure of relative compensation influences participant behavior, while Hirota and Sunder (2005) have found that short horizons are important factors in the emergence of bubbles. Three types of compensation structure are used in these experiments: a linear compensation scheme based on portfolio performance, and two versions of compensation based on relative performance in a short-term horizon. 2. Wealth constraint (tight/loose); supply of securities. An infinite number of trades (e.g., overlapping generations and the availability of credit) is often cited as a prerequisite for bubbles. Ricke (2004) discusses how credit made available from margin could generate bubbles. High liquidity leads to bubbles in the work of Caginalp et al. (2001). Scheinkman and Xiong, (2003), Hong et al. (2006) analyze the effect of a short sales constraint on the formation of bubbles. Therefore, experimenting with wealth constraints may provide valuable insights into the effectiveness of certain policies (such as margin rule change, credit availability, and so forth) to control bubbles. 3. The type of investors in the market (speculative/conservative). This variable tests the Keynes-Hicks theory of speculation where differences in traders’ willingness to take risks is the foundation of speculative markets. Traders in the experiments are pre-tested for attitudes toward risk taking. Bubbles are observed under the following conditions: 1. A market of speculators with short-term investment horizon. 2. A market of mixed conservative and speculative traders with short-term investment horizon. 3. A market of mixed trader types with short-term investment horizon using their own money. On the other hand, bubbles are dampened under the following investment environments: 1. A market of conservative traders with a short-term horizon. 2. A market of mixed trader types with a long-term investment horizon. 3. A market of mixed trader types when the wealth of the traders, especially the bulls, is constrained. 4. A single-period trading environment.

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

The remaining part of this paper is organized into four sections. Section 9.2 presents the hypotheses to be tested by incorporating them into the experimental design, which is discussed in greater detail in Sect. 9.3. The results are reported in Sect. 9.4 with Sect. 9.5 summarizing and concluding the paper.

9.2 Bubbles in the Asset Markets The possibility of asset bubbles has long been recognized; however, more formal theoretical development is of relatively recent vintage. Harrison and Kreps (1978), for instance, suggest that in general the right to resell the asset makes traders willing to pay more for it than they would if obliged to hold it forever. Thus, market price could exceed fundamental value. Literature on rational bubbles emphasizes that once a bubble is started, it would be rational to price the bubble component even if it is expected to burst with positive probability. Brunnermeier and Nagel (2004) examine stock holdings of hedge funds during the recent NASDAQ tech bubble and find that the portfolios of these sophisticated investors were heavily tilted towards (overpriced) technology stocks. However, this does not seem to be the result of unawareness of the bubble on the part of hedge funds.4 At an individual stock level, hedge funds reduced their exposure before prices collapsed suggesting awareness and implicit pricing of the bubble component. On the other hand, whether bubbles can even get started has been questioned 64 Diba and Grossman (1987). Essentially, if there is a finite number of periods, starting from the next to the last period, the expectation that the bubble might end may be sufficient to keep it from ever starting. By the process of backward induction or an unraveling argument, bubbles will not exist. Moreover, if the number of trades is finite, withdrawal of early trades at a profit means the remaining traders would be at a negative sum game; that is, with finite trades, will there be a “greater fool” who gets stuck when the bubble bursts. Still, perturbing the model by adding uncertainties on the length of the horizon among traders or market size may preserve the possibility of asset bubbles. Smith et al. (1988) are among the first to investigate the incidence of bubbles. Their design was to give traders common beliefs (according to one of Tirole’s requirements) and long horizons of up to 15 trading periods. Bubbles are observed in 4

Griffin et al. examine the extant theoretical literature about bubbles that includes models where naive individuals cause excessive price movements and smart money trades against (and potentially eliminates) a bubble versus models where sophisticated investors follow market prices and help drive a bubble. In considering these competing views over the tech bubble period on NASDAQ they find evidence that supports the view that institutions contributed more than individuals to the spectacular NASDAQ rise and fall.

139

several of their experimental markets. It is unclear, however, what institutional setting, other than long trading periods, induces bubbles in their study. In a speculative market where bubbles could be present, speculative traders are more likely to purchase shares (and even more so, if bubbles are rationally priced) than riskaverse traders. Not only are they more willing to put a higher value on risky assets, they are also more likely to take the chance that they might not be able to sell out their inventory before the bubble bursts. Therefore, our first hypothesis is that bubbles are more likely to be formed in a market of risk taking traders (speculators). The compensation scheme could also affect the behavior of traders. For instance, Allen and Gorton (1988), and Allen and Gale (2000) show that an option-type compensation scheme for portfolio managers could induce speculative bubbles in asset prices. Portfolio managers are encouraged via incentive rewards to generate short-term trading gains even in a finite horizon world. The current practice of publishing and ranking the short-term investment performance of portfolio managers and the very substantial incentives to hedge fund managers’ performance that may be based on unrealized gains on illiquid assets could give rise to adverse incentives. Portfolio managers who are concerned about these rankings will either take on a riskier strategy for the possibility of outshining their peers or they will simply play it safe and follow the crowd. Both portfolio strategies could lead to the formation of bubbles. The play safe by “following the herd” strategy will cause asset prices to have a strong positive correlation in the short term. Portfolio managers would be buying when others are buying, thus creating an upward price trend, and selling when others were selling, thus bursting the bubble it created. On the other hand, pursuit of a risky strategy may be sufficient to create price leadership that is followed by others in the market. This would be more likely in an uncertain valuation environment. Voth and Temin (2003) suggest that riding the bubble may actually be a profitable strategy. Portfolio managers are subject to an occupational hazard: unless they produce winning results, they stand a good chance of being fired. On the other hand, star performers receive seven or even eight figure incomes as new cash flows into the funds they manage. This compensation system is similar to tournament models where participants are paid according to their relative performance among a group of peers rather than on their absolute performance. Tournament systems are likely to produce increased performance when (1) there is difficulty in monitoring the activities of the agent, (2), when the agent possesses valuable information (Baker 1992), and (3) when good performance measures are available (Baker 1992). All three of these conditions exist in the realm of professional money management, and therefore it is probable that a relative performance compensation system

140

will be effective in increasing manager performance.5 Thus, it is hypothesized that a tournament incentive scheme that encourages a short-term horizon for portfolio managers is more likely to create bubbles.6 Finally, an important policy question has been whether restricting the availability of credit or the supply of securities in a market (by raising the margin requirement or allowing short sales) could reduce or even eliminate the formation of speculative bubbles. (Ackert et al. 2006 find that price run ups and crashes are moderated when traders are allowed to short sell). Most countries, including the U.S. and Japan, have adjusted these conditions in the recent past through adjustments in the use of stock index futures and by easing credit conditions. These changes have tended to occur subsequent to large declines in the country’s equity markets. With limited or asymmetric ability to go short versus long, speculators on the long side have an advantage in acquiring funds for investment. Additionally, if the life of a bubble is uncertain and relatively long lasting, costly short sell will not be profitable even if the bubbles eventually burst. The usual experimental design often endows traders with a relatively large initial wealth such that the budget constraint is not binding. This experiment will test the effect of a tighter budget constraint by both reducing the initial endowment and increasing the supply of securities. It is hypothesized that a wealth constraint and/or relative increase in the supply of securities will reduce the incidence of bubbles. To summarize, the effect of three factors: attitude toward risk, investment horizon, and wealth constraint are examined as to their contribution to the creation and control of asset bubbles. They are tested by incorporating them into the experimental design of a laboratory setting described below, the importance of these factors in the formation and control of asset price bubble, singly and jointly, can now be formally examined.

9.3 Experimental Design The evolutionary nature of laboratory experimental research is such that the results of any study acts as a catalyst for new questions and therefore new experiments. As with Smith

5

Becker and Huselid (1992) and Ehrenberg and Bognanno (1990) have documented in field studies that such tournament compensation systems are effective in raising performance in professional golf and auto racing competitions. 6 It is possible that, if there is sufficient number of short horizon portfolio managers herding in the manner described by Froot et al. (1992), a bubble can start on basis of any information. Shleifer and Vishny (1990) also propose that the portfolio managers have short horizon; however, it is the risk of uncertain return from investing in the longer horizon that prevented disequilibrium to be arbitraged away.

J.S. Ang et al.

et al. (1988), we note that many of our latter experiments were directly motivated by the results obtained from our earlier ones. This progression of thought and analysis will be apparent in the later section on results. Herein, however, we present the method of our investigation in comprehensive form. The creation of “bubbles” within asset markets is examined under the control of three primary factors: (1) the degree of trader risk aversion, (2) trader investment horizon, and (3) available investment capital/supply of securities. Table 9.1 summarizes the design of 14 experiments used to investigate these factors on the presence of asset bubbles. Each of these experiments uses a common market mechanism that builds on the earlier work of Forsythe et al. (1982), Plott and Sunder (1982), and Ang and Schwarz (1985). These common features are summarized below.

9.3.1 General Market Design 1. A double-oral auction, similar to that used on the floor of major U.S. exchanges, is replicated. The recruited traders are physically present within a single room during the course of trading. These traders are independent and trade solely for their own account. There are no specialists or other privileged traders. 2. Only those shares of a single generic security are traded. The sole attribute of these shares is the payment of dividends at the end of each period. 3. Each market (experiment) has ten trading periods. These periods are further categorized into five trading years each of which consists of two contiguous trading periods (A and B). Endowments (discussed below) are reinitialized at the end of the second period of each year. Thus, the initial market represents a two-period model with each security entitled to two payoffs (dividends), one at the end of period A and the other at the end of period B.7 4. Each trading period last for 6 min, with opening, warning (at 5 and 51=2 min), and closing bells. Consequently, each experiment has a total of 60 (6 min 10 periods) trading minutes. During the 6-min periods, traders can observe the continually updated bid/ask and past transacted prices. 7

In the experiment, a trader has at least the following choices available: (a) Maintain the endowed position by not trading and receiving the stochastic payoffs at the end of each period; (b) Hold the securities through period A and sell in period B, in which case the investor will receive the first period dividend and the selling price; (c) Sell the initial holdings in period A to receive the sale price; (d) Buy additional shares in period A, receive dividends at the end of the period, and then sell the securities in period B; (e) Sell the securities in period A and then buy back securities in period B in order to receive the dividends; (f) Purchase a net amount of shares in both periods; (g) Purchase and sell shares within each period.

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

141

Table 9.1 Experimental Design This table categories five designs of fourteen experiments used to examine the impact of risk aversion, investment horizon, and credit/supply constraints (initial endowment) upon the formation and control of asset bubbles. Participant Groupsa Las Vegas

Design 1

Initial Endowmentb 2 securities 10,000 francs

Investment Horizonc Two period

Risk Aversiond Mixed

Las Vegas

2

2 securities 10,000 francs

ShortenedMixed

4m, 6m$

Las Vegas

3

5 securities 3,000 francs

ShortenedMixed

7x,8x,9t,10t

FSU1

4

2 securities 10,000 francs

Two period

FSU1

5

2 securities 10,000 francs

ShortenedSingle

Single Type

Experimentse 1,2,3,5$

11s, 13c 12sm, 14cm

Type

FSU2

2

2 securities 10,000 francs

ShortenedMixed

15, 16

FSU2

6

10 securities 1,000 francs

ShortenedMixed

17, 18

Albania

1

2 securities 10,000 francs

Two period

Albania

2

2 securities 10,000 francs

ShortenedMixed

24,25

Albania

6

10 securities 1,000 francs

ShortenedMixed

26

Albania

7

2 securities 5,000 francs

Single Period

Mixed

19

Albania

8

2 securities 5,000 francs

Single Period/Tournament

Mixed

20,21

Albania

9

20 securities 500 francs

Single Period/Tournament

Mixed

22

a

Mixed

23

The participant groups consist of the following: Las Vegas represents students from the University of Las Vegas at Nevada. FSU1 and FSU2 represent students from the Florida State University at two different time periods. Albania represents students from the University of Tirana in Albania.

b

The initial endowment refers to traders wealth position at the beginning of each trading year of an experiment. This endowment allows traders to sell (using provided securities) or buy (using francs, the currency used in these experiments). The additional securities and reduced currency endowments provided in Design 3 serves to better equate relative purchase and selling abilities. c Investment horizon refers to the horizon within which traders effectively operate. A two-period horizon refers to a market where period A securities are based on the dividends paid in both periods (A & B) of a trading year. In a shortened investment horizon, the trader is induced (via the tournament compensation schedule of Table 9.3) to operate with a horizon which is shorter than the two-period environment in which the securities will pay dividends. d Mixed risk aversion means that traders with various risk preferences were participants within the same market. Single type means that only speculative (s) or conservative (c) traders made up that market. The designations (s) and (c) appear next to experiments 11–14 in the last column. e Notation is as follows: $ represents a market where traders provided $20 of their own money to trade, the sum of which became the pool of money dispersed according to relative profit performance. m,x,t represents the number of traders receiving the tournament prize as outlined in Table 9.3. This tournament compensation was used to induce a shortened horizon market and was differentially paid to the top two (t) or the top six (x) traders. In experiments marked (m), the first three trading years paid a bonus to the top six traders followed by years where only the top two traders received bonuses.

9.3.2 Dividend Design 1. At the beginning of each year, each trader is endowed with trading capital and shares of the generic security. Each share pays dividends at the end of the first (A) and second (B) periods. The second period dividend is a liquidating

dividend. Reinitializing of position at the beginning of each year allows for replication of decision making in experimental markets.8 8 See Smith et al. (1988) for an example of when reinitialization is not used.

142

J.S. Ang et al.

Table 9.2 Dividend Design This table presents the cash flow payoffs which a single asset will provide to its owner. This payoff is different for Trader Types I, II, and III and therefore provides for different fundamental valuations. Rational Expectations Equilibrium are determined by the trader type with the highest valuation for that period.

I II III

Period A Dividend Statec G B 350 110 250 150 200 140

Expected Dividendd

Yearly Expected Dividende

230 200 170

Period B Dividend Statec G B 250 150 200 140 350 110

200 170 230

430 370 400

2

I II III

200 350 250

140 110 150

170 230 200

350 250 200

110 150 140

230 200 170

400 430 370

3

I II III

250 200 350

150 140 110

200 170 230

200 350 250

140 100 150

170 230 200

370 400 430

4

I II III

350 250 200

110 150 140

230 200 170

250 200 350

150 140 110

200 170 230

430 370 400

5

I II III

200 350 250

140 110 150

170 230 200

350 250 200

110 150 140

230 200 170

400 430 370

Yearsa

Trader Typeb

1

Expected Dividendd

a Each experiment is composed of five trading years, each of which contains two trading periods A and B. Ownership of an asset in period A entitles the bidder of both period A and Period B dividends (two-period valuation) whereas period B ownership merits only that period’s dividend (single-period valuation). b There are three trader types in each trading year with four traders in each category. These trader types only differ by the amount of dividend cashflows that the single traded asset will provide its holder. The four traders within each category are rotated within the other categories so as to maintain an uncertain valuation environment. c Dividend States refer to the stochastic payoff that will be provided to specific trader types given the occurrence of the G (Good) or B (Bad) state. The realization of the state of nature is determined at the end of each trading period by flipping a fair coin. d Given equal fifty percent probability of occurrence of G or B, the expected dividend is the simple average of period G and B payoffs. e The yearly expected dividend represents the summation of expected dividend for both periods A and B.  Signifies trader type with the highest expected value

2. The dividends to be paid at the end of each period are stochastic. Two equally likely dividend outcomes are possible, the good (G) state and the bad (B) state. The realized state is announced at the end of each 6-min trading period as determined by the flip of a coin by the experimenter. 3. The dollar amount of the dividends paid at the end of each period depends on the trader’s type and the realized state. The 12 traders who make up each market are classified into three types to allow for differences in induced values. The dividend payouts for these three trader types are summarized in Table 9.2. As an example, at the beginning of year 1, the four traders of type I are privately informed that for period A they will receive either 350 or 110 for the good (G) and bad (B) states, respectively, and for period B either 250 or 150 for the good and bad states, respectively. 4. The trader type with the highest expected dividend (0.5

Good Dividend C 0.5 Bad Dividend) is rotated each period so as to enhance trader uncertainty about equilibrium prices. Virtually all previous experimental studies have documented that given sufficient learning (through repeated trading) in a stationary dividend payout environment, prices will rather quickly approach the rational equilibrium level. This learning has two sources: (a) observation that one’s own payouts are not changing, and

(b) observation that market generated bids, offers, and transacted prices are not changing. Our expectation is that the greater the trader’s reliance on market generated (as opposed to prior dividend) information, the more likely bubbles are to occur due to bandwagon and other crowd psychologies. If, instead of bubbles, we should observe that prices converge to rational equilibrium prices (as in the constant dividend studies), then this would strengthen our knowledge concerning efficiency in these laboratory markets. This result would also suggest that trading methods based on historical prices alone would not have value.

9.3.3 Investment Horizon 1. Three types of investment horizon are provided within these experiments: a single-period, a two-period, and a shortened-horizon. Initially, at the beginning of each trading year, a trader is entitled to two stochastic dividends for each security held, one each at the end of periods A and B. Therefore, at the beginning of period A, a rational trader will value the security for both its period A and period B stochastic dividends. Hence, all A period

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

pricing should reflect a two-period investment horizon. Subsequent to the termination of period A trading and the announcement and payment of the period A dividend, period B trading proceeds. As the security is now only entitled to the B period dividend, a single-period investment horizon results for all B periods. Our hypothesis is that a shortened investment horizon increases the possibility of an asset pricing bubble. We test for this by creating a tournament compensation package in period A. The incentive for traders is to concentrate upon their single (A) period performance over the concerns of a rational twoperiod price. This incentive results in a shortened investment horizon.9 2. In this study, Initially, a trader’s dollar compensation is defined by the following: we alter the compensation structure to induce a change in the length of a trader’s investment horizon profit function: Pi D fŒdi;j;A  Xi;A C di;j;B  Xi;B C .Ri  Ci /

(9.1)

where Pi , dollar profit per trading year for trader i. It consists of dividend income and trading gains (losses) from both periods; f, the conversion rate of francs into dollars.10 ; di;j;t , the dividend paid in francs to trader i, given state j occurs in period t; j D G or B; t D A or B; Xi;t , the number of shares held by trader i at the end of period t; t D A or B; Ri , revenues in francs for trader i for all shares sold during periods A and B; Ci , costs in francs for trader i for all shares purchased during periods A and B. In order to induce pressure for a shortened investment horizon, an additional compensation package is introduced in Period A of some experiments as identified in Table 9.1. This tournament compensation system is based on the traders’ relative performance as measured by the Tournament Performance Index (TPI) below: TPIi D Ri  Ci C MXi;A

(9.2)

9 It is important to note that there is a difference between a one- and two-period horizon and a shortened-horizon. In a one-period model, only a single dividend is valued. In a two-period model, two dividends are valued. In our shortened-investment horizon, the trader is induced to operate within a horizon that is different from that of his operating environment. That is, within a two-period operating environment, the trader is given an incentive to operate with a shorter (possibly single) period horizon. This is quite different from a single-period model. This shortened horizon is a stronger test of market efficiency, in that the pressures are away from rather toward rational equilibrium prices, (as defined in Equation (9.4), subsequently). The methodology is meant to emulate modern portfolio managers operating in an environment of perpetual horizon stock securities yet receiving tournament incentives to outperform colleagues on a short-term basis. 10 Francs are the currency used within this study. They have been used successfully by Plott and Sunder (1982), Ang and Schwarz (1985), as well as others. Their primary benefit is to avoid the technical problem of dealing with small dollar amounts.

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where Ri ; Ci , and Xi;A are as previously defined, and M represents the closing market value of the shares. This closing market value is taken to be the price of the second to last transacted price for that period. This procedure is introduced in order to reduce the possibility of manipulating market value by collaboration on a final transaction. It represents a simplified version of the priceaveraging process that takes place on most organized exchanges for the setting of opening and closing prices. 3. The tournament compensation system provides traders with an incentive to outperform each other in period A only. This incentive system increases the importance of single-period performance (in A) over two-period concerns; that is, it induces a shorter investment horizon in period A. A trader’s compensation is dependent upon his relative rank as summarized in Table 9.3. In Schedule Six, the top six (of twelve) traders are rewarded with francs ranging from 1,500 to 200. Schedule Two is an alternative schedule that is hypothesized to induce even greater competitive pressure as only the top two traders are compensated greatly.11 Ehrenberg and Bognanno (1990) and Table 9.3 Tournament Compensation Schedule This table presents the additional tournament compensation schedule provided to traders based on their relative profitability in period A of certain experiments (see shortened investment horizon listed in Table 1). Relative profitability is measured by: TPIi D Ri  Ci C M  Xi;A where TPIi is the tournament performance index for trader i, Ri is the revenues received from the sale of assets in period, Ci is the cost of assets purchased, M is the closing market value for the period, and Xi;A represents the end-of-period asset holdings. Together the index measures the total of realized and unrealized capital gains. The addition of the tournament compensation to period A provides an incentive for traders to prefer period A capital gains over equivalent period B dividends and thereby induces a shortened (from the two period model) investment horizon. Schedule Six (s)

Two (t)

Rank 1 2 3 4 5 6 7 to 12

Lowest

1

Highest

3000 francs

Lowest

1000 0

2 3 to 12

TPI Highest

Compensation 1500 francs 1000 700 400 200 200 0

a

Two compensation schedules are introduced. The first provides for those traders who do better than the average (i.e., the top six) receive the additional compensation list. In the second schedule, only the top two “superstars” are richly rewarded. The tournament literature (e.g. Baker 1991) suggests that tournament systems, and especially schedule two, provide effective incentive systems to increase performance. This design is meant to emulate the short-term performance pressures faced by professional money managers.

11

The compensation schemes depict the different ways portfolio managers are being rewarded: those who are above the average or beaten

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Becker and Huselid (1992) find that the reward spread does cause increased performance incentives. Therefore, we expect Schedule Two to increase incentives for shortterm pricing behavior. Table 9.1 summarizes the experimental use of the performance reward schedule.

9.3.4 Risk Aversion 1. Prior to selection, each potential trader was given a lengthy questionnaire. Intermingled within this material were two psychological tests on risk taking: the Jackson Personality Inventory (1976) and the Jackson et al. (1972) tests.12 These two tests have been applied in laboratory (Ang and Schwarz 1985) and field studies (Durand et al. 2006) and are more practical to administer than the theoretical risk measures found in the economics literature.13 Those persons who score in the top 12, signifying the least risk averse, and the bottom 12, or the most risk averse, are invited to participate in the second stage of the experiment. 2. Traders for experiments 1–10 were students from the University of Las Vegas at Nevada and were recruited from a senior level options class. These students had all taken two statistics, a corporate finance, a valuation, a portfolio analysis, and an options course. They were well trained in arbitrage, present value, and expected value. From this pool of students, 12 were chosen to participate based on their attribute ranking in risk aversion. Participants were chosen so that a mix of risk aversion types were represented in the same market. Included were those the market (Schedule Six), and those who are the superstars (Schedule Two). 12 The authors are aware of the work of Holt and Laury (2002), which was not available at the time of this study. According to Holt and Laury their experiment shows that increases in the payoff level increase RRA. However, when estimating RRA, Holt and Laury assume that subject’s utilities depend only on payments in the experiment. They fail to account for the wealth subjects have from other sources (see Heinemann 2003). 13 The Jackson Personality Inventory is scientifically designed questionnaire for the purpose of measuring a variety of traits of interest in the study of personality. It was developed for use on populations of average or above average ability. Jackson states (1976), p.9, “It is particularly appropriate for use in schools, colleges, and universities as an aid to counseling, for personality research in a variety of settings, and in business and industry.” Of the 16 measurement scales of personality presented, one scale directly measures monetary risk-taking using a set 20 true and false questions. Mean and standard deviation measures for 2,000 male and 2,000 female college students are provided. Jackson et al. (1972) demonstrate four facets of risk taking: physical, monetary, social, and ethical. The authors’ questionnaires are situational in that the respondent is asked to choose the probability that would be necessary to induce the respondent to choose a risky over a certain outcome. Jackson (1977) presents high internal consistency correlation between the risk measurement techniques.

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who ranked at all levels of the scale, from high to low risk aversion. This was done so that differences in individual risk behavior could be tracked within an identical market environment. 3. Experiments 11–14 were conducted at Florida State University (FSU1) and as summarized in Table 9.1, these experiments were designed so that an experimental market consisted entirely of traders that were either relatively more risk averse (conservatives) or less risk averse (speculators). This experimental form allowed for evaluation of whether risk aversion is uniquely a necessary or sufficient condition for the presence of bubbles. 4. Experiments 15–26 were conducted at a later date at Florida State University (FSU2) and the University of Tirana in Albania. This was done to confirm the robustness of our results. We intentionally chose students from two different universities with different backgrounds to represent the two extremes in our test. Experiments 19–26 were administrated to subjects in Albania, who have a low degree of familiarity with capital markets while experiments 15–18 were administered to Florida State University students (FSU2) who had taken a financial engineering course and completed another more involved laboratory asset market experiment. Hence, we consider these FSU2 students to be super experienced relative to the students from Albania.

9.3.5 Validation Procedures The following procedures are incorporated into the experimental design to insure the reliability and external validity of the results: 1. To guard against the possibility that subjects’ experience with trading could change their attitudes toward risk taking, they were retested. Subsequent to the first four experiments, additional risk questionnaires were given to the participants. A Spearman rank correlation (with initial risk rankings) was 902 with a t-statistic of 6.61 indicating that there had been no significant change in the relative risk attributes of the traders. 2. Videos were used to verify recorded information, identify possible irregularities, and to train new subjects. 3. Subjects were given extensive training on the operation of the game; the main experiments were conducted on groups of experienced, if not super-experienced, traders. 4. Lengthy post experiment questionnaires were also given to the subjects. Among other things, these were used to verify that the traders considered their trading strategies taken at the time of trade to be rational.

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

9.4 Results and Analysis 9.4.1 Control Experiments For experiments 1–14, five experimental designs were used to test for the effects of risk aversion, investment horizon, and capital endowment on the presence of asset bubbles. These designs are summarized in Table 9.1. The first design consisting of experiments 1, 2, 3, and 5 were control markets where the two-period model was tested without extrane-

Fig. 9.1 Experiment #1

Fig. 9.2 Experiment #2

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ous influence from the three treatment variables mentioned above. Figures 9.1–9.5 plot the series of resulting prices. Bid and ask prices are represented by a “C” symbol and are connected by a vertical solid line. Transacted prices are identified by a solid horizontal line connecting each trade. From earlier laboratory studies, we would expect prices to converge to rational expectation equilibrium levels after an initial period of learning. Two relevant concepts of equilibrium prices in these markets have been proposed (see Forsythe et al. 1982). The first is the Naive Equilibrium (NE) Price. The NE is the highest

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Fig. 9.3 Experiment #3

Fig. 9.4 Experiment #4

price any trader in the market is willing to pay based on his individual valuation of the expected dividends for the two periods or: NE D Max kŒE.DA;k / C E.DB;k /

(9.3)

where k classifies the trader type (1, 2, or 3) based on prior expected dividend valuations (see Table 9.2). E.DA;k / and E.DB;k / are the values of expected dividends in periods A and B to the k’th trader type.

The NE price is the market price that will prevail if the traders use only their private information to determine value. It is naïve in the sense that traders do not learn about the valuations of other traders from the market trading information. These traders also ignore the option value to trade (e.g., hold a security for a period and then sell it to another trader who would value it most in the remaining period). The second is the Perfect Foresight Equilibrium (PFE) price. It is equal to the highest total value that successive owners of the same share will pay; or, in the experiment, the

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

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Fig. 9.5 Experiment #5

sum of the highest expected payoffs for periods A and B for all traders or: PFE D Maxk E.DA;k / C Max E.DB;k /

(9.4)

These prices represent two extreme benchmarks in the continuum of the value of capital market in discovering information through trading. The NE price gives no role to capital market in price discovery, while the PFE price assumes full discovery (i.e., trading in capital market can correctly identify the share’s highest value in each future holding period). They define, respectively, the lower and upper bounds of the share’s fundamental value. Thus with payoffs to traders and across holding periods under the control of the experimenters, we can now identify with certainty whether a stock is undervalued (when price is below NE), or overvalued (when price is above PFE), or is in a bubble (when price is grossly below NE or above PFE, as in a negative or positive bubble). If the experimental market captures a well functioning capital market, learning and repeated trails would cause prices to converge toward PFE. There are two properties in (Equations (9.3) and (9.4)) that are worth noting. First, NE and PFE prices are identical in a one period world when price determination is closer to a simple auction of a single period payoff. Second, when the payoff in the equations are dollars, as in cash dividends and capital gains or losses, NE and PFE give the risk neutral prices. In the absence of risk neutrality, a negative difference between observed prices and these prices may be interpreted as a risk premium. The results illustrated in Figs. 9.1–9.3 establish the validity of our experimental design as we are able to produce results similar to those obtained in previous experimental

studies. In particular, we are able to reproduce the result that prices converge to PFE with learning and repeated trials. These prices are plotted as a solid horizontal line and are greater than the NE prices.14 The inexperience of traders in experiment 1 is greatly reduced in experiments 2 and 3 as traders learn to cope with the large uncertainty in valuations (introduced by design). This pricing uncertainty increases the traders’ reliance on “market generated information” in order to determine valuation. A micro-analysis of traders’ accounts in experiment 2 shows that some traders became actively involved in arbitrating between the A and B periods of a trading year. As a consequence these prices tended toward their PFE equilibrium levels. Traders’ learning contributed to further pricing efficiencies in experiment 3. Some earlier “irrational” trades by selected individuals had resulted in substantial losses creating a “once-bitten” effect and more rational decisions were 14 Note that all odd numbered experiments used the dividend design in Table 9.2. In order to differentiate between (1) learning about a stationary environment and (2) learning efficient valuation within laboratory markets, we created nonstationarity in equilibrium prices across experiments. In particular, for all even numbered experiments, the dividend payoffs of Table 9.2 were simply cut in half so that rational equilibrium prices were also one half that of the odd numbered experiments. When this equilibrium dividend rotation is viewed in conjunction with the previously mentioned rotation of trader types, it becomes apparent that each individual trader was likely to view the environment (at least initially) as nonstationary. Consequently, any results that we show regarding equilibrium pricing and convergence would suggest that learning about valuation methods rather than a stationary environment creates rational valuation. That is, we are concerned about learning that takes place within the trader (how he values) not about the environment (stationary value). We are able to pursue this expanded question due to our debt to earlier authors who have already well established the presence of the latter.

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followed subsequently. By the end of this experiment, prices in both the A and B periods were close to the PFE price.15 A final examination of the validity of the experimental design was performed by requiring each trader from Las Vegas experiments 5 and 6 to “invest” his own money ($20) into the markets. As a result, it was possible for traders to lose as well as to win. The results, illustrated in Fig. 9.5, show continued price convergence toward equilibrium levels.16 Of interest is the pattern of the bid-ask spread within a period. The data suggests that the primary resolution of uncertainty is obtained during the first transaction of a period. Subsequent trading tends to vary little from earlier levels with subsequently smaller bid-ask spread levels. We conclude the control section noting that the experimental design creates price-revealing trades that foster PFE equilibrium pricing. While consistent with earlier research, these results extend our knowledge into a much more uncertain (nonstationary) valuation environment more typical of real world asset markets. In addition, the validity of these results are not affected by whether or not a dollar investment is required from traders; trading behavior is similar under both environments.

9.4.2 The Formation of Bubbles With a well-functioning experimental design established, we now sequentially introduce our hypothesized treatment variables. In experiment 4, we introduce the shortened trading horizon with a tournament prize as described in the experimental design. At this point, we have an advantage over previous studies in that we were able to recruit the identical 12 traders back. This level of experience will lead to converging equilibrium prices as opposed to bubble formation.17 The effect of the tournament compensation is to shorten the traders’ investment horizon in period A from a PFE twoperiod model. By providing tournament payment based on period A relative ranking, there is an increased incentive to generate period A capital gains over equivalent period B div15

While period A prices exceeded the calculated PFE price of 460, this price is somewhat unknown to traders at this point. Prior trading results had created a history of B period prices averaging 320. Consequently, it was rational for a PFE trader to pay up to 550 (230 for A period plus 320 for B period sales price). The last trade in period 5A of 505 was well below that level. A more detailed presentation of the experimental results further reveals the rationality of these prices and is available from the authors upon request. 16 Again, period A prices seem to drift upward due to initial excess pricing in period B. 17 Our design is to eliminate the bubbles effect of miscalculation caused by inexperienced traders as suggested by White (1990) and King et al. (1990). It is more useful and realistic to study the formation and control of bubbles in markets of experienced traders.

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idends. The tournament compensation, while increasing the incentive to win, does not necessarily equate to higher equilibrium prices. The prize is paid to the largest (realized and unrealized) relative capital gains that can be achieved in either a bull or bear market. Extraordinary results are shown in Fig. 9.4 where five massive price bubbles are observed in each of the A periods. At this point, a new learning phase was initiated as traders competed strategically for the tournament prize. The dominant initial strategy centered on buying all available assets at increasing price levels thereby creating artificial price support for capital gains. While this often resulted in achieving the prize, it also meant dealing with an inventory of overvalued assets in period B. Some traders actually lost money for the year even though they obtained the prize. It is important to note that the bubbles did not discourage the traders from participating, and at least for awhile the number willing to participate actually increased. Examination of asset holdings reveals that there were four to five active prize seekers in later bubbles versus one to two initially. In addition, seven to nine traders continued to hold securities at the periods’ end rather than to sell out at extremely high bubble levels. The much higher increased tournament reward structure for “superstar” performers of periods 4 and 5 (see Schedule Two of Table 9.3) resulted in the largest bubbles (consistent with our predictions) and with the greatest variability in prices and bid-ask spreads. Again, the buying frenzy in period 5 was led by different traders than those in period 4. This continued rotation in trading leadership highlights that the results are not driven by a few misinformed traders. In fact, period B prices are very stable and efficiently priced. Furthermore, a trader questionnaire survey at the end of experiment 4 revealed that traders were fully cognizant of expected dividend value, yet they looked to both dividends and market generated information to determine value. Traders stated that they were influenced by the behavior of their peers and were motivated to earn as much as possible. Several traders noted that the introduction of the tournament compensation stimulated them to take on more risk. The net result of these effects was to create a herd or bandwagon effect centered on market-generated information. Despite the earlier findings of experiment 5, we tested the validity of these bubbles in an environment where traders used their own money rather than the experimenters.18 Would such wild speculation occur when a trader’s own money was at risk? Figure 9.6 clearly shows this answer to be yes. In all five A periods, average prices are more than twice the equilibrium value. As before, period B pricing is very efficient

18

To the authors’ knowledge, this is the first time traders in an experimental market of this type have used their own money to trade and still produced bubbles.

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149

Fig. 9.6 Experiment #6

and stable. That is, although our traders engaged in bubble pricing, they arrived at it through rational means. The traders in these markets had now participated in six experiments, the most of any research to date. Yet, even in the presence of super-experienced traders we continue to find bubble formation. In addition, these traders were aware of the situation and made every opportunity to profit from the bubble.19

9.4.3 The Control of Bubbles It became readily apparent from the earlier experiments that restrictions on the supply side of the market were having an influence on market prices. Many traders found themselves bound in their actions by the institutional makeup of the experimental markets. Many of the traders suggested that they be allowed to short-sell in future experiments so as to implement sell strategies in overvalued markets. 19 For instance, new strategies were employed at various stages (which perpetuated continuing uncertainty in the markets). At one time, the market actually stood still for an extended period. Then traders began to liquidate at any price rather than to replicate their earlier strategy of waiting until late in the period to sell out at bubble prices. Other traders began to try and scalp the market by driving prices both up and down thereby generating capital gains in both price directions. Even others began to try and force losses on traders with large inventories and thereby improve their relative ranking. This was accomplished successfully in period 2A by selling at a loss (at a price below market prices) in order to create a low settle price, M (the second to last trade). Other attempts at this strategy followed in all remaining A periods. Nevertheless, bubbles persisted and many traders were frustrated in their inability to arbitrage them away.

As previously mentioned, the tournament compensation system does not alter PFE prices as the prize can be achieved in any type of market environment and with any type of price pattern. Given the results of our previous experiments as well as traders’ comments,20 it appeared that buyers (longs) had an advantage over sellers (shorts). Is it possible that the bubbles we observe were due to differential market position in addition to a shortened investment horizon? In order to answer this question, we conducted four more experiments that provided traders with initial endowments and better equated the position of buyers and sellers. Rather than being endowed with two securities and 10,000 francs of trading capital as before, each trader is initially endowed with five securities and 3,000 francs (see Table 9.3).21 The price patterns of experiments 7–10 are as startling as the dramatic bubbles earlier. We find that the market is immediately priced at a discount to PFE.22 This had never 20 Traders completed survey questionnaire at the completion of experiments 4, 6, and 10. 21 Given that experiment 6, period A prices averaged around 600, initial trading capital of 3,000 francs would provide buying power of roughly five securities. Consequently, the new buying power and selling power were a prior relatively equal. Even though period A prices turned out to be quite a bit lower in experiments 7–10, this did not create a great advantage to buyers since the supply of securities (5 traders 12 traders D 60) was relatively large for a 6-min trading period. As such, there was an ample supply of securities relative to buying power in order to drive prices down should traders turn bearish. 22 We are unable to recruit all 12 traders back for experiments 7–10 due to graduation, taking of jobs, etc. We were, however, able to retain 7 of the original 12 traders. These traders had now participated in six previous experiments. The five replacements were drawn from the original pool of subjects that had completed the risk attribute questionnaires. These new traders were chosen to replace the risk types that had

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happened in any of the tournament periods before. If this had been simply the result of learning, we would have expected a gradual decline from the lofty levels of experiments 4 and 6. Rather, we see an immediate discount price that generally remains at a discount throughout all four experiments.23 Overall, we consider this to be strong evidence that a necessary condition for the creation of the large bubbles of these markets is that the institutional environment be biased toward more purchasing ability relative to that of selling. In summary, experiments 7–10 highlight the importance of the supply of securities and the supply of investable funds that may be augmented by short selling. Bubbles observed in experiments 4 and 6 are immediately eliminated when the relative purchasing advantage of long traders is removed. Rational pricing reflecting a modest risk premium results even when traders are faced with a shortened investment horizon.

9.4.4 The Impact of Risk Aversion The results of previous experiments, especially 6, showed that trader risk aversion was an important factor in determining trader strategy and therefore price patterns. In general, it was found that speculative traders were more likely to seize

vacated, so that, in general, we maintained a wide dispersion of risk types within the market. In addition, some of these new traders had sat in as observers to previous experiments. Others viewed videos of the earlier experiments. All were instructed in the past experimental results and the various strategies previously used were explained. As such, we do not believe that this change is a critical factor in the continuation of our investigation. 23 An analysis of many of the last trades of period A for experiments 7–10 often shows either a sharp spike up or down. This illustrates that the traders had become very efficient (through learning) in their manipulation of closing prices. Given the large supply of securities available to squelch a price bubble, speculators were no longer singularly (due to large initial endowments of trading capital) able to create capital gains by driving market prices up. With this constraint, they quickly learned that all they needed to accomplish was to purchase the most securities at current prices and then drive the market up on the final few trades. This was often easily accomplished in that (1) only the second to last trade needed to be higher in line with the calculation rules of the TPI, and (2) as no surprise, there was always many traders who were willing to sell their securities at a price above the current level. The art to this strategy became a matter of timing; do not try to buy the market too early lest you run out of capital, and do not be too late lest you be unable to make the second to the last trade. There did not appear to be too much of a problem for buyers in accomplishing this in experiments 7 and 8; however, starting in experiment 9, some short traders, annoyed at bullish traders getting the tournament prize, began jockeying in these last seconds with the long traders to drive prices down. The results of such feuds appear in periods 3A, 4A, and 5A of experiment 9 and each A period of experiment 10. The winner of these duels increasingly became the trader who was best able to execute his trade. Eventually, trading activity become so frantic in the last 15 s of trading that the open outcry systems of double oral auction began to break down.

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upon the opportunity created by the introduction of uncertainty (via the tournament period) in search of capital gains. In contrast, the more conservative traders were likely to allow the speculators to act first by creating a positive price trend and would simply sell at inflated prices or they would allow speculators to first initiate the “burst” of the bubble and then follow in their footsteps. Consequently, the conservative traders were often those responsible for the perpetuation of a direction initially set by speculators. The purpose of experiments 11–14 were to further test these relationships.24 We chose at this time to create two separate trading groups according to risk aversion, each composed of 12 traders. These 24 traders were chosen from a pool of 70 students that completed the risk-ranking questionnaire described earlier. The 70 respondents were rank ordered from highest to lowest in risk aversion. The top 12 and bottom 12 students were chosen to participate in the experiments. This method allows us to obtain good separation according to risk aversion. Contrary to our previous experiments, these markets would be made up entirely of one risk aversion class. We label these two risk classes as speculators and conservatives. This is a relative nomenclature as all of these traders are considered to be risk averse, and we only presume to provide an ordinal measure of risk aversion Figs. 9.7–9.10. The design of these experiments follows that of experiments 1–6, as we wish to test for the presence of bubbles and the initial endowments of experiments 7–10 have already been shown to eliminate bubbles. All of these traders had previously participated in two experimental markets and therefore can be considered experienced. Nevertheless, we test for rationality of pricing in experiments 11 and 13 before introducing the shortened horizons in experiments 12 and 14. Figures 9.11–9.13 reveal that both markets are quite rational in that they charge a discount from PFE as a risk premium. As expected, the conservative traders of experiment 13 charge a larger risk premium than the speculative traders of experiment 11. This result provides strong evidence in support of our measure/separation of risk aversion. We also note that the speculative group exhibits prices above the PFE levels of period B. This is consistent with our earlier results where this was found in the single period case Fig. 9.14. Experiments 12 and 14 introduce the tournament compensation schedule to induce a shorter investment horizon. As expected, the speculative group seizes upon the opportunity and price bubbles are generated in the latter periods. Also to no surprise, the conservative group does not create

24

Experiments 11–14 were conducted at a second university and, therefore, the results provide information about the external validity of our experiments outside the setting of a single university.

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Fig. 9.7 Experiment #7

Fig. 9.8 Experiment #8

the pressure necessary to cause bubbles to form. As a result, we conclude that a necessary condition for asset bubbles is the presence of speculators.25

9.4.5 The Formation of Negative Bubbles We have just learned that the effect of the reduced investment horizon is to increase the incentive for short-term speculative

25

A detailed examination of individual trades reveals the speculative group of traders are found to be more innovative in designing new trading strategies both in the creating and bursting of bubbles. The finding is consistent with the observation made by Friedman (1992) in his review of a dozen NBER working papers on asset pricing. He finds these recent research results demonstrate that rational speculative behaviors such as

an attempt by investors to learn from other investors, to affect another’s opinion, or to simply engage in protective trading could in some context, such as imperfect information, magnify price fluctuations.

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Fig. 9.9 Experiment #9

Fig. 9.10 Experiment #10

gains and that speculative traders are those most eager to earn these profits. We now extend the research design to investigate the question of whether negative bubbles are also possible. We test this proposition by conducting four new experiments (labeled as experiments 15–18 in Table 9.1). We conduct experiments 15 and 16 as “controls” to replicate the positive bubble environment found in experiments 4, 6, and 12. Experiments 15 and 16 validate our previous results with a new set of experimental subjects while Figs. 9.15 and 9.16

plot the pattern of close prices relative to the equilibrium level (horizontal line). In both experiments large positive bubbles emerge in most trading years. We now pose the following question, “Would an environment opposite to that of Design 2 lead to negative bubbles?” We keep the structure of Design 2, but since it was the unequal endowment effect (more purchasing power versus selling pressure, under 2 securities, 10,000 francs) that created the ability to pursue profits in a positive bubble

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Fig. 9.11 Experiment #11

Fig. 9.12 Experiment #12

environment, we reverse the endowment effect in experiments 17 and 18 by providing 10 securities and 1,000 francs to each trader. This one change provides traders in experiments 17 and 18 with a much greater ability to buy relative to sell. The results plotted in Figs. 9.17 and 9.18 show a preponderance for negative bubbles. While the initial four years of experiment 17 show some learning adjustment to this new and difficult trading scheme, large price discounts emerge to the extent that period 5A’s closing price is insignificantly different than period 5B’s, which is a single period receiving only a single dividend. By experiment 18, each year shows

downward trending markets in each A period. The reader may notice that the positive bubbles seem to burst while the negative bubbles don’t. However, since our design did not allow more cash to be made available through borrowing or infusion, correction may not be observed in the short trading period.

9.4.6 Statistical Analysis Table 9.4 summarizes regression analyses of the impact of the variables just discussed upon the divergence of asset

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Fig. 9.13 Experiment #13

Fig. 9.14 Experiment #14

prices from their PFE levels in period A. In particular, we test the following relation: PL PFE D f .I; E; I E; T; I E T; S; I S; A; I A; $/ (9.5) where PL -PFE, the deviation from equilibrium for period A of each trading year where PL is the last trade of the period and PFE is the Perfect Foresight Equilibrium price; f , a linear additive model; I, a dummy variable representing the shortened Investment horizon according to Table 9.1. I D 1 for shortened horizon, and 0 otherwise (i.e., exper-

iments 4,610,12,14); E, a dummy variable representing the Endowment effect according to Table 9.1. E D 1 when 2 securities are issued, and 0 otherwise (i.e., experiments 1–6, 11–14); I E, an interaction dummy variable representing both a shortened investment horizon and two security endowment (i.e., experiments 4, 6, 12, 14); T, a dummy variable representing the Tournament effect according to Table 9.1. T D 1 when there is a tournament prize for two traders only, and 0 otherwise (i.e., experiments 4, 6, 12, 14 (years 4 and 5) and 9, 10); I E T, an interaction dummy variable representing a shortened investment horizon, a two security

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting Fig. 9.15 Experiment #15

Fig. 9.16 Experiment #16

Fig. 9.17 Experiment #17

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Fig. 9.18 Experiment #18

endowment, and a tournament effect (i.e., experiments 4, 6, 9, 10 (years 4 and 5)); S, a dummy variable representing the extent to which speculators participated in the experiments according to Table 9.1. S D 1 for experiments 11, 12 and 0 otherwise; I S, an interaction dummy variable representing the shortened investment horizon and a pure speculative trader market (i.e., experiment 12); A, the ratio of end-ofperiod asset inventory for speculative traders to total asset holdings. Speculative traders are those who scored in the top one-half of the risk measurement questionnaires; I A, an interaction variable for shortened investment horizon and ratio asset holdings for speculators (experiments 4, 6–10, 12, 14); $, a dummy variable representing experiments where traders risked their own money according to Table 9.1. $ D 1 when own money is used, and 0 otherwise (i.e., experiments 5 and 6). Due to their differential design, the results for experiments 1–10 appear separately in Panel A and those for experiments 11–14 in Panel B. Model 1 of Panel A tests the impact of (1) I D 1, a shortened horizon, (2) E D 1, a restricted endowment effect (wealth and supply effects), and (3) I D 1; E D 1, an interaction of a shortened horizon with restricted initial endowment. Given that the regression was run with no intercept, the coefficients represent estimates of each variable’s independent impact. The results suggest that neither a shortened investment horizon nor a biased endowment effect (advantage to “bulls” versus “bears”) is sufficient to induce bubble behavior. However, the interaction of these two variables is highly significant in explaining the bubble results of these experiments. That is, an environment that provides both the incentive and the ability to profit from a bubble will likely result in positive price divergence.

As hypothesized earlier, we test for the heightened effect of tournament incentives (i.e., T D 1) by examining the effect of “superstar” prizes paid to only the top two traders (as outlined in Tables 9.1 and 9.3). We also test for an interaction effect with a shortened horizon .I D 1/ and restricted endowment .E D 1/. Model 2 results are consistent with Model 1 in that a tournament effect is not sufficient in itself (t D 0:13 on T variable); however, in conjunction with a reduced horizon and restricted endowment, the tournament interacts to explain a significant part (t D 4:57 on I E T) of the bubbles in these experiments. In Model 3, we observe the impact of speculative traders vis à vis conservatives by introducing a measure of asset purchase activity. The end-of-period asset holdings for the speculative group (the top one half of traders in risk ratings) is compared to the total asset endowment for all traders. In the absence of any effect, assets should be evenly divided and this ratio, A, should be equal to 5. The results of Model 3 indicate that speculators independently do not impact the presence of a bubble (t D 0:01 for A); however, when speculators operate within a shortened horizon .I A/ they do significantly differentiate themselves from conservatives by buying more and contributing to positive price bubbles. Finally, the impact of the use of the trader’s own money is shown not to significantly alter the effects of the price bubbles (t D 0:54 for $). The R2 of 80 suggest that the vast majority of price deviation from PFE levels can be explained by investment horizon, endowment effects, and risk aversion. Panel B reports the results for experiments 11–14 where markets were either composed of all speculators (11, 12) or all conservatives (13, 14). Due to this makeup, variables A and I*A are not defined in these regressions, although S and I*S are substituted in their place and represent the speculative

represents the deviation from equilibrium for Period A of each trading year where PL is the last trade of the period and PFE is the Perfect Foresight Equilibrium price. a dummy variable representing the shortened Investment horizon according to Table 9.1. I D 1 for shortened horizon, and 0 otherwise (i.e., experiments 4,6–10,12,14) is a dummy variable representing the Endowment effect according to Table 9.1 E D 1 when 2 securities are issued, and 0 otherwise. (i.e., experiments 1–6, 11–14) is an interaction dummy variable representing both a shortened investment horizon and two security endowment (i.e., experiments 4,6,12,14) is a dummy variable representing the Tournament effect according to Table 9.1. T D 1 when there is a tournament prize for two traders only, and 0 otherwise (i.e., experiments 4,6,12,14 (years 4&5) and 9,10) is an interaction dummy variable representing a shortened investment horizon, a two security endowment, and a tournament effect (i.e., experiments 4,6 (years 4&5), and 9, 10). is a dummy variable representing experiments composed entirely of Speculators according to Table 9.1. S D 1 for experiments 11 & 12 and 0 otherwise. is an interaction dummy variable representing the shortened investment horizon and a pure speculative trader market (i.e., experiment 12) represents the ratio of end-of-period Asset inventory for speculative traders to total asset holdings. Speculative traders are those who scored in the top one-half of the risk measurement questionnaires. is an interaction variable for shortened investment horizon and ratio asset holdings for speculators (experiments 4,6–10,12,14) is a dummy variable representing experiments where traders risked their own money according to Table 9.1. $ D 1 when own money is used, and 0 otherwise (i.e., experiments 5 & 6)

b As reported in Table 9.1, experiments 1–10 consisted of traders with a wide range (mixed) of risk aversion. Experiments 11 & 12 were composed only of speculative traders with experiments 13 & 14 composed of conservatives. c NOINT means the regression was run by suppressing the intercept.    , , signify statistical significance levels at .10, .05, and .01, respectively.

I A $

I E T S I S A

PL -PFE I E I E T

Table 9.4 The Impact of Investment Horizon, Credit/Supply Constraints, Risk Aversion, and Other Variables This table shows the extent to which certain variables cause a deviation from perfect foresight equilibrium values.a The following regression is estimated separately for experiments 1–10 and 11–14 according to the experimental design of Table 1. PL -PFE D f(I, E, I E, T, I E T, S, I S, A, I A, $)  Model Intercept I E I E T I E T S I S A I A $R2 Panel A Experiments 1–10 .n D 50/b 1 NOINTc 45:1 .1:48/ 38:5 .1:26/ 512.3 (8.40) .67 .79 2 48:1 .1:43/ 9.6 (.23) 311.8 (6.28) 6.1 (0.13) 382.3 (4.57) 145.1 (1.80) 195.9 (2.23) 17.3 (0.38) 282.5 (3.08) 0.8 (0.01) 292.4 (1.77) 23:6 .0:54/ .80 3 178:1 .2:73/ Panel B Experiments 11–14 .n D 20/ 4 150:5 .11:17/ 109.1 (6.98) 89.9 (4.92) .75 61.1 (2.83) 67.0 (3.17) 45.8 (1.53) .83 5 139:0 .9:30/ 61.8 (2.71) a t-values in parentheses. Variables defined as follows:

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Table 9.5 Negative Bubble and Single Period Results This table shows the extent to which endowment in conjunction with other variables causes a deviation from perfect foresight equilibrium values.a The following regression is estimated separately for experiments 15–18, 19–22, and 23–26 according to the experimental design of Table 9.1.  PL -PFE D f .NI; E; I E; EN ; I EN / Model

Intercept

I E

NI

I EN

R2

201:3 .2:34/

.55

92:9 .6:89/

.57

208:0 .4:08/

.80

Panel A Experiments 15–18 .n D 20/ 6

NOINTb

45.0 (0.37)

400.6 (4.67)

Panel B Experiments 19–22 .n D 40/ 7

NOINTb

22:8.1:60/

22.5 (2.36)

Panel C Experiments 23–26 .n D 20/ 8 a

I E I E EN I EN 

174.5 (3.95)

t-values in parentheses. Variables defined as follows:

PL -PFE

b

130:0 .3:61/

represents the deviation from equilibrium for Period A of each trading year where PL is the last trade of the period and PFE is the Perfect Foresight Equilibrium price. is a dummy variable representing the shortened Investment horizon according to Table 9.1. I D 1 for shortened horizon, and 0 otherwise. is a dummy variable representing the Endowment effect according to Table 9.1. E D 1 when 2 securities are issued, and 0 otherwise. is an interaction dummy variable representing both a shortened investment horizon and two security endowment. is a dummy variable representing the sell side of the Endowment effect hypothesized to lead Negative bubbles. E D 1 when 10 securities are issued and 0 otherwise. is an interaction dummy variable representing both a shortened investment horizon and a ten security environment.

NOINT means the regression was run by suppressing the intercept. ;  ;  signify statistical significance levels at .10, .05, and .01, respectively.

markets (11 and 12) and the interaction of shortened horizon with a speculative market (12). In addition, a restricted endowment effect .E D 1/ is imposed for experiments 11–14 since experiments 7–10 clearly established their necessity in creating bubbles. Model 4 results highlight the significant positive effect of the combined shortened horizon/restricted endowment effect (t D 6:98 for I). More importantly, the speculative group statistically differs from conservatives with an additional mean price difference of 89.9 .t D 4:92/. Model 5 supports the results of experiments 1–10 in that: (1) a shortened investment horizon with restricted endowments leads to price bubbles .t D 2:71/ for I, (2) a heightened tournament incentive will heighten short-term horizons and lead to positive price effects (t D 2:83 for I T), and (3) speculators contribute to positive price bubbles in restricted endowment environments (t D 3:17 for S).26 The visual analysis of experiments 15–18 (negative bubble experiments) is confirmed by the regression results reported in Table 9.5. The variables are as defined earlier under equation 5 albeit the EN representing a dummy variable for the negative endowment effect. EN D 1 when the initial endowment equals 10 securities and 1,000 firms and 0 otherwise. In addition, since the shortened horizon variable I occurs for all years except 1A of each experiment, I and Furthermore, although insignificant, the p-value for I  S is equal to 14 suggesting that the speculative difference may be even greater under a shortened investment horizon.

26

E are highly correlated. The design is therefore set to only measure the interaction effects of a shortened horizon and endowment. The four periods (1A of each experiment) are the control periods where a shortened horizon is not present (dummy NI D 1 for not I). The parameter estimates of Model 6 show significant positive results for both positive and negative bubbles. The joint presence of a shortened horizon induced by a tournament payoff along with a buy side endowment (2 securities, 10,000 firms); that is, I E D 1, leads to an average increase of 400.6 francs in price levels. The single alteration of the endowment to sell side (10 securities, 1,000 francs) in the presence of a tournament leads to an average decrease in price of 201.3 francs. The estimate for NI reflects the insignificant impact of the control periods where the endowment effect is present but without the tournament payoff inducing a shortened horizon. So as in the earlier results, the combined effect of the incentive (i.e., the tournament) and the ability (i.e., the endowment) work to create both positive and negative price bubbles.

9.4.7 Further Tests To check the robustness of our results we conducted eight final experiments in a unique and different setting, the former communist country of Albania. Of its many unique

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

characteristics, one of the most important is its history of being the most isolated (politically and economically) country in Europe since World War II. Since democratic reforms began in 1991, a new business school was opened in the second largest city of Albania, Shkodra, where the third year students served as traders. Would the students whose country didn’t have a securities market or a history of free market trade show the same results as we had found at U.S. universities? While our previous experiments had the most experienced traders ever used in a study, these Albanian students may indeed represent the least experienced traders examined to date, which may be regarded as an extreme test of the validity of our results. Because of the newness of the trading experience for these students, a single period design was used in the first four experiments. For each experiment’s ten trading years (no period B), asset payoffs were for a single dividend payoff. The amounts used were the same as those of Table 9.2 so that equilibrium levels remained at 230 for each year. As shown in Table 9.1, Design 7 (experiment 19) consists of a single period security without a tournament effect. Design 8 (experiments 20 and 21) introduces the tournament payoff of Table 9.3 (Schedule Two) within the single period environment. This allows us to test for the presence of bubbles in the simpler pricing environment while also easing the learning experience of the Albanian students toward two-period tournament pricing. The pricing results for these three experiments can be seen in Figs. 9.19–9.21. Without the tournament in experiment 19, pricing is rational and typical showing a discount (risk pre-

Fig. 9.19 Experiment #19

159

mium) of about 30 francs from the equilibrium level of 230. Near the end of experiment 20, the tournament effect appears to have created some price movement above equilibrium. This pressure continues into experiment 21 where prices trade at an average premium of 30 francs. Although these premiums do not constitute a bubble, it is clear they had a significant positive impact on pricing levels. Would this effect be eliminated (reversed) by changing the buy/sell pressure as was done earlier under Design 6 where negative bubbles were induced? Design 9 tests this proposition by changing the endowment from 2 securities and 5,000 francs to 20 securities and 500 francs. The results, reported in Fig. 9.22, show that even in these simple markets the endowment effect combined with tournament payoff leads to pricing away from equilibrium. These observations are confirmed by the regression results of Model 7 in Table 9.5 where buy side preference .I E D 1/ leads to significant increase in prices while sell side preference .I EN D 1/ leads to lower prices. The absence of a tournament payoff .NI D 1/ leads to insignificant price effects as investment horizon cannot be altered in a single period market. The Albanian students had now participated in four single period experiments and were ready to attempt two-period pricing. Experiment 23 was a simple two period pricing environment without any tournament payoff as in Design 1 (control). The plot of prices in Fig. 9.23 shows that the students initially struggled with two-period pricing since period A prices (two payoffs) differed little from period B prices (single payoff), although by the end of the experiment enough learning had developed.

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Fig. 9.20 Experiment #20

Fig. 9.21 Experiment #21

Experiments 24 and 25 introduce the shortened investment horizon (tournament effect) within the two-period framework as in Design 2 earlier. The price patterns in Figs. 9.24 and 9.25 show the creation of positive price bubbles to levels approaching 650 francs. Despite the histor-

ical background of this country and these students, they responded to market pressures in the same bubble-like manner. The last experiment, 26, alters the endowment to the sell side as before to see if negative bubbles can also be obtained. Price paths in Fig. 9.26 show a general downward trend of

9 The Creation and Control of Speculative Bubbles in a Laboratory Setting

161

Fig. 9.22 Experiment #22

Fig. 9.23 Experiment #23

prices. The prices in period A show significant and growing discounts from the equilibrium levels of 460. These observations are confirmed by the regression results reported in Panel C of Table 9.5 with buy side endowment contributing 174.5 francs and sell side endowment reducing levels by 208.0 francs.

9.5 Conclusions The results of this study have a number of implications for real world markets. Experiments 1–6 seem to imply that within an environment that restricts selling pressures,

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Fig. 9.24 Experiment #24

Fig. 9.25 Experiment #25

a shortened investment horizon is sufficient to create asset bubbles. In application to the real world, short-term performance of traders, portfolio managers, and so forth could create pressures leading to price bubbles. Experiments 7–10 provide restrictions to the previous conclusion in that a shortened investment horizon creates bubble pressure only

when the market environment favors buyers over sellers. Unfortunately, most of our real world securities markets do have such a bias via restricted short sales, asymmetric leverage for longs versus shorts, restricted options and futures, and the like. Experiments 11–14 add to the puzzle by demonstrating the role of speculators within bubble formation. As a whole,

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163

Fig. 9.26 Experiment #26

the study suggests that necessary and sufficient conditions for the formation of asset bubbles are a shortened investment horizon, restricted selling activity relative to buyers, and the presence of speculators. We have also shown that repeated replication of these experiments under different settings still produce robust results. The first and third variables are a matter of fact within U.S. securities markets while restricted selling activity relative to buyers can take many forms. Either enhancing the buyer’s position or restricting the seller’s position is sufficient. Examples include increasing purchasing (speculative) ability through reduced stock margin levels, introduction of high leverage stock index futures, and in macroeconomic terms, a growing money supply or savings level. This latter variable may help explain the previous high levels of the Japanese equity market. The high level of Japanese savings creates very large endowments available for investment purchase. Given a limited supply of securities, our experimental markets show that these conditions will lead to a bubble. They also suggest that the bubble will burst when there is greater equating between the supply and demand. Recent changes in the Japanese institutional framework may, as predicted by this study, have lead to the bursting of that bubble. The primary prescription put forth for regulatory authorities to eliminate unnecessary market volatility resulting from asset bubbles is to create an institutional environment that does not restrict the transfer of information to the market. Structure the variables so that both bulls and bears have equal costs in executing their trades.

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164 DeLong, J. B., A. Schleifer, L. H. Summers and R. T. Waldmann. 1989. “The size and incidence of the losses from noise trading.” Journal of Finance 44, 681–696. Diba, B. T. and H. Grossman. 1987. “On the inception of rational bubbles.” Quarterly Journal of Economics 102, 197–700. Dufwenberg, M., T. Lindqvist and E. Moore. 2005. “Bubbles and experience: an experiment.” The American Economic Review 95, 1731–1737. Durand, R., R. Newby and J. Sanghani. 2006. “An intimate portrait of the individual investor.” Working paper. Ehrenberg, R. G. and M. L. Bognanno. 1990. “Do tournaments have incentive effects?” Journal of Political Economy 98, 1307–1324. Flood, R. P. and R. J. Hodrick. 1990. “On testing for speculative bubbles.” Journal of Economic Perspectives 4, 85–102. Forsythe, R., T. Palfrey and C. Plott. 1982. “Asset valuation in an experimental model.” Econometrica 50, 537–563. Friedman, B. M. 1992. “Monetary economics and financial markets: retrospect and prospect.” NBER Reporter, 1–5. Froot, K. A., D. S. Scharfstein and J. C. Stein. 1992. “Herd on the street: informational inefficiencies in a market with short-term speculation.” Journal of Finance 47, 1461–1484. Garber, P. 1990. “Famous first bubbles.” Journal of Economic Perspectives 4, 35–54. Gurkaynak, R. S. 2005. “Econometric tests of asset price bubbles: taking stock,” REDS Working paper No. 2005–2004. Harrison, M. and D. Kreps. 1978. “Speculative investor behavior in a stock market with heterogeneous expectations.” Quarterly Journal of Economics 92, 323–336. Hart, O. D. and D. M. Kreps. 1986. “Price destabilizing speculation.” Journal of Political Economy 94, 927–953. Heinemann, F. 2003. “Risk aversion and incentive effects: comment.” Working paper (Ludwig-Maximilians-Universitat Munchen). Hirota and Sunder. 2005. “Price bubbles sans dividend anchors: evidence from laboratory stock markets.” Working paper. Holt, C. A. and S. K. Laury. 2002. “Risk aversion and incentive effects.” American Economic Review 92, 1644–1655. Hong, H. G., J. A. Scheinkman and W. Xiong. 2006. “Asset float and speculative bubbles.” Journal of Finance 61, 1073–1117. Jackson, D. 1976. “Jackson personality inventory manual,” Research Psychologists Press, Goshen, NY. Jackson, D. 1977. “Reliability of the Jackson personality inventory.” Psychological Reports 40, 613–614. Jackson, D., D. Hourany and N. Vidmar. 1972. “A four dimensional interpretation of risk taking.” Journal of Personality 40, 433–501. James, D. and R. M. Isaac. 2000. “Asset markets: how they are affected by tournament incentives for individuals.” American Economic Review 90, 995–1004.

J.S. Ang et al. King, R. R., V. L. Smith, A. Williams and M. Van Boening. 1990. “The robustness of bubbles and crashes,” Working paper, University of Washington, Seattle, WA. Lei, V., C. Noussair and C. Plott. 2001. “Nonspeculative bubbles in experimental asset markets: lack of common knowledge of rationality vs. actual irrationality.” Econometrica 69(4), 831–859. Plott, C. and S. Sunder. 1982. “Efficiency of experimental markets with insider information: an application of rational expectations models.” Journal of Political Economy 90, 663–698. Porter, D. P. and V. L. Smith. 2003. “Stock market bubbles in the laboratory.” Journal of Behavioral Finance 4(1), 7–20. Ricke, M. 2004. “What is the link between margin loans and stock market bubbles?” University of Muenster, Department of Banking No. 03–01. Robin, S. and B. Ruffieux. 2001. “Price bubbles in laboratory asset markets with constant fundamental values.” Experimental Economics 4, 87–105. Roll, R. 1989. “Price volatility, international market links, and their implications for regulatory policy.” Journal of Financial Services Research 3, 211–246. Scheinkman, J. A. and W. Xiong. 2003. “Overconfidence and speculative bubbles.” Journal of Political Economy 111, 1183–1219. Shiller, R. J. 1988. “Fashions, fads, and bubbles in financial markets,” in Knights, raiders, and targets, J. Coffee Jr., L. Lowenstein and S. Ross-Ackerman (Eds.). Oxford Press, New York. Shleifer, A. and R. W. Vishny. 1990. “Equilibrium short horizons of investors and firms.” American Economic Review 80, 148–153. Siegel, J. J. 2003. “What is an asset price bubble? an operational definition.” European Financial Management 9, 11–24. Smith, V., G. Suchanek and A. Williams. 1988. “Bubbles, crashes, and endogenous expectations in experimental spot asset markets.” Econometrica 56, 1119–1151. Stiglitz, J. 1990. “Symposium on bubbles.” Journal of Economic Perspective 4, 13–18. Sunder, S. 1992. “Experimental asset markets: a survey,” in Handbook of experimental economics, J. H. Kagel and A. E. Roth (Eds.). Princeton University Press, Princeton, NY. Voth, H.-J. and Temin, P. 2003. “Riding the South Sea Bubble,” MIT Economics, Working Paper No. 04–02. West, K. 1988. “Bubbles, fads, and stock price volatility tests: a partial evaluation.” Journal of Finance 43, 639–655. White, E. 1990. “The stock market boom and crash of 1929 revisited.” Journal of Economic Perspectives 4, 67–84.

Chapter 10

Portfolio Optimization Models and Mean–Variance Spanning Tests Wei-Peng Chen, Huimin Chung, Keng-Yu Ho, and Tsui-Ling Hsu

Abstract In this chapter we introduce the theory and the application of the computer program of modern portfolio theory. The notion of diversification is age-old: “don’t put your eggs in one basket,” obviously predates economic theory. However, a formal model showing how to make the most of the power of diversification was not devised until 1952, a feat for which Harry Markowitz eventually won the Nobel Prize in economics. Markowitz portfolio shows that as you add assets to an investment portfolio the total risk of that portfolio – as measured by the variance (or standard deviation) of total return – declines continuously, but the expected return of the portfolio is a weighted average of the expected returns of the individual assets. In other words, by investing in portfolios rather than in individual assets, investors could lower the total risk of investing without sacrificing return. In the second part we introduce the mean–variance spanning test that follows directly from the portfolio optimization problem. Keywords Minimum variance portfolio r Optimal risky portfolio r Capital allocation line r Mean–variance spanning tests

10.1 Introduction of Markowitz Portfolio-Selection Model Harry Markowitz (1952, 1959) developed his portfolioselection technique, which came to be called modern portfolio theory (MPT). Prior to Markowitz’s work, securityselection models focused primarily on the returns generated W.-P. Chen () Department of Finance, Shih Hsin University, Taipei, Taiwan e-mail: [email protected] H. Chung and T.-L. Hsu Graduate Institute of Finance, National Chiao Tung University, Hsinchu, Taiwan e-mail: [email protected]; [email protected] K.-Y. Ho Department of Finance, National Taiwan University, Taipei, Taiwan e-mail: [email protected]

by investment opportunities. Standard investment advice was to identify those securities that offered the best opportunities for gain with the least risk and then construct a portfolio from these. Following this advice, an investor might conclude that railroad stocks all offered good risk-reward characteristics and compile a portfolio entirely from these. The Markowitz theory retained the emphasis on return; but it elevated risk to a coequal level of importance, and the concept of portfolio risk was born. Whereas risk has been considered an important factor and variance an accepted way of measuring risk, Markowitz was the first to clearly and rigorously show how the variance of a portfolio can be reduced through the impact of diversification. He proposed that investors focus on selecting portfolios based on their overall risk-reward characteristics instead of merely compiling portfolios from securities that each individually have attractive risk-reward characteristics. A Markowitz portfolio model is one where no added diversification can lower the portfolio’s risk for a given return expectation (alternately, no additional expected return can be gained without increasing the risk of the portfolio). The Markowitz Efficient Frontier is the set of all portfolios of which expected returns reach the maximum given a certain level of risk. The Markowitz model is based on several assumptions concerning the behavior of investors and financial markets:

1. A probability distribution of possible returns over some holding period can be estimated by investors. 2. Investors have single-period utility functions in which they maximize utility within the framework of diminishing marginal utility of wealth. 3. Variability about the possible values of return is used by investors to measure risk. 4. Investors care only about the means and variance of the returns of their portfolios over a particular period. 5. Expected return and risk as used by investors are measured by the first two moments of the probability distribution of returns-expected value and variance.

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10.2.2 Risk

6. Return is desirable; risk is to be avoided.1 7. Financial markets are frictionless.

10.2 Measurement of Return and Risk Throughout this chapter, investors are assumed to measure the level of return by computing the expected value of the distribution, using the probability distribution of expected returns for a portfolio. Risk is assumed to be measurable by the variability around the expected value of the probability distribution of returns. The most accepted measures of this variability are the variance and standard deviation.

The variance of a single security is the expected value of the sum of the squared deviations from the mean, and the standard deviation is the square root of the variance. The variance of a portfolio combination of securities is equal to the weighted average covariance2 of the returns on its individual securities: n n X X     wi wj Cov ri ; rj Var rp D p2 D

(10.2)

i D1 j D1

Covariance can also be expressed in terms of the correlation coefficient as follows:   Cov ri ; rj D ij i j D ij

(10.3)

10.2.1 Return Given any set of risky assets and a set of weights that describe how the portfolio investment is split, the general formula of expected return for n assets is: E.rP / D

n X

wi E.ri /

where ij D correlation coefficient between the rates of return on security i; ri , and the rates of return on security j; rj I i , and j represent standard deviations of ri and rj respectively. Therefore: n n X   X wi wj ij i j Var rp D

(10.1)

i D1

where

n P

wi D 1:0; n D the number of securities, wi the

i D1

proportion of the funds invested in security i; ri ; rP the return on i th security and portfolio p; E. / the expectation of the variable in the parentheses. The return computation is nothing more than finding the weighted average return of the securities included in the portfolio.

1 Markowitz model assumes that investors are risk averse. This means that given two assets that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher returns must accept more risk. The exact tradeoff will differ by investor based on individual risk aversion characteristics. The implication is that a rational investor will not invest in a portfolio if a second portfolio exists with a more favorable risk-return profile (i.e., if for that level of risk an alternative portfolio exists that has better expected returns). Using risk tolerance, we can simply classify investors into three types: risk-neutral, risk-averse, and risk-lover. Riskneutral investors do not require the risk premium for risk investments; they judge risky prospects solely by their expected rates of return. Riskaverse investors are willing to consider only risk-free or speculative prospects with positive premium; they make investment according the risk-return tradeoff. A risk-lover is willing to engage in fair games and gambles; this investor adjusts the expected return upward to take into account the “fun” of confronting the prospect’s risk.

(10.4)

i D1 j D1

Overall, the estimate of the mean return for each security is its average value in the sample period; the estimate of variance is the average value of the squared deviations around the sample average; the estimate of the covariance is the average value of the cross-product of deviations.

10.3 Efficient Portfolio Efficient portfolios may contain any number of asset combinations. We examine efficient asset allocation by using two risky assets for example. After we understand the properties of portfolios formed by mixing two risky assets, it will be easy to see how the portfolio of many risky assets might best be constructed.

2

High covariance indicates that an increase in one stock’s return is likely to correspond to an increase in the other. A low covariance means the return rates are relatively independent and a negative covariance means that an increase in one stock’s return is likely to correspond to a decrease in the other.

10 Portfolio Optimization Models and Mean–Variance Spanning Tests

167

10.3.1 Two-Risky-Assets Portfolio

E.rP / D wA E.rA / C wB E.rB /

asset B

r=0

r=1 0  when the horizon is long. The benefit of the optimal strategy is a non-monotone function of the persistence parameter , tending to be greatest for  D 0:10 at the longer horizons. The short run predictability of returns as measured by the 1-year R2 is not, however, necessarily associated with higher certainty equivalent for the optimal strategy relative to the unconditional strategy. This is illustrated in Fig. 19.3, where the certainty equivalent ratio under the optimal and the unconditional strategies for a 20-year horizon, CEW Rou , is plotted against the 1-year R2 for the nine scenarios. Interestingly, Table 19.3a shows that there is also no clear relation between the advantage of the optimal strategy and the long run predictability of returns (20-year regression R2 ) when  D : the correlation between the 20-year CEW R across scenarios and the 20-year R2 is 0:05. Less surprisingly, since  carries more information about future investment opportunities when the R2 is high, the 20-year CEW R has a correlation of 0.48 with the 20-year R2 when  is one standard deviation below its mean and a correlation of 0.57 when  is one standard deviation above its mean.

301

Finally, it is interesting to note that there is also no relation between CEW Rou and the monthly or annual out-of-sample predictability as measured by the root mean squared error (RMSE).21 In summary, the gains to the optimal strategy can be very large for long horizon investors. For scenarios (v) and (vi), which correspond closely to the behavior of the expected rate of return that we shall extract from forecasts of long run rates of return, the gains run from 16% to 125% over a 20-year horizon, depending on the initial value of . The gains of the optimal strategy come from both market timing and hedging. Market timing is simply the variation in the equity allocation with the equity premium. Hedging is the additional allocation to equities that results from the negative correlation between innovations to rates of return and returns on the equity security, which implicitly recognizes that equities are not so risky in the long run.22 The benefits of market timing, but not of hedging, are captured by the myopic rule, x m : xm 

˛ C ˇt  r yt D : 2 P P2

(19.21)

The certainty equivalent wealth associated with the myopic strategy is calculated by evaluating the expected utility associated with the myopic strategy, and the details are given in Appendix 19C. Table 19.3b reports the certainty equivalent wealth ratios between the optimal and the myopic strategies CEW Rom . The results show that the hedging gains offered by the optimal strategy but not by the myopic strategy, are zero when P D 0 and of the order of 6% at the 20-year horizon when P D 0:50, but are as high as 38–66% when P D 0:90. For comparison, the certainty equivalent wealth associated with the optimal buy-and-hold strategy, x b , which is described in Appendix 19D, was calculated numerically. The initial equity allocation of the buy-and-hold strategy depends on the value of t , but subsequent changes in the allocation are determined entirely by the realized asset returns. The certainty equivalent wealth ratios between the optimal and the buy-and-hold strategies CEW Rob , reported in Table 19.3c and Fig. 19.4g–i, show that the buy-and-hold strategy is

21 The out of sample RMSE is calculated from the simulated data used for Table 19.1; the predictive relation is estimated from the first 65 years of data, and the out-of-sample RMSE is calculated from the differences between the predicted and the realized returns over the following five years. The results, which are not reported here, are available on request. Goyal and Welch (2003, 2004) use the out-of-sample RMSE as a measure of the value of a predictive instrument. 22 Stambaugh (1999) and Barberis (2000) compare myopic and buyand-hold strategies.

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Table 19.3b Long run return predictability and the value of dynamic vs. myopic strategies Horizon

(v)

(vi)

(vii)

(viii)

(ix)

›  P  P

Scenarios

(i) 0:02 0:008 0:00 0:140

(ii) 0:02 0:008 0:50 0:142

(iii) 0:02 0:008 0:90 0:144

(iv) 0:10 0:018 0:00 0:140

0:10 0:018 0:50 0:144

0:10 0:018 0:90 0:148

0:50 0:040 0:00 0:139

0:50 0:040 0:50 0:147

0:50 0:040 0:90 0:155

1 year

R2 t D 0:05 t D 0:09 t D 0:13

7:4% 1:00 1:00 1:00

7:4% 1:00 1:00 1:00

7:4% 1:00 1:00 1:00

6:9% 1:00 1:00 1:00

6:9% 1:00 1:00 1:00

6:9% 1:00 1:00 1:00

4:8% 1:00 1:00 1:00

4:8% 1:00 1:00 1:00

4:8% 1:00 1:00 1:00

5 years

R2 t D 0:05 t D 0:09 t D 0:13

26:5% 1:00 1:00 1:00

28:8% 1:00 1:00 1:00

31:0% 1:00 1:00 1:00

18:8% 1:00 1:00 1:00

22:4% 1:00 1:00 1:00

26:8% 1:00 1:01 1:02

4:6% 1:00 1:00 1:00

5:9% 1:00 1:00 1:00

7:8% 1:01 1:01 1:02

10 years

R2 t D 0:05 t D 0:09 t D 0:13

38:0% 1:00 1:00 1:00

44:1% 1:00 1:00 1:01

50:7% 1:00 1:02 1:04

20:4% 1:00 1:00 1:00

27:8% 1:00 1:01 1:02

39:8% 1:02 1:06 1:12

2:6% 1:00 1:00 1:00

3:6% 1:01 1:01 1:01

5:6% 1:04 1:05 1:06

20 years

R2 t D 0:05 t D 0:09 t D 0:13

45:6% 1:00 1:00 1:00

56:8% 1:01 1:02 1:05

71:1% 1:04 1:16 1:41

15:9% 1:00 1:00 1:00

24:6% 1:02 1:04 1:06

46:1% 1:20 1:38 1:66

1:3% 1:00 1:00 1:00

1:9% 1:02 1:02 1:02

3:3% 1:12 1:13 1:14

40 years

R2 t D 0:05 t D 0:09 t D 0:13

43:7% 1:00 1:00 1:00

58:0% 1:04 1:09 1:20

79:3% 1:35 2:32 5:56

8:8% 1:00 1:00 1:00

15:2% 1:08 1:11 1:14

38:3% 2:10 2:76 3:86

0:6% 1:00 1:00 1:00

1:0% 1:05 1:05 1:05

1:8% 1:31 1:33 1:34

This table reports the theoretical values under different scenarios from regressions of R2 long run returns on the value of  at the beginning of the period. It also reports the ratios of the certainty equivalent wealth for an optimal dynamic strategy to the certainty equivalent p wealth under a myopic strategy for different horizons and initial values of t . The exogenous parameters are v D 4%;  D 9%; r D 3%; V .1/ D 14%; ˛ D 0, and ˇ D 1. The risk aversion parameter is  D 5

Table 19.3c Long run return predictability and the value of dynamic vs. buy-and-hold strategies Horizon

(v)

(vi)

(vii)

(viii)

(ix)

›  P  P

Scenarios

(i) 0:02 0:008 0:00 0:140

(ii) 0:02 0:008 0:50 0:142

(iii) 0:02 0:008 0:90 0:144

(iv) 0:10 0:018 0:00 0:140

0:10 0:018 0:50 0:144

0:10 0:018 0:90 0:148

0:50 0:040 0:00 0:139

0:50 0:040 0:50 0:147

0:50 0:040 0:90 0:155

1 year

R2 t D 0:05 t D 0:09 t D 0:13

7:4% 1:00 1:01 1:02

7:4% 1:00 1:01 1:02

7:4% 1:00 1:01 1:02

6:9% 1:00 1:01 1:02

6:9% 1:00 1:01 1:02

6:9% 1:00 1:02 1:03

4:8% 1:01 1:01 1:03

4:8% 1:01 1:01 1:03

4:8% 1:01 1:02 1:03

5 years

R2 t D 0:05 t D 0:09 t D 0:13

26:5% 1:02 1:07 1:14

28:8% 1:02 1:08 1:15

31:0% 1:01 1:09 1:16

18:8% 1:04 1:08 1:13

22:4% 1:04 1:10 1:16

26:8% 1:05 1:12 1:18

4:6% 1:08 1:10 1:12

5:9% 1:09 1:12 1:15

7:8% 1:10 1:14 1:17

10 years

R2 t D 0:05 t D 0:09 t D 0:13

38:0% 1:04 1:16 1:29

44:1% 1:05 1:19 1:35

50:7% 1:05 1:24 1:43

20:4% 1:10 1:17 1:26

27:8% 1:13 1:23 1:36

39:8% 1:18 1:36 1:53

2:6% 1:19 1:21 1:24

3:6% 1:24 1:27 1:30

5:6% 1:32 1:36 1:42

20 years

R2 t D 0:05 t D 0:09 t D 0:13

45:6% 1:11 1:31 1:56

56:8% 1:15 1:47 1:88

71:1% 1:21 1:83 2:57

15:9% 1:24 1:34 1:48

24:6% 1:38 1:56 1:81

46:1% 1:76 2:27 2:92

1:3% 1:44 1:46 1:51

1:9% 1:59 1:63 1:68

3:3% 1:90 1:96 2:05

40 years

R2 t D 0:05 t D 0:09 t D 0:13

43:7% 1:27 1:55 2:00

58:0% 1:47 2:12 3:28

79:3% 2:11 5:24 15:66

8:8% 1:58 1:70 1:92

15:2% 2:09 2:40 2:91

38:3% 4:81 6:73 10:28

0:6% 2:11 2:15 2:22

1:0% 2:65 2:71 2:81

1:8% 3:97 4:07 4:25

This table reports the theoretical values under different scenarios of R2 from regressions of long run returns on the value of  at the beginning of the period. It also reports the ratios of the certainty equivalent wealth for an optimal dynamic strategy to the certainty equivalent wealth under an optimal buy-and-hold strategy for different horizons and initial values of t . The exogenous parameters are v D 4%;  D 9%; p r D 3%; V .1/ D 14%; ˛ D 0, and ˇ D 1. The risk aversion parameter is  D 5

19 Persistence, Predictability, and Portfolio Planning

303

Fig. 19.5 The certainty equivalent wealth ratio under the optimal and the unconditional strategies for a 20-year horizon vs. the 1-year predictive regression R2 . The figure plots the certainty equivalent wealth ratio (CEWRou ; CEWRom ; CEWRob ) between the optimal and the unconditional for a 20-year horizon investor CEWRou , as a function of 1 year for the nine scenarios reported in Table 19.1. The initial value of  is set at respectively, 5%, 9% and 13%. The exogenous parameters are v D 4%;  D 9%; r D p 3%; V .1/ D 14%; ˛ D 0, and ˇD1

extremely inefficient.23 At the 20-year horizon the gains ofthe optimal strategy when  D 0:1 are of the order of 38– 81% when D D 0:50 and 76–192% when D D 0:90 Fig. 19.5. To this point, we have shown that time-variation in expected stock returns, which may be very difficult to detect by standard regression methods, may nevertheless imply both significant variation in stock prices that is unrelated to changes in cash flow expectations, and substantial potential gains to the use of dynamic portfolio strategies for long horizon investors. However, the gains that we have calculated assume that it is possible to observe the instantaneous expected return on the stock and, as we have seen, regression estimates of the relation between expected returns and even perfect instruments of it are likely to be very imprecise. Therefore, 23

This is in contrast to the findings of Brennan and Torous (1999) who show that a buy-and-hold strategy performs well relative to a rebalancing strategy when the investment opportunity set is treated as constant.

in the next section, we explore the use of estimates of long run expected returns derived from a dividend discount model (DDM) as inputs to dynamic portfolio models.

19.5 Forecasts of Equity Returns Regressions that attempt to predict stock returns from instruments such as the dividend yield or the interest rate lack strong theoretical restrictions on the regression coefficients and, as we have seen in Sect. 19.2, the data are likely to yield very imprecise estimates of the coefficients even when the instruments are perfect. It is not surprising therefore that such regressions have essentially no out-of-sample predictive power. An alternative to the simple regression approach is to estimate the expected return on stocks by comparing the current level of the stock market with forecasts of future

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dividends on the market portfolio – the Dividend Discount Model (DDM) approach. The advantage of this approach, which has long been employed in “Tactical Asset Allocation” models, is that the expected return is estimated directly and that there is no need to estimate a regression coefficient relating the stock (excess) return to the predictor instruments. The offsetting disadvantage is that the rate of return estimated from the DDM is a long run internal rate of return and there is no reason to believe that this will be equal to the instantaneous expected rate of return even if the dividend forecasts are unbiased. Therefore, it is necessary to develop a model to derive the expected instantaneous rate of return from the long run expected rate of return.

19.5.1 Models and Estimation Procedure We employ two models to convert the estimated DDM long run expected rate of return into an estimate of t , the instantaneous expected rate of return. The first model assumes that the dividend growth rate g is a known constant. The second model assumes that the growth rate follows an OrnsteinUhlenbeck process.24 Both methods assume that the instantaneous expected rate of return follows an O-U process as in Equation (19.2). Our basic input data are direct estimates of the DDM long run “expected rate of return,” t ,25 as defined by the discounted cash flow model: Pt D

1 X Et ŒDt C  D1

1 C t

;

(19.22)

where Pt is the level of the stock price index at time t, and Et ŒDt C is the dividend on the index expected at time t to be paid at time t C .

Note that Equation (19.23) rests on the assumption that the dividend expectations in (19.22) form a geometric series. It was shown in Sect. 19.3 that if the expected rate of return, , follows an O-U process and the dividend growth rate is constant, then the price-dividend ratio v./ also satisfies the ordinary differential equation (ODE) (pde). Since v is a monotonic function of , at each point in time there exists a t whose implied v from the ODE is equal to the pricedividend ratio associated with t from Equation (19.23) for a given value of g. This implies a (nonlinear) one-to-one mapping between t and t for a given set of parameters for the stochastic process for ;   .; v ; ; D /. As a result, these parameters and the time series of t can be estimated by an iterative process: starting with an initial value of  D  0 ; t is calculated from the time series of t ; then  is estimated from the time series of t , and a new t series is re-calculated from t ; the process continues until convergence is achieved. We denote the resulting Model 1 estimates by 1 and 1 . This iterative procedure is essentially a non-linear Kalman filter in which the latent variable t is a nonlinear function of the observable variable t . The transition equation for t is the discrete-time equivalent of the O-U process: t C t D .1  e  t / C e  t t C "

(19.24)

and the observation equation (which contains no observation error) is: t D f .t / D g C

D D g C .v.t ; //1 : P

The mapping of v implied by the DDM to the v implied by the ODE is equivalent to solving the nonlinear function f numerically.

19.5.1.2 Model 2: Stochastic g 19.5.1.1 Model 1: Constant g When the dividend growth rate, g, is constant, there is a oneto-one correspondence between t and the price-dividend ratio, P =D, which is given by the (discrete time version of the) Gordon growth model: vt 

24

Pt 1Cg ; D Dt t  g

(19.23)

Brennan and Xia (2001) assume a similar model for the dividend growth rate. 25 There is no assurance that t , the solution to Equation (19.22), will be equal to the expected rate of return except when all future dividends are known and discount rates are constant.

To allow for the possibility that dividend growth rate expectations are stochastic, the instantaneous dividend growth rate is assumed to follow an Ornstein-Uhlenbeck process: dg D g .g  g/dt C g dzg :

(19.25)

Two steps are then required to derive  from : first the instantaneous dividend growth rate, g, and the parameters of its stochastic process, g  .g ; g ; g/, are estimated and then, given the time series of gt and g , the time series of t and the parameters of its stochastic process,  are estimated.

19 Persistence, Predictability, and Portfolio Planning

305

With stochastic dividend growth, the DCF valuation formula t can be written as: X …s EŒ.1 C gt Ci /jgt Pt i D1 D ; Dt .1 C kt /s sD1 1

(19.26)

For a given set of parameters of the dividend growth process g ; EŒ.1Cgt Ci /jgt is a known linear function of gt . That is, Equation (19.26) is used to solve for the time series gt , given g , the observed series of price-dividend ratios P =D, and the DDM expected rates of return t . The estimated time series gt is then used to estimate g , and theprocedure  iterates until there is a consistent set of gt and g ; gt ; t . This procedure can again be interpreted as a Kalman filter in which g is the latent variable with the transition equation gt C t D .1  e g t /g C e g t gt C "; while the observation equation relating the observable variables  and P =D to the unobservable variable g is given implicitly by (19.26).   Conditional on gt ; t from the first step, t is determined step. The price-dividend ratio

in the second  v  v ; gI  ; g now satisfies the following two-statevariable partial differential equation: 0D

1 2 1  vgg C g  g vg C 2 v 2 g 2 CŒg .g  g/ C g D gD vg C Œ .  / C D D v C Œg   v C 1:

(19.27)

Equation (19.27) depends on the set of parameters g , which was estimated in the first step, and the unknown  , which is to be estimated together with t in the second step. For a given value of   , the PDE (19.27) is solved numerically

for v ; gI  ; g . Given gt from the first step, a mapping between t and t ; t D f .t ; gt /, is defined by setting v ; gt I  ; g equal to the observed price dividend ratio P/D. The time series of t estimates are then used to estimate a new  and the process is iterated until convergence. This can be interpreted as another Kalman filter in which the transition equation of the latent variable t is given by (19.24) while the nonlinear observation equation is: P =D D v.; gt I  ; g /: The resulting estimates from Model 2 are denoted by 2 and 2 .

19.5.2 Estimates Quarterly data on real dividends and the price-dividend ratio for the S&P 500 index are obtained for the sample period 1950.1 to 2002.2. Four different DDM discount rate series, t , are used for illustrative purpose. The first two series are estimates of the real long run expected rate of return on equities that would have been assessed by investors at each date. The Arnott and Bernstein (2002) (A&B) series,  1 , is constructed by adding to the current dividend yield an estimate of the expected long run real growth rate in dividends that in turn is equal to a forecast of real GNP per capita growth less a “dilution factor” – these are estimated using an average of the experience over the previous 40 years and the experience since the series began in year 1810. The Ilmanen (2003) (IL) series,  2 , is constructed by adding together estimates of the “dividend yield” and the long run growth rate of dividends. The former is calculated from a smoothed earnings yield multiplied by 59%. The latter is an average of 2% and the past 10, 20, 30, 40 and 50 years’ geometric average real growth rate in corporate earnings. Both the A&B and the IL series were constructed around 2002 for research purposes and use data or parameter values that may not have been available to market participants at the date of the forecast. However, the next two series,  3 and  4 , were constructed in real time for use by investment professionals in asset allocation strategies. The series  3 is provided by Wilshire Associates (WA), an investment management consulting firm, and the series  4 is from Barclays Global Investors (BGI), a global investment management firm. They are both in nominal terms and are constructed by first aggregating I/B/E/S consensus estimates of growth rates in earnings per share out for 5 years for individual firms in the S&P 500 index and then letting the growth rates converge linearly to the economy-wide average growth rate over the next 10 years. The resulting growth rates are then used to project current dividends and to calculate an implied DDM rate of return from the current level of the S&P 500 index. These two nominal series are obtained from real-time forecasts of fundamentals that do not use any future information. Table 19.4 reports the estimated parameters of the  process for each series under the two assumptions about dividend growth. Several features stand out in the real series. First, the estimates of  ;  , and  are largely unaffected by whether the expected dividend growth rate is assumed to be constant or not. Second, the estimates of both  and  are somewhat higher for the Ilmanen series than for the A&B series; the effect of these differences on v , the standard deviation of the stationary distribution of , are largely offsetting

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Table 19.4 Parameter estimates of the  series Parameters    Scenario (vi) Table 19.2 0.100 0:0180 0:090 Real models A&B 1;1 0.085 0:0173 0:047 A&B 1;2 0.083 0:0172 0:045 IL 2;1 0.122 0:0196 0:066 IL 2;2 0.115 0:0224 0:066 BGI 3;1 BGI 3;2 WA 4;1 WA 4;2

Nominal models 0.091 0:0239 0.085 0:0214 0.122 0:0336 0.095 0:0266

0:133 0:113 0:137 0:111

v

P

D

D

0:0400

0:900

0:1200

0:879

0:0419 0:0423 0:0397 0:0467

0:977 0:981 0:884 0:885

0:0852 0:0776 0:0852 0:0822

0:126 0:106 0:117 0:066

0:0560 0:0519 0:0680 0:0611

0:812 0:657 0:682 0:714

0:0859 0:2076 0:0859 0:2202

0:234 0:249 0:095 0:209

g

g

g

Dg

0:103

0:0090

0:328

0:413

0:025

0:0034

0:379

0:055

0:209

0:0294

0:378

0:326

0:220

0:0339

0:402

0:417

This table reports the parameter estimates associated with the instantaneous expected return, , series, which are derived from the long run discount rate ›. The two real › series are calculated in Arnott and Bernstein (2002) (A&B) and Ilmanen (2002) (IL), while the two nominal › series are provided by Barclays Global Investors (BGI) and Wilshire Associates (WA). In each › series, we derive  under two cases. In case I, the dividend growth rate g is assumed to be a constant. In case II, the dividend growth rate g is assumed to follow a mean-reverting process. When the real › from Ilmanen or A&B is used as the long run discount rate, g is set to 0.86% in the first case, while g is set to 0.86% in the second case. When the nominal › is used, g is set to 4.82% in the first case, while g is set to 4.82% in the second case

and the estimates are very close to the value of 4% that we have assumed in our simulations in Sects. 19.2 and 19.3. Third, the estimates of the correlation P are all between 0:884 and 0:981; this is what we should expect since  is a proxy for the discount rate. Finally, most of the parameters are quite close to the values that were assumed or derived for Scenario (vi) in Table 19.2 (which assumes a constant expected dividend growth rate). The exception is D , which is much higher in Scenario (vi). We suspect that this is because D was set equal to 12.02% in Table 19.2 while its estimates is only 8.52% in Table 19.4.26 The estimates of the parameters of the nominal  process are broadly similar to those estimated for the real models except that in both cases the Model 2D is over 20%, but this is offset by the much higher level of mean reversion of the dividend growth rate g ; in addition, D is positive in both nominal models, while it is negative in the two real models. Figure 19.6 shows that when the A&B series  1 is used as the input, the estimates of  from the two models, 1;1 and 1;2 , are virtually coincident, suggesting that there is little advantage in allowing for a stochastic dividend growth rate. Note that both estimates of the instantaneous expected rate of return, , are much more volatile than the underlying DDM  series from which they are derived: the maximum value of 1;2 is over 16% in June 1982 and the minimum is around 3% in March 2000; in contrast, the maximum and minimum values of the DDM , which occur in the same 2 months, were respectively 7.9% and 2.1%. 26

As noted above, 12.02% is the sample volatility of annual real dividend growth over the period 1872–2001, while 8.52% is the estimate of D in Equation (19.25) derived from quarterly data for the period 1950.1 to 2002.2 using the algorithm described under Model 2 above.

Figure 19.7 plots the corresponding  estimates from the Ilmanen series  2 . As in the previous case, the 2;1 and 2;2 series track each other closely except around the end of 1999 when the decline in the Model 2 estimate, 2;2 , is much more dramatic than that in 2;1 : 2;2 reaches a minimum of 3:2% just before the end of the bull market in December 2000. For comparison, Figure 19.8 plots the  i and the i;2 .i D 1; 2/ for both the A&B and the Ilmanen series. While the two series of i;2 .i D 1; 2/ generally track each other quite well, there are periods of significant difference. In the 1950s the Ilmanen estimate exceeds the A&B estimate by up to 3.8%. Significant differences of the opposite sign occur during the period 1983–1996; in September 1987, the Ilmanen estimate was less than 1% while the A&B estimate was over 6.5%. Figures 19.9 and 19.10 plot the quarterly nominal estimates of  i ; i;1 and i;2 .i D 3; 4/ from BGI and WA for the shorter periods for which these series are available. These last two series differ from the previous two in three significant ways. First, they are truly ex-ante; second, they are nominal expected rates of return; third, they are based on analysts’ forecasts of earnings growth rates that are known to be upward biased. Not surprisingly, these nominal  series lie everywhere above the two real  series. Nevertheless, they show a similar pattern of increase during the 1970s, the BGI series  3 (the WA series  4 ) reaching a peak of 19.1% (18.8%) in the third quarter of 1981 (second quarter 1982). After the peak, there is a prolonged decline in both series. Although for both series, i;1 and i;2 .i D 3; 4/ move largely in parallel, i;1 is everywhere above i;2 .i D 3; 4/ and the difference between them is much larger than that observed for the A&B and the IL real series. Figure 19.11 shows  i and the corresponding instantaneous expected return i;2 .i D 3; 4/ together. The two i;2

19 Persistence, Predictability, and Portfolio Planning Fig. 19.6 Estimates of the real 1;1 and 1;2 series (1950.1 to 2002.2). The figure plots the long run expected real return ›1 from Arnott and Bernstein (2002) (A&B) together with the estimated series of 1;1 and 1;2 , corresponding to the two cases of constant g and mean-reverting g

Fig. 19.7 Estimates of the real 2;1 and 2;2 series (1950.1 to 2002.2). The figure plots the long run expected real return ›2 from Ilmanen (2003) (IL) together with the estimated series of 2;1 and 2;2 , corresponding to the two cases of constant g and mean-reverting g

307

308 Fig. 19.8 Estimates of the real 1;2 and 2;2 Series (1950.1 to 2002.2). The figure plots the long run expected real return ›1 and ›2 together with their corresponding 1;2 and 2;2 series under the assumption of mean-reverting g

Fig. 19.9 Estimates of the nominal 3;1 and 3;2 Series (1972.4 to 2002.1). The figure plots the long run expected real return ›3 from Barclays Global Investors (NGI) together with the estimated series of 3;1 and 3;2 , corresponding to the two cases of constant g and mean-reverting g

M.J. Brennan and Y. Xia

19 Persistence, Predictability, and Portfolio Planning Fig. 19.10 Estimates of the nominal 4;1 and 4;2 series (1973.1 to 2004.1). The figure plots the long run expected real return ›4 from Wilshire Associate (WA) together with the estimated series of 4;1 and 4;2 , corresponding to the two cases of constant g and mean-reverting g

Fig. 19.11 Estimates of the nominal 3;2 and 4;2 series (1973.1 to 2002.2). The figure plots the long run expected real return ›3 and ›4 together with their corresponding 3;2 and 4;2 series under the assumption of mean-reverting g

309

310

series track each other closely except for the period 1980– 1982 as well as 1999–2000, when the WA estimate 4;2 exceeds the BGI estimate 3;2 by 1–3%, and the period 2001–2002 when the WA estimate 4;2 is below the BGI estimate 3;2 by 2–5%.

19.5.3 Return Prediction We have already shown that even when return predictability is of great economic importance, the statistical evidence of predictability may be weak and hard to detect using standard statistical methods. Therefore, even if the  series contain valuable information for portfolio planning, the statistical evidence of their predictive power may be weak. Panel A of Table 19.5 reports the results of regressions of quarterly real returns on the S&P 500 index on values of i;2 .i D 1; 2/ derived from the A&B and IL real DDM series, while Panel B reports the results of regressing quarterly nominal returns on the estimates of the nominal i;2 .i D 3; 4/ derived from the BGI and WA nominal DDM series. We report results for the whole sample period from 1950.2 to 2002.2, and also the two approximate halves of the period, omitting the influential 1974.3 quarter when the real return on the S&P 500 was approximately 28%. For the two real  series, the effect of omitting this quarter is to raise both the estimated coefficient towards its theoretical value of unity and to raise the regression R2 . While the regression coefficients are not significantly different from either zero or their theoretical value of unity, the point estimates are close to unity. The explanatory power of the predictive variable in both models is considerably greater in the first half of the sample period where the regression R2 is around 4–5% and the predictive coefficient is statistically significant at the 5% level. In all other periods, however, regression R2 ’s are all below 2% and none of the predictive coefficients is significant at the 5% level. Panel B reports the corresponding results for nominal returns using the predictors derived from the BGI and WA nominal DDM models. These models have greater predictive power than the two real models for the relevant sample periods, with R2 at around 2–3%, despite the fact that they were made in real time and contain no “look-ahead” bias. The estimates of the coefficients of the predicted return are little affected by the omission of 1974.3 and, as in the two previous cases, are close to the theoretical value of unity but are not significantly different from zero either.

M.J. Brennan and Y. Xia

The evidence from Table 19.5 is thus broadly consistent with the earlier observation that the predictive coefficient estimates, while close to their theoretical values, are associated with large estimation error and that the weak statistical evidence of predictive power of  series at a quarterly horizon does not provide much information on their economic importance to investors with horizons of 20 years or longer.

19.5.4 Historical Simulations In this section we report the results of simulating the optimal and unconditional policies using each of four  series, A&B 1;1 , IL 2;1 , BGI 3;2 , and WA 4;2 , for a long horizon investor with a relative risk aversion coefficient,  , of 5 when the equity allocation is constrained to lie between 0 and 100%. Allocations to stocks were revised quarterly, and borrowing and short sale constraints were imposed. Under the unconditional constrained strategy, the fraction of the portfolio that is allocated to the risky asset is determined from Equation (19.18) subject to the constraint that 0  x u  1, and is constant over time. Under the constrained optimal strategy, the fraction of wealth allocated to the risky asset is determined by solving an optimal problem whose value function depends on ; it is described in Appendix 19A. The return on the market portfolio is taken as the historical real (nominal) return on the S&P 500 index for each quarter and the riskless interest rate is taken as the realized real (nominal) return on a 30-day Treasury from CRSP for the real 1;1 and 2;1 (the nominal 3;2 and 4;2 ). The instantaneous return volatility, P D 15:7% (17.1%), is determined from equation (var) by setting V .0:25/ equal to the sample volatility, 7.74% (8.40%), of quarterly S&P 500 index real (nominal) return, when either 1;1 or 2;1 (3;2 or 4;2 ) is used. The same value of P is used in both the optimal and the unconditional policies. The other parameters for the optimal policies are taken from the appropriate line of Table 19.4, and the constant unconditional equity premium  for the unconditional policy is calculated from the corresponding  from Table 19.4.27 For both the optimal and the 27

The unconditional equity premium  for the unconditional strategy was calculated in three different ways: (1) using the same  as that used in calculating the optimal strategy; (2) setting  to the sample mean of the S&P 500 Index excess return during the whole sample period of 1929 to 2002; and (3) setting  to the gradually updated sample mean excess return with the initial value calculated from 1929 to 1949 (or 1972 for the BGI or WA series). The first approach ensures that differences between the wealth realized under the optimal and the unconditional strategies are not caused by different assumptions about the

19 Persistence, Predictability, and Portfolio Planning

311

Table 19.5 Quarterly return prediction A. Real return predictive regressions

1

Sample period 1950.2–2002.2

Obs. 209

2

1950.2–1974.2, 1974.4–2000.2

208

3

1950.2–1974.2

97

4

1974.4–2002.2

111

1;2 as Predictor a0 a1 0.005 0.874 (0.46) (1.74) [0.43] [1.60]

R2 (%) 1.43

2;2 as Predictor a0 a1 0.009 0.701 (0.86) (1.53) [0.79] [1.27]

R2 (%) 1.12

0.005 (0.43) [0.38]

0.981 (2.02) [1.83]

1.94

0.008 (0.85) [0.75]

0.800 (1.81) [1.50]

1.57

0:019 (0.92) [1.04]

2.157 (2.08) [2.45]

4.37

0:020 (1.09) [1.06]

1.961 (2.42) [2.21]

5.79

0.011 (0.82) [1.07]

0.708 (1.22) [1.86]

1.35

0.017 (1.40) [2.36]

0.494 (0.88) [1.80]

0.70

B. Nominal return predictive regressions

1

Sample period 1973.2–2002.2

Obs. 117

2

1973.2–1974.2, 1974.4–2000.2

116

3;2 as Predictor a0 a1 0.003 1.026 (0.20) (1.86) [0.18] [1.69] 0.008 0.953 (0.48) (1.80) [0.41] [1.60]

2

R (%) 2.89

2.75

4;2 as Predictor a0 a1 0.006 0.924 (0.37) (1.80) [0.34] [1.66] 0.010 0.852 (0.67) (1.73) [0.54] [1.42]

R2 (%) 2.74

2.57

This table reports the results of regressing real and nominal quarterly returns of the S&P 500 stock index on forecasts of the return at the beginning of the quarter calculated from the estimated value of i;2 .i D 1; 2; 3; 4/ and the estimated parameters of the joint stochastic process using equation   1:0  e =4 R.t; t C 0:25/ D a0 C a1 i;2 t C "t ; i D 1; 2; 3; 4  where R.t; t C 0:25/ is the one quarter real return on the S&P 500 stock index in Panel A and is the corresponding nominal return in Panel B. In Panel A, the real return on the S&P 500 index is regressed on the estimated real A&B and IL i;2 .i D 1; 2/ series. In Panel B, nominal returns on the index is regressed on the estimated nominal BGI and WA i;2 .i D 3; 4/ series. The OLS t -ratios are reported in parenthesis and Newey-West adjusted t -ratios are in brackets

unconditional strategies under the real  (nominal ), the risk free rate  is set at a constant 1.1% (4.1%), which is the sample mean of the realized real (nominal) return of the 30-day Treasury bill rate from 1950 to 2002. Figures 19.12 and 19.13 summarize the results of the simulations under the real A&B 1;1 and IL 2;1 . For each figure the investor is assumed to be concerned with maximizing the expected utility of wealth on the last date included in the figure. For example, Fig. 19.12a describes the evolution of wealth of an investor who starts investing at the end of the first quarter (or equivalently the beginning of the second quarter) of 1950 with a horizon of the end of the first quarter of 1970; thus his initial horizon is 20 years and deunconditional equity premium. Unconditional strategies based on (2) and (3) have similar realized wealth as that based on (1) except for the period 1950–1970, during which unconditional strategies based on (2) and (3) significantly outperform that based on (1) but still underperform the optimal strategy. Since the relative performance of the optimal strategy is consistent across the three unconditional strategies, we only report results of the unconditional strategy based on (1) and omit those based on (2) and (3) for brevity.

creases each period. Investment decisions are assumed to be made at the beginning of each quarter based on the current value of . The figure shows that an investor, who would have followed the optimal strategy based on 1;1 over this period, would have vastly outperformed his unconditional counterpart. Much the same pattern is visible in Fig. 19.13a, which is based on 2;1 . Figures 19.12b and 19.13b show that the optimal strategies continue to outperform the unconditional strategies but by a smaller margin over the subsequent 20-year period 1970–1990. Both the 1;1 and 2;1 based strategies lose more in the oil-price related bear market of 1974 on account of their more aggressive stock positions but more than make up for this by the end of the period. A 12year horizon investor over the period 1990–2002 does better (about as well) under the optimal strategies than (as) under the unconditional strategy using 1;1 .2;1 / series. Finally, the simulations for an investor with a 52-year horizon starting in 1950 show that the “optimal” investor ends up well ahead. Since the i;1 and i;2 .i D 1; 2/ series are trivially different we do not report the results for the 2 series.

312 Fig. 19.12 Cumulative real wealth under the optimal and unconditional strategies for a long horizon. Investor (1;1 as the predictor). The figure plots the cumulative real wealth under the optimal and unconditional strategies for long horizon investors with a risk aversion parameter  D 5. The optimal strategy is based on the estimated real A&B 1;1 series and its associated parameter estimates. Both the optimal and the unconditional strategies are constrained to have allocations between 0 and 1. In panel a, the investment horizon is 20 years and the investors starts investing in 1950.06 with a terminal date in 1970.03. In panel b, the investment horizon is also 20 years and the investor starts investing in 1970.06 with a terminal date in 1990.03. In panel c, the investment horizon is 13 years and the investor starts investing in 1990.06 with a terminal date in 2002.06. In panel d, the investment horizon is 53 years and the investor starts investing in 1950.06 with a terminal date in 2002.06 Fig. 19.13 Cumulative real wealth under the optimal and unconditional strategies for a long horizon. Investor (2;1 as the predictor). The figure plots the cumulative real wealth under the optimal and unconditional strategies for long horizon investors with a risk aversion parameter  D 5. The optimal strategy is based on the estimated real IL 2;1 series and its associated parameter estimates. Both the optimal and the unconditional strategies are constrained to have allocations between 0 and 1. In panel a, the investment horizon is 20 years and the investors starts investing in 1950.06 with a terminal date in 1970.03. In panel b, the investment horizon is also 20 years and the investor starts investing in 1970.06 with a terminal date in 1990.03. In panel c, the investment horizon is 13 years and the investor starts investing in 1990.06 with a terminal date in 2002.06. In panel d, the investment horizon is 53 years and the investor starts investing in 1950.06 with a terminal date in 2002.06

M.J. Brennan and Y. Xia

19 Persistence, Predictability, and Portfolio Planning Fig. 19.14 Cumulative nominal wealth under the optimal and unconditional strategies for a long horizon. Investor (3;2 as the Predictor). The figure plots the cumulative real wealth under the optimal and unconditional strategies for long horizon investors with a risk aversion parameter  D 5. The optimal strategy is based on the estimated real BGI 3;2 series and its associated parameter estimates. Both the optimal and the unconditional strategies are constrained to have allocations between 0 and 1. In panel a, the investment horizon is 20 years and the investors starts investing in 1973.03 with a terminal date in 1992.12. In panel b, the investment horizon is around 10 years and the investors start investing in 1993.03 with a terminal date in 2002.03. In panel c, the investment horizon is about 30 years and the investor starts investing in 1973.03 with a terminal date in 2002.03

Fig. 19.15 Cumulative nominal wealth under the optimal and unconditional strategies for a long horizon. Investor (4;2 as the predictor). The figure plots the cumulative real wealth under the optimal and unconditional strategies for long horizon investors with a risk aversion parameter  D 5. The optimal strategy is based on the estimated real WA 4;2 series and its associated parameter estimates. Both the optimal and the unconditional strategies are constrained to have allocations between 0 and 1. In panel a, the investment horizon is 20 years and the investors starts investing in 1973.06 with a terminal date in 1993.02. In panel b, the investment horizon is around 10 years and the investors start investing in 1993.06 with a terminal date in 2002.06. In panel c, the investment horizon is about 30 years and the investor starts investing in 1973.06 with a terminal date in 2002.06

313

314

Both the A&B and Ilmanen DDM series are constructed by projecting historical growth rates of aggregate series such as profits or GDP. In contrast, the nominal BGI and WA DDM series constructed by investment professionals relies on bottom-up forecasts of individual firm growth rates. Figures 19.14 and 19.15 compare the (nominal) wealth outcomes from the unconditional vs. the optimal strategy derived from BGI 3;2 and WA 4;2 series for the 20-year period 1973–1993 and the 11-year period 1993–2003. In the first period, the wealth outcome under the optimal strategy is 53–67% higher than that under the unconditional strategy: it is noteworthy that these strategies do not suffer the large losses of the A&B and Ilmanen-based strategies during 1974. For the 11-year investment period 1993–2003, the optimal strategies based on both 3;2 and 4;2 series underperform the unconditional ones by 10–20%. It is interesting to note that the unconditional strategy out performs the optimal strategies during the bubble period at the end of the 1990s, but that it substantially under performs as the bubble collapses during the period 2000–2003: it is precisely in such circumstances that we would expect that a dynamic strategy that takes account of the long run expected rate of return implicit in asset prices to do well. Figures 19.14c and 19.15c compare the outcome of the unconditional strategies with those of the optimal ones that start with a 30-year horizon in 1973. Over this longer horizon, the optimal strategies show a consistent advantage. While we should be careful from inferring too much from these historical simulations that represent only a single sample path of stock prices for a single level of risk aversion, it is encouraging that the optimal dynamic strategies tend to outperform naive unconditional strategies, even when they are based on real time data.28

19.6 Conclusion There is considerable disagreement about whether or not stock returns are time-varying and predictable. On the one hand, there is ample evidence of in-sample return predictability from regressions of stock returns on instruments such as the dividend yield and short-term interest rate, and from variance ratio tests. On the other hand, it has been argued 28

The instantaneous expected returns estimated from the DDM long run expected returns depend on parameters of the expected return  and the dividend growth g processes,  ;  , and g etc. These parameters were estimated using data from the whole sample period so that our estimates of instantaneous expected returns, even when they are based on real time DDM estimates, rely on future data. For the A&B and IL series, we also estimated  for the period of 1950–2002 by first estimating the parameters using data only from 1900 to 1949, and the superior performance of the optimal strategy remains unchanged. We do not have long enough sample for the BGI and WA series, which are only available starting from 1973, to carry out this robustness check.

M.J. Brennan and Y. Xia

that in-sample predictive regressions perform poorly out of sample, so that the evidence of time variation should be discounted. In this paper, we have shown that time variation in expected returns that implies both large variation in stock market valuation ratios and substantial gains to long-term dynamic investment strategies is likely to be hard to detect by standard statistical methods. As a result, weak statistical evidence for return predictability does not in itself imply that return predictability is economically insignificant. In view of the difficulty of estimating expected returns from regressions of realized returns on instruments such as the interest rate or dividend yield, we suggest that it is likely to be more productive to estimate the expected long run rate of return by comparing the current level of stock prices with forecasts of expected future dividends in the dividend discount model (DDM) paradigm. This forwardlooking approach has the advantage that it does not rely on hard-to-estimate regression coefficients from past data. The disadvantage is that the rate of return that emerges from the dividend discount rate model is a long run expected rate of return. In order to use the DDM expected rate of return estimate in the dynamic portfolio planning, we show how the instantaneous expected rates of return can be estimated from the DDM long run expected rate of return under the assumption that the instantaneous expected rate of return follows an Ornstein-Uhlenbeck process; the technique is also extended to a setting in which the expected growth rate of dividends, instead of being constant, also follows an Ornstein-Uhlenbeck process. Time series of expected rates of return are estimated for four time series of DDM expected rates of return. Two of them are historical “back-casts,” and two are real time estimates provided by investment professionals. Simulations using realized S&P 500 index returns and the 30-day Treasury bill rates suggest that there may be significant benefits from the use of expected instantaneous rates of return derived from dividend discount model expected returns, even when the expected returns are estimated in real time. In this paper, however, we have examined the benefit of the optimal market timing strategy without taking account of misspecification of, or errors in estimating, the stochastic process for the expected instantaneous rate of return. Determining the sensitivity of DDM-based dynamic portfolio strategies to errors in specifying and estimating the stochastic process for the instantaneous expected return is a task for future work.

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Appendix 19A The Optimal Strategy The investor is assumed to have an iso-elastic utility function defined over end-of-period wealth at the investment horizon T : 8 3 2 < T WT1r  > 15 max E0 4u.WT / D e ; 1r x  D1 : e T ln WT subject to the following dynamic budget constraint: dW D Œx.˛ C ˇ  r/ C r dt C xP dzP ; W

316

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where x is defined as the proportion of wealth invested in the single risky asset whose stochastic process was given in Equations (19.1–19.2). The risk free interest rate, r, is assumed to be constant for simplicity. Under the iso-elastic utility function, the indirect utility function, J.W; ; t/  maxxt Et Œu.WT / , is homogeneous in W : W 1r J.W; ; t/ D e t T .; t/; 1r where satisfies the following Bellman equation:  0 D max x

  1 2  ' C .  / C .1   /x P ' 2  C .1   /Œx.˛ C ˇ  r/ C r 

1   .1   /x 2 P2 Ct 2



a3 D

C./ D

2a3 .1  e  / ; .  a2 / C . C a2 /e 

B./ D

4a3 .˛ C ˇ  r/.1  e  2 /2 ; Œ.  a2 / C . C a2 /e 

P  yt C ŒB./ C C./yt ; P P2

and the function .; t/ under the unconstrained optimal strategy is reduced to: 1

.; t/ D exp A./ C B./yt C C./yt2 2

(19A.5) 

" A./ D a3

.ˇS /2 2 2 .˛ C ˇ  r/2 C   a2 2

(19A.6) !

# C r.1   /   

(19A.1)

where,  D @2 ;  D @ , and t D @ are the partial @ @t derivatives of with respect to  or t. Denote the equity premium by yt  ˛ C ˇt  r and the remaining investment horizon by   T  t, then the investor’s unconstrained optimal dynamic policy xt is given by:



(19A.4)

then q D a23 4a1 a3 > 0 for all  1, which is the condition for the well-behaved normal case. In this paper, we focus on dynamic strategies under  > 1, so A./; B./, and C./ are given by the normal solution:

C

4a3 .˛ C ˇ  r/Œ.2a2 C /e   4a2 e  3 Œ.  a2 / C . C a2 /e 

C

ˇ ˇ 2a3 .ˇs /2 ˇˇ .  a2 / C . C a2 /e  ˇˇ ln ˇ ˇ; 2 2  a22

@2

xt D

1 ; P2

 2

C 2a2  

(19A.7)

p where  D q. If the optimal allocation x  is constrained to be between zero and one (i.e., no borrowing or short-sale is allowed), then no closed-form solution is available. Both the constrained optimal policy x  and the associated function are solved numerically from the Bellman Equation (19A.1) subject to the constraint 0  x   1.



o

where A./; B./, and C./ are solutions to a system of three ordinary differential equations with boundary conditions of A./ D 0; B./ D 0; and C./ D 0 at  D 0: The details of the equations are contained in Kim and Omberg (1996) for the general HARA utility and in Xia (2001) for the CRRA utility. In particular, let

2  1 2 P ; 1C (19A.2) a1 D ˇ 

.1   /ˇ P a2 D 2  ; (19A.3) P

Appendix 19B The Unconditional Strategy If the investor believes that the equity premium is constant, then the investment opportunity set is constant from the investor’s perspective. The unconditional strategy, xu D

y ˛ C ˇ  r D ; 2 P P2

is based on the long run mean of the equity premium. However, the wealth process of the investor evolves according to the true dynamics of the equity premium given by Equations (19.1–19.2): dW D Œx u .˛ C ˇ  r/ C r dt C x u P dzP ; W

(19B.1)

19 Persistence, Predictability, and Portfolio Planning

317

The indirect utility function J u .W; ; / is then given by:   W 1r W 1r J u .W; ; t/ D Et e T T D e T T Et 1r 1r  

  WT exp .1   / ln Wt  

 1r WT T WT exp Et .1   / ln De 1r Wt 

 WT C Vart .1   / ln : Wt Solving for ln WWTt and its first two conditional moments from Equation (19B.1) gives the following result: J.W; ; t/ D e t

Wt1r D. /CE. /t e ; 1r

(19B.2)

where "

C.1   /

2 "

2ˇP C .1   / 

.1   /x u ˇ  C .1   / x u ˇP C.1   / 



 J .W; ; t/ D Et e m

!#

x u ˇ2 2

 D e T

WT1r m

.; t/ 1r (19C.2)

where WT is determined via the wealth dynamics (19C.1). Similar to the indirect utility function under the optimal strategy, we conjecture that the function m is of the form m . /CB m . /yC 1 C m . /y 2 2

m D eA

:

(19C.3)

where Am ./; B m ./, and C m ./ are solutions to a system of ordinary differential equations (ODE) similar to the case of the optimal strategy: dCm D a1m .C m /2 C a2m C m C a3m ; d

(19C.4)

with the boundary conditions:

1  e 

Am ./ D 0; B m ./ D 0; and C m ./ D 0 at  D 0: This system of ODEs has exactly the same form of solutions as that under the optimal strategy, but with slightly different coefficients, .a1m ; a2m ; a3m /. In particular, 

a1m

Appendix 19C The Myopic Strategy

D

ˇ 2 2

¤

a1 ; a2m

D a2 and a3m D

If the investor has a short investment horizon, then it is optimal to adopt the myopic market-timing strategy, ˛ C ˇt  r ; P2

The wealth process of the investor evolves according to the following dynamics: dW D Œx m .˛ C ˇt  r/ C r dt Cx m P dzP ; W

WT1r 1r

 1  dAm D a1m C m C .B m /2 C .˛ C ˇ  r/B m d 2 C.1   /r  ; (19C.6)



xu2 ˇ 2 2 1  e 2 1  ; (19B.3)  C .1   /2 2 2 2 1  e  E./ D .1   /x u ˇ : (19B.4) 

xm;t D

T

dBm 1 D a1m B m C m C a2m B m C .˛ C ˇ  r/C m ; d 2 (19C.5)

1 D./ D .1   / r C .x u /2 P2 2 ˇ 2 2

and the investor’s indirect expected utility function is given by

(19C.1)

.1   /ˇ P D2  P



1 D a3 P2

where a2 and a3 are given in (19A.2–19A.3) in Appendix 19A. Therefore, the solution to Am ./; B m ./, and C m ./ has exactly the same expression as that given by equations (19A.5–19A.7) in Appendix 19A except that a1 is replaced by a1m and  is replaced by m 

q a22  4a1m a3 :

318

M.J. Brennan and Y. Xia

Appendix 19D The Optimal Buy-and-Hold Strategy The dynamics of the stock price (with dividends reinvested) given in Equations (19.1–19.2) implies that the stock price at the time T conditional on information at t is:

 1 (19D.1) PT D Pt exp ˛ C ˇ  r  P2 .T  t/ 2 Cˇ.t  / Z

T

Cˇ t

1  e .T t / 

1  e .T s/ dz .s/ C P 

Z

T t

 dzP .s/ :

The optimal buy-and-hold strategy, x b , which is the proportion of current wealth Wt invested in the stock, then solves the following optimization problem by normalizing Wt to one dollar: 2

1 3 x b PPTt C .1  x b /e r.T t / 6 7 max Et 4 5 : (19D.2) b 1   x The strategy x b , the indirect utility J b , and the certainty equivalent wealth CEW b are all solved numerically.

Chapter 20

Portfolio Insurance Strategies: Review of Theory and Empirical Studies Lan-chih Ho, John Cadle, and Michael Theobald

Abstract A portfolio insurance strategy is a dynamic hedging process that provides the investor with the potential to limit downside risk while allowing participation on the upside so as to maximize the terminal value of a portfolio over a given investment horizon. First, this paper introduces the basic concepts and payoffs of a portfolio insurance strategy. Second, it describes the theory of alternative portfolio insurance strategies. Third, it empirically compares the performances of various portfolio insurance strategies during different markets and time periods. Fourth, it summaries the recent market developments of portfolio insurance strategies, especially in terms of the variations of features in CPPI investments. Finally, it addresses the impacts of these strategies on financial market stability. Keywords CPPI r OBPI r Risk-based portfolio insurance

20.1 Introduction Portfolio insurance refers to any strategy that protects the value of a portfolio of risky assets. The risky assets can be stocks, bonds, currencies, or even alternative assets, such as commodities, real assets, hedge funds, credits and so forth. If the value of the risky asset declines, the insurance or hedge will increase in value to help offset the decline in price of the hedged risky assets. If the price of the risky asset increases, the increase of the insured portfolio will be less than the increase in the risky asset itself but will nevertheless still increase.

L.-c. Ho () Department of Foreign Exchange, Central Bank of the Republic of China (Taiwan), 2, Roosevelt Road, Sec. 1, 10066 Taipei, Taiwan, ROC e-mail: [email protected] J. Cadle and M. Theobald Accounting and Finance Subject Group, University of Birmingham, Birmingham, UK

Table 20.1 illustrates how portfolio insurance works. In this example, the underlying risky asset is purchased for $95 and $5 is spent on portfolio insurance. The minimum amount that the insured investor can realize is $95, but the uninsured portfolio can fall in value to a low of $75 if the market falls. If the value of the risky asset increases, the value of the insured portfolio will increase, but at a smaller rate. Figure 20.1 illustrates the profit and loss of the insured and uninsured portfolio. Portfolio insurance allows market participants to alter the return distribution to fit investors’ needs and preferences for risk. Figure 20.2 shows the effect of insurance on the expected returns of a portfolio. Notice that the uninsured portfolio has greater upside potential as well as greater downside risk, whereas the insured portfolio limits the downside loss to the cost of the hedge. The upside potential of the insured portfolio is always below that of the uninsured portfolio. The cost of the insurance is the lower return for the insured portfolio should prices increase. While some investors would prefer the greater upside potential that the uninsured portfolio offers, risk-averse investors would prefer the limited-risk characteristics that the hedged portfolio offers.

20.2 Theory of Alternative Portfolio Insurance Strategies This section introduces the mechanisms behind various portfolio insurance strategies. These strategies share a common characteristic, a convex payoff function, which implies a “buy high and sell low” rule for the risky asset. If the market level declines an investor sells a fraction of the risky asset and buys the riskless asset; if the market level rises an investor switches from the riskless to the risky asset and rallies with the market. For illustrative purposes, we construct a portfolio that comprises a risky currency, the euro (EUR), and a risk-free currency, the US dollar (USD) overnight deposit account. For comparison purposes, we assume that the

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_20, 

319

320

Table 20.1 Mechanics of portfolio insurance: an example

L.-c. Ho et al.

Initial investment Cost of portfolio insurance

$100 $5

Amount of investment going toward securities

$95

Amount invested D 100 Value of portfolio at year end ($)

Return on uninsured portfolio (%)

Return on insured portfolio ($)

Net return on insured portfolio (%)

75

25

95

5

80

20

95

5

85

15

95

5

90

10

95

5

95

5

95

5

100

0

95

5

105

5

100

0

110

10

105

5

115

15

110

11

120

20

115

15

125

25

120

20

130

30

125

25

Fig. 20.1 Gains and losses of insured and uninsured portfolios: an example

initial value of EUR in the portfolio is 25%, short positions in EUR are not allowed, but that an investor can borrow via USD to buy EUR.

20.2.1 Portfolio Insurance with Synthetic Put (Option-Based Portfolio Insurance) The most popular portfolio insurance strategy is the synthetic put approach of Rubinstein and Leland (1981), also

referred to as an option-based portfolio insurance (OBPI) strategy. If an investor holds a risky asset and buys one atthe-money put option on that asset, the value of the resultant portfolio will not be less than the exercise price net of the premium at expiration. Thus, the investor effectively hedges the portfolio against downside risk. However, the implied volatility is usually higher than the historical volatility in option markets, which indicates that an actual option is more expensive than a synthetic one. Therefore, a synthetic option is often used as an alternative hedging vehicle in practice.

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies

321

Fig. 20.2 Expected returns on insured and uninsured portfolios

In a currency portfolio, a synthetic put can be created by dynamically rebalancing the portfolio between the risky (foreign) and riskless (domestic) currencies according to the delta of a put option, D D e rf T ŒN.d1 /  1 :

(20.1)

Specifically, the investment proportion of the risky currency in the portfolio value can be expressed as: WSynP ut D

Se rf T N.d1 / S C D S D ; V S CP

(20.2)

where P is the premium of a European put option on foreign currency from Garman-Kohlhagen (1983), P D Xe rT N.d2 /  Se rf T N.d1 /;

determined by the changes of delta. Table 20.2 illustrates the adjustment process. One important parameter in the synthetic put approach is the volatility. Rendleman and O’Brien (1990), Do (2002), Do and Faff (2004), and Bertrand and Prigent (2005) use different measures of volatility, such as implied, historical and Leland’s (1985) adjusted historical volatility, and discuss the effects of volatility estimation on the synthetic put strategy. Additionally, in order to reduce transaction costs, some revisions on the rebalancing discipline are adopted, such as the time discipline with weekly/monthly rebalancing, the price discipline with adjustments according to changes in the value of the risky asset itself, or the volume discipline corresponding to changes in the number of shares that should be held.

(20.3)

p ln.S=X /C.rrf C 2 =2/T p and d2 D d1   T ; S is and d1 D  T the spot exchange rate, r is the domestic risk-free interest rate, rf is the foreign risk-free interest rate, ¢ is the volatility of spot exchange rate return, X is the exercise price, T is the time to expiration, and N. / is the cumulative standard normal distribution function. Assume that the initial portfolio contains 1,000,000 units of EUR. In order to make the initial value of the EUR position in the portfolio equal to 25%, an investor needs to create a synthetic put with a delta of 0:75 and sell 750,000 units of the EUR. The dollar income from the EUR sold is kept in an USD cash account and earns the overnight deposit rate. As the EUR depreciates, the synthetic put moves further in-the-money and an investor sells more EUR; conversely, an investor withdraws money from the USD cash account to buy EURs as the EUR appreciates. The daily adjustment is

20.2.2 Constant Proportion Portfolio Insurance Another simplified approach, not involving complex “Greeks,” is the constant proportion portfolio insurance (CPPI) strategy developed by Black and Jones (1987) and Perold (1986), and extended to fixed-income instruments by Hakanoglu et al. (1989). In CPPI, the exposure to the risky asset, E, is always kept at the cushion times a multiplier, m. The cushion, C , is the difference between the portfolio value, V , and a protected floor value, FL, and the multiplier is constant throughout the investment period: C D V  FL;

(20.4)

E D m C:

(20.5)

2007/1/1 2007/1/2 2007/1/3

1.3201 1.3272 1.3170

(1) 0.750000 0.729593 0.764881

(2) 750,000 20,407 35,288

(3) $330,025 $358,884 $309,651

.4/ D .3/=.9/ 25.0 27.1 23.5

Table 20.2 Daily adjustment process of portfolio insurance with synthetic put EUR asset EUR amt. USD Synthetic put EURUSD Delta EUR asset (%) sold ./ value (5) $990,075 $27,084 $46,474

$ Income of EUR sold

USD Cum. Interest income income .6/ D .5/C.6/C.7/ (7) $990,075 $145 $963,137 $142 $1,009,753 $148

.9/ D .3/ C .6/ .10/ D d.9/=.9/ $1,320,100 $1,322,020 0.15 $1,319,404 0.20

.8/ D .7/=.9/ 75.0 72.9 76.5

Portfolio return (%)

Portfolio USD value

USD asset (%)

322 L.-c. Ho et al.

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies

The investment proportion of the risky currency in the portfolio value can be expressed as: WCPPI D

E : V

323

20.2.3 Portfolio Insurance with Downside Risk Control (Risk-Based Portfolio Insurance)

(20.6)

In a portfolio of securities, the floor is usually set as the discount price of a zero-coupon bond, which approaches the par value at the end of the investment horizon. In a currency portfolio, the floor is set as the forward rate, with a maturity equal to the investment horizon, that appreciates with time; that is, Ft D F0  exp.r0  t/, where F0 D S0  exp..r0  rf;0 / T /. Note that when the domestic riskfree rate is higher than the foreign risk-free rate, the forward rate is at a premium, which would result in short-selling of the risky asset at the beginning of the investment horizon; that is, E D m C D m .V  FL/ D m Units .S  F / < 0. In order to make the initial value of the EUR position in the portfolio equal to 25%, a discount factor (DF) is needed, and the multiplier1 also needs to be trimmed at the beginning of investment horizon. Thus, the synthetic floor is set as: Ft D DF  F0  exp.r0  t/:

(20.7)

Table 20.3 illustrates the adjustment process. As EUR appreciates, the portfolio value increases more than the floor level increases, the cushion increases, and investors buy EUR, vice versa. There are various market developments in CPPI mechanism with additional features, such as constraints on the investment level, constraints on leverage, variable floors, and variable multipliers and so forth, which will be introduced in later section.

The idea of risk management has become widespread and has been applied to the asset allocation problem. By dynamically controlling for downside risk while maximizing the expected return of a portfolio, the concept of the modern riskbased asset allocation process is analogous to that of the classic portfolio insurance strategy. Zhao and Ziemba (2000) are the first to explore the portfolio payoffs between the two approaches. There are basically two popular measures of downside risk, one is the Value-at-Risk (VaR) and the other more coherent one is the Expected Shortfall (ES). Accordingly, there are two risk-based portfolio insurance (RBPI) strategies. VaR-based portfolio insurance is a strategy that permanently controls the shortfall risk of the portfolio. The allocation to the risky asset is adjusted each day so that the shortfall probability – the probability of realizing a portfolio return .RP / that falls below a pre-specified threshold .R/ at the end of the investment period – does not exceed a target value, ˛ (say 5%): Prob.RP < R/  ˛: (20.8) To operationalize the shortfall risk, Herold et al. (2005) use the lower partial moment of order zero, which is equivalent to VaR, and assume a normally distributed portfolio returns: LPM 0 .RP / D N

  u ; 

RP D W  RA C .1  W / RF ;

(20.9) (20.10)

where u and  are the mean and the volatility of portfolio returns, respectively, and  is the required portfolio return. RA and RF are the return of the risky and the riskless asset, respectively. Controlling for a fixed shortfall probability, the dynamic allocation process is solved for the weight of the risky asset so that the portfolio return is always higher than the required return (i.e., VaR): 1

The magnitude of the multiplier depends on the discount factor. The lower the discount factor, the lower the multiplier in order to keep the initial exposure fixed at 25%; that is, E D m .V  FL/.

NVaR˛;t .RP / D ut .RP /  N 1 .˛/t .RP /:

(20.11)

Table 20.3 Daily adjustment process of constant proportion portfolio insurance Floor discount factor .DF/ D 0:94; multiplier .m/ D 5:24592 CPPI

2007/1/1 2007/1/2 2007/1/3

EURUSD

1.3201 1.3272 1.3170

Portfolio USD value

Floor level

EUR asset USD value

EUR amount

(1)

(2)

.3/ D ..1/–.2// m

(4)

$1,320,100 $1,322,020 $1,319,562

1.2572 1.2574 1.2577

$330,025 $338,752 $324,510

250,000 255,238 246,401

EUR asset (%) .5/ D .3/=.1/ 25.0 25.6 24.6

USD asset USD value

USD asset (%)

Portfolio return(%)

.6/ D .1/–.3/

.7/ D .6/=.1/

.8/ D d.1/=.1/

$990,075 $963,137 $1,009,753

75.0 74.4 75.4

0.15 0.19

324

L.-c. Ho et al.

Table 20.4 illustrates the daily adjustment process. Since the VaR of a portfolio return does not change dramatically each day during a certain period of time, the frequency of adjustment is much less than that in traditional portfolio insurance strategies. In comparison with the traditional portfolio insurance strategy, Herold et al. (2005) show that the VaR-based model can be regarded as a generalized version of CPPI with a dynamic and time-varying implied multiplier. The risky currency investment proportion is then given by: WVaR D

 ; Z˛  

(20.12)

where Z˛ is the ˛-quantile of the standard normal distribution and the process drift is assumed to be equal to zero at the daily differencing interval. The implied multiplier of the VaR model is obtained by: W V C =V V 1 1 E D  D D ; mimpl D D C C Z˛   C Z˛   VaR .RP / (20.13) where  D C=V is expressed in percentage terms. Thus, the inverse of the CPPI multiplier can be interpreted as the maximum loss or worst case return that is allowed to occur over the next period. Bertrand and Prigent (2001) propose that the upper bound on the multiplier can be determined by the extreme value theory (EVT). ES-based portfolio insurance is a strategy that permanently controls the expected shortfall of the portfolio. The expected shortfall is defined as the expected value of the loss of a portfolio in a certain percentage of worst cases within a given holding period. The allocation to the risky asset is adjusted each day so that the expected shortfall does not exceed a target value. Following the idea of Herold et al. (2005), Hamidi et al. (2007) propose a conditional CPPI multiplier that links the traditional CPPI strategy with the RBPI strategy. The conditional CPPI multiplier is determined by keeping the risk exposure of a portfolio constant, and the risk exposure can be defined either by the VaR or by the ES: mcond D

1 1 or D VaR˛ .RP / ES˛ .RP /

where HVaR˛;t .RP / D Percentile.RPt 260 ; : : : ; RPt I 5%/  R, among which HVaR˛;t .RP / is the historical VaR of portfolio returns at date t at the 1  ’% confidence level, RPt 260 ; : : : ; RPt indicates the daily portfolio return series given a certain weight of the risky asset during the past year, and EŒ is the expectation operator. Table 20.5 illustrates the daily adjustment process. Since the ES of a portfolio return does not change dramatically each day during a certain period of time, the frequency of adjustment is also much less than that in the traditional portfolio insurance strategies.

(20.14)

By assuming a normally distributed portfolio return, the expected shortfall and the VaR are scalar multiples of the standard deviation. There will be no differences across portfolio strategies employing variance, VaR and ES risk measures. Therefore, Ho et al. (2008) use the historical distribution of portfolio returns to operate the ES-based portfolio insurance strategy: HES˛;t .RP / D EŒRP jRP < H VaR˛;t .RP /  R; (20.15)

20.3 Empirical Comparison of Alternative Portfolio Insurance Strategies In the late 1980s and early 1990s, much of the research in this area focused on simulated comparisons of the performances among alternative portfolio insurance strategies. In the 2000s, most studies use empirical data to analyze the circumstances where one strategy would outperform the others based on some modifications on certain portfolio insurance mechanisms, such as volatility input, rebalancing intervals and so forth. The performances are typically evaluated by floor protection – whether they achieve the desired floor at the end of the investment horizon, the cost of insurance – the opportunity cost of forfeiting upward markets, and the Sharpe ratio. Listed below are some of the studies that have appeared in academic journals.

20.3.1 Zhu and Kavee (1988) Zhu and Kavee evaluate and compare the performances of the two traditional portfolio strategies, the synthetic put approach of Rubinstein and Leland (1981) and the constant proportion approach of Black and Jones (1987). They employ a Monte Carlo simulation methodology, assuming lognormally distributed daily returns with an annual mean return of 15% and paired with different values for market volatility, in order to discover whether these strategies can really guarantee the floor return and how much investors have to pay for the protection. Both strategies are able to reshape the return distribution so as to reduce downside risk and retain a certain part of the upside gains. However, they demonstrate that a certain degree of protection can be achieved at a considerable cost. There are two types of costs in implementing a portfolio insurance strategy. The first is the explicit cost, that is, the transactions costs. The other is the implicit cost, which is the average return forgone in exchange for protection against

EURUSD

1.3201 1.3272 1.3170

Control VaR

2007/1/1 2007/1/2 2007/1/3

EUR adj. rule (2) 25.0 25.0 25.0

Portfolio USD value

(1) $1,320,100 $1,322,020 $1,319,617

EUR asset USD value .4/ D .3/ S $330,025 $331,800 $329,250

EUR amount .3/ D .2/ IniAmt 250,000 250,000 250,000

Table 20.4 Daily adjustment process of VaR-based portfolio insurance ’ D 5.%/

.5/ D .4/=.1/ 25.0 25.1 25.0

EUR asset (%)

$0 $0

.6/ D d.3/ S

USD deposit adjust .C/

.7/ D .6/ C .7/ C .8/ $990,075 $990,220 $990,367

Cumulated USD amount

(8) $145.2 $146.3 $145.3

Interest income

.9/ D .7/=.1/ 75.0 74.9 75.0

USD asset (%)

0.15 –0.18

.10/ D d.1/=.1/

Portfolio return (%)

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies 325

EURUSD

1.3201 1.3272 1.3170

Control ES

2007/1/1 2007/1/2 2007/1/3

EUR adj. rule (2) 25.0 25.0 25.2

Portfolio USD value

(1) $1,320,100 $1,322,020 $1,319,617

EUR asset USD value .4/ D .3/ S $330,025 $331,800 $331,374

EUR amount .3/ D .2/ IniAmt 250,000 250,000 251,613

Table 20.5 Daily adjustment process of ES-based portfolio insurance ’ D 5.%/

.5/ D .4/=.1/ 25.0 25.1 25.1

EUR asset (%)

$0 $2;124

.6/ D d.3/ S

USD deposit adjust .C/

.7/ D .6/ C .7/ C .8/ $990,075 $990,220 $988,242

Cumulated USD amount

(8) $145.2 $146.3 $145.0

Interest income

.9/ D .7/=.1/ 75.0 74.9 74.9

USD asset (%)

0.15 –0.18

.10/ D d.1/=.1/

Portfolio return (%)

326 L.-c. Ho et al.

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies

327

Using simulated stocks and bills prices, Perold and Sharpe examine and compare how the four dynamic asset allocation strategies – namely, the buy-and-hold, constant mix, CPPI, and OBPI – perform in bull, bear, and flat markets and in volatile and not-so-volatile markets. CPPI and OBPI strategies sell stocks as the market falls and buy stocks as the market rises. This dynamic allocation rule represents the purchase of portfolio insurance and has a convex payoff function, which results in a better downside protection and a better upside potential than a buy-andhold strategy. However, they do worse in relatively trendless, volatile markets. Conversely, a constant mix strategy – holding a constant fraction of wealth in stocks – buys stocks as the market falls and sells them as it rises. This rebalancing rule effectively represents the sale of portfolio insurance and has a concave payoff function, which leads to less downside protection than, and not as much upside as, a buy-and-hold strategy. However, it does best in relatively trendless and volatile markets. Perold and Sharpe suggest that no one particular type of dynamic strategy is best in all situations. Financial analysts can help investors understand the implications of various strategies, but they cannot, and should not, choose a strategy without a substantial understanding of an investor’s circumstances and desires.

where St is the price of the risky asset at date t; k0 the strike price, r0 the annual continuously compounded riskless rate of interest, ¢0 the ex ante volatility parameter at the beginning of the insurance period, and T the maturity of the option. Assume that at the beginning of the insurance period, one manager predicts a high-volatile market and a 20% annualized volatility, while a second manager believes a lowvolatile market and a 10% volatility. The delta for the first manager is higher, which means he would buy more insurance and allocate less to a position in the risky asset. Assume that the ex-post volatility turns out to be 10%. The second manager will have made the proper allocation between risky and riskless assets. In contrast, the high-estimate manager will have invested too little in the risky asset. This misallocation would lose the opportunity of participating in the price appreciation in a strong market. Thus, a manager who underestimates volatility will typically end up buying less insurance than is necessary to ensure a given return, while a manager who overestimates volatility will buy more insurance than is necessary and forgo gains. As for the issue of portfolio rebalancing, Rendleman and O’Brien examine adjustment frequencies by time intervals; namely, daily, weekly, monthly, and bi-monthly. The effect (error) is measured as the difference of the horizon insurance values between noncontinuous and continuous trading. They suggest that weekly rebalancing produces an amount of error that appears to be tolerable by most portfolio managers. More importantly, they address the biggest potential risk of implementing portfolio insurance strategies – the gap risk, by simulating the performance of the OBPI strategy over the period of the October 1987 market crash. They indicate that most insured portfolios would have fallen short of their promised values because the managers would not be able to adjust the portfolio in time before a big drop in the market. The gap risk will be discussed further in the financial stability section.

20.3.3 Rendleman and O’Brien (1990)

20.3.4 Loria et al. (1991)

Rendleman and O’Brien address the issue that the misestimation of volatility input can have a significant impact on the final payoffs of a portfolio using a synthetic put strategy. In an OBPI strategy, the daily portfolio adjustments depend on the delta of the put option on the risky asset. In the original Black and Scholes (1973) valuation equation, the delta at each day is a function of the following variables:

Loria, Pham, and Sim simulate the performance of a synthetic put strategy using futures contracts based on the Australian All Ordinaries Index for the period April 1984–March 1989. Their study contains 20 consecutive nonoverlapping 3-month insurance periods whose expiration dates coincide with the expiration dates of each SPI futures contract traded on the Sydney Futures Exchange. Four implementation scenarios are examined: a zero floor versus 5% floor that correspond to a portfolio return of 0% and 5% annually,

the downside risk. When the market becomes volatile, the protection error of the synthetic put approach increases, and the transaction costs may be unbearable. On the other hand, while the constant proportion approach may have lower transactions costs, its implicit cost may still be substantial.

20.3.2 Perold and Sharpe (1988)

Dt D f .St ; kN0 ; rN 0 ; 0 ; T /;

(20.16)

328

respectively; and a realized volatility versus Leland’s (1985) modified volatility. They report that there is no perfect guarantee of loss prevention under any scenario. Even in the scenario with a 5% floor and modified volatility, 2 out of 20 contracts do not meet the desired floor. In addition, the OBPI strategy is most effective under severe market under severe market conditions. In other periods characterized by insignificant market declines, the value of the insured portfolio is below that of the market portfolio. Loria, Pham, and Sim suggest futures mispricing may be one potential culprit for this outcome.

L.-c. Ho et al.

have a realized volatility of less than 15%; violation of either one or both of these conditions is regarded as indicating a turbulent period. All portfolio insurance strategies achieve 100% floor protection during tranquil periods whereas the futures-based OBPI approach records the highest portfolio return. During turbulent times, futures-based portfolio insurance continues to perform quite well. The 1987 stock market crash makes the assessment difficult because of trading halts. However, assuming the futures continued to trade during that crisis, from the algorithm’s perspective, the futures-based CPPI maintains a positive return, while the OBPI results in a negative return.

20.3.5 Do and Faff (2004) 20.3.6 Cesari and Cremonini (2003) This empirical paper is an extension of Loria et al. (1991). Do and Faff examine two approaches (the OBPI and CPPI) and conduct simulations across two implementation strategies (via Australia stock index and bills, and via SPI futures and stock index). Furthermore, they consider the use of implied volatility as an input into the model, and the use of a zero floor versus 5% floor. The dataset consists of 59 nonoverlapping 3-month insurance periods, which span from October 1987 to December 2002. Thus, their key contributions relative to Loria et al. (1991) are the examination of a futures-based CPPI, a fine-tuned algorithm that allows for dividend payments, the consideration of ex ante volatility information, and more up to date data. In terms of floor protection, the futures-based portfolio insurance implementation generally dominates its index-andbill rival in both floor specifications, which reflects the low transaction costs in the futures market. Furthermore, the perfect floor protection is possible when implied volatility is used rather than using ex-post volatilities. From the cost of insurance perspective, the futures-based strategy generally induces a lower cost of insurance than its index-and-bill rival in both floor specifications, which reflects the same reason as above. However, the cost of insurance is higher when implied volatility is used compared to ex-post volatilities. The possible explanation is that the implied volatility is often higher than the same period ex-post volatility, which results in overhedging. As for the performances between the OBPI and CPPI approaches, there is no strong evidence to distinguish between them. Within the futures-based implementation, the synthetic put appears to dominate the CPPI with respect to floor protection, while the latter appears to slightly outperform in terms of upside participation. They also examine whether portfolio insurance strategies work under stress conditions by assessing the strategies’ effectiveness during tranquil and turbulent periods. Tranquil periods are defined as ones that return more than 4% and

This study is an extensive comparison of a wide variety of traditional portfolio insurance strategies. There are basically five dynamic asset allocation strategies: (1) buy-and-hold (BH); (2) constant mix (CM); (3) constant proportion (without and with the lock-in of profits, CP and CPL); (4) the option-based approach (with three variations, BCDT, NL, PS); and (5) technical strategy (with two kinds of stop-loss mechanism, MA and MA2). Therefore, nine strategies in total are considered. For each strategy, eight measures for risk, return and risk-adjusted performance are calculated; namely, mean return, standard deviation, asymmetry, kurtosis, downside deviation, Sharpe ratio, Sortino ratio, and return at risk. The strategies are then compared in different market situations (bear, no-trend, bull markets) and with different market volatility periods (low, medium and high periods), taking into account transaction costs and discrete rebalancing of portfolios. The three market situations are defined accordingly if market average returns fall into the three ranges: .30%; 5%/; .5%; C5%/; .C5%; C30%/, and the three ranges for the volatile periods are (10%, 15%), (15%, 25%), (25%, 30%). Transaction costs are treated in two ways: a proportional cost to the value traded and a correction to the Leland’s (1985) option volatility. Two main rebalancing disciplines are used: a time discipline with weekly adjustment, and a price discipline with adjustment only when prices are increased/decreased by 2.5% with respect to the previous rebalance time. Monte Carlo simulations of MSCI World, North America, Europe, and Pacific stock market returns show that no strategy is dominant in all market situations. However, in bear and no-trend markets, CP and CPL strategies appear to be the best choice. In a bull market or in no-trend but with high volatility market, the CM strategy is preferable. If the market phase is unknown, CP, CPL, and BCDT strategies are recommended.

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies

In addition, these results are independent of the volatility level and the risk-adjusted performance measure adopted.

20.3.7 Herold et al. (2005) By constructing a fixed income portfolio, in which the risky asset is the JPMorgan government bond index and the riskless asset is cash (1-month yield), Herold, Maurer, and Purschaker compare the hedging performances between the traditional CPPI strategy and the risk-based (specifically, VaR-based) strategy with a 1-year investment horizon that begins at each year from 1987 to 2003. CPPI avoids losses in the bear years of 1994 and 1999. The mean return is inferior to (about 40 base points below) that of the risk-based strategy. CPPI also produces a higher turnover.

20.3.8 Hamidi et al. (2007) Although Hamidi, Jurczenko, and Maillet propose a conditional CPPI multiplier determined either by VaR or by ES, only a VaR-based measure is studied in this empirical work, and the ES measure is absent for future research. The data set contains 29 years of daily returns of the Dow Jones Index, from January 2, 1987 to May 20, 2005, 4,641 observations in total. They use a rolling window of 3,033 returns to estimate the VaR, and there are 1,608 VaRs in the out-of-sample period. They resort to eight methods of VaR calculation: one non-parametric method using the historical simulation approach; three parametric methods based on distributional assumptions: namely, the normal VaR, the RiskMetrics VaR based on the normal distribution, and the GARCH VaR based on the Student-t distribution; four semi-parametric methods using quantile regression to estimate the conditional autoregressive VaR (CAViaR): namely, the symmetric Absolute Value CAViaR, the Asymmetric Slope CAViaR, the IGARCH(1,1) CAViaR, and the Adaptive CAViaR. According to the 1,608 back-testing results, the Asymmetric Slope CAViaR is the best model to fit the data. After having calculated the VaR values, the conditional multipliers can be determined. The estimations spread between 1.5 and 6, which are compatible with multiple values used by practitioners in the market (between 3 and 8). Using the time-varying CPPI multipliers estimated by different methods, and a “multi-start” analysis – the fixed 1-year investment horizon beginning at every day of the out-of-sample period – they find that the final returns of these insured portfolios are not significantly different.

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20.3.9 Ho et al. (2008) This empirical study presents a complete structure of comparing traditional portfolio insurance strategies (OBPI, CPPI) with modern risk-based portfolio insurance strategies (VaR-, ES-based RBPI). By constructing a currency portfolio, in which the risky asset is the Australian dollar and the riskless asset is a US dollar overnight deposit, Ho, Cadle, and Theobald compare the dynamic hedging performances between the traditional and the modern strategies with a 1-year investment horizon that begins at each year from 2001 to 2007. When implementing the OBPI strategy, the delta is calculated based on the modified Black-Scholes formula, Dt D f .St ; kN0 ; rt ; t ; T /. That is, daily annualized historical volatilities are used as inputs in the put-replication process instead of the constant ex ante or the implied volatility in the original model to mitigate the volatility misestimation problem. Besides, the latter two parameters would make the portfolio daily returns more volatile. The interest rates are also updated daily for the same reason. When CPPI is implemented, the possible upper bounds of the multiplier are examined via the EVT, which ranges from 4 to 6 corresponding to different confidence levels. When risk-based approaches are employed, both a historical distribution and a normal distribution with exponentially weighted volatility of risky asset returns are assumed. A daily rebalancing principle is adopted in their research without any modification to show the original results of hedging. The performances are evaluated from six differing perspectives. In terms of the Sharpe ratio and the volatility of portfolio returns, the CPPI is the best performer, while the VaR based upon the normal distribution is the worst. From the perspective that the return distribution of the hedged portfolio is shifted to the right and in terms of both the average and the cumulative portfolio returns across years, the ES-based strategy using the historical distribution ranks first. Moreover, the ES-based strategy results in a lower turnover within the investment horizon, thereby saving transaction costs.

20.4 Recent Market Developments 20.4.1 Type of Risky Asset Over the past few years, portfolio insurance reemerged due to low structuring and trading costs. According to Pain and Rand (2008), OBPI investments have not been popular, which reflects the difficulty in explaining options to investors. However, the CPPI strategy is much more prevalent, stemming from a broadening in the asset classes

L.-c. Ho et al.

B/W in Print

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Fig. 20.3 Structured note issuance by type of risky asset

for which investors need principal protection. Many of principal-protected structured notes are not only designed to link the performance of traditional assets, such as stocks, currencies, interest rates and bonds, but also to invest in alternative risky assets, such as hedge funds, fund of hedge funds, credit derivatives, commodities, real estates, private equities, and any combination of the above assets.

Fig. 20.4 Structured note issuance on alternative assets

institutional and retail clients, such as pension funds, insurance companies, private banks who purchase products for onward sale to their clients, and high net wealth individuals. Among which, the institutional investors are especially important in continental Europe.

20.4.4 Modified CPPI Mechanisms 20.4.2 Market Size Figure 20.3 shows the evolution of shares of structured note issuance by type of risky asset. The share of notes linked to equity and alternative assets increased significantly in 2002–2007. In 2007, the shares of currencies and equities are both about 40%, alternative assets occupy more than 10%, and bonds have the least share. Figure 20.4 illustrates the evolution of issuance of structured notes linked to alternative assets. The total issuance has grown five times, from about US $5 billion in 2002 to more than US $45 billion in 2007.

20.4.3 Market Participants According to Pain and Rand (2008), portfolio insurance products are more prevalent in Europe than in the United States. The key issuers (sellers) are typically large investment banks that can provide the necessary structuring and marketing expertise. The main investors include both

Since the inception of portfolio insurance strategies, CPPI investments have evolved to incorporate various different features. Of particular importance are the key components in the design of CPPI investments (i.e., the floor, the multiplier, and the exposure to risky assets).

20.4.4.1 Variations in Floor Ratchet floor: The performance of CPPI investments is price dependent; any gains at a particular point in time may still be lost if the underlying asset price subsequently falls. To address this, products with a “ratchet” mechanism have been introduced to lock in a proportion of upside performance. More specifically, the floor can be set as a percentage of the historical maximum net asset value of the portfolio. That is, the floor jumps up with the portfolio value in order to reduce the risky asset allocation when the market peaks. The time-invariant portfolio protection (TIPP) strategy proposed by Estep and Kritzman (1988) reflects this variation.

20 Portfolio Insurance Strategies: Review of Theory and Empirical Studies

Margin floor: The converse to the above case is when the market falls substantially for the cushion to approach zero, the exposure to the risky asset also approaches zero. Under a conventional CPPI, this leads to “cash-out” where the portfolio is fully invested in the risk-free asset, with no possibility of regaining any exposure. One method to counter this is to adjust the starting floor value, by augmenting this by a margin, which can later be used to help regain some exposure after a near cash-out situation. Straight-line floor: The floor in a conventional CPPI is a zero-coupon bond, which is sensitive to the level of interest rates. When interest rates fall, the floor would rise and the allocation switches away from the risky asset. This in turn would limit the subsequent potential upside from the CPPI. However, this sensitivity can be removed by allowing the floor to vary linearly with time.

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offers the flexibility to build in additional features, such as zero-coupon, fixed or contingent coupons, and any variations mentioned above. Swaps: An alternative idea in the structured products area could be a collateralized swap. The client has a leveraged exposure to the risky asset and receives a leveraged return on the performance of the asset with no upfront payment required. In return, the client will accrue a financing fee on the leverage amount, payable periodically or at maturity. The swap normally requires the client to have a pre-agreed credit line with the issuer or pledge shares or cash to the issuer as collateral.

20.5 Implications for Financial Market Stability

20.4.4.2 Variations in Multiplier

20.5.1 Amplification of Market Price Movements Rather than having a constant multiplier in a conventional CPPI, dynamic portfolio insurance (DPI) allows for the multiplier to vary over time in relation to the volatility of the risky asset and reflects investors’ appetite for risk. In fact, DPI is a cushion management technique that quantifies the level of risk born by the investment in the risky asset. There is often a maximum level for the multiplier, which can be determined by stress testing on the returns of the risky asset using EVT. The risk-based portfolio insurance strategies in the earlier section reflect this feature. 20.4.4.3 Variations in the Exposure to Risky Assets Constraints on the investment level: Another method to avoid the “cash-out” situation when a market falls is to incorporate a minimum level of investment in the risky asset. Conversely, to avoid unbounded investment in the risky asset as a market rallies, a maximum investment level can also be imposed. Constraints on leverage: Exposure to the risky asset of more than the initial available funds can be achieved by allowing borrowing. There may be constraints on how much can be borrowed. Volatility caps: Some CPPI products include mechanisms that allow the percentage exposure to the risky asset to be reduced if its realized volatility exceeds a certain level.

The portfolio insurance strategies mentioned in this chapter share a common “buy high and sell low” rule for the risky asset. When the underlying asset markets move in one particular direction, either trending up or down, these actions could conceivably have feedback effects in markets that amplify price developments. If the underlying asset markets are deep and liquid, the feedback effects may be limited. However, if the underlying asset markets are less liquid (for example, CPPI products linked to hedge funds on credit derivatives), even small actions may reinforce market prices. Moreover, some underlying assets that appear less correlated in normal conditions may become more correlated in stress situations when CPPI products all look to reduce risky positions simultaneously. Taken together the stress situations and the inherently less liquid nature of certain underlying asset markets, the feedback would have a large impact. Fortunately, according to Pain and Rand (2008), market contacts reveal that the amount of portfolio insurance-related dynamic hedging flows still remain modest relative to the size of the corresponding underlying asset markets, even in those new asset classes. Moreover, market contacts did not perceive portfolio insurance as a significant factor during the 2007 financial market turmoil whereas it was perceived as the driving force during the 1987 stock market crash.

20.4.5 Structured Products 20.5.2 Gap Risk Structured notes: Many of the issuances of principalprotected products are designed as medium-term structured notes, with maturities ranging from 5 to 15 years. The model

The biggest potential risk of implementing a portfolio insurance strategy is that the payoffs are particularly

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vulnerable to sudden huge drops in the risky asset prices before the portfolio can be rebalanced. In such cases, the value of the insured portfolio would fall below the floor; however, a hard guarantee, such as 100% principal protection, to the investors must be fulfilled at the end of investment horizon. Therefore, the issuers of CPPI products would suffer a gap risk. In particular, if the issuers issue several CPPI products written on different underlying assets, and those seemingly uncorrelated assets suddenly become much more correlated in stressed conditions (such as the credit crunch led by the subprime mortgage crisis since September 2007), then the scale of the gap risk may be very much underestimated. The unexpected huge losses may jeopardize the issuers and the stability of financial markets. In theory, the issuer of CPPI products can hedge such exposure to gap risk by options. The issuer needs to model the likely worst-case move in the risky asset price before the next rebalancing opportunity and build the cost of this implicit option into the premiums and fees charged to the investors or provided by capital. But, in practice, the pricing of such options can be quite complex, because the issuer does not exactly know the underlying asset price processes and their correlations in stressed markets. Furthermore, it is difficult for the issuers to find available options through which to hedge their exposures. Pain and Rand (2008) report that some issuers of CPPI products create securities that package up the gap risk and sell these to investors, including private banks and funds. But these structured derivatives are not popular due to the complicated nature of risk. Other issuers seek to limit their exposure to gap risk by other imprecise hedging vehicles or so-called proxy hedging. But, this in turn, exposes the issuers to the basis risk.

20.6 Conclusion A portfolio insurance strategy is a dynamic hedging process that provides the investor with the potential to limit downside risk while allowing participation on the upside so as to maximize the terminal value of a portfolio over a given investment horizon. This paper looks at the basic concept of portfolio insurance, the alternative dynamic hedging strategies available to the portfolio manger, the empirical comparison of the performances of various portfolio insurance strategies, the recent market development of portfolio insurance techniques, and finally, the impact of portfolio insurance on financial market stability.

L.-c. Ho et al.

References Bertrand, P. and J. L. Prigent. 2001. Portfolio insurance: the extreme value approach to the CPPI method, Working paper, Thema University of Cergy. Bertrand, P. and J. L. Prigent. 2005. “Portfolio insurance strategies: OBPI versus CPPI.” Finance 26(1), 5–32. Black, F. and R. Jones. 1987. “Simplifying portfolio insurance.” Journal of Portfolio Management 14, 48–51. Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 81, 637–659. Cesari, R. and D. Cremonini. 2003. “Benchmarking, portfolio insurance and technical analysis: a Monte Carlo comparison of dynamic strategies of asset allocation.” Journal of Economic Dynamics and Control 27, 987–1011. Do, B. H. 2002. “Relative performance of dynamic portfolio insurance strategies: Australian evidence.” Accounting and Finance 42, 279–296. Do, B. H. and R. W. Faff. 2004. “Do futures-based strategies enhance dynamic portfolio insurance?” The Journal of Futures Markets 24(6), 591–608. Estep, T. and M. Kritzman. 1988. “TIPP: Insurance without complexity.” Journal of Portfolio Management 14(4), 38–42. Garman, M. B. and S. W. Kohlhagen. 1983. “Foreign currency option values.” Journal of International Money and Finance 2, 231–237. Hakanoglu, E., R. Kopprasch, and E. Roman. 1989. “Constant proportion portfolio insurance for fixed-income investment: a useful variation on CPPI.” Journal of Portfolio Management 15(4), 58–66. Hamidi, B., E. Jurczenko, and B. Maillet. 2007. An extended expected CAViaR approach for CPPI. Variances, Working paper, University of Paris-1. Herold, U., R. Maurer, and N. Purschaker. 2005. “Total return fixedincome portfolio management: a risk-based dynamic strategy.” Journal of Portfolio Management 31, 32–43. Ho, L. C., J. Cadle and M. Theobald. 2008. An analysis of risk-based asset allocation and portfolio insurance strategies, Working paper, Central Bank of the Republic of China (Taiwan). Leland, H. 1985. “Option pricing and replication with transaction costs.” Journal of Finance 40, 1283–1301. Loria, S., T. M. Pham, and A. B. Sim. 1991. “The performance of a stock index futures-based portfolio insurance scheme: Australian evidence.” Review of Futures Markets 10(3), 438–457. Pain, D. and J. Rand. 2008. “Recent developments in portfolio insurance.” Bank of England Quarterly Bulletin Q1, 37–46. Perold, A. R. 1986. Constant proportion portfolio insurance, Working paper, Harvard Business School. Perold, A. R. and W. F. Sharpe. 1988. “Dynamic strategies for asset allocation.” Financial Analysts Journal 44(1), 16–27. Rendleman, R. J. Jr. and T. J. O’Brien. 1990. “The effects of volatility misestimation on option-replication portfolio insurance.” Financial Analysts Journal 46(3), 61–70. Rubinstein, M. and H. E. Leland. 1981. “Replicating options with positions in stock and cash.” Financial Analysts Journal 37, 63–72. Zhao, Y. and W. T. Ziemba. 2000. “A dynamic asset allocation model with downside risk control.” Journal of Risk 3(1), 91–113. Zhu, Y. and R. C. Kavee. 1988. “Performance of portfolio insurance strategies.” Journal of Portfolio Management 14(3), 48–54.

Chapter 21

Security Market Microstructure: The Analysis of a Non-Frictionless Market Reto Francioni, Sonali Hazarika, Martin Reck, and Robert A. Schwartz

Abstract The Capital Asset Pricing Model describes a frictionless world characterized by infinite liquidity. In contrast, trading in an actual marketplace is replete with costs, blockages, and other impediments. Equity market microstructure focuses on how orders are handled and turned into trades in the non-frictionless environment. For over three decades, the literature has grown while, concurrently, trading systems around the world have been reengineered. After depicting the frictionless CAPM, we consider the development of microstructure analysis, concentrating on issues germane to market architecture. We then consider the design of one facility, Deutsche Börse’s electronic platform, Xetra. Important insights were gained from the microstructure literature during Xetra’s planning period (1994–1997), and Xetra’s implementation marked a huge step forward for Germany’s equity markets. Nevertheless, academic research and the design of a real world marketplace remain works in progress. Keywords Market microstructure r CAPM r Market makers r Liquidity r Price volatility r Price discovery r Trading costs r Electronic markets

21.1 Introduction Security market microstructure addresses issues that involve the implementation of portfolio (investment) decisions in a marketplace. Implementation entails the placement and handling of orders in a securities market, and their translation into trades and transaction prices. The process 

This chapter includes material from Francioni et al. (2008) and from Schwartz (1988), which was reprinted in Schwartz and Francioni (2004).

R.A. Schwartz and S. Hazarika () Zicklin School of Business, Baruch College, CUNY, New York, NY, USA e-mail: [email protected] R. Francioni and M. Reck Deutsche Börse, Frankfurt, Germany

links fundamental information concerning equity valuation (which is of primary concern to portfolio managers) to prices and transaction volumes that are realized in the marketplace. The quality of the link depends on the rules, procedures, and facilities of a securities market, and on the broader regulatory and competitive environment within which the market operates. Widespread interest on the part of the securities industry, government, and academia is testimony to the importance of market microstructure analysis. The subject addresses issues that concern investors, broker/dealer intermediaries, market regulators, exchanges, and other trading venues as well as the broad economy. Interest in microstructure has increased sharply over the past three and a half decades, spurred in particular by three events: the U.S. Securities and Exchange Commission’s (SEC) Institutional Investor Report (1971), the passage by the U.S. Congress of the Securities Acts Amendments of 1975, and the sharp stock market drop on October 19, 1987. Further, the advent of computer-driven trading in recent years has enabled researchers to capture electronically the full record of all trades and quotes, and this has provided empirical researchers with far richer data (referred to as “high frequency data”) for analyzing trading and price setting. Over the years, microstructure analysis has expanded and, concomitantly, exchange structure has strengthened. We consider both of these developments in this chapter. First, we set forth the major challenges that the microstructure literature addresses. Second, we consider the properties of a frictionless trading environment. Third, we present a broad view of the direction in which microstructure analysis has been and is evolving. Fourth, we turn to one application – the design of an actual marketplace: Deutsche Börse’s electronic trading system, Xetra. The German market was the last of the major European bourses to introduce an electronic trading platform, and it is state of the arts, which makes Deutsche Börse a particularly interesting case in point. Fifth, in the concluding section, we consider the bumpy and hazardous road that takes us from theory to the development of an actual marketplace.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_21, 

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21.2 Microstructure’s Challenge Microstructure analysis has four broad applications. First (and this is a key focus of the chapter), it gives guidance to market structure development. The link with market structure is straightforward: the critical factor that drives microstructure analysis is friction in the marketplace (i.e., the explicit and implicit costs of implementing portfolio decisions), and trading costs dependent on the architecture of the marketplace, which determines how orders are handled and turned into trades. The flipside of friction is illiquidity, and a primary function of a market center is to amass liquidity. Microstructure’s second application is to facilitate the development of trading strategies and algorithms for asset managers and broker/dealer intermediaries. The importance of this application is evident in the current development of computer-driven algorithmic trading. Algorithms can be finetuned to take account of, for example, the probability of a limit order executing, time of day effects such as market openings and closings, the search for liquidity in a fragmented environment, and the choice of a trading modality (e.g., a continuous limit order book market, a quote-driven dealer market, a periodic call auction, a block trading facility, or hybrid combinations of the above). The third application of microstructure analysis concerns tests of market efficiency. In the 1970s, at a time when the subject was first emerging, the Efficient Markets Hypothesis (EMH) was widely accepted by financial economists as a cornerstone of modern portfolio theory, and it continues to receive broad academic support today. The hypothesis addresses informational as distinct from operational efficiency (the latter refers to the containment of transaction costs by superior market design). According to the EMH, a market is informationally efficient if no participant is able to achieve excess risk adjusted returns by trading on currently available information. Many of the EMH tests have considered one major part of the information set – market information (e.g., recent quotes, trading volume, and transaction prices). If prices properly reflect all known information, then (in a frictionless market at least) they must change randomly over time; hence the term “random walk.” Earlier studies, based on daily data, generally supported the random walk hypothesis. However, with the advent of high frequency data, the footprints of complex correlation patterns have been detected. This observation, along with superior knowledge of the impact of trading costs on returns behavior, is casting a new light on market efficiency. Whether inefficiency is thought of in operational or informational terms, the EMH is not as stellar as it once was. In its fourth application, microstructure analysis sheds light on how new information is incorporated into security prices. In a zero cost, frictionless environment, share values would be continuously and instantaneously updated with the release of new information. In actual markets, however, information must be received and assessed, traders’ orders

R. Francioni et al.

must be placed and processed, and executions must be delivered and accounts cleared and settled. Costs, both explicit (e.g., commissions) and implicit (e.g., market impact), are incurred throughout this chain of events. Highlighted in much microstructure literature are the costs that some participants incur when, in an asymmetric information environment, other participants receive information first and trade on it to the disadvantage of the uninformed. Asymmetric information is not the only reality, however. In light of the size, complexity, and imprecision of much publicly available information, one might expect that investors in possession of the same (large) information set will form different expectations about future risk and return configurations. This situation is referred to as “divergent expectations.”1 Asymmetric information and divergent expectations together reflect a rich set of forces that impact the dynamic behavior of security prices. This overview of microstructure’s four broad applications underscores that trading frictions are the subject’s raison d’être. Participant orders cannot be translated into trades at zero cost (markets are not perfectly liquid), and trades typically are not made at market clearing (i.e., equilibrium) prices. Trading decision rules (algorithms) are needed because the costs of implementing portfolio decisions can sharply lower portfolio performance. In fact, much algorithmic trading is designed to control trading costs, rather than to exploit profitable trading opportunities. Today, trading is recognized as an activity that is both distinct from investing and equivalently professional. Market structure is of concern to the buyside desks precisely because markets are not perfectly liquid, and neither are they perfectly efficient, either informationally or operationally. Consequently, better market structure can deliver superior portfolio performance for participants. What is the economic service, one might ask, that an equities market provides? The fuzzy link that connects information and prices in the non-frictionless environment underscores two major market functions – price discovery and quantity discovery. Price discovery refers to participants collectively searching for equilibrium prices. Quantity discovery refers to the difficulty that participants who would be willing to trade with each other actually have finding each other and trading when markets are fragmented. This difficulty is accentuated because some participants (primarily institutional investors) do not immediately reveal the total size of their orders (doing so would unduly drive up their market impact costs). Market structure affects both the accuracy of price discovery and the completeness of quantity discovery. The link between market structure and price discovery depends on the environment within which participants are operating. At one end of the spectrum, investors can be equally 1

For a recent discussion, see Davis et al. (2007).

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

informed and form homogeneous expectations based on the information they all possess. At the other end, they can be differentially informed and form divergent expectations with regard to commonly shared information. When investors who share common information all agree on share values (i.e., have homogeneous expectations), prices can be “discovered” in the upstairs offices of research analysts. When investors are not equally informed, and when they form different expectations based on common information, prices must be discovered in the marketplace. In this second environment, the economic service provided by an exchange is clear – it “produces the price.” Regarding quantity discovery, handling the orders of large institutional customers is a challenge. It is not at all uncommon for an institution to want to buy or to sell, for instance, 500,000 shares of a company that has an average daily trading volume of 300,000 shares. Executing an order of this size can easily drive prices away from the trader before the job has been completed. The adverse price move is a market impact cost. Institutions attempt to control their market impact costs by trading patiently and, as much as possible, invisibly. Good market structure can help. To this end, a number of alternative trading systems (ATSs) have been formed in recent years, and dark (i.e., non-transparent) liquidity pools have emerged. With prices discovered in the marketplace, participants employ trading strategies when they come to the market to implement their portfolio decisions. Participants with differential information that will soon become public determine how best to meter their orders into the market so as to move prices to new levels with minimal speed. Additional questions that any trader might ask include: “If I trade now, at the current moment, how will the price that I will receive compare with the average price that shares are trading at today?” “Is price currently at a sustainable, validated level, or is it likely to move higher or lower in the coming hours, minutes, or even seconds?” “Would I do better to be patient and place a limit order, or submit a market order and get the job done right away?” “Should I attempt to trade now in the continuous market, or wait for a closing call?” The orders that a set of participants reveal to the market depend on how questions such as these are answered, and prices that are set and trading volumes that are realized depend on the orders that are revealed. The categories of trading costs that receive the most attention on the part of exchanges, regulators, and academicians are generally those that are the most straightforward to measure: commissions and bid-ask spreads. Increasingly precise measures of market impact are also becoming available, and this cost too is being widely taken into account. On the other hand, the opportunity cost of a missed trade, being far more difficult to quantify, is often overlooked. Also more challenging is quantifying a cost that has received little formal attention: realizing executions at poorly discovered

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prices. The problem, of course, is that equilibrium values are not observable and appropriate benchmark values are not easily defined.

21.3 The Perfectly Liquid Environment of CAPM Peter Bernstein’s (2007) piece in the Journal of Portfolio Management has the intriguing title, “The Surprising Bond Between CAPM and the Meaning of Liquidity.” In it he wrote, “The more liquid an asset, the greater the dominance of systematic risk over stock specific risk.” We build on this insight in this section. In so doing, we formalize the fact that the Capital Asset Pricing Model (CAPM) describes an extreme case, a totally frictionless world where liquidity is infinite and systematic risk has complete dominance over stock specific risk. The analysis provides a good platform from which to launch a discussion of market microstructure, the study of a non-frictionless environment. CAPM models the prices of the individual equity shares that, in aggregate, comprise the market portfolio. Following standard methodology, we start our analysis of the frictionless environment by taking the market portfolio to be one single asset (e.g., an all encompassing exchange traded fund). We consider the demand of an agent to hold shares of this one risky asset when the only alternative is the riskless asset. We show that an individual agent’s demand curve to hold shares of the risky asset is downward sloping, and then use this curve to re-derive certain key CAPM equations to show that the associated demand to hold shares of each individual equity issue in that portfolio is infinitely elastic, and that therefore, the market for the individual shares is infinitely liquid. In the CAPM world, each individual equity issue in the market portfolio has an intrinsic value that is given by the parameter that locates the height (on the price vector) of that infinitely elastic demand. In the section that follows, we turn to the non-frictionless environment of microstructure analysis where individual stock demand curves are downward sloping, the liquidity of individual shares is, therefore, finite, and individual shares do not have intrinsic values. To obtain the representative investor’s demand curve to hold shares of the risky market portfolio, first we state the agent’s utility (of wealth) function. The demand curve to hold shares of the market portfolio may then be obtained directly from the utility function. The derivation follows Ho et al. (1985). We make the following assumptions: The investor’s portfolio comprises a risk-free asset and

one risky asset (shares of the market portfolio). Share price and share holdings are continuous variables.

336 Short selling is unrestricted. The existence of a brief trading period, T0 to T1 , which is

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of P1 , the expected utility of end of period wealth, written as a function of P and Q, given N0 and C0 , would be

followed by a single investment period, T1 to T2 . All transactions made during the trading period are settled

at point in time T1 . The investor seeks a portfolio at the beginning of the investment period (at time T1 ) that will maximize the expected utility of wealth to be realized at the end of the investment period (at time T2 ). Investor expectations with respect to the share price at the end of the investment period (at time T2 ) are exogenously determined (expectations are independent of the current price of shares). Investors are risk averse. The following variables are used: C0 D holdings of the risk-free asset at the beginning of the trading period .T0 / C1 D holdings of the risk-free asset at the beginning of the investment period .T1 / N0 D number of shares of the market portfolio held at the beginning of the trading period .T0 / N1 D number of shares of the market portfolio held at the beginning of the investment period .T1 / R0  1 D risk-free rate of interest over the trading period R1  1 D risk-free rate of interest over the investment period P1 D price at which shares of the market portfolio are purchased or sold during the trading period P2 D price at which shares of the market portfolio can be sold at the end of the investment period .T2 / rm D P2 =P1  1 D return on the market portfolio Q D number of shares traded by the investor at the beginning of the investment period .T1 /I Q > 0 indicates a purchase; Q < 0 indicates a sale

21.3.1 The Expected Utility of End of Period Wealth The participant starts the investment period with C1 dollars of the risk-free asset and N1 shares of the market portfolio (the risky asset). Therefore, wealth at T2 is given by C1 R1 CN1 P2 . As of T1 , this wealth is uncertain because P2 is uncertain. As of T1 , the expected utility of end of period wealth can be written as (21.1) EU.C1 R1 C N1 P2 / The investor starts the trading period with C0 dollars of the risk-free asset and N0 shares of the risky asset. If during the trading period the decision maker were to exchange holdings of the risk-free asset for Q shares of the risky asset at a price

h.P1 ; QjN0 ; C0 / D EU Œ.C0 R0  QP1 /R1 C .N0 C Q/P2 (21.2) where C0 R0  QP1 D C1 and N0 C Q D N1 . Equation (21.2) can be rewritten as h.P1 ; QjN0; C0 / D c C gQ.a  bQ  P1 /

where

(21.3)

c D U.W/ N20 U 0 .W /=R1 g D U0 .W/R1 a D ŒE.P2 /  2 N0 =R1 b D =R1 1 D  ŒU00 .W/=U0 .W/ Var.P/ 2

The step from Equation (21.2) to Equation (21.3) involves a Taylor expansion of the investor’s utility around the expected value of wealth if the investor does not trade.2 The procedure is a convenient way of introducing the variance term into the utility function.3

21.3.2 The Reservation Demand Curve Equation (21.3) can be further assessed with the use of risk aversion and risk premium measures that are defined in Appendix A. Specifically, using Equation (21.3), we now obtain both a reservation price demand curve and an ordinary demand curve to hold shares of the risky asset. We consider the reservation demand curve first. The reservation price for a purchase or a sale is the maximum price the decision maker would be willing to pay to buy a given number of shares .Q > 0/, or the minimum price the decision maker would be willing to receive to sell a given number of shares .Q < 0/ when the only alternative is not to trade at all. Equation (21.3) shows that, if no trade is made (that is, if Q D 0), the decision maker’s expected utility is equal to c. The reservation price for any value of Q is the price that equates the expected utility Œh.Pl ; QjN0 ; C0 / if 2 For a discussion of the Taylor procedure see, for example, R. G. D. Allen, Mathematical Analysis of Economists, London, England: Macmillan, 1960. 3 Two further assumptions are required to obtain Equation (21.3): (1) the third derivative of utility with respect to wealth is small enough to ignore; and (21.2) the squared deviation of the expected rate of return on the risky asset from the risk-free rate is small enough to ignore.

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

the trade is made, with the expected utility (c) if no trade is made. Thus, the reservation price for any value of Q is given by (21.4) h.P R ; QjN0 ; C0 / D c where PR is the reservation price associated with the trade of Q shares. Given Equation (21.3), for Equation (21.4) to be satisfied, we must have a  bQ  P1 D 0. Hence the reservation price demand curve is P R D a  bQ

(21.5)

21.3.3 The Ordinary Demand Curve Using Equation (21.3), we can also obtain the ordinary demand curve. At any value of P1 , the decision maker selects the value of Q that maximizes expected utility. Hence, the ordinary price demand curve is given by @h 0 .P ; QjN0 ; C0 / D 0 @Q

We thus see that the investor achieves an optimal holding of the risky asset by obtaining the number of shares that equates the marginal risk premium with the market price of risk.

21.3.5 The Investor’s Optimal Point on the Capital Market Line The demand model can be used to assess the investor’s optimal point on the capital market line. Let rp be the return on the combined portfolio (N1 shares of the market portfolio and C1 dollars of the risk-free asset). From Appendix Equation (21.7) we have D rp

D

D

The derivation in this section through Sect. 21.3.6 follows Schwartz (1988). When the investor has traded the optimal number of shares of the market portfolio at the market determined price per share, his or her risk premium can be related to the market price of risk. Assessing, the ordinary demand curve at P0 D P1 gives P1 D

E.P2 / 2 N1  R1 R1

(21.8)

Multiplying by R1 =P1 , rearranging, and recognizing that ŒE.P2 /=Pl  1 D E.rm / and R1  1 D rf , we get 2 N1 D E.rm /  rf P1

(21.9)

M % D E.rm /  rf

(21.10)

Therefore, we have

where M % is the marginal risk premium (see Appendix 21A). Note that the right-hand side is the price of risk.

2

2 1 W RA Var.rp / 2 N1

(21.11)

Because ¢p D .NP=W/¢m , we have Var .rp / D ¢p .NP=W/¢m and can write Equation (21.11) as

(21.7)

21.3.4 The Risk Premium and the Market Price of Risk

W N1

which, using RA D U00 .W/=U0 .W/, the measure of absolute risk aversion, can be written as

(21.6)

where P0 is the “ordinary” price associated with the trade of Q shares. Differentiating h in Equation (21.3) with respect to Q, setting the derivative equal to zero, and rearranging gives P 0 D a  2bQ

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1 RA p 2



PW N1

m

(21.12)

Substituting (21.12) into (21.9) and simplifying gives RR P D

E.rm /  rf m

(21.13)

where RR .D WRA / is the measure of relative risk aversion. Equation (21.13) shows that for the investor to hold an optimal combined portfolio, the market price of risk per standard deviation of the market portfolio must be equal to the investor’s coefficient of relative risk aversion times the standard deviation of the combined portfolio’s return. Letting w D N1 P1 =W, substituting w¢m ; D ¢p into Equation (21.13), and rearranging gives wD

E.rm /  rf Var.rm /RR

(21.14)

Equation (21.14) shows that the percentage of wealth that the risk averse participant invests in the market portfolio is positively related to the expected return E.rm /, and negatively related to rf ; Var.rm /, and RR . Investors all face the same values of E.rm /; rf and Var.rm /, but differ according to their degree of risk aversion. More risk averse investors (larger RR ) have smaller optimal values of w and hence are more

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apt to lend at the risk-free rate (which implies w < 1); less risk averse investors (smaller RR ) have larger optimal values of w and hence are more likely to borrow at the risk-free rate (which implies w > 1). The right-hand side of Equation (21.13) is the market price of risk per standard deviation of the market portfolio. The total compensation for risk taking is the price of risk times the standard deviation that the investor accepts (here, the standard deviation of the combined portfolio). Multiplying both sides of Equation (21.13) by p , we obtain 

E.rm /  rf RR Var.rp / D m

 p

(21.15)

Adding rf to both sides of Equation (21.15) gives the investor’s total compensation for waiting and for risk taking: 

E.rm /  rf E.rp / D rf C RR Var.rp / D rf C m

 p (21.16)

Equation (21.16) shows that the location of the investor’s optimal point on CAPM’s capital market line depends on his or her measure of relative risk aversion .RR /.

21.3.6 The ith Risky Asset’s Point on the Security Market Line We now assess the demand model to show the location of an ith risky asset on the security market line. In so doing, we establish that the demand for the ith risky asset is infinitely elastic. Equation (21.10) shows that the marginal risk premium for each investor, as a percentage of P1 , will equal E.rm /  rf . Therefore, for each investor, RA Var.P2 /N1 D E.rm /  rf P1

(21.17)

It follows from Equation (21.17) that investors with lower values of RA hold a larger number of shares, such that the product RA N1 is the same for all investors. Because rm D .P2 =P1 /  1; Var.rm / D Var.P2 /=P1 2 . Substituting Var.rm /P1 2 D Var.P2 / into Equation (21.17) and simplifying gives RA Var.rm /P1 N1 D E.rm /  rf

(21.18)

Using P1 N1 D wW we obtain wRR Var.rm / D E.rm /  rf

(21.19)

Equation (21.19) can be interpreted as an equilibrium condition for each investor. Because wRR D RA N1 P1 , and given that the product RA N1 is constant across investors, RR w is constant across all investors. [It is also clear from Equation (21.19) that the product wRR must be constant across all investors, because E.rm /; rf , and Var.rm / are the same for all.] The equilibrium condition for each investor with respect to the market portfolio implies an equilibrium condition for each investor with respect to any ith risky asset in the market portfolio. The CAPM shows that the relevant measure of risk for the ith risky asset is “i D ¢im =Var.rm/. Therefore, writing Var.rm / D ¢im =“i , substituting into Equation (21.19), and multiplying both sides by “i we get wRR m D ˇi ŒE.rm /  rf

(21.20)

Adding rf to both sides of Equation (21.20) gives CAPM’s security market line, rf C wRR i m D rf C ˇi ŒE.rm /  rf D E.ri /

(21.21)

Where E.ri / is the expected return on the ith stock in the market portfolio. Equation (21.21), assessed at w D 1, shows that the expected return for the ith risky asset depends on its covariance with the market return, and on the measure of relative risk aversion for an investor whose optimal combined portfolio contains the market portfolio only. The equation also shows that the ith risky asset’s specific location on the security market line depends on the covariance of the asset’s return with the return on the market portfolio, and hence that its expected return depends only on ˇi , its systematic risk. It follows from the above discussion that the demand to hold shares of the market portfolio is downward sloping, while the demand for each individual stock in the market portfolio is infinitely elastic. The reason is that perfect substitutes do not exist for the aggregate portfolio, but they do exist for the individual stocks. Only one factor characterizes any ith stock – ˇi , its covariance with the market. But the covariance for any stock can be duplicated exactly by an appropriate combination of two or more other stocks, and all holdings that have the same covariance must yield the same expected return. If they were to yield different expected returns, an unlimited number of shares of the higher yielding position would be bought, and an unlimited number of shares of the lower yielding position would be sold short until, with costless trading, the buying and selling pressures bring the two prices into exact equality. Unlimited buying (selling) at any price lower (higher) than the beta appropriate, CAPM price manifests an infinitely elastic demand to hold shares. That is, at an infinitesimally higher price no shares will be held, and at an infinitesimally lower price demand will be unlimited.

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

Bernstein’s (2007) two insights immediately follow: a stock’s systematic risk totally dominates its specific risk, and the market for each ith stock is infinitely liquid at the price which translates into E.ri /, its systematic risk appropriate return. As we turn to the non-frictionless market, the infinitely liquid, infinitely elastic property of CAPM is a good point of departure from the frictionless world. A common denominator in many microstructure analyses is that the demand to hold shares of individual stocks is downward sloping (which means that shares do not have intrinsic values). Market makers post bid and ask quotes that, when raised, result in more public sales to the market maker and, when lowered, result in more public purchases from the market maker. Bid and ask quotes can be distributed over multiple price points in competitive dealer markets as well as on public limit order books. Trading is not costless. Both explicit costs (e.g., commissions and taxes) and implicit costs (e.g., market impact costs) are incurred. Information is complex and imprecise, and thus investors commonly disagree about its interpretation. Arbitrage is not costless, and perfect substitutes for individual issues do not exist. Share values depend not only on the calculations of systematic risk in the upstairs markets, but also on how orders interact in the marketplace. As a consequence of all of this, trades that are made and the transaction prices that they are made at also depend on the structure of a marketplace. Microstructure analyses address these realities that CAPM does not comprehend.

21.4 What Microstructure Analysis Has to Offer: Personal Reflections In this section we review the development of microstructure analysis. Our objective is not to provide a comprehensive survey of the literature, but to highlight some of the important themes that have given guidance to market structure development. More detailed information can be obtained from Cohen et al. (1979) who have provided an early survey of the field; from O’Hara (1997) who discusses important theoretical microstructure models; from Madhavan (2000); Biais et al. (2005); Parlour and Seppi (2008) who have provided more recent surveys; and from Hasbrouck (2007) who deals with empirical microstructure research and research methodology. We first focus on the early literature, next turn to more recent developments, and finally present our thoughts concerning an important direction in which future microstructure research ought to head.

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21.4.1 The Early Focus The first contributions to the new field in financial economics that came to be called “microstructure” were made by a couple of people who participated in the SEC’s Institutional Investor Report (1971). A handful of others independently started to focus on microstructure topics in the early 1970s. Eventually a few of the early researchers came to recognize the commonality of their interests and, applying the title of Garman’s (1976) well-known paper, “Market Microstructure,” they gave the field its name. Much of the early literature focused on dealers and exchange specialists. These market makers were viewed as the suppliers of immediacy to investors, and the spread was considered the price they charge for providing this service in an environment where order arrival is non-synchronous. Of key importance was the relationship between spreads and the costs of market making. The earlier market maker studies were in large part motivated by a desire to determine whether or not these intermediaries were realizing monopoly profits and, if so, whether or not their profits were attributable to market making being a natural monopoly. Spreads that are greater than the costs of market making would be taken as an indication of monopoly power on the part of the dealers, and spreads that were negatively related to trading volumes would indicate economies of scale in market making, which could imply a natural monopoly (Stigler 1964). Spreads were indeed found to decrease with transactions volume, but reasons other than market making being a natural monopoly were advanced (Smidt 1971; Tinic 1972). The general picture that emerged was that the trading costs incurred by investors could be lowered by strengthening competition between market maker intermediaries. In particular, competition in the NYSE market was deemed inadequate, as specialists and the Exchange itself were viewed as having monopoly positions: each stock was assigned to just one specialist; the NYSE’s order consolidation rule (Rule 390) precluded in-house executions by requiring that exchange members send their orders for NYSE-listed securities to an exchange; and commissions were fixed and unjustifiably high (Tinic and West 1980).4 Not surprisingly, the focus on the market maker firms led several researchers to model market maker pricing decisions (i.e., the setting of their bid and ask quotes). These 4

Another major issue addressed by the microstructure literature at that time was the impact of information on trading volume and price (Copeland 1976; Beja and Hakansson 1977; Beja and Goldman 1980).

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included Bagehot (1971); Stoll (1978); Amihud and Mendelson (1980); Ho and Stoll (1980, 1981, 1983); Mildenstein and Schleef (1983). With one exception (Bagehot 1971), the early formulations dealt with inventory considerations. A market maker firm holding an undesirably long position would lower the quotes (i.e., lower the offer so as to sell more shares, and reduce the bid so as to discourage others from selling shares to it). Reciprocally, a market maker who was short would raise the quotes. This response on the part of the public (buy more shares when the market maker’s offer is lower, and sell more shares when the market maker’s bid is higher) is evidence that the public’s demand to hold shares of any specific stock was taken to be downward sloping. A variety of mathematical tools were used to solve for optimal market maker quotes. These models also gave further insight into the cost components of the market maker’s spread (Stoll 1989). While insightful, the early inventory-based pricing models suffered from some shortcomings. First, the early formulations for the most part assumed monopoly market makers, although some of these models were applied to markets such as the New York Stock Exchange where exchange specialists were in fact competing with other floor brokers and customer limit orders (Demsetz 1968). The application of theory further suffered from the reality that the price of immediacy for an investor is not the spread of an individual market maker, or even an average market maker spread, but the inside spread (i.e., the lowest ask across all market makers minus the highest bid).5 It is important to note that dealer spreads could individually remain relatively invariant with respect to transaction volume while the inside spread fell appreciably. A further shortcoming of most of these earlier models is that they did not take account of a major cost incurred by market makers: the losses generated by trading with better informed investors. Recognition of this reality (which is also outside the scope of the frictionless world of CAPM) led to a development that did much to establish microstructure as an important new field in financial economics – the introduction of market maker models that were based, not on inventory management, but on controlling the cost incurred when some investors are in possession of information that the market maker and other investors have not yet received. Bagehot (1971) was the first to embark on this line of thought. He was later followed by, among others, Gloston and Milgrom (1985) and Kyle (1985). With information asymmetries, the market maker always looses when trading with a better informed participant. For microstructure theorists at the time, this meant that, for the dealer market not to fail, some investors must trade for

5

For further discussion, see Cohen et al. (1979).

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reasons that are not related to information.6 Liquidity considerations (i.e., an investor’s personal cash flow needs) was one such motive for public buying and selling. A third participant type was also introduced along with the liquidity traders – noise traders (participants who trade on price moves as if they contain information when in fact they do not). This trio of informed traders, liquidity traders, and noise traders was used to show how markets could function and, in so doing, enable new information to be incorporated into security prices (Grossman and Stiglitz 1980; Milgrom and Stokey 1982; Kyle 1985; Glosten and Milgrom 1985; Copeland and Galai 1983; Easley and O’Hara 1987, 1991, 1992). At this stage in its early development, the microstructure pricing models were predominantly market maker models. One exception should be noted, however: a National Book System proposed by Mendelson et al. (1979) contained a comprehensive description of an order driven automated trading system that provided guidance for designing the first exchange-based electronic trading systems. For a more recent discussion of automated trading systems, see Domowitz and Steil (1999). Most equity markets around the globe are now order driven, limit order book markets that might include market makers in a hybrid structure (as does the NYSE), but are not basically quote driven (i.e., dealer) markets (as was the old Nasdaq and London Stock Exchange). The limit order book markets are driven by the orders placed by the investors themselves, not by market maker intermediaries.

21.4.2 The Current Focus Over the years, microstructure analysis has grown extensively on both the theoretical and empirical fronts. Concomitantly, the securities markets themselves have evolved, becoming ever more technologically developed, more global in outreach, but also more fragmented between different trading facilities. One important new direction microstructure research has taken is to further model the order driven market, an environment where natural buyers and sellers provide immediacy to each other because some, who are patient, are willing to post limit orders while others, who demand immediacy, choose to submit market orders that execute against the posted limit orders. Understanding the costs of, and motives for, placing limit orders as distinct from market orders was called for.

6

A market supported by informational trading only can indeed function if agents trade with each other because their expectations are divergent. When the information that triggers trading is common knowledge, the condition may be thought of as one where agents are agreeing to disagree.

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

With limit orders, the very existence of the bid-ask spread has to be explained. That is, with a sufficiently large number of participants placing priced orders, one might expect that orders would be posted at virtually every available price point in the neighborhood of equilibrium, and that the spread would disappear. Cohen et al. made this point in their review paper (Cohen et al. 1979), and they analyzed the existence of the spread in Cohen et al. (1981).7 They further wrote, “With regard to modeling the market spread, we suggest that a straightforward aggregation from individual spreads is not possible in a system where there is no clear distinction between demanders and suppliers of immediacy, and where traders meet in a dynamic, interactive environment that incorporates the impact of investor order placement strategies.” Strategic order placement clearly required further analysis. The task, however, was not simple. Some of the first papers in this area assumed, as is true for a dealer market, that limit order and market order participants are two separate, exogenously fixed groups that are separated by a firewall (Glosten 1994). This assumption, while simplifying mathematical modeling, unfortunately distills out much of the richness of an order driven market. More recent models have eliminated the firewall (Handa and Schwartz 1996a; Foucault 1999; Parlour 1998; Handa et al. 2003; Faucault et al. 2005; Goettler et al. 2005). With the choice between limit order and market order endogenous, for any market to function, participants must divide naturally into four groups that reflect two dichotomies (one between buyers and sellers and the other between limit order and market order placers), not the standard two (buyers and sellers). With order type selection endogenous in the order driven market, the balance between immediacy demanders and immediacy suppliers becomes a second equilibrium that must be understood. That is, one needs to recognize the conditions under which some participants will choose to be liquidity demanders (place market orders) while others choose to be liquidity suppliers (place limit orders). If a reasonable balance is not achieved between these two groups, the order driven market will fail (as indeed it does for thinner, small cap stocks). Increasingly, these issues have been handled, and some sophisticated limit order models have been developed.8 Microstructure analysis of trading systems has expanded to include periodic call auctions.9 The economics of a call auction are quite different from those of continuous trading and, consequently, so too are the order placement strategies

7

Cohen et al. (1981) describe the tradeoff between execution probability and price improvement in the optimal choice between limit and market orders. 8 See Bach and Baruch (2007) for a recent discussion and further references. 9 See Economides and Schwartz (1995) for a description of alternative call market structures.

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that participants should employ when they approach a call market. Call auctions do not, by their very nature, supply immediacy. Rather, orders that are entered during a call’s bookbuilding phase are held for a periodic crossing at a single clearing price at the (generally predetermined) time of the market call. Consequently, buy and sell orders submitted to a call do not execute when they arrive even if they match or cross in price (matching and crossing orders execute immediately in a continuous trading environment). This being the case, limit and market orders have a different meaning in a call: limit orders do not supply immediacy to market orders, and market orders are simply extremely aggressively priced limit orders (i.e., a market order to sell in a call effectively has a limit price of zero, and a market order to buy effectively has a limit price of infinity). Today, virtually all modern, electronic exchanges open and close their continuous markets with call auctions. Consequently, participants face further decisions when operating in a call plus continuous, hybrid market: how to submit an order to a call auction that is followed by continuous trading (e.g., an opening call), and how to submit an order to a continuous trading environment that is followed by a call auction (e.g., a closing call). Taking these tactical decisions into account is part of the complexity of microstructure analysis. Technological developments have simultaneously enabled new trading venues to emerge (which can fragment markets) while providing connectivity between them (which can consolidate markets). Concurrently, regulatory initiatives have been motivated by the desire to intensify inter-market competition. Questions can be raised, however, concerning fragmentation of the order flow. The conventional wisdom has been that the consolidation of order flow improves liquidity, and exposing each order to all other displayed orders gives investors the best prices for their trades. Consolidating trading in a single market provides incentives to liquidity suppliers to compete aggressively for market orders by revealing their trading interest, and by being the first to establish a more favorable price (if time is used as a secondary priority rule). On the other hand, arguments in favor of trading on multiple markets include the benefits of inter-market competition, and the fact that traders with disparate motives for trading may want different marketplaces to trade in (i.e., the “one size does not fit all” argument). And so, different markets develop to serve diverse investor needs (such as achieving a faster execution versus obtaining a better price). One growing need among large institutional investors, the ability to trade large orders with minimal market impact, has led to the advent of dark pool, block trading facilities such as Liquidnet, Pipeline, and ITG’s Posit that aid in quantity discovery. This development in the industry has spawned a related line of research on off-exchange and upstairs trading (Seppi 1990; Grossman 1992; Keim and Madhavan 1996; Madhavan and Cheng 1997).

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A spectrum of market quality issues have been of long and continuing importance to microstructure researchers. These include market transparency,10 both pre- and posttrade, (Porter and Weaver 1998), the accentuation of intraday price volatility, and correlation patterns that have been observed in high frequency data (Engle and Granger 1987). Other important issues include price clustering and tick sizes (Harris 1991, 1994; Angel 1997). Applications such as transaction cost analysis (TCA) and algorithmic trading have received increasing attention (Domowitz et al. 2001). The relative performance of floor based versus electronic trading is another important issue (Domowitz and Steil 1999). A major line of empirical research was pioneered by Hasbrouck 1993 who decomposes transaction prices into two components: a random walk component and a stationary component. The random-walk component is identified with an efficient price that the market is trying to discover. The stationary component is viewed as microstructure noise. Microstructure noise is commonly explained by features such as the bid-ask spread, market impact, and the discreteness of the pricing grid. The noise component has also been attributable to price discovery itself being a dynamic process (Menkveld et al. 2007 and Paroush et al. 2008).11 Numerous empirical studies have focused on two of the world’s premier markets, the New York Stock Exchange and NASDAQ (Hasbrouck 1991, 1995; Hasbrouck and Sofianos 1993; Christie and Schultz 1994; Christie et al. 1994; Bessembinder and Kaufman 1997a, b; Bessembinder 1999, 2003; Barclay et al. 2003, among others). Many other studies have considered European markets, Asian markets, and other markets around the world (e.g., Biais et al. 1995; Sandas 2001; Ozenbas et al. 2002).12 Across all of these markets, structural and performance differences have been noted, but also major similarities have been observed. It is apparent that, despite the influence of historic and cultural considerations, trader behavior and market performance around the globe depend largely on microstructure realities. Alternatively stated, trading rooms and markets around the world bear striking resemblances to each another. Another recent line of research has considered how search costs affect bid-ask spreads in financial markets. To this end, Duffie et al. (2005) present a dynamic model of market makers under the assumption of no inventory risk and information that is symmetrically distributed. They show that sophisticated investors who have better search and bargaining

10

Trading systems differ in their degree of transparency. Pagano and Röell (1996) investigate whether greater transparency enhances market liquidity by reducing the opportunities for taking advantage of uninformed participants. 11 Also see Hasbrouck (1995), Harvey and Huang (1991) and Jones et al. (1994). Further references are provided by Menkveld et al. (2007). 12 Also see Bessler (2006) for discussion and further references.

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abilities face tighter bid-ask spreads. This is in contrast to traditional information-based models that imply that spreads are wider for more sophisticated (i.e., better informed) investors. As we have noted, unlike in the frictionless market arena of CAPM, amassing liquidity is a primary function of a marketplace and market structure features are generally designed with liquidity implications in mind. Asset managers also take liquidity into account, along with the two other standard variables of modern portfolio theory, risk and return. Difficulties in defining, measuring, and modeling liquidity are formidable, however, and the literature that deals with it directly is relatively sparse (Bernstein 1987). Nevertheless, liquidity considerations have permeated the microstructure literature, both explicitly and implicitly.13 Looking back over the development of microstructure analysis, two observations stand out. First, microstructure studies have in multiple ways given direction to market structure development. Second, to a remarkable extent, the various theoretical microstructure models that are center stage today, and many empirical analyses that are based upon them, share a common structural framework – the asymmetric information paradigm. This consistency is desirable in that it implies that the field has grown by accretion rather than by replacement. Consequently, new insights are more apt to refine than to contradict old conclusions. Consistency, however, is not desirable if the common structural framework becomes overly rigid and restrictive, and if it yields incomplete and/or misleading answers to questions involving trader behavior, market structure, and regulatory policy. At times, a literature starts to advance along new fronts. We consider this possibility next for the microstructure literature.

21.4.3 Future Directions As we have noted, the current focus in the literature is on asymmetric information-based models, which are characterized as follows. Trading is driven by informational change, liquidity needs, and noise trading. The information motive for trading is the first mover of the three (liquidity and noise trading are required so that a market will not fail). Further, order arrival in the continuous environment is generally taken to be asynchronous. For a continuous trading regime to function with asynchronous order arrival, the presence of a limit order book and/or a market maker intermediary is required.

13

For further discussion and references regarding liquidity see Amihud and Mendelson (1986), Chordia et al. (2000, 2008), Hasbrouck and Seppi (2001), Amihud (2002) and Pastor and Stambaugh (2003).

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

Information trading is of keen interest because it represents the process by which new information is reflected in share values. In the standard asymmetric information models, it is assumed that all participants in possession of the same information form equivalent expectations concerning future risk and return configurations. When information changes, however, participants may not all receive the news at the same time; some receive it before others, a reality that, at any point in time, can divide traders into two groups – the informed and the uninformed. Informed participants will never trade with each other; consequently, liquidity and noise traders must be present for a market to function. As noted, asymmetry of information, for the most part, lies at the heart of the standard microstructure models of today. The homogeneous expectations assumption has been tempered of late. As a further departure from the infinitely liquid, zero cost environment of CAPM, it is being recognized that some participants produce “private information” (namely, that they further process information so as to gain insights that are not immediately available to others). Whether participant expectations differ because of the actual production of private information, or simply because different people interpret the same information or news announcement differently, the expectations of a group of investors can be divergent. Also at the heart of the asymmetric information models is the presumption that a stock has a fundamental value that bears a unique relationship, not to trader activity in the marketplace, but to the fundamental information that informed traders possess. The process of information being fully reflected in prices under asymmetric information is the act of informed and uninformed agents trading with each other until any discrepancy between a market price and a fundamental value is eliminated. The process can be viewed as arbitrage. In the earlier dealer models, the market maker was assumed to know a stock’s fundamental value. In later models, informed traders but not the market maker knows the fundamental values (Kyle 1985). Especially in the later models, price discovery is not instantaneous; rather, it is a protracted process that depends on the individual strategies employed by the informed and uninformed agents. In recent years, an alternative paradigm has been emerging: a divergent expectations environment (Miller 1977). While institutionally realistic, this paradigm has met with considerable academic resistance. For one thing, homogeneous expectations environments are far easier to deal with mathematically and homogeneity has, in many applications, proven to be a useful modeling assumption. The assumption has also been retained for another reason. As an attribute of individual rationality, it is presumed that intelligent agents facing the same information and applying the same

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(correct) analytic techniques will reach the same conclusions and, therefore, will have homogeneous expectations. Fundamental information, however, is enormous in scope. It is complex and imprecise, and our tools for analyzing it are relatively crude. In the presence of fuzzy information, expectations can be divergent. Allowing for divergent expectations opens another path for microstructure analysis, and it introduces new questions concerning agent behavior, market structure, and regulatory policy. Moreover, a further element can enter the analysis in a divergent expectations environment: along with forming their own opinions, agents may also respond to the opinions of others; that is, exhibit adaptive valuation behavior (Paroush et al. 2008; Davis et al. 2007).14 Just how agents communicate with each other and respond to each others’ opinions is a subject for ongoing research. The topic also opens another interface with behavioral finance. Price discovery acquires a different meaning in a divergent expectations environment, and this has important implications for market structure. When asymmetric information characterizes a community of investors, the strategic behavior of informed agents can affect the path that price takes when news moves a share value from one equilibrium to another, but the new equilibrium is path independent. With divergent expectations, the new equilibrium is path dependent – it depends on how the opinions of a diverse set of agents are integrated (Paroush et al. 2008). Alternatively stated, with divergent expectations, price discovery is a coordination process and, as such, is directly effected by market structure. In the standard asymmetric information environment, the key dichotomy is between informed and uninformed participants. But a second dichotomy also exists – one that separates large institutional customers from small retail customers. One might expect that the informed investor set would largely comprise the institutional customers. After all, the institutions are professional; they can afford to continuously monitor information and respond to news, and their very size (all else constant) reduces their per share cost of doing so. With divergent expectations, however, there is no presumption that institutional customers can, because of their size, consistently evaluate shares more accurately. On the contrary, institutions commonly disagree with each other and, as a consequence, commonly trade with each other. In the divergent expectations environment, institutional investors do not necessarily have an advantage over retail customers as fundamental analysts. In fact, their size makes 14 Adaptive valuation behavior refers to individual agents becoming more bullish (bearish) when learning of the relatively bullish (bearish) attitudes of others.

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trading more difficult and they incur higher transaction costs. So what accounts for their popularity? The value added by the mutual funds, pension funds, and so forth comes largely from their ability to facilitate diversification. Further, they can bring a systematic, professional, and disciplined approach to portfolio management (Davis et al. 2007).

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Big Board from a floor-based “slow” market into a hybrid that includes a “fast market” electronic venue. As of this writing, the floor-based component of the NYSE’s hybrid has been markedly reduced in importance. Several specialist firms have ceased operations, other floor brokers have departed, and the trading room areas have collapsed from five to two.

21.5 From Theory to Application 21.5.2 Regulatory Initiatives Microstructure analysis is inherently involved with analyzing the detailed functioning of a marketplace. The literature has a strong theoretical component and, to a large extent, is structured to yield insights into the effect of market design (structure and regulation) on market performance. Hopefully, theory can provide a broad roadmap for real world market architects to follow. In this section we provide a broad overview of major technology and regulatory changes that have taken place in the United States and Europe.15

21.5.1 Technological Developments Two exogenous forces have driven market structure change: technology and regulation. Regarding technology, the first big step was taken in 1971 in the U.S. when the National Association of Securities Dealers (NASD) introduced an electronic automated quotation (AQ) display system called NASDAQ. The Toronto Stock Exchange was the first exchange to introduce an electronic order driven platform – its Computer Assisted Trading System (CATS) in 1977. Following in Toronto’s footsteps, London instituted SEAQ in 1986, Paris rolled out its Cotation Assistée en Continu (CAC) in 1986, and Deutsche Börse’s Xetra came to life in 1997. Also in 1997, the London Stock Exchange introduced its Stock Exchange Trading System (SETS) limit order platform. By the end of the twentieth century most of the exchanges in Europe had converted to electronic limit order book platforms. Change came more slowly in the U.S. Instinet introduced an electronic platform in 1969. Nearly 30 years later, Instinet became known as an Electronic Communications Network (ECN). In short order, a slew of other ECNs emerged, led most prominently by Archipelago and Island. In 2002, NASDAQ implemented its own electronic platform, which, at the time, was called “SuperMontage.” Most recently, in the spring of 2006, the newly privatized NYSE Group initiated its Hybrid Market, a facility that has transformed the 15

Further discussion of market structure development is provided by Harris (2003).

Major regulatory initiatives have played an important role in jump-starting these market structure changes. The 1975 Congressional Securities Acts Amendments was the first sizable regulatory foray into market structure development. The Amendments precluded the fixing of commission rates and mandated the development of a National Market System (NMS). In 1997, the U.S. Securities and Exchange Commission instituted its new Order Handling Rules (OHRs), which require that market makers holding customer limit orders display those orders in their quotes, and that dealers at least match any quotes that they themselves display on an ECN (either by bettering the quotes that they offer customers or by posting their superior quotes in NASDAQ’s SuperMontage). Following the OHRs, four other regulatory initiatives were introduced in the U.S. in relatively fast succession. In 1998, the SEC release Regulation of Exchanges and Alternative Trading Systems (RegATS), set forth rules that allow ATSs the alternative of registering as national exchanges or as broker-dealers. In 2000, the NYSE, under pressure from the SEC, rescinded its order consolidation rule (Rule 390). In 2001, the U.S. markets completed the transition from fractional to decimal pricing, which resulted in the minimum tick size decreasing from 1/16 of a dollar or 6.25 cents (it had earlier been 1/8 of a dollar or 12.5 cents) to one cent. In 2005, the SEC adopted Regulation NMS, the key provision of which is that better-priced limit orders cannot be traded through (the trade-through rule was fully implemented in 2007). On the eastern side of the Atlantic, the first major regulatory initiative was taken in 1993 when the Investment Services Directive opened the door for cross-border trading by introducing the single European passport. As discussed in Schwartz and Francioni (2004), “Passporting defines a system of mutual acceptance of other EU countries’ rules without truly harmonizing all of the details of the various rules.” Major regulatory change has come again to the European arena in the form of the Markets in Financial Instruments Directive (MiFID). Key provisions in MiFID include a best execution requirement (echoes of the 1975 U.S. Securities Acts Amendments), a quote disclosure requirement for upstairs broker/dealers (echoes of the U.S. Order Handling Rules),

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

and the disallowance of order focusing rules (echoes of the U.S. SEC pressuring the withdrawal of NYSE Rule 390). A major regulatory difference is that, no trade-through rule has been imposed on the European markets (unlike under the U.S. SEC’s Reg NMS).

21.6 Deutsche Börse: The Emergence of a Modern, Electronic Market We turn in this section to the designing of an actual marketplace. Our focus is on Deutsche Börse: it is the dominant stock exchange in Germany, the last of the major European Borses to go electronic, and its technology is state of the art. Important insights were gained from the microstructure literature during Xetra’s planning period and the system’s implementation has marked a huge step forward for Germany’s equity markets. But our roadmap, which is undoubtedly incomplete today, was even more limited in the 1994–1997 years when Xetra was being designed. And, there is always the danger that the cartographer whose map is being used has some misconceptions (e.g., believes in the existence of the Northwest Passage).

21.6.1 The German Equities Market in the Mid-1990s As recently as the mid-1990s, the German market had major structural defects that would undermine its competitiveness in the European arena. In recognition of this, Deutsche Börse, the newly founded exchange operator of the Frankfurter Wertpapierbörse (FWB), became the leading force for change.16 In the mid-1990s, Frankfurt’s trading floor was the major marketplace for German stocks, but the German market was badly fragmented. Kursmaklers, the equivalent of specialists, concentrated much of the liquidity in their order books. A primitive (by today’s standards) electronic trading system, IBIS (which was owned by FWB), operated in parallel with the floor trading. IBIS’s central component was an open limit order book that had hit and take functionality, but did not match orders automatically. The electronic system captured about 40% of the trading volume in the 30 large-cap DAX stocks, but no link existed between IBIS and the floor. Seven other floor-based regional exchanges were also operating in Germany with technical infrastructures that were similar to 16

FWB also owned the futures and options exchange Deutsche Termine Börse. After the 1997 merger with SOFFEX, DTB became Eurex.

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those in Frankfurt. In total, the regionals at that time were attracting roughly 10% of German exchange-based trading volume. Moreover, off-board trading has been (and still is) prevalent in Germany (Davis et al. 2006). Transparency for floor trading (pre-trade transparency in particular) was low. Quotes were not distributed publicly (they were available on the floor only). Price priority between different trading venues was not enforced and orders executed in one market commonly traded through orders waiting to be executed in another market. Market manipulation and other abuses of power and position were believed to be rife on the old Frankfurt floor. Given the appreciable market fragmentation, poor transparency, imperfect inter-market linkages, and dubious floor behavior, transaction costs were high. Changes, both structural and regulatory, were called for. The result was the development of Xetra, an electronic orderdriven trading system that comprises two principal modalities – a continuous order book platform and periodic singleprice call auctions.17

21.6.2 Designing a New Trading System Xetra’s development started in 1994, and the system was launched in 1997.18 Strong external forces also motivated this reengineering of Deutsche Börse’s market structure: regulatory reform, soaring trading volumes, pan-European harmonization of the exchange industry, vibrant cross-border competition for order flow, and the rising concerns of market participants about the future performance of Germany’s financial markets. Through Xetra’s design stage, microstructure theory, even as it existed at the time, was an indispensable guide. This new field in financial economics, with its origin in issues concerning the competitive and architectural structure of an equity market, should have been able to give guidance to the development of an actual marketplace such as Xetra. To an extent, it has fulfilled its promise. The literature gave Deutsche Börse a broad roadmap, and it has highlighted underlying relationships and other important considerations that a market architect should be aware of. Building the Xetra model involved specifying principles that the new market should implement, and the system’s functionality also had to be defined. Most importantly, the new market system was to provide equal and decentralized access to all of its participants. Further, the system’s functionality and the market information delivered to users (both pre- and post-trade) were to be the same for all traders.

17 18

For further discussion and descriptions, see Francioni et al. (2008). Appendix B provides details of Xetra’s design.

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A trader’s location should not matter. With this in mind, Deutshe Börse’s fundamental architectural decision was to structure a hybrid market that included two major modalities – a continuous electronic order driven platform, and periodic call auctions that were used primarily for market openings and closings.19 An absolutely critical attribute of an order driven trading system is its ability, vis-à-vis its competitors, to win the battle for liquidity. Regarding this matter, the earlier microstructure literature has given some guidance, but liquidity is a complex attribute to deal with. As it is not easy to define and measure, liquidity has been very difficult to model and assess. However, as noted above, the measurement and analysis of liquidity are currently attracting considerably more attention in the microstructure literature. Price discovery and transparency are two other issues for which the microstructure literature has provided valuable guidance. The architects at Deutsche Börse recognized that price discovery is a primary function of a market center, and their major reason for introducing the call auctions was to sharpen its accuracy, particularly at market openings and closings. Understanding that transparency is important while recognizing that it should not be excessive, the decision was made to disclose only the indicative clearing price (not the full book of orders) in the pre-call, bookbuilding period. Microstructure literature has given insights into the operations of the public limit order book for continuous trading. At the time, recognition was also emerging of periodic call auctions, a modality that was clearly differentiated from, but could effectively be used with, the continuous market. With regard to continuous trading, microstructure analyses of the use of limit and market orders and of the interaction between these two order types proved to be most valuable. However, a deeper understanding of the economics of an order driven market now exists than was the case in the 1994– 1997 period when Xetra was being designed. Another important contribution of microstructure theory has been the classification of traders according to their needs for immediacy and their propensities to be either givers or takers of liquidity. The differentiation between informed and uniformed traders also proved to be valuable, particularly with respect to the market maker role that has been incorporated into Xetra. Specifically, market makers, referred to as “designated sponsors,” were included to bolster liquidity provision for smaller cap stocks. A balance had to be achieved between the obligations imposed on the designated sponsors and the privileges granted to them. To accomplish this, information had to be assessed concerning the role of dealers in general (e.g., NASDAQ-Type market makers) and special19

Interestingly, the microstructure literature on call auctions was relatively sparse at that time. For an early discussion, see Handa and Schwartz (1996b).

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ists in particular (e.g., NYSE-Type specialists). That balance defined the designated sponsors’ role in Xetra, and secured their willingness to accept it. Market microstructure insights also yielded the understanding needed to transform the specialist role into the newly designed designated sponsor role. Interestingly, the specialist role in the NYSE has now been changed to better harmonize with the Exchange’s electronic platform, and these intermediaries are now referred to as “designated market makers.” But designing an automated trading system is indeed a complex task, and the gap between theory and implementation is both large and intricate. Trading decisions can be made in a large variety of ways that run the gamut from humans interacting directly with humans without computers, to humans trading via electronic order handling and execution systems, to computers making trading decisions that are sent electronically to a computerized market (e.g., computerdriven algorithmic trading). Since the mid-1990s, market structure development has involved mainly the design of an electronic trading facility. Deutsche Börse took account of the fact that automation impacts both the way in which trading decisions are made, and the process by which prices are determined and trades executed in a market center. An electronic market requires the specification of an array of critical features (e.g., the trading modalities employed, rules of price and quantity determination, and basic features such as order types and trading parameters). With an electronic market, the software that implements a desired market structure must be specified on a level of detail that far exceeds what is required for human intermediated trading. For instance, a human agent (specialist) has historically handled price determination at NYSE openings. This function is performed with reference to various rules, but the specialist is also free to exercise reasonable judgment. Further, human-to-human interactions can evolve naturally as problems, opportunities, and new competitive pressures arise. In contrast, with a fully electronic opening, every possible condition that can occur must be recognized and a rule for dealing with it specified, and electronic interaction can be changed only by rewriting the code that specifies with stepby-step precision just how orders are handled and turned into trades and transaction prices. How does one achieve the precise specifications that a computerized trading system must have? In 1994, the market architects at Deutsche Börse could study the operations of other electronic platforms (e.g., CATS in Toronto and CAC in Paris). Doing so was helpful but of limited value given that Deutsche Börse was looking to develop a distinctive system. When moving into new territory, market structure development is a venture. How does one know in advance whether or not it will work? How can one determine whether or not the new system will be viable from a business perspective?

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

Nevertheless, design decisions have to be made, technical requirements must be specified, and the system must be built. The decisions involved represent huge financial bets on whether or not a new market structure will attract sufficient liquidity. Prototyping a new market in the design phase helps the assessment process, but doing so was considerably more difficult in 1994 than it is today with the advent of superior information technology and testing capabilities. In 1994, the architects were forced to rely more on their own educated judgment and on any insights they might gain from microstructure research. Those who are involved in the design of an actual market realize that the devil is in the details. Consider, for instance, the specification of a call auction. A call has excellent theoretical properties, but how should an actual auction be designed? It is straightforward to say that the market clearing price in a call auction should be the value that maximizes the number of shares that trade. But what should the specific rule be for selecting the clearing price if two prices both result in the same maximum trade size? Additionally, how transparent should the book be in the pre-call, order entry period? Are further design features needed to counter the possibility of gaming? And so on. Other considerations that for the most part are outside the scope of the microstructure literature also came into play during the design of Xetra. Information technology issues such as scalability, open architecture, and system reliability are of critical importance. So too are procedures for post-trade clearing and settlement. One of the final steps in the structural design of the new German market was the introduction in 2003 of a central counterparty (with a CCP, counterparty risk management was centralized and trading became fully anonymous, both pre- and post-trade). Electronic trading is also a prerequisite for highly efficient straight-through processing (STP involves all stages of a trade’s life cycle). Information technology has further facilitated the timely capture of market data (all trades, quotes, market index values, etc.) and has expedited its delivery to users. With regard to these diverse applications, Deutsche Börse has achieved a closer integration between trading on Xetra and the broader market infrastructure.

21.7 Conclusion: The Roadmap and the Road A market architect must have a roadmap that, broadly speaking, says where one ought to head and roughly how to get there. To this end, the microstructure literature has added clarity, articulation, and intellectual support. Briefly stated, the objective is to reduce trading frictions (costs), sharpen price discovery, and facilitate quantity discovery. The means of achieving this broad objective involve the amassing of

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liquidity. This is done through the appropriate use of limit order books for both continuous and call auction trading and, where appropriate, the inclusion of broker/dealer intermediaries. Further insights are gained from microstructure’s indepth analyses of trading motives (new information, liquidity needs, and technical trading signals). The literature has also provided guidance with regard to issues such as transparency and the consolidation (fragmentation) of order flow. But theory, even if it does provide a good roadmap, can take one only so far. The closer one gets to the design of an actual system, the more apparent the complexities of trading and trading systems become. The road actually traveled is indeed bumpy and hazardous. System designers know that “the devil is in the details.” They have to grapple with issues ranging from scalability, reliability, and other IT requirements, to business considerations concerning the ultimate profitability of a trading venue. The market architects at Deutsche Börse recognized these issues and their new system, Xetra, has marked a huge step forward for the German equity market. Today, important problems persist with regard to market design in Germany (and in all other markets around the world). Two fundamental questions concerning market architecture that have yet to be adequately answered: (1) What is the best way to deal with large, institutional orders? (2) How is liquidity creation best handled for mid-cap and smallcap stocks? At the same time, important microstructure topics continue to emerge at the academic research desks. Are there limits beyond which microstructure theory cannot provide guidance? Are there limits to the level of efficiency that a real world market can ever achieve? Undoubtedly, both answers are “yes” but, without question, neither of these limits has as of yet been reached. Quite clearly, microstructure research and the design of an actual marketplace remain works in progress.

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21 Security Market Microstructure: The Analysis of a Non-Frictionless Market Jones, C., G. Kaul, and M. Lipson. 1994. “Information, trading and volatility.” Journal of Financial Economics 36, 127–154. Keim, D., and A. Madhavan. 1996. “The upstairs market for large-block transactions: analysis and measurement of price effects.” Review of Financial Studies 9, 1–36. Kyle, A. 1985. “Continuous auctions and insider trading.” Econometrica 53, 1315–1335. Madhavan, A. 2000. “Market microstructure.” Journal of Financial Markets 3, 205–258. Madhavan, A. and M. Cheng. 1997. “In search of liquidity: an analysis of upstairs and downstairs trades.” Review of Financial Studies 10, 175–204. Mendelson, M., J. Peake, and T. Williams. 1979. “Toward a modern exchange: the Peake-Mendelson-Williams proposal for an electronically assisted auction market,” in Impending changes for securities markets: what role for the exchange?, E. Bloch and R. Schwartz (Eds.). JAI Press, Greenwich, CT. Menkveld, A., S. Koopman, and A. Lucas. 2007. “Modelling aroundthe-clock price discovery for cross-listed stocks using state space methods.” Journal of Business and Economic Statistics 25(2), 213–225. Mildenstein, E., and H. Schleef. 1983. “The optimal pricing policy of a monopolistic marketmaker in the equity market.” Journal of Finance 38, 218–231. Milgrom, P. and N. Stokey. 1982. “Information, trade and common knowledge.” Journal of Economic Theory 26(1), 17–27. Miller, E. 1977. “Risk, uncertainty and divergence of opinion.” Journal of Finance 32, 1151–1168. O’Hara, M. 1997. Market microstructure theory, Basil Blackwell, Cambridge, MA. Ozenbas, D., R. Schwartz, and R. Wood. 2002. “Volatility in U.S. and European equity markets: an assessment of market quality.” International Finance 5(3), 437–461. Pagano, M. and A. Röell. 1996. “Transparency and liquidity: a comparison of auction and dealer markets with informed trading.” Journal of Finance 51, 579–611. Parlour, C. 1998. “Price dynamics in limit order markets.” Review of Financial Studies 11, 789–816. Parlour, C. and D. Seppi. 2008. “Limit order markets: a survey,” Handbook of financial intermediation & banking, A. W. A. Boot and A. V. Thakor (Eds.), Elesevier, Amsterdam. Paroush, J., R. Schwartz, and A. Wolf. 2008. The dynamic process of price discovery in an equity market, Working paper, Baruch College, CUNY. Pástor, L. and R. Stambaugh. 2003. “Liquidity risk and expected stock returns.” Journal of Political Economy 113, 642–685. Porter, D. and D. Weaver. 1998. “Post-trade transparency on Nasdaq’s national market system.” Journal of Financial Economics 50(2), 231–252. Sandas, P. 2001. “Adverse selection and competitive market making: empirical evidence from a limit order market.” Review of Financial Studies 14, 705–734. Schwartz, R. 1991. Reshaping the equity markets: a guide for the 1990s, Harper Business, New York. Schwartz, R. and R. Francioni. 2004. Equity markets in action, Wiley, New York. Seppi, D. 1990. “Equilibrium block trading and asymmetric information.” Journal of Finance 45, 73–94. Smidt, S. 1971. “Which road to an efficient stock market: free competition or regulated monopoly?” Financial Analyst Journal, 27, 18–20, 64–69. Stigler, G. 1964. “Public regulation of the securities markets.” Journal of Business 37, 117–142. Stoll, H. 1978. “The supply of dealer services in securities markets.” Journal of Finance 33, 1133–1151.

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Appendix 21A Risk Aversion and Risk Premium Measures Our analysis of the perfectly liquid CAPM environment makes reference to two measures of risk aversion and to several dimensions of a risk premium. We provide details concerning both of these in this appendix.

21A.1 Risk Aversion We use two risk aversion measures: (1) RA D U00 .W/=U0 .W/ is a measure of absolute risk aversion, and (2) RR D WRA is a measure of relative risk aversion. Because U00 < 0 for a risk averse decision maker, RA ; RR > 0 for risk aversion. Larger values of RA and RR indicate higher degrees of risk aversion. RA is a measure of absolute risk aversion because it reflects the decision maker’s reaction to uncertainty in relation to the absolute (dollar) gains/losses in an uncertain situation. RR is a measure of relative risk aversion because it reflects the decision maker’s reaction to uncertainty in relation to the percentage gains/losses in an uncertain situation.20

21A.2 Risk Premiums A risk premium is the minimum dollar compensation a decision maker requires to hold a risky asset in place of an alternative that involves no risk. Specifically, a decision maker would be indifferent between a riskless investment with a certain return of D dollars, and a risky investment with an expected dollar return of E(Z) equal to D plus the investor’s risk premium. In general, the investor’s risk premium depends upon his or her utility function and initial wealth, and upon the distribution of Z. Pi . / in Equation (21.3) is a risk premium: equals onehalf of RA (the measure of the investor’s absolute risk aversion) times Var.P2 /, which measures the absolute (dollar) risk attributable to holding one share of the market portfolio. The uncertainty associated with holding N shares of the risky 20

For further discussion, see J. Pratt, “Risk Aversion in the Small and the Large,” Econometrica, January 1964.

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asset is Var.NP2 / D N2 Var.P2 /; thus the total risk premium for holding N shares is T D N12

(21A.1)

Dividing Equation (21A.1) by N1 .D N0 C Q/ gives the risk premium per share (the average risk premium): A D N1

(21A.2)

Differentiating Equation (21A.1) with respect to N1 gives the risk premium for a marginal share (the marginal risk premium): (21A.3) m D 2 N1 Dividing Equation (21A.3) by P1 expresses the marginal risk premium as a percentage of current price: M % D

M 2 N1 D P1 P1

(21A.4)

The return on the combined portfolio of N1 shares of the market portfolio and C1 dollars of the risk-free asset is rP D

P2 1 P1



P1 N1 W





P1 N1 C 1 rf W

(21A.5)

and the variance of the return on the combined portfolio is  Var

P2 P1



P1 N1 W



D

N1 W

2 Var.P2 /

(21A.6)

Thus the investor’s risk premium associated with the uncertain return realized from the combined portfolio is rp D

N1 W

2

(21A.7)

Appendix 21B Designing Xetra This appendix provides further detail on the development and design of Deutsche Börse’s electronic trading platform, Xetra. The first steps in designing Xetra involved specifying principles that the new market should implement, and defining the system’s functionality. This was done by Deutsche Börse working together with key market participants. Most importantly, the new market system was to provide equal and decentralized access to all its participants. Further, the system’s functionality and the market information delivered to users (whether pre- or post-trade) were to be the same for all traders. A trader’s location should not matter. Equity trading in the German market has been and continues to be order driven. This was true both for IBIS

and for floor trading that was managed by a Kursmakler specialist acting in the capacity of auctioneer, broker, and dealer. It was clear from the beginning that Xetra should run an open limit order book (open in the sense that aggregated order volume is displayed at all price points in the order book). Additionally, order matching was automated and trader anonymity ensured. Core features of an electronic trading system are determined by the market structure that it implements. The structure defines how orders are handled and translated into trades and transaction prices. Xetra’s market model comprises diverse sub-models, each with a single trading modality, or a combination of multiple modalities (i.e., it is a hybrid). Most importantly, Xetra implements both continuous trading and periodic call auction trading. This differentiation is required to cope with liquidity differences among stocks, and different liquidity needs among users depending on the size of their orders and motives for trading. The market for all stocks opens and closes with a call auction, while less liquid stocks trade in multiple call auctions per day. Once the building-blocks were defined (i.e., continuous trading and call auctions), and their combinations specified, the next design step was to detail the specific features of each of the modalities. Those features are either static (i.e., represent basic structures such as the order book), or dynamic (i.e., define processes and behavior such as order matching). The next two sections of this appendix consider the systems design in more detail for continuous trading and periodic call auction trading, respectively.

21B.1 Continuous Trading By the mid-1990s, order books for continuous trading with price and time priorities had been implemented around the globe. In designing Xetra, Deutsche Börse’s market architects could refer to a wide range of existing examples, and to a broad microstructure literature. Once the eligible order types were identified, the center piece of the development was the definition of the detailed rules of price-time matching. The complexity of this definition was broken down into a finite set of individual cases that involved various order book situations combined with various incoming orders, for which the trading outcome was to be defined by a rule. All rules collectively described the dynamics of order matching. A major challenge in designing continuous trading involves the measures that should be taken to provide an orderly market in periods of sharply elevated price volatility. To deal with this, the concept of a “price corridor” was formulated. Diverse corridors around historical prices were defined that set the benchmark for an “orderly” price for the next

21 Security Market Microstructure: The Analysis of a Non-Frictionless Market

trade. If a price occurred that lay outside its corridor, trading was to be halted (briefly) with the entire order book transported into a call auction. The purpose of the call was to allow the market to consolidate in both space and time. Trading in the continuous market was resumed upon completion of the call. Lastly, all trading parameters for the continuous platform had to be determined. This included specifying tick-sizes, breadth of the price corridors, and durations and timings. Together, this provided a comprehensive overview of the “steering wheels” for the newly designed market.

21B.2 Call Auction Trading The purpose of Xetra’s call auctions is threefold: (1) to open and close continuous trading, (2) to trade less liquid stocks in multiple calls per day with no continuous trading offered, and (3) to stabilize the market in times of large price moves. Despite those multiple purposes, a single design was defined for the auctions. Additionally, certain key consistencies between continuous trading and the call had to be achieved. For example, both limit and market orders that could be submitted to continuous trading were allowed entry into the call order book. This seemingly simple requirement was difficult to implement because it expanded the universe of possible order book configurations (and therefore necessitated more complex matching rules). Additional procedures for setting the clearing price were also required to guard against erroneous pricing that could be caused by market orders overpowering an insufficient number of limit orders. As with continuous trading, price and time priority execution rules were stipulated. Most crucial was the degree of transparency that the calls would offer. Sufficient information about the order book had to be delivered for market participants to have relevant price and quantity information concerning actual market situations, but detailed information was suppressed to inhibit excessive information leakage and gaming. The pre-call information now available in Xetra is the highest bid and the lowest offer posted in the call when these orders do not cross, or the indicative call auction price that is calculated when the order book is crossed. In other words, the full order book content is not visible – pre-call, the Xetra screen displays only the potential outcome of the call at each point in time. When Xetra was under development, call auction trading at pre-specified times was managed on the floor by specialists who were responsible for price determination, timing,

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and the provision of dealer liquidity. The challenge was to reengineer the call so that it could be run by a computer, not by a human intermediary. The issue that Deutche Börse was facing was also grappled with by market microstructure academicians and other market architects. Substantial external guidance was received in the planning process. In particular, important inputs were obtained concerning the optimal degree of transparency for the call’s anti-gaming measures. The availability at the time of a variety of different call auction designs (both used and proposed) enabled Xetra’s calls to be designed relatively quickly.

21B.3 Electronic Trading for Less Liquid Stocks Kursmaklers (specialists) on the Frankfurt floor (both today and in the past) provide immediate liquidity at times when external liquidity is insufficient. The desire was strongly expressed, with two provisos, for a market maker to be incorporated into Xetra’s order driven model for less liquid stocks. The two provisos were (1) market participants must all have equal access to information, and (2) equal access to functionality must be maintained at a maximum level. Consequently, any changes that would favor the dealers were kept to a minimum. The dealers were referred to as “designated sponsors.” Like market makers in general, the designated sponsors were given both privileges and obligations. The primary obligation is that, on request of other market participants, the designated sponsor must provide quotes for a minimum volume and maximum spread in a stock during continuous trading. Additionally, multiple designated sponsors were included, so that they might compete with each other. Concurrently, the fulfillment of each sponsor’s obligation is measured, and the results published. The designated sponsors’ primary privilege is that they can see the identity of the quote requesters in an environment that otherwise ensures complete anonymity. Further, a sponsor balances the order book in all call auctions for the stocks that it is registered in. This gives the designated sponsors a last mover advantage (the freedom to trade against any imbalance that might exist at the market clearing price). With this privilege, a designated sponsor can influence the clearing price so as to execute orders that otherwise would not have transacted in that call. Lastly, the designated sponsors, depending on their measured performance, receive fee reductions.

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21B.4 Xetra’s Implementation and the Migration of Liquidity to Xetra Since 1997 Xetra went operational in fall 1997. At the beginning, the new system attracted roughly 60% of trading in the most liquid segment of the market, the 30 DAX stocks. Trading on Xetra for mid-cap stocks was not as successful – market share for this segment was about 20%, as the less liquid stocks largely continued at that time to trade on the floor. But the 1997 launch was just the start of a sequence of releases that have continued through the current time. One more recent innovation was the “continuous call auction.” With this facility, calls are not held at pre-specified

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times but are triggered by the occurrence of a “critical” liquidity situation. The continuous call comprises a dealerauctioneer who is responsible for providing a base level of liquidity in each call, as well as controlling its timing. Additionally, Xetra allows internalization of trading by member firms. Consequently, Xetra, which originally started as an exchange trading system, now also serves as the technical platform for OTC trading. Major innovations have benefited a broad range of cap sizes and, across the board, floor trading has continued to decline. Xetra has now been rolled out to 260 member firms in Europe, and its market share currently stands at 95% of all on-exchange trading in Germany today.

Chapter 22

Options Strategies and Their Applications Cheng Few Lee, John Lee, and Wei-Kang Shih

Abstract In this chapter we introduce different types of options and their characteristics. Then, we develop put-call parity theorems for European, American, and futures options. Finally, we discuss option strategies and their investment applications. Keywords Call option r Put option r American option r European option r Exercise price r Time to maturity r Futures option r Long straddle r Short straddle r Long vertical spread r Short vertical spread r Calendar spread r Protective put r Covered call r Collar

22.1 Introduction The use of stock options for risk reduction and return enhancement has expanded at an astounding pace over the last 20 years. Among the causes of this growth, two are most significant. First, the establishment of the Chicago Board Option Exchange (CBOE) in 1973 brought about the liquidity necessary for successful option trading, through public listing and standardization of option contracts. The second stimulus emanated from academia. In the same year that the CBOE was established, Professors Fischer Black and Myron Scholes published a paper in which they derived a revolutionary option-pricing model. The power of their model to predict an option’s fair price has since made it the industry standard. The development of option-valuation theory shed new light on the valuation process. Previous pricing models such as CAPM were based on very stringent assumptions, such as there being an identifiable and measurable market portfolio, as well as various imputed investor attributes, such

C.F. Lee and W.-K. Shih () Rutgers University, Newark, NJ, USA e-mail: [email protected] J. Lee Center for PBBEF Research, New York, NY, USA

as quadratic utility functions. Furthermore, previous theory priced only market risk since investors were assumed to hold well-diversified portfolios. The strength of the Black-Scholes and subsequent option-pricing models is that they rely on far fewer assumptions. In addition, the option-valuation models price total risk and do not require any assumptions concerning the direction of the underlying securities price. The growing popularity of the option concept is evidenced by its application to the valuation of a wide array of other financial instruments (such as common stock and bonds) as well as more abstract assets including leases and real estate agreements. This chapter aims to establish a basic knowledge of options and the markets in which they are traded. It begins with the most common types of options, calls, and puts, explaining their general characteristics and discussing the institutions where they are traded. In addition, the concepts relevant to the new types of options on indexes and futures are introduced. The next focus is the basic pricing relationship between puts and calls, known as put-call parity. The final study concerns how options can be used as investment tools. The chapter on option valuation that follows utilizes all these essential concepts to afford a deeper conceptual understanding of valuation theory.

22.2 The Option Market and Related Definitions This section discusses option-market and related definitions of options, which are needed to understand option valuations and option strategies.

22.2.1 What Is an Option? An option is a contract conveying the right to buy or sell a designated security at a stipulated price. The contract

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_22, 

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normally expires at a predetermined time. The most important element of an option contract is that there is no obligation placed upon the purchaser: it is an “option.” This attribute of an option contract distinguishes it from other financial contracts. For instance, while the holder of an option has the opportunity to let his or her claim expires unused if so desired, futures and forward contracts obligate their parties to fulfill certain conditions.

22.2.2 Types of Options and Their Characteristics A call option gives its owner the right to buy the underlying asset while a put option conveys to its holder the right to sell the underlying asset. An option is specified by five essential parts: 1. 2. 3. 4. 5.

The type (call or put) The underlying asset The exercise price The expiration date The option price

While the most common type of underlying asset for an option is an individual stock, other underlying assets for options exist as well. These include futures contracts, foreign currencies, stock indexes, and U.S. debt instruments. In the case of common stock options (on which this discussion is exclusively centered), the specified quantity to which the option buyer is entitled to buy or sell is one hundred shares of the stock per option. The exercise price (also called the strike price) is the price stated in the option contract at which the call (put) owner can buy (sell) the underlying asset up to the expiration date, the final calendar date on which the option can be traded. Options on common stocks have expiration dates 3 months apart in one of three fixed cycles. 1. January/April/July/October 2. February/May/August/November 3. March/June/September/December The normal expiration date is the third Saturday of the month. (The third Friday is the last trading date for the option.) As an example, an option referred to as an “ABC June 25 call” is an option to buy one hundred shares of the underlying ABC stock at $25 per share, up to its expiration date in June. Option prices are quoted on a per-share basis. Thus, a stock option that is quoted at $5 would cost $500 ($5 100 shares), plus commission and a nominal SEC fee. A common distinction among options pertains to when they can be exercised. Exercising an option is the process of

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carrying out the right to buy or sell the underlying asset at the stated price. American options allow the exercise of this right at any time from when the option is purchased up to the expiration date. On the other hand, European options allow their holder the right of exercise only on the expiration date itself. The distinction between an American and European option has nothing to do with the location at which they are traded. Both types are currently bought and sold in the United States. There are distinctions in their pricing and in the possibility of exercising them prior to expiration. Finally, when discussing options, the two parties to the contract are characterized by whether they have bought or sold the contract. The party buying the option contract (call or put) is the option buyer (or holder), while the party selling the option is the option seller (or writer). If the writer of an option does not own the underlying asset, he or she is said to write a naked option. Table 22.1 shows a listing of publicly traded options for Johnson & Johnson at September 21, 2007.

22.2.3 Relationships Between the Option Price and the Underlying Asset Price A call (put) option is said to be in the money if the underlying asset is selling above (below) the exercise price of the option. An at-the-money call (put) is one whose exercise price is equal to the current price of the underlying asset. A call (put) option is out of the money if the underlying asset is selling below (above) the exercise price of the option. Suppose ABC stock is selling at $30 per share. An ABC June 25 call option is in the money .$30  25 > 0/, while an ABC June 35 call option is out of the money .$30  35 < 0/. Of course, the expiration dates could be any month without changing the option’s standing as in, at, or out of the money. The relationship between the price of an option and the price of the underlying asset indicates both the amount of intrinsic value and time value inherent in the option’s price, as shown in Equation (22.1): Intrinsic Value D Underlying Asset Price Option Exercise Price

(22.1)

For a call (put) option that is in the value (underlying asset price > exercise price), its intrinsic value is positive. And for at-the-money and out-of-the-money options the intrinsic value is zero. An option’s time value is the amount by which the option’s premium (or market price) exceeds its intrinsic value. For a call or put option: Time Value D Option Premium  Intrinsic Value

(22.2)

22 Options Strategies and Their Applications

Table 22.1 Options quotes for Johnson & Johnson at 09/21/2007

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Stock price at 9/21/2007 D $65.12 Call options expiring Fri Jan 18, 2008 Strike

Symbol

Last

Bid

Ask

Vol

Open int

40 45 50 55 60 65 70 75 80 85 90

JNJAH.X JNJAI.X JNJAJ.X JNJAK.X JNJAL.X JNJAM.X JNJAN.X JNJAO.X JNJAP.X JNJAQ.X JNJAR.X

24 20 15:5 11:1 6:4 2:65 0:55 0:1 0:05 0:05 0:05

25.5 20.5 15.7 10.9 6.4 2.65 0.55 0.05 N/A N/A N/A

25:6 20:7 16 11:1 6:5 2:7 0:6 0:1 0:05 0:05 0:05

10 3 11 33 275 1;544 845 2 10 0 0

5;427 2;788 8;700 10;327 32;782 70;426 48;582 13;629 4;497 3;275 3;626

Put options expiring Fri Jan18, 2008 Strike Symbol Last 40 JNJMH.X 0:05 45 JNJMI.X 0:1 50 JNJMJ.X 0:12 55 JNJMK.X 0:25 60 JNJML.X 0:7 65 JNJMM.X 1:8 70 JNJMN.X 5 75 JNJMO.X 13:3

Bid N/A 0.05 0.1 0.25 0.65 1.85 4.9 9.7

Ask 0:05 0:1 0:15 0:3 0:7 1:95 5 9:9

Vol

Open int 1;370 5;002 14;004 31;122 69;168 46;774 1; 582 20

where intrinsic value is the maximum of zero or stock price minus exercise price. Thus an option premium or market price is composed of two components, intrinsic value and time value. In-the-money options are usually the most expensive because of their large intrinsic-value component. An option with an at-the-money exercise price will have only time value inherent in its market price. Deep out-of-the-money options have zero intrinsic value and little time value and consequently are the least expensive. Deep in-the-money cases also have little time value, and time value is the greatest for at-the-money options. In addition, time value (as its name implies) is positively related to the amount of time the option has to expiration. The theoretical valuation of options focuses on determining the relevant variables that affect the time-value portion of an option premium and the derivation of their relationship in option pricing. In general, the call price should be equal to or exceed the intrinsic value: C Max.S  E; 0/ where: C D the value of the call option; S D the current stock price; and E D the exercise price. Figure 22.1 illustrates the relationship between an option’s time value and its exercise price. When the exercise

0 3 1 99 227 30 20 0

Fig. 22.1 The relationship between an option’s exercise price and its time value

price is zero, the time value of an option is zero. Although this relationship is described quite well in general by Fig. 22.1, the exact relationship is somewhat ambiguous. Moreover, the identification of options with a mispriced time-value portion in their total premium motivates interest in a theoretical pricing model. One more aspect of time value that is very important is the change in the amount of time value an option has as its duration shortens. As previously mentioned, options with a

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Exercise price X January 15 $5.50 20 25

April Max.20  15; 0/ C1:25 D $6:25 Max.20  20; 0/ Max.20  20; 0/ C1:00 D $1:00 C2:00 D $2:00 Max.20  25; 0/ Max.20  25; 0/ C5:0 D $0:50 C1:25 D $1:25

July Max.20  15; 0/ C3:50 D $8:50 Max.20  20; 0/ C5:00 D $5:00 Max.20  25; 0/ C3:50 D $3:50

Other values are shown in the table above.

22.2.3.1 Additional Definitions and Distinguishing Features Fig. 22.2 The relationship between time value and time to maturity for a near-to-the-money option (assuming a constant price for the underlying asset)

longer time to maturity and those near to the money have the largest time-value components. Assuming that a particular option remains near to the money as its time to maturity diminishes, the rate of decrease in its time value, or what is termed the effect of time decay, is of interest. How time decay affects an option’s premium is an important question for the valuation of options and the application of option strategies. To best see an answer to this question refer to Fig. 22.2. In general, the value of call options with the same exercise price increases as time to expiration increases: C .S1 ; E1 ; T1 /  C .S1 ; E1 ; T2 / where T1  T2 and T1 and T2 are the time to expiration. Note that for the simple case in Fig. 22.2 the effect of time decay is smooth up until the last month before expiration, when the time value of an option begins to decay very rapidly. This effect is made clearer if we refer to Chap. 23, in which the components of the option-pricing model was examined. Example 22.4 shows the effect of time decay. Example 22.1. It is January 1, the price of the underlying ABC stock is $20 per share, and the time premiums are shown in the following table. What is the value for the various call options? Exercise price X 15 20 25

January $0:50 1:00 0:50

April $1:25 2:00 1:25

July $3:50 5:00 3:50

Solution 22.1. Call premium D Intrinsic value C Time premium C15;Jan D Max.20  15; 0/ C 0:50 D $5:50 per share or $550 for 1 contract

Options may be specified in terms of their classes and series. A class of options refers to all call and put options on the same underlying asset. For example, all AT&T call and put options at various exercise prices and expiration months form one class. A series is a subset of a class and consists of all contracts of the same class (such as AT&T) having the same expiration date and exercise price. When an investor either buys or sells an option (that is, is long or short) as the initial transaction, the option exchange adds this opening transaction to what is termed the open interest for an option series. Essentially, open interest represents the number of contracts outstanding at a particular point in time. If the investor reverses the initial position with a closing transaction (that is, sells the option if he or she originally bought it or vice versa) then the open interest for the particular option series is reduced by one). While open interest is more of a static variable, indicating the number of outstanding contracts at one point in time, volume represents a dynamic characteristic. More specifically, volume indicates the number of times a particular option is bought and sold during a particular trading day. Volume and open interest are measures of an option’s liquidity, the ease with which the option can be bought and sold in large quantities. The larger the volume and/or open interest, the more liquid the option. Again, an option holder who invokes the right to buy or sell is exercising the option. Whenever a holder exercises an option, a writer is assigned the obligation to fulfill the terms of the option contract by the exchange on which the option is traded. If a call holder exercises the right to buy, a call writer is assigned the obligation to sell. Similarly, when a put holder exercises the right to sell, a put writer is assigned the obligation to buy. The seller or writer of a call option must deliver 100 shares of the underlying stock at the specified exercise price when the option is exercised. The writer of a put option must purchase 100 shares of the underlying stock when the put option is exercised. The writer of either option receives the premium or price of the option for this legal obligation. The

22 Options Strategies and Their Applications

maximum loss an option buyer can experience is limited to the price of the option. However, the maximum loss from writing a naked call is unlimited; the maximum loss possible from writing a naked put is the exercise price less the original price of that put. To guarantee that the option writer can meet these obligations, the exchange clearinghouse requires margin deposits. The payment of cash dividends affects both the price of the underlying stock and the value of an option on the stock. Normally, no adjustment is made in the terms of the option when a cash dividend is paid. However, strike price or number of shares may be adjusted if the underlying stock realizes a stock dividend or stock split. For example, an option on XYZ Corporation with an exercise price of $100 would be adjusted if XYZ Corporation stock split two for one. The adjustment in this case would be a change in the exercise price from $100 to $50, and the number of contracts would be doubled. In the case of a noninteger split (such as three for two), the adjustment is made to the exercise price and the number of shares covered by the option contracts. For example, if XYZ Corporation had an exercise price of $100 per share and had a three-for-two split, the option would have the exercise price adjusted to $662=3 , and the number of shares would be increased to 150. Notice that the old exercise value of the option, $10,000 ($100 100 shares) is maintained by the adjustment ($662=3 150 shares).

22.2.4 Types of Underlying Assets Although most people would identify common stocks as the underlying asset for an option, a variety of other assets and financial instruments can assume the same function. In fact, options on agricultural commodities were introduced by traders in the U.S. as early as the mid-l800s. After a number of scandals, agricultural commodity options were banned by the government. They were later reintroduced under tighter regulations and in a more standardized tradable form. Today, futures options on such agricultural commodities as corn, soybeans, wheat, cotton, sugar, live cattle, and live hogs are actively traded on a number of exchanges. The biggest success for options has been realized on financial futures. Options on the S&P 500 index futures contracts, NYSE index futures, foreign-currency futures, 30-year U.S. Treasury bond futures, and gold futures have all realized extraordinary growth since their initial offerings back in 1982. Options on futures are very similar to options on the actual asset, except that the futures options give their holders the right (not the obligation) to buy or sell predetermined quantities of specified futures contracts at a fixed price within a predetermined period. Options on the actual asset have arisen in another form as well. While a number of options have existed for various

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stock-index futures contracts, options now also exist on the stock index itself. Because of the complexity of having to provide all the stocks in an index at the spot price should a call holder exercise his or her buy right, options on stock indexes are always settled in cash. That is, should a call holder exercise his or her right to buy because of a large increase in the underlying index, that holder would be accommodated by a cash amount equal to the profit on his contract, or the current value of the option’s premium. Although the options on the S&P 100 stock index at the Chicago Board Options Exchange (CBOE) are the most popular among traders, numerous index options are now traded as well. These include options on the S&P 500 index, the S&P OTC 250 index, the NYSE composite and AMEX indexes (computer technology, oil and gas, and airline), the Philadelphia Exchange indexes (gold/silver), the Value Line index, and the NASDAQ 100 index.

22.2.5 Institutional Characteristics Probably two of the most important underlying factors leading to the success of options have been the standardization of contracts through the establishment of option exchanges and the trading anonymity brought about by the Option Clearing Corporations and clearinghouses of the major futures exchanges. An important element for option trading is the interchangeability of contracts. Exchange contracts are not matched between individuals. Instead, when an investor or trader enters into an option contract, the Option Clearing Corporation (or clearinghouse for the particular futures exchange) takes the opposite side of the transaction. So rather than having to contact a particular option writer to terminate an option position, a buyer can simply sell it back to the exchange at the current market clearing price. This type of anonymity among option-market participants is what permits an active secondary market to operate. The sources of futures options traded on the various futures exchanges mentioned earlier are determined by the open-auction bidding, probably the purest form of laissezfaire price determination that can be seen today. With the open-auction-bidding price mechanism there are no market makers, only a large octagonal pit filled with traders bidding among themselves to buy and sell contracts. While some traders buy and sell only for themselves, many of the participants are brokers representing large investment firms. Different sides of the pit usually represent traders who are dealing in particular expiration months. As brokers and other pit participants make trades they mark down what they bought or sold, how much, at what price, and from whom. These cards are then collected by members of the exchange

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who record the trades and post the new prices. The prices are displayed on “scoreboards” surrounding the pit. While stock options and options on commodities and indexes are traded in a similar fashion, one major difference prevails – the presence of market makers. Market makers are individuals who typically trade one type of option for their own account and are responsible for ensuring that a market always exists for their particular contract. In addition, some option exchanges utilize board brokers as well. These individuals are charged with the maintenance of the book of limit orders (orders from outside investors that are to be executed at particular prices or when the market goes up or down by a prespecified amount). Essentially, market makers and board brokers on the options exchanges share the duties performed by the specialists on the major stock exchanges. Although stocks can be bought with as little as 50% margin, no margin is allowed for buying options – the cost of the contract must be fully paid. Because options offer a high degree of leverage on the underlying asset, additional leveraging through margins is considered by regulators to be excessive. However, if more than one option contract is entered into at the same time – for instance, selling and buying two different calls – then, of course, a lower cost is incurred, since the cost of one is partially (or wholly) offset by the sale of the other.

22.3 Put-Call Parity This section addresses a most important concept, called put-call parity (for option valuation). The discussion includes European options, American options, and future options.

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Theorem 22.1. Put-Call Parity for European Options with No Dividends. Ct;T D Pt;T C St  EBt;T where: Ct;T D value of a European call option at time t that matures at time T .T > t/; Pt;T D value of a European put option at time t, that matures at time T ; St D value of the underlying stock (asset) to both the call and put options at time t; E D exercise price for both the call and put options; Bt;T D price at time t of a default-free bond that pays $1 with certainty at time T (if it is assumed that this risk-free rate of interest is the same for all maturities and equal to r – in essence a flat-term structure – then Bt;T D e r.T t / , under continuous compounding), or Bt;T D 1= .1 C r/T t for discrete compounding. Equation (22.3) uses the following principle. If the options are neither dominant nor dominated securities, and if the borrowing and lending rates are equal, then the return patterns of a European call and a portfolio composed of a European put, a pure discount bond with a face value equal to the options exercise price E, and the underlying stock (or asset) are the same.1 In understanding why the put-call parity theorem holds, and to support the theorem, two additional properties of option pricing must be provided: Property 22.1. At maturity (time T ) the call option is worth the greater of ST  E dollars or zero dollars: CT D Max.0; ST  E/

22.3.1 European Options As an initial step to examining the pricing formulas for options, it is essential to discuss the relationships between the prices of put and call options on the same underlying asset. Such relationships among put and call prices are referred to as the put-call parity theorems. Stoll (1969) was the first to introduce the concept of put-call parity. Dealing strictly with European options he showed that the value of a call option would equal the value of a portfolio composed of a long put option, its underlying stock, and a short discounted exercise price. Before stating the basic put-call parity theorem as originally devised by Stoll, it must be assumed that the markets for options, bonds, and stocks (or any other underlying asset we choose) are frictionless.

(22.3)

(22.4)

As an example, suppose that the call option has an exercise price of $30. At maturity, if the stock’s (asset’s) price is $25, then the value of the call is the maximum of (0, 25–30) or (0, 5), which of course is zero. If an option sells for less than (ST  E) its intrinsic value, an arbitrage opportunity will exist. Investors would buy the option and short sell the stock, forcing the mispricing to correct itself. Consequently, this first property implies that a call option’s value is always greater than zero. An equivalent property and argument exist for the value of a put option as well.

1

Any security x is dominant over any security y if the rate of return on x is equal to or greater than that of y for all states of nature and is strictly greater for at least one state. For an expanded discussion of this subject, see Merton (1973) and Smith (1976).

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Property 22.2. At maturity, the value of a put option is the greater of (E  ST ) dollars or zero dollars: PT D Max.0; E  ST /

P0;1yr D $5 C $110

Example 22.2. A call option with 1 year to maturity and exercise price of $110 is selling for $5. Assuming discrete compounding, a risk-free rate of 10%, and a current stock price of $100, what is the value of a European put option with a strike price of $110 and 1-year maturity?

.1:1/1

 $100

P0;1yr D $5

(22.5)

Using the same line of reasoning and argument as for the call option, the second property also implies that the value of a put option is never less than zero. Table 22.2 provides proof of this first put-call parity theorem. Suppose at time t, two portfolios are formed: portfolio B is just a long call option on a stock with price St an exercise price of E, and a maturity date at T . Portfolio A consists of purchasing one hundred shares of the underlying stock (since stock options represent one hundred shares), purchasing (going long) one put option on the same stock with exercise price E and maturity date T , and borrowing at the risk-free rate an amount equal to the present value of the exercise price or EBt;T with face value of E. (This portion of the portfolio finances the put, call, and stock position.) At maturity date T , the call option (portfolio B) has value only if ST > E, which is in accordance with Property 1. For portfolio A, under all these conditions the stock price and maturing loan values are the same, whereas the put option has value only if E > ST . Under all three possible outcomes for the stock price ST , it can be seen that the values of portfolios A and B are equal. Proof has been established for the first put-call parity theorem. Example 22.2 provides further illustration.



1

22.3.2 American Options Of course, this first put-call parity theorem holds only under the most basic conditions (that is, no early exercise and no dividends). Jarrow and Rudd (1983) give an extensive coverage of the effects of more complicated conditions on put-call parity. These authors demonstrate that the effect of known dividends is simply to reduce, by the discounted value (to time t) of the dividends, the amount of the underlying stock purchased. In considering stochastic dividends, the exactness of this pricing relationship breaks down and depends on the degree of certainty that can be maintained about the range of future dividends. Put-call parity for American options is also derived under various dividend conditions. Jarrow and Rudd demonstrate that as a result of the American option’s early exercise feature, strict pricing relationships give way to boundary conditions dependent on the size and certainty of future dividends, as well as the level of interest rates and the size of the exercise price. To summarize, they state that for sufficiently high interest rates and/or exercise prices it may be optimal to exercise the put prior to maturity (with or without dividends). So the basic put-call parity for an American option with no dividends and constant interest rates is described by the following theorem. Theorem 22.2. Put-Call Parity for an American Option with No Dividends Pt;T C S  EBt;T > Ct;T > Pt;T C St  E

(22.6)

Solution 22.2. Increasing the generality of conditions results in increasing boundaries for the equilibrium relationship between put and

Pt;T D Ct;T C EBt;T  St Table 22.2 Put-call parity for a European option with no dividends

Time T (maturity) ST > E

ST D E

ST < E

1. Buy 100 shares of the stock .St /

ST

ST

ST

2. Buy a put (Pt , maturity at T with exercise price E)

0

0

E  ST

3. Borrow EBt;T dollars

E .ST E/

E 0

E 0

.ST  E/

0

0

Time t strategy Portfolio A

Portfolio A value at time T Portfolio B 1. Buy a call (Ct maturing at T with exercise price E)

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call options. The beauty of these arguments stems from the fact that they require only that investors prefer more wealth to less. If more stringent assumptions are made, then the bounds can be made tighter. For an extensive derivation and explanation of these theorems see Jarrow and Rudd (1983). Example 22.3 provides further illustration. Example 22.3. A put option with 1 year to maturity and an exercise price of $90 is selling for $15; the stock price is $100. Assuming discrete compounding and a risk-free rate of 10%, what are the boundaries for the price of an American call option? Solution 22.3. Pt;T C S  EBt;T > Ct;T > Pt;T C St  E

1 > Ct;T > $15 C $100  $90 $15 C $100  $90 .1:1/1 $33:18 > Ct;1yr > $25

Futures Options As a final demonstration of put-call parity the analysis is extended to the case where the underlying asset is a futures contract. The topic of futures contracts and their valuation will be more fully examined. Nevertheless, this chapter takes time to apply put-call parity when the options are on a futures contract because of the growing popularity and importance of such futures options. A futures contract as described in as a contract in which the party entering into the contract is obligated to buy or sell the underlying asset at the maturity date for some stipulated price. While the difference between European and American options still remains, the complexity of dividends can be ignored since futures contracts do not

pay dividends. Put-call parity for a European futures option (when interest rates are constant) is as follows: Theorem 22.3. Put-Call Parity for a European Futures Option. Ct;T D Pt;T C Bt;T .Ft;T  E/

(22.7)

where Ft;T is the price at time t for a futures contract maturing at time T (which is the underlying asset to both the call and put options). Option pricing Properties 1 and 2 for call and put options apply in an equivalent sense to futures options as well. However, to understand this relationship as stated in Equation (22.7) it must be assumed that the cost of a futures contract is zero. While a certain margin requirement is required, the majority of this assurance deposit can be in the form of interest-bearing securities. Hence as an approximation a zero cost for the futures contract is not unrealistic. Again, the easiest way to prove this relationship is to follow the same path of analysis used in proving Theorem 1. Table 22.3 indicates that the argument for this theorem’s proof is similar, with only a few notable exceptions. The value of the futures contract at time T (maturity) is equal to the difference between the price of the contract at time T and the price at which it was bought, or FTT Ft;T . This is an outcome of the fixed duration of a futures contract as opposed to the perpetual duration of common stock. Second, because no money is required to enter into the futures contract, the exercise price is reduced by the current futures price and the total is lent at the risk-free rate. (Actually this amount is either lent or borrowed depending on the relationship between Ft;T and E at time t. If Ft;T  E < 0, then this amount will actually be borrowed at the risk-free rate.) Why are there options on spot assets as well as options on futures contracts for the spot assets? After all, at expiration the basis of a futures contract goes to zero and futures prices

Table 22.3 Put-call parity for a European futures option

Time T (maturity) FTT > E

FTT D E

1. Buy a futures contract .Ft;T /

FTT  Ft;T

FTT  Ft;T

FTT  Ft;T

2. Buy a put (Pt;T on Ft;T with exercise price E and maturity T )

0

0

E  FTT

3. Lend Bt;T .FtT  E/ dollars

Ft;T E .FTT E/

Ft;T E 0

Ft;T E 0

FTT  E

0

0

Time t strategy

FTT < E

Portfolio A

Portfolio A’s value at time T Portfolio B 1. Buy a call (Ct;T on Ft;T with exercise price E and maturity T )

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equal spot prices; thus, options in the spot and options on the future are related to the same futures value, and their current values must be identical. Yet a look at the markets shows that options on spot assets and options on futures for the same assets sell at different prices. One explanation for this is that investors who purchase options on spot must pay a large sum of money when they exercise their options, whereas investors who exercise an option on a future need only pay enough to meet the initial margin for the futures contract. Therefore, if the exercise of the option is important to an investor, that investor would prefer options on futures rather than options on spot and would be willing to pay a premium for the option on the future, whereas the investor who has no desire to exercise the option (remember, the investor can always sell it to somebody else to realize a profit) is not willing to pay for this advantage and so finds the option on spot more attractive.

22.3.3 Market Application Put options were not listed on the CBOE until June 1977. Before that time, brokers satisfied their clients’ demands for put option risk-return characteristics by a direct application of put-call parity. By combining call options and the underlying security, brokers could construct a synthetic put. To illustrate, the put-call parity theorem is used when a futures contract is the underlying asset. Furthermore, to simulate the option broker’s circumstances on July 1, 1984, the equation is merely rearranged to yield the put’s “synthetic” value: Pt;T D Ct;T  Bt;T Ft;T C Bt;T E

(22.8)

So instead of a futures contract being purchased, it is sold. Assume the following values and use the S&P 500 index futures as the underlying asset. Ct;T D $ 3:35; Ft;T D 154:85 (September contract); E D 155:00; and Bt;T D 0:9770 (current price of a risk-free bond that pays $1 when the option and futures contract expire, average of bid and ask prices for T -bills from The Wall Street Journal). According to Equation (22.8), the put’s price should equal the theorem price: Pt;T D $3:497. The actual put price on this day (July 1, 1984) with the same exercise price and expiration month was Pt;T D $3:50. With repeated comparisons of the theorem using actual prices, it becomes clear that put-call parity is a powerful equilibrium mechanism in the market.

22.4 Risk-Return Characteristics of Options One of the most attractive features of options is the myriad ways in which they can be employed to achieve a particular combination of risk and return. Whether through a straight option position in combination with the underlying asset or some portfolio of securities, options offer an innovative and relatively low-cost mechanism for altering and enhancing the risk-return tradeoff. In order to better grasp these potential applications this section analyzes call and put options individually and in combination, relative to their potential profit and loss and the effects of time and market sentiment.

22.4.1 Long Call The purchase of a call option is the simplest and most familiar type of option position. The allure of calls is that they provide the investor a great deal of leverage. Potentially, large percentage profits can be realized from only a modest price rise in the underlying asset. In fact, the potential profit from buying a call is unlimited. Moreover, the option purchaser has the right but no obligation to exercise the contract. Therefore, should the price of the underlying asset decline over the life of the call, the purchaser need only let the contract expire worthless. Consequently, the risk of a long call position is limited. Figure 22.3 illustrates the profit profile of a long call position. The following summarizes the basic risk-return features for a long-call position. Profit potential: unlimited Loss potential: limited (to cost of option) Effect of time decay: negative (decrease option’s value) Market expectation: bullish As the profit profile indicates, the time value of a long call declines over time. Consequently, an option is a wasting asset. If the underlying asset’s price does not move above the exercise price of the option E by its expiration date T , the buyer of the call will lose the value of his initial investment (the option premium). Consequently, the longer an investor holds a call, the more time value the option loses, thereby, reducing the price of the option. This leads to another important point – taking on an option position. As with any other investment vehicle, the purchaser of a call expresses an opinion about the market for the underlying asset. Whereas an investor can essentially express one of three different sentiments (bullish, neutral, or bearish) about future market conditions, the long call is strictly a bullish position. That is, the call buyer only wins if the underlying asset rises in price. However, depending on the exercise price of the call, the

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Fig. 22.3 Profit profile for a long call

Fig. 22.4 Profit profile for a short call

buyer can express differing degrees of bullishness. For instance, since out-of-the-money calls are the cheapest, a large price increase in the underlying asset will make these calls the biggest percentage gainers in value. So an investor who is extremely bullish would probably go with an out-of-themoney call, since its intrinsic value is small and its value will increase along with a large increase in the market.

22.4.2 Short Call Selling a call (writing it) has risk-reward characteristics, which are the inverse of the long call. However, one major distinction arises when writing calls (or puts) rather than buying them. That is, the writer can either own the underlying asset upon which he or she is selling the option (a covered write), or simply sell the option without owning the asset (a naked write). The difference between the two is of consid-

erable consequence to the amount of risk and return taken on by the seller. Let us first examine the profit profile and related attributes of the naked short call, displayed in Fig. 22.4. When the writer of a call does not own the underlying asset, his or her potential loss is unlimited. Why? Because if the price of the underlying asset increases, the value of the call also increases for the buyer. The seller of a call is obliged to provide a designated quantity of the underlying asset at some prespecified price (the exercise price) at any time up to the maturity date of the option. So if the asset starts rising dramatically in price and the call buyer exercises his or her right, the naked-call writer must go into the market to buy the underlying asset at whatever the market price. The naked-call writer suffers the loss of buying the asset at a price S and selling it at a price E when S > E (less the original premium collected). When common stock is the underlying asset, there is no limit to how high its price could go. Thus, the naked-call writer’s risk is unlimited as well. Of course, the naked-call writer could have reversed position by buying

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back the original option he sold – that is, zeroing out the position – however, this also done at a loss. The following summarizes the basic risk-return features for a naked shortcall position. Profit potential: limited (to option premium) Loss potential: unlimited Effect of time decay: positive (makes buyer’s position less valuable) Market expectation: bearish to neutral The naked short-call position is obviously a bearish position. If the underlying asset’s price moves down, the call writer keeps the entire premium received for selling this call, since the call buyer’s position becomes worthless. Once again, the naked-call writer can express the degree of bearishness by the exercise price at which he or she sells the call. By selling an in-the-money call, the writer stands to collect a higher option premium. Conversely, selling an out-of-themoney call conveys only a mildly bearish to neutral expectation. If the underlying asset’s price stays where it is, the value of the buyer’s position, which is solely time value, will decay to zero; and the call writer will collect the entire premium (though a substantially smaller premium than for an in-the-money call). While the passing of time has a negative effect on the value of a call option for the buyer, it has a positive effect for the seller. One aspect of an option’s time value is that in the last month before the option expires, its time value decays most rapidly. Why? Time value is related to the probability that the underlying asset’s price will move up or down enough to make an option position increase in value. This probability declines at an accelerating (exponential) rate as the option approaches its maturity date. The consideration of time value, then, is a major element when investing in or hedging with options. Unless an investor is extremely bullish,

Fig. 22.5 Profit profile for a covered short call

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it would probably be unwise to take a long position in a call in its last month before maturity. Conversely, the last month of an option’s life is a preferred time to sell since its time value can more easily and quickly be collected. Now consider the other type of short-call position, covered-call writing. Because the seller of the call owns the underlying asset in this case, the risk is truncated. The purpose of writing a call on the underlying asset when it is owned is twofold. First, by writing a call option, one always decreases the risk of owning the asset. Second, writing a call can increase the overall realized return on the asset. The profit profile for a covered short call (or a covered write) in Fig. 22.5 provides further illustration. The following summarizes the basic risk-return features for the covered short-call position. Profit potential: limited (exercise price  asset price C call premium) Loss potential: limited (asset price  call premium) Effect of time decay: positive Market expectation: neutral to mildly bullish By owning the underlying asset, the covered-call writer’s loss on the asset for a price decline is decreased by the original amount of the premium collected for selling the option. The total loss on the position is limited to the extent that the asset is one of limited liability, such as a stock, and cannot fall below zero. The maximum profit on the combined asset and option position is higher than if the option was written alone, but lower than simply owning the asset with no short call written on it. Once the asset increases in price by a significant amount the call buyer will very likely exercise the right to purchase the asset at the prespecified exercise price. Thus, covered-call writing is a tool or strategy for enhancing an asset’s realized return while lowering its risk in a sideways market.

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22.4.3 Long Put Again, the put option conveys to its purchasers the right to sell a given quantity of some asset at a prespecified price on or before its expiration date. Similar to a long call, a long put is also a highly leveraged position, but the purchaser of the put makes money on the investment only when the price of the underlying asset declines. While a call buyer has unlimited profit potential, a put buyer has limited profit potential since the price of the underlying asset can never drop below zero. Yet like the long-call position, the put buyer can never lose more than the initial investment (the option’s premium). The profit profile for a long put is shown in Fig. 22.6. The following summarizes the basic risk-return features for the profit profile of a long-put position. Profit potential: limited (asset price must be greater than zero) Loss potential: limited (to cost of put) Effect of time decay: negative or positive Market expectation: bearish An interesting pricing ambiguity for this bearish investment is how the put’s price is affected by the time decay. With the long call there is a clear-cut relation – that is, the effect of the time decay is to diminish the value of the call. The relationship is not so clear with the long put. Although at certain prices for the underlying asset the value of the long-put position decreases with time, there exist lower asset prices for which its value will increase with time. It is the put’s ambiguous relationship with time that makes its correct price difficult to ascertain. (This topic has been explored in Chap. 5.) One uniquely attractive attribute of the long put is its negative relationship with the underlying asset. In terms of the capital asset pricing model, it has a negative beta (though usually numerically larger than that of the underlying asset, due to the leverage affect). Therefore, the long put is an ideal hedging instrument for the holder of the underlying asset who wants to protect against a price decline. If the investor is

Fig. 22.6 Profit profile for a long put

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wrong and the price of the asset moves up instead, the profit from the asset’s price increase is only moderately diminished by the cost of the put. (See Chap. 23 for more on hedging and related concepts.)

22.4.4 Short Put As was true for the short-call position, put writing can be covered or uncovered (naked). The risk-return features of the uncovered (naked) short put are discussed first. For taking on the obligation to buy the underlying asset at the exercise price, the put writer receives a premium. The maximum profit for the uncovered-put-writer is this premium, which is initially received. Figure 22.7 provides further illustration. While the loss potential is limited for the uncoveredput-writer, it is nonetheless still very large. Thus, someone neutral on the direction of the market would sell out-of-themoney (lower exercise price) puts. A more bullish sentiment would suggest that at-the-money options be sold. The investor who is convinced the market will go up should maximize return by selling a put with a larger premium. As with the long put, the time-decay effect is ambiguous and depends on the price of the underlying asset. The following summarizes the basic risk-return features for the profit profile of an uncovered short-put position. Profit potential: limited (to put premium) Loss potential: limited (asset price must be greater than zero) Effect of time decay: positive or negative Market expectation: neutral to bullish Referring again to Fig. 22.5 for the combined short-call and long-asset position, notice the striking resemblance of its profit profile at expiration to that for the uncovered short put. This relationship can be seen mathematically by using

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Fig. 22.7 Profit profile for an uncovered short call

Fig. 22.8 Profit profile for a long straddle

put-call parity. That is, the synthetic put price PT D E C CT ST , or at expiration the value of the put should equal the exercise price of the call option plus the call option’s value minus the value at time T of the underlying asset. Buying (writing) a call and selling (buying) the underlying asset (or vice versa) allows an investor to achieve essentially the same risk-return combination as would be received from a long put (short put). This combination of two assets to equal the risk and return of a third is referred to as a synthetic asset (or synthetic option in this case). Synthesizing two financial instruments to resemble a third is an arbitrage process and is a central concept of finance theory. Now a look at covered short puts is in order to round out the basics of option strategies. For margin purposes and in a theoretical sense, selling a put against a short-asset position would be the sale of a covered put. However, this sort of position has a limited profit potential if the underlying asset is anywhere below the exercise price of the put at expiration. This position also has unlimited upside risk, because the short position in the asset will accrue losses while the profit from the put sale is limited. Essentially, this position is equivalent to the uncovered or naked short call, except that the latter has less expensive transaction costs. Moreover, because the time value for put options is generally less than that of calls, it will be advantageous to short the call.

Strictly speaking, a short put is covered only if the investor also owns a corresponding put with exercise price equal to or greater than that of the written put. Such a position, called a spread, is discussed later in this chapter.

22.4.5 Long Straddle A straddle is a simultaneous position in both a call and a put on the same underlying asset. A long straddle involves purchasing both the call and the put. By combining these two seemingly opposing options an investor can get the best risk-return combination that each offers. The profit profile for a long straddle in Fig. 22.8 illustrates the nature of this synthetic asset. The following summarizes the basic risk-return features for the profit profile of a long-straddle position. Profit potential: unlimited on upside, limited on downside Loss potential: limited (to cost of call and put premiums) Effect of time decay: negative Market sentiment: bullish or bearish The long straddle’s profit profile makes clear that its riskreward picture is simply that of the long call overlapped by the long put, with each horizontal segment truncated

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(represented by the horizontal dashed lines on the bottom). An investor will profit on this type of position as long as the price of the underlying asset moves sufficiently up or down to more than cover the original cost of the option premiums. Thus, a long straddle is an effective strategy for someone expecting the volatility of the underlying asset to increase in the future. In the same light, the investor who buys a straddle expects the underlying asset’s volatility of price to be greater than that imputed in the option price. Since time decay is working against the value of this position, it might be unwise to purchase a straddle composed of a call and put in their last month to maturity when their time decay is greatest. It would be possible to reduce the cost of the straddle by purchasing a high-exercise-price call and a low- exercise put (out-of-the-money options); however, the necessary up or down movement in the asset’s price in order to profit is larger. Example 22.4 provides further illustration. Example 22.4. Situation: An investor thinks the stock market is going to break sharply up or down but is not sure which way. However, the investor is confident that market volatility will increase in the near future. To express his position the investor puts on a long straddle using options on the S&P 500 index, buying both at-the-money call and put options on the September contract. The current September S&P 500 futures contract price is 155.00. Assume the position is held to expiration. Transaction: 1. Buy 1 September 155 call at $2.00 ( 500 per contract) 2. Buy 1 September 155 put at $2.00 Net initial investment (position value) Results: 1. If futures price D 150.00: (a) 1 September call expires at $0 (b) 1 September put expires at $5.00 (c) Less initial cost of put Ending position value (net profit) 2. If futures price D 155.00: (a) 1 September call expires at $0 (b) 1 September put expires at $0 Ending position value (net loss) 3. If futures price D 160.00: (a) 1 September call expires at $5.00 (b) 1 September call expires at $0 (c) Less initial cost of put Ending position value (net profit)

Maximum loss potential: $2,000, the initial investment. Breakeven points: 151.00 and 159.00, for the September S&P 500 futures contract.2 Effect of time decay: negative, as evidenced by the loss incurred, with no change in futures price (result 2).

22.4.6 Short Straddle For the most part, the short straddle implies the opposite riskreturn characteristics of the long straddle. A short straddle is a simultaneous position in both a short call and a short put on the same underlying asset. Contrary to the long-straddle position, selling a straddle can be an effective strategy when an investor expects little or no movement in the price of the underlying asset. A similar interpretation of its use would be that the investor expects the future volatility of the underlying asset’s price that is currently impounded in the option premiums to decline. Moreover, since the time decay is a positive effect for the value of this position, one appropriate time to set a short straddle might be in the last month to expiration for the combined call and put. Figure 22.9 shows the short straddle’s profit profile, and Example 22.5 provides further illustration. The following summarizes the basic risk-return features for the profit profile of a short-straddle position. Profit potential: limited (to call and put premiums) Loss potential: unlimited on upside, limited on downside Effect of time decay: positive Market expectation: neutral

($1,000) ($1,000) ($2,000)

($1,000) $2,500 ($1,000) $500 ($1,000) ($1,000) $2,000 $2,500 ($1,000) ($1,000) $500

Summary: Maximum profit potential: unlimited. If the market had contributed to move below 150.00 or above 160.00, the position would have continued to increase in value.

Example 22.5. Situation: An investor thinks the market is overestimating price volatility at the moment and that prices are going to remain stable for some time. To express his opinion, the investor sells a straddle consisting of at-the-money call and put options on the September S&P 500 futures contract, for which the current price is 155.00. Assume the position is held to expiration. Transaction: 1. Sell 1 September 155 call at $2.00 ( 500 per contract) 2. Sell 1 September 155 put at $2.00 Net initial inflow (position value)

2

$1,000 $1,000 $2,000

Breakeven points for the straddle are calculated as follows: Upside BEP D Exercise price C Initial net investment (in points) 159:00 D 155:00 C 4:00 Downside BEP D Exercise price  Initial net investment (in points) 159:00 D 155:00 C 4:00 151:00 D 155:00  4:00

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Fig. 22.9 Profit profile for a short straddle

Results: 1. If futures price D 150.00: (a) 1 September 155 call expires at 0 (b) I September 155 put expires at $5.00 (c) Plus initial inflow from sale of put Ending position value (net loss) 2. If futures price D 155.00: (a) 1 September 155 call expires at 0 (b) I September 155 put expires at 0 Ending position value (net profit) 3. If futures price D 160.00: (a) 1 September 155 call expires at $5.00 (b) I September put expires at 0 (c) Plus initial inflow from sale of call Ending position value (net loss)

$1,000 ($2,500) $1,000 ($500) $1,000 $1,000 $2,000 ($2,500) $1,000 $1,000 ($500)

Summary: Maximum profit potential: $2,000, result 2, where futures price does not move. Maximum loss potential: unlimited. If futures price had continued up over 160.00 or down below 145.00, this position would have kept losing money. Breakeven points: 151.00 and 159.00, an eight-point range for profitability of the position.3 Effect of time decay: positive, as evidenced by result 2.

22.4.7 Long Vertical (Bull) Spread When dealing strictly in options, a spread is a combination of any two or more of the same type of options (two calls or two puts, for instance) on the same underlying asset. A vertical spread specifies that the options have the same 3

Breakeven points for the short straddle are calculated in the same manner as for the long straddle: exercise price plus initial prices of options.

maturity month. Finally, a long vertical spread designates a position for which one has bought a low-exercise-price call (or a low-exercise-price put) and sold a high-exercise-price call (or a high-exercise-price put) that both mature in the same month. A long vertical spread is also known as a bull spread because of the bullish market expectation of the investor who enters into it. Actually, the long vertical spread (or bull spread) is not a strongly bullish position, because the investor limits the profit potential in selling the highexercise-price call (or high-exercise-price put). Rather, this is a popular position when it is expected that the market will more likely go up than down. Therefore, the bull spread conveys a bit of uncertainty about future market conditions. Of course, the higher the exercise price at which the call is sold, the more bullish the position. An examination of the profit profile for the long vertical spread (see Fig. 22.10) can tell more about its risk-return attributes. The following summarizes the basic risk-return features for the profit profile of a long-vertical-spread position. Profit potential: limited (up to the higher exercise price) Loss potential: limited (down to the lower exercise price). Effect of time decay: mixed Market expectation: cautiously bullish Although profit is limited by the shorted call on the upside, the loss potential is also truncated at the lower exercise price by the same short call. There are other reasons for this being a mildly bullish strategy. The effect of time decay is ambiguous up to the expiration or liquidation of the position. That is, if the asset price, St , is near the exercise price of the higher-price option EH , then the position acts more like a long call and time-decay effect is negative. Conversely, if St is near the exercise price of the lower-price option EL , then the bull spread acts more like a short call and the time-decay effect is neutral. Consequently, unless an investor is more than mildly bullish, it would probably be unwise to put on a bull spread with the low exercise price call near the current price of the

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Fig. 22.10 Profit profile for a long vertical spread

asset while both options are in their last month to expiration. Example 22.6 provides further illustration. Example 22.6. Situation: An investor is moderately bullish on the West German mark. He would like to be long but wants to reduce the cost and risk of this position in case he is wrong. To express his opinion, the investor puts on a long vertical spread by buying a lower-exercise-price call and selling a higher-exercise-price call with the same month to expiration. Assume the position is held to expiration. Transaction: 1. Buy 1 September 0.37 call at ($ 587.50) $0.0047 ( 125000 per contract) 2. Sell 1 September 0.38 call at $ 1 62.50 $0.0013 Net initial investment ($ 425.00) (position value) Results: 1. If futures price D 0.37: (a) 1 September 0.37 call expires at $0 (b) 1 September 0.38 call expires at $0 Ending position value (net loss) 2. If futures price D 0.3800: (a) 1 September 0.37 call expires at $0.01 (b) 1 September 0.38 call expires at $0 Less initial cost of 0.37 call Ending position value (net profit) 3. If futures price D 0.39: (a) 1 September 0.37 call expires at $0.02 (b) 1 September 0.38 call expires at $0.01 Less initial premium of 0.37 call Plus initial premium of 0.38 call Ending position value (net profit)

($587.50) $162.50 ($425.00) $1,250.00 $162.50 ($587.50) $825.00

Summary: Maximum profit potential: $825.00, result 2. Maximum loss potential: $425.00, result 1. Breakeven point: 0.3734.4 Effect of time decay: mixed. positive if price is at high end of range and negative if at low end.

22.4.8 Short Vertical (Bear) Spread The short vertical spread is simply the reverse of the corresponding long position. That is, an investor buys a highexercise-price call (or put) and sells a low-exercise-price call (or put), both having the same time to expiration left. As the more common name for this type of option position is bear spread, it is easy to infer the type of market sentiment consistent with this position. The profit profile for the short vertical spread is shown in Fig. 22.11. As the profit profile indicates, this strategy is profitable as long as the underlying asset moves down in price. Profit is limited to a price decline in the asset down to the lower exercise price, while risk is limited on the upside by the long-call position. From the time-decay effects shown, a mildly bearish investor might consider using options in the last month to expiration with the EL option near the money. The following summarizes the basic risk-return features for the profit profile of a short-vertical-spread position. Profit potential: limited (down to EL ) Loss potential: limited (up to EH ) Effect of time decay: mixed (opposite to that of long vertical spread) Market sentiment: mildly bearish

$2,500.00 ($1,250.00) ($587.50) $162.50 $825.00

4

Breakeven point for the long vertical spread is computed at a lower exercise price plus price of long call minus price of short call .0:3734 D 0:3700 C 0:0047  0:0013/.

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Fig. 22.11 Profit profile for a short vertical spread

Fig. 22.12 Profit profile for a neutral calendar spread

22.4.9 Calendar (Time) Spreads A calendar spread (also called a time or horizontal spread) consists of the sale of one option and the simultaneous purchase of another option with the same exercise price but a longer term to maturity. The objective of the calendar spread is to capture the faster erosion in the time-premium portion of the shorted nearer-term-to-maturity option. By taking a position in two of the same type options (two calls or two puts), both with the same exercise price, the investor utilizing this strategy expresses a neutral opinion on the market. In other words, the investor is interested in selling time rather than predicting the price direction of the underlying asset. Thus, a calendar spread might be considered appropriate for a sideways-moving or quiet market. However, if the underlying asset’s price moves significantly up or down, the calendar spread will lose part of its original value. Figure 22.12 displays the calendar spread’s profit profile and related riskreturn attributes.

The profit profile shows that this strategy will make money for a rather narrow range of price movement in the underlying asset. While similar in nature to the short straddle (both are neutral strategies), the calendar spread is more conservative. The reason? It has both a lower profit potential and lower (limited) risk than the short straddle. The lower potential profit is the result of only benefiting from the time decay in one option premium instead of two (the call and the put) for the short straddle. Moreover, taking opposite positions in the same type of option at the same exercise price adds a loss limit on each side against adverse price moves. The following summarizes the basic risk-return features for the profit profile of a neutral calendar-spread position. Profit potential: limited Loss potential: limited (to original cost of position) Effect of time decay: positive. (Option sold loses value faster than option bought.) Market sentiment: neutral

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The calendar spread does not have to be neutral in sentiment. By diagonalizing this spread it is possible to express an opinion on the market. For instance, by selling a near-term higher-exercise-price option and purchasing a longer-term lower-exercise-price option, the investor is being bullish in position. Such a position is thus referred to as a bullish calendar spread. Why is it bullish? Remember that with the neutral calendar spread we are concerned solely with benefiting from the faster time decay in the premium of the shorted near-term option. Any significant movement in price upwards, for instance, would have not been profitable because it would have slowed the time decay and increased the intrinsic value of the shorted near-term option. In fact, we would eventually lose money because the difference in premiums between the near-term and the longer-term options (the spread) would narrow as the underlying asset’s price increased. However, the bullish calendar spread is much like a long vertical (or bull) spread in that a modest increase in price for the asset up to the higher exercise price will be profitable. At the same time, though, the bullish calendar spread also reaps some of the benefits from the greater time decay in the nearer-term option’s premium. While this strategy might sound superior to the straight bull spread, it really depends on market conditions. With a bullish calendar spread, its gain from time decay will probably not be as great as that from a neutral calendar spread, nor will its bullish nature be as profitable as a straight bull spread in the event of a modest price increase for the underlying asset. The real world application will be discussed in the next section.

22.5 Examples of Alternative Option Strategies In this section we are considering several other option strategies and using the options of Constellation Energy Group, Inc. (CEG) in the following examples. Below is the information published on July 13, 2007 for all options that will expire in October 2007. CEG stock closed at $94.21 on July 13, 2007 (Table 22.4).

22.5.1 Protective Put Assume that an investor wants to invest in the CEG stock on July 13, 2007, but does not desire to bear any potential loss for prices below $95. The investor can purchase CEG stock and at the same time buy the put option CEGVS.X with a strike price of $95. Let S0 ; ST , and X

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Table 22.4 Call and put option quotes for CEG at 07/13/2007 Call option expiring close Fri Oct 19, 2007 Strike

Symbol

Bid

Ask

70 75 80 85 90 95 100 105 110 115

CEGJN.X CEGJO.X CEGJP.X CEGJQ.X CEGJR.X CEGJS.X CEGJT.X CEGJA.X CEGJB.X CEGJC.X

23:5 19 14:6 11:4 8 5 2:85 1:5 0:65 0:2

25:5 20:9 16:4 12:2 8:5 5:4 3:2 1:75 0:9 0:45

Bid 0:15 0:4 0:9 1:6 2:95 4:9 7:7

Ask 0:35 0:65 1:15 1:85 3:4 5:5 8:6

Put option expiring close Fri Oct 19, 2007 Strike Symbol 70 CEGVN.X 75 CEGVO.X 80 CEGVP.X 85 CEGVQ.X 90 CEGVR.X 95 CEGVS.X 100 CEGVT.X

denote the stock purchase price, future stock price at the expiration time T, and the strike price, respectively. Given S0 D $94:21, X D $95, and the premium for the put option $5.5, Table 22.5 shows the values for Protective Put at different stock prices at time T. The profit profile of the Protective Put position is constructed in Fig. 22.13.

22.5.2 Covered Call This strategy involves investing in a stock and selling a call option on the stock at the same time. The value at the expiration of the call will be the stock value minus the value of the call. The call is “covered” because the potential obligation of delivering the stock is covered by the stock held in the portfolio. In essence, the sale of the call sold the claim to any stock value above the strike price in return for the initial premium. Suppose a manager of a stock fund holds 1,000 shares of CEG stock on July 13, 2007 and she plans to sell the CEG stock if its price hits $100. Then she can write 1,000 share of the call option CEGJB.X with a strike price of $100 to establish the position. She shorts the call and collects premiums. Given that S0 D $94:21; X D $100, and the premium for the call option $2.85, Table 22.6 shows the values for Covered Call at different stock prices at time T. The profit profile of the Covered Call position is constructed in Fig. 22.14.

22 Options Strategies and Their Applications

Table 22.5 Value of protective put position at option expiration

373

Long a put at strike price

$95:00

Premium

Buy one share of stock

$5:50

Price

Stock Price

Payoff

Profit

Payoff

Profit

Payoff

Profit

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

$24:21 $19:21 $14:21 $9:21 $4:21 $0:79 $5:79 $10:79 $15:79 $20:79 $25:79

$25:00 $20:00 $15:00 $10:00 $5:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00

$19:50 $14:50 $9:50 $4:50 $0:50 $5:50 $5:50 $5:50 $5:50 $5:50 $5:50

$95:00 $95:00 $95:00 $95:00 $95:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

$4:71 $4:71 $4:71 $4:71 $4:71 $4:71 $0:29 $5:29 $10:29 $15:29 $20:29

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

Long put .X D $95/

$94:21

One share of stock

Fig. 22.13 Profit profile for protective put

Protective put value

Protective Put : Profit $30 $20

Profit

$10

One Share of Stock

$0 Long Put (X=$95)

−$10 −$20

Protective Put Value

−$30 $70

$75

$80

$85

$90

$95

$100 $105 $110 $115 $120

Stock Price

Table 22.6 Value of covered call position at option expiration

Write a call at strike price Buy one share of stock Stock

One share of stock

Price

Payoff

Profit

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

$24:21 $19:21 $14:21 $9:21 $4:21 $0:79 $5:79 $10:79 $15:79 $20:79 $25:79

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

Premium Price

$100.00 Written call .X D $100/ Payoff $0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $5:00 $10:00 $15:00 $20:00

$2:85 $94:21

Covered call Profit

Payoff

Profit

$2:85 $2:85 $2:85 $2:85 $2:85 $2:85 $2:85 $2:15 $7:15 $12:15 $17:15

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $100:00 $100:00 $100:00 $100:00

$21:36 $16:36 $11:36 $6:36 $1:36 $3:64 $8:64 $8:64 $8:64 $8:64 $8:64

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Fig. 22.14 Profit profile for covered call

Covered Call : Profit $30

Profit

$20 One Share of Stock

$10

Written Call (X=$100)

$0 -$10

Covered Call

-$20 -$30 $70

$75

$80

$85

$90

$95 $100 $105 $110 $115 $120

Stock Price

Table 22.7 Value of collar position at option expiration Long a put at strike price $85:00 Write a call at strike price

Premium

$105:00

$1:85

Premium

Buy one share of stock

$1:50

Price

$94:21

Stock

One share of stock

Long put .X D $85/

Write call .X D $105/

Collar value

Price

Payoff

Profit

Payoff

Profit

Payoff

Payoff

Profit

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

$70:00 $75:00 $80:00 $85:00 $90:00 $95:00 $100:00 $105:00 $110:00 $115:00 $120:00

$24:21 $19:21 $14:21 $9:21 $4:21 $0:79 $5:79 $10:79 $15:79 $20:79 $25:79

$15:00 $10:00 $5:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00

$13:15 $8:15 $3:15 $1:85 $1:85 $1:85 $1:85 $1:85 $1:85 $1:85 $1:85

$85:00 $85:00 $85:00 $85:00 $90:00 $95:00 $100:00 $105:00 $105:00 $105:00 $105:00

$9:56 $9:56 $9:56 $9:56 $4:56 $0:44 $5:44 $10:44 $10:44 $10:44 $10:44

Profit

$0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $0:00 $5:00 $10:00 $15:00

Fig. 22.15 Profit profile for collar

$1:50 $1:50 $1:50 $1:50 $1:50 $1:50 $1:50 $1:50 $3:50 $8:50 $13:50

Collar : Profit

Profit

$30 $20

One Share of Stock

$10

Long put (X=$85)

$0

Write Call (X=$105) Collar Value

−$10 −$20 −$30 $70

$75

$80

$85

$90

$95

$100

$105

$110

$115

$120

Stock Price

22.5.3 Collar A collar combines a protective put and a short call option to bracket the value of a portfolio between two bounds. For example, an investor holds the CEG stock selling at $94.21. Buying a protective put using the put option CEGVQ.X with an exercise price of $85 places a lower bound of $85 on the

value of the portfolio. At the same time, the investor can write a call option CEGJA.X with an exercise price of $105. The call and the put sell at $1.50 and $1.85, respectively, making the net outlay for the two options to be only $0.35. Table 22.7 shows the values of the Collar position at different stock prices at time T. The profit profile of the Collar position is shown in Fig. 22.15.

22 Options Strategies and Their Applications

22.6 Conclusion This chapter has introduced some of the essential differences between the two most basic kinds of options: calls and puts. A delineation was made of the relationship between the option’s price or premium and that of the underlying asset. The option’s value was shown to be composed of intrinsic value, or the underlying asset price less the exercise price, and time value. Moreover, it was demonstrated that for an option the time value decays over time, particularly in the last month to maturity. Index and futures options were studied to introduce these important financial instruments. Put-call parity theorems were developed for European, American, and futures options in order to show the basic valuation relationship between the underlying asset and its call and put options. Finally, investment application of options and related combinations were discussed, along with relevant risk-return characteristics.

References Amram, M. and N. Kulatilaka. 2001. Real options, Oxford University Press, USA. Ball, C. and W. Torous. 1983. “Bond prices dynamics and options.” Journal of Financial and Quantitative Analysis 18, 517–532. Bhattacharya, M. 1980. “Empirical properties of the Black-Scholes formula under ideal conditions.” Journal of Financial and Quantitative Analysis 15, 1081–1106. Black, F. 1972. “Capital market equilibrium with restricted borrowing.” Journal of Business 45, 444–445. Black, F. 1985. “Fact and fantasy in the use of options.” Financial Analysts Journal 31, 36–72. Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 31, 637–654. Bodhurta, J. and G. Courtadon. 1986. “Efficiency tests of the foreign currency options market.” Journal of Finance 41, 151–162. Bookstaber, R. M. 1981. Option pricing and strategies in investing, Addison-Wesley Publishing Company, Reading, MA. Bookstaber, R. M. and R. Clarke. 1983. Option strategies for institutional investment management, Addison-Wesley Publishing Company, Reading, MA. Brennan, M. and E. Schwartz. 1977. “The valuation of American put options.” Journal of Finance 32, 449–462. Cox, J. C. and M. Rubinstein. 1979. “Option pricing: a simplified approach.” Journal of Financial Economics 8, 229–263. Cox, J. C. and M. Rubinstein. 1985. Option markets, Prentice-Hall, Englewood Cliffs, NJ. Eckardt, W. and S. Williams. 1984. “The complete options indexes.” Financial Analysts Journal 40, 48–57. Ervine, J. and A. Rudd. 1985. “Index options: the early evidence.” Journal of Finance 40, 743–756.

375 Finnerty, J. 1978. “The Chicago board options exchange and market efficiency.” Journal of Financial and Quantitative Analysis 13, 28–38. Galai, D. and R. W. Masulis. 1976. “The option pricing model and the risk factor of stock.” Journal of Financial Economics 3, 53–81. Galai, D., R. W. Masulis, R. Geske, and S. Givots. 1988. Option markets, Addison-Wesley Publishing Company, Reading, MA. Gastineau, G. 1979. The stock options manual, McGraw-Hill, New York. Geske, R. and K. Shastri. 1985. “Valuation by approximation: a comparison of alternative option valuation techniques.” Journal of Financial and Quantitative Analysis 20, 45–72. Hull, J. 2005. Options, futures, and other derivatives, 6th Edition, Prentice Hall, Upper Saddle River, NJ. Jarrow, R. A. and A. Rudd. 1983. Option pricing, Richard D. Irwin, Homewood, IL. Jarrow, R. A. and S. Turnbull. 1999. Derivatives securities, 2nd Edition, South-Western College Pub, Cincinnati, OH. Lee, C. F. and A. C. Lee 2006. Encyclopedia of finance, Springer, New York. Liaw, K. T. and R. L. Moy. 2000. The Irwin guide to stocks, bonds, futures, and options, McGraw-Hill Companies, New York. Macbeth, J. and L. Merville. 1979. “An empirical examination of the Black-Scholes call option pricing model.” Journal of Finance 34, 1173–1186. McDonald, R. L. 2005. Derivatives markets, 2nd Edition, Addison Wesley, Boston, MA. Merton, R. 1973. “Theory of rational option pricing.” Bell Journal of Economics and Management Science 4, 141–183. Rendleman, R. J. Jr. and B. J. Barter. 1979. “Two-state option pricing.” Journal of Finance 34, 1093–1110. Ritchken, P. 1987. Options: theory, strategy and applications, Scott, Foresman. Rubinstein, M. and H. Leland. 1981. “Replicating options with positions in stock and cash.” Financial Analysts Journal 37, 63–75. Rubinstein, M., H. Leland, and J. Cox. 1985. Option markets, PrenticeHall, Englewood Cliffs, NJ. Sears, S. and G. Trennepohl. 1982. “Measuring portfolio risk in options.” Journal of Financial and Quantitative Analysis 17, 391–410. Smith, C. 1976. “Option pricing: a review.” Journal of Financial Economics 3, 3–51. Stoll, H. 1969. “The relationships between put and call option prices.” Journal of Finance 24, 801–824. Summa, J. F. and J. W. Lubow. 2001. Options on futures, John Wiley & Sons, New York. Trennepohl, G. 1981. “A comparison of listed option premium and Black-Scholes model prices: 1973–1979.” Journal of Financial Research 4, 11–20. Weinstein, M. 1983. “Bond systematic risk and the options pricing model.” Journal of Finance 38, 1415–1430. Welch, W. 1982. Strategies for put and call option trading, Winthrop, Cambridge, MA. Whaley, R. 1982. “Valuation of American call options on dividend paying stocks: empirical tests.” Journal of Financial Economics 10, 29–58. Zhang, P. G. 1998. Exotic options: a guide to second generation options, 2nd Edition, World Scientific Pub Co Inc, Singapore.

Chapter 23

Option Pricing Theory and Firm Valuation Cheng Few Lee, Joseph E. Finnerty, and Wei-Kang Shih

Abstract In this chapter, we introduce the basic concepts of call and put options. Second, we discuss the Black-Scholes option pricing model and its application. Third, we discuss how to apply the option pricing theory in capital structure issue. Finally, the warrant, one type of equity options, is discussed in detail. Keywords Warrants r Executive stock option r Publicly traded option r Call option r Put option r American option r European option r Exercise price r Time to maturity r Black-Scholes option pricing model r Capital structure

23.1 Introduction We discuss the emergence of options and option pricing, including several types of options and how their value is determined. We begin looking at the basic concepts of options in Sect. 23.2, then go on to discuss factors that affect the value of options in Sect. 23.3. Hedging, hedge ratio, and option valuation are discussed in Sect. 23.4. Section 23.5 discusses how option pricing theory is used to investigate the capital structure question. We close the chapter with a look at the type of option called the warrant in Sect. 23.6 and a summary of this chapter is discussed in Sect. 23.7.

23.2 Basic Concepts of Options In general, there are three types of equity options: (1) warrants, (2) executive stock options, and (3) publicly traded options. A warrant is a financial instrument issued by a corporation that gives the purchaser the right to buy a fixed number of shares at a set price for a specific period. There C.F. Lee and W.-K. Shih () Rutgers University, Newark, NJ, USA e-mail: [email protected] J.E. Finnerty University of Illinois at Urbana-Champaign, Champaign, IL, USA

are two major differences between a warrant and a publicly traded option. The first is that the maturity of the warrant is normally less than 9 months. The second difference is that the warrant is an agreement between the corporation and the warrant’s buyer. This means that if the warrant’s owner decides to exercise his right and purchase stock, the corporation issues new shares and receives the cash from the sale of those shares. The publicly traded option is an agreement between two individuals who have no relationship with the corporation whose shares are being optioned. When the publicly traded option is exercised, money is exchanged for shares between individuals and the corporation receives no funds, only a new owner. Executive stock options are a means of compensation for corporate employees. For services rendered, the manager or the employee has the right to buy a specific number of shares for a set price during a given period. Unlike warrants and publicly traded options, executive stock options cannot be traded. The option’s owner has only two choices: exercise the option or let it expire. Like a warrant, should the owner decide to exercise the option, the corporation receives money and issues new shares. The use of executive stock options for management compensation raises an interesting agency question. The firm’s managers may make investment and financing decisions that increase the firm’s risk in order to increase the value of their stock options. Such action could have a detrimental effect on the bondholders and other creditors of the firm. Thus, we will see that the value of an option is directly related to the variability or riskiness of the underlying asset, which in this case is the firm. Publicly traded options are probably the most widely known of the three types of equity option instruments. An important date in the history of these options is 1973, when the Chicago Board Options Exchange was founded. Although it was possible to trade options over the counter before that time, trading volume was relatively low. This date marks the beginning of a phenomenal growth in the popularity of options as a financial instrument. Indeed, in terms of the value of securities traded, the Chicago Board Options Exchange is running neck and neck with the New York and Tokyo stock

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exchanges as the world’s largest securities market. There are now five options-trading centers in the United States – the Chicago Board Option Exchange, the American Stock Exchange, the Philadelphia Stock Exchange, the Pacific Stock Exchange, and the New York Stock Exchange – and there is a steady stream of proposals for new listings on these exchanges. The share volume of trading and the general acceptance of these financial instruments renders options valuation an important subject for study. However, even more significant for our purposes, the theory of options valuation has important applications in financial management and in the valuation of other financial instruments. Black and Scholes (1973) point out those corporate securities can be viewed as options or contingent claims on firm value. In viewing a firm’s securities as options, we need to evaluate the interdependencies of bonds and stocks. As we have seen, a bond’s price is the present value of future interest and principal payments, and a stock’s price is determined by future dividends. The contingent-claims approach to security valuation differs in that it considers the valuation of all of the firm’s classes of securities simultaneously. Bonds and stocks are valued in terms on the value placed on the firm’s assets; that is, their value is contingent on the firm’s assets or investments. Banz and Miller (1978) developed an approach to making capital budgeting decisions based on a contingent claims framework. They created a method for calculating the NPV of an investment based on estimates derived from using an option valuation model. In this chapter, we consider the question of option valuation and how options are used to make financial management decisions. However, before doing so, we must become acquainted with some of the terminology of options trading. There are two basic types of options: puts and calls. A call gives the holder the right to buy a particular number of shares of a designated common stock at a specified price, called the exercise price (or strike price), on or before a given date, known as the expiration date. Hence, the owner of shares of common stock can create an option and sell it in the options market, thereby increasing the return or income on his or her stock investment. Such an option specifies both an exercise price and an expiration date. On the Chicago Board of Options Exchange, options are typically created for 3, 6, or 9 months. The actual expiration date is the third Saturday of the month of expiration. A more venturesome investor may create an option in this fashion without owning any of underlying stock. This is called naked option writing and can be very risky, especially if the value of the underlying asset has a high degree of variability. A put gives the holder the right to sell a certain number of shares of common stock at a set price on or before the expiration date of the option. In purchasing a put, the owner of shares has bought the right to sell these shares by the expira-

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tion date at the exercise price. As with calls, the creator can own the underlying shares (covered writing) or not (naked writing). The owner of a put or call is not obligated to carry out the specified transaction, but has the option of doing so. If the transaction is carried out, it is said to have been exercised. For example, if you hold a call option on a stock that is currently trading at a price higher than the exercise price, because the stock could be immediately resold at a profit. (You also could sell the option or hold it in the hope of further gains.) This call option is said to be “in the money.” On the other hand, if the call option is “out of the money” – that is, the stock is trading at a lower price – you certainly would not want to exercise the option, as it would be cheaper to purchase the stock directly. An American option can be exercised at any time up to the expiration date. A simpler instrument to analyze is the European option, which can only be exercised on the expiration date. In this case, the term to maturity of the option is known. Because of this simplifying factor, we will concentrate on the valuation of the European option. The factors determining the values of the two types of options are the same, although, all other things equal, an American option is worth more than a European option because of the extra flexibility permitted the option holder. Although our discussion is limited to equity options, many kinds of publicly traded options are available. Those include options in stick indexes, options on treasury bonds, options on future contracts, options on foreign currencies, and options on agricultural commodities. The proceeding discussion presented quite a lot of new terminology. For convenience, we list and define those terms below. Call

Put Exercise price of the contract Expiration date Exercise option American option

European option

Call option “in the money”

An option to purchase a fixed number of shares of common stock An option to sell a fixed number of shares of common stock Trading price set for the transaction as specified in an option Time by which the option transaction must be carried out Carrying out the transaction specified in an option An option in which the transaction can be carried out at any time up to the expiration date An option in which the transaction can only be carried out at the expiration date If the stock price is above the exercise price

23 Option Pricing Theory and Firm Valuation

Put option “in the money” Call option “out of the money” Put option “out of the money”

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If the stock price is below the exercise price If the stock price is below the exercise price If the stock price is above the exercise price

23.2.1 Option Price Information Each day the previous day’s options trading are reported in the press. Exhibit 23.1 is a partial listing of equity options traded on the Chicago Board of Options Exchange. In this exhibit, we have highlighted the option for Johnson & Johnson (JNJ). The first 3 months (September, October, and January) are for a call option contract on Johnson & Johnson for 100 shares of stock at an exercise price of 65.12 represents the closing price of Johnson & Johnson shares on the New York Stock Exchange. The numbers in the September column represents the price of one call option with a September expiration date. Since each option is the right to buy 100 shares, the cost of the call option with the strike price $55 is $1030 or $10:3 100. The values of the call options for September is somewhat higher, reflecting the fact that if you owned those Exhibit 23.1 Listed options quotations

Close price JNJ 65.12 65.12 65.12 65.12 65.12 65.12 MRK 51.82 51.82 51.82 51.82 51.82 51.82 51.82 PG 69.39 69.39 69.39 69.39 69.39 69.39 69.39 69.39

options you would have a longer time to exercise them. In general, this time premium is reflected in higher prices for options with longer lives. The last three columns (September, October and January) represent the value of put options on Johnson & Johnson stock. Again, each put option contract is for 100 shares and worth 100 times the price, or $2 for the October 55 Johnson & Johnson puts. The specific factors that determine option value are discussed next. How much should you pay for a put or call option? The answer is not easy to determine. However, it is possible to list the various factors that determine option value. We begin with a simple question. How much is a call option worth on its expiration date? The question is simple because we are in a deterministic world. The uncertain future movement of the price of the stock in question is irrelevant. The call must be exercised immediately or not at all. If a call option is out of the money on its expiration date, it will not be exercised and the call becomes a worthless piece of paper. On the other hand, if the call option is in the money on its expiration date, it will be exercised. Stock can be purchased at the exercise price, and immediately resold at the market price, if so desired. The option value is the difference between these two prices. On the call option’s expiration date, its value will be either 0 or some positive amount equal to the difference between the market price of the stock and the

Strike price

Calls Sep

Oct

Jan

Puts Sep

Oct

Jan

45.00 50.00 55.00 60.00 65.00 70.00

20.40 N/A 10.30 5.30 0.10 N/A

N/A N/A 10.40 N/A 1.15 0.05

N/A N/A 11.10 6.40 2.60 0.55

N/A N/A N/A N/A 0.05 4.80

N/A N/A 0.02 N/A 0.85 4.50

N/A N/A 0.30 0.70 1.95 5.00

45.00 47.50 50.00 52.50 55.00 57.50 60.00

6.70 4.34 1.85 0.03 0.01 N/A N/A

7.08 4.90 2.70 1.10 0.35 N/A N/A

N/A 6.04 4.40 2.85 1.70 0.95 0.50

N/A N/A 0.03 0.55 3.20 5.70 8.20

0.15 N/A 0.65 1.50 3.30 N/A N/A

0.85 1.30 2.10 3.10 4.60 N/A 8.40

40.00 45.00 55.00 60.00 65.00 70.00 75.00 80.00

29.70 24.60 14.50 9.70 4.50 0.05 N/A N/A

29.70 24.70 N/A 9.80 4.90 0.90 0.05 N/A

30.10 N/A 15.20 10.70 6.19 2.75 0.70 0.15

N/A N/A N/A N/A N/A 0.60 5.50 10.40

N/A N/A N/A 0.08 0.21 1.50 5.70 10.50

N/A N/A 0.15 0.42 1.00 2.65 5.70 N/A

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exercise price of the option. Symbolically, let P D price of stock, Vc D value of call option, and E D exercise price. Then Vc D P  E if E P , and Vc D 0 if P < E. This relationship can be written as Vc D MAX.0; P  E/ where MAX denoted the larger of the two bracketed terms. For a put option .Vp /; Vp D E  P if E > P , and Vp D 0 if E  P . This can be written as Vp D MAX.0; E  P /: The call position is illustrated in Fig. 23.1a, where we consider a call option with an exercise price of $50. The figure shows the option value as a function of stock price to the option holder. For any price of the stock up to $50, the call option is worthless. The option’s value increases as the stock price rises above $50. Hence, if on the expiration date the stock is trading at $60, the call option is worth $10. Figure 23.1b, which is the mirror image of (a), shows the option position from the view point of the writer of the call option. If the stock is trading below the exercise price on the expiration date, the call option will not be exercised and the seller incurs no loss. However, if the stock is trading at $60,

Fig. 23.1 Value of $50 exercise price call option (a) to holder, (b) to seller

the seller of the call option will be required to sell this stock for $50, which is $10 below the price that could be obtained on the market. We have seen that once a call option have been purchased; the holder of the option has the possibility of obtaining gains but cannot incur losses. Correspondingly, the writer of the option may incur losses but cannot achieve any gains after receiving the premium. Further, there is no net gain in the sense that the profit incurred by one will balance the loss of the other. Thus, options are zero sum securities – they merely transfer wealth rather than create it. To acquire this instrument, a price must be paid to the writer, and it is this price that corresponds to the call value. This value is the price paid to acquire the chance of future profit and will therefore reflect uncertainty about the future market prices on the common stock. We will now discuss how to determine the value of an option before its expiration date.

23.3 Factors Affecting Option Value 23.3.1 Determining the Value of a Call Option Before the Expiration Date Suppose that you are considering the purchase of a call option, with an exercise price of $50, on a share of common stock. Let us try to determine what factors should be taken into account in trying to assess the value of such an option prior to the expiration date. As we have seen, the problem is trivial at the expiration date. However, determining the value of an option prior to expiration is more complex. An important factor in this determination is the current price per share of the stock. Indeed, it is most straightforward to think of option value as a function of the market price. Given this framework, we can see fairly quickly how to set bounds on the value of a call option. Lower bound: The value of a call cannot be less than the payoff that would accrue if it were exercised immediately. We can think of Fig. 23.1 as showing the payoff from immediate exercise of a call as a function on the current price of the stock. In our example, this payoff is zero for any market price below $50 and equal to the market price less for the $50 exercise price when market price is above $50. Suppose that the current stock price is $60 per share, and you are offered a call option for $8. This is certain to be profitable because you could immediately exercise the call and then resell the stock at a $10 gain. Subtracting the cost of the call option would yield a profit of $2. Of course, market participants are well aware of this, so that the excess of demand oversupply for such immediate profits are unattainable.

23 Option Pricing Theory and Firm Valuation

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Upper bound: The value of a call cannot be more than the market price of the stock. Suppose that, at the same cost, you are offered two alternatives: (a) purchase a share of stock (b) purchase an option on the same share of stock Looking into the future, option (b) will either be exercised or discarded as worthless. If the option is exercised, its value will be the difference between the future stock prices less the exercise price. This will be less than the future stock price, which is the value derived from option (a). Similarly, if the option turns out to have no value, this cannot be preferable to holding the stock, which, while it may have fallen in price, will have retained some value. Therefore, option (b) cannot be preferable to (a), so that the option value cannot exceed the market price of the stock. These conclusions are illustrated in Fig. 23.2, which relates the call-option value to the market price of the stock. The ACD line shows the payoff that would be obtained if the option were exercised immediately. The AB line is the set of points at which the call value equals the market price of the stock. The call value must lie between these boundaries; that is, in the shaded area depicted in Fig. 23.2. So far, we have been able to determine a fairly wide range in which the option value must lie for any given value of the market price of the stock. Let us now see if we can be more precise in formulating the shape of the relationship between call-option value and market price stock. The following considerations should help in forming the appropriate picture of Fig. 23.3. 1. If the market price of the stock is 0, the call value will also be 0, as indicated by point A. This extreme case arises only when there is no hope that the stock will ever have

Fig. 23.2 Value of call option

Fig. 23.3 Call option value as a function of stock price

any value; otherwise, investors would be prepared to pay some price, however small, for the stock. It follows, in such a dire circumstance, that the call option will never be exercised, and so it too has no value. 2. All other things equal, as the stock price increases, so does the call value (see lines AB and CD). A call with an exercise price of $50 would be worth more if the current market price were $40 than if it were $35, all else equal. The probability that the market price will eventually exceed the exercise price by any given amount is higher in the former case than in the latter, and so, consequently, is the payoff expected from the holding call. 3. If the price of the stock is high in relation to the exercise price, each dollar increase in price induces an increase of very nearly the same amount in the call value (shown by the slope of the line at point F). Suppose that the market price of the stock exceeds the exercise price of a call by such a large amount that it is virtually impossible for the stock price to fall below the exercise price before the expiration date. Then, any change in stock price will induce a change of the same amount in the payoff expected from the call, since it is certain that the option will be exercised. The point is that if it is known that an option will eventually be exchanged for stock, this is tantamount to already owning stock. The call holder has effectively purchased the stock without paying the full amount for it right away. The balance owned is the exercise price to be paid at the expiration date. It follows that the value of the call option is the market value of the stock less the present value of the exercise price. Notice that in our previous notation, this will exceed P-E if the exercise price does not have to be paid until some time in the future, because of the time value of money.

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4. The call value rises at an increasing rate as the value of the stock price rises, as shown by curvature AFG. The segment AF shows that as the stock price increases, the call value increases by a lesser amount. The segment FE shows the call price increasing by a larger amount than the increase in the stock price. Theoretically, this is impossible, because if it occurs, eventually the call price would exceed the price of the underlying stock. Putting these four considerations together, we show in Fig. 23.3a typical curve relating call-option value to the market price of a stock. Beginning at the origin, when both quantities are 0, the curve shows call value increasing as market price increases. Eventually, the curve becomes virtually parallel to the two 45-degree lines, forming the boundaries if the possible-values region. This is a result of the fourth consideration listed above. The curve graphed in Fig. 23.3 shows the relationship between the call-option value and the market price of the stock, when all other relevant factors influencing call-option value are constant. Next, we must try to see what these other factors might be and assess their impact on call value. We consider, in turn, five factors that influence the value of a call option: 1. Market price of the stock. In Fig. 23.3, we have seen the curvilinear relationship between call-option value and market price of the stock. All other things equal, the higher the market price, the higher the call-option value on the stock. As already noted, the slope of this relationship increases as market price becomes higher, to the point where, eventually, each dollar increase in the stock price translates into an increase of about the same amount in the call option value on that stock. 2. The exercise price. The exercise price offered two otherwise identical call options on the same stock; you would prefer the call with the lower exercise price. This would involve larger gains from any favorable movement in the price of the stock than would an option with a higher exercise price. Therefore, we conclude that the lower the exercise price, the higher the call-option value, all other things equal. 3. The risk-free interest rate. If a call option is eventually exercised, the holder of the option will reap some of the benefits of an increase in the market value of the stock. However, the holder will do so without having to immediately pay the exercise price. This payment will only be made at some future time when the call option is actually exercised. In the meantime, this money can be invested in government securities to earn a no-risk return. This opportunity confers an increment of value on the call option; the higher the risk-free rate of interest, the greater this incremental value or lower the present value of the exercise price. Therefore, we would expect to find, all else equal,

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Table 23.1 Probabilities for future prices of two stocks Less volatile stock More volatile stock Future price($)

Probability

Future price ($)

Probability

42 47 52 57 62

.10 .20 .40 .20 .10

32 42 52 62 72

.15 .20 .30 .20 .15

that the higher the risk-free rate of interest, the greater the call-option value. Moreover, the longer the exercise of the option is postponed, the greater the risk-free interest earnings. Accordingly, we would expect the risk-free interest rate to determine call-option value in conjunction with the time remaining before the expiration date. 4. The volatility of the stock price. Suppose that you are offered opportunity to purchase one of two call options, each with an exercise price of $50. Table 23.1 lists probabilities for different market prices on the expiration date for each of the two stocks. In each case, the mean (expected) future price is the same. However, for the second stock, prices that differ substantially from the mean are far more likely than for the first stock. The price of such a stock is said to be relatively volatile. And the expected price of such a stock is said to be relatively volatile. We now show that the expected payoff on the expiration date is higher for a call option on the more volatile stock than for a call option on the less volatile stock. For less volatile stock, the option will not be exercised for prices below $50, but for the three higher prices its exercise will result in payoffs of $2, $7, and $12. Therefore, we find expected payoff from call option on less volatile stock D .0/.:10/ C .0/.:20/ C .2/.:40/ C .7/.:20/ C .12/.:10/ D $3:40. Similarly, for a call option on more volatile stock, expected payoff from call option on more volatile stock D .0/.:15/ C .0/.:20/ C .2/.:30/ C .12/.:20/ C .22/.:15/ D $6:30. We find, then, that although the expected future price is the same for the two stocks, the expected payoff is higher from a call option on the more volatile stock. This conclusion is quite general. For example, it does not depend on our having set the exercise price below the expected future stock price. The reader is invited to verify that the same qualitative finding would emerge if the exercise price were $55. We can conclude that, all other things remaining the same, the greater the volatility in the price of the stock, the higher the call-option value on that stock. One useful way to measure volatility is through the variance in day-to-day changes in the stock price. Figure 23.4 illustrates our assertion about the influence of stock-price volatility on call-option value. The figure shows

23 Option Pricing Theory and Firm Valuation

Fig. 23.4 Call-option value as function of stock price for high-, moderate-, and low-volatility stocks

the relationship between call value and current market price for three stocks. The three curves have the same general form depicted in Fig. 23.3. However, notice that for any given current market price of the stock, the higher the variance of dayto-day price changes, the greater the call-option value. The notion of volatility in future stock market prices is related to the length of the time horizon being considered. Specifically, if the variance of day-to-day changes is  2 , the variance in the change from the present to t days is  2 t if the price changes are serially independent. Therefore, the further ahead the expiration date, the greater the volatility in price movements. This suggests that for a European option, which can only be exercised on the expiration date, the relevant measure of volatility is  2 t where t is the number of days remaining to the expiration date. The larger this quantity, the greater the call-option value. 5. Time remaining to expiration date. We have seen in factors 3 and 4 above two reasons to expect that the longer the time remaining before the expiration date, the higher the call-option value, all else equal. The reason is that the extra time allows larger gains to be derived from postponing payment of the exercise price and permits greater volatility in price movements of the stock. These two considerations both operate in the same direction – toward increasing the call-option value. We have seen that five factors influence the value of a call option. The interaction of these various factors in determining option value is rather complex.1 We present a formula 1

Compared to the CAPM, the option pricing model (OPM) is a relative pricing approach. In the CAPM, the major determinants are risk and return. In the OPM, the return of the asset or the market does not affect the option value.

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that, under certain assumptions, can be shown to determine a call-option value as a function of these five parameters. However, before proceeding to this somewhat complicated relationship, we discuss a simple situation in which option valuation is more straightforward. This is useful in explaining further the dependence of option value on the risk-free rate of interest. Our aim here is to show, for a special set of assumptions, that an investor can guarantee a particular return from a combined strategy of holding shares in a stock and writing call options on that stock, even though there is some uncertainty about the future price of the stock. The essential simplifying assumption required to generate this result is that there are only two possible values for the future price per share of the stock. In addition, for convenience we will use European options, with an expiration date in 1 year. Further, we assume that the stock pays no dividend and that there are no transactions costs. Within this framework, consider the factors in Table 23.2. An investor can purchase shares for $100 each and write European call options with an exercise price of $100. There are two possible prices of the stock on the expiration date, with uncertainty as to which will materialize. In Table 23.3, we list the consequences to the investor for the two expiration-date stock prices. If the higher price prevails, each share of stock will be worth $125. However, the call options will be exercised, at a cost per share to the writer of the difference between the stock price and the exercise price. In Table 23.3, this is a negative value of $25 from having written the call option. If the lower of the two possible stock prices materializes on the expiration date, each share owned will be worth $85, and it will have cost nothing to write the option because it will not be exercised. Suppose that the investor wants to form a hedged portfolio by both purchasing stock and writing call options, so as to guarantee as large a total value of holdings per dollar invested

Table 23.2 Data for a hedging example Current price per share: Future price per share: Exercise price of call option:

$100 $125 with probability .6 $85 with probability .4 $100

Table 23.3 Possible expiration-date outcomes for hedging example Expiration-date Value per share of Value per share of stock price stock holdings options written $125 $85

$125 $85

$25 $0

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as possible, whatever the stock price on the expiration date. The hedge is constructed to be riskless, since any profit (or loss) from the stock is exactly offset with a loss (or profit) from the call option. This is called a perfect or riskless hedge and can be accomplished by purchasing H number of shares for each option written. We now determine H . If H shares are purchased and one option written, the total value on the expiration date will be $125H  $25 at the higher market price and $85H  $0 at the lower price. Suppose we choose H so that these two amounts are equal; that is 125H  25 D 85H or H D

5 25 D : 40 8

Then, the same total value results whatever the stock’s expiration-date market price. This ratio is known as the hedge ratio of stocks to options, the implication being that a hedged portfolio is achieved by writing eight options for each five shares purchased. More generally, it follows from the above argument that the hedge ratio is given by H D

PU  E PU  PL

where PU D upper share price; PL D lower share price; E D exercise price of option; and E is assumed to be between PU and PL . Returning to our example, suppose that five shares are purchased and an option on eight shares is written. If the expiration-date price is $125, then the total value of the investor’s portfolio is .5/.125/  .8/.25/ D $425:

On the expiration date, an initial investment of $500  $8 Vc will be worth $1:08.500  8 Vc /. If this is to be equal to the value of the hedged portfolio, then 1:08.500  8Vc / D 425 so that Vc D

.1:08/ .500/  425 D $13:31: .1:08/ .8/

Therefore, we conclude that if the price per share for the call option is $13.31, the hedging strategy will yield an assured rate of return equal to the risk-free interest rate. In a competitive market, this is the price that will prevail, and therefore is the value of the call option. Suppose that the price of the call option was above $13.31. By forming a hedged portfolio in the manner just described, investors could ensure a return in excess of the risk-free rate. Such an opportunity would attract many to sell options, thus driving down the price. Conversely, if the price of the option was below $13.31, it would be possible to achieve a return guaranteed to be in excess of the risk-free rate by both purchasing call options and selling short the stock. The volume of demand thus created for the call option would drive up its price. Hence, in a competitive market, $13.31 is the only sustainable price for this option. Our example shows that a hedged portfolio can achieve an assured rate of return of 8%. Notice that our analysis depends on the level of the risk-free rate. If that rate is 10% annually, the call-option value is $14.20. This illustrates the dependence of call-option value on the risk-free interest rate.

23.4 Determining the Value of Options 23.4.1 Expected Value Estimation

If the expiration-date stock price is, $85, total value is .5/.85/  .8/.0/ D $425: As predicted, the two are identical, so that this value is assured. Next, we must consider the investor’s income from the writing of call options. Let Vc denote the price per share of a call option. Then, the purchase of five shares costs $500, but $8 Vc is received from writing call options on eight shares, so that the net outlay will be $500  $8 Vc . For this outlay, a value 1 year hence of $425 is assured. However, there is another simple mechanism for guaranteeing such a return. An investment could be made in government securities at the risk-free interest rate. Suppose that this rate is 8% annually.

A higher expected rate of return, as compared with the hedged portfolio, can be achieved by the exclusive purchase either of shares or of call options. Suppose that a single share is purchased for $100. One year hence, the expected value of that share is expected value of share D .:6/.125/ C .:4/.85/ D $109 Hence, the expected rate of return from a portfolio consisting entirely of holdings of this stock is 9%. Similarly, suppose that a call option is purchased for $13.31. The expected value of this option on the expiration date is expected value of call D .:6/.25/ C .:4/.0/ D $15:

23 Option Pricing Theory and Firm Valuation

Fig. 23.5 Put-option value

Hence, the expected rate of return is expected rate of return on call D

1513:31

100%D12:7%: 13:31

The increased expected rates of return from exclusively holding these two types of securities should not be surprising. They simply represent increases required by investors for assuming additional risk. We have shown how call-option valuation can be determined within a simple framework in which only two values for the future market price of a stock are possible. We must move on to more realistic situations that involve a range of future market prices. The value of a put option is determined by the same factors that determine the value of a call option, except that the factors have different relationships to the value of a put than they have to the value of a call. Figure 23.5 shows the relationship between put-option value and stock price. The value of a put option is VP D MAX .E  P; 0/ where VP D value of the put; E D exercise price; and P D value of the underlying stock. The maximum value of a put is equal to the exercise price of the option when the stock price is 0. This is shown by the line “Maximum value of the put” in Fig. 23.5. The minimum value of the put is 0, which occurs when the stock price exceeds the exercise price.

23.4.2 The Black-Scholes Option Pricing Model The notion that the price of a call option should be such that the rate of return on a fully hedged portfolio is equal to the risk-free rate of interest has been used by Black and Scholes

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(1973) to derive a more generally applicable procedure for valuing an option. The assumption that only two future prices are possible is dropped for a more realistic view of future price movements. The complete set of assumptions on which the Black-Scholes formula is based are given below. Only European options are considered. Options and stocks can be traded in any quantities in a perfectly competitive market; there are no transactions costs and all relevant information is freely available to market participants. Short-selling of stocks and options in a perfectly competitive market is possible. The risk-free interest rate is known and is constant up to the expiration date. Market participants are able to borrow or lend at this rate. No dividends are paid on the stock. The stock price follows a random path in continuous time such that the variance of the rate of return is constant over time and known to market participants. The logarithm of future stock prices follows a normal distribution. Under these assumptions, Black and Scholes show that the call-option value on a share of stock is given by Vc D P ŒN.d1 /  e rt EŒN.d2 /

(23.1)

where V D value of option; P D current price of stock; r D continuously compounded annual risk-free interest rate; t D time in years to expiration date; E D exercise price; e D 2:71828 : : :is a constant; and N.d1 / D probability that a standard normal random variable is less than or equal to di .i D 1; 2/, with

d1 D

P ln E





2 C rC t 2 p  t

(23.2a)

and

d2 D

P ln E





2 C r t p 2 p D d1   t  t

(23.2b)

where  2 D variance of annual rate of return on the stock, continuously compounded; and logarithms are to base e. A binomial distribution approach to derive the OPM can be found in Appendix 2A. Although the specific form of the Black-Scholes option pricing formula is complicated, its interpretation is straightforward. The first term of Equation (23.1) is the value of an investor’s right to that portion of the probability distribution of the stock’s price that lies above the exercise price. This term equals the expected value of this portion of the stock’s price distribution, as shown in Fig. 23.6.

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Fig. 23.6 Probability distribution of stock prices

The second term in Equation (23.1) is the present value of the exercise price times the probability that the exercise price will be paid (the option will be exercised at maturity). Overall, the Black-Scholes model involves the present value of a future cash flow, a common concern in finance. The BlackScholes model equates option value to the present expected value of the stock price minus the present value of the cost of exercising the option. More important for our purposes than the particular formula is the manner in which the model relates option value to the five factors discussed in the previous section. These relationships are as follows: 1. All other things equal, the higher the current market price of the stock, the higher the call-option value 2. All other things equal, the higher the exercise price, the lower the option value 3. All other things equal, the higher the risk-free interest rate, the higher the option value 4. All other things equal, the greater the time to the expiration date, the higher the option value 5. All other things equal, the greater the volatility of the stock price (as measured by  2 ), the higher the option value According to the Black-Scholes model, although the calloption value depends on the variance of rate of return, it does not depend on the stock’s expected rate of return. This is because a change in expected rate of return affects stock price but not the relative value of the option and the stock. Recall from our earlier discussion that formation of the hedged portfolio was independent of expected return. For example, changing the two probabilities in Table 23.2 has no effect on call-option value. The following example illustrates the computation of option value using the Black-Scholes formula. Example 23.1. Suppose that the current market price of a share of stock is $90 and an option is written to purchase the stock for $100 in 6 months. The current risk-free rate of

C.F. Lee et al.

interest is 8% annually. Since the time to the expiration date is half a year, we have, in the notation of the Black-Scholes model, P D 90I E D 100I r D :08; and t D :50. All of this information is readily available to market participants. More problematic is an assessment of the likely volatility of returns on the stock over the next 6 months. One possible approach is to estimate this volatility using data on past price changes. For instance, we could compute the variance of daily changes in the logarithm of price (daily rate of return) over the last 100 trading days. Multiplying the result by the number of trading days in the year would then yield an estimate of the required variance on an annual basis.2 For this stock, suppose that such a procedure yields the standard deviation  D :6 This value or some alternative (perhaps developed subjectively) can then be substituted, together with the values of the other four factors, in Equation (23.1) to obtain an estimate of the market value of the call option. We now illustrate these calculations. First, we need the natural logarithm of P/E. This is

P D ln .:9/ D :1054: ln E It follows that   ln PE C rt C  2 2t d1 D p  t :1054 C .:08/ .:50/ C 12 .:36/ .:50/ D :06 p :6 :50   ln PE C rt   2 2t p d2 D  t D

D

:1054 C .:08/ .:50/  12 .:36/ .:50/ p :6 :50

D :37 Using the table of the cumulative distribution function of the standard normal distribution in the appendix at the back of this book, we find the probability that a standard normal random variable less than .06 is N .d1 / D N .:06/ D :5239 and the probability that a standard normal random variable is less than :37 is N .d2 / D N .:37/ D :3557:

2

If we use monthly rate of return to calculate the variance, then the annualized variance will be 12 times the monthly variance.

23 Option Pricing Theory and Firm Valuation

These numbers represent the probability that the stock price will be at least equal to the exercise price – 52% of the time over the life of the option – and the probability of exercise – 35%. We also require e rt D e .:08/.:5/ D :9608: Finally, on substitution into Equation (23.1), we find Vc D PN .d1 /  e rt EN .d2 / D .90/ .:5239/  .:9608/ .100/ .:3557/ D $12:98 Therefore, according to the Black-Scholes model, the call option should be priced at $12.98 per share. A further implication of the Black-Scholes model is that the quantity N .d1 / provides the hedge ratio for a hedged portfolio of stocks and written options. We see, then, that to achieve such a portfolio, .5239 shares of stock should be purchased for each option written.

23.4.3 Taxation of Options The taxation of option gains, losses, and income are a fairly complex and constantly changing part of the tax law. However, tax treatment can have a large impact on the usefulness of options for the individual investor. Income (premiums) received by the option writer is taxed as normal income, just as if the option writer were providing a service. Options can be used to defer gains into the future, which reduces the investor’s current tax liability. For example, an investor with a short-term gain on a stock can purchase a put to protect against a drop in the stock price. This allows the investor to hold the stock and not realize the gain until sometime in the future, perhaps the next year, thereby deferring the payment of tax from the present until some future period. Hence, the tax position of the investor is also a factor in determining the value or usefulness of options. However, because we do not have a homogeneous tax structure, it is difficult to incorporate the tax effect into a formulation such as the Black-Scholes model.

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American call option before the expiration date. Therefore, if American and European call options on a nondividendpaying stock are otherwise identical, they should have the same value. The effect of dividends on a call option depends on whether the firm is expected to pay dividends before the option’s expiration date. Such action should reduce the option’s value – the greater the expected dividend, the larger the reduction in option value. The reason is that a dividend payment amounts to the transfer to stockholders of part of the firm’s value, a distribution in which option holders do not share. The larger this reduction in firm value, all other things equal, the lower the expected future price of the stock, because that price reflects firm value at any point in time. Therefore, the higher the dividend, the smaller the probability of any gains from exercising a call option on common stock and, hence, the lower the value of the option. The extreme case occurs if the firm is liquidated and the entire amount of funds realized from the liquidation is distributed as dividends to common stockholders. Once such a distribution is made, the stock and the call option have zero value. If the firm pays dividends, the value of an American call option on its stock will exceed that of an otherwise identical European option. The reason is that the holder of the American call can exercise the option just before the exdividend date and thus receive the dividend payment. If the benefits from such an exercise outweigh the interest income that would have been earned on the exercise price had the option been held to the expiration date, it benefits the holder of an American call to exercise the option. Since this opportunity is not available to the holder of a European option, the American option should have a higher value. The solution to valuing an option on a stock that pays dividends can be approximated by replacing P , the stock price in the Black-Scholes formula, with the stock price minus the present value of the known future dividend. This is justified because if the option is not exercised before maturity, the option holder will not receive the dividends, in which case the holder should subtract the present value of the dividend from the stock price.

23.4.4 American Options

23.5 Option Pricing Theory and Capital Structure

In our analysis of options, we have assumed the options are European and thus may only be exercised on the expiration date, and that no dividends are paid on the stock before the expiration date. Merton (1973) has shown that if the stock does not pay dividends, it is suboptimal to exercise an

In this section we show how option pricing theory can be used to analyze the value of the components of a firm’s capital structure. Suppose that management of an unlevered corporation decides to issue bonds. For simplicity, we assume these to be zero coupon bonds with face value B to be paid

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at the maturity date.3 When the bonds mature, bondholders must be paid an amount B, if possible. If this money cannot be raised, bondholders have a first claim on the firm’s assets and will be paid the firm’s entire value, Vf , which in this case will be less than B. Let us look at this arrangement from the point of view of the firm’s stockholders. By using debt, stockholders can be regarded as having sold the firm to bondholders, with an option to purchase it back by paying an amount B at the maturity date of the bonds. Thus, the stockholders hold a call option on the firm, with expiration date at the date of maturity of the bonds. If the firm’s value exceeds the value of the debt on the expiration date, the debt will be retired, so that stockholders will have exercised their call option. On the other hand, if the firm’s value is less than the face value of the debt, stockholders’ limited liability allows them to leave the firm in the hands of the bondholders. The call option will not be exercised, so that the stock will be worth nothing. However, stockholders will have no further obligations to bondholders. Therefore, on the date of expiration, the value of this call option, or, equivalently, the value of stockholders’ equity, will be V D Vf  B if Vf > B and 0 if Vf < B. Thus, we can write   V D MAX 0; Vf  B

This analysis indicates that the stockholders own a call option on the firm’s assets. At any time, they could exercise this option by delivering the face value of the bonds to pay off the bondholders and realize the value of their option. This point is illustrated in the following example, where we employ the Black-Scholes model. Example 23.2. An unlevered corporation is valued at $14 million. The corporation issues debt, payable in 6 years, with a face value of $10 million. The standard deviation of the continuously compounded rate of return on the total value of this corporation is .2. Assume that the risk-free rate of interest is 8% annually. In our previous notation, the current market price of the asset (in this case, the firm) is $14 million, and the exercise price of the option is the face value of the debt. Therefore, we have P D 14; E D 10; r D :08; t D 6, and  D :2, where we are measuring value in units of a million dollars. We will use the Black-Scholes formula, Equation (23.1), to compute the value of stockholders’ equity; that is, the value of the stockholders’ call option on the firm. First, we require the natural logarithm of P=E, so that

(23.3)

where V D value of stockholders’ call option on the firm; Vf D value of the firm; and B D face value of debt. This relationship is shown in Fig. 23.7. As we can see, this is exactly the relationship shown in Fig. 23.1a for the value of a call option. We see, then, that the theory of options can provide us with insights into the valuation of debt and stockholders’ equity.

ln

P E

D ln.1:4/ D :3365:

Next, we find d1 D

D

ln

P  E

C rt C  2 2t p  t

:3365 C .:08/ .6/ C 12 .:2/ .6/ p D 1:91 :2 6

and d2 D

D

ln

P  E

C rt   2 2t p  t

:3365 C .:08/ .6/  12 .:2/ .6/ p D 1:42 :2 6

From tabulated values of the cumulative distribution function of the standard normal random variable, we find N .d1 / D N .1:91/ D :9719 and Fig. 23.7 Option approach to capital structure

N .d2 / D N .1:42/ D :9222: We also require

3

These bonds will sell for less than the face value, the difference reflecting interest to be paid on the loan.

e rt D e .:08/.6/ D :6188:

23 Option Pricing Theory and Firm Valuation

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On substitution in Equation (23.1), we find the value of stockholders’ equity V D PN.d1 /  e rt EN.d2 / D .14/.:9719/  .:6188/.10/.:9222/ D $7:90 million Since the total value of the firm is $14 million, the value of the debt is value of debt D $14  $7:9 D $6:10 million: Assuming the market is efficient, this firm could sell debt with a face value of $10 million for $6.1 million. These receipts could be distributed to stockholders, leaving them with an equity worth $7.9 million. We now turn to examine the effects of two factors on our calculations.

23.5.1 Proportion of Debt in Capital Structure Let us compare our results of Example (23.3) with an otherwise identical situation, but where now only $5 million face value of debt is to be issued. With E D 5 so that P=E D 2:8, we find In(P/E) to be 1.0296. Using the same procedure as in Example (23.3), d1 D 3:33 and d2 D 2:84, so that N .d1 / D :9996 and N .d2 / D :9977. Using Equation (23.1), the value of stockholders’ equity is V D PN.d1 /  e rt EN.d2 / D .14/.:9996/  .:6188/.5/.:9977/ D $10:91 million Therefore, the value of the debt is value of debt D 14  10:91 D $3:09 million: In Table 23.4, we compare, from the point of view of bondholders, the cases where the face values of issued debt are $5 million and $10 million. We see from the table that increasing the proportion of debt in the capital structure decreases

Table 23.4 Effect of different levels of debt on debt value Actual value per dollar debt face Face value of debt Actual value of debt value of debt ($ millions) ($ millions) 5 10

3.09 6.10

$.618 $.610

the value of each dollar of face value of debt. If all the debt is issued at one time, this phenomenon, which simply reflects the increase in bondholders’ risk as debt increases, will result in the demand for correspondingly higher interest rates. As our example illustrates, if the corporation sells only $5 million of face-value bonds, a price of $618,000 per million dollars in face value will be paid for these bonds. However, if bonds with face value of $10 million are to be issued, this price will fall to $610,000. In such a case, the market for bonds operates such that the risk to bondholders of future default is considered in establishing the price of the zero coupon bonds, or, more generally the interest rate attached to any bonds. Suppose, however, that our company issues bonds with face value of $5 million, at a cost to bond purchasers of $3.09 million. One year later, this corporation decides to issue more debt. In valuing this new debt, potential purchasers of bonds will assess the risk of default. One factor taken into consideration will be the total level of debt of the corporation; that is, the amount of existing debt as well as the size of the new issue. Hence, the market price for the new bonds will reflect riskiness. Consider, however, the position of existing bondholders. The interest rates to be paid to these holders of old debt has already been established. The issue of new debt is going to increase the chance of default on all loans. Consequently, the value of existing bonds must fall when new bonds are issued. Therefore, the issue of new bonds entails a decrease in the wealth of existing bondholders. The beneficiaries are the stockholders of the company whose wealth increases by a corresponding amount. We see, then, that there is a conflict of interest between existing bondholders and holders of common stock. All other things equal, existing bondholders will prefer that no further debt be issued, while stockholders will prefer to see more debt issued. This provides an illustration of the agency problem discussed in Chap. 23. Purchasers of bonds will require covenants restricting the freedom of action of corporate management in order to protect the value of their investment.

23.5.2 Riskiness of Business Operations Let us return to the corporation of Example 23.2, which is about to issue debt with face value of $10 million. Leaving the other variables unchanged, suppose that the standard deviation of the continuously compounded rate of return on the corporation’s total value is .4, rather than .2. This implies that the corporation is operating in an environment of greater business risk. Setting  2 D :4, with all other relevant variables as specified in Example 23.2, we find d1 D 1:32 and d2 D :34, so that N .d1 / D :9066 and N .d2 / D :6331. From

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Table 23.5 Effect of different levels of business risk on the value of $10 million face value of debt Variance of rate of Value of equity Value of debt return ($ millions) ($ millions) .2 .4

7.90 8.77

6.10 5.23

Equation (23.1), we find the value of stockholders’ equity to be V D PN.d1 /  e rt EN.d2 / D .14/.:9066/  .:6188/.10/.:6331/ D $8:77 million Thus, the value of the debt is value of debt D 14  8:77 D $5:23 million: Table 23.5 summarizes the comparison between these results and those of Example 23.2. We see that this increase in business risk, with its associated increase in the probability of default on the bonds, leads to a reduction from $6.10 million to $5.23 million in the market value of the $10 million face value of debt. To the extent that potential purchasers of bonds are able to anticipate the degree of business risk, the higher interest rate on the bonds reflects the risk involved. The greater the degree of risk, the higher the interest rate that must be offered to sell a particular amount of debt. However, suppose that having issued debt, corporate management embarks on new projects with a higher level of risk than could have been foreseen by the purchasers of the bonds. As we have just seen, this will lower the value of existing bonds. This decrease in bondholders’ wealth accrues as a gain in wealth to holders of common stock. Once again, we find a conflict of interest between stockholders and bondholders. Once debt has been sold, it will be in the interests of stockholders for the firm to operate with a high degree of business risk, while bondholders will prefer lower levels of risk. Thus, the degree of business risk represents another example of an agency problem. Bondholders will want protection against the possibility that management will take on riskier than anticipated projects, and will demand protective covenants against such actions. Because of this factor, the issue of a large volume of debt is likely to be accompanied by constraints on the freedom of management action in the firm’s operation.

23.6 Warrants A warrant is an option issued by a corporation to individual investors to purchase, at a stated price, a specified number of

the shares of common stock of that corporation. Warrants are issued in two principal sets of circumstances: 1. In raising venture capital, either to start a new company or to substantially increase the scope of operations of an existing company, warrants are often issued to lenders as an additional inducement to the promised interest payments or to purchasers of new stock issues. 2. Often, when issuing bonds, a company will increase the bonds’ attractiveness by attaching warrants. Thus, as well as receiving interest payments on the bonds, their purchasers obtain an option to buy stock in the corporation at a specified price. As we have seen, such options will have some value, and so their attachment to bonds should lead to a lowering of the interest rate paid on a fixed quantity of bonds or an increase in the number of bonds that can be sold at a given interest rate. Since a warrant is essentially a call option on a specified number of shares of common stock, the principles underlying the valuation of call options are also applicable to the valuation of warrants. Suppose that a warrant entitles its holder to purchase N shares of common stock at a total cost E. If, at the expiration date, the price per share of common stock is P , then shares of total value NP can be purchased at cost E. If NP exceeds E, the option to purchase will be exercised, and the difference between these quantities is the value of the warrant. On the other hand, if NP is less than E, it will not pay to exercise the option, so that the warrant will be worthless. The warrant’s value on the expiration date can then be expressed as Vw D MAX .0; NP  E/ where Vw D value of warrant; N D number of shares that can be purchased; P D market price per share of stock; and E D exercise price for the purchase of N shares of stock. Prior to the expiration date, for reasons discussed earlier in this chapter, the warrant’s value will exceed this theoretical value. The same factors affecting the value of an ordinary call option are also relevant here. Thus, all other things equal warrant value will increase with increases in the volatility of the stock price, in the risk-free interest rate, and in the time to the expiration date. However, the basic version of the BlackScholes model generally will not be directly applicable, as it pertains to the valuation of warrants. The reason is that, generally, warrants differ in important respects from ordinary call options. These factors are as follows: 1. The life of a call option typically is just a few months. However, the life of a warrant is several years. While it may not be unreasonable to expect volatility of rate of return to remain constant for a few months, it is less likely that it will do so for several years.

23 Option Pricing Theory and Firm Valuation

2. Over a period of several years, it is likely that dividends will be paid. As we have seen, dividend payments, which do not accrue to warrantholders, reduce the value of options to purchase stock. 3. For many warrants, the exercise price is not fixed to the expiration date, but changes at designated points in time. It may well pay to exercise the option to purchase shares immediately before such a change. 4. It may be that the number of shares that all warrantholders are entitled to purchase represents a considerable fraction of the total number of shares of the corporation. Thus, if these options are all exercised, and total earnings are unaffected, then earnings per share will be diluted. Let us look at this last point in more detail. Since by using warrants, a corporation can extract more favorable terms from bondholders, it follows that, in return, the company must have transferred something of value to these bondholders. This transfer can be visualized as giving the bondholders a stake in the corporation’s equity. Thus, we should regard equity comprising both stockholdings and warrant value. We will refer to this total equity, prior to the exercise of the options, as old equity so that Old Equity D Stockholders’ Equity C Warrants: Suppose that the warrants are exercised. The corporation then receives additional money from the purchase of new shares, so that total equity is New Equity D Old Equity C Exercise Money: We denote by N the number of shares outstanding and by Nw the number of shares that warrant holders can purchase. If the options are exercised, there will be a total of N C Nw shares, a fraction Nw =.Nw C N / of which is owned by former warrantholders. These holders then own this fraction of the new equity; that is H.New Equity/ D H.Old Equity/ C H.Exercise Money/ w . where H D NwNCN Thus, a fraction, Nw =.Nw C N /, of the exercise money is effectively returned to the former warrant holders. In fact, they have really spent Œ1Nw =.Nw CN / D N=.Nw CN / of the exercise money to acquire a fraction, Nw =.Nw C N /, of the old equity. Therefore, in valuing the warrants, the BlackScholes formula must be modified. We need to make the appropriate substitutions in Equation (23.1) for the current stock price, P , and the exercise price of the option, E.

P D

Nw Nw C N

.Value of old equity/

391

ED

N Nw C N

.Exercise Money/

It also follows that the appropriate measure of volatility,  2 , is the variance of rate of return on the total old equity (including the value of warrants), not simply on stockholders’ equity. Suppose that a firm has one million shares outstanding, currently selling at $100 per share. There are also 500,000 warrants with an exercise price of $80 per share. The warrants are worth $20, or the current stock price, $100, less the exercise price of $80. The value of the old equity is Old Equity D $100 .1m/ C $20 .0:5m/ D $110 million: If the warrants are exercised, the firm will receive $40 million .$80 :5 m/ of new equity, so that the new equity is New Equity D $110m C $40m D $150 million: When they exercise their warrants, the warrantholders will own one-third of the shares outstanding; that is, H D

Nw Nw C N

or

500;000 500;000 C 1;000;000

and the old shareholders will own the remaining two-thirds of the shares outstanding. The warrantholders now have an investment worth $50 million, or 1 1 1 .$150m/ D .$110m/ C .$40m/ D 50 million: 3 3 3 It makes sense for the warrantholders to exercise their warrant; they spend $40 million for shares that are worth $50 million. In terms of the warrant value, the market should be willing to pay $20 per warrant for 500,000 warrants, or $10 million. A convertible bond is a security that gives its owner the right to exchange it for a given number of shares of common stock any time before the maturity date of the bond. Hence, a convertible bond is actually a portfolio of two securities: a bond and a warrant. The value of a convertible bond is the value of the bond portion of the portfolio plus the value of the warrant.

23.7 Conclusion In Chap. 23, we have discussed the basic concepts of call and put options and have examined the factors that determine the value of an option. One procedure used in option valuation is

392

the Black-Scholes model, which allows us to estimate option value as a function of stock price, option-exercise price, timeto-expiration date, and risk-free interest rate. The option pricing approach to investigating capital structure is also discussed, as is the value of warrants.

References Amram, M. and N. Kulatilaka. 2001. Real options, Oxford University Press, USA. Banz R. and M. Miller. 1978. “Prices for state contingent claims: some estimates and applications.” Journal of Business 51, 653–672. Bhattachayra, M. 1980. “Empirical properties of the Black-Scholes formula under ideal conditions.” Journal of Financial and Quantitative Analysis 15, 1081–1105. Black, F. 1972. “Capital market equilibrium with restricted borrowing.” Journal of Business 45, 444–445. Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 31, 637–659. Bookstaber, R. M. 1981. Option pricing and strategies in investing, Addison-Wesley, Reading, MA. Cox, J. C. and M. Rubinstein. 1985. Option markets, Prentice-Hall, Englewood Cliffs, NJ. Finnerty, J. 1978. “The Chicago board options exchange and market efficiency.” Journal of Financial and Quantitative Analysis 13, 29–38.

C.F. Lee et al. Galai, D. and R. W. Masulis. 1976. “The option pricing model and the risk factor of stock.” Journal of Financial Economics 3, 53–81. Hull, J. 2005. Options, futures, and other derivatives, 6th Edition, Prentice Hall, Upper Saddle River, NJ. Jarrow R. A. and S. Turnbull. 1999. Derivatives securities, 2nd Edition, South-Western College Pub, Cincinnati, OH. Lee, C. F. 2009. Handbook of quantitative finance and risk management, Springer, New York. Lee, C. F. and A. C. Lee. 2006. Encyclopedia of finance, Springer, New York. Liaw, K. T. and R. L. Moy. 2000. The Irwin guide to stocks, bonds, futures, and options, McGraw-Hill Companies, New York. MacBeth, J. and L. Merville. 1979. “An empirical examination of the Black-Scholes call option pricing model.” The Journal of Finance 34, 1173–1186. McDonald, R. L. 2005. Derivatives markets, 2nd Edition, Addison Wesley, Boston, MA. Rendleman, R. J., Jr. and B. J. Barter. 1979. “Two-state option pricing.” Journal of Finance 24, 1093–1110. Ritchken, P. 1987. Options: theory, strategy, and applications, Scott, Foresman, Glenview, IL. Summa, J. F. and J. W. Lubow. 2001. Options on futures. John Wiley & Sons, New York. Trennepohl, G. 1981. “A comparison of listed option premia and Black-Scholes model prices: 1973–1979.” Journal of Financial Research, 11–20. Zhang, P. G.. 1998. Exotic options: a guide to second generation options, 2nd Edition, World Scientific Pub Co Inc, Singapore.

Chapter 24

Applications of the Binomial Distribution to Evaluate Call Options Alice C. Lee, John Lee, and Jessica Shin-Ying Mai

Abstract In this chapter, we first introduce the basic concepts of call and put options. Then we show how the simple one period binominal call option pricing model can be derived. Finally, we show how a generalized binominal option pricing model can be derived. Keywords Binomial distribution r Option r Simple binomial option pricing model r Generalized binomial option pricing model r Hedge ratio r Cumulative binomial function r Exercise price r Decision tree

purchaser is under no obligation to buy; it is, indeed, an “option.” This attribute of an option contract distinguishes it from other financial contracts. For instance, whereas the holder of an option may let his or her claim expire unused if he or she so desires, other financial contracts (such as futures and forward contracts) obligate their parties to fulfill certain conditions. A call option gives its owner the right to buy the underlying security, a put option the right to sell. The price at which the stock can be bought (for a call option) or sold (for a put option) is known as the exercise price.

24.1 Introduction

24.3 The Simple Binomial Option Pricing Model In this chapter, we will show that how the binomial distribution can be used to derive the call option pricing model. In the second section of this chapter, we will define the basic definition of option. In the third section, we will define and examine the simple binominal option pricing model. In the forth section, we will derive the generalized n period binominal option pricing model. Finally, in the fifth section, we will summarize our findings and make concluding remarks.

24.2 What Is an Option? In the most basic sense, an option is a contract conveying the right to buy or sell a designated security at a stipulated price. The contract normally expires at a predetermined date. The most important aspect of an option contract is that the

A.C. Lee State Street Corp., Boston, MA, USA

Before discussing the binomial option model, we must recognize its two major underlying assumptions. First, the binomial approach assumes that trading takes place in discrete time; that is, on a period-by-period basis. Second, it is assumed that the stock price (the price of the underlying asset) can take on only two possible values each period; it can go up or go down. Say we have a stock whose current price per share S can advance or decline during the next period by a factor of either u (up) or d (down). This price either will increase by the proportion u  1 0 or will decrease by the proportion 1  d; 0 < d < 1. Therefore, the value S in the next period will be either uS or dS. Next, suppose that a call option exists on this stock with a current price per share of C and an exercise price per share of X and that the option has one period left to maturity. This option’s value at expiration is determined by the price of its underlying stock and the exercise price X . the value is either

J. Lee Center for PBBEF Research, New York, NY, USA J.S.-Y. Mai () Rutgers University, Newark, NJ, USA e-mail: [email protected]

Cu D Max .0; uS  X/

(24.1)

Cd D Max .0; dS  X/

(24.2)

Or

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Why is the call worth Max (0, uS – X) if the stock price is uS? The option holder is not obliged to purchase the stock at the exercise price of X, so she or he will exercise the option only when it is beneficial to do so. This means the option can never have a negative value. When is it beneficial for the option holder to exercise the option? When the price per share of the stock is greater than the price per share at which he or she can purchase the stock by using the option, which is the exercise price, X. Thus, if the stock price uS exceeds the exercise price X, the investor can exercise the option and buy the stock. Then he or she can immediately sell it for uS, making a profit of uS – X (excluding commission). Likewise, if the stock price declines to dS, the call is worth Max (0, dS – X). Also for the moment, we will assume that the risk-free interest rate for both borrowing and lending is equal to r percent over the one time period and that the exercise price of the option is equal to X. To intuitively grasp the underlying concept of option pricing, we must set up a risk-free portfolio – a combination of assets that produces the same return in every state of the world over our chosen investment horizon. The investment horizon is assumed to be one period (the duration of this period can be any length of time, such as an hour, a day, a week, etc.). To do this, we buy h share of the stock and sell the call option at its current price of C. Moreover, we choose the value of h such that our portfolio will yield the same payoff whether the stock goes up or down. h .uS/  Cu D h .dS/  Cd

(24.3)

By solving for h, we can obtain the number of shares of stock we should buy for each call option we sell. hD

Cu  Cd .u  d /S

(24.4)

Here h is called the hedge ratio. Because our portfolio yields the same return under either of the two possible states for the stock, it is without risk and therefore should yield the risk-free rate of return, r percent, which is equal to the riskfree borrowing and lending rate. In essence this condition must be true; otherwise, it would be possible to earn a riskfree profit without using any money. Therefore, the ending portfolio value must be equal to .1 C r/ times the beginning portfolio value, hS – C. .1 C r/ .hS  C/ D h .uS/  Cu D h .dS/  Cd

 C D

Rd ud



Cu C

 uR Cd R ud

where d < r < u. To simplify this equation, we set pD

Rd so 1  p D ud



uR ud

 (24.7)

Thus we get the option’s value with one period to expiration: C D ŒpCu C .1  p/Cd =R

(24.8)

This is the binomial call option valuation formula in its most basic form. In other words, this is the binomial valuation formula with one period to expiration of the option. To illustrate the model’s qualities, let’s plug in the following values, while assuming the option has one period to expiration. Let X D $100 S D $100 U D .1:10/: so uS D $110 D D .:90/; so dS D $90 R D 1 C r D 1 C :07 D 1:07 First we need to determine the two possible option values at maturity, as indicated in Table 24.1. Next, we calculate the value of p as indicated in Equation (24.7). pD

1:07  :90 1:10  1:07 D :85 so 1  p D D :15 1:10  :90 1:10  :90

Solving the binomial valuation equation as indicated in Equation (24.8), we get C D Œ:85.10/ C :15.0/ =1:07 D $7:94 Table 24.1 Possible option value at maturity l Stock (S)

Option (C)

Next period (maturity) uS D $110

Cu D Max .0; uS  X/ D Max .0; 110  100/ D Max .0; 10/ D $10

dS D $90

Cd D Max .0; dS  X/ D Max .0; 90–100/ D Max .0; 10/ D $0

(24.5) $100

Note that S and C represent the beginning values of the stock price and the option price, respectively. Setting R D 1 C r, rearranging to solve for C, and using the value of h from Equation (24.4), we get

(24.6)

C

24 Applications of the Binomial Distribution to Evaluate Call Options

395

The correct value for this particular call option today, under the specified conditions, is $7.94. If the call option does not sell for $7.94, it will be possible to earn arbitrage profits. That is, it will be possible for the investor to earn a risk-free profit while using none of his or her own money. Clearly, this type of opportunity cannot continue to exist indefinitely.

190.61

162.22

137.89

138.06

137.89

117.35

117.50

99.75 137.89

117.35

24.4 The Generalized Binomial Option Pricing Model

99.75

99.88

99.75

84.90

Suppose we are interested in the case where there is more than one period until the option expires. We can extend the one-period binomial model to consideration of two or more periods. Because we are assuming that the stock follows a binomial process, from one period to the next it can only go up by a factor of u or go down by a factor of d . after one period the stock’s price is either uS or dS. Between the first and second periods, the stock’s price can once again go up by u or down by d , so the possible prices for the stock two periods from now are uuS, udS, and ddS. This process is demonstrated in tree diagram from (Fig. 24.1) in Example 24.1 later in this appendix. Note that the option’s price at expiration, two periods from now, is a function of the same relationship that determined its expiration price in the one-period model, more specifically, the call option’s maturity value is always

72.16

$100.00

137.89

117.35

99.75

99.88

99.75

84.90

72.16

85.00

99.75

84.90

72.16

72.25

72.16

61.41

52.20

11 0

1

2

3

4

(24.9)

Fig. 24.1 Price path of underlying stock (Source: Rendelman, R. J., Jr. and B. J. Bartter. 1979. “Two-state option pricing.” Journal of Finance 34(5), 1093–1110.)

where T designated the maturity date of the option. To derive the option’s price with two periods to go .T D 2/, it is helpful as an intermediate step to derive the value of Cu and Cd with one period to expiration when the stock price is either uS or dS, respectively.

In Equation (24.12), we used the fact that Cud D Cdu because the price will be the same in either case. Following Equations (24.10) and (24.11), we can obtain

CT D Max Œ0; ST  X

Cu D ŒpCuu C .1  p/Cud =R

Cuu D ŒpCuuu C .1  p/Cuud =R

(24.13)

(24.10)

Cd D ŒpCd u C .1  p/Cd d =R

Cdd D ŒpCddu C .1  p/Cddd =R

(24.14)

(24.11)

Cud D ŒpCudu C .1  p/Cudd =R

(24.15)

Equation (24.10) tells us that if the value of the option after one period is Cu , the option will be worth either Cuu (if the stock price goes up) or Cud (if stock price goes down) after one more period (at its expiration date). Similarly, Equation (24.11) shows that the value of the option is Cd after one period, the option will be worth either Cdu or Cdd at the end of the second period. Replacing Cu and Cd in Equation (24.8) with their expressions in Equations (24.10) and (24.11), respectively, we can simplify the resulting equation to yield the two-period equivalent of the one-period binomial pricing formula, which is ı  C D p 2 Cuu C 2p.1  p/Cud C .1  p/2 Cd d R2 (24.12)

Substituting Cuu ; Cud and Cdd in Equation (24.12), we can obtain the three-period option pricing model  C D p 3 Cuuu C 3p 2 .1  p/Cuud C3p.1  p/2 Cudd C .1  p/3 Cddd

ı

R3 (24.16)

Equation (24.16) can also be written in terms of a generalized form as follows: C D

3 1 X 3Š 3 R kŠ.3  k/Š kD0

p .1  p/3k MaxŒ0; uk d 3k S  K k

(24.17)

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C1 D Max Œ0; .1:1/3 .:90/0 .100/  100 D 33:10

Following Equations (24.13), (24.14), and (24.15), we can obtain Cuuu D ŒpCuuuu C .1  p/Cuuud =R

(24.18)

C3 D Max Œ0; .1:1/ .:90/2 .100/  100 D 0

Cddd D ŒpCdddu C .1  p/Cdddd =R

(24.19)

C4 D Max Œ0; .1:1/0 .:90/3 .100/  100 D 0

Cuud D ŒpCuudu C .1  p/Cuudd =R

(24.20)

Cudd D ŒpCuddu C .1  p/Cuddd =R

(24.21)

Substituting Cuuu ; Cuud ; Cudd and Cddd in Equation (24.16), we can obtain the four-period option pricing model C D Πp 4 Cuuuu C 4p 3 .1  p/Cuuud C 6p 2 .1  p/2 Cuudd C 4p.1  p/3 Cud d d C .1  p/4 Cd d d d =R4 (24.22) Equation (24.22) can also be written in terms of a generalized form as follows: C D

4 4Š 1 X R4 kŠ.4  k/Š kD0

p .1  p/ k

4k

k

MaxŒ0; u d

4k

S  K

(24.23)

We know the values of the parameters S and X. If we assume that R, u, and d will remain constant over time, the possible maturity values for the option can be determined exactly. Thus deriving the option’s fair value with two periods to maturity is a relatively simple process of working backwards from the possible maturity values. Finally, using this same procedure of going from a oneperiod model, two-period model, three-period model and to a four-period model, we can extend the binomial approach to its more generalized form, with n-period maturity: C D

C2 D Max Œ0; .1:1/2 .:90/ .100/  100 D 8:90

1 Rn

n X kD0

nŠ kŠ.n  k/Š

p k .1  p/nk M axŒ0; uk d nk S  X

(24.24)

To actually get this form of the binomial model, we could extend the two-period model to three periods, then from three periods to four periods, and so on. Equation (24.24) would be the result of these efforts. To show how Equation (24.24) can be used to assess a call option’s value, we modify the example as follows: S D $100; X D $100; R D 1:07; n D 3; u D 1:1 and d D :90. First we calculate the value of p from Equation (24.7) as.85, so 1  p is .15. Next, we calculate the four possible ending values for the call option after three periods in terms nk of Max Œ0; uk d S  X .

Now we insert these numbers (C1 ; C2 ; C3 , and C4 ) into the model and sum the terms.  3Š 3Š 1 .:85/0 .:15/3 X 0 C .:85/1 .:15/2 X 0 C D 3 .1:07/ 0Š3Š 1Š2Š  3Š 3Š 2 1 3 0 .:85/ .:15/ X 8:90 C .:85/ .:15/ X 33:10 C 2Š1Š 3Š0Š  3X 2X1 1 0C0C .:7225/.:15/.8:90/ D 1:225 2X1X1  3X 2X1 C X.:61413/.1/.33:10/ 3X 2X1X1 1 Œ.:32513X 8:90/ C .:61413X 33:10/ 1:225 D $18:96 D

As this example suggests, working out a multiple-period problem by hand with this formula can become laborious as the number of periods increases. Fortunately, programming this model into a computer is not too difficult. Now let’s derive a binomial option pricing model in terms of the cumulative binomial density function. As a first step, we can rewrite Equation (24.24) as # k nk nŠ K nk u d p .1  p/ C DS kŠ.n  K/Š Rn kDm " n # X X nŠ (24.25) p k .1  p/nk  n R kŠ.n  k/Š "

n X

kDm

This formula is identical to Equation (24.24) except that we have removed the Max operator. In order to remove the Max operator, we need to make uk d nk S  X positive, which we can do by changing the counter in the summation from k D 0 to k D m. What is m? It is the minimum number of upward stock movements necessary for the option to terminate “in the money” (that is, uk d nk S  X > 0). How can we interpret Equation (24.25)? Consider the second term in brackets; it is just a cumulative binomial distribution with parameters of n and p. Likewise, via a small algebraic manipulation we can show that the first term in the brackets is also a cumulative binomial distribution. This can be done by defining P 0  .u=R/p and 1  P 0  .d =R/.1  p/. Thus P k .1  p/nk

uk d nk D p rk .1  p 0 /nk Rn

24 Applications of the Binomial Distribution to Evaluate Call Options

Therefore, the first term in brackets of Equation (24.25) is also a cumulative binomial distribution with parameters of n and p 0 . Using the definition of cumulative binomial function, we can write the binomial call option model as C D SB1 .n; p 0 ; m/ 

X B2 .n; p; m/ Rn

(24.26)

397

if the stock went down in the first period, it can go down again to $72.25 or up in the second period to $99.88. Using the same argument, we can trace the path of the stock’s price for all four periods. If we are interested in forecasting the stock’s price at the end of period 4, we can find the average price of the stock for the 16 possible outcomes that can occur in period 4.

where

16 P

B1 .n; p ; m/ D 0

n P kDm

B2 .n; p; m/ D

n P kDm

Ckn p 0k .1

p /

0 nk

P D

Example 24.1. A Decision Tree Approach to Analyzing Future Stock Price By making some simplifying assumptions about how a stock’s price can change from one period to the next, it is possible to forecast the future price of the stock by means of a decision tree. To illustrate this point, let’s consider the following example. Suppose the price of Company A’s stock is currently $100. Now let’s assume that from one period to the next, the stock can go up by 17.5% or go down by 15%. In addition, let us assume that there is a 50% chance that the stock will go up and a 50% chance that the stock will go down. It is also assumed that the price movement of a stock (or of the stock market) today is completely independent of its movement in the past; in other words, the price will rise or fall today by a random amount. A sequence of these random increases and decreases is known as a random walk. Given this information, we can lay out the paths that the stock’s price may take. Figure 24.1 shows the possible stock prices for company A for four periods. Note that in period 1 there are two possible outcomes: the stock can go up in value by 17.5% to $117.50 or down by 15% to $85.00. In period 2 there are four possible outcomes. If the stock went up in the first period, it can go up again to $138.06 or down in the second period to $99.88. Likewise,

16

D

190:61 C 137:89 C : : : C 52:20 16

D $105:09

Ckn p k .1  p/nk

and m is the minimum amount of time the stock has to go up for the investor to finish in the money (that is, for the stock price to become larger than the exercise price). In this chapter, we showed that by employing the definition of a call option and by making some simplifying assumptions, we could use the binomial distribution to find the value of a call option. In Chap. 26, we will show how the binomial distribution is related to the normal distribution and how this relationship can be used to derive one of the most famous valuation equations in finance, the Black-Scholes option pricing model.

Pi

i D1

We can also find the standard deviation for the stock’s return. 

.190:61  105:09/2 C : : : C .52:20  105:09/2 P D 16

1=2

D $34:39 P and P can be used to predict the future price of stock A.

24.5 Conclusion In this chapter, we demonstrate how can an option pricing model be derived in a less mathematically fashion using binominal option pricing model. We have first discussed the basic concepts of call options then we show how the decision trees can be used to derive binominal call option pricing model. The binominal call option pricing model can be used to derived Black-Scholes Option Pricing Model (1973) as shown by Rendleman and Barter (RB 1979), and Cox et al. (CRR 1979), which we will discuss in detail in the next chapter.

References Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 31, 637–659. Cox, J. C. and M. Rubinstein. 1985. Option markets, Prentice-Hall, Englewood Cliffs, NJ. Cox, J., S. A. Ross, and M. Rubinstein. 1979. “Option pricing: a simplified approach.” Journal of Financial Economics 7, 229–263. Hull, J. 2005. Options, futures, and other derivatives, 6th Edition, Prentice Hall, Upper Saddle River, NJ. Jarrow R. A. and S. Turnbull. 1999. Derivatives securities, 2nd Edition, South-Western College Pub, Cincinnati, OH. Lee, C. F. 2009. Handbook of quantitative finance and risk management, Springer, New York. Lee, C. F. and A. C. Lee. 2006. Encyclopedia of finance, Springer, New York.

398 Lee, C. F., J. C. Lee, and A. C. Lee. 2000. Statistics for business and financial economics, World Scientific Publishing Co. Singapore. Lee, Jack C., C. F. Lee, R. S. Wang, and T. I. Lin. 2004. “On the limit properties of binomial and multinomial option pricing models: review and integration.” Advances in Quantitative Analysis of Finance and Accounting New Series Volume 1, World Scientific, Singapore, Vol. 1. Lee, John C. 2001. “Using microsoft excel and decision trees to demonstrate the binomial option pricing model.” Advances in Investment Analysis and Portfolio Management 8, 303–329.

A.C. Lee et al. MacBeth, J. and L. Merville. 1979. “An empirical examination of the Black-Scholes call option pricing model.” The Journal of Finance 34, 1173–1186. Rendleman, R. J., Jr. and B. J. Barter. 1979. “Two-state option pricing.” Journal of Finance 34(5), 1093–1110.

Chapter 25

Multinomial Option Pricing Model Cheng Few Lee and Jack C. Lee

Abstract In this chapter, we extend the binomial option pricing model to a multinomial option pricing model. Then we derive the multinomial option pricing model and apply it to the limiting case of Black and Scholes model. Finally, we introduce a lattice framework for option pricing model and its application to option valuation.

the current stock price is S, the stock price at the end of the period will be one of fi S ’s. We can represent this movement with the following diagram:

S

Keywords Multinomial process r Black and Scholes Model r Cox r Ross and Rubinstein (CRR) lattice binomial approach r Lattice model

25.1 Introduction Instead of two possible movements for the stock price, as considered by Cox et al. (1979) and Rendleman and Barter (1979), it is natural to extend it to the situation in which there are k C 1 possible price movements. In this section we will present the extension proposed by Madan et al. (1989). More details are provided and should be helpful to the readers. We will derive the multinomial option pricing model in Sect. 25.2 and the Black and Scholes model as a limiting case in Sect. 25.3.

C.F. Lee () Rutgers University, Newark, NJ, USA e-mail: [email protected] J.C. Lee National Chiao Tung University, Hsinchu, Taiwan

f2S

w. p. q2

⋅

⋅

⋅

⋅ w. p. qk+1

where w.p. denotes with probability. We need the following definitions and notations in our presentation. Let X D .x1 ; x2 ; : : : xk ; xkC1 /T , and q D .q1 ; q2 ; : : : qk ;



qkC1 /T ; 0 < qj < 1; j D 1; 2; : : : k C 1. X is said to have

the multinomial distribution, or X  Mult.n; q /, if and only



if the joint probability density is f .X / D f .x1 ; x2 ; : : : ; xkC1 /

nŠ D q x1 q2 : : : qkC1 xkC1 x1 Šx2 Š : : : xkC1 Š 1 x2  kC1  Q xj n D q x1 ; x2 ; : : : xkC1 j D1 j for all 0  xj  n, where

kC1 P j D1

1 T q D 1, where 1

Suppose that the stock price follows a multiplicative multinomial process over discrete periods. The rate of return on the stock over each period can have .k C 1/ possible values: fi  1 with probability qi ; i D 1; : : : k C 1. Thus, if

w. p. q1

fk+1S

25.2 Multinomial Option Pricing Model 25.2.1 Derivation of the Option Pricing Model

f1S





.kC1/

xj D 1 T X D n and



(25.1)

kC1 P

qj D

j D1

W .k C 1/-dimensional vector of unit

entries. Let C D the current value of the n-period option price; K D the option exercise price; S  D the stock price at the end of the n-period; n D the number of periods to maturity; and rO D one plus the riskless rate per period. The following theorem gives the option price for the multinomial case. This theorem extends the option price formula from the binomial case to the multinomial case.

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Theorem 1. The current value of the n-period multinomial option price C is given by C D

X X 2A

n x1 ; x2 ; : : : ; xkC1

Now, let  D .1 ; 2 ; : : : k ; kC1 /T ; j D

If investors were risk-neutral, that is,



kC1 X



.

.fj S /  qj D rO S;

j D1

8 9 kC1 < kC1 Y fj qj xj Y =  K rO n qj S : ; rO j D1

fi qj rO

(25.2)

j D1

or

kC1 P

fj qj D rO , implying

j D1

kC1 P j D1

fi qj rO

D

kC1 P

j D 1 T  D 1,

j D1



then  and q are both the probability vectors in multinomial where A D fX j X D .x1 ; x2 ; : : : ; xkC1 /T ; xj 2 N [





f0g ; 1 T X D n; S > Kg and S and S  are the current



stock price and the stock price at the end of the n-period, respectively.





distribution. So we can rewrite Equation (25.2) as C D SP  .A /  K rO n P q .Aq /; 

(25.3)



where 82 9 3 kC1 n = Y X< 5 4 P  .A / D (25.4) j x j  : x1 ; x2 ; : : : ; xkC1 ;

Proof. Since 0 S D 

f1x1



f2x21



xkC1 fkC1

S D@

kC1 Y

1 x fj j A

X 2A

j D1



 S;

82 9 3 kC1 n = X< Y 5 4 P q .Aq / D qj xj : (25.5)  : x1 ; x2 ; : : : ; xkC1 ;

j D1

C D rO n EŒmax.S   K; 0/ , and

X 2A

j D1



  A D X j 1 T X D n; S  > K ;





we have C D

8 X 0, compute the bivariate normal probability, N2 .a; bI /, as:

y

x where x 0 D x ; y 0 D y y and p is the correlation between x the random variables x 0 and y 0 . The first step in the approximation of the bivariate normal probability N2 .a; bI / is as follows:

a a1 D p 2.1  2 /

The second step in the approximation involves computing the product ab. if ab  0, compute the bivariate normal probability, N2 .a; bI /, using the following rules: 1: If a  0; b  0; and   0; then N2 .a; bI /D '.a; bI / 2: If a  0; b 0; and  > 0; then N2 .a; bI /D N1 .a/  '.a; bI / 3: If a 0; b  0; and  > 0; then N2 .a; bI /D N1 .b/  '.a; bI / 4: If a 0; b 0; and   0; then N2 .a; bI /D N1 .a/ C N1 .b/  1 C '.a; bI / (28.13)

N2 .a; bI / D N2 .a; 0I ab / C N2 .b; 0I ab /  ı

(28.12) where

h 

 f xi0 ; xj0 D exp a1 2xi0  a1 C b1 2xj0  b1

i   C2 xi0  a1 xj0  b1 ;

(28.14)

where the values of N2 ./ on the right-hand side are computed from the rules, for ab  0 .a  b/Sgn.a/ ; ab D p a2  2ab C b 2 ıD

5 X 5

X p '.a; bI /  :31830989 1  2 wi wj f xi0 ; xj0 ; i D1 j D1

b and b1 D p 2.1  2 /

.b  a/Sgn.b/ ba D p a2  2ab C b 2

1  Sgn.a/ Sgn.b/ ; 4 

and Sgn.x/ D

1 x 0 1 x < 0

N1 .d / is the cumulative univariate normal probability.

28.4 Valuating American Options The pairs of weights, (w) and corresponding abscissa values .x 0 / are as follows: i; j 1 2 3 4 5

w 0.24840615 0.39233107 0.21141819 0.033246660 0.00082485334

x0 0.10024215 0.48281397 1.0609498 1.7797294 2.6697604

An American call option whose exercise price is $48 has an expiration time of 90 days. Assume the risk-free rate of interest is 8% annually, the underlying price is $50, the standard deviation of the rate of return of the stock is 20%, and the stock pays a dividend of $2 exactly 50 days; (a) What is the European call value? (b) Can the early exercise price predicted? (c) What is the value of the American call?

432

C.F. Lee et al.

(a) The current stock net price of the present value of the promised dividend is

Since d D 2 > 0:432, therefore, the early exercise is not precluded. (c) The value of the American call is now calculated as

S x D 50  2e 0:08.50=365/ D 48:0218:



i h p C D 48:208 N1 .b1 / C N2 a1 ; b1 I 50=90  48e 0:08.90=365/ N1 .b2 /e 0:08.40=365/ i p CN2 .a2 ; b2 I  50=90

The European call value can be calculated as C D .48:0218/N.d1/  48e 0:08.90=365/ N.d2 /

C2e 0:08.50=365/ N1 .b2 /

where

(28.15)

Œln.48:208=48/ C .:08 C :5.:20/2 /.90=365/ p d1 D :20 90=365 D 0:25285 d2 D 0:292  0:0993 D :15354

Since both b1 and b2 depend on the critical ex-dividend stock price St , which can be determined by C.St ; 40=365I 48/ D St C 2  48; by using trial and error, we find that St D 46:9641. An Excel program used to calculate this value is presented in Fig. 28.1. Substituting Sx D 48:208; X D $48 and St  into Equations (28.9b) and (28.9c) we can calculate a1 a2 b1 and b2 :

From standard normal table, we obtain N.0:25285/ D :5 C :3438 D :599809 N.:15354/ D :5 C :3186 D :561014

a1 D d1 D 0:25285

So the European call value is

a2 D d2 D 0:15354

   48:208  2 50 ln 46:9641 C 0:08 C 0:22 365 b1 D p D 0:4859 .:20/ 50=365

C D .48:516/.0:599809/  48.0:980/.0:561014/ D 2:40123: (b) The present value of the interest income that would be earned by deferring exercise until expiration is

p In addition, we also know  D  50=90 D 0:7454

X.1  e r.T t / / D 48.1  e 0:08.9050/=365 / D 48.1  :991/ D :432



Calculation of St (critical ex-dividend stock price) 46.962 46.963 46.9641 46.9 47

S (critical ex-dividend stock price) X (exercise price of option) r (risk-free interest rate) volatility of stock T -t (expiration date-exercise date) d1

48 0.08 0.2 0.10959

48 0.08 0.2 0.10959

48 0.08 0.2 0.10959

48 0.08 0.2 0.10959

48 0.08 0.2 0.10959

48 0.08 0.2 0.10959

48 0.08 0.2 D .90  50/=365

0.4773

0.1647

0.1644

0.164

0.1846

0.1525

d2

0.5435

0.2309

0.2306

0.2302

0.2508

0.2187

D (dividend) c (value of European call option to buy one share) p (value of European put option to sell one share)  C.S t ; T  tI X/  St  D C X

2 0.60263

2 0.96319

2 0.96362

2 0.9641

2 0.93649

2 0.9798

2.18365

1.58221

1.58164

1.58102

1.61751

1.56081

0.60263

0.00119

0.00062

2.3E-06

0.03649

0.0202

D .LN.C3=C4/ C .C5 C C6^ 2=2/ .C7//=.C6 SQRT.C7// D .LN.C3=C4/ C .C5  C6^ 2=2/ .C7//=.C6 SQRT.C7// 2 D C3 NORMSDIST.C8/  C4 EXP.C5 C7/ NORMSDIST.C9/ D C4 EXP.C5 C7/ NORMSDIST .C9/  C3 NORMSDIST.C8/ D C12  C3  C10 C C4

Fig. 28.1

46

b2 D 0:485931  0:074023 D 0:4119

46

28 Bivariate Option Pricing Models

433

From the above information, we now calculate related normal probability as follows:

and p.S; T I X / maxŒ0; Xe rT  S

(28.16b)

N1 .b1 / D N1 .0:4859/ D 0:6865 respectively, and the lower price bounds for the American call and put options are

N1 .b2 / D N1 .0:7454/ D 0:6598 Following Equation (28.14),We now calculate the value of N2 .0:25285; 0:4859I 0:7454) and N2 .0:15354; 0:4119I 0:7454/ as follows: Since ab¡ > 0 for both cumulative bivariate normal density function, therefore, we can use Equation N2 .a; bI ¡/ D N2 .a; 0I ¡ab / C N2 .b; 0I ¡ba /  • to calculate the value of both N2 .a; bI ¡/ as follows: Œ.0:7454/.0:25285/ C 0:4859 .1/ ab D p 2 .0:25285/  2.0:7454/.0:25285/.0:4859/ C .0:4859/2 D 0:87002 Œ.0:7454/.0:4859/  0:25285 .1/ ba D p .0:25285/2  2.0:7454/.0:25285/.0:4859/ C .0:4859/2 D 0:31979

(28.17a)

P .S; T I X / maxŒ0; Xe rT  S

(28.17b)

and

respectively. The put-call parity relation for non-dividendpaying European stock options is c.S; T I X /  p.S; T I X / D S  Xe rT ;

(28.18a)

and the put-call parity relation for American options on nondividend-paying stocks is S  X  C.S; T I X /  P .S; T I X /  S  Xe rT : (28.18b)

• D .1  .1/.1//=4 D 1=2

N2 .0:292; 0:4859I 0:7454/ D N2 .0:292; 0:0844/ C N2 .0:5377; 0:0656/  0:5 D N1 .0/ C N1 .0:5377/ ˆ.0:292; 0I 0:0844/  ˆ.0:5377; 0I 0:0656/  0:5 D 0:07525 Using a Microsoft Excel program in the following Figures 28.2 and 28.3, we obtain N2 .0:1927; 0:4119I 0:7454/ D 0:06862 Then substituting the related information into the Equation (28.15), we obtain: C D $3:08238 All related results are shown in the second column of Table 28.1.

For non-dividend-paying stock options, the American call option will not rationally be exercised early, while the American put option may be.

28.6 Dividend-Paying Stocks If dividends are paid during the option’s life, the above relations must reflect the stock’s drop in value when the dividends are paid. To manage this modification, we assume that the underlying stock pays a single dividend during the option’s life at a time that is known with certainty. The dividend amount is D and the time to ex-dividend is t. If the amount and the timing of the dividend payment is known, the lower price bound for the European call option on a stock is c.S; T I X / maxŒ0; S  De rt  Xe rT :

28.5 Non-Dividend-Paying Stocks To derive the lower price bounds and the put-call parity relations for options on non-dividend-paying stocks, simply set the cost-of-carry rate, b, equal to the risk-less rate of interest, r. Note that the only cost of carrying the stock is interest. The lower price bounds for the European call and put options are c.S; T I X / maxŒ0; S  Xe rT

C.S; T I X / maxŒ0; S  Xe rT

(28.16a)

(28.19a)

In this relation, the current stock price is reduced by the present value of the promised dividend. Because a Europeanstyle option cannot be exercised before maturity, the call option holder has no opportunity to exercise the option while the stock is selling cum dividend. In other words, to the call option holder, the current value of the underlying stock is its observed market price less the amount that the promised dividend contributes to the current stock value; that is, S Dert . To prove this pricing relation, we use the same arbitrage

434

C.F. Lee et al.

Option Explicit Public Function Bivarncdf (a As Double, b As Double, rho As Double) As Double Dim rho_ab As Double, rho_ba As Double Dim delta As Double If (a  b  rho) D 0 And rho D 0, 1, 1)) / Sqr(a ^ 2  2  rho  a  b C b ^ 2) rho_ba D ((rho  b  a)  IIf(b >D 0, 1, 1)) / Sqr(a ^ 2  2  rho  a  b C b ^ 2) delta D (1  IIf(a >D 0, 1, 1)  IIf(b >D 0, 1, 1)) / 4 Bivarncdf D Bivarncdf(a, 0, rho_ab) C Bivarncdf(b, 0, rho_ba)  delta End If End Function Public Function Phi(a As Double, b As Double, rho As Double) As Double Dim a1 As Double, b1 As Double Dim w(5) As Double, x(5) As Double Dim i As Integer, j As Integer Dim doublesum As Double a1 D a / Sqr(2  (1  rho ^ 2)) b1 D b / Sqr(2  (1  rho ^ 2)) w(1) D 0.24840615 w(2) D 0.39233107 w(3) D 0.21141819 w(4) D 0.03324666 w(5) D 0.00082485334 x(1) D 0.10024215 x(2) D 0.48281397 x(3) D 1.0609498 x(4) D 1.7797294 x(5) D 2.6697604 doublesum D 0 For i D 1 To 5 For j D 1 To 5 doublesum D doublesum C w(i)  w(j)  Exp(a1  (2  x(i)  a1) C b1  (2  x(j)  b1) C 2  rho  (x(i)  a1)  (x(j)  b1)) Next j Next i Phi D 0.31830989  Sqr(1  rho ^ 2)  doublesum End Function Fig. 28.2 Microsoft excel program for calculating function phi

28 Bivariate Option Pricing Models

435

Option Pricing Calculation S(current stock price)D St (critical ex-dividend stock price)D S(current stock price NPV of promised dividend)D X(exercise price of option)D r(risk-free interest rate)D ¢(volatility of stock)D T(expiration date)D t(exercise date)D D(Dividend)D d1(non-dividend-paying)D

50 46.9641 48.0218 48 0.08 0.2 0.24658 0.13699 2 0.65933

d2(non-dividend-paying)D d1 (critical ex-dividend stock price)D

0.56001 0:16401

d2 (critical ex-dividend stock price)D d1(dividend-paying)D

0:23022 0.25285

d2(dividend-paying)D a1 D

0.15354 0.25285

a2 D b1 D

0.15354 0.48593

b2 D

0.41191



D B3–B11 EXP.B7 B10/

D .LN.B3=B6/ C .B7 C 0:5 B8^ 2/ B9/=.B8 SQRT.B9// D B12  B8 SQRT.B9/ D .LN.B4=B6/ C .B7 C 0:5 B8^ 2/ .B9–B10//=.B8  SQRT.B9–B10// D B14  B8 SQRT.B9–B10/ D .LN.B5=B6/ C .B7 C 0:5 B8^ 2/ .B9//=.B8 SQRT.B9// D B16  B8 SQRT.B9/ D .LN..B3–B11 EXP.B7 B10//=B6/ C .B7 C 0:5 B8^ 2/ .B9//=.B8 SQRT.B9// D B18  B8 SQRT.B9/ D .LN..B3–B11 EXP.B7 B10//=B4/ C .B7 C 0:5 B8^ 2/ .B10//=.B8  SQRT.B10// D B20  B8 SQRT.B10/

C.St ; T  tI X/ D

0.9641

C.St ; T  tI X/St D C X D

2.3E-06

D B4 NORMSDIST.B14/  B6 EXP.B7 .B9–B10// NORMSDIST.B15/ D B23  B4  B11 C B6

N1.a1/ D N1.a2/ D N1.b1/ D N1.b2/ D N1.b1/ D N1.b2/ D ¡D a D a1I b D b1 ˆ.a; bI ¡/ D ˆ.a; bI ¡/ D ¡ab D

0.59981 0.56101 0.68649 0.6598 0.31351 0.3402 0:74536

DNORMSDIST(B18) DNORMSDIST(B19) DNORMSDIST(B20) DNORMSDIST(B21) D NORMSDIST.B20/ D NORMSDIST.B21/ D SQRT.B10=B9/

0.20259 0.04084 0.87002

¡ba D

0:31979

N2.a; 0I ¡ab/ D N2.b; 0I ¡ba/ D •D

0.45916 0.11092 0.5

D phi.B20; 0; B37/ D phi.B18; 0; B36/ D ..B32 B18  .B20// IF.B18 >D 0; 1; 1//=SQRT.B 18^ 2  2 B32 B18  B20 C .B20/^ 2/ D ..B32  B20  .B18// IF.B20 >D 0; 1; 1//=SQRT.B18^ 2  2 B32 B18  B20 C .B20/^ 2/ Dbivarncdf(B18,0,B36) D bivarncdf.B20; 0; B37/ D .1  IF.B18 >D 0; 1; 1/ IF.B20 >D 0; 1; 1//=4

a D a2I b D b2 ˆ.a; bI ¡/ D ˆ.a; bI ¡/ D

0.24401 0.02757

Fig. 28.3 Microsoft excel program for calculating two alternative American call options

D phi.B21; 0; B45/ D phi.B19; 0; B44/

436

C.F. Lee et al.

Option Pricing Calculation ¡ab D

0.94558

¡ba D

0:48787

N2.a; 0I ¡ab/ D N2.b; 0I ¡ba/ D •D N2.a1; b1I ¡/ D N2.a2; b2I ¡/ D c(value of European call option to buy one share) p(value of European put option to sell one share) c(value of American call option to buy one share)

0.47243 0.09619 0.5 0.07007 0.06862 2.40123 1.44186 3.08238

D ..B32 B19  .B21// IF.B19 >D 0; 1; 1//=SQRT.B19^ 2  2 B32 B19  B21 C .B21/^ 2/ D ..B32  B21  .B19// IF.B21 >D 0; 1; 1//=SQRT.B19^ 2  2 B32 B19  B21 C .B21/^ 2/ Dbivarncdf(B19,0,B44) D bivarncdf.B21; 0; B45/ D .1  IF.B19 >D 0; 1; 1/ IF.B21 >D 0; 1; 1//=4 D bivarncdf.B18; B20; B32/ D bivarncdf.B19; B21; B32/ D B5 NORMSDIST.B16/  B6 EXP.B7 B9/ NO RMSDIST.B17/ D B5 NORMSDIST.B16/ C B6 EXP.B7 B9/ NO RMSDIST.B17/ D .B3–B11 EXP.B7 B10// .NORMSDIST.B20/ C bivarncdf.B18; B20; SQRT.B10=B9///  B6 EXP.B7 B9/ .NORMSDIST.B21/  EXP.B7 .B9  B10// C bivarncdf.B19; B21; SQRT.B10=B9/// C B11 EXP.B7 B10/ NORMSDIST.B21/

Fig. 28.3 (continued)

transactions, except we use the reduced stock price S Dert in place of S. The lower price bound for the European put option on a stock is: p.S; T I X / maxŒ0; Xe rT  S  De rt :

(28.19b)

Again, the stock price is reduced by the present value of the promised dividend. Unlike the call option case, however, this serves to increase the lower price bound of the European put option. Because the put option is the right to sell the underlying stock at a fixed price, a discrete drop in the stock price such as that induced by the payment of a dividend serves to increase the value of the option. An arbitrage proof of this relation is straightforward when the stock price, net of the present value of the dividend, is used in place of the commodity price. The lower price bounds for American stock options are slightly more complex. In the case of the American call option, for example, it may be optimal to exercise just prior to the dividend payment because the stock price falls by an amount D when the dividend is paid. The lower price bound of an American call option expiring at the ex-dividend instant would be 0 or S  Xert , whichever is greater. On the other hand, it may be optimal to wait until the call option’s expiration to exercise. The lower price bound for a call option expiring normally is shown in Equation (28.19a). Combining the two results, we get C.S; T I X / maxŒ0; S  Xe rt ; S  De rt  Xe rT (28.20a) The last two terms on the right-hand side of Equation (28.20a) provide important guidance in deciding

whether to exercise the American call option early, just prior to the ex-dividend instant. The second term in the squared brackets is the present value of the early exercise proceeds of the call. If the amount is less than the lower price bound of the call that expires normally; that is, if S  Xe rt  S  De rT  Xe rt ;

(28.21)

the American call option will not be exercised just prior to the ex-dividend instant. To see why, simply rewrite Equation (28.21) so it reads D < X Œ1  e r.T t /

(28.22)

In other words, the American call will not be exercised early if the dividend captured by exercising prior to the exdividend date is less than the interest implicitly earned by deferring exercise until expiration. Figure 28.4 depicts a case in which early exercise could occur at the ex-dividend instant, t. Just prior to ex-dividend, the call option may be exercised yielding proceeds St C D  X , where St , is the ex-dividend stock price. An instant later, the option is left unexercised with value c.St ; TtI X/, where c./ is the European call option formula. Thus, if the ex-dividend stock price, St is above the critical ex-dividend stock price where the two functions intersect, St , the option holder will choose to exercise her option early just prior to the ex-dividend instant. On the other hand, if St  St , the option holder will choose to leave her position openuntil the option’s expiration. Figure 28.5 depicts a case in which early exercise will not occur at the ex-dividend instant, t. Early exercise will not

28 Bivariate Option Pricing Models

437

Fig. 28.4 American call option price as a function of the ex-dividend stock price immediately prior to the ex-dividend instant. Early exercise may be optimal

Fig. 28.5 American call option price as a function of the ex-dividend stock price immediately prior to the ex-dividend instant. Early exercise will not be optimal

occur if the functions, St C D  X and c.St ; T  tI X / do not intersect, as is depicted in Fig. 28.5. In this case, the lower boundary condition of the European call, St  Xer.T t / , lies above the early exercise proceeds, St C D  X , and hence the call option will not be exercised early. Stated explicitly, early exercise is not rational if

time t, early exercise is suboptimal, where .X  S /e r.t tn / is less than .X  S C D/. Rearranging, early exercise will not occur between tn and t if1

St C D  X < St  Xe r.T t /

Early exercise will become a possibility again immediately after the dividend is paid. Overall, the lower price bound of the American put option is

This condition for no early exercise is the same as Equation (28.21), where St is the ex-dividend stock price and where the investor is standing at the ex-dividend instant, t. The condition can also be written as D < X Œ1  e r.T t /

(28.22)

In words, if the ex-dividend stock price decline – the dividend – is less than the present value of the interest income that would be earned by deferring exercise until expiration, early exercise will not occur. When condition Equation (28.22) is met, the value of the American call is simply the value of the corresponding European call. The lower price bound of an American put option is somewhat different. In the absence of a dividend, an American put may be exercised early. In the presence of a dividend payment, however, there is a period just prior to the ex-dividend date when early exercise is suboptimal. In that period, the interest earned on the exercise proceeds of the option is less than the drop in the stock price from the payment of the dividend. If tn represents a time prior to the dividend payment at

tn > t 

ln 1 C r

D X S

:

P .S; T I X / maxŒo; X  .S  De rt / :

(28.23)

(28.20b)

Put-call parity for European options on dividend-paying stocks also reflects the fact that the current stock price is deflated by the present value of the promised dividend; that is, c.S; T I X /p.S; T I X / D S De rt Xe rT :

(28.24)

That the presence of the dividend reduces the value of the call and increases the value of the put is again reflected here by the fact that the term on the right-hand side of 1

It is possible that the dividend payment is so large that early exercise prior to the dividend payment is completely precluded. For example, consider the case where X D 50; S D 40; D D 1; t D 0:25 and r D 0:10. Early exercise is precluded if r, D 0:25  lnŒ1  l=.50  40/ =0:10 D 0:7031. Because the value is negative, the implication is that there is no time during the current dividend period (i.e., from 0 to t) where it will not pay the American put option holder to wait until the dividend is paid to exercise his option.

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C.F. Lee et al.

Table 28.1 Arbitrage transactions for establishing put-call parity for American stock options S  Dert  X  C.S; T I X/  P .S; T I X/

Position

Initial Value

Ex-Dividend Day(t)

Buy American Call Sell American Put Sell Stock Lend D e rt Lend X Net Portfolio Value

C P S Dert X C C P C S  Dert  X

D D

Put Exercised normally(T) Terminal Value SQT  X SQT > X 0 SQT  X Q .X  ST / 0 SQT SQT XerT

XerT

X.e rT  1/

X.e rT  1/

r

Xe CQ C X.e r  1/

0

Equation (28.24) is smaller than it would be if the stock paid no dividend. Put-call parity for American options on dividend-paying stocks is represented by a pair of inequalities; that is, S  De rt  X  C.S; T I X /  P .S; T I X /  S  De rt  Xe rT

Put Exercised Early.”/ Intermediate Value CQ .X  SQ / SQ

(28.25)

To prove the put-call parity relation Equation (28.25), we consider each inequality in turn. The left-hand side condition of Equation (28.25) can be derived by considering the values of a portfolio that consists of buying a call, selling a put, selling the stock, and lending X C Dert risklessly. Table 28.1 contains these portfolio values. In Table 28.1, it can be seen that, if all of the security positions stay open until expiration, the terminal value of the portfolio will be positive, independent of whether the terminal stock price is above or below the exercise price of the options. If the terminal stock price is above the exercise price, the call option is exercised, and the stock acquired at exercise price X is used to deliver, in part, against the short stock position. If the terminal stock price is below the exercise price, the put is exercised. The stock received in the exercise of the put is used to cover the short stock position established at the outset. In the event the put is exercised early at time T, the investment in the riskless bonds is more than sufficient to cover the payment of the exercise price to the put option holder, and the stock received from the exercise of the put is used to cover the stock sold when the portfolio was formed. In addition, an open call option position that may still have value remains. In other words, by forming the portfolio of securities in the proportions noted above, we have formed a portfolio that will never have a negative future value. If the future value is certain to be nonnegative, the initial value must be nonpositive, or the left-hand inequality of Equation (28.25) holds.

The right-hand side of Equation (28.25) may be derived by considering the portfolio used to prove European put-call parity. Table 28.1 contains the arbitrage portfolio transactions. In this case, the terminal value of the portfolio is certain to equal zero, should the option positions stay open until that time. In the event the American call option holder decides to exercise the call option early, the portfolio holder uses his long stock position to cover his stock obligation on the exercised call and uses the exercise proceeds to retire his outstanding debt. After these actions are taken, the portfolio holder still has an open long put position and cash in the amount of X Œ1  e r.T t / . Since the portfolio is certain to have nonnegative outcomes, the initial value must be nonpositive or the right-hand inequality of Equation (28.25) must hold.

28.7 Conclusion In this chapter reviewed the basic concept of the bivariate normal density function and presented the bivariate normal CDF. The theory of American call stock option pricing model for one dividend payment was also presented. The evaluations of stock option models without dividend payment and with dividend payment were also discussed. Finally, we provided an Excel program for evaluating American option pricing model with one dividend payment.

References Stoll, H. R. 1969. “s” Journal of Finance 24, 801–824. Whaley, Robert E. 1981. “On the valuation of American call options on stocks with known dividends.” Journal of Financial Economics 9, 207–211.

Chapter 29

Displaced Log Normal and Lognormal American Option Pricing: A Comparison Ren-Raw Chen and Cheng-Few Lee

Abstract This paper compares the American option prices with one known dividend under two alternative specifications of the underlying stock price: displaced log normal and log normal processes. Many option pricing models follow the standard assumption of the Black–Scholes model (Journal of Political Economy 81:637–659, 1973) in which the stock price, follows a log normal process. However, in order to reach a closed form solution for the American option price with one known dividend, Roll (Journal of Financial Economics 5:251–258, 1977), Geske (Journal of Financial Economics 7: 63–81, 1979), and Whaley (Journal of Financial Economics 9:207–211, 1981) assume a displaced lognormal process for the cum-dividend stock price which results in a lognormal process for the ex-dividend stock price. We compare the two alternative pricing results in this paper. Keywords Displaced log normal r Log normal r Option pricing r Black–Scholes model

29.1 Introduction The American option pricing model with one known dividend was first derived by Roll (1977), improved by Geske (1979a), and corrected by Whaley (1981). The Geske-RollWhaley model (GRW model hereafter) is a closed form model with bivariate normal probability functions that are easy to implement. Roll (1977) adopts a portfolio that contains European and compound options to replicate the probabilistic cash flows of the American option with one known dividend. Geske (1979a) and Whaley (1981) assume the exdividend stock price process follows a lognormal diffusion

R.-R. Chen () Graduate School of Business Administration, Fordham University, New York, NY 10019, USA e-mail: [email protected]

and arrive at a closed form solution. If we assume the cum-dividend stock price process to follow a lognormal process, the price will be higher.

29.2 The American Option Pricing Model Under the Lognormal Process As Geske pointed out (1979a), the American option formula can be directly derived by taking the risk neutral expectation of the option payoff. Follow the GRW notation and let P be the cum-dividend price at an arbitrary time t, and S be the ex-dividend price. No arbitrage pricing states that P should yield the risk free rate under the equivalent martingale measure: dP D rdt C dW (29.1) P Note that Pt is log normally distributed with log mean and log variance:

2 .s  u/ Eu Œln Ps D ln Pu C r  2 Vu Œln Ps D  2 .s  u/

(29.2)

where u < s are two arbitrary times. The underlying stock pays a known dividend of D at time t and the option expires at T . At time t; St D Pt  ˛D where ˛ is a constant. Note that the option will be exercised if the exercise value is greater than the option value after ex-dividend; that is the American option will be exercised if Pt  K > C.St / where C.St / is the European value of the option at time t (which is a function of the ex-dividend stock price St ). In this case, the payoff is Pt  K. Otherwise, that is, Pt  K < C.St /, the call continues to be held. The trigger of early exercise is when the stock price satisfies the following equation:

C.-F. Lee Rutgers University, New Brunswick, NJ, USA e-mail: [email protected]

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_29, 

8 < Pt  K D C.St / or : St C ˛D  K D C.St / 439

440

R.-R. Chen and C.-F. Lee

Like CRW, we follow the second expression of the above equation. Since St appears on both sides and is the only unknown in this equation, we can solve for St for the equality: St C ˛D  K D C.St /.1 In other words, St > St (which implies St C ˛D  K > C.St /) and the option will be early exercised, otherwise, it does not. Hence, the pricing formula can be derived by implementing the following integrals: 2 6 C0 D e rt 4

 ZPt

Z1 .Pt  K/f .Pt /dPt C

Pt

3

Note that if Pt ˛D is and f .Pt / were both log normal, then the product of two log normal variables, 

ZPt Z1 f .ST j Pt  ˛D /f .Pt /dST dPt 0

K Pt Z˛DZ1

D

f .ST jPt  ˛D/f .Pt  ˛D/dST dPt 0

7 C.St /f .Pt /dPt 5

K

Pt ˛D

Z

Z1

D

0

(29.3) where Pt is the critical value over which triggers early exercise. Note that the call price (under no early exercise) in the above equation, C.St /, is a function of the ex-dividend stock price and hence can be written as the following integration: 21 3 Z C.St / D e r.T t / 4 .ST  K/f .ST jSt /dST 5

f .ST ; Pt  ˛D/dST dPt 0

(29.6)

K

would result in a bivariate normal probability. Unfortunately, since Pt  ˛D is a displaced lognormal, its distribution is unknown and certainly cannot be log normal. Consequently, Equation (29.6) (and hence Equation (29.5)) will not have a closed form solution.

(29.4)

K

29.3 The Geske-Roll-Whaley Model Plugging (29.4) into (29.3), we have: The GRW model, in order to arrive at a closed form solution, assumes the following:

2

Z1 rt 6 C0 D e 4 .Pt  K/f .Pt /dPt

dS D rdt C dW: S

Pt

(29.7)

Pt

Z Z1 C 0

K  ZPt Z1

K 2 De

rt

6 4

In this case, the ex-dividend price follows a log normal distribution with log mean and variance:

ST f .ST jSt /f .Pt /dST dPt

0

3

2 Eu Œln Ss D ln Su C r  .s  u/ 2

7 f .ST jSt /f .Pt /dST dPt 5

Vu Œln Ss D  2 .s  u/:

K Pt

Z1 Z .Pt  K/f .Pt /dPt C

Pt

Under the circumstance, Equation (29.5) becomes: 2

0

C0 D e

Z1 ST f .ST jPt  ˛D/f .Pt /dST dPt K Pt

Z Z1 K 0

rt

6 4

Z1 .Pt  K/f .Pt /dPt

Pt

3 7 f .ST jPt  ˛D/f .Pt /dST dPt 5



ZPt Z1 C

ST f .ST jSt /f .Pt /dST dPt 0 K

K



ZPt Z1 (29.5)

K 0

Note that the call option has a lower bound of S  e r.T t/ K. As a result, ˛D > K.1  e r.T t/ /.

1

(29.8)

3 7 f .ST jSt /f .Pt /dST dPt 5 (29.9)

K

A simple change of variable, St D Pt  ˛D, simplifies the above equation into:

29 Displaced Log Normal and Lognormal American Option Pricing: A Comparison

2 Z1 6 C0 D e rt 4 .St C ˛D  K/f .St /dSt

441

where ln.S0 =St / C .r C  2 =2/t p  t p a2 D a1   t

a1 D

St 

ZSt Z1 C

ST f .ST jSt /f .St /dST dSt 0 K

3



ZSt Z1 K

7 f .ST jSt /f .St /dST dSt 5 (29.10)

ln.S0 =K/ C .r C  2 =2/T p  T p b2 D b1   T b1 D

0 K

This would lead to a nice bivariate solution (shown below) if St and ST are jointly log normal. However, it is not possible that both Pt and St be log normal at the same time since a displaced log normal variable (i.e. St ) can no longer be log normally distributed. Assuming that St can be log normally distributed, GRW continue with the following derivation and arrive at their option pricing formula2: 2 3  Z1 ZSt 6 7 e rt 4 .St C ˛D  K/f .St /dSt C C.St /f .St /dSt 5 St

0

D S0 N.a1 / C e rt .˛D  K/N.a2 / 9 8 ZSt < Z1 = e r.T t / .ST  K/f .ST jSt /dST f .St /dSt C e rt : ; 0 K „ ƒ‚ … X

(29.11) We can then simplify the second term of the right hand side of (29.11), X , as: X D e rT

211 Z Z 4 .ST  K/f .ST jSt /f .St /dSt dST 0 K

3 Z1 Z1 7  .ST  K/f .ST jSt /f .St /dSt dST 5 St K

p D S0 N.b1 /  e r.T t / KN.b2 /  S0 M a1 ; b1 I p t T p Ce r.T t / KM a2 ; b2 I p t T p p t D S0 M a1 ; b1 I  pT  e r.T t / KM a2 ; b2 I  pTt

The derivation of (29.11) is provided in the Appendix. Putting (29.11) and (29.12) together, we arrive at the GeskeRoll-Whaley formula: h h p i S0 N.a1 /CM a1 ; b1 I  p t e rT K e r.T t / N.a2 / T p i C e rt ˛DN.a2 / (29.13) CM a1 ; b1 I  pTt where M.a; bI / is a bivariate normal probability function with correlation coefficient , and N.a/ is a univariate formal probability function. Note that this result is consistent with the GRW model only if S0 is used in computing the option price, and not today’s stock price, P0 . This implies that it is S (the ex-dividend price) but not P (the cum-dividend price) that follows the log normal diffusion. Whaley (1981, p. 209, last two lines) and Geske (1979b, p. 375, definition of S / already point out that the result is consistent with the exdividend process being log normal. Note that while both exdividend and cum-dividend processes are martingales (after deflated by the risk free rate), the actual observed price series are not. The actual observed prices are cum-dividend prior to the dividend and ex-dividend after the dividend. As a result, the observed price series,  Pt t <  (29.14) Vt D St t >  but each ex or cum dividend series is a martingale:  St D  Pt D

Pt  e r. t / ˛D t <  t  St

and

t X DN p Friday, October 20, 2007 with an exercise price $115, the  t  t riskless rate and the instantaneous variance need to be esti(30.60) mated. The riskless rate is estimated either by using the aver  ˇ   age of the bid and ask quotes on U.S. Treasury bills or from p  E S t  ˇ S  > X D S .t/ er.t t / N the money market quotes for deposits of approximately the " S .t /  #  ln X C r C 12  2 .t   t/ same maturity as the option. The interest rate for a deposit

p : (30.61) of 1 month was 5.14875% on that particular day. The only  t  t missing piece of information is the instantaneous variance of  Combining Equations (30.59) and (30.60) with T D t  t the stock price. Several different techniques have been suggested for estiinto Equation (30.61) yields Equation (30.53). It is worth observing that two terms of Equation (30.53) mating the instantaneous variance. In this regard the work of have economic meaning. The first term, SN .d1 /, denotes Latané and Rendleman (1976) must be mentioned; they dethe present value of receiving the stock provided that rive standard deviations of continuous price-relative returns S .t  / > X . The second term gives the present value of pay- that are implied in actual call-option prices on the assumping the striking price provided that S .t  / > X . In the special tion that investors behave as if they price options according to the Black–Scholes model. In the example, the implicit varicase when there is no uncertainty and  D 0, observe that: ance is calculated by using the actual October 20, 2007 call price of an IBM option with an exercise price of $120 to solve N .d1 / D N .d2 / D N .1/ D 1I and for an estimate of the instantaneous variance. More specif (30.62) ically, a numerical search is used to approximate the stanC D S .t/  er.t t / X I dard derivation implied by the Black–Scholes formula with that is, a call is worth the difference between the current value these parameters: stock price S D 116:67, exercise price of the stock and the discounted value of the striking price pro- X D 120, time to expiration T D 30=365 D 0:0822, riskvided S .t  / > X ; otherwise the call price would be zero. less rate r D 0:0514875, and call-option price C D 2:00. When  ¤ 0 that is, when uncertainty exists and the stock The approximated implicit standard deviation is found to be price is volatile – the two terms in Equation (30.61) are mul-  D 23:82%. A simple spreadsheet like Excel for example tiplied by N .d1 / and N .d2 /, respectively, to adjust the call can be used to calculate this “implied volatility,” using the price for the prevailing uncertainties. These two probabilities “solver” or the “goal seek” tool. That way, one can simply can also be given an economic interpretation. As mentioned write the Black–Scholes formula and set as “changing variearlier, N .d1 / is called delta; it is the partial derivative of able” the volatility, so that the “target variable” – the calcuthe call price with respect to the stock price. N .d2 / gives lated option price – becomes equal to its market price. the probabilities that the call option will be in the money, as After the clarifications are made, the example is this: Equation (30.59) shows. given S D 116:67; X D 115; T D 0:0822; r D Assuming that investors in the economy have risk-neutral 0:0514875, and  D 23:82%, we use Equation (30.53) preferences, it will be possible to derive the Black–Scholes to compute C . Using Equations (30.54) and (30.55) we formula without using stochastic differential equations. calculate: Garven (1986) has shown that to derive Equation (30.53)



0:23822 knowledge is required of normal and log normal distributions 116:67  0:0822 ln =115 C 0:0514875 C 2 and basic calculus, as presented in Appendix 30A. Sample p d1 D Problem 30.6.1 provides further illustration. 0:2382  0:0822

Sample Problem 30.4. Equation (30.53) indicates that the Black–Scholes option-pricing model is a function of only five variables: T , the time to expiration, S , the stock price,  2 , the instantaneous variance rate on the stock price, X , the exercise price, and r, the riskless interest rate. Of these five variables, only the variance rate must be estimated; the other four variables are directly observable. A simple example is

D 0:30722

0:23822 116:67  0:0822 ln =115 C 0:0514875  2 d2 D p 0:2382  0:0822

D 0:238922:

458

G. Chalamandaris and A.G. Malliaris

Using the Excel function Normsdist() we can calculate the cumulative standardized normal distribution. Thus, we calculate for the call option premium,

The search for these new models moved in the direction of the local volatility models (see for example Derman et al. (1996)), asset jump models (see Merton (1992)) or stochastic volatility models (Dupire 1994; Heston 1991). UnfortuC D 116:67  N .0:30722/115  exp .0:0514875  0:0822/ nately, the larger scope of these models that is required to encompass for example the more general and more realistic  N .0:238922/ D 4:34: case of incomplete markets (see Pliska 1997; Britten-Jones and Neuberger 1996) adds significantly to their complexity The calculated call-option price of $4.34 is between the acand thus obscures their intuitive content: instead of a sintual bid-offer for the call price of $4.30–$4.50 reported in gle non-observable parameter – the volatility – on which the Bloomberg. Black–Scholes model depends, these models require the esThis simple example shows how to use the Black–Scholes timation (“calibration”) of more parameters that relate to the model to price a call option under the assumptions of the price in usually untractable ways. model. The example is presented for illustrative purposes This fact alone, and considering the conceptual transonly, and it relies heavily on the implicit estimate of the parency and robustness of the Black–Scholes model, kept it variance, its constancy over time, and all the remaining asto this day still very popular among practitioners in many opsumptions of the model. The appropriateness of estimating tion markets, especially in the more liquid ones that trade the the implicit instantaneous variance is ultimately an empirisimpler options in very large volumes. Traders that are aware cal question, as is the entire Black–Scholes pricing formula. of its limitations can be very often much more effective in Boyler and Ananthanarayanan (1977) studied the implicadealing with their “out of model” unpredictable risks than tions of using an estimate of the variance in option-valuation the ones who can be lulled in false security behind a more models and showed that this procedure produces biased opsophisticated but much more obscure model. Taleb (1997) tion values. However, the magnitude of this bias is not large. relates many cases that provide evidence of this fact. Chiras and Manaster (1978) compared implied volatilities Several textbooks such as Hull (2006), Chance (2006), from option prices with volatilities estimated from historical and McDonald (2005) extend the discussion above of the data and found the latter much better forecasts. Black–Scholes model and give numerous applications and One additional remark must be made. The closeness in examples. this example of the calculated call option price to the actual call price is not necessarily evidence of the validity of the Black–Scholes model. Earlier extensive empirical work that took place to investigate how market prices of call options compare with price predicted by Black–Scholes showed positive evidence of the model’s robustness in the market (see 30.7 Extending the Risk-Neutral Argument: MacBeth and Merville (1979) and Bhattacharya (1980)). For The Martingale Approach example, Chiras and Manaster (1978) also studied whether a strategy of buying low-volatility options and sell highFollowing the work of Harrison and Kreps (1979), Harrison volatility options produced excess returns. Their evidence and Pliska (1981), and Cox and Ross (1976) a more general was supportive of the Black–Scholes model. approach in pricing derivatives was devised than setting up However, the market crash of 1987 revealed its limitaeach time a riskless replicating portfolio that we described tions in predicting abnormal market conditions. Indeed, the above. Although the mathematics for the development of the appearance of phenomena like the pronounced “volatility Martingale method demand some basic knowledge of ab3 smile,” or daily price jumps that defied the assumption of stract measure theory that is outside the scope of this chapter, a lognormal distribution for the asset returns, called for alterwe will attempt to give a simple and intuitive explanation of native models that explain in a more parsimonious and genthis approach in this section. eral way all market prices. We have already mentioned the concept of a Martingale process. Let’s assume that at every time t, we can collect the entire “flow of information” about the asset in question; that is everything that it is known up to time t. This flow of 3 Volatility smile is a pattern of consistently higher implied volatilities information is formally symbolized by the so-called filtration for strikes set far from the at-the-money strike relative to the ATM im- fFt gt 0 . A stochastic process X .t/ is called a Martingale if plied volatility. That is, using the Black-Scholes equation in reverse, so it satisfies the relation as to recover the implied volatility from the market prices of options with different strikes, one would discover that each strike will yield a different number– see Das and Sundaram (1997) for an empirical study.

E ŒX .t/ jFs D X .s/ :

(30.63)

30 Itô’s Calculus and the Derivation of the Black–Scholes Option-Pricing Model

In other words, a process is a Martingale, if after collecting all the available information up to time s, the best forecast of the value of the process at any future time is no other than its current value. Of course, this fact is reminiscent of the main feature of the (weak form) of the efficient market hypothesis: past and present information is all included at the current price of an asset. Now let’s also assume that we have a stock whose price S .t/ is the underlying of the derivative we wish to price. For simplicity, we let S .t/ follow a Geometric Brownian motion described by the Itô stochastic differential equation dS .t/ D   S .t; w/  dt C   S .t; w/  dZ .t; w/

(30.64)

Dividing by S .t; w/ we get the elementary proportional change of its price equal to

459

In a risk-neutral world, the investors that populate it care only about expected returns and ignore the different risks that accompany the different assets, and so all the expected returns of the assets that span that economy should be equal. This means that in that world, the investors will expect no future growth (nor diminishment) of the stock value relative to the numeraire asset (assuming that both assets provide no income), since they bear no risk to justify any expected growth relative to this and therefore the expected value of the stocknumeraire ratio is equal to its current value. More formally, in a risk neutral world ˇ   S .0/ S .T / ˇˇ F (30.68) D E 0 ˇ P .T; T / P .0; T / and because in our example P .T; T / D 1,

dS .t; w/  d .ln S .t; w// D   dt C   dZ .t; w/ : S .t; w/ (30.65) Assuming that we calculate its price in U.S. dollars, its expected proportional change (log-change) in price will be 

 dS .t/ E D   dt S .t; w/

(30.66)

E ΠS .T / j F0 D

S .0/ : P .0; T /

(30.69)

Hence, we gather that in this risk-neutral world the normalized stock price follows a Martingale process, which translates into the following Itô process d

S .t/ P .t; T /



D 0  dt C  

S .t/ P .t; T /



 d ZQ .t; w/ (30.70)

and its variance equal to 

 dS .t/ Var D  2  dt: S .t; w/

(30.67)

Also let’s assume that we don’t measure the stock price in current dollars, but we measure it relative to a tradable asset that grows deterministically in time. For example, that could be a zero coupon bond that is traded today, it pays 1 U.S. dollar at time T , has constant yield R, and trades today at P .0; t /. Hence, instead of looking at the dollar stock price S .t/ at time t we look at the normalized “stock per zero coupon .t / bond price” PS.t;T / , and the zero coupon bond is called the numeraire asset. The difference between measuring the stock price in zero coupon bonds instead of in dollars is the following: A dollar today is (almost certainly) more valuable than a dollar in the future time T , simply because they represent different cashflows in the time/cashflow 2-dimensional space, in fact they are different assets. On the other hand, our zero coupon bond represents at all times until and including time T , the same claim at the same cashflow; that is, at all times remains one single asset. Hence, in the first case we measure the same instrument relative to two different assets, whereas in the second case we measure it consistently relative to a single asset.

where dZQ .t; w/ is a Wiener process. Of course, we do not assume that all investors are risk neutral or that all the assets of the economy grow at the same rate or that the stock price grows according to Equation (30.70). This is more like a “trick” to simplify valuation. Indeed, we can write Equation (30.68) like  E

Q

ˇ  Z C1 S .T / ˇˇ S .T; w/ F0 D ˇ P .T; T / 1 P .T; T /

fw;T jF0 .T; wjF0 /  dw D

S .0/ P .0; T /

(30.71)

Where fw;T jF0 .T; wjF0 / is the conditional density function of the stock price S .T; w/ at time T ; that is, integrated across all possible random events w 2 . It can be proven that if there is a unique density function that satisfies Equation (30.71) then the market is arbitrage-free and complete – that is, all the contingent claims on the stock with a payoff at time T can be replicated by unique dynamic trading strategies between the two assets. What’s more, if such a unique density function exists, then the replicating dynamic trading strategy is “previsible” – it is decided at each step t ! t C dt conditioned on variables whose value is known at that time (for

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G. Chalamandaris and A.G. Malliaris

example, S .t/ ;  .S; t// – and “self-financing” – it requires no insertion of new wealth as time evolves from time t D 0 towards the maturity of the claim. The interested reader can look at Musiela and Rutkowski (1997), Karatzas and Shreve (2005) or Lamberton and Lapeyre (2007) for a rigorous treatment of the Martingale methods for derivative pricing. Also Rebonato (1998, 2002) produces a very extensive exposition of the Martingale approach for the construction of relatively more advanced interest rate models. Equation (30.68) is the cornerstone of the Martingale approach and the modern methods of pricing. In general, this method consists of the following steps: 1. First, we search for a numeraire asset with regard to which one can express the future payoff of the function. For example, if the option has a cash payoff at time T , a natural choice for the numeraire asset is a money market account that is continuously rolled over, or a zero coupon bond maturing at time T . On the other hand, if the deliverability of the option is a swap starting at time T , then the natural choice should be a forward starting annuity, the payments of which coincide with the payments of the swap. 2. Next, we apply Equation (30.68) looking for the density function that satisfies it. That is, we look for the “measure” that makes the discounted price process for the underlying stock of the option, a Martingale process. This density function/measure is also called risk neutral measure and is denoted by a capital letter, say Q. Then we write all expectations with regard to this measure as  E

Q

ˇ  S .0/ S .T / ˇˇ F0 D ˇ P .T; T / P .0; T /

d

S .t/ P .t; T /



D S .t/  d

(30.73)

It is obvious that for different assets one defines as the numeraire asset, the induced measure will be different. However, in a complete market all the measures will calculate the same price for the contingent claim.

1 P .t; T /

C

1  dS .t/ : P .t; T / (30.74)

Of course, in the general case the numeraire asset needs not be deterministic. The application of Itô’s Lemma with the multiplication rules of Equations (30.8) and (30.9) helps us handle the general case. For reason of simplicity, however, we choose in our exposition to employ a deterministic numeraire asset. We write Equation (30.74) using the convention of continuous compounding for the price of the zero coupon .P .t; T / D e R.T t / /, d

S .t/ P .t; T /





R 1 D S .t/   dt C  dS .t/ P .t; T / P .t; T /

and by substituting for dS .t/ according to (30.64) we finally get d

S .t/ P .t; T /





R P .t; T /

D S .t/  C

 dt

1  .  S .t/  dt P .t; T /

C   S .t/  dZ .t// :

(30.72)

3. If this measure Q exists and is unique then the market is complete as in Pliska (1997). Once it is discovered, then one can price all the contingent claims on the discounted asset by calculating a simple expectation. Furthermore, the price process of each contingent claim can be replicated in a unique way by dynamically trading the two S .T / assets. Indeed, all the contingent claims X P .T;T / that have a payoff at time T are just a deterministic function of the underlying instrument. This leads us to the conclusion that the discounted payoff of the claim is also a Martingale process. Hence, it can be priced by   X .S .T // ˇˇ V .0; S .0// D EQ ˇF0 : P .0; T / P .T; T /

The way to find this risk-neutral measure is derived from the following argument: we want to build the stochastic dif.t / ferential equation for the normalized stock price PS.t;T / that

S .t / is, we want to calculate d P .t;T / . In our case, the zero coupon price is deterministic; that is, it can be regarded as a stochastic process with zero volatility, and by applying the product rule we get

Or d

S .t/ P .t; T /



D

S .t/ P .t; T /

C D

 .  R/  dt

S .t/ P .t; T /

S .t/ P .t; T/

   dZ .t/

 



.  R/  dt C dZ .t/  (30.75)

It then follows that the so-called “change of measure,” is nothing other than requiring that .R/ dtCdZ .t/ D d ZQ .t/  is a Wiener process, so that Equation (30.75) is a Martingale process according to the above. From Equation (30.75), one can see that the difference between Equations (30.64) and (30.75) is only the drift of the process, since the volatilities of the two processes are identical.

30 Itô’s Calculus and the Derivation of the Black–Scholes Option-Pricing Model

Also, because the normalized stock price is a Martingale, Equation (30.64) becomes

  2 S .0/ Q D  exp  t C   Z .t/ P .0; 2

 T/ 2 S .0/ Q  exp  t C   Z .t/ ) S.t/ D P .t; T /  P .0; T / 2 (30.76)



S .t/ P .t; T /





461

Now ZQ .t/ D ZQ .t/  ZQ .0/ is the increment of a Wiener process; that is, it follows the normal distribution with zero mean and variance equal to T ; that is, C D P .0; T / 8

 Z1 < 2 T max S .0/  exp R   : 2 1

It is worth noting here that the generality of the Martingale approach to pricing lies on the fact that the numeraire asset – a tradable instrument linked to deterministic rates in the Black–Scholes economy – can be in general stochastic. In truth, any asset can be used as the numeraire asset and provided that the market is complete and offers no opportunity of arbitrage, Equation (30.73) will yield a unique probability density function that satisfies it. One can observe the similarity between fixed income pricing and the Martingale method: in a complete and arbitragefree fixed income market, all bonds, swaps, FRAs, and other linear instruments imply a unique term structure that prices exactly all of them. The extension of this view to the options market is that, if similar conditions apply to this market too (complete and arbitrage-free), then there exists a unique density function that prices all the market options exactly. Sample Problem 30.5. Deriving the Black–Scholes Formula for a Call Option with the Martingale Approach. Let us define the numeraire asset to be a zero-coupon bond P .t; T / D e R.T t / with maturity, which coincides with the maturity T of the option. The payoff of the option is max .S .T /  K; 0/. According to the above, let Q be the risk-neutral measure .t / such that the discounted stock process PS.t;T / is a Martingale, such that   S .T / S .0/ Q jF0 D : (30.77) E P .T; T / P .0; T / Under this measure, all the discounted contingent claims on the stock price are also Martingales, which by also applying (30.76) gives us   C max .S .T /  K; 0/ D EQ jF0 ) P .0; T / P .T;  T/ S .0/ C D P .0; T /  E Q max  exp P .0; T /   2 

 T C   ZQ .T /  K; 0 jF0 2  D P .0; T /  E Q max .S .0/  exp



  2 T C   ZQ .T /  K; 0 jF0

R 2 (30.78)

C  y 

p

 T

9

exp  y 2 = 2  K; 0  p dy: 2 ; (30.79)

The option is exercised only when the final stock price is greater than the strike; that is 

 p 2 S .0/  exp R T C y T K 2

  p K 2 T C y T ) exp R 2 S .0/

2

p  S .0/ T    y  T  ln ) R  2 K

2

S .0/  ln C R T K 2 )y  p D d2  T (30.80) We can then write Equation (30.79) as 8 ˆ ˆ 

>   exp  = p 2 dy C  y  T  K  p > 2 > ; Z1

 Z1  1 y2 D P .0; T /  p  S .0/  exp  2 2

 d 2 2 p  T dyC C  y  T C R  2 2

Zd 2 1 y P .0; T /  p  dy K  exp  2 2 D I1  I2

1

(30.81) The first integral is calculated by “completing the square.” Indeed,

462



G. Chalamandaris and A.G. Malliaris

p y2 C y  T 2

p 1 h 2 T D   y2  2    y  T C R 2 2 i 2  R  T C  2  T   p 2 1 D   y  T 2RT 2 p

(30.82)

We employ the change of variable x D y   T ) y D p x C  T and I1 becomes 1 I1 D P .0; T /  S .0/  exp .R  T / p 2

Z1 1 exp   x 2 dx  2 p .d 2C T / p d 2CZ T Dd1

1 1 2 D S .0/ p exp   x dx  2 2 D S .0/  N .d1 /

 Y .0/ Y .T / jF0 D : X .T / X .0/

(30.87)

It is also very easy to check that the sum of the two Wiener processes is a Wiener process itself dZ .t/ D   dZ 1 .t/ C N  dZ 2 .t/

(30.88)

(30.84)

are independent with the previous one, they follow the normal distribution, and p they have a mean 0, and standard deviation  t .

thus reaching the Black–Scholes result. Sample Problem 30.6. Pricing an Option to Exchange One Asset for Another Let us consider two tradable assets with respective prices X .t/ and Y .t/ that provide no income. In this section we are pricing the option that permits us, if we hold the asset X .t/, to exchange it for Y .t/; that is, the payoff of the option is max.Y .t/  X .t/ ; 0/. Let’s also assume that the maturity of the option is T . The natural choice for the numeraire asset is X .t/ – the asset we hold. Since both assets are tradable without any dividends, Equation (30.72) tells us that there exists a unique density function/measure Q that satisfies 

X  Y   D : X  Y

(30.83) The second integral is simply equal to

EQ

D

p with N D 1  2 , since the successive increments of the new process

1

I2 D P .0; T /  K  N .d2 / ;

In Equation (30.86) the driving Wiener processes Z1 .t/ and Z2 .t/ are independent. However, we observe that although X .t/ is driven only by Z1 .t/ ; Y .t/ is driven by both. The result is that the returns of Y .t/ become correlated with the returns of X .t/. Indeed, it very easy to check from the definition of correlation that   dX .t/ dY .t/ E  X .t/ Y .t/ v " " u

2 #

2 # u dX .t/ dY .t/ tE  X dt  Y dt E X .t/ Y .t/

According to the previous, we do know that the solutions to these stochastic differential equations are

  2 x  x  t C x  Z1 .t/ 2

 2 Y .t/ D Y .0/  exp Y  Y  t C Y    Z1 .t/ 2  (30.89) CY  N  Z2 .t/

X .t/ D X .0/  exp

By dividing the Equations in 30.89, we get the ratio 

2 Y 2 Y  x   x t 2 2  C .Y    x /  Z1 .t/ C Y  N  Z2 .t/ :

Y .t/ Y .0/ D  exp X .t/ X .0/

(30.85)

(30.90) This is the density function that ensures that the current price of this ratio is fair. For reasons of simplicity we let the dynamics of the two assets be described by the following Itô processes



dX .t/ D X  X .t/  dt C CX  X .t/  dZ 1 .t/

d

dY .t/ D Y  Y .t/  dt C CY    Y .t/  dZ 1 .t/ CY  N  Y .t/  dZ 2 .t/

If we consider Equation (30.90) as a bivariate function u D f .Z1 ; Z2 / and we apply the Itô’s Lemma 30.18 with the multiplication rules of Equation (30.19) we have

(30.86)

Y .t/ X .t/

D

@f @f @f  dt C  dZ 1 C  dZ 2 @t @Z1 @Z2 C

1 @2 f 1 @2 f  dt C  dt 2 2 @Z1 2 @Z2 2

(30.91)

30 Itô’s Calculus and the Derivation of the Black–Scholes Option-Pricing Model

The partial derivatives are Equations (30.92a) to (30.92b):

presented

below

from



2 Y .t/ @f Y x2  D Y  x   (30.92a) @t 2 2 X .t/ Y .t/ @f D .Y    x /  @Z1 X .t/

(30.92b)

463

According to the Martingale approach, we need a density probability function f Q .t; w//measure makes Equation (30.95) a Martingale process, or lently, makes the following normalized process, a process h

Y  x 

(30.92c)

Y .t/ @2 f D .Y    x /2  2 X .t/ @Z1

(30.92d)

@2 f Y .t/ D Y2  N2  2 X .t/ @Z2

(30.92e)



Y .t/ X .t/ Y .t/ X .t/

2  x 2 Y  D Y  x  2 2  1 1 2 2 2 C .Y    x / C Y    dt 2 2 C .Y    x /  dZ 1 C Y    dZ 2 (30.93)

N 2 2

i  dt C dZ .t/ D d ZQ .t/ (30.96)

  V .0/ max .Y .T /  X .T / ; 0/ D EQ X .0/ X .T /

  Y .T /  1; 0 D E Q max X .T /

(30.97)

This last equation denotes a simple call option on the asset Y .t / with strike equal to one. X .t / Furthermore, the dynamics of this asset satisfies the assumptions of the Black–Scholes model, which means that we can apply the formula if we make the following substitutions: Y .t/ ! S .t/ 1 ! K 1 ! P .t; T / X .t/  !

(30.98)

0!R

That way we reach the result

D .Y    x /  E ŒdZ 1 C Y  N  E ŒdZ 2 D 0 h i E ..Y    x /  dZ 1 C Y  N  dZ 2 /2 i h   D .Y    x /2  E dZ 1 2 C Y 2  N2  E dZ2 2 C 0  D Y 2  2 C x 2  2  Y    x   CY 2 1  2  dt   D Y 2 C x 2  2  Y    x  dt D N 2  dt

V .0/ V .0/ D N .d1 /   N .d2 / X .0/ X .0/

(30.99)

with ln d1 D

2 Y .0/ C T X .0/ 2 p  T

ln d2 D

2 Y .0/  T X .0/ 2 p :  T (30.100)

(30.94)

We can then turn Equation (30.93) into a simpler equation of only one stochastic factor Y .t / X .t / Y .t / X .t /

C

  V .0/ Q V .X .T / ; Y .T // DE X .0/ X .T /



E Œ.Y    x /  dZ 1 C Y  N  dZ 2

d

Under this measure Q, the prices of all the contingent claims on this ratio of the assets are Martingales and can each be replicated with a unique, self-financing and previsible trading strategy between the two assets. In other words,

)

Since we have only one asset under consideration – the ratio Y .t / of X .t / – it is perhaps redundant to describe its dynamics with two driving Wiener processes. The sum of the increments of the two processes follows a normal distribution with mean and variance



Y 2 x 2 2

N

@f Y .t/ D Y  N  @Z2 X .t/

We substitute in Equation (30.91) to get the stochastic differential equation of the ratio of the assets; that is, d



to find Q that equivaWiener

 D Y  x 



Y 2 x 2  2 2



 N 2 C dt CN dZ: 2 (30.95)

30.8 Remarks on Option Pricing For a review on the early literature on option pricing, see the two papers by Smith (1976, 1979). It is appropriate here to make a few remarks on the Black–Scholes option-pricing model to clarify its significance and its limitation.

464

First, the Black–Scholes model for a European call as originally derived, and as reported here, is based on several simplifying assumptions. 1. The stock price follows an Itô equation. 2. The market operates continuously. 3. There are no transaction costs in buying or selling the option or the underlying stock. 4. There are no taxes. 5. The riskless rate is known and constant. 6. There are no restrictions on short sales. Several researchers have extended the original Black– Scholes model by modifying these assumptions. Merton (1973) generalizes the model to include dividend payments, exercise-price changes, and the case of a stochastic interest rate. Roll (1977) has solved the problem of valuing a call option that can be exercised prior to its expiration date when the underlying stock is assumed to have made known dividend payments before the option matures. Ingersoll (1976) studies the effect of differential taxes on capital gains and income while Scholes (1976) determines the effects of the tax treatment of options on the pricing model. Furthermore, Merton (1976) and Cox and Ross (1976) show that if the stock-price movements are discontinuous, under certain assumptions the valuation model still holds. Derman et al. (1996) replaced the assumption of constant volatility with the concept of the local volatility – volatility that is stochastic in the sense that depends on the asset price – but also not fully stochastic since this dependence is deterministic. This assumption helped explain the existence of volatility smiles but with results that lacked economic intuition. Following a different direction Hull and White (1987) examined the effects of stochastic volatility and the capacity of the extended model to fit parsimoniously market prices. Also, Leland (1985) studied the consequences of introducing transaction costs in the replication of options. These and other modifications of the original Black–Scholes model are quite robust with respect to the relaxation of its fundamental assumptions. Second, it is worth repeating that the use of Itô’s calculus and the important insight concerning the appropriate concept of an equilibrium by creating a risk-free hedge portfolio have let Black and Scholes obtain a closed-form solution for option pricing. In this closed-form solution several variables do not appear, such as (1) the expected rate of return of the stock, (2) the expected rate of return of the option, (3) a measure of investor’s risk preference, (4) investor expectations, and (5) equilibrium conditions for the entire capital market. Third, the Black–Scholes pricing model has found numerous applications. Among these are the following: (1) pricing the debt and equity of a firm; (2) the effects of corporate policy and, specially, the effects of mergers, acquisitions, and scale expansions on the relative values of the debt and equity

G. Chalamandaris and A.G. Malliaris

of the firm; (3) the pricing of convertible bonds; (4) the pricing of underwriting contracts; (5) the pricing of leases; and (6) the pricing of insurance. Smith (1976, 1979) summarizes most applications and indicates the original reference. See also Brealey and Myers (1996). Fourth, Black (1976) shows that the original call-option formula for stocks can be easily modified to be used in pricing call options on futures. The formula is   C T; F;  2 ; X; r D erT ŒFN .d1 /  XN .d2 / (30.101) d1 D

ln .F =X / C 12  2 T p  T

(30.102)

d2 D

ln .F =X /  12  2 T p  T

(30.103)

In Equation (30.101) F now denotes the current futures price. The other four variables are as before – time to maturity, volatility of the underlying futures price, exercise price, and risk-free rate. Note that Equation (30.101) differs from Equation (30.53) only in one respect: by substituting erT F for S in the original Equation (30.53) Equation (30.101) is obtained. This holds because the investment in a futures contract is zero, which causes the interest rate in Equations (30.102) and (30.103) to drop out. Another easy way to reach the same conclusion is to define as the numeraire asset the zero coupon bond that matures at the expiry of the option. With this numeraire asset, the process of the discounted futures price shows no drift under the risk-neutral measure. Fifth, three important papers by Harrison and Kreps (1979) and Kreps (1981, 1982) consider some foundational issues that arise in conjunction with the arbitrage theory of option pricing. The important point to consider is this: the ability to trade securities frequently can enable a few multiperiod securities to span many states of nature. In the Black–Scholes theory there are two securities and many uncountable states of nature, but because there are infinitely many trading opportunities and because uncertainty resolves nicely, markets are effectively complete. Thus, even though there are far fewer securities than states of nature, markets are complete and risk is allocated efficiently. An interesting result of Harrison and Kreps (1979) is that certain self-trading strategies can create something out of nothing when there are infinitely many trading opportunities. The doubling strategies are the well-known illustrations of this phenomenon. Harrison and Kreps introduce the concept of a simple strategy to eliminate free lunches and conjecture that a nonnegative wealth constraint could rule out the doubling strategies. Duffie and Huang (1985) give an interpretation of admissible strategy as a limit of a sequence of simple strategies and use an integrability condition on the trading strategies. These latter papers helped develop the Martingale approach and established the equivalence of the absence of

30 Itô’s Calculus and the Derivation of the Black–Scholes Option-Pricing Model

arbitrage in complete markets with the existence of a unique Martingale measure for a given numeraire asset. See Pliska (1997) for a very clear discussion on this issue and the effects of market incompleteness in pricing. The first results in this direction, in a continuous-time setup, were obtained by Stricker (1990) and Ansel and Stricker (1992). These results were extended in various directions by Delbaen (1992), Schweizer (1992), Lackner (1993), Delbaen and Schachermayer (1994) and others. A very extensive and rigorous exposition of the Martingale approach is given by Musiela and Rutkowski (1997), whereas a more heuristic application of the method can be found in Rebonato (1998, 2004). For a detailed survey of numerous empirical tests, see Galai (1983) for stock options, Shastri and Tandon (1986) and Bodurtha and Courtadon (1987) for currency options and Chance (1986) for the market prices of index options. Finally for analytical results regarding the pricing of numerous exotic options, see Haug (1998).

30.9 Conclusion This chapter has discussed the basic concepts and equations of stochastic calculus (Itô’s calculus), which has become a very useful tool in understanding finance theory and practice. By using these concepts and equations, the manner in which Black and Scholes derived their famous optionpricing model, together with the more modern approach of the Martingale measure, was also illustrated. Although this chapter is not required to understand the basic ingredients of security analysis and portfolio management, it is useful for those with training in advanced mathematics to realize how advanced mathematics can be used in finance.

References Ansel, J.P. and C. Stricker. 1992. “Lois de martingale, densities et decompositions de Foellmer Schweizer.” Annales de l’Institut Henri Poincare (B) Probability and Statistics 28, 375–392. Arnold, L. 1974. Stochastic differential equation: theory and applications, Wiley, New York. Baxter, M. and A. Rennie. 1996. Financial calculus, Cambridge University Press, Cambridge. Bhattacharya, M. 1980. “Empirical properties of the Black–Scholes formula under ideal conditions.” Journal of Financial Quantitative Analysis 15, 1081–1105. Bjork, T. 1998. Arbitrage theory in continuous time, Oxford University Press, Oxford. Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 81, 637–654. Black, F. 1976. “The pricing of commodity contracts.” Journal of Financial Economics 3, 167–178.

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Bodurtha, J.N. and G.R. Courtadon. 1987. “Tests of an American option pricing model on the foreign currency options market.” Journal of Financial and Quantitative Analysis 22, 153–168. Boyler, P.P. and A. L. Ananthanarayanan. 1977. “The impact of variance estimation in option valuation models.” Journal of Financial Economics 5, 375–387. Brealey, R., and S. Myers. 1996. Principles of corporate finance, 5th ed., McGraw-Hill, New York. Britten-Jones, M. and A. Neuberger. 1996. “Arbitrage pricing with incomplete markets.” Applied Mathematical Finance 3, 347–363. Chance, D. M. 1986. “Empirical tests of the pricing of index call options.” Advances in Futures and Options Research 1(A), 141–166. Chance, D. 2006. An introduction to derivatives and risk management, Thompson South Western, Mason. Chiras, D. and S. Manaster. 1978. “The information content of option prices and a test of market efficiency.” Journal of Financial Economics 6, 213–234. Cox, D.R. and H.D. Miller. 1990. The theory of stochastic processes, Chapman & Hall, New York. Cox, J. C. and S. A. Ross. 1976. “The valuation of options for alternative stochastic processes.” Journal of Financial Economics 3, 145–226. Cox, J. C. and M. Rubinstein. 1985. Options markets, Prentice-Hall, New Jersey. Das, S.R. and R.K. Sundaram. 1997. Of smiles and smirks: a term structure perspective, working paper, Harvard Business School and Stern School of Business, NYU, New York. Delbaen, F. 1992. “Representing martingale measures when asset prices are continuous and bounded.” Mathematical Finance 2, 107–130. Delbaen, F. and W. Schachermayer. 1994. “A general version of the fundamental theorem of asset pricing.” Mathematical Annals 300, 463–520. Derman, E., I. Kani, and J. Zhou. 1996. “The local volatility surface: unlocking the information in index option prices.” Financial Analyst’s Journal 52, 25–36. Duffie, D. 1996. Dynamic asset pricing theory, 2nd ed., Princeton University Press, New Jersey. Duffie, D. and C. Huang. 1985. “Implementing arrow-debreu equilibria by continuous trading of few long-lived securities.” Econometrica 53, 1337–1356. Dupire, B. 1994. “Pricing with a smile.” Risk 7, 32–39. Fama, E. 1970. “Efficient capital markets: a review of theory and empirical work.” Journal of Finance 25, 383–417. Fama, E. 1991. “Efficient capital markets: II.” Journal of Finance 46, 1575–1617. Galai, D. 1983. “A survey of empirical tests of option-pricing models.” in Option pricing, M. Brenner (Ed.). Lexington Books, Lexington, pp. 45–80. Garven, J. R. 1986. “A pedagogic note on the derivation of the Black– Scholes option pricing formula.” The Financial Review 21, 337–344. Gikhman, I. and A. V. Skorokhod. 1969. Investment to the theory of random processes, Saunders, New York. Harrison, J. M. and D. M. Kreps. 1979. “Martingales and arbitrage in multiperiod securities markets.” Journal of Economic Theory 20, 381–408. Harrison, J. and S. Pliska. 1981. “Martingales and stochastic integrals in the theory of continuous trading.” Stochastic Processes and Their Application 2, 261–271. Haug, E.G. 1998. Option pricing formulas, McGraw-Hill, New York. Heston, S.L. 1991. “A closed form solution for options with stochastic volatility with application to bond and currency options.” Review of Financial Studies 6, 327–343. Hull, J. and A. White. 1987. “The pricing of options on assets with stochastic volatilities.” Journal of Finance 42, 281–300. Hull, J. 2006. Options, futures and other derivatives, 6th ed., Pearson Prentice Hall, New Jersey.

466 Ingersoll, J. E. 1976. “A theoretical and empirical investigation of the dual purpose funds: an application of contingent claims and analysis.” Journal of Financial Economics 3, 83–123. Jarrow, R. and A. Rudd. 1983. Option pricing, R. D. Irwin, IL. Karatzas, I. and S. Shreve. 2005. Brownian motion and stochastic calculus, Corrected Eighth Printing, Springer, New York. Kreps, D. M. 1981. “Arbitrage and equilibrium in economics with infinitely many commodities.” Journal of Mathematical Economics 8, 15–35. Kreps, D. M. 1982. “Multiperiod securities and the efficient allocation of risk: a comment on the Black–Scholes Option pricing model.” in The economics of information and uncertainty, J. J. McCall (Ed.). University of Chicago Press, Chicago. Lackner, P. 1993. “Martingale measure for a class of right continuous processes.” Mathematical Finance 3, 43–53. Lamberton, D. and B. Lapeyre. 2007. Introduction to stochastic calculus applied to finance, Chapman & Hall, New York. Latané, H. A. and R. J. Rendleman, Jr. 1976. “Standard derivations of stock price ratios implied in option prices.” Journal of Finance 31, 369–381. Leland, H.E. 1985. “Option pricing and replication with transaction costs.” Journal of Finance 40, 1283–1301. MacBeth, J. D. and L. J. Merville. 1979. “An empirical examination of the black–scholes call option pricing model.” Journal of Finance 34, 1173–1186. Malliaris, A. G. 1983 “Itô’s calculus in financial decision making.” Society of Industrial and Applied Mathematical Review 25, 482–496. Malliaris, A. and W. A. Brock. 1982. Stochastic methods in economics and finance, North-Holland, Amsterdam. McDonald, R. 2005. Derivatives markets, Pearson Addison Wesley, Boston. Merton, R. C. 1973. “The theory of rational option pricing.” Bell Journal of Economics and Management Science 4, 141–183. Merton, R. C. 1976. “Option pricing when underlying stock returns are discontinuous.” Journal of Financial Economics 3, 125–144. Merton, R. C. 1982. “On the mathematics and economics assumption of continuous-time model.” in Financial Economics: Essay in Honor of Paul Cootner, W. F. Sharpe and C. M. Cootner (Ed.). Prentice-Hall, New Jersey, 19–51. Merton, R. C. 1992. Continuous-time finance, Wiley-Blackwell, New York. Musiela, M. and M. Rutkowski 1997. Martingale methods in financial modeling, Springer, New York. Neftci, S. 2000. Introduction to the mathematics of financial derivatives, Academic Press, New York. Oksendal, B. 2007. Stochastic differential equations, 5th ed., Springer, New York. Pliska, S.R. 1997. Introduction to mathematical finance, Blackwell, Oxford. Rebonato, R. 1998. Interest rate option models, 2nd ed., Wiley, New York. Rebonato, R. 2002. Modern pricing of interest rate derivatives, Princeton University Press, New Jersey. Rebonato, R. 2004. Volatility and correlation: the perfect hedger and the fox, 2nd ed., Wiley, New York. Roll, R. 1977. “An analytic valuation formula for unprotected American call options on stocks with known dividends.” Journal of Financial Economics 5, 251–281. Rubinstein, M. 1976. “The valuation of uncertain income streams and the pricing of options.” Bell Journal of Economics 7, 407–425. Scholes, M. 1976. “Taxes and the pricing of options.” Journal of Finance 31, 319–332. Schweizer, M. 1992. “Martingale densities for general asset prices.” Journal of Mathematical Economics 21, 363–378.

G. Chalamandaris and A.G. Malliaris Shastri, K. and K. Tandon. 1986. “Valuation of foreign currency options: some empirical results.” Journal of Financial and Quantitative Analysis 21, 377–392. Smith, C. W., Jr. 1976. “Option pricing: a review.” Journal of Financial Economics 3, 3–51. Smith, C. W., Jr. 1979. “Applications of option pricing analysis.” in Handbook of financial economics, J. L. Bicksler (Ed.). North-Holland, Amsterdam, pp. 79–121. Stricker, C. 1990. “Arbitrage et Lois de Martingale.” Annales de l’Institut Henri Poincare (B) Probability and Statistics 26, 451–460. Taleb, N. 1997. Dynamic hedging, Wiley, New York. Wilmott, P. 2000. Paul Wilmott on quantitative finance, Wiley, New York.

Appendix 30A An Alternative Method To Derive the Black–Scholes Option-Pricing Model Perhaps it is unclear why it is assumed that investors have risk-neutral preferences when the usual assumption in finance courses is that investors are risk averse. It is feasible to make this simplistic assumption because investors are able to create riskless portfolios by combining call options with their underlying securities. Since the creation of a riskless hedge places no restrictions on investor preferences other than nonsatiation, the valuation of the option and its underlying asset will be independent of investor risk preferences. Therefore, a call option will trade at the same price in risk-neutral economy as it will in a risk-averse or risk-preferent economy.

30A.1 Assumptions and the Present Value of the Expected Terminal Option Price To derive the Black–Scholes formula it is assumed that there are no transaction costs, no margin requirements, and no taxes; that all shares are infinitely divisible; and that continuous trading can be accomplished. It is also assumed that the economy is risk neutral. In the risk-neutral assumptions of Cox and Ross (1976) and Rubinstein (1976), today’s option price can be determined by discounting the expected value of the terminal option price by the riskless rate of interest. As was seen earlier, the terminal call-option price can take on only two values: St  X if the call option expires in the money, or 0 if the call expires out of the money. So today’s call option price is C D exp .rt/ Max .St  X; 0/ where: C D the market value of the call option; r D riskless rate of interest; t D time to expiration;

(30A.1)

30 Itô’s Calculus and the Derivation of the Black–Scholes Option-Pricing Model

St D the market value of the underlying stock at time t; and X D exercise or striking price. Equation (30A.1) says that the value of the call option today will be either St  X or 0, whichever is greater. If the price of stock at time t is greater than the exercise price, the call option will expire in the money. This simply means that an investor who owns the call option will exercise it. The option will be exercised regardless of whether the option holder would like to take physical possession of the stock. If the investor would like to own the stock, the cheapest way to obtain the stock is by exercising the option. If the investor would not like to own the stock, he or she will still exercise the option and immediately sell the stock in the market. Since the price the investor paid .X / is lower that the price he or she can sell the stock for .St /, the investor realizes an immediate the profit of St  X . If the price of the stock .St / is less than the exercise price .X /, the option expires out of the money. This occurs because in purchasing shares of the stock the investor will find it cheaper to purchase the stock in the market than to exercise the option. Assuming that the call option expires in the money, then the present value of the expected terminal option is equal to the present value of the difference between the expected terminal stock price and the exercise price, as indicated in Equation (30A.2): C D exp .rt/ E ŒM ax .St  X; 0/ Z 1 .St  X/ h .St / dSt (30A.2) D exp .rt/ x

where h .St / is the log normal density function of St . To evaluate the integral in (30A.2) rewrite it as the difference between two integrals: Z

Z

1 x



1

St h .St / dSt  X

C D exp .rt /

h .St / dSt x

D Ex .St /  exp .rt /  X  exp .rt/  Œ1  H .X / (30A.3) where: Ex .St / D the partial expectation of St ; truncated from below at xI and H .X / D the probability that St  X: Equation (30A.3) says that the value of the call option is present value of the partial expected stock price (assuming the call expires in the money) minus the present value of the exercise price (adjusted by the probability that the stock’s price will be less than the exercise price at the expiration of the option). The terminal stock price St , can be rewritten

467

as the product of the current price .S / and the t-period log normally distributed price ration St =S , so St D S .St =S /. Equation (30A.3) can also be rewritten:

St dSt St g S S x=s S

 Z 1 St dSt g X S S x=s

St D S exp .rt / Ex=S S   X X exp .rt / 1  G (30A.4) S

 Z C D exp .rt/ S

1

where:

St g D log normal density function of St =S I S

St D the partial expextation of St =S; Ex=S S truncated from below at x=S I X D the probability that St =S  X =S: G S

30A.2 Present Value of the Partial Expectation of the Terminal Stock Price The right-hand side of Equation (30A.4) is evaluated by considering the two integrals separately. The first integral, S exp .rt/ Ex=S .St =S /, can be solved by assuming the return on the underlying stock follows a stationary random walk. That is, St D exp .Kt/ (30A.5) S where K is the rate of return on the underlying stock per unit of time. Taking the natural logarithm of both sides of Equation (30A.5) yields:

St D .Kt/ ln S Since the ratio St =S is log normally distributed, it follows that Kt is log normally distributed with density f .Kt/, mean K t, and variance K2 t. Because St =S D exp .Kt/, the differential can be rewritten, d St =S D exp .Kt/ tdK. g .St =S / is a density function of a log normally distributed variable St =S ; so it can be transformed into a density function of a normally distributed variable Kt according to the relationship St =S D exp .Kt/ as: g

St S



D f .Kt/

St S

(30A.6)

468

G. Chalamandaris and A.G. Malliaris

These transformations allow the first integral in Equation (30A.4) to be rewritten: S exp .rt / Ex=S Z

St S



f .Kt/ exp .Kt/ t dK:



ln.x=S /

Because Kt is normally distributed, the density f .Kt/ with mean K t and variance K2 t is     1=2 1 f .Kt/ D 2 K2 t exp  .Kt  K t/2 K2 2 Substitution yields: S exp .rt / Ex=S Z

St S



 1=2 D S exp .rt / 2 K2 t

1



exp ŒKt exp ln.x=S /

   1 2 2

 .Kt  K t/ K t t dK 2

St S



S exp .rt / Ex=S

 1=2 St D SE exp .rt/ 2 K2 t   Z 1S   2. 2 1 2

exp .Kt/exp  Kt  K C K t K t 2 ln.x=S / (30A.9) Since the equilibrium rate of return in a risk-neutral economy is the riskless rate, E .St =S / may be rewritten as exp .rt/:

St SE S

exp .rt/ D S exp .rt / exp .rt / D S

So Equation (30A.9) becomes (30A.7)

Equation (30A.7)’s integrand can be simplified by adding the terms in the two exponents, multiplying and dividing the re sult by exp  12 K2 t . First, expand the term .Kt  K t/2 and factor out t so that:    1 2 2 exp ŒKt exp  .Kt  K t / K t 2 Next, factor out t so:  ı  1  exp .Kt/ exp  t K 2  2K K C 2K K2 2 

Now combine the two exponents: 

(30A.8)

   In Equation (30A.8), exp K C 12 K2 t D E .St =S /, the mean of the t-period log normally distributed price ration St =S . So, Equation (30A.7) becomes:

1

D S exp .rt /

     2 1 K2 t

exp  Kt  K C K2 t 2

ı  1  exp  t .K 2  2K K C 2K  2K2 K/ K2 2

 1=2 St D S 2 K2 t S exp .rt / Ex=S S   Z 1   2 . 2 1 2 K t t dK

exp  Kt  K C K t 2 ln.x=S/

(30A.10) To complete the simplification of this part of the Black– Scholes formula, define a standard normal random variable y:    . y D Kt  K C K2 t K2 t 1=2 : Solving for Kt yields:   Kt D K C K2 t C K t 1=2 y and therefore: t dK D K t 1=2 dy:



  Now, multiply and divide this result by exp  12 K2 t to get:

By making the transformation from Kt to y the lower limit of integration becomes 

  . ln .x=S /  K C K2 t K t 1=2 :

  ı 2 1  2 2 2 4 4 exp  t .K  2K K C K  2K K C K  K / K Further simplify the integrand by noting that the assumption 2 of a risk neural economy implies: Next, rearrange and combine terms to get:

  1 2 

h  exp K C K t D exp .rt / i.  2 1 2 exp  t K2 K  K  K2  K4  2K K2 2 Taking the natural logarithm of both sides yields:

  1 2 D exp K C K t

2 1 2 K C K t D .rt/ 2

30 Itô’s Calculus and the Derivation of the Black–Scholes Option-Pricing Model

    Hence, K C 12 K2 t D r C 12 K2 t. The lower limit of integration is now:

  1 K t 1=2 D d1 :  ln .S =x/ C r C K2 t 2 Substituting this into Equation (30A.10) and making the transformation to y yields:

,

Z 1 St 1 2 .2 /1=2 dy DS S exp .rt/ Ex=S exp  y S 2 d1 Since y is a standard normal random variable (distribution is symmetric around zero) the limits of integration can be exchanged: S exp .rt / Ex=S

 St  S

DS

R d1 1

.  exp  12 y 2 .2 /1=2 dy

D SN .d1 / (30A.11)

469

The integrand is now simplified by following the same procedure used in simplifying the previous integral. Define a standard normal random variable Z: 

Kt  K t ZD K t 1=2



Solving for Kt yields: Kt D K t C K t 1=2 Z and t dK D K t 1=2 dZ. Making the transformation from Kt to Z means the lower limit of integration becomes ln .X =S /  K t : K t 1=2 Again, note that the assumption of a risk-neutral economy implies:

1 2 exp K C K t D exp .rt / 2

where N .d1 / is the standard normal cumulative distribution function evaluated at y D d1 .

Taking the natural logarithm of both sides yields:

1 K C K2 t D rt 2

30A.3 Present Value of the Exercise Price under Uncertainty To complete the derivation, the integrals that corresponds to the term X exp .rt/ Œ1  G .X =S / must be evaluated. Start by making the logarithmic transformation: ln

St S



or:

1 2 K t D r  K t: 2

Therefore, the lower limit of integration becomes:

D Kt

This transformation allows the rewriting of g .St =S / to .S =St / f .Kt/ as mentioned previously. The differential can be written: St D exp .Kt/ t dK d S Therefore,

       ln .S =x/ C r  12 K2 t D  d1  K t 1=2 1=2 K t D d2 Substitution yields: Z

1

x exp .rt/ Œ1  G .X =S/ D x exp .rt /

exp d2

   Z d2 1

 Z 2 .2 /1=2 dZ D x exp .rt/ exp 2 1    1 2 1=2

 Z .2 / dZ D x exp .rt/ N .d2 / 2

X exp .rt/ Œ1  G .X =S / Z

1

D X exp .rt /

f .Kt/ t dK ln.X =S/

D X exp .rt / 



1=2 2 K2 t

1  .Kt  K t/2 2

Z

1

exp

(30A.13)

ln.X =S /

  K2 t t dK

(30A.12)

where N .d2 / is the standard normal cumulative distribution function evaluated at Z D d2 .

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G. Chalamandaris and A.G. Malliaris

Substituting, Equation (30A.11) and (30A.13) into Equation (30A.4) completes the derivation of the Black– Scholes formula: C D SN .d1 /  X exp .rt / N .d2 /

(30A.14)

This appendix provides a simple derivation of the Black– Scholes call-option pricing formula. Under an assumption of risk neutrality the Black–Scholes formula was derived using only differential and integral calculus and a basic knowledge of normal and log normal distributions.

Chapter 31

Constant Elasticity of Variance Option Pricing Model: Integration and Detailed Derivation Y.L. Hsu, T.I. Lin, and C.F. Lee

Abstract In this paper we review the renowned Constant Elasticity of Variance (CEV) option pricing model and give the detailed derivations. There are two purposes of this article. First, we show the details of the formulae needed in deriving the option pricing and bridge the gaps in deriving the necessary formulae for the model. Second, we use a result by Feller to obtain the transition probability density function of the stock price at time T given its price at time t with t < T . In addition, some computational considerations are given which will facilitate the computation of the CEV option pricing formula. Keywords Constant elasticity of variance model r Noncentral Chi-square distribution r Option pricing

t with t < T . We also showed the details of the formulae needed in deriving the option pricing. A proof of Feller’s result is given in the Appendix.

31.2 The CEV Diffusion and Its Transition Probability Density Function The CEV option pricing model assumes that the stock price is governed by the diffusion process dS D Sdt C S ˇ=2 dZ;

Y.L. Hsu () and T.I. Lin Department of Applied Mathematics and Institute of Statistics, National Chung Hsing University, Taichung, Taiwan e-mail: [email protected] C.F. Lee Graduate Institute of Finance, National Chaio Tung University, Hsinchu, Taiwan and Department of Finance, Rutgers University, New Brunswick, NJ, USA

(31.1)

where dZ is a Wiener process and  is a positive constant. The elasticity is ˇ  2 since the return variance .S; t/ D  2 S ˇ2 with respect to price S has the following relationship

31.1 Introduction Cox (1975) has derived the renowned Constant Elasticity of Variance (CEV) option pricing model and Schroder (1989) has subsequently extended the model by expressing the CEV option pricing formula in terms of the noncentral Chi-square distribution. However, neither of them has given details of their derivations as well as the mathematical and statistical tools in deriving the formulae. There are two purposes of this article. First, we integrated the results obtained by Cox (1975) and Schroder (1989) and bridged the gaps in deriving the necessary formulae for the model. Second, we use a result by Feller (1951) to obtain the transition probability density function of the stock price at time T given its price at time

ˇ < 2;

d .S; t/=dS D ˇ  2; .S; t/=S which implies that d .S; t/=.S; t/ D .ˇ  2/dS=S . Upon integration on both sides, we have log .S; t/ D .ˇ  2/ log S C log  2 , or .S; t/ D .S; t/ D  2 S ˇ2 . If ˇ D 2, then the elasticity is zero and the stock prices are lognormally distributed as in the Black and Scholes model. If ˇ D 1, then Equation (31.1) is the model proposed by Cox and Ross (1976). In this article, we will focus on the case of ˇ < 2 since many empirical evidences (see Campbell 1987, Glosten et al. (1993), Brandt and Kang 2004) have shown that the relationship between the stock price and its return volatility is negative. The transition density for ˇ > 2 is given by Emanuel and Macbeth (1982) and the corresponding CEV option pricing formula can be derived through a similar strategy. For more details, see Chen and Lee (1993).



This paper is dedicated to honor and memory my Ph.D. advisor, Prof. Jack C. Lee, who died from cardiovascular disease on 2 March 2007. This paper is reprinted from Mathematics and Computers in Simulation, 79 (2008), pp. 60–71.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_31, 

471

472

Y.L. Hsu et al.

In order to derive the CEV option pricing model, we need the transition probability density function f .ST jSt ; T > t/ of the stock price at time T given the current stock price St . For the transition probability density function f .ST jSt /, we will start with the Kolmogorov forward and backward equations. Assume Xt follows the diffusion process dX D .X; t/dt C .X; t/dZ;

(31.2)

and P D P .Xt ; t/ is the function of Xt and t, then P satisfies the partial differential equations of motion. From Equation (31.2), we have the Kolmogorov backward equation, @2 P @P @P 1 2  .X0 ; t0 / 2 C .X0 ; t0 / C D 0; 2 @X0 @t0 @X0

(31.3)

and the Kolmogorov forward (or Fokker–Planck) equation i @P h i @P 1 @2 h 2  .Xt ; t/P  .Xt ; t/P  D 0: 2 2 @Xt @Xt @t (31.4) Consider the following parabolic equation

.P /t D .axP /xx  .bxCh/P ; x

0 < x < 1;

(31.5)

where P D P .x; t/, and a; b; h are constants with a > 0, .P /t is the partial derivative of P with respect to t, ./x and ./xx are the first and second partial derivatives of ./ with respect to x. This can be interpreted as the Fokker–Planck equation of a diffusion problem in which bx C h represents the drift, and ax represents the diffusion coefficient. Lemma 1. (Feller (1951)): Let f .x; t j x0 / be the probability density function for x and t conditional on x0 . The explicit form of the fundamental solution to the above parabolic equation is given by f .t; x j x0 / D

b



e bt x x0

Before pursuing further, we will first consider the special case in which ˇ D 1 which is the model considered by Cox and Ross (1976). In this situation we have dS D .S; t/dt C .S; t/dZ;

@P @P 1 2 @2 P  S 2 C Œ.r  a/S  h C D rP; 2 @S @S @t

(31.9)

and the corresponding Kolmogorov forward equation for the diffusion process (Equation 31.8) is  i @P @ h 1 @2 2 D 0; . S P / C .r  a/S  h P  T 2 @St2 @ST @t (31.10) which is obtained by using (31.4) with .xt ; t/ D .r a/S h. Comparing with Equation (31.6), we set a D  2 =2, x D ST , x0 D St , b D r   2 =2, h D h and t D  D .T  t/. Thus, we have the following transition probability density function for the Cox–Ross model: f .ST jSt ; T > t/ D

exp a.e bt  1/   b.x C x0 e bt /

a.e bt  1/

2b bt 1=2

I1h=a .e ; xx / 0 a.1  e bt /

where Ik .x/ is the modified Bessel function of the first kind of order k and is defined as

(31.8)

p where .S; t/ D  S. Now suppose also that each unit of the stock pays out in dividends in the continuous stream b.S; t/ so that the required mean becomes .S; t/ D rS  b.S; t/ D rS  .aS C h/, where b.S; t/ D aS C h and r isp the risk-free interest rate. Then dS D Œ.r  a/S  h dt C  S dZ and the differential option price equation becomes

.ha/=2a

(31.6)



Proof. See the Appendix.

2.r   2 =2/  2 Œe .r 2 =2/  1 !.1C2h= 2 /=2 2 St e .r =2/

exp ST ( ) 2 2.r   2 =2/ŒST C St e .r =2/

 2 Œe .r 2 =2/  1

I1C2h= 2 2

4.r   2 =2/.St ST e .r =2/ /1=2

 2 Œe .r 2 =2/  1  2

! :

(31.11) We next consider the constant elasticity of variance diffusion, dS D .S; t/ C .S; t/dZ; (31.12) where

Ik .x/ D

X1

.x=2/2rCk : rD0 rŠ.r C 1 C k/

(31.7)

.S; t/ D rS  aS;

(31.13)

31 Constant Elasticity of Variance Option Pricing Model

473

and .S; t/ D S ˇ=2 ;

0  ˇ < 2:

(31.14)

Then dS D .r  a/Sdt C S ˇ=2 dZ: Let Y D Y .S; t/ D S

2ˇ

(31.15)

. By Ito’s Lemma with

For a proof of the above formula, see Chen and Lee (1993). We next present the detailed derivations of the option pricing formula as presented by Schroder (1989). Since the option pricing formula is expressed in terms of the noncentral Chi-square complementary distribution function, a brief review of the noncentral Chi-square distribution is presented in the next section.

@2 Y @Y @Y D .2ˇ/S 1ˇ ; D 0; D .2ˇ/.1ˇ/S ˇ ; @S @t @S 2 we have h i 1 d Y D .r  a/.2  ˇ/Y C  2 .ˇ  1/.ˇ  2/ dt 2 C 2 .2  ˇ/2 Y dZ:

(31.16)

31.3 Review of Noncentral Chi-Square Distribution If Z1 ; : : : ; Z are standard normal random variables, and ı1 ; : : : ; ı are constants, then

The Kolmogorov forward equation for Y becomes  1 @2  2 @P D  .2  ˇ/YP 2 @t 2 @Y    1 @ .r  a/.2  ˇ/Y C  2 .ˇ  1/.ˇ  2/ P :  @Y 2 (31.17) Then f .ST j St ; T > t/ D f .YT j yt ; T > t/ j J j where J D .2ˇ/S 1ˇ . By Feller’s Lemma with a D 12  2 .2ˇ/2 , b D .r  a/.2  ˇ/, h D 12  2 .ˇ  2/.1  ˇ/, x D 1=T , x0 D 1=t and t D  D .T  t/, we have  1=.2.2ˇ// f .ST j St ; T > t/ D .2  ˇ/k 1=.2ˇ/ xz1ˇ  

e xz I1=.2ˇ/ 2.xz/1=2 ; (31.18)

Y D

 X .Zi C ıi /2

(31.20)

i D1

is the noncentral Chi-square distribution with  degrees of P freedom and noncentrality parameter  D j D1 ıj2 , and is 2

denoted as 0 ./. When ıj D 0 for all j , then Y is distributed as the central Chi-square distribution with  degrees of freedom, and is denoted as 2 . The cumulative distribution 2 function of 0 ./ is 2

F .xI ; / D P .0 ./  x/ D e =2 Z

x



where

1 X

.=2/j j Š2=2 .=2 C j / j D0 y

y =2Cj 1 e  2 dy;

x > 0:

0

k D

2.r  a/ ;  2 .2  ˇ/Œe .ra/.2ˇ/  1 2ˇ .ra/.2ˇ/

x D k  St

e

;

(31.21) 2ˇ

z D k  ST

An alternative expression for F .xI ; / is

:

Cox (1975) obtained the following option pricing formula:   1 X e x x n G n C 1 C 1=.2  ˇ/; k  K 2ˇ r C D St e .n C 1/ nD0   1 X e x x nC1=.2ˇ/ G n C 1; k  K 2ˇ r   Ke ;  n C 1 C 1=.2  ˇ/ nD0 (31.19) R1 where G.m; / D Œ.m/ 1  e u um1 du is the standard complementary gamma distribution function.

F .xI ; / D

1 X .=2/j e =2 j D0

jŠ

P .2C2j  x/: (31.22) 2

The complementary distribution function of 0 ./ is Q.xI ; / D 1  F .xI ; /;

(31.23)

where F .xI ; / is given in either Equation (31.21) or Equation (31.22). 2 The probability density function of 0 ./ can be expressed as a mixture of central Chi-square probability density functions.

474

Y.L. Hsu et al.

p02 ./ .x/ D e =2 

1 X . 1 /j 2

jŠ

j D0

D

e

p2

1 .xC/=2 X

C2j

31.4.1 Detailed Derivations of C 1 and C 2

.x/

2ˇ

Making the change of variable w D k  ST

x  : 2j j Š .=2 C j /2 j D0

2=2

(31.24) An alternative expression for the probability density function 2 of 0 ./ is

dST D .2  ˇ/1 k 

Z C1 D e

I

Ik .z/ D

1

p .2 xw/.w=x/1=.2ˇ/ .x=k  /1=.2ˇ/ d w Z 1 D e r .x=k  /1=.2ˇ/ e xw .x=w/1=.42ˇ/

k X

2

i D1

1 2ˇ

p

I 1 .2 xw/d w 2ˇ Z 1 D e r St e .ra/ e xw .w=x/1=.42ˇ/

(31.26)

y

e z cos.k / cos.k/d D Ik .z/:

i D1

e xw .x=w/1=.42ˇ/ y

p

I 1 .2 xw/d w 2ˇ Z 1 a D e St e xw .w=x/1=.42ˇ/ I

(31.27)

0

Xi  0Pk

1

y

The noncentral Chi-square distribution satisfies the reproductivity property with respect to n and . If X1 ; : : : ; Xk are 2 independent random variables with Xt distributed as 0ni .i /, then Y D

p .2 xw/.w=k  /1=.2ˇ/ d w

Z



k X

1 2ˇ

D e r

It is noted that for integer k, Z

e xw .x=w/1=.42ˇ/ y

where Ik is the modified Bessel function of the first kind of order k and is defined as Ik .z/ D

w.ˇ1/=.2ˇ/ d w:

1

r

I

 2 j

1 z =4 1 kX z : 2 j Š.k C j C 1/ j D0

1=.2ˇ/

Thus, with y D k  K 2ˇ , we have,

  1 x .2/=4 1 p02 ./ .x/ D exp  . C x/  2  2 p

I.2/=2 . x/; x > 0; (31.25)



, we have

=2Cj 1 j

ni

i :

y

p .2 xw/d w; (31.30)

and Z

(31.28)

1 2ˇ

C2 D Ke

1

r

1=.2ˇ/

.2  ˇ/k 

1

.xw12ˇ / 42ˇ e xw

y

i D1

1=.2ˇ/

ˇ1 p k w 2ˇ d w .2 xw/ 2ˇ Z 1 1 D Ke r x 42ˇ w.12ˇC2ˇ2/=.42ˇ/ e xw

I

31.4 The Noncentral Chi-square Approach to Option Pricing Model

1 2ˇ

y

Following Schroder (1989), with the transition probability density function given in (31.18), the option pricing formula under the CEV model is

 DT t C D E max.0; ST  K/ ; Z 1 D e r f .ST j St ; T > t/.ST  K/dST K Z 1

D e r

ST f .ST j St ; T > t/dST K

e

r

Z

1

y

(31.29)

1 2ˇ

p .2 xw/d w: (31.31)

Recall that the probability density function of the noncentral Chi-square distribution with noncentrality  and degree of freedom  is



K

D C1  C2 :

1 2ˇ

p 1 .x=/.2/=4 I.2/=2 . x/e .Cx/=2 2 D P .xI ; /:

p02 ./ .x/ D

f .ST j St ; T > t/dST

K

p .2 xw/d w Z 1 D Ke r e xw .x=w/1=.4ˇ/ I

I

31 Constant Elasticity of Variance Option Pricing Model

475

R1 Let Q.xI ; / D x p02 ./ .y/dy. Then letting w0 D 2w  and x 0 D 2x, we have Z C1 D St e

1

a

e .xCw/=2

w 1=.42ˇ/ x

y

Z

1

D St e a

e .x

0

0

Cw /=2

1 2ˇ

C2 D Ke

x 1=.42ˇ/

e

w

y

Z

1

D Ke r

e .x

0

I

 p  2 xw d w

1 2ˇ

!1=.42ˇ/ 0

0

Cw /=2

2y

(31.35)

(31.32)

obtained by noting that .  2/=2 D 1=.2  ˇ/, implying 0 0  D 2 C 2=.2  ˇ/. Analogously, with w D 2w, x D 2x and In .z/ D In .z/, we have xw

D

C D St e

Z

1

1

P .2yI 2; 2k/d k D y

Z

nD0 1

D y

D

1 X nD0

z 1 z

g.i; y/:

i D1

(31.36)

p 1 y .2/=4 I 2 ky e .kCy/=2 dy 2 2 k

.2/=2 1 y .2/=4 1 p ky 2 k 2





Z

1





D

1 X

e y=2 y

2

e k=2

Z

1

z

D



 C2n  dy 2

n

nD0

1 X

C2n 2 1

 1 .C2n/=2

z

2 ; 2x : (31.34) 2ˇ

.k=2/  .n C 1/

.1=2/.C2n/=2 y=2 C2n 1  e  y 2 dy  C2n 2

e k=2

nD0

.k=2/n Q.zI  C 2n; 0/; (31.37)  .n C 1/

n

.zk/ dk nŠ.n C   1 C 1/

e z znC1 .n C /

D

1 X .ky=4/n .kCy/=2 e  dy nŠ C2n 2 nD0   1 n 1 n X k=2 k 2 e D  .n C 1/ nD0

(31.33)

y



1

D

p 1 e zk .z=k/1 kz

1 X

n X

Next, applying the monotone convergence theorem, we have

It is noted that 2  2=.2  ˇ/ can be negative for ˇ < 2. Thus further work is needed. Using the monotone convergence theorem and the integration by parts, we have Z

g.i; y/, which

G.m C 1; t/ D g.m C 1; t/ C G.m; t/:

Z

r Ke Q 2yI 2 

i D1

The above result can also be expressed as

Q.zI ; k/ D

2 ; 2x 2ˇ

.i /

i D1

obtained by noting that .   2/=2 D 1=.2  ˇ/, implying   D 2  2=.2  ˇ/. Thus, Q 2yI 2 C

P1

1

n X y i 1 e y

Z

x w0

p

1 0 dw

I 1 2 x 0 w0 2ˇ 2

2 ; 2x ; D Q 2yI 2  2ˇ

a

g.i; y/:

Z 1 n1 e k k n1 k G.n; y/ D dk D  de k .n/ .n/ y y Z 1 n2 k k e y n1 e y C dk D .n/ .n  1/ y

0

1

1 X i D1

nD0

Z

D St e a Q.2yI ; x /;

2 a ; 2x ; D St e Q 2yI 2 C 2ˇ

Z

g.n C   1; z/

Now we also have the result G.n; y/ D can be shown by observing that

p

1 0 dw 2 x 0 w0 2

r

1 X

 p  2 xw d w

1 2ˇ

!1=.42ˇ/ 0 w x0

2y

I

I

D

Z

1 y

e k k n dk .n C 1/

g.n C ; z/G.n C 1; y/

where Z

1

Q.zI  C 2n; 0/ D Z

z

.1=2/.C2n/=2 y=2 C2n 1  e  y 2 dy  C2n 2

1

D z=2



C2n 1  C2n  e y y 2 1 dy

2

D G .n C =2; z=2/ :

476

Y.L. Hsu et al.

Furthermore, from the property of G.; / as shown in Equation (31.36), we have Q.zI ; k/ D

1 X

We conclude that Q.2zI 2; 2k/ D 1 

g .n C 1; k=2/ G .n C =2; z=2/

1 X

g .n C ; z/

n X

g .i; k/ :

i D1

nD1

(31.39)

nD0

D

2 ; z=2 : g .n; k=2/ G n C 2 nD0

1 X

(31.38) Hence

From (31.35) and (31.39) we observe that Z

1

P .2zI 2; 2k/d k D 1  Q .2zI 2.  1/; 2y/ : y

(31.40) Thus, we can write C2 as

Q.2zI 2; 2k/ D

1 X

g .n; k/ G .n C   1; z/ :

Z C2 D Ke r

P 2xI 2 C

y

nD0

D Ke r Q 2yI 2 

Again from the property of G.; / as given by (31.36), we have Q.2zI 2; 2k/ D g .1; k/ G .; z/ C g .2; k/ G . C 1; z/

D Ke

Cg .3; k/ G . C 2; z/ C    D g .1; k/ ŒG .  1; z/ C g .; z/ Cg .2; k/ ŒG .  1; z/ C g . C 1; z/ Cg .3; k/ ŒG .  1; z/ C g .; z/

D ŒG .  1; z/ C g .; z/

1 X nD1

1 X

g .n; k/

r

1  Q.2xI

2 ; 2w d w 2ˇ

2 ; 2x 2ˇ



2 ; 2y/ : 2ˇ

(31.41)

From (31.41) we immediately obtain Q 2yI 2 

Cg . C 1; z/ C g . C 2; z/ C   

Cg . C 1; z/

1



2 2 ; 2x C Q 2xI ; 2y D 1 2ˇ 2ˇ (31.42)

implying g .n; k/ Q.zI 2n; k/ C Q.kI 2  2n; z/ D 1;

(31.43)

with degrees of freedom 2  2n of Q.kI 2  2n; z/ can be a non-integer. 1 X From Equation (31.42), we can obtain that the noncentral g .n; k/ C    Cg . C 2; z/ Chi-square Q.2yI 2  2=.2  ˇ/; 2x/ with 2  2=.2  ˇ/ nD3 degrees of freedom and the noncentrality parameter 2x can D G .  1; z/ C g .; z/ be represented by another noncentral Chi-square distribution Cg . C 1; z/ Œ1  g .1; k/ 1  Q.2xI 2=.2  ˇ/; 2y/ with degrees of freedom 2=.2  ˇ/ Cg . C 2; z/ Œ1  g .1; k/  g.2; k/ C    and the noncentrality parameter 2y. The standard definition of noncentral Chi-square distribution in Sect. 31.3 has inte1 X ger degrees of freedom. If the degree of freedom is not an D G .  1; z/ C g . C n; z/ integer, we can use Equation (31.43) to transfer the original nD0 noncentral Chi-square distribution into another noncentral g . C 1; z/ Œg .1; k/ Chi-square distribution. Thus, we obtain an option pricing formula for the CEV model in terms of the complementary g . C 2; z/ Œg .1; k/ C g .2; k/ C    noncentral Chi-square distribution function Q.zI ; k/ which D 1  g . C 1; z/ Œg .1; k/ is valid for any value of ˇ less than 2, as required by the model. g . C 2; z/ Œg .1; k/ C g .2; k/     : nD2

31 Constant Elasticity of Variance Option Pricing Model

477

Substituting Equation (31.41) into Equation (31.34), we obtain

2 a C D St e Q 2yI 2 C ; 2x 2ˇ

2 r ; 2y/ ; (31.44) 1  Q.2xI Ke 2ˇ 2ˇ

 2ˇ where K , x D k St e .ra/.2ˇ/, k  D 2.r a/=  2 y D k .ra/.2ˇ/  .2  ˇ/.e  1/ and a is the continuous proportional dividend rate. The corresponding CEV option pricing formula for ˇ > 2 can be derived through a similar manner. When ˇ > 2 (see, Emanuel and Macheth (1982), Chen and Lee (1993)), the call option formula is as follows:

C D St e

a

Ke

Q 2xI

r

2 ; 2y ˇ2

1  Q.2yI 2 C



2 ; 2x/ :(31.45) ˇ2

where  D 2

2 ; 2ˇ 2ˇ .ra/.2ˇ/

 D 2k  St

e

:

Thus, the option pricing formula for the CEV model as given in (31.44) that can be obtained directly from the payoff function  ST  K; if ST > K max.ST  K; 0/ D (31.47) 0; otherwise by taking the expectation of (31.47), with ST having the distribution given by (31.46). Before concluding this subsection we consider that the noncentral Chi-square distribution will approach log-normal as ˇ tends to 2. Since when either  or  approaches to infinity, the standardized variable 2

0 ./  . C / p 2. C 2/

We note that from the evaluation of the option pricing formula C , especially C2 , as given in (31.34), we have 2ˇ

2k  ST

2

 0 ./;

(31.46)

2ˇ

2

lim

ˇ!2

tends to N(0,1) as either  ! 1 or  ! 1. Using the fact that .x a  1/=a will approach to ln x as a ! 0, it can be verified that

0 ./  . C / 2k  ST  . C / D lim p p ˇ!2 2. C 2/ 2. C 2/ 2ˇ

D lim ˇ!2

s

 D

2r ? ST

?

2ˇ r ?  .2ˇ/

 .1  ˇ/ 2 .e r  .2ˇ/  1/  2r ? St  2 .e r ?  .2ˇ/  1/

e

 2 .e r ?  .2ˇ/  1/=.2  ˇ/ 2ˇ r ?  .2ˇ/ e

.1  ˇ/ 2 .e r ?  .2ˇ/  1/ C 4r ? St

ln ST  Œln St C .r ?   2 =2/ p ;  

where r ? D r  a. Thus, ln ST j ln St  N.ln St C .r  a   2 =2/;  2 /

31.4.2 Some Computational Considerations (31.48)

as ˇ ! 2 . Similarly, (Equation 31.48) also holds when 2ˇ  ˇ ! 2C . From (Equation 31.45), we have 2k  ST 02  ./, where  D 2 C 2=.ˇ  2/ if ˇ > 2. Thus, we clarify the result of (Equation 31.48).

As noted by Schroder (1989), (31.39) allows the following iterative algorithm to be used in computing the infinite sum when z and k are not large. First initialize the following four variables (with n D 1) gA D

e z z D g.1 C ; z/; .1 C /

478

Y.L. Hsu et al.

gB D e k D g.1; k/;

Acknowledgements We gratefully acknowledge the editor and an anonymous referee for his insightful comments and suggestions of the paper. Research Supported in Part by NSC grant 95-2118-M-005-003.

Sg D gB; R D 1  .gA/.Sg/: Then repeat the following loop beginning with n D 2 and increase increment n by one after each iteration. The loop is terminated when the contribution to the sum, R, is declining and is very small.

z D g.n C ; z/; nC1

k gB D gB D g.n; k/; n1 gA D gA

Sg D Sg C gB D g.1; k/ C g.n; k/ R D R  .gA/.Sg/ D the nth partial sum. As each iteration, gA equals g.n C ; z/, gB equals g.n; k/ and Sg equals g.1; k/ C    C g.n; k/. The computation is easily done. As for an approximation, Sankaran (1963) showed that the  2 h distribution of 0 =. C k/ is approximately normal with the expected value  D 1 C h.h  1/P  h.2  h/mP 2 =2 and variance  2 D h2 P .1CmP /, where h D 1 23 .Ck/.C3k/ . C2k/2 , P D . C2k/=. Ck/2 and m D .h1/.13h/. Using the approximation, we have approximately

References Brandt, M. W. and Q. Kang. 2004. “On the relationship between the conditional mean and volatility of stock returns: a latent VAR approach.” Journal of Financial Economics 72, 217–257. Campbell J. 1987. “Stock returns and the term structure.” Journal of Financial Economics 18, 373–399. Chen, R. R. and C. F. Lee. 1993. “A constant elasticity of variance (CEV) family of stock price distributions in option pricing: review and integration.” Journal of Financial Studies 1, 29–51. Cox, J. 1975. Notes on option pricing I: constant elasticity of variance diffusion, Unpublished Note, Standford University, Graduate School of Business. Also, Journal of Portfolio Management (1996) 23, 5–17. Cox, J. and S. A. Ross. 1976. “The valuation of options for alternative stochastic processes.” Journal of Financial Economics 3, 145–166. Emanuel, D. and J. MacBeth. 1982. “Further results on the constant elasticity of variance call option pricing formula.” Journal of Financial and Quantitative Analysis 17, 533–554. Feller, W. 1951. Two singular diffusion problems. Annals of Mathematics 54, 173–182. Glostern, L., R. Jagannathan, and D. Runkle. 1993. “On the relation between the expected value and the volatility of the nominal excess returns on stocks.” Journal of Finance 48, 1779–1802. Sankaran, M. 1963. “Approximations to the non-central Chi-square distribution.” Biometrika 50, 199–204. Schroder, M. 1989. “Computing the constant elasticity of variance option pricing formula.” Journal of Finance 44 211–219.

2

Q.zI ; k/ D P r.0 > z/ 2

0 z > Ck Ck

D Pr 0 D Pr @

2

0 Ck

!h

>

!

z  Ck

h

Appendix 31A Proof of Feller’s Lemma

1 A

1hP Œ1hC0:5.2h/mP  Dˆ p h 2P .1 C mP /

We need some preliminary results in order to prove Equation (31.6). 

h! z Ck

:

Proposition 1. f .z/ D e A=zz1 is the Laplace transformation of I0 .2.Ax/1=2 /, where Ik .x/ is the Bessel function Ik .x/ D

rD0

31.5 Conclusion The option pricing formula under the CEV model is quite complex because it involves the cumulative distribution function of the noncentral Chi-square distribution Q.zI ; k/. Some computational considerations are given in the article which will facilitate the computation of the CEV option pricing formula. Hence, the computation will not be a difficult problem in practice.

1 X

.x=2/2rCk : rŠ.r C 1 C k/

Proof. By the definition of Laplace transformation and the monotone convergence theorem, we have Z

1

f .z/ D

e zx I0 .2.Ax/1=2 /dx Z

0 1

D

e 0

zx

1 X .Ax/1=2 rŠ.r C 1/ rD0

31 Constant Elasticity of Variance Option Pricing Model

Z D



1

e

zx

0

479

A1h=a e A D .1  h=a/

.Ax/2 .Ax/ C C 1C .2/ 2Š.3/  .Ax/n C C nŠ.n C 1/

A 1 .A/2 .A/n C 2 C C    C C z z 2Šz3 nŠznC1   A .A/2 1 .A/n 1C C D CC C z z 2Šz2 nŠzn

w.t; sI x0 / D

D e A=z z1 :



Z

1

e sx f .t; xI x0 /dx 0



h=a



sbx0 e bt D exp sa.e bt  1/ C b ! h b 2 x0 e bt   ;

 1  I a a.e bt  1/ sa.e bt C 1/ C b b sa.e bt  1/ C b



(31.50) where .nI z/ D  1 .n/

Rz 0

e x x n1 dx.

Proof. The proof of the lemma is too tedious and hence is omitted. For more details, please see Lemma 7 of Feller (1951).  We now turn to prove Equation (31.6). From Equation (31.50), let AD

 1 bx0 and z D sa.e bt  1/ C b : bt a.1  e / b

The w.t; sI x0 / in Equation (31.50) can be rewritten as zh=a e .11=z/A w.t; sI x0 / D .1  h=a/ D

A=z

e

x h=a

x

dx

0

zh=a e .11=z/A .1  h=a/ Z 1 0 e Ax =z .Ax 0 =z/h=a .A=z/1h=a dx 0

0

D

Z

1h=a A

e A .1  h=a/

Z

1

0

e A.1x /=z .x 0 /h=a z1 dx 0 0

D

(31.49)

where a; b; h are constants, 0 < h < a, then the Laplace transformation of f .t; x; x0 / with respect to x takes the form w.t; sI x0 / D

e A=z .1  /h=a z1 d:

0

Z A1h=a e A 1 .1  /h=a .1  h=a/ 0  Z 1   zx 1=2 dx d

e I0 2.Ax/ 0

Proposition 2. Consider the parabolic differential equation 0 < x < 1;

1

By Proposition 31.1, we know .z/ D e A=z z1 is the  that f1=2 and by the Fubini Laplace transformation of I0 2.Ax/ theorem, we have

D

  Pt D .axP /xx  .bx C h/P x ;

Z

1h=a A

e A .1  h=a/ Z 1Z 1   .1  /h=a e zx I0 2.Ax/1=2 dxd

0

D

0 2h=a A

A e .1  h=a/x0 e bt Z 1Z 1 0 0 bt

.1  /h=a e sx e Ax =.x0 e / 0

0

 

I0 2A.e bt x 0 =x0 /1=2 ddx 0 : Hence, upon comparing the two formulae for w.t; sI x0 / and by the monotone convergence theorem, we have  1h=a b bx0 .1  h=a/a.e bt  1/ a.e bt  1/ Z 1  b.x C x0 e bt /

exp .1  /h=a I0 a.e bt  1/ 0

2b.e bt xx0 /1=2 d

a.1  e bt /  1h=a bx0 b D .1  h=a/a.e bt  1/ a.e bt  1/ Z 1  b.x C x0 e bt /

exp .1  /h=a a.e bt  1/ 0  bt   1 1=2 X b.e xx0 / = a.1  e bt / 2r d

rŠ.r C 1/ rD0

f .t; xI x0 / D

D

 1h=a bx0 b .1  h=a/a.e bt  1/ a.e bt  1/   b.x C x0 e bt /

exp a.e bt  1/  2r 1  X b.e bt xx0 /1=2 = a.1  e bt /

rŠ.r C 1/ rD0 Z

1



.1  /h=a  r d 0

480

Y.L. Hsu et al.

 1h=a bx0 b D .1  h=a/a.e bt  1/ a.e bt  1/   b.x C x0 e bt /

exp a.e bt  1/   2r 1 X b.e bt xx0 /1=2 = a.1  e bt /

rŠ.r C 1/ rD0

.r C 1/.1  h=a/ .r C 1 C 1  h=a/

D

b



e bt x x0

.ha/=2a

a.e bt  1/   b.x C x0 e bt /

exp a.e bt  1/

2b bt 1=2

I1h=a .e xx0 / : a.1  e bt /

This completes the proof.

Chapter 32

Stochastic Volatility Option Pricing Models Cheng Few Lee and Jack C. Lee

Abstract In this chapter, we assume that the volatility of option price model is stochastic instead of deterministic. We apply such assumption to the nonclosed-form solution developed by Scott (Journal of Finance and Quantitative Analysis 22(4):419–438, 1987) and the closed-form solution of Heston (The Review of Financial Studies 6(2):327–343, 1993). In both cases, we consider a model in which the variance of stock price returns varies according to an independent diffusion process. For the closed form option pricing model, the results are expressed in terms of the characteristic function. Keywords Nonclosed-form option pricing model r Closed– forem option pricing model r Heston model r Itô’s Lemma r Characteristic function r Moment generating function

32.1 Introduction The variance of stock returns plays a vital role in option pricing, as evidenced in the Black-Scholes pricing formula. Also, it has long been observed that stock price changes over time. Thus, the assumption of a constant variance of stock price returns does not seem to be reasonable in option price. In this chapter, we will consider the situation in which the variance of stock price returns is not a constant. Instead, we will assume the variance of stock price returns is random and follows some distributions. Emphasis will be placed on the nonclosed-form solution developed by Scott (1987) and the closed-form solution of Heston (1993). In both cases, we consider a model in which the variance of stock price returns varies according to an independent diffusion process.

32.2 Nonclosed-Form Type of Option Pricing Model In the Black-Scholes model, the variance of stock returns was assumed to be constant. However, empirical studies show that volatility seems to change day by day. Its behavior looks like it is performing a random walk. Thus, it is intuitional to assume the following stochastic processes for stock prices with random variance: dS D ˛S dt C S dz1 d D ˇ.N  /dt C  dz2 ;

(32.1)

where dz1 and dz2 are Wiener processes. Since both stock price and volatility are assumed random variables, we need the following generalization of Ito’s Lemma. Lemma 32.1. Generalization of Ito’s Lemma Let f be a function of variables x1 ; x2 ; : : : ; xn and time t. Suppose that xi ’s follow Ito’s process: dxi D ai dt C bi d zi ;

i D 1; 2;    ; n;

where dzi ’s are Wiener processes, instantaneous correlation coefficients between dzi and dzj is ¡ij , and ai ’s and bi ’s are functions of x1 ; x2 ; : : : ; xn and t. Then 1 X @f X @2 f 1 @f C ai C bi bj ij A dt df D @ @x @t 2 @x @x i i j i i;j 0

C

X @f bi d zi @xi i

Proof. A Taylor series expansion of f gives

C.F. Lee () Rutgers University, New Brunswick, NJ, USA e-mail: [email protected] J.C. Lee National Chiao Tung University, Taiwan, ROC

f D

X @f 1 X @2 f @f

t C

xi C

xi xj @xi @t 2 i;j @xi @xj i C

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_32, 

1 X @2 f 1 @2 f

xi t C . t/2 C R2 2 i @xi @t 2 @t 2 481

482

C.F. Lee and J.C. Lee

 1 D C1 ˛S C C2 ˇ.N  /  C3 C C11  2 S 2 2  1 1 C C22  2 C C12 S  dt 2 2

A discrete form for dx can be written as p

xi D ai t C bi t"i ; where ©i ’s are standard normal variables. Then

CC1 Sd z1 C C2 d z2

3

xi xj D ai aj . t/2 C ai bj . t/ 2 "j 3

Caj bi . t/ 2 "i C bi bj t "i "j : The first term on the right is of order .dt/2 and the second and third terms are distributed with mean 0 and a standard deviation of order .dt/3=2 . As t tends to 0, lim xi xj D ij bi bj ;

where ¡ is the instantaneous correlation between dz1 and dz2 , and the subscripts on C indicate partial derivatives and  D T  t so that d  D dt. To eliminate the risk from the portfolio, we observe that the coefficients for dz1 and dz2 in the return of the portfolio, dC.; ; 1 / C w2 dC.; ; 2 / C w3 dS, are respectively

t !0

where ¡ij is the correlation coefficient of ©j and ©j . Similarly the terms, xi t, are all distributed with mean 0 and a standard deviation of order .dt/3=2 , so they are comparably small with t as t ! 0. It can be seen that all the terms in R2 are at most as large as . t/3=2 , thus we can collect all terms of order t and write down the differential df as: df D

X @f @f dt .ai dt C bi d zi / C @x @t i i

1 X @2 f C bi bj ij dt 2 i;j @xi @xj 3 2 X @2 f X @f 1 @f C D4 ai C bi bj ij 5 dt @x @t 2 @x @x i i j i i;j C

X @f bi d zi @xi i

This completes the proof. A call option price on the stock shall have the form C.S; ¢; £/, a function of the stock price, volatility, and time to expiration. With the introduction of a random variance and thus two sources of uncertainty, a portfolio with only one option and one stock is not sufficient for creating a riskless investment strategy. A portfolio consisting of one stock and two options having different expiration dates is required, or it should be C.; ; 1 / C w2 C.; ; 2 / C w3 S: Now we may use Ito’s Lemma above to derive the stochastic differential: dC D C1 dS C C2 d C C3 d  1 1 1 C C11 .dS /2 C H22 .d/2 C C33 .d /2 2 2 2 CC12 dSd C C13 dSd  C C23 dd 

(32.2)

C1 .; ; 1 /S C w2 C1 .; ; 2 /S C w3 S and C2 .; ; 1 / C w2 C2 .; ; 2 /: Set both the two quantities to be 0, we get wO 2 D C2 .; ; 1 /=C2 .; ; 2 /: and wO 3 D C1 .; ; 1 / C C2 .; ; 1 /C1 .; ; 2 /=C2 .; ; 2 /: Then the return of the portfolio becomes: dC.; ; 1 / C wO 2 dC.; ; 2 / C wO 3 dS  1 1 D C3 .1 / C C11 .1 / 2 S 2 C C12 .1 /S  ı 2 2  1 C2 .1 / 1 C3 .2 / C C11 .2 / 2 S 2 C C22 .1 / 2  2 C2 .2 / 2  1 1 C C12 .2 /S  C C22 .2 / 2 dt 2 2 Since it is a riskless return, it should be equal to the riskfree rate; that is, dC.; ; 1/ C wO 2 dC.; ; 2 / C wO 3 dS D r ŒC.; ; 1 / C wO 2 C.; ; 2 / C wO 3 S dt: Substituting w O 2 and wO 3 into the above two differential equations we get 1 1 1 C3 .1 / C C11 .1 / 2 S 2 C C12 .1 /S  ı C C22 .1 / 2 2 2 2  C2 .1 / 1 1 C3 .2 / C C11 .2 / 2 S 2 C C12 .2 /S   C2 .2 / 2 2  1 C C22 .2 / 2 2

32 Stochastic Volatility Option Pricing Models

483

1 C1 ˛S C C2 ˇ.N  /  C3 C C11  2 S 2 2

 1 1 C C22  2 C C12 S  ı C; 2 2   C2  C1 S .˛  r/ C 2 ; D rC C C

D rŒC.; ; 1 /  C2 .; ; 1 /C.; ; 2 /=C2 .; ; 2 / C .C1 .; ; 1 / C C2 .; ; 1 /C1 .; ; 2 /=C2 .; ; 2 // S ; implying 1 1 C3 .1 /  C11 .1 / 2 S 2  C12 .1 /S  ı 2 2 C2 .1 / 1  C22 .1 / 2 C C.1 /r  C1 .1 /S r  2 C2 .2 /  1 1 C3 .2 /  C11 .2 / 2 S 2  C12 .2 /S  2 2  1  C22 .2 / 2 C C.2 /r  C1 .2 /S r D 0 (32.3) 2

or 1 1 1 C3  C11  2 S 2  C12 S  ı  C22  2 2 2 2 CC r  C1 S r  C2 Œˇ.N  /   D 0: (32.4)

This equation with the boundary conditions has a unique solution and it is easy to show that this solution also satisfies It is easily observed that the solutions of the following Equation (32.3). It is obvious that the expected return on the equation are also solutions of Equation (32.3): stock does not influence the option price, but the expected change and the risk premium associated with the volatilC3  12 C11  2 S 2  12 C12 S   12 C22  2 C C r  C1 S r D 0: ity do. The solution for the option price function follows from But further investigations show that a more general solution lemma 32.1 of Cox et al. (1985): can be obtained from 1 1 1 C3  C11  2 S 2  C12 S   C22  2 2 2 2  CC r  C1 S r  C2 b D 0 where b is an arbitrary function or constant. This means that a unique solution for the option price function in the random variance model cannot be obtained only through arbitrage. An alternative view on this problem is that we may form a duplicating portfolio for an option in this model that which contains the stock, the riskless bond, and another option. We cannot determine the price of a call option without knowing the price of any other call on the same stock. However, this is just what we are trying to do with the equation. In other words, only with a predetermined market price of the risk we can obtain a unique solution for the option price function. Some aspects to the market price of the risk are shown in Appendix 32A. To derive a unique option pricing function, the following equation based on the arbitrage pricing theory by S. Ross in 1976 is introduced, E

dC C



  C1 C2 D r C 1 S C 2  dt C C   C1 S C2   rC .˛  r/ C  dt C C 2

where .’  r/ is the risk premium on the stock and œ2  is the risk premium associated with d¢. Equating these expressions we may get:

O rt maxf0; St  KgjS0; 0 /; C.S; ; tI r; K/ D E.e (32.5) where EO represents a risk-adjusted expectation. For the riskadjustment, we reduce the mean parameter of dP and d† by the corresponding risk premiums. For the stock return, ’ is replaced by the risk-free rate, r, and for the standard deviation, Œˇ.N  /   is used in place of ˇ.N  /. Following Karlin and Taylor (1981), the backward equation for the function can be derived and it can be shown that it solves the Equation (32.4) with the adjustments on the dS and d¢ processes. The option pricing function in Equation (32.5) is a general solution to this random variance model. For operational issues, parameters of the ¢ process, the risk premium œ , and the instantaneous correlation coefficients between the stock return and d¢ are all required. With these parameters and the current value of ¢ given, we do simulations to compute the option prices. The model can be simplified with œ and • set as 0. The risk premium could be zero if, for example, the volatility risk of the stock can be diversifiable or if d¢ is uncorrelated with the marginal utility of wealth. Now we develop the distribution of the stock price function at expiration with œ D 0 and • D 0. Consider the following solution conditional on the process f¢s W 0 < s < tg: St D S0 exp

8 t 0, we have 1 A!1

ZA

F .x C ı/  F .x  ı/ D lim

sin ı t itx e 'x .t/dt t

A

(32.10)

Proof. For the proof of the theorem, see Wilks (1963, 116–117). A very useful result for obtaining tail probability is given in the following theorem.

Theorem 2 (Gil-Pelaez 1951). Let '.t/ be the characteristic function of the random variable X. Then 1  F .x/ D P .X > x/ 1 1 D C 2

Z1

Œe itx .t/  e i tx .t/ dt 2i t

0

1 1 D C 2

Z1

 e itx .t/ dt Re it 

(32.12)

0

Proof. see Gil-Pelaez (1951). The above theorems will be useful in deriving the desired probabilities in Sect. 32.4.

32.4 Closed-Form Type of Option Pricing Model Instead of the non-closed solution of option pricing model with stochastic volatility as proposed by Scott (1987) and discussed in Sect. 32.2, we now discuss the closed-form solution of option pricing model when the volatility of stock returns is assumed stochastic, with a special emphasis on the model proposed by Heston (1993). The results are expressed in terms of the characteristic function, which is a close-form mathematically. Of course, numerical solution is still needed in practice. Since there are two stochastic partial differential equation as given in Equation (32.1), we will need Lemma 32.1 with n D 2. We next assume that the stock price at time t follows the diffusion dS .t/ D  .t/ S .t/ dt C

p .t/S.t/dZ1 .t/

(32.13)

Where .t/ is the instantaneous drift of stock return and V .t/ is the variance, and Z1 .t/ is a Wiener process.

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C.F. Lee and J.C. Lee

Furthermore, we assume that V .t/ follows dV .t/ D ˛.S; ; t/dt C ˇ.S; ; t/dZ2 .t/

(32.14)

where Z2 is Wiener process with dZ 1 dZ 2 D dt; 1    1. It is noted that  is the volatility of volatility and  is the correlation between dZ 1 and dZ 2 . In Black-Scholes case, there is only one source of randomness – the stock price, which can be hedged with stock.

In the present case, random changes in volatility also need to be hedged in order to form a riskless portfolio. So we set up a portfolio … containing the option being priced whose value we denote by U.S; v; t/, a quantity  of the sock and a quantity  1 of another asset with value U1 .S; v; t/, we have … D U.s; v; t/  S  1 U1 .s; v; t/ By Lemma 32.1, we have

  @U @U 1 1 @2 U @2 U @U @2 U dS C dv C C vS 2 2 C vˇS C 2 vˇ 2 2 dt; @S @v @t 2 @S @v@S 2 @v   @U1 @U1 1 1 2 2 @2 U1 @2 U1 @U1 @2 U1 dS C dv C C vS 2 C  dt; d U1 D C vˇS vˇ @S @v @t 2 @S 2 @v@S 2 @v2 dU D

d… D d U  dS  1 d U1   v @2 U 1 @2 U @U @2 U C S 2 2 C vˇS C 2 vˇ 2 2 dt D @t 2 @S @S @v 2 @v   v @2 U1 1 2 2 @2 U1 @U1 @2 U1 C S2 C  dt  1 C vˇS vˇ @t 2 @S2 @S @v 2 @v2   @U @U1 C  1  dS @S @S   @U @U1 C d v:  1 @v @v

To eliminate all randomness from the portfolio, we must choose @U1 @U   1 D0 @S @S to eliminate dS terms, and

 1 1 @2 U @2 U @U @2 U C vS 2 2 C vˇS C 2 vˇ 2 2 dt @t 2 @S @v@S 2 @v   1 1 2 2 @2 U1 @U1 @2 U1 @2 U1 C vS 2 C  dt  1 C vˇS vˇ @t 2 @S2 @v@S 2 @v2

 d… D

@U1 @U  1 D0 @S @S to eliminate dv terms. This leaves us with

D r…dt D r.U S 1 U1 /dt;

32 Stochastic Volatility Option Pricing Models

487

where we have used the fact that the return on a risk-free portfolio must equal the risk-free rate r. Collecting all U

terms on the left-hand side and all U1 terms on the right-hand side, we find that

@U @2 U @2 U 1 1 @2 U @U C vS 2 2 C vˇS C 2 vˇ 2 2 C rs  rU @t 2 @S @v@S 2 @v @S @U @v 1 1 @U1 @2 U1 @2 U1 @2 U1 U1 C vS 2 2 C vˇS C 2 vˇ 2 2 C rS  rU1 @t 2 @S @v@S 2 @v @S D : @U1 @v

The left-hand side is a function of U and the right-hand side is a function of U1 only. The only way that this can be true is for both sides to be equal to some function f of the variables S,v,t. 1 1 @2 U @2 U @2 U @U C vS 2 2 C vˇS C 2 vˇ 2 2 @t 2 @S @v@S 2 @v @U U  rU D .˛  ˇ/ Crs @S @v

(32.15)

where, without loss of generality, we have written the arbitrary function f of S,v,t as .˛  ˇ/. Conventionally, .S; v; t/ is called the market price of volatility risk because it tells us how much of the expected return of U is explained by the risk of v in the capital asset pricing model framework.

32.4.1 The Heston Model The Heston model (Heston 1993) corresponds to choosing ˛.S; v; t/ D k.  v.t//; ˇ.S; v; t/ D 1;  D  in Equation (32.14). These equations become p v.t/ S.t/ dZ1 .t/ p d v.t/ D Œ  v.t/ dt C  v.t/ dZ2 .t/

dS.t/ D  Sdt C

1 @U @2 U @2 U 1 2 @2 U vS C  2v C rS C vS 2 2 @S @S @v 2 @v @S @U @U  rU C D0 CfΠ v.t/  .S; v; t/g @v @t (32.17) A European call option with strike price K and maturing at time T satisfies the PDE Equation (32.19) subject to the following boundary conditions: U.S; v; T / D max.0; S  k/; U.0; v; T / D 0; @U .1; v; T / D 1; @S @U @U rS .S; 0; t/ C k .S; 0; t/  rU.S; 0; t/ @S @v @U .S; 0; t/ D 0; C @t U.S; 1; t/ D S: By analogy with the Black-Scholes formula, we guess a solution of the form U .S; v; t/ D SP1  Ke r.T t / P2

(32.16)

with dZ1 dZ2 D dt where  is the speed of reversion of v.t/ and  represents the long-term mean of volatility process. We substitute the above values for ˛.S; v; t/ and ˇ.S; v; t/ into the general valuation equation to obtain

(32.18)

where the first term is the present value of the spot asset upon optimal exercise, and the second term is the present value of the strike price payment. Let x D ln S , then U .x; v; t / D e x P1  Ker.T t / P2 From Equation (32.20), we have @U @P1 @P2 D ex  rKe r.T t / P2  Ke r.T t / ; @t @t @t (32:180 a)

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C.F. Lee and J.C. Lee

@U @x 1 @U @U D D @S @x @S S @x   @P2 1 x @P1 e C e x P1  ke r.T t / D S @x @x

(32.180b)

where @P1 @P2 @U D ex C e x P1  Ke r.T t / ; @x @x @x @U @P1 @P2 D ex  ke r.T t / ; @v @v @v

(32.180c)

For the option price to satisfy the boundary condition in Equation (32.10), these PDEs Equation (32.12) are subject to the terminal condition.  1 if x > lnŒk Pj .x; v; T I lnŒk / D 1.x>lnŒk / D 0 o:w: (32.20) Let fj . ; v; t/ denote the characteristic function of Pj .x; v; t/. The inversion of fj . ; v; t/ is given by 1 Pj .x; v; t / D 2

2 2 @2 U x @ P1 r.T t / @ P2 D e  ke ; (32.180d) @v2 @v2 @v2



1 1 @U @2 U @ @U D D @S @v @v @S S @v @x   1 x @2 P1 @P1 @2 P2 e C ex  ke r.T t / 2 ; D S @x@v @v @x (32.180e) 2

2



.a/





1 @S

@C 1 @x S

  @Pj @ Pj @ Pj 1 1 @ Pj v 2 C  v C  2 v 2 C r C uj v 2 @x @x@v 2 @v @x   @Pj @Pj C D0 for j D 1; 2 (32.19) C a  bj v @v @t 2

i e i x fj . ; v; t/ d

2

where u1 D 1=2; u1 D 1=2; a D ; b1 D C; b2 D C:

Z1

2 e i x fj . ; v; t / d 1

Z1

@2 Pj 1 D .c/ @x@v 2

i e i x

@fj . ; v; t / d @v

1

D

Substituting Equation (32:180 a)–(32.180g) into Equation (32.16), we obtain 2

Z1

@Pj 1 D @x 2

@2 Pj 1 .b/ D 2 @x 2



1 @C 1 @2 C C 2 2 2 S @x S @x   1 @P1 @P2 D 2 ex C e x P1  Ke r.T t / S @x @x  2 1 @ P1 @P1 @P1 C ex C 2 ex C ex 2 S @x @x @x  2 x r.T t / @ P2 Ce P1  Ke @x 2  1 @2 P2 @P1 @P2 C Ke r.T t / D 2 ex C ex S @x 2 @x @x  @2 P2 (32.180g) Ke r.T t / 2 : @x D

(32.21)

1

2

@C @S

e i x fj . ; v; t / d 1

@ U @ P1 @P1 @ P2 C e x P1  ke r.T t / 2 ; D ex C 2e x @x 2 @x 2 @x @x (32.180f) @2 C @ D 2 @S @S

Z1

Z1

@2 Pj 1 .d/ D @v2 2

e i x

@2 fj . ; v; t/ d @v2

1

1 @Pj D .e/ @t 2

Z1 e i x

@fj . ; v; t / d @t

1

Substituting Equation (32.21) into Equation (32.19) gives  @fj 1 2  2 @2 fj  v fj  iv C v 2  r C uj  i fj 2 @v 2 @v   @fj @fj C D 0: C a  bj v @v @t The above equation can be rewritten as 

@2 fj @fj C 2 v ˛fj  ˇ @v @v

 Ca

@fj @fj C  i r fj D 0 @v @t (32.22)

where ˛D

'2 2 C uj i'; ˇ D bj  i';  D ; a D # 2 2

To solve for the characteristic function explicitly, we guess the function form

32 Stochastic Volatility Option Pricing Models

fj D exp fC .'; t / C D .'; t/ vg fj .'; v; T / D

1 i' lnŒk exp fC .'; t/ C D .'; t/ vg ; e i'

where the characteristic function of Pj .x; v; T / through Equation (32.20) Z1 fj .'; v; T / D

e i'x Pj .x; v; T / dx

489

D .'; t/ D m

1  e dt ; 1  ge dt

bj C 'i  d m D ; mC bj C 'i C d    2 1  ge dt C i'r: C .'; t / D a m t  2 ln  1g where g D

Using the result of Gil-Pelaez (1951), one can invert the characteristic function to get the desired probabilities. Therefore,

1

Z1 D

1 i' lnŒk e e i x dx D : i'

1 1 Pj .x; v; t / D C 2



Z1 Re

 e i'x fj .'; v; t/ d': i'

0

lnŒk

It follows that @fj D @t



@C @D Cv @t @t

32.5 Conclusion

fj ;

@fj D Dfj ; @v @2 fj D D 2 fj : @v2 Then . / can be repressed as

@C C aD  i' fj @t

@D 2 C ˛  ˇD C D fj D 0 Cv @t

Then Equation (32.22) is satisfied if @C D aD C i'; @t   @D D  ˛ˇD C  D2 D  .DmC / .Dm / ; @t p ˇ˙d ˇ ˙ ˇ 2  4˛ m˙ D D ; (32.23) 2 2 where p ˇ 2  4˛ s

2

2  

2 D bj   i  4 uj i'  2 2 q 2   D  i  bj   2 2  2 uj i :

In this chapter, we assume that the volatility of underlying asset is a stochastic process. Emphasis will be placed on the nonclosed-form solution developed by Scott (1987) and the closed-form solution of Heston (1993). In both cases, we consider a model in which the variance of stock price returns varies according to an independent diffusion process. For the closed form option pricing model, the results are expressed in terms of the characteristic function.

References Cox, J. C., J. E. Ingersoll and S. A. Ross. 1985. “A Theory of the Term Structure of Interest Rates,” Econometrica 53, 385–407. Gil-Pelaez, J. 1951. “Note on the inversion theorem.” Biometrika 38, 481–482. Heston, S. 1993. “A closed-form solution for options with stochastic volatility with application to bond and currency options.” The Review of Financial Studies 6(2), 327–343. Karlin, S. and H. Taylor. 1981. A Second Course in Stochastic Processes. New York: Academic Press. Levy, P. 1925. Calcul des probabilities, Gauthier-Villars, Paris. Ross, S. 1976. “The arbitrage theory of capital asset pricing.” Journal of Economic Theory, 13, 341–360. Scott, L. 1987. “Option pricing when the variance changes randomly: theory, estimation, and an application.” Journal of Finance and Quantitative Analysis 22(4), 419–438. Wilks, S. S. 1963. Mathematical statistics, 2nd Printing, Wiley, NY.

d D

Integrating Equation (32.23) with the terminal conditions C . ; 0/ D 0 and D . ; 0/ D 0 gives

Appendix 32A The Market Price of the Risk The equation we used to derive a unique solution for the stock price function is closely related to arbitrage pricing theory developed by S. Ross in 1976. Consider some derivative depending on n state variables and time t. Suppose that there

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are a total of at least n C 1 traded securities. Some assumptions are made as:

where kj ’s satisfy X

The short selling of securities with full use of proceeds is

permitted. There are no transaction costs and taxes. All securities are perfectly divisible. There are no riskless arbitrage opportunities. Security trading is continuous.

Denote the state variables as ™i and suppose that they follow ItoO 0 s processes: di D m i dt C si i d zi and the correlation between dzi and dzj is ¡ij . Let fj be the price of ith traded security. With ItoO 0 s Lemma, X ij fj d zi dfj D j fj dt C

j

The cost of forming such a portfolio is

i

ij fj D

@i

m i i C

P

kj fj and if there

j

are no arbitrage opportunities, the return rate of portfolio must be equal to the riskfree interest rate, so that X

kj j fj D r

X

j

or

X

kj fj

j

kj fj .j  r/ D 0:

(32A.2)

j

where X @fj

(32A.1)

The return of the portfolio is then X kj j fj dt: d… D

i

j fj D

kj ij fj D 0:

j

1 X @2 fj @f C i k si sk mi mk @t 2 @i @k i;k

Equations (32A.1) and (32A.2) can be regarded as n C 1 homogeneous linear equations in kj ’s. The kj ’s are not all zero if and only if the equations are consistent. In other words, fj .j  r/ is a linear combination of ¢ij fj ’s, so that

@fj si i : @i

fj .j  r/ D

X

i ij fj

i

In these equations, j is the instantaneous mean rate of return provided by fj . Since there are n C 1 securities and n sources of variation, an instantaneous riskless portfolio … can be obtained as …D

X j

kj fj ; i D 1;   ; n

or fj .j  r/ D

X

i ij fj ;

i

in which œi is the market price of the risk for ™i .

(32A.3)

Chapter 33

Derivations and Applications of Greek Letters: Review and Integration Hong-Yi Chen, Cheng-Few Lee, and Weikang Shih

Abstract In this chapter, we introduce the definitions of Greek letters. We also provide the derivations of Greek letters for call and put options on both dividends-paying stock and non-dividends stock. Then we discuss some applications of Greek letters. Finally, we show the relationship between Greek letters, with one of the examples from the Black– Scholes partial differential equation.

@… @S where … is the option price and S is underlying asset price. We next show the derivation of delta for various kinds of stock option.

Keywords Greek letters r Delta r Theta r Gamma r Vega r Rho r Black–Scholes option pricing model r Black–Scholes partial differential equation

33.2.1 Derivation of Delta for Different Kinds of Stock Options

D

From Black–Scholes option pricing model, we know the price of call option on a non-dividend stock can be written as:

33.1 Introduction “Greek letters” are defined as the sensitivities of the option price to a single-unit change in the value of either a state variable or a parameter. Such sensitivities can represent the different dimensions to the risk in an option. Financial institutions who sell options to their clients can manage their risk by Greek letters analysis. In this chapter, we will discuss the definitions and derivations of Greek letters. We will also specifically derive Greek letters for call (put) options on non-dividend stock and dividends-paying stock. Some examples are provided to explain the application of Greek letters. Finally, we will describe the relationship between Greek letters and the implication in a delta neutral portfolio.

Ct D St N.d1 /  Xe r N.d2 /

(33.1)

and the price of put option on a non-dividend stock can be written as: Pt D Xer£ N .d2 /  St N .d1 /

(33.2)

where

s2 C r C X 2  d1 D p s 

 St  ¢2 ln X C r  2s £ p d2 D p D d 1  ¢s £ ¢s £ ln

 St 

 DT t

33.2 Delta () The delta of an option, , is defined as the rate of change of the option price with respect to the rate of change of underlying asset price:

N ./ is the cumulative density function of normal distribution. Z N .d1 / D

d1

1

Z f .u/ du D

d1

1

1 u2 p e  2 du: 2

First, we calculate H.-Y. Chen (), C.-F. Lee, and W. Shih Rutgers University, Newark, NJ, USA e-mail: [email protected]

N 0 .d1 / D

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_33, 

d12 @N .d1 / 1 D p e 2 @d1 2

(33.3)

491

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H.-Y. Chen et al.

N 0 .d2 / D

d2 1 St 1 1 D Xer£ p e 2   er£  p X St ¢s £ 2 

@N .d2 / @d2

d22 1 D p e 2 2

.d1 s 1 2 D p e 2

d2 1 1 1  .1  N .d1 // C St p e 2  p St ¢s £ 2 

p 2 /

D St

p d12 s2  1 D p e  2  e d1 s   e  2 2 d12 ln. SXt /C 1 D p e 2  e 2

1 Dp e 2

d2  21



2 rC 2s 

C St  e

s2  2

St r e X

(33.4)

and

The derivation of Equation 33.5 is in the following:

where

@Ct @N .d1 / @N .d2 / D N .d1 / C St  Xer£ @St @St @St D N .d1 / C St

1 D N .d1 / C St p e 2   Xe

r£



1 p St ¢s £

d21 St 1 1 p e 2   er£  p X St ¢s £ 2 

d21 d2 1 1 1 p p D N .d1 / C St e 2  St e 2 St ¢s 2 £ St ¢s 2 £

D N .d1 / For a European put option on a non-dividend stock, delta can be shown as (33.6)

D N.d1 /  1 The derivation of Equation 33.6 is

D

@N .d2 / @N .d1 / @Pt D Xer£  N .d1 /  St @St @St @St D Xe

r£ @.1

 St

Ct D St eq£ N .d1 /  Xer£ N .d2 /

(33.7)

Pt D Xer£ N .d2 /  St eq£ N .d1 /

(33.8)

d1 D

@N .d1 / @d1 @N .d2 / @d2  Xer£ @d1 @St @d2 @St d2  21

 N .d2 // @d2  .1  N .d1 // @d2 @St

@.1  N .d1 // @d1 @d1 @St

d2 1 1 e 2 D N .d1 /  1 p St ¢s 2 £

If the underlying asset is a dividend-paying stock providing a dividend yield at rate q, Black–Scholes formulas for the prices of a European call option on a dividend-paying stock and a European put option on a dividend-paying stock are

Equations (33.2) and (33.3) will be used repetitively in determining following Greek letters when the underlying asset is a non-dividend paying stock. For a European call option on a non-dividend stock, delta can be shown as (33.5)

D N.d1 /

D

d2 1 1 p e 2  N .d1 /  1 St ¢s 2 £

d2 D

ln

 St 

ln

 St 

X

X

C rqC p ¢s £ C rq p ¢s £

¢s2 2

¢s2 2

£

£

p D d 1  ¢s £

To make the following derivations more easily, we calculate Equations (33.9) and (33.10) in advance. N0 .d1 / D

d12 @N .d1 / 1 D p e 2 @d1 2

N0 .d2 / D

@N .d2 / @d2

(33.9)

d2 1 2 D p e 2 2 

.d1 ¢s 1 2 D p e 2 

p 2 £/

p d2 ¢s2 £ 1 1 D p e 2  ed1 ¢s £  e 2 2  d2 ln. SXt /C 1 1 D p e 2  e 2 

¢2 rqC 2s £

d2 1 St 1 D p e 2   e.rq/£ X 2 

 e

¢s2 £ 2

(33.10)

33 Derivations and Applications of Greek Letters: Review and Integration

For a European call option on a dividend-paying stock, delta can be shown as (33.11)

D eq£ N.d1 / The derivation of Equation (33.11) is

D

@N .d1 / @N .d2 / @Ct D eq£ N .d1 / C St eq£  Xer£ @St @St @St D eq£ N .d1 / C St eq£  Xer£

@N .d1 / @d1 @d1 @St

@N .d2 / @d2 @d2 @St

d21 1 1 D eq£ N .d1 / C St eq£ p e 2  p S ¢ 2  t s £ d21 1 St 1  Xer£ p e 2   e.rq/£  p X S ¢ 2  t s £

D eq£ N .d1 / C St eq£  St eq£

1 p

St ¢s 2 £

d21

e 2

d21 1 p e 2 St ¢s 2 £

D eq£ N .d1 / For a European call option on a dividend-paying stock, delta can be shown as

D eq£ ŒN.d1 /  1 :

(33.12)

The derivation of Equation (33.12) is

D

@N .d2 / @Pt D Xer£  eq£ N .d1 / @St @St  St eq£ D Xer£

@N .d1 / @St

@.1  N .d2 // @d2  eq£ .1  N .d1 // @d2 @St

 St eq£

33.2.2 Application of Delta Figure 33.1 shows the relationship between the price of a call option and the price of its underlying asset. The delta of this call option is the slope of the line at the point of A corresponding to current price of the underlying asset. By calculating delta ratio, a financial institution that sells options to a client can make a delta neutral position to hedge the risk of changes of the underlying asset price. Suppose that the current stock price is $100, the call option price on stock is $10, and the current delta of the call option is 0.4. A financial institution sold 10 call option to its client, so that the client has right to buy 1,000 shares at time to maturity. To construct a delta hedge position, the financial institution should buy 0:4 1;000 D 400 shares of stock. If the stock price goes up to $1, the option price will go up by $0.4. In this situation, the financial institution has a $400 ($1 400 shares) gain in its stock position, and a $400 ($0:4 1;000 shares) loss in its option position. The total payoff of the financial institution is zero. On the other hand, if the stock price goes down by $1, the option price will go down by $0.4. The total payoff of the financial institution is also zero. However, the relationship between option price and stock price is not linear, so the delta changes over different stock prices. If an investor wants his portfolio to remain delta neutral, he should adjust his hedged ratio periodically. The more frequent adjustments he makes, the better delta-hedging he gets. Figure 33.2 exhibits the change in delta affects the deltahedges. If the underlying stock has a price equal to $20, then the investor who uses only delta as a risk measure will consider that his portfolio has no risk. However, as the underlying stock prices changes, either up or down, the delta changes as well, and thus the investors will have to use different delta hedging. Delta measure can be combined with other risk measures to yield better risk measurement. We will discuss it further in the following sections.

@.1  N .d1 // @d1 @d1 @St

d21 1 St 1 D Xer£ p e 2   e.rq/£  p X St ¢s £ 2  d21 1 1 p  eq£ .1N .d1 //CSt eq£ p e 2  St ¢s £ 2 

D St eq£

d21 1 e 2 C eq£ .N .d1 /  1/ p St ¢s 2 £

C St eq£ De

q£

1 p

St ¢s 2 £

.N .d1 /  1/

493

d21

e 2

Fig. 33.1

494

H.-Y. Chen et al.

@N.d1 / @d1 @N.d2 / @d2  rX  er£ N.d2 / C Xer£ @d1 @£ @d2 @£ !   2 ¢ ¢2 r C 2s r C 2s ln Xst 1  d21 p   p D St p e 2  2¢s £3=2 2¢s £ ¢s £ 2 

D St



 rX  er£ N.d2 / C Xer£ 

d2 1 1 D St p e 2  2 

!   ¢2 ¢2 r C 2s ln Xst r C 2s p   p 2¢s £3=2 2¢s £ ¢s £

 rX  er£ N.d2 /

33.3 Theta (‚) The theta of an option, ‚, is defined as the rate of change of the option price respected to the passage of time:

where … is the option price and t is the passage of time. If £ D T  t, theta .‚/ can also be defined as minus one timing the rate of change of the option price respected to time to maturity. The derivation of such transformation is easy and straightforward: @… @£ @… @… D D .1/ ‚D @t @£ @t @£ where £ D T  t is time to maturity. For the derivation of theta for various kinds of stock option, we use the definition of negative differential on time to maturity.

33.3.1 Derivation of Theta for Different Kinds of Stock Options For a European call option on a non-dividend stock, theta can be written as: St ¢s ‚ D  p  N0 .d1 /  rX  er£ N.d2 /: 2 £ The derivation of Equation (33.13) is @N.d1 / @Ct D St @£ @£

C .r/  X  er£ N.d2 / C Xer£

d2 1 1 C St p e 2  2 

d2 1 1 D St p e 2  2 

@… ‚D @t

‚D



  ln Xst r C 2s r  p  p ¢s £ 2¢s £3=2 2¢s £



Fig. 33.2

d2 1 St 1 p e 2   er£ X 2  ! ¢2

@N.d2 / @£

(33.13)

!   ¢2 r C 2s ln Xst r p   p ¢s £ 2¢s £3=2 2¢s £ ¢s2 2

p ¢s £

!  rX  er£ N.d2 /

St ¢s D  p  N0 .d1 /  rX  er£ N.d2 / 2 £ For a European put option on a non-dividend stock, theta can be shown as St ¢s ‚ D  p  N0 .d1 / C rX  er£ N.d2 /: 2 £

(33.14)

The derivation of Equation (33.14) is @Pt D  .r/  X  er£ N.d2 / @£ @N.d2 / @N.d1 / C St  Xer£ @£ @£

‚D

D .r/X  er£ .1  N.d2 //  Xer£

@.1  N.d2 // @d2 @d2 @£

@.1  N.d1 // @d1 D .r/X  er£ .1  N.d2 // @d1 @£

1  d21 St r£ r£ 2  e C Xe  p e X 2  !   ¢2 r C 2s ln Xst r  p   p ¢s £ 2¢s £3=2 2¢s £ 0 1  st  ¢s2 ¢s2 2 d rC rC 2 ln X 1 1  St p e 2  @ p 2   p A 3 ¢ £ 2¢ 2  s =2 s £ 2¢s £ C St

D rX  er£ .1  N.d2 //

33 Derivations and Applications of Greek Letters: Review and Integration

!   ¢2 r C 2s ln Xst r p   p ¢s £ 2¢s £3=2 2¢s £ !   ¢2 ¢2 r C 2s r C 2s ln Xst 1  d21 p   p  St p e 2  2¢s £3=2 2¢s £ ¢s £ 2  ! ¢s2 d21 1 2 p D rX  er£ .1  N.d2 //  St p e 2  ¢s £ 2  d21 1 C St p e 2  2 

St ¢s D rX  er£ .1  N.d2 //  p  N0 .d1 / 2 £ St ¢s D rX  er£ N.d2 /  p  N0 .d1 / 2 £

495

D q  St e

q£

N.d1 /  St e

q£

d2 1 1 p e 2  2 

¢s2 2

!

p ¢s £

 rX  er£ N.d2 / D q  St eq£ N.d1 / 

St eq£ ¢s  N0 .d1 /  rX  er£ N.d2 / p 2 £

For a European call option on a dividend-paying stock, theta can be shown as ‚ D rX  er£ N.d2 /  qSt eq£ N.d1 / 

St eq£ ¢s p  N0 .d1 /: 2 £ (33.16)

The derivation of Equation (33.16) is For a European call option on a dividend-paying stock, theta can be shown as ‚ D q  St eq£ N.d1 / 

St eq£ ¢s p  N0 .d1 /  rX  er£ N.d2 /: 2 £ (33.15)

The derivation of Equation (33.15) is @Ct @N.d1 / D q  St eq£ N.d1 /  St eq£ @£ @£ @N.d2 / C .r/  X  er£ N.d2 / C Xer£ @£ @N.d1 / @d1  rX  er£ N.d2 / D q  St eq£ N.d1 /  St eq£ @d1 @£

‚D

@N.d2 / @d2 D q  St eq£ N.d1 /  St eq£ @d2 @£ !   ¢2 ¢2 r  q C 2s r  q C 2s ln Xst 1  d21

p e 2  p  p  2¢s £3=2 ¢s £ 2¢s £ 2 

d21 1 St  rX  er£ N.d2 / C Xer£  p e 2   e.rq/£ X 2  !  st  2 ¢ r  q C 2s ln X rq  p   p ¢s £ 2¢s £3=2 2¢s £ C Xer£

D q  St eq£ N.d1 /  St eq£ d2 1 1

p e 2  2 

!   2 ¢2 r C 2s r C 2s ln Xst p   p 2¢s £3=2 2¢s £ s £

 rX  er£ N.d2 /

0 1  st  ¢s2 d21 r C ln 1 r X  p2 A C St eq£ p e 2  @ p  3= ¢s £ 2¢ £ 2  s 2¢s £ 2

@N.d2 / @Pt D  .r/  X  er£ N.d2 /  Xer£ @£ @£ @N.d1 / C .q/St eq£ N.d1 / C St eq£ @£

‚D

D rX  er£ .1  N.d2 //  Xer£  qSt eq£ N.d1 / C St eq£

@.1  N.d2 // @d2 @d2 @£

@.1  N.d1 // @d1 @d1 @£

D rX  er£ .1  N.d2 // C Xer£

d2 1 St 1  p e 2   e.rq/£ X 2  !  st  ¢2 ln X r  q C 2s rq   p  p ¢s £ 2¢s £3=2 2¢s £ d2 1 1  qSt eq£ N.d1 /  St eq£ p e 2 2  !   2 ¢ ¢2 r  q C 2s r  q C 2s ln Xst  p  p  2¢s £3=2 ¢s £ 2¢s £ d2 1 1 D rX  er£ .1  N.d2 // C St eq£ p e 2 2  !  st  ¢2 r  q C 2s ln X rq  p   p ¢s £ 2¢s £3=2 2¢s £ d2 1 1  qSt eq£ N.d1 /  St eq£ p e 2 2  !   2 ¢ ¢2 r  q C 2s r  q C 2s ln Xst  p  p  2¢s £3=2 ¢s £ 2¢s £

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D rX  er£ .1  N.d2 //  qSt eq£ N.d1 / ! ¢s2 d2 q£ 1  21 2  p  St e p e ¢s £ 2 

The derivation of Equation (33.17) is t @ @C @N .d1 / @d1 @St @2 Ct D D  D 2 @St @d1 @St @St

D rX  er£ .1  N.d2 //  qSt eq£ N.d1 /

1 St

1 p D p N0 .d1 / ¢s £ St ¢s £

D N0 .d1 / 

q£

St e ¢s  p  N0 .d1 / 2 £ D rX  er£ N.d2 /  qSt eq£ N.d1 / 

St eq£ ¢s p  N0 .d1 / 2 £

For a European put option on a non-dividend stock, gamma can be shown as 1 p N0 .d1 / : St ¢s £

D

33.3.2 Application of Theta (‚)

(33.18)

The derivation of Equation (33.18) is The value of option is the combination of time value and stock value. When time passes, the time value of the option decreases. Thus, the rate of change of the option price with respective to the passage of time, theta, is usually negative. Because the passage of time on an option is not uncertain, we do not need to make a theta hedge portfolio against the effect of the passage of time. However, we still regard theta as a useful parameter, because it is a proxy of gamma in the delta neutral portfolio. For the specific detail, we will discuss in the following sections.

2

@ Pt D @St 2

D

D N0 .d1 / 

@

@Pt @St

D

@St

@.N .d1 /  1/ @d1  @d1 @St

1 St

1 p D p N0 .d1 / ¢s £ St ¢s £

For a European call option on a dividend-paying stock, gamma can be shown as eq£ p N0 .d1 / : St ¢s £

D

33.4 Gamma ()

(33.19)

The derivation of Equation (33.19) is The gamma of an option, , is defined as the rate of change of delta respected to the rate of change of underlying asset price: D

D

@2 … @

D @S @S2

@

2

@ Ct D @St 2

@Ct @St

@St

@ .eq£ N.d1 // @St

D

@N .d1 / @d1 St  D eq£  N0 .d1 /  p @d1 @St ¢s £

eq£ p N0 .d1 / St ¢s £

For a European call option on a dividend-paying stock, gamma can be shown as D

33.4.1 Derivation of Gamma for Different Kinds of Stock Options

eq£ p N0 .d1 / : St ¢s £

The derivation of Equation (33.20) is

For a European call option on a non-dividend stock, gamma can be shown as 1 p N0 .d1 / : St ¢s £

D

1

D eq£ 

where … is the option price and S is the underlying asset price. Because the option is not linearly dependent on its underlying asset, delta-neutral hedge strategy is useful only when the movement of underlying asset price is small. Once the underlying asset price moves wider, gamma-neutral hedge is necessary. We next show the derivation of gamma for various kinds of stock options.

D

(33.17)

D

2

@ Pt D @St 2

D eq£ 

@

@Pt @St

@St

D

@ .eq£ .N .d1 /  1// @St

@ Œ.N .d1 /  1/ @d1  @d1 @St

(33.20)

33 Derivations and Applications of Greek Letters: Review and Integration

D eq£  N0 .d1 /  D

1 St

p ¢s £

eq£  0 p N .d1 / St ¢s £

33.4.2 Application of Gamma () One can use delta and gamma together to calculate the changes of the option due to changes in the underlying stock price. This change can be approximated by the following relations.

497

The above analysis can also be applied to measure the price sensitivity of interest rate- related assets or portfolio to interest rate changes. Here we introduce Modified Duration and Convexity as risk measure corresponding to the above delta and gamma. Modified duration measures the percentage change in asset or portfolio value resulting from a percentage change in interest rate. Modified Duration D

Change in price Change in interest rate

 Price

D  =P Using the modified duration,

changeinoptionvalue  changeinstockprice 1 C  .changeinstockprice/2 2 From the above relation, one can observe that the gamma makes the correction for the fact that the option value is not a linear function of underlying stock price. This approximation comes from the Taylor series expansion near the initial stock price. If we let V be option value, S be stock price, and S0 be initial stock price, then the Taylor series expansion around S0 yields the following. @V .S0 / 1 @2 V .S0 / .S  S0 / C .S  S0 /2 @S 2Š @S 2 1 @n V .S0 / CC .S  S0 /n 2Š @S n @V .S0 / .S  S0 /  V .S0 / C @S

V .S /  V .S0 / C

C

1 @2 V .S0 / .S  S0 /2 C o.S / 2Š @S 2

If we only consider the first three terms, the approximation is then, @V .S0 / 1 @2 V .S0 / .S  S0 / C .S  S0 /2 @S 2Š @S 2 1  .S  S0 / C .S  S0 /2 2

V .S /  V .S0 / 

For example, if a portfolio of options has a delta equal to $10,000 and a gamma equal to $5,000, the change in the portfolio value if the stock price drops to $34 from $35 is approximately change in portfolio value  .$10000/ .$34  $35/ C

1

.$5000/ .$34  $35/2 2

 $7500

Change in Portfolio Value D Change in interest rate D .Duration P/

Change in interest rate we can calculate the value changes of the portfolio. The above relation corresponds to the previous discussion of delta measure. We want to know how the price of the portfolio changes given a change in interest rate. Similar to delta, modified duration only shows the first order approximation of the changes in value. In order to account for the nonlinear relation between the interest rate and portfolio value, we need a second order approximation similar to the previous gamma measure; this can become the convexity measure. Convexity is the interest rate gamma divided by price, Convexity D =P and this measure captures the nonlinear part of the price changes due to interest rate changes. Using the modified duration and convexity together allow us to develop first as well as second order approximation of the price changes similar to previous discussion. Change in Portfolio Value  Duration P

.change in rate/ C

1

Convexity P 2

.change in rate/2 : As a result, .duration P/ and (convexity P) act like the delta and gamma measure, respectively, in the previous discussion. This shows that these Greeks can also be applied in measuring risk in interest rate-related assets or portfolio. Next we discuss how to make a portfolio gamma neutral. Suppose the gamma of a delta-neutral portfolio is , the gamma of the option in this portfolio is o , and ¨o is the

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number of options added to the delta-neutral portfolio. Then, the gamma of this new portfolio is

h 0 St 2 3=2 1  d21 @ ¢s £  ln X C r C D St p e 2  ¢s2 £ 2 

¨o o C :

0

To make a gamma-neutral portfolio, we should trade ¨o  D = o options. Because the position of option changes, the new portfolio is not in the delta-neutral. We should change the position of the underlying asset to maintain delta-neutral. For example, the delta and gamma of a particular call option are 0.7 and 1.2, respectively. A delta-neutral portfolio has a gamma of 2;400. To make a delta-neutral and gamma-neutral portfolio, we should add a long position of 2;400=1:2 D 2;000 shares and a short position of 2;000

0:7 D 1;400 shares in the original portfolio.

1  St p e 2 

d2  21

d2 1 1 D St p e 2  2 

@



h  ln SXt C r C

¢s2 2

¢s2 £

¢s2 £3=2 ¢s2 £



¢s2 2

i 11 £  £2 A

i 11 £  £2 A

p D St £  N0 .d1 /

For a European put option on a non-dividend stock, vega can be shown as p  D St £  N0 .d1 / : (33.22) The derivation of Equation (33.22) is

33.5 Vega ( ) D The vega of an option, , is defined as the rate of change of the option price respected to the volatility of the underlying asset: @… D @¢ where … is the option price and ¢ is volatility of the stock price. We next show the derivation of vega for various kinds of stock option.

33.5.1 Derivation of Vega for Different Kinds of Stock Options For a European call option on a non-dividend stock, vega can be shown as p (33.21)  D St £  N0 .d1 / : The derivation of Equation (33.21) is D



d21 1 St p e 2   er£ X 2 

i 11 0 h ¢2  ln SXt C r C 2s £  £ 2 A @ ¢s2 £

D Xer£

@.1  N.d2 // @d2 @.1  N.d1 // @d1  St @d2 @¢s @d1 @¢s

1  d21 St r£ 2 D Xe p e  e X 2 

i 11 0 h ¢2  ln SXt C r C 2s £  £ 2 A @ ¢s2 £ r£

0 1 C St p e 2 

d2  21

@

1 C St p e 2 

¢s2 2

i 11 £  £2 A

h ¢s2 £3=2  ln SXt C r C

d2  21

d2 1 1 D St p e 2  2 

@

¢s2 2

¢s2 £

0 h St 1  d21 @  ln X C r C D St p e 2  ¢s2 £ 2  0

@N.d1 / @N.d2 / @Ct D St  Xer£ @¢s @¢s @¢s

@N.d1 / @d1 @N.d2 / @d2 D St  Xer£ @d1 @¢s @d2 @¢s h 0 St 2 3=2 1  d21 @ ¢s £  ln X C r C D St p e 2  ¢s2 £ 2 

@N.d2 / @N.d1 / @Pt D Xer£  St @¢s @¢s @¢s

¢s2 2

i 11 £  £2 A

h ¢s2 £3=2  ln SXt C r C

¢s2 £3=2 ¢s2 £

i 11 £  £2 A

¢s2 £

¢s2 2

i 11 £  £2 A



p D St £  N0 .d1 /

 Xer£

For a European call option on a dividend-paying stock, vega can be shown as p  D St eq£ £  N0 .d1 / :

(33.23)

33 Derivations and Applications of Greek Letters: Review and Integration

0

The derivation of Equation (33.23) is

@

@N.d1 / @N.d2 / @Ct D St eq£  Xer£ D @¢s @¢s @¢s

d2  21

¢s2 2

i 11 £  £2 A

d21 St 1  Xer£ p e 2   e.rq/£ X 2 

i 11 0 h ¢2  ln SXt C r  q C 2s £  £ 2 A @ ¢s2 £

¢s2 2

¢s2 2

i 11 £  £2 A

p  D St eq£ £  N0 .d1 / :

(33.24)

The derivation of Equation (33.24) is @N.d2 / @N.d1 / @Pt D Xer£  St eq£ @¢s @¢s @¢s @.1  N.d2 // @d2 @d2 @¢s

@.1  N.d1 // @d1 @d1 @¢s

2 d 1  1 St .rq/£ r£ 2 D Xe p e  e X 2   St eq£

i 11 £  £2 A

i 11 £  £2 A

¢s2 2

i 11 £  £2 A

2 3=2

1  d21 ¢ £ 2  s2 D St e p e ¢s £ 2  p D St eq£ £  N0 .d1 /

33.5.2 Application of Vega ( )

For a European call option on a dividend-paying stock, vega can be shown as

D Xer£

¢s2 2

¢s2 2

q£

2 3=2

d2 ¢ £ 1 1 D St p e 2  s 2 ¢s £ 2  p D St eq£ £  N0 .d1 /

D

i 11 £  £2 A

d2 1 1 C St eq£ p e 2 2  h 0 ¢s2 £3=2  ln SXt C r  q C @ ¢s2 £

i 11 £  £2 A

0 h  ln SXt C r  q C d21 1  St eq£ p e 2  @ ¢s2 £ 2 

¢s2 2

¢s2 £

d2 1 1 D St eq£ p e 2 2  0 h St  ln X C r  q C @ ¢s2 £



d21 1 D St eq£ p e 2 2  h 0 ¢s2 £3=2  ln SXt C r  q C @ ¢s2 £

h  ln SXt C r  q C

d2 1 1 C St eq£ p e 2 2  h 0 ¢s2 £3=2  ln SXt C r  q C @ ¢s2 £

@N.d1 / @d1 @N.d2 / @d2 D St eq£  Xer£ @d1 @¢s @d2 @¢s 1 D St eq£ p e 2  h 0 ¢s2 £3=2  ln SXt C r  q C @ ¢s2 £

499

Suppose a delta-neutral and gamma-neutral portfolio has a vega equal to  and the vega of a particular option is o . Similar to gamma, we can add a position of =o in option to make a vega-neutral portfolio. To maintain delta-neutral, we should change the underlying asset position. However, when we change the option position, the new portfolio is not gamma-neutral. Generally, a portfolio with one option cannot maintain its gamma-neutral and vega-neutral at the same time. If we want a portfolio to be both gamma-neutral and vega-neutral, we should include at least two kind of options on the same underlying asset in our portfolio. For example, a delta-neutral and gamma-neutral portfolio contains option A, option B, and underlying asset. The gamma and vega of this portfolio are 3; 200 and 2; 500, respectively. Option A has a delta of 0.3, gamma of 1.2, and vega of 1.5. Option B has a delta of 0.4, gamma of 1.6 and vega of 0.8. The new portfolio will be both gamma-neutral and vega-neutral when adding ¨A of option A and ¨B of option B into the original portfolio.

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p

d2 £ 1 St 1 p e 2   er£  X ¢ 2  s p

2 d 1 £ 1 C X£  er£ N.d2 / D St p e 2  ¢s 2  p

d2 1 £ 1  St p e 2  ¢s 2 

Gamma Neutral W  3200 C 1:2¨A C 1:6¨B D 0



Vega Neutral W  2500 C 1:5¨A C 0:8¨B D 0 From two equations shown above, we can get the solution that ¨A D 1; 000 and ¨B D 1; 250. The delta of new portfolio is 1; 000 0:3 C 1; 250 0:4 D 800. To maintain delta-neutral, we need to short 800 shares of the underlying asset.

D X£  er£ N.d2 / For a European put option on a non-dividend stock, rho can be shown as

33.6 Rho (ρ) The rho of an option is defined as the rate of change of the option price respected to the interest rate: rho D

@… @r

33.6.1 Derivation of Rho for Different Kinds of Stock Options For a European call option on a non-dividend stock, rho can be shown as (33.25)

The derivation of Equation (33.25) is @N.d1 / @Ct D St  .£/  X  er£ N.d2 / rho D @r @r @N.d2 /  Xer£ @r @N.d1 / @d1 C X£  er£ N.d2 / D St @d1 @r @N.d2 / @d2 @d2 @r p

d21 1 £ D St p e 2  ¢ 2  s

(33.26)

The derivation of Equation (33.26) is @Pt D .£/  X  er£ N.d2 / @r @N.d2 / @N.d1 /  St C Xer£ @r @r @.1  N.d2 // @d2 D X£  er£ .1  N.d2 // C Xer£ @d2 @r

rho D

where … is the option price and r is interest rate. The rho for an ordinary stock call option should be positive because higher interest rate reduces the present value of the strike price, which in turn increases the value of the call option. Similarly, the rho of an ordinary put option should be negative by the same reasoning. We next show the derivation of rho for various kinds of stock option.

rho D X£  er£ N.d2 /:

rho D X£  er£ N.d2 /:

 St

@.1  N.d1 // @d1 @d1 @r

D X£  er£ .1  N.d2 //  Xer£

p

£ 1  d21 St r£   p e 2  e X ¢s 2  p

2 d1 1 £ C St p e 2  ¢s 2  d2 1 1 D X£  e .1  N.d2 //  St p e 2  2  p

2 d 1 £ 1 C St p e 2  ¢ 2  s

r£

p

£ ¢s

D X£  er£ N.d2 / For a European call option on a dividend-paying stock, rho can be shown as rho D X£  er£ N.d2 /:

(33.27)

The derivation of Equation (33.27) is

 Xer£

rho D

@N.d1 / @Ct D St eq£ @r @r  .£/  X  er£ N.d2 /  Xer£

C X£  er£ N.d2 /  Xer£ D St eq£

@N.d2 / @r

@N.d1 / @d1 C X£  er£ N.d2 / @d1 @r

33 Derivations and Applications of Greek Letters: Review and Integration

@N.d2 / @d2 @d2 @r p

d2 £ q£ 1  21 C X£  er£ N.d2 /  D St e p e ¢s 2 

p

1  d21 St .rq/£ £ r£ 2  Xe  p e  e  X ¢s 2  p

2 d 1  1 D St eq£ p e 2  C X£  er£ N.d2 / ¢ 2  s p

d2 £ q£ 1  21   St e p e ¢ 2  s  Xer£

D X£  er£ N.d2 / For a European put option on a dividend-paying stock, rho can be shown as rho D X£  er£ N.d2 /:

(33.28)

501

Rhoput D X£er£ N.d2 / D .$58/.0:25/e.0:05/.0:25/N !   ln.65=58/ C 0:05  12 .0:3/2 .0:25/ p Š 3:168: .0:3/ 0:25 This calculation indicates that given 1% change increase in interest rate, say from 5% to 6%, the value of this European call option will decrease 0.03168 .0:01 3:168/. This simple example can be further applied to stocks that pay dividends using the derivation results shown previously.

33.7 Derivation of Sensitivity for Stock Options Respective with Exercise Price For a European call option on a non-dividend stock, the sensitivity can be shown as

The derivation of Equation (33.28) is @Pt D .£/  X  er£ N.d2 / rho D @r @N.d2 / @N.d1 /  St eq£ C Xer£ @r @r @.1  N.d2 // @d2 D X£  er£ .1  N.d2 // C Xer£ @d2 @r  St eq£

@.1  N.d1 // @d1 @d1 @r

D X£  er£ .1  N.d2 //

p

d21 St £ 1  Xer£  p e 2   e.rq/£  X ¢s 2  p

d21 1 £ C St eq£ p e 2  ¢s 2  p

d21 1 £ D X£  er£ .1  N.d2 //  St eq£ p e 2  ¢s 2  p

2 d1 1 £ C St eq£ p e 2  ¢s 2 

@Ct D er£ N.d2 /: @X The derivation of Equation (33.29) is

@N.d1 / @N.d2 / @Ct D St  e r N.d2 /  Xe r @X @X @X @N.d1 / @d1 @N.d2 / @d2  e r N.d2 /  Xe r D St @d1 @X @d2 @X

2 d1 1 1 1 e r N.d2 /  Xe r D St p e  2  p   X s  2



d12 1 1 St r 1 e

p e 2  p   X X s  2

2 d1 St 1  e r N.d2 / D p e 2  X s 2 

d2 St 1  21 e D e r N.d2 /   p X s 2  For a European put option on a non-dividend stock, the sensitivity can be shown as

D X£  er£ N.d2 /

33.6.2 Application of Rho (ρ) Assume that an investor would like to see how interest rate changes affect the value of a 3-month European put option she holds with the following information. The current stock price is $65 and the strike price is $58. The interest rate and the volatility of the stock is 5% and 30% per year, respectively. The rho of this European put can be calculated as follows:

(33.29)

@Pt D er£ N.d2 /: @X

(33.30)

The derivation of Equation (33.30) is @Pt @N.d2 / @N.d1 / D er£ N.d2 / C Xer£  St @X @X @X @.1  N.d2 // @d2 D er£ .1  N.d2 // C Xer£ @d2 @X  St

@.1  N.d1 // @d1 @d1 @X

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H.-Y. Chen et al.

D er£ .1  N.d2 //



d21 1 1 St 1 p    Xer£ p e 2   er£ X X ¢s £ 2 

2 d1 1 1 1 C St p e 2  p   X ¢s £ 2 

d21 St 1 D er£ .1  N.d2 // C p e 2 X ¢s 2 £

2 d1 St 1 e 2  p X ¢s 2 £ D er£ N.d2 / For a European call option on a dividend-paying stock, the sensitivity can be shown as @Ct D er£ N.d2 /: @X

(33.31)

The derivation of Equation (33.31) is @N.d1 / @N.d2 / @Ct D St eq£  er£ N.d2 /  Xer£ @X @X @X @N.d1 / @d1 @N.d2 / @d2  er£ N.d2 /  Xer£ D St eq£ @d1 @X @d2 @X

2 d1 1 1 1 D St eq£ p e 2  p   X ¢s £ 2   er£ N.d2 /  Xer£



1 1 1  d21 St .rq/£ 2

p e  e p   X X ¢s £ 2 

2 q£ d1 1 St e e 2  D p  er£ N.d2 / X ¢s 2 £

d2 1 St eq£  21  p  e X ¢s 2 £

 St eq£

@.1  N.d1 // @d1 @d1 @X

D er£ .1  N.d2 //  Xer£



1  d21 St .rq/£ 1 1 2 p e  e p   X X ¢s £ 2 

d2 1 1 1 1 C St eq£ p e 2  p   X ¢ £ 2  s q£

d2 St e 1 1 D er£ .1  N.d2 // C p e 2 X ¢s 2 £

d2 St eq£ 1 1  p e 2 X ¢s 2 £ D er£ N.d2 /

33.8 Relationship Between Delta, Theta, and Gamma So far, the discussion has introduced the derivation and application of each individual Greeks and how they can be applied in portfolio management. In practice, the interaction or tradeoff between these parameters is of concern as well. For example, recall the partial differential equation for the Black–Scholes formula with non-dividend paying stock can be written as @… 1 @2 … @… C rS C  2 S 2 2 D r…: @t @S 2 @S Where … is the value of the derivative security contingent on stock price, S is the price of stock, r is the risk free rate,  is the volatility of the stock price, and t is the time to expiration of the derivative. Given the earlier derivation, we can rewrite the Black–Scholes PDE as 1 ‚ C rS C  2 S 2  D r…: 2

D er£ N.d2 / For a European put option on a dividend-paying stock, the sensitivity can be shown as @Pt D er£ N.d2 /: @X

(33.32)

The derivation of Equation (33.32) is @Pt @N.d2 / @N.d1 / D er£ N.d2 / C Xer£  St eq£ @X @X @X @.1  N.d // @d 2 2 D er£ .1  N.d2 // C Xer£ @d2 @X

This relation gives us the tradeoff between delta, gamma, and theta. For example, suppose there are two delta neutral . D 0/ portfolios, one with positive gamma . > 0/ and the other one with negative gamma . < 0/ and they both have value of $1.… D 1/. The tradeoff can be written as 1 ‚ C  2 S 2  D r: 2 For the first portfolio, if gamma is positive and large, then theta is negative and large. When gamma is positive, change in stock prices results in higher value of the option. This

33 Derivations and Applications of Greek Letters: Review and Integration

means that when there is no change in stock prices, the value of the option declines as we approach the expiration date. As a result, the theta is negative. On the other hand, when gamma is negative and large, change in stock prices result in lower option value. This means that when there are no stock price changes, the value of the option increases as we approach the expiration and theta is positive. This gives us a tradeoff between gamma and theta and they can be used as proxy for each other in a delta neutral portfolio.

33.9 Conclusion In this chapter we have shown the derivation of the sensitivities of the option price to the change in the value of state variables or parameters. The first Greek is delta . /, which is the rate of change of option price to change in price of underlying asset. Once the delta is calculated, the next step is the rate of change of delta with respect to underlying asset price, which gives us gamma ./. Another two risk measures are theta .‚/ and rho ./, which measure the change in option value with respect to passing time and interest rate, respectively. Finally, one can also measure the change in option value with respect to the volatility of the underlying asset and this gives us the vega .v/. The relationship between these risk measures are shown, including an example from the Black–Scholes partial differential equation. Furthermore, the applications of these Greeks letters in portfolio management have also been discussed. Risk management is one of the important topics in

503

finance today, both for academics and practitioners. Given the recent credit crisis, one can observe that it is crucial to properly measure the risk related to the ever-more complicated financial assets.

References Bjork, T. 1998. Arbitrage theory in continuous time, Oxford University Press, Oxford. Boyle, P. P. and D. Emanuel. 1980. “Discretely adjusted option hedges.” Journal of Financial Economics 8(3), 259–282. Duffie, D. 2001. Dynamic asset pricing theory, Princeton University Press, Princeton, NJ. Fabozzi, F. J. 2007. Fixed income analysis, 2nd ed., Wiley, Hoboken, NJ. Figlewski, S. 1989. “Options arbitrage in imperfect markets.” Journal of Finance 44(5), 1289–1311. Galai, D. 1983. “The components of the return from hedging options against stocks.” Journal of Business 56(1), 45–54. Hull, J. 2006. Options, futures, and other derivatives, 6th ed., Pearson, Upper Saddle River, NJ. Hull, J., and A. White. 1987. “Hedging the risks from writing foreign currency options.” Journal of International Money and Finance 6(2), 131–152. Karatzas, I. and S. E. Shreve. 2000. Brownian motion and stochastic calculus, Springer, New York. Klebaner, F. C. 2005. Introduction to stochastic calculus with applications, Imperial College Press, London. McDonlad, R. L. 2005. Derivatives markets, 2nd ed., Addison-Wesley, Boston, MA. Shreve, S. E. 2004. Stochastic calculus for finance II: continuous time model, Springer, New York. Tuckman, B. 2002. Fixed income securities: tools for today’s markets, 2nd ed., Wiley, Hoboken, NJ.

Chapter 34

A Further Analysis of the Convergence Rates and Patterns of the Binomial Models San-Lin Chung and Pai-Ta Shih

Abstract This paper extends the generalized Cox-RossRubinstein (hereafter GCRR) model of Chung and Shih (2007). We provide a further analysis of the convergence rates and patterns based on various GCRR models. The numerical results indicate that the GCRR-XPC model and the GCRR-JR .p D 1=2/ model (defined in Table 34.1) outperform the other GCRR models for pricing European calls and American puts. Our results confirm that the node positioning and the selection of the tree structures (mainly the asymptotic behavior of the risk-neutral probability) are important factors for determining the convergence rates and patterns of the binomial models. Keywords Binomial model r Rate of convergence r Monotonic convergence

34.1 Brief Review of the Binomial Models Ever since the seminal works of Cox et al. (1979, hereafter CRR) and Rendleman and Bartter (1979), many articles have extended the binomial option pricing models in many aspects. One stream of the literature modifies the lattice or tree type to improve the accuracy and computational efficiency of binomial option prices. The pricing errors in the discretetime models are mainly due to “distribution error” and “nonlinearity error” (see Figlewski and Gao (1999) for thorough discussions). Within the literature, there are many proposed solutions that reduce the distribution error and/or nonlinearity error. It is generally difficult to reduce the distribution error embedded in the binomial model when the number of time steps .n/ is small, probably due to the nature of the problem. Nevertheless there are two important works that might shed some S.-L. Chung () and P.-T. Shih Department of Finance, National Taiwan University, No. 85, Section 34.4, Roosevelt Road, Taipei 106, Taiwan, R.O.C. e-mail: [email protected]

light on understanding or dealing with distribution error. First, Omberg (1988) developed a family of efficient multinomial models by applying the Gauss-Hermite quadrature1 to the integration problem (e.g., N.d1 / in the Black-Scholes formula) presented in the option pricing formulas. Second, Leisen and Reimer (1996) modified the sizes of up- and down-movements by applying various normal approximations (e.g., the Camp-Paulson inversion formula) to the binomial distribution derived in the mathematical literature. Concerning the techniques to reduce nonlinearity error, there are least three important works (methods) worth noting in the literature. First, Ritchken (1995) and Tian (1999) showed that one can improve the numerical accuracy of the binomial option prices by allocating one of the final nodes on the strike price or one layer of the nodes on the barrier price. Second, Figlewski and Gao (1999) proposed the so-called adaptive mesh method that sharply reduces nonlinearity error by adding one or more small sections of fine high-resolution lattice onto a tree with coarser time and price steps. Third, Broadie and Detemple (1996) and Heston and Zhou (2000) suggested modified binomial models by replacing the binomial prices prior to the end of the tree by the Black-Scholes values, or by smoothing payoff functions at maturity, and computing the rest of binomial prices as usual. Another stream of research focuses on the convergence rates and patterns of the binomial models. For example, Leisen and Reimer (1996) proved that the convergence is of order O.1=n/ for the CRR model, the Jarrow and Rudd (1983) model, and the Tian (1993) model. Heston and Zhou (2000) also showed that the rate of convergence for the lattice approach can be enhanced by smoothing the option payoff p functions and cannot exceed 1= n at some nodes of the tree. Concerning the convergence patterns of the binomial models, it is widely documented that the CRR binomial prices converge to the Black-Scholes price in a wavy, erratic way (for example, see Broadie and Detemple 1996; Tian 1999).2 1

The Gauss-Hermite quadrature is a well developed and highly efficient method in the numerical integration literature. 2 Also see Diener and Diener (2004) for a complete discussion of the price oscillations in a tree model.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_34, 

505

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Several remedies have been proposed in the literature, e.g., see Broadie and Detemple (1996), Leisen and Reimer (1996), Tian (1999), Heston and Zhou (2000), Widdicks et al. (2002), and Chung and Shih (2007). In a recent article, Chung and Shih (2007) proposed a generalized CRR (hereafter GCRR) model by adding a stretch parameter to the CRR model. With the flexibility from the stretch parameter, the GCRR model provides a numerically efficient method for pricing a range of options. Moreover, Chung and Shih (2007) also showed that (1) the convergence pattern of the binomial prices to the accurate price depends on the node positioning (see their Theorem 2), and (2) the rate of convergence depends on the selection of the tree structures (i.e., the risk-neutral probability p and jump sizes u and d ). In this paper, we provide a further analysis of the convergence rates and patterns based on the GCRR models. Our results confirm that the node positioning and the selection of the tree structures are important factors for determining the convergence rates and patterns of the binomial models. The rest of the paper is organized as follows. Section 34.2 discusses the importance of node positioning for monotonic convergence of binomial prices. We also summarize a complete list of binomial models with monotonic convergence in Sect. 34.2. Section 34.3 introduces the GCRR model and discusses its flexibility for node positioning. Various GCRR related models are also defined in Sect. 34.3. Section 34.4 presents the numerical results of various GCRR models (such as the GCRR-XPC and GCRR-JR models) against other binomial models for pricing a range of options. The numerical analysis is mainly focused the convergence rates and patterns of various binomial models. Section 34.5 concludes the paper.

period price jumps either upward to uS with probability q or downward to dS with probability 1  q, where 0 < d < e r t < u and 0 < q < 1. Again the standard dynamic hedge approach ensures that the risk-neutral probability p, instead of q, is relevant for pricing options. The binomial option pricing model is completely determined by the jump sizes u and d and the risk-neutral probability p. In the CRR model, three conditions; that is, the risk-neutral mean and variance of the stock price in the next period and ud D 1, are utilized to determine u; d , and p as follows:

34.2 The Importance of Node Positioning for Monotonic Convergence

Therefore, Theorem 1 shows that the node positioning (i.e., the relative distance of the strike price between the most adjacent nodes) is the main reason why the CRR binomial prices converge to the Black-Scholes price in a wavy, erratic way. To get a monotonic convergent pattern, we can (1) use the CRR model corrected with the quadratic function (see Equation (34.2)) as suggested by Chung and Shih (2007); (2) fine tune the binomial model such that ".n/ is a constant (see Tian 1999); (3) choose the appropriate number of time steps n such that ".n/ is a constant (see WAND 2002); (4) smooth the option payoff function to be a continuous and differentiable function (see Heston and Zhou 2000); (5) replace the binomial prices prior to the end of the tree by the BlackScholes values and computing the rest of binomial prices as usual (see Broadie and Detemple 1996); or (6) place the strike price in the center of the final nodes (see Leisen and Reimer 1996; Chung and Shih 2007). Once the monotonic convergent pattern is derived, various extrapolation techniques are applicable to the above binomial

Following the standard option pricing model, we assume that the underlying asset price follows a geometric Brownian motion; that is, dS=S D  dt C  dW; (34.1) where  is the drift rate,  is the instantaneous volatility, and W is a Wiener process. Standard dynamic hedge and complete market argument assures that options can be valued by assuming risk neutrality; that is,  D r. For pricing plain vanilla European options, Black-Scholes had derived closedform solutions. For pricing other types of options such as American options, we have to rely on numerical approximation methods, such as a binomial model. In an n-period binomial model, the time to maturity of the option .T / is divided into n equal time steps; that is,

t D T =n. If the current stock price is S , then the next

u D e

p

t

; d D e 

p

t

; and p D .e r t  d /=.u  d /:

It is widely documented that the CRR binomial prices converge to the Black-Scholes price in a wavy, erratic way (for example, see Broadie and Detemple 1996; Tian 1999). This kind of convergent pattern can be explained by the following theorem. Theorem 1. Let Cn and CBS be the n-period CRR binomial and continuous-time prices of a standard European call option with the terminal payoff max.ST  X; 0/. Therefore, Cn D CBS C

ˇ.0:5  ".n//2 C O.1=n/ n

(34.2)

p 2 =2/T p where ˇ D  2XerT .d2 /   T ; d2 D ln.S=X /C.r ;  T m nm x =X / , and m is the smallest ".n/ D ln.S ln.u=d / ; Sx D Su d m nm > X. integer that satisfies Su d

Proof. Please see Chung and Shih (2007).

34 A Further Analysis of the Convergence Rates and Patterns of the Binomial Models

models and the option price can be more efficiently calculated. The extrapolation formula used in this article is briefly summarized in the Appendix 38A.

34.3 The Flexibility of GCRR Model for Node Positioning In the GCRR model of Chung and Shih (2007), a stretch parameter ./ is incorporated into the CRR model so that the shape (or spanning) of the lattice can be flexibly adjusted. In other words, the condition that ud D 1, which is used in the traditional CRR model, is relaxed in the GCRR model. As a result, the GCRR model has one more degree of freedom to adjust the shape of the binomial lattice. We restate the GCRR model of Chung and Shih (2007) in Theorem 2. Theorem 2. In the GCRR model the jump sizes and probability of going up are as follows: u D e 

p

t

1

; d D e  

p

t

; and p D

e r t  d ; ud

(34.3)

507

the GCRR-XPC binomial prices converges to the accurate price smoothly and monotonically. A straightforward extension of Chung and Shih (2007) is to place the strike price on a certain place of binomial tree; for example, on the 1/3-th or the 2/3-th nodes. Moreover, the idea of the GCRR model potentially can be applied to other binomial models. For example, we can generalize the Jarrow and Rudd (1983, hereafter JR) model by setting pD

2

p 1 1 2 ; u D e .r 2  / t C t ; C1

1 2 1 and d D e .r 2  / t   

p

t

:

(34.4)

By changing the stretch parameter , the shape (or spanning) of the lattice can be flexibly adjusted so that one of the final nodes can be allocated on the strike price. In the following section, we will investigate the numerical performances of various GCRR models where one of the final nodes can be allocated on the strike price and therefore the convergent patterns are monotonic. In addition to CRR and JR models, six GCRR related models are investigated in the next section and their definitions are given in Table 34.1.

where  2 RC is a stretch parameter that determines the shape (or spanning) of the binomial tree. Moreover, when

t ! 0; that is, the number of time steps n grows to infinity, the GCRR binomial option prices converge to the BlackScholes formulae for plain vanilla European options.

34.4 Numerical Results of Various GCRR Models

The stretch parameter ./ in the GCRR model gives us the flexibility for node positioning. For instance, Chung and Shih (2007) placed the strike price in the center of the final nodes (termed GCRR-XPC model in their paper) and showed that

In this section, we will investigate the convergence pattern, the rate of convergence, and the accuracy of six GCRR models defined in Table 34.1 against the CRR model and the

Table 34.1 The definitions of various GCRR models

Model

Definition

GCRR-XPC

A GCRR model where the stretch parameter  is chosen such that the strike price is placed in the center of final nodes

GCRR-X on the 1/3-th node

A GCRR model where the stretch parameter  is chosen such that the number of nodes above the strike price is one third of the total final nodes and one of the final nodes can be allocated on the strike price

GCRR-X on the 2/3-th node

A GCRR model where the stretch parameter  is chosen such that the number of nodes above the strike price is two thirds of the total final nodes and one of the final nodes can be allocated on the strike price

GCRR-JR .p D 1=2/

A GCRR-JR model where the stretch parameter  is chosen such that the up probability is closest to one half and one of the final nodes can be allocated on the strike price

GCRR-JR .p D 1=3/

A GCRR-JR model where the stretch parameter  is chosen such that the up probability is closest to one third and one of the final nodes can be allocated on the strike price

GCRR-JR .p D 2=3/

A GCRR-JR model where the stretch parameter  is chosen such that the up probability is closest to two thirds and one of the final nodes can be allocated on the strike price

508

S.-L. Chung and P.-T. Shih

JR model. Figure 34.1 shows the convergence pattern of the CRR model, the GCRR-XPC model, GCRR-X on the 1/3-th node model, and GCRR-X on the 2/3-th node model for pricing an in-the-money call option. The parameters in Fig. 34.1 are adopted from Tian (1999): the asset price is 100, the strike price is 95, the volatility is 0.2, the maturity of the option is 6 months, and the risk-free rate is 0.06. The pricing error is defined as the difference between the binomial price and the Black-Scholes price. In Fig. 34.1, the pricing errors are plotted against an even number of time steps ranging from 10 to 100. It is apparent from Fig. 34.1 that the convergence of the CRR model is not monotonic and smooth. Moreover, the pricing errors oscillate between positive and negative values. In contrast, the GCRR-XPC model, GCRR-X on the 1/3-th nodes model, and GCRR-X on the 2/3-th nodes model converge monotonically to the Black-Scholes price.

Therefore, various extrapolation techniques are applicable to the GCRR-XPC model, GCRR-X on the 1/3-th node model, and GCRR-X on the 2/3-th node model. A similar convergence pattern is also embedded in GCRR-JR .p D 1=2/ model, GCRR-JR .p D 1=3/ model and GCRR-JR .p D 2=3/ model as shown in Fig. 34.2. Table 34.2 reports the prices of European call and American put options calculated using the CRR model, the GCRR-XPC model, GCRR-X on the 1/3-th nodes model, and GCRR-X on the 2/3-th nodes model. The parameters are the following: the asset price is 40, the strike price is 45, the asset price volatility is 0.4, the maturity of the option is 6 months, and the risk-free rate is 0.6. The errors of the CRR model do not necessarily go down as n increases. For example, the error for the CRR model is 0.0001215 when n D 6400, worse than an error of 0.0000966 when n D 3200. In contrast, the error decreases

0.15 CRR GCRR-XPC GCRR- X on the 1/3-th nodes GCRR- X on the 2/3-th node

Pricing Error

0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 10

28

46

64

82

100

Number of Time Steps

Fig. 34.1 The convergence patterns of the CRR model and three related GCRR models. This figure shows the convergence patterns of the CRR model, GCRR-XPC model, GCRR-X on the 1/3-th node model,

and GCRR-X on the 2/3-th node model. The asset price is 100, the strike price is 95, the volatility is 0.2, the maturity of the option is 6 months, and the risk-free rate is 0.06

0.05 JR GCRR-JR (p=1/3) GCRR-JR (p=1/2) GCRR-JR (p=2/3)

Pricing Error

0

-0.05

-0.1

-0.15

-0.2 10

28

46 64 Number of Time Steps

Fig. 34.2 The convergence patterns of the JR model and three related GCRR models. This figure shows the convergence patterns of the JR model, GCRR-JR .p D 1=2/ model, GCRR-JR .p D 1=3/ model and

82

100

GCRR-JR .p D 2=3/ model. The asset price is 100, the strike price is 95, the volatility is 0.2, the maturity of the option is 6 months, and the risk-free rate is 0.06

0:0197709 0:0057017 0:0037874 0:0000405 0:0004341 0:0004564 0:0000966 0:0001215

Error

0:020305 0:0009942 0:0025996 0:0008327 0:0005569 0:0003715 0:0000189

3:1127553 3:0872827 3:089197 3:0929439 3:0925503 3:0934409 3:0930811 3:0931059 3:0929844

Price

7:0271842 7:0068793 7:0058851 7:0084847 7:007652 7:008209 7:0078375 7:0078186

50 100 200 400 800 1600 3200 6400 B-S price

N

50 100 200 400 800 1600 3200 6400 20:4232027 0:3824407 3:1221175 1:4950395 1:4993231 19:6141612

Error ratio

3:467556 1:5054227 93:5404085 0:0932659 0:9511233 4:722664 0:7954713

Error ratio

6:9997733 7:0036989 7:0057155 7:0067137 7:0071955 7:0074457 7:0075681 7:0076292

Price

3:0761791 3:0845771 3:0887797 3:0908818 3:091933 3:0924587 3:0927216 3:092853 3:0929844

Price

GCRR-XPC

0:0039255 0:0020166 0:0009982 0:0004819 0:0002501 0:0001224 0:0000611

Error

1:9988942 1:9994844 1:9997516 1:9998781 1:9999397 1:9999699 1:9999849

0:0168054 0:0084073 0:0042047 0:0021026 0:0010514 0:0005257 0:0002629 0:0001314

1:9465921 2:0202401 2:0714921 1:9264892 2:04292 2:0050691

Error ratio

American put

Error ratio

Error

7:0165994 7:0174403 7:0155716 7:0139161 7:012333 7:0111153 7:0101693 7:0094751

Price

3:1172274 3:1164068 3:1109797 3:1070312 3:1033497 3:1006093 3:0984943 3:0969498 3:0929844

Price

Error ratio

1:0350322 1:3015833 1:2810991 1:3551761 1:3594105 1:3838448 1:3895148

Error ratio

0:0008409 0:0018687 0:4499987 0:0016555 1:1287822 0:0015831 1:0457441 0:0012178 1:2999743 0:000946 1:2873403 0:0006942 1:3626916

Error

0:024243 0:0234224 0:0179953 0:0140468 0:0103653 0:0076248 0:0055099 0:0039653

Error

GCRR-X on the 1/3-th nodes

6:9792354 6:9880395 6:9950865 6:9990891 7:0018798 7:0036826 7:0049213 7:00576

Price

3:0294775 3:0495034 3:065062 3:0739472 3:0801303 3:0841135 3:0868519 3:0887076 3:0929844

Price

0:0088041 0:007047 0:0040026 0:0027906 0:0018028 0:0012387 0:0008387

Error

0:063507 0:0434811 0:0279225 0:0190372 0:0128541 0:008871 0:0061326 0:0042768

Error

GCRR-X on the 2/3-th node

1:2493368 1:7606073 1:4343098 1:5479088 1:4553711 1:477035

Error ratio

1:4605662 1:557207 1:4667279 1:48102 1:4490152 1:4465349 1:4339211

Error ratio

Notes: The Parameters are: the asset price is 40, the strike price is 45, the asset price volatility is 0.4, the maturity of the option is 6 months, the risk-free rate is 0.6, and the number of time steps starts at 50 and doubles each time subsequently. This table reports price, pricing errors, and error ratios from the CRR model, the GCRR-XPC model, the GCRR-X on the 1/3-th nodes model, the GCRR-X on the 2/3-th nodes model

Error

Price

N

CRR

Table 34.2 Price and error ratios for European calls and American puts using the CRR model and three related GCRR models. European call

34 A Further Analysis of the Convergence Rates and Patterns of the Binomial Models 509

510

S.-L. Chung and P.-T. Shih

as n increases for the the GCRR-XPC model, GCRR-X on the 1/3-th nodes model, and GCRR-X on the 2/3-th nodes model. Thus, one can apply the Richardson extrapolation method to increase the accuracy for these models. Let e.n/ be the pricing error of the n-step binomial model; that is, (34.5) e.n/ D Cn  CBS ; where Cn is the n-step binomial price and CBS is the BlackScholes price.3 Define the error ratio as: .n/ D e.n/=e.2n/:

(34.6)

The error ratio is a measure of the improvements in accuracy as the number of time steps doubles. It is clear from Table 34.2 that the pricing error of the GCRR-XPC model is almost exactly halved when the number of time steps doubles; that is, the error ratio almost equals two. In contrast, Table 34.2 indicates that the GCRR-X on the 1/3-th nodes model and the GCRR-X on the 2/3-th nodes model have p a slower convergence rate of order O.1= n/. Similarly, in Table 34.3, the pricing error of the GCRR-JR .p D 1=2/ model is almost exactly halved when the number of time steps doubles; that is, the error ratio almost equals two. In contrast, it seems that the GCRR-JR .p D 1=3/ model and the GCRR-JR .p D 2=3/ model have a slower convergence p rate of order O.1= n/. The extrapolation formulas when the error ratio is known are suggested in the Appendix. We also compare the numerical efficiency of the various GCRR models including GCRR-XPC model, GCRR-X on the 1/3-th nodes model, GCRR-X on the 2/3-th nodes model, GCRR-JR .p D 1=2/ model, GCRR-JR .p D 1=3/ model, and GCRR-JR .p D 2=3/ model. The above binomial methods are appropriate for applying the Richardson extrapolation to increase the accuracy. All the above methods are applied to price a range of European and American options. The accuracy of each method is measured using the root mean squared errors (RMSE) with respect to the closed-form solution, or the GCRR-XPC model with a two-point extrapolation .n D 50;000/ in the case of American puts. The parameters of these options are adopted from WAND (2002) as follows: the stock price is 40, the risk free rate is 0.06, the strike prices are 35 and 45, the volatilities are 0.2 and 0.4, and the maturities are 0.25 and 0.5.

3 Since there is no exact solution for the American put, following Tian (1999) and Heston and Zhou (2000), the pricing errors of the n-step binomial model is defined as: e.n/ D Cn  Cn=2 .

To clarify the role played by Richardson extrapolation, we first compare the numerical efficiency of each method on a raw basis without any extrapolation procedure. The results are presented in Table 34.4. It is clear from Table 34.4 that GCRR-XPC outperforms the other methods for pricing European calls and American puts and GCRR-JR .p D 1=2/ model is the second best model. Moreover, the results of Table 34.4 also show that for European call options, GCRR-XPC and GCRR-JR .p D 1=2/ model monotonically converge to the Black-Scholes value at the 1=n rate. On the other hand, GCRR-X on the 1/3-th nodes model, GCRR-X on the 2/3-th nodes model, GCRR-JR .p D 1=3/ model and GCRR-JR .p D 2=3/ modelımonotonically converge to the p n rate. Black-Scholes value at the 1 We compare the numerical efficiency of the various GCRR models when the Richardson extrapolation technique is applied to all methods. The number of time steps used for all models with a two-point extrapolation is .n; 2Œn=4 / where Œx denotes the Gaussian symbol. In Table 34.5, it is very clear that GCRR-XPC also outperforms the other methods for pricing European calls and American puts, and GCRR-JR .p D 1=2/ model is the second best model when the Richardson extrapolation formula is applied to all binomial models.

34.5 Conclusion This paper provides a further analysis of the convergence rates and patterns based on the GCRR model of Chung and Shih (2007). We discuss how to correct the oscillatory behavior of the CRR model by various GCRR models so that the binomial prices converge to its continuous time price monotonically and smoothly, and thus one can apply the Richardson extrapolation method to increase the accuracy for these models. It is shown that when the risk-neutral probability p equals or converges to 0.5, then the convergence rate is order of O.1=n/. For example, GCRR-XPC and GCRR-JR .p D 1=2/ model monotonically converge to the Black-Scholes value at the 1=n rate. However, when p does note equal or p converge to 0.5, the convergence rate is order of O.1= n/. For example, GCRR-X on the 1/3-th nodes model, GCRR-X on the 2/3-th nodes model, GCRR-JR .p D 1=3/ model, and GCRR-JR .p D 2=3/ model monotonically converge to the p Black-Scholes value at the 1= n rate. Finally, the numerical results indicate that the GCRR models with various modifications are efficient for pricing a range of options. GCRR-XPC outperforms the other GCRR models for pricing European calls and American puts and GCRR-JR .p D 1=2/ model is the second best model.

0:0176285 0:0041689 0:0005397 0:0023722 0:0010473 0:0006063 0:0000826 0:0001073

Error

0:0106746 0:0043763 0:0002667 0:0012808 0:0013773 0:0003649 0:0001685

3:110613 3:0971533 3:0935241 3:0953566 3:0940317 3:0923781 3:093067 3:0928771 3:0929844

Price

7:0249121 7:0142375 7:0098612 7:0101279 7:008847 7:0074697 7:0078346 7:0076661

50 100 200 400 800 1600 3200 6400 B-S price

N

50 100 200 400 800 1600 3200 6400 2:4391967 16:4106629 0:2082054 0:9299303 3:7744614 2:1656175

Error ratio

4:2285686 7:7248534 0:2274975 2:2650576 1:7273667 7:3408261 0:7696826

Error ratio

6:9948508 7:0020738 7:0048184 7:0061896 7:0069831 7:0073317 7:0075096 7:0076004

Price

3:064688 3:0808786 3:0867817 3:0896514 3:0914335 3:0921893 3:0925832 3:0927856 3:0929844

Price

0:007223 0:0027446 0:0013712 0:0007935 0:0003486 0:000178 0:0000908

Error

2:3374246 1:9516768 1:8609907 2:1490994 1:9505708 1:9816991 2:0184492

0:0282964 0:0121058 0:0062028 0:003333 0:0015509 0:0007951 0:0004012 0:0001988

2:6316614 2:0016892 1:7280424 2:276293 1:9588599 1:9605003

Error ratio

American put

Error ratio

Error

GCRR-JR .p D 1=2/

7:0162674 7:0171254 7:0154509 7:0139527 7:0125108 7:011281 7:0103178 7:0095929

Price

3:1266496 3:1229406 3:1159382 3:1107778 3:1063754 3:1028496 3:1001488 3:0981502 3:0929844

Price

0:000858 0:0016745 0:0014982 0:0014419 0:0012298 0:0009631 0:0007249

Error

0:0336651 0:0299562 0:0229538 0:0177934 0:0133909 0:0098652 0:0071644 0:0051658

Error

GCRR-JR .p D 1=3/

0:5123652 1:117721 1:0390038 1:1724385 1:2768974 1:328679

Error ratio

1:123812 1:305067 1:2900141 1:3287666 1:3573885 1:3769746 1:3869016

Error ratio

6:970908 6:9846444 6:9931022 6:9979603 7:0011871 7:0032567 7:0046428 7:0055771

Price

3:000282 3:0342243 3:0554242 3:0676809 3:0759322 3:081286 3:0849041 3:0873602 3:0929844

Price

0:01373646 0:00845776 0:00485807 0:0032268 0:00206967 0:00138609 0:0009343

Error

0:09270245 0:05876007 0:03756026 0:02530355 0:01705221 0:0116984 0:00808034 0:00562422

Error

GCRR-JR .p D 2=3/

1:62412507 1:74097083 1:50553798 1:55909214 1:4931684 1:48356412

Error ratio

1:5776437 1:56442129 1:48438717 1:48388688 1:45765261 1:44776122 1:43670451

Error ratio

Notes: The Parameters are: the asset price is 40, the strike price is 45, the asset price volatility is 0.4, the maturity of the option is 6 months, the risk-free rate is 0.6, and the number of time steps starts at 50 and doubles each time subsequently. This table reports price, pricing errors, and error ratios from the JR model, the GCRR-JR .p D 1=3/ model, the GCRR-JR .p D 1=2/ model, the GCRR-JR .p D 2=3/ model

Error

Price

N

JR

Table 34.3 Price and error ratios for European calls and American puts using the JR model and three related GCRR models European call

34 A Further Analysis of the Convergence Rates and Patterns of the Binomial Models 511

512

S.-L. Chung and P.-T. Shih

Table 34.4 A Comparison of the performance of various models over a range of option types and parameters (without extrapolation) GCRR-X GCRR-X GCRR-JR GCRR-JR GCRR-JR N GCRR-XPC on the 1/3-th nodes on the 2/3-th nodes .p D 1=2/ .p D 1=3/ .p D 2=3/ Approximation RMS error for European calls 50 100 200 400 800 1600 3200 6400

9.17E-03 4.59E-03 2.29E-03 1.15E-03 5.74E-04 2.87E-04 1.43E-04 7.17E-05

2.55E-02 1.94E-02 1.34E-02 9.77E-03 6.91E-03 4.94E-03 3.51E-03 2.49E-03

3.17E-02 2.26E-02 1.50E-02 1.06E-02 7.32E-03 5.15E-03 3.61E-03 2.54E-03

1.50E-02 7.05E-03 3.61E-03 1.85E-03 9.06E-04 4.56E-04 2.28E-04 1.14E-04

2.18E-02 1.67E-02 1.22E-02 9.13E-03 6.72E-03 4.90E-03 3.53E-03 2.54E-03

4.40E-02 2.82E-02 1.80E-02 1.21E-02 8.23E-03 5.64E-03 3.91E-03 2.72E-03

2.12E-02 1.41E-02 9.36E-03 6.43E-03 4.46E-03 3.13E-03 2.20E-03 1.55E-03

1.89E-02 1.30E-02 8.72E-03 6.15E-03 4.34E-03 3.06E-03 2.17E-03 1.53E-03

Approximation RMS error for American puts 50 100 200 400 800 1600 3200 6400

4.46E-03 2.24E-03 1.11E-03 5.50E-04 2.75E-04 1.37E-04 6.81E-05 3.39E-05

1.70E-02 1.26E-02 8.51E-03 6.08E-03 4.25E-03 3.01E-03 2.12E-03 1.50E-03

1.67E-02 1.25E-02 8.47E-03 6.06E-03 4.24E-03 3.00E-03 2.12E-03 1.50E-03

8.29E-03 3.84E-03 1.98E-03 9.76E-04 4.86E-04 2.45E-04 1.22E-04 6.07E-05

Note: The options test set for both European and American options is from WAND (2002) and consists of combinations of strike prices of 35 and 45; maturities of 0.25 and 0.5, and volatilities of 0.2 and 0.4. RMS errors averaged over the eight different sets of parameters are reported. GCRR-XPC, GCRR-X on the 1/3-th nodes, GCRR-X on the 2/3-th nodes, GCRR-JR .p D 1=2/, GCRR-JR .p D 1=3/, and GCRR-JR .p D 2=3/ models are as described in the text Table 34.5 A comparison of the performance of various models over a range of option types and parameters (with extrapolation) GCRR-X GCRR-X GCRR-JR GCRR-JR GCRR-JR N GCRR-XPC on the 1/3-th nodes on the 2/3-th nodes .p D 1=2/ .p D 1=3/ .p D 2=3/ Approximation RMS error for European calls 50 100 200 400 800 1600 3200 6400

7.61E-04 2.95E-05 7.47E-06 1.88E-06 4.72E-07 1.18E-07 2.96E-08 7.39E-09

1.65E-02 9.91E-03 3.72E-03 2.16E-03 9.25E-04 5.08E-04 2.35E-04 1.24E-04

2.05E-02 7.73E-03 4.68E-03 1.85E-03 1.05E-03 4.67E-04 2.51E-04 1.18E-04

5.24E-03 1.54E-03 3.56E-04 1.80E-04 8.82E-05 2.15E-05 4.45E-06 2.83E-06

2.52E-02 1.23E-02 5.14E-03 3.12E-03 1.52E-03 7.66E-04 3.75E-04 1.90E-04

2.68E-02 1.20E-02 7.33E-03 2.78E-03 1.49E-03 7.63E-04 3.70E-04 1.84E-04

1.71E-02 7.64E-03 3.55E-03 1.85E-03 9.01E-04 4.23E-04 2.13E-04 1.02E-04

1.36E-02 6.59E-03 3.28E-03 1.62E-03 7.81E-04 3.90E-04 1.99E-04 9.96E-05

Approximation RMS error for American puts 50 100 200 400 800 1600 3200 6400

1.05E-03 1.03E-04 1.58E-04 2.09E-05 2.26E-05 5.94E-06 2.68E-06 7.21E-07

1.14E-02 5.43E-03 2.48E-03 1.09E-03 5.44E-04 2.48E-04 1.25E-04 5.93E-05

9.06E-03 4.40E-03 2.04E-03 9.43E-04 4.61E-04 2.27E-04 1.15E-04 5.71E-05

3.83E-03 1.09E-03 4.38E-04 1.28E-04 4.56E-05 1.48E-05 3.67E-06 2.29E-06

Note: The options test set for both European and American options is from WAND (2002) and consists of combinations of strike prices of 35 and 45; maturities of 0.25 and 0.5, and volatilities of 0.2 and 0.4. RMS errors averaged over the eight different sets of parameters are reported. GCRR-XPC, GCRR-X on the 1/3-th nodes, GCRR-X on the 2/3-th nodes, GCRR-JR .p D 1=2/, GCRR-JR .p D 1=3/, and GCRR-JR .p D 2=3/ models are as described in the text

34 A Further Analysis of the Convergence Rates and Patterns of the Binomial Models

References Broadie, M. and J. Detemple. 1996. “American option valuation: new bounds, approximations, and a comparison of existing methods.” Review of Financial Studies 9, 1211–1250. Chung, S. L. and P. T. Shih. 2007. “Generalized Cox-Ross-Rubinstein binomial models.” Management Science 53, 508–520. Cox, J. C., S. A. Ross, and M. Rubinstein. 1979. “Option pricing: a simplified approach.” Journal of Financial Economics 7, 229–263. Diener, F. and M. Diener. 2004. “Asymptotics of the price oscillations of a European call option in a tree model.” Mathematical Finance 14, 271–293. Figlewski, S. and B. Gao. 1999. “The adaptive mesh model: a new approach to efficient option pricing.” Journal of Financial Economics 53, 313–351. Heston, S. and G. Zhou. 2000. “On the rate of convergence of discretetime contingent claims.” Mathematical Finance 10, 53–75. Jarrow, R. and A. Rudd. 1983. Option pricing, Richard D. Irwin, Homewood, IL. Leisen, D. P. J. and M. Reimer. 1996. “Binomial models for option valuation – examining and improving convergence.” Applied Mathematical Finance 3, 319–346. Omberg, E. 1988. “Efficient discrete time jump process models in option pricing.” Journal of Financial Quantitative Analysis 23, 161–174. Rendleman, R. J. and B. J. Bartter. 1979. “Two-state option pricing.” Journal of Finance 34, 1093–1110. Ritchken, P. 1995. “On pricing barrier options.” The Journal of Derivatives 3, 19–28. Tian, Y. 1993 “A modified lattice approach to option pricing” Journal of Futures Markets 13, 563–577. Tian, Y. 1999. “A flexible binomial option pricing model.” Journal of Futures Markets 19, 817–843. Widdicks, M., A. D. Andricopoulos, D. P. Newton, and P. W. Duck. 2002. “On the enhanced convergence of standard lattice methods for option pricing.” Journal of Futures Markets 22, 315–338.

513

Appendix 34A Extrapolation Formulas for Various GCRR Models Since the convergence pattern of the GCRR model is monotonic and smooth, we apply the extrapolation technique to enhance its accuracy. We first establish a two-point extrapolation formula using the GCRR binomial prices with geometrically increasing time steps n and 2n. The Black-Scholes price can be exactly calculated as follows: CBS D

.n/C2n  Cn ; .n/  1

(34.7)

where C2n and Cn are the binomial option prices according to the number of time steps equal to 2n and n, respectively. Although the error ratio .n/ is generally unknown, we can still apply the following extrapolation formula to improve accuracy as long as the error ratio converges to a constant quickly enough: CO 2n D .C2n  Cn /=.  1/, where  is the limit of the error ratio series. It is straightforward to prove that the pricing error remaining in the extrapolated price follows:   .n/ e.2n/: CO 2n  CBS D 1

(34.8)

In other words, the pricing error reduces from e.2n/ to .n/ 1 e.2n/ after a two-point extrapolation technique is applied.

Chapter 35

Estimating Implied Probabilities from Option Prices and the Underlying Bruce Mizrach

Abstract This paper examines a variety of methods for extracting implied probability distributions from option prices and the underlying. The paper first explores nonparametric procedures for reconstructing densities directly from options market data. I then consider local volatility functions, both through implied volatility trees and volatility interpolation. I then turn to alternative specifications of the stochastic process for the underlying. I estimate a mixture of log normals model, apply it to exchange rate data, and illustrate how to conduct forecast comparisons. I finally turn to the estimation of jump risk by extracting bipower variation. Keywords Options r Implied probability densities r Volatility smile r Jump risk r Bipower variation

35.1 Introduction Traders, economists and central bankers all have interest in assessing market beliefs. The problem is that markets speak to us in the language of prices, and extracting their views can often be a difficult inference problem. This paper surveys and occasionally extends a variety of methods for obtaining not just market expectations, but the full range of predictive densities. I examine and implement procedures for both options and the underlying. A mastery of these approaches can help banks, trading firms and their regulators avoid excessive risks. The paper begins by explaining the theoretical connection between option prices and probability measures. The relationship linking the option price and the risk neutral density is established, under very general conditions, using the Feynman–Kac lemma. I then establish empirically that the log normality restrictions implied by Black–Scholes are too severe. Our evidence,

B. Mizrach () Department of Economics, Rutgers University, New Jersey Hall, New Brunswick, NJ, USA e-mail: [email protected]

drawn from an exchange rate example, illustrates the volatility smile and skew. The paper then moves beyond the log-normal and extracts densities using histogram estimators. These approaches perform fairly poorly, but I introduce a well-behaved method that can be solved in a spreadsheet. A closely related approach using Rubinstein’s (1994) implied binomial trees follows. We use the modifications of Barle and Cakici (1998) and Cakici and Foster (2002) in a numerical example. The most popular approach in the practitioner literature, local volatility modeling, is discussed next. We follow Shimko (1993) and Dumas, Fleming and Whaley (1998) and interpolate the volatility surface. These techniques are computationally straightforward, but are limited by potential arbitrage violations in the tails, precisely the region of greatest interest to policy authorities. I next turn to alternative stochastic processes for the underlying. I parameterize the spot price process as a mixture of log normals, as in Ritchey (1990) and Melick and Thomas (1997), and fit the model to options prices. This model has proven useful in analyzing exchange rate crises in Haas, Mittnik and Mizrach (2006) and, in Mizrach (2006), predicting Enron’s bankruptcy risk prior to its collapse. Econometric issues are considered in the stochastic process section. These include the choice of metric. The paper endorses the view that matching option volatility is the preferred distance measure. Hypothesis testing against simpler alternatives is not straightforward due to nuisance parameters, and a bootstrap approach is discussed. Comparing value-at-risk over a specific interval is done using the approach of Christoffersen (1998), and comparison of the entire predictive density is also done using Berkowitz (2001). I then examine jump processes and introduce the crucial recent results of Barndorff-Neilsen and Shephard (2006). These enable a straightforward extraction of the jump risk component in the underlying. I find that, in exchange rates, jump risk does not move in lockstep with stochastic volatility. I now begin in the obvious place, with the Black–Scholes model as a baseline.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_35, 

515

516

B. Mizrach

35.2 Black Scholes Baseline

This can be expressed as a partial differential equation (PDE),

I specify the price process and the hedge portfolio in the first part. The link between the risk neutral measure and the solution to the portfolio differential equation is in the second part.

@Y 1 @2 Y 2 2 @Y C rSt C  St D rYt : @t @S 2 @Y 2

This equation was initially proposed and solved by Black and Scholes (1973) in their seminal paper. A number of derivative securities can be priced using the solution to (35.9) along with the appropriate boundary conditions,

35.2.1 The B–S Differential Equation We assume that the spot price St follows a geometric Brownian motion, St D S0 exp.˛t C Wt /

(35.1)

where S0 is an initial condition, ˛ and  are constants and Wt is a standard Brownian motion. This process is log normal because log.St / D log.S0 / C ˛t C Wt is normally distributed. Ito’s lemma implies that dSt D St dt C St dW t ;

(35.2)

with drift  D .˛ C  2 =2/ and deterministic volatility . Suppose that Yt is the price of a derivative security contingent on S and t. From Ito’s lemma, we obtain, d Yt D

1 @2 Y 2 2 @Y @Y @Y St C C St dW t :  S dt C t @S @t 2 @S 2 @S (35.3)

Consider next a portfolio, …, short one derivative security and long an amount @Y =@S of the underlying, …t D Yt C

@Y St : @S

(35.4)

The change in value of this portfolio in a short interval is d…t D d Yt C

@Y dSt : @S

(35.5)

Substituting (35.2) and (35.3) into (35.5) yields,

YT D exp.r/ .ST /

.St / D exp.r/ maxŒST  K; 0 :

d…t D r…t dt

(35.7)

BSC .St ; K; T; r; / D St ˆ.d1 /  K exp.r/ˆ.d2 /; (35.12) with ln.St =K/ C .r C  2 =2/ p ;   p d2 D d1   ; d1 D

and ˆ./ denotes the distribution.





1 @2 Y 2 2 @Y @Y C S dt D r Y  S  t t dt: t @t 2 @S 2 @S

cumulative standard

normal

35.2.2 A Probability Density Approach The link between the typically numerical PDE approach and stochastic processes is provided by the Feynman–Kac analysis. Consider again a spot price process that solves (35.1), and a derivative security that follows the PDE, 1 @2 Y 2 2 @Y @Y St C C  St D 0; @S @t 2 @S 2

(35.13)

and satisfies the boundary condition (35.11). Applying Ito’s lemma to (35.13), we again obtain (35.3). The first term in (35.3) drops out from (35.13), and then integrating on both sides, Z

Z

T

T

d Yt D YT  Yt D t

Substituting from (35.4) and (35.6), this becomes

(35.11)

We then obtain the well-known Black–Scholes formula,

(35.6)

Since this equation does not involve d Wt , the portfolio is riskless during the interval dt. Under a no arbitrage assumption, this portfolio can only earn the riskless rate of interest r,

(35.10)

where T is the date of expiry, and  D T  t. For a European call with strike price K, the boundary condition would be



@Y 1 @2 Y 2 2 d…t D   S C dt: t @t 2 @S 2

(35.9)

t

@Y St dW t : @S

(35.14)

Taking expectations, the right hand side is zero, leaving (35.8)

Yt D EŒYT D exp.r/EQ Œ .ST /jSt D s

(35.15)

35 Estimating Implied Probabilities from Option Prices and the Underlying

Note that we have replaced the solution of the PDE with a conditional expectation taken under the risk neutral measure Q. Regularity conditions are established in Karatzas and Shreve (1991). Hambly (2006) shows how we can use the Feynman– Kac to construct probability densities for the spot price. Set .s/ D I .s/, the indicator function of a set . The solution to the PDE (35.13) with this boundary condition is EŒI .ST /jSt D s D PrŒST 2 jSt D s

(35.16)

The probability density f obeys the Kolmogorov (or Fokker–Planck) forward equation, 1 @2 f 2 2 @f @f St C C  St D 0 @S @t 2 @S 2

(35.17)

where we impose a terminal delta function. This equation describes the probability density of being at ST given that we are at St . In the case where the spot price process follows a Brownian process with drift r, dSt D rSt dt C St d Wt ;

(35.18)

we calculate that the transition density is the log-normal, f .ST / D

"

1

ST

p exp  2  2 

Œln. SSTt /  .r   2 =2/ 2 2 2 

# :

517

35.3 Empirical Departures from Black Scholes Data on option prices appear to be inconsistent with the Black–Scholes assumptions. In particular, volatility seems to vary across strike prices – often with a parabolic shape called the volatility “smile.” The smile is often present on only one part of the distribution giving rise to a “smirk.” Tompkins (2001) shows that these patterns are quite typical in a wide range of options markets, including stock indices, Treasury bonds and exchanges rates for Germany, the U.K. and the U.S. The motivating examples in this section are drawn from the foreign exchange market and draw upon the work of Haas, Mittnik and Mizrach (2006). I analyze the US Dollar/ British Pound (US$/BP) option which trades on the Philadelphia exchange. Both American and European options are traded. The BP options are for 31,250 Pounds. I use daily closing option prices that are quoted in cents. Spot exchange rates are expressed as US$ per unit of foreign currency and are recorded contemporaneously with the closing trade. Foreign currency appreciation (depreciation) will increase the moneyness of a call (put) option. Interest rates are the Eurodeposit rates closest in maturity to the term of the option. To obtain a rough idea about the implied volatility pattern in the currency options, I look at sample averages. I sort the data into bins based on the strike/spot ratio, S=K, and compute implied volatilities using the Black–Scholes formula. In Fig. 35.1, I plot the data for all of 1992 and 1993, for

Averages of Implied Volatility BP/US$ Options 1992-93 17

Implied Volatility (% Per Annum)

16

15

14

13

12

11

10 1.08

Strike Price/Spot Price

Fig. 35.1 Averages of implied volatility BP/US$ options 1992–1993. Notes: The figure reports implied volatility for BP/US$ options from the Philadelphia Exchange for 1992–1993

518

B. Mizrach

the British Pound. It displays the characteristic pattern, with the minima of the implied volatility at the money, and with higher implied volatilities in the two tails. Black–Scholes cannot account for this pattern in implied volatility. The next three sections discuss how to move beyond the restrictive Black–Scholes assumptions.

35.4 Beyond Black Scholes I extend the Black–Scholes model by allowing volatility to vary with the strike price. The formal link is established using the forward Kolmogorov equation.

35.4.1 How Volatility Varies with the Strike The Feynman–Kac analysis enables us to define a risk neutral probability in which we can price options. Let f .ST / denote the terminal risk neutral (Q-measure) probability at time T , and let F .ST / denote the cumulative probability. A European call option at time t, expiring at T , with strike price K, is priced Z

.ST  K/f .ST /dST ;

Dupire (1994) clarifies the isomorphism between the approaches that specify the density and those that specify price process. He shows that for driftless diffusions, there is a unique stochastic process corresponding to a given implied probability density. From the Kolmogorov equation (35.17) for this density, I rearrange 1 @2 C @C @C D  2 K 2 2  rK @T 2 @K @K

(35.22)

and obtain the local volatility formula, @C @C C rK @K :  2 .K; t/ D @T 1 2 @2 C K 2 @K 2

(35.23)

The principal problem in estimating f is that one does not observe a continuous function of option prices and strikes. My first attempt to estimate f utilizes a histogram method to try and trace out the density.

35.5 Histogram Estimators

1

C.K; / D exp.r/

35.4.2 The Kolmogorov Equation

(35.19)

K

In the case where f ./ is the log-normal density and volatility  is constant with respect to K, this yields the Black–Scholes formula, In this benchmark case, implied volatility is a sufficient statistic for the entire implied probability density which is centered at the risk-free rate. In the baseline case, practitioners often try to invert (35.2) to estimate “implied volatility.” The academic literature has focused on whether volatility in the physical measure is better predicted by implied volatility, GARCH or stochastic volatility. Under basic no-arbitrage restrictions, we can consider more general densities than the log-normal for the underlying. Breeden and Litzenberger (1978) show that the first derivative is a function of the cumulative distribution, @C =@K jKDST D e r .1  F .ST //:

(35.20)

The second derivative then extracts the density, @2 C =@K 2 jKDST D e r f .ST /:

(35.21)

This section develops two nonparametric estimators of the risk neutral density.

35.5.1 A Crude Histogram Estimator This section describes a method first proposed by Longstaff (1990) and then modified by Rubinstein (1994). Let S be the current price of the underlying asset. Let C1 to Cn be the prices of n associated call options with striking prices K1 < K2 <    < Kn all with the same time to expiration. Denote by ST the price of the asset at expiration, and let r be the riskless rate of interest, and ı be the payout rate on the underlying asset (e.g. a dividend for a stock or in the case of a foreign exchange option, the interest differential). As ST goes from Kn to Kn1 the payoff goes from 0 to Kn  Kn1 with an average payoff of 0:5.Kn  Kn1 /. It follows that Cn1 D 0:5pn .Kn  Kn1 /=.1 C r/ with pn the risk neutral probability associated with the histogram bin n. Consider now the price of the call with the next lowest striking price. There are three outcomes one must consider.

35 Estimating Implied Probabilities from Option Prices and the Underlying

519

Histogram Estimators Probability 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 Longstaff/Rubinstein

Normal Distribution

Optimized Histogram

−0.4 75

80

85

90

95

100

105

110

115

120

125

140

Strike

Fig. 35.2 Histogram estimators Notes: The figure reports various different histogram estimators

If ST < Kn2 then the payoff is zero. In the interval between Kn1 and Kn2 the average payoff is 0:5.Kn1  Kn2 /. Above Kn1 the payoff is pn Œ.Kn1  Kn2 / C 1=2.Kn  Kn1 / . The latter part is easily seen as just .1 C r/Cn1 . Adding up these components, gives us the recursion, h Xj Cnj D 0:5pnj C1 C

i

kD2

pnj Ck

.Knj C1  Knj /=.1 C r/ C Cnj : (35.24) I can view the underlying asset as a call with strike K0 D 0. As before, I have three outcomes to consider. If ST < 0 then the payoff is zero. For 0  ST < K1 , then the payoff is 1=2.K1  0/p1 , and for ST > K1 , ı .K1  0/Œp2 C    C pn .1 C r/ C C1 :

(35.25)

Adding up these payoff components gives us, h i Xn pk .1 C r/ C C1 : S=.1 C ı/ D K1 0:5p1 C

Pj 1 More generally, I can write pj D 2Œ1  kD1 pk .1 C r/ .Cj 1  Cj /.Kj  Kj 1 /1 . I implement this histogram method in Fig. 35.2. The baseline is a normal distribution for which I have priced 11 Black–Scholes calls for strikes from 75 to 125, with a spot price of 100. For this example, the Longstaff–Rubinstein estimator does not provide even a good rough approximation. The probabilities whipsaw back and forth; every other bin turns negative. After some trial and error, it was apparent this property was not uncommon. I turn next to an improved estimator.

35.5.2 An Improved Histogram Estimator Let bi > 0; i D 0; : : : ; n C 1 be the cardinality of a set uniformly distributed with mean Ki C1 Ki for i D 1 to n1, and KN i at the end points. To transform bi into a probability measure, define

kD2

pj D b j

(35.26) Exploiting the fact that the probabilities sum to one, ı S=.1Cı/ D K1 Œ0:5p1 C.1p/1 .1Cr/CC1 ;

(35.27)

p1 D 2Œ1  .1 C r/.S=.1 C ı/  C1 /K11 :

b0 ;:::;bnC1

(35.28)

j D1

bj

(35.29)

The improved estimator minimizes the percentage deviation from the simulated and observed call prices, min

I can rearrange (35.27) and solve for P1 ,

ı XnC1

Xn j D1

.CO j  Cj /=Cj ;

(35.30)

where the c C 0 s can be determined using the recursion (35.24).

520

B. Mizrach

I used this optimized histogram estimator to fit the same 11 Black–Scholes calls from the prior example. For a small number of calls, this algorithm can be solved in a spreadsheet. I plot the risk neutral probabilities in Fig. 35.2. The optimized histogram is a substantial improvement over the crude estimator. There are no negative probabilities (by construction) and with the exception of the long bin from 0 to 75, it provides a good approximation to the underlying normal. This method is quite reliable for back of the envelope calculations, but for more complicated processes, one needs more powerful techniques.

and Cakici and Foster (2002). The first key modification is to center the tree at the forward price Fi;j , Fi;j D pi;j Si;j C1 C .1  pi;j /Si;j :

(35.34)

The other change is to value options using Black–Scholes with interpolated volatilities rather than the Arrow–Debreu prices i;j , BSCi;j .S0;0 ; K; T; / Xn i;j exp.r t/ max.Si;j  K; 0/; (35.35) D j D1

which is a call with strike price K, expiration t, and volatility . To grow the tree from the initial spot price (or any odd number step in the tree), we have two even nodes. The upper central node is

35.6 Tree Methods 35.6.1 A Standard Tree I begin the discussion by defining a notation for a standard binomial tree. Let S0;0 denote the current spot price. Any element of the spot price tree is a sequence of recombining up and down moves u; d :

Si C1;j C1 D Fi;j

i;j Fi;j C Ci;j

(35.36)

i;j Fi;j  Ci;j

where Si;j D S0;0 uj d i j ; i D 0; : : : ; N; j D 0; : : : i; (35.31) where p u D exp T =N ; d D 1=u; N is the number of steps in the tree, T is the time to expiration, and  is the annualized volatility. The risk neutral probability of an up move is pD

exprT =N d ud

(35.32)

where r is the risk free rate of return. The main advantage of the standard approach is simplicity. One of the important limitations for our purposes is the assumption that the probabilities are constant throughout the tree. This implies that the nodal probability of reaching any particular terminal point SN;j is just the binomial probability

Ci;j D exp.r t/Ci;j .Fi;j ; t/ 

Xi kDj C1

i;k .Fi;k  Fi;j /; (35.37)

The lower central node must then satisfy Si C1;j Si C1;j C1 D 2 . Fi;j We then move up from the central node using the recursions, Si;j C1 D

Ci1;j Si;j  i;j Fi 1;j .Fi 1;j  Si;j /

Ci1;j  i;j .Fi 1;j  Si;j /

:

(35.38) Moves down from the central node are given by Si;j D

i 1;j Fi 1;j .Si;j C1  Fi 1;j /  Pi1;j Si;j C1 i 1;j .Si;j C1  Fi 1;j /  Pi1;j (35.39)

where pN;j D PrŒST D SN;j D ˆ.j; N; p/; j Š.N Š  j /Š

p j .1  p/N j : D NŠ

(35.33)

Pi;j D exp.r t/Pi;j .Fi;j ; t/ 

Xj 1 kD1

i;k .Fi;k  Fi;j / (35.40)

The Arrow–Debreu probabilities are then updated,

35.6.2 Implied Binomial Trees The implied binomial tree approach of Rubinstein (1994) and Jackwerth and Rubinstein (1996) relaxes the assumption of constant probabilities as in (35.32). Our discussion here follows the developments in Barle and Cakici (1998)

i C1;j exp.r t/ 9 8 for j D n C 1 = pi;j i;j < D pi;j 1 i;j 1 C .1  pi;j /i;j for 2  j  n : ; : for j D 1 .1  pi;1 /i;1 (35.41)

35 Estimating Implied Probabilities from Option Prices and the Underlying

35.6.3 Numerical Example We illustrate the implied binomial trees using a numerical example. Consider an 2-step tree, 6 months from expiration, which implies t D 0:25. The initial price S0;0 D 100. Volatility is a smirk, rising when the strike price is above 100, K D max.20; 20 Fi;j =S0;0 /. The continuously compounded interest rate is r D ln.4:8790/ D 5%. Starting from S0;0 , we compute the forward price, F0;0 D 100 exp.0:04879 0:25/ D 101:2272. The Arrow–Debreu price at the origin is 0;0 D 1:0. The Black–Scholes call struck at the futures price, expiring next period and volatility  D 20 101:2272=100 D 20:2454 is BSCi;j .S D S0;0 ; K D F0;0 ; T D 0:25;  D 20:2454/ D 4:0367; which implies

Ci;j D exp.0:04879 0:25/4:0367 D 4:0862:

Having found the new spot prices, we can update the transition probabilities, p0;0 D

.F0;0  S1;0 / D 0:4798; .S1;1  S1;0 /

and the Arrow–Debreu prices, 1;1 D 0;0 p0;0 exp.r t/ D 0:4740; 1;0 D 0;0 .1  p0;0 / exp.r t/ D 0:5139: The next central node is at the 2-step ahead forward price, S2;1 D 100 exp.0:04879 0:25 2/ D 102:4695: We work up from there using (35.38). This requires the forward prices at the prior node, F1;1 D exp.r t/S1;1 D 111:0902; and the call prices. We compute the new local volatility  D 22:2180, and price the call

We then use (35.36) to find S1;1 D 101:2272

521

.1:0/.101:2272/ C 4:0862 D 109:7434: .1:0/.101:2272/  4:0862

We find the lower central node by equating spot and futures prices, 2 S1;0 D F0;0 =S1;1 D 93:3719:

S2;2 D

BSCi;j .S0;0 ; K D F1;1 ; T D 0:5;  D 22:2180/ D 3:1597: Since this call is at expiry, we don’t need to discount,

Ci;j D BSCi;j . Substituting into (35.38),

.3:1597/.102:4695/  .0:4740/.111:0902/.111:0902  102:4695/ 3:1597  .0:4740/.111:0902  102:4695/

D 140:4882:

Moving down from the central node requires (35.39). We use the futures price F1;0 D exp.r t/S1;0 D 94:5178;

and the volatility  D 20, to price the put, BSPi;j .S0;0 ; K D F1;0 ; T D 0:5;  D 20/ D 2:3970: Substituting into (35.39),

S2;0 D

.0:5139/.94:5178/.102:4695  94:5178/  .2:3970/.102:4695/ .0:5139/.102:4695  94:5178/  2:3970

D 83:2339:

522

B. Mizrach

This completes this simple example. A simpler approach which interpolates the implied volatility surface is my next topic.

35.8 PDF Approaches

35.7 Local Volatility Functions The most popular practitioners method is the use of local volatility functions. The seminal references are Shimko (1993) and Dumas, Fleming and Whaley (1998). Shimko (1993) proposed to fit a polynomial to the smile so as to infer how volatility varied with the striking price. He chose to use a quadratic, O .K; t/ D a0 C a1 K C a2 K 2 :

(35.42)

In the case where volatility varies in this fashion, the density f .ST / can be written, N 0 .d2 / Œd2K  .a1 C 2a2 K/.1  d2 d2K /  2a2 K : (35.43) Shimko’s method has several advantages. It can be used to generate call prices for a continuum of strikes. In this way, I can fill in the histogram proposed by Longstaff (1990) and Rubinstein (1994). The downside is that it can easily generate either negative probabilities or pricing inconsistencies. I can illustrate this in an empirical example now. The difficulties with Shimko’s method arise when @=@K is too steep. At some point, the volatility may become so high that is implies a more deeply out of the money option has a higher price than those close to being in the money. I utilize data from the Philadelphia Exchange for June 25, 1992 DM/$ puts and calls. I fit a quadratic to the smile, 0:4035 .%M /2 C 0:288 .%M / C 10:362 in Fig. 35.3(a). At 4% out of the money, the implied volatility rises to 15.67 and at 6%, 23.16. Once I translate these volatilities into prices in Fig. 35.3(b), one can see that an arbitrage exists. The 6% call is more valuable than the 4%, 0.16 versus 0.10. A further difficulty is that even when this arbitrage is not a problem, there is no guarantee that the bracketed term in (35.58) will remain positive for all choices of a. I also conjecture that one set of restrictions does not necessarily imply the other, making this a fairly complicated constrained optimization. Brunner and Hafner (2003) is a promising step in this direction. Bliss and Panigirtzoglou (2002) endorse the use of local volatility functions1 over the mixture-of-normals approach in

1

the next section, but this conclusion does not seem to hold when we look just at the tails of the distribution.

They use natural splines in the implied volatility-delta space. When fitting the mixture of normals, they use the call price-strike metric.

35.8.1 Mixture-of-Log-Normals Specification I assume that the stock price process is a draw from a mixture of three (nonstandard) normal distributions, ˆ.j ; j /, j D 1; 2; 3; with 3 2 1 . Three additional parameters 1 ; 2 and 3 define the probabilities of drawing from each normal. To nest the Black–Scholes, we restrict the mean to equal the interest differential, 2 D id  if . Risk neutral pricing then implies restrictions on either the other means or the probabilities. I chose to let 1 , 1 and 3 vary, which implies 1 1 ; (35.44) 3 D 3 and 2 D 1  1  3 :

(35.45)

For estimation purposes, this leaves six free parameters  D .1 ; 2 ; : : : ; 6 /. To keep the estimates positive, I embed the parameters into the exponential function. The left-hand tail distribution is given by ˆ.1 ; 1 / D ˆ.id  if  e 1 ; 100 e 2 /:

(35.46)

The only free parameter of the middle normal density is the standard deviation, ˆ.2 ; 2 / D ˆ.id  if ; 100 e 3 /:

(35.47)

I use the logistic function for the probabilities to bound them on [0, 1], 1 D

e 4 ; 1 C e 4

(35.48)

3 D

e 5 : 1 C e 5

(35.49)

The probability specification implies the following mean restrictions on the third normal,

e 4 =.1 C e 4 / 6 ˆ.3 ; 3 / D ˆ .id  if C e 1 /  : ; 100

e e 5 =.1 C e 5 / (35.50)

Mizrach (2006) shows that this data generating mechanism can match a wide range of shapes for the volatility smile.

35 Estimating Implied Probabilities from Option Prices and the Underlying

a

523

DM/$ Options June 25, 1992

Implied Volatility 15.0 14.5 14.0

Option Volatilities Estimated Volatility

13.5 13.0 12.5 12.0 11.5 11.0 10.5 10.0 4.0

3.0

2.0

1.0

−1.0

0.0

−2.0

-3.0

% In (+) or Out(-) of the Money

b DM/$ Options June 25, 1992 Prices from Shimko's Method

Option Prices 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 2.0

1.0

0.0

−1.0

−2.0

−3.0

−4.0

−5.0

−6.0

−7.0

−8.0

% In (+) or Out (-) of the Money Fig. 35.3 Local volatility functions and arbitrage violations Notes: The dark line in panel (a) is a quadratic function 0:4035.%M /2 C0:288 .%M / C 10:362 fit to the implied volatility of the DM/US$ options.

As we extrapolate the local volatility surface further out of the money, the model produces an arbitrage violation in panel (b) for the calls more than 5% out of the money

35.8.2 Data and Estimation Results

American options which can be exercised before expiration. The second is choosing the loss function for estimation. I approximate American puts and calls using the Bjerksund and Stensland (1993) approach. Hoffman (2000) shows that the Bjerksund–Stensland algorithm compares favorably in accuracy and computational efficiency to the Barone-Adesi and Whaley (1987) quadratic approximation. Our estimates were also quite similar using implied binomial trees.2

I focus on a subset of the results from Haas, Mittnik and Mizrach (2006). I utilize the British Pound/US$ option data described in Sect. 35.3. For estimation purposes, I excluded options that were less than 5 or more than 75 days to maturity, more than 10% in or out of the money, and with volumes less than 5 contracts. This seemed to eliminate most data points with unreasonably high implied volatilities. There are two key issues in fitting a stochastic process to the options data. The first is to extend the analysis to

2 We can also price exotics in this framework, pricing options using Monte Carlo or other numerical procedures. Conditions under

524

B. Mizrach

35.9 Inferences from the Mixture Model

Let fdj;t gnj D1 D Œc.1 ; K1 /; : : : ; c.m ; Km /; p.mC1 ; KmC1 /; : : : ; p.n ; Kn / denote a sample of size n of the calls c and puts p traded at time t, with strike price Kj and expiring in j years, and denote the pricing estimates from the model by fdj;t ./g. In estimation, Christoffersen and Jacobs (2004) emphasize that the choice of loss function is important. Bakshi, Cao and Chen (1997), for example, match the model to data using option prices. This can lead to substantial errors among the low priced options though. Since these options are associated with tail probability events, this is not the best metric for our exercise. We obtained the best fit overall using the implied Bjerksund–Stensland implied volatility, j;t D BJS T 1 .dj;t ; St ; it /:

j;t ./ D BJS T 1 .dj;t ./; St ; it /:

(35.52)

We then minimize the sum of squared deviations from the implied volatility in the data,



Xn j D1

35.9.1 Tests of the Adequacy of Black–Scholes Hansen (1996) considers the case of additive nonlinear models like our mixture model, j;t D .1  1  3 /ˆ.id  if ; 2 .3 // C1 ˆ.1 .1 /; 1 .2 // C 3 ˆ.3 ; 3 1 .6 //

(35.51)

Let the estimated volatility be denoted by

min

There is often a belief that the standard model is sufficient in all but the most extreme circumstances. It turns out that a formal test of this kind in a mixture model is not at all straightforward. The forecast comparisons discussed in the second and third parts may make more sense for practitioners.

.j;t ./  j;t /2 :

(35.53)

As Christoffersen and Jacobs note, this is just a weighted least squares problem that, with the monotonicity of the option price in , satisfies the usual regularity conditions. I report estimates for 2 sample days around the ERM crisis in Table 35.1. Haas, Mittnik and Mizrach (2006) establish that the model fits well both before and after the crisis. Inferences drawn from the BP/US$ options are analyzed in the next section.

Under Black–Scholes, 1 D 3 D 0, and it is tempting to compare the models using the likelihood ratio. Unfortunately, under the Black–Scholes alternative, the parameters in the two tail log normals, 1 , 2 and 6 , are nuisance parameters. Formally, the derivative of the likelihood function is flat with respect to these parameters. Their nonidentification invalidates the distribution theory for the standard LR test. Hansen (1996) reports severe size distortions in several cases for the standard test. Fortunately, Hansen also offers a constructive alternative. In our present setting, we can take minimizing the squared residuals, "2t D .j;t ./  j;t /2 as the objective function. The restricted alternative is given by "2R;t D .j;t .3 /  j;t /2 Construct the F -test,

which these simulated moments models are appropriate is considered in Mizrach (2006).

Table 35.1 Estimates of the mixture of log normals model Date 1 2 3 4

Fn D n

Xn j D1

."2R;t  "2t /=

Xn

"2 : j D1 R;t

5

6

%VaR0:05

%VaR0:01

20-Aug-1992

2:179 (0.01)

1:908 (0.15)

0:816 (0.01)

1:855 (0.01)

1:908 (0.07)

2:440 (0.63)

4:84%

7:14%

17-Sep-1992

0:882 (0.07)

0:943 (0.48)

3:634 (0.48)

3:108 (0.02)

2:051 (0.09)

1:933 (2.12)

6:11%

8:73%

(35.54)

LR0:05

LR0:01

11.87 (0.00)

0.09 (0.76)

Notes: The  0 s are estimates of the model (35.14). t -ratios are in parentheses. %VaR0:05 and %VaR0:01 are the value at risk, in percent of long position in the BP over a 4-week time horizon at the 95 and 99% confidence level. The LR statistic is given by (35.55) and compares the VaR across the two dates. The simulation size is N D 250 observations. The LR test, with p-values underneath, is distributed 2 .1/.

35 Estimating Implied Probabilities from Option Prices and the Underlying

Hansen then shows that Fn D sup Fn ./:  2‚

whose distribution is not asymptotically F and must be obtained by bootstrap. Since coefficients are likely to change daily, this is potentially tedious procedure.

525

under the August loss coverage rejects too often, 26 times, rather than the expected 0:05 250 D 12:5 times under the null. The likelihood ratio is 11.87 which rejects the null at better than 1% significance, At the 99% level, I find n0 D 3, and the test, with LR0:01 D 0:094, is not nearly powerful enough to reject that the forecasts are different.

35.9.3 Comparing the Entire Density 35.9.2 Hypothesis Tests on the Forecast Intervals To compare the value-at-risk on the 2 days at both confidence levels. I adapt the framework of Christoffersen (1998). In each of the three cases, I construct the test of the null hypothesis from a sequence of Bernoulli trials,  It˛ D

1; PrŒ.ST  St /=St < %VaR˛ > ˛ 0 PrŒ.ST  St /=St < %VaR˛ < ˛

Next, we evaluate the forecast densities produced across their entire support. The approach we take is the one originally proposed by Berkowitz (2001). He notes that the probability integral transform b .st / D F



st

f .u/d u: 1

; (35.55)

where St is the current spot price, T  t is 4 weeks, and ˛ is the critical level of the VaR. I take the VaR loss intervals from August 20 as the null, and compute %VaR0:05 D 4:84% and %VaR0:01 D 7:14%. Since the VaR is assessing tail risk, we are concerned with the coverage of the forecast interval from .1; %VaR˛ /. Under what Christoffersen calls unconditional coverage, I test

generates uniform, independent and identically distributed estimates under fairly weak assumptions. Testing for an independent uniform density in small samples can be problematic, so Berkowitz suggests transforming the data into normal random variates, b .st //; zt D ˆ1 .F A simple test of the null hypothesis that the transformed forecast statistics, zt , have mean zero can be performed using the likelihood ratio,

H0 W EŒIt˛ D ˛; LR D

using the likelihood ratio ˛ n0 .1  b ˛ /n1 / ; LR˛ D 2 lnŒ˛ n0 .1  ˛/n1 =.b

Z

(35.56)

where b ˛ D n0 =.n1 C n0 /; is the maximum likelihood estimator of ˛. Under H0 , LR˛ is distributed 2 .1/. As one might expect, the VaR rose substantially with the higher volatility during the crisis. Values at risk rises to %VaR0:05 D 6:11% and %VaR0:01 D 8:73%. To assess the statistical significance of this, I implement the test in (35.56) in Table 35.1. I simulate N D 250 forecasts from the September 17 parameterization. Let n0 be the number of times that .ST  St /=St is less than %VaR˛ . I find n0

XT t D1



z2t  1 ; O 2

(35.57)

where O is the forecast standard deviation. LR is then approximately distributed 2 .1/. I graph the forecast density for August 20, 1992 in Fig. 35.4. The realized 4-week returns are then plotted in comparison to the density. To the naked eye, the left-tail, associated with BP depreciation, is substantially longer. This was indeed detected in our VAR exercise in the prior section. When looking across the entire density, the power of the test is substantially weaker. The average forecast is 0.9756 or approximately a 2:5% decline. The average z-value is 0:4261, and O 2 D 2:4195. I compute LR D 1:5008 which has a p-value of only 0:2206. Clearly, when the risks are onesided, you want to exploit this information.

526

B. Mizrach

Forecast Density Comparison 9.00 8.00

20-Aug-92 Forecast Density Realized Data

7.00 6.00 Frequency (%)

Fig. 35.4 Forecast density comparison Notes: The light bars are the forecast density for the 4-week return on the spot exchange rate using the mixture model parameters from August 20, 1992 in Table 35.1. The dark bars are the subsequent 20 realizations. A formal statistical comparison is conducted using (35.57)

5.00 4.00 3.00 2.00 1.00 0.00 0.88

0.91

0.93

0.96

0.98

1.01

1.03

1.06

1.08

1.11

1.13

S(t+20)/S(t)

where 0 D .1 C k/ and fn is the Black–Scholes option price when the variance is

35.10 Jump Processes I now move back to the case of a single process for the underlying. The key step is to introduce discontinuities in the price process through jumps. I begin with the Merton (1976) model as a baseline, and then allow for stochastic volatility in the second part.

 2 C nı 2 =T; and the risk free rate is r  k C n ln.1 C k/=T: Carr and Wu (2004) consider the generalization of the jump diffusion model to Levy processes. Bates (1991) has estimated this model to infer the risk in options prior to the 1987 stock market crash.

35.10.1 Merton Model Merton (1976) has proposed a jump diffusion model dSt D .  k/St dt C St dW t C dqt ;

(35.58)

35.10.2 Bipower Variation where dW t is a Wiener process, dqt is the Poisson process generating the jumps, and  is the volatility. dW and dq are considered independent. This assumption is important because we cannot apply risk-neutral valuation to situations where the jump size is systematic.  is the rate at which jumps happen,  is the expected return, and k is the average jump size. This model gives rise to fatter left and right tails than Black–Scholes. If we assume that the log of k is normal with standard deviation ı, the European call option price is X1 e 0 T .0 /n fn ; C D nD0 nŠ

I follow Andersen, Bollerslev and Diebold (2007) and consider a stochastic volatility model with jumps, dpt D t dt C t d Wt C t dqt

where pt D ln.St /, qt is a counting process with intensity t , and t is the jump size with mean  and standard deviation  . The quadratic variation for the cumulative return process, rt D pt  p0 is then Z

(35.59)

(35.60)

t

Œr; r t D 0

s2 ds C

X 0<st

 2 .s/:

(35.61)

35 Estimating Implied Probabilities from Option Prices and the Underlying

Estimation of the quadratic variation proceeds with discrete sampling from the log price process. Denote the

-period returns by rt C1; D pt C1  pt C1 . The realized volatility is RVt C1; D

X1=

r2 : j D1 t Cj  ;

X

!0

t <st C1

 2 .s/:

(35.66)

I next turn to an application of this analysis in the ERM exchange rate context

35.10.3 An Application I will now take the sampling interval to be daily changes, and compute 50-period rolling sample estimates of realized volatility, RVt

1=2

D

X 50

To extract the integrated volatility from (35.60), BarndorffNielsen and Shephard have also introduced the realized bipower variation, X1= ˇ ˇˇ ˇ ˇrt Cj  ; ˇ ˇrt C.j 1/ ; ˇ

1=2 rt2j =50

(35.67)

X50 ˇ ˇˇ ˇı ˇrt j ˇ ˇrt j 1/ ˇ 50:

(35.68)

j D1

t <st C1

t

(35.63)

BVt C1; D 2 1

p lim.RVt C1;  BVt C1; / D

(35.62)

In the now vast literature on the stochastic volatility model,  D 0, researchers have employed realized R t 2 volatility as an estimator of the integrated volatility, 0 s ds. In the case of discontinuous price paths, BarndorffNeilsen and Shephard (2006) show that the realized volatility will also include the jump component, Z t C1 X s2 ds C  2 .s/: p limRVt C1; D

!0

527

and bipower variation, BVt D . =2/

j D1

We constrain the jump risk to be positive,

j D2

p where 1 D 2= . It is then possible to show Z

t C1

p limBVt C1; D

!0

(35.64)

t

s2 ds:

(35.65)

By comparing (35.63) and (35.65), we have the estimate of just the jump portion of the process,

t D .maxŒRVt  BVt ; 0 /1=2

(35.69)

I again use the British Pound/US$ exchange rate sample for January to October 1992 analyzed in Mizrach (1995), and plot the realized standard deviation and jump risk in Fig. 35.5. The figure has several features worth noting. The first is that while realized volatility for the British Pound was quite

Jump Risk Estimates 1.20

Daily % Return

1.00

Realized Volatility Jump Risk

0.80

0.60

0.40

Fig. 35.5 Jump risk estimates Notes: The figure plots realized daily standard deviation (35.67) and jump risk (35.32) for the British Pound spot rate between March 18 and September 30, 1992

0.20

0.00 12-Mar-1992

1-May-1992

20-Jun-1992

Date

9-Aug-1992

28-Sep-1992

528

high leading into the Spring of 1992, jump risk is zero until April 24. Jump risk rises rather steadily through the rest of the spring, peaking at 0:2836% on June 1. Jump risk then falls through the summer, returning to zero from July 24 to August 24. On August 25, the jump risk spikes to 0:2188% and then falls back until September 14th, when it spikes again to 0:4155%. Following a third local maximum of 0:3850% on Britain’s September 17th ERM withdrawal, jump risk falls back to zero by September 22. This is true even though the realized volatility remains as high as it was during the crisis. Although not shown in the figure, both the volatility and jump risks remain high into 1993. At the end of 1992, more than 3 months after the crisis, the realized volatility is at 0:9007%. The jump risk spikes twice above 0:30%, October 19 and December 2.

35.11 Conclusion It is important to stress what this paper has not discussed: the preservation of moments into the change to the risk neutral measure. While straightforward in the Black–Scholes case, the results do not hold in general. This paper has only shown, empirically, that there is useful information in the risk neutral measure. The stochastic volatility model was not addressed in great detail; my emphasis on tail behavior gave priority to jumps. I also have not discussed approaches like Bates (2006) that rely on the characteristic function. The paper has covered ground in three important areas: (1) nonparametric approaches estimating implied densities directly from option prices; (2) parametric modeling of the local volatility surface; (3) generalizations of the PDEs for the underlying process. I also evaluate and implement tools for hypothesis testing and evaluating forecasts from alternative models. The nuisance parameter issue is addressed with a bootstrap. Value-at-risk and forecast comparison relies on the formal statistical procedures introduced by Christoffersen (1998) and Berkowitz (2001). I last considered the very recent work of BarndorffNeilsen and Shephard on bipower variation. This enables researchers to easily extract the jump component from the underlying. These discontinuities are unhedgeable risks that may be systemic. Policy makers may find these tools and inference worthwhile in a variety of contexts. Their subjective weights between type I and type II errors should not only be tested ex-post but incorporated directly in the estimation. Both Skouras (2007) and Christoffersen and Jacobs (2004) have made progress along these lines.

B. Mizrach Acknowledgements I would like to thank seminar participants at the OCC, Federal Reserve Bank of Philadelphia, and the FDIC Derivative Securities and Risk Management Conference for helpful comments.

References Andersen, T. G., T. Bollerslev, and F. X. Diebold. 2007. “Roughing it up: including jump components in the measurement, modeling and forecasting of return volatility.” Review of Economics and Statistics 89, 701–720. Bakshi, C., C. Cao, and Z. Chen. 1997. “Empirical performance of alternative option pricing models.” Journal of Finance 52, 2003–2049. Barle, S. and N. Cakici. 1998. “How to grow a smiling tree.” Journal of Financial Engineering 7, 127–146. Barndorff-Nielsen, O. and N. Shephard. 2006. “Econometrics of testing for jumps in financial economics using bipower variation.” Journal of Financial Econometrics 4, 1–30. Barone-Adesi, G. and R. E. Whaley. 1987. “Efficient analytic approximation of American option values.” Journal of Finance 42, 301–320. Bates, D. 1991. “The crash of ’87: was it expected? the evidence from options markets.” Journal of Finance 46, 1009–1044. Bates, D. 2006. “Maximum likelihood estimation of latent affine processes.” Review of Financial Studies, 19, 909–965. Berkowitz, J. 2001. “Testing density forecasts with applications to risk management.” Journal of Business and Economic Statistics 19, 466–474. Bliss, R. and N. Panigirtzoglou. 2002. “Testing the stability of implied probability density functions.” Journal of Banking and Finance 26, 381–422. Bjerksund, P. and G. Stensland. 1993. “Closed-form approximation of American options.” Scandinavian Journal of Management 9, 87–99. Breeden, D. and R. Litzenberger. 1978. “State contingent prices implicit in option prices.” Journal of Business 51, 621–651. Brunner, B. and R. Hafner. 2003. “Arbitrage-free estimation of the riskneutral density from the implied volatility smile.” Journal of Computational Finance 7, 76–106. Cakici, N. and K. R. Foster. 2002. “Risk-neutralized at-the-money consistent historical distributions in currency options pricing.” Journal of Computational Finance 6, 25–47. Carr, P. and L. Wu. 2004. “Time-changed Levy processes and option pricing.” Journal of Financial Economics 71, 113–141. Christoffersen, P. 1998. “Evaluating interval forecasts.” International Economic Review 39, 841–862. Christoffersen, P. and K. Jacobs. 2004. “The importance of the loss function in option valuation.” Journal of Financial Economics 72, 291–318. Dumas, B., J. Fleming, and R. Whaley. 1998. “Implied volatility functions: empirical tests.” Journal of Finance 53, 2059–2106. Dupire, B. 1994. “Pricing with a smile.” Risk 7, 18–20. Haas, M., S. Mittnik, and B. Mizrach. 2006. “Assessing central bank credibility during the ems crises: comparing option and spot marketbased forecasts.” Journal of Financial Stability 2, 28–54. Hambly, B. 2006. Lecture notes on financial derivatives, Oxford University, New York. Hoffman, C. 2000. Valuation of American options, Oxford University Thesis. Karatzas, I. and S. Shreve. 1991. Brownian motion and stochastic calculus, 2nd ed. Springer, New York. Melick, W. and C. Thomas. 1997. “Recovering an asset’s implied PDF from option prices: an application to crude oil during the Gulf crisis.” Journal of Financial and Quantitative Analysis 32, 91–115.

35 Estimating Implied Probabilities from Option Prices and the Underlying Merton, R. 1976. “Option pricing when underlying stock returns are discontinuous.” Journal of Financial Economics, 3, 124–144. Mizrach, B. 1995. “Target zone models with stochastic realignments: an econometric evaluation.” Journal of International Money and Finance 14, 641–657. Mizrach, B. 2006. “The Enron bankruptcy: when did the options market lose its smirk.” Review of Quantitative Finance and Accounting 27, 365–382. Ritchey, R. 1990. “Call option valuation for discrete normal mixtures.” Journal of Financial Research 13, 285–295. Shimko, D. 1993. “Bounds of probability.” Risk 6, 33–37.

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Skouras, S. 2007. “Decisionmetrics: a decision-based approach to econometric modelling.” Journal of Econometrics 127, 414–440. Stein, E. M. and J. C. Stein. 1991. “Stock price distributions with stochastic volatility: an analytic approach.” Review of Financial Studies 4, 727–752. Tompkins, R. 2001. “Implied volatility surfaces: uncovering regularities for options on financial futures.” The European Journal of Finance 7, 198–230.

Chapter 36

Are Tails Fat Enough to Explain Smile Ren-Raw Chen, Oded Palmon, and John Wald

Abstract It has been well documented that using the Black-Scholes model to price options with different strikes generates the so-called volatility smile. Many previous papers have attributed the smile to the normality assumption in the Black-Scholes model. Hence, they generalize the Black-Scholes model to incorporate a richer distribution. In contrast to previous studies, our model allows for not only a richer distribution, but also the relaxation of another crucial assumption in the Black-Scholes – continuous trading. We show, using S&P 500 call options, how relaxation of continuous trading explains a non-trivial change in the volatility. When an empirical distribution is considered, the smile is almost completely removed. Market prices of options that differ only in their strike prices are inconsistent with a single volatility for the underlying asset, and this well-known feature is called the volatility smile or smirk. The volatility smile is often attributed to either the non-normality of stock returns or to the impossibility of performing costless arbitrage (rebalancing) in continuous time. In this paper, we consider option pricing models that incorporate no rebalancing and/or a nonparametrically estimated density. We attempt to empirically identify which of these two factors contributes more to the smile bias. Using S&P 500 index options, we find that the model with no rebalancing but with normality has a somewhat diminished smile, and the model with a nonparametric density, but with continuous trading, has a slight smile. Only the model with both a nonparametric density and no rebalancing has no perceptible smile bias. R.-R. Chen () Graduate School of Business Administration, Fordham University, New York, NY 10019, USA e-mail: [email protected] O. Palmon Rutgers Business School, Rutgers University, Piscataway, NJ 08854, USA e-mail: [email protected] J. Wald Department of Finance, College of Business, University of Texas at San Antonio, San Antonio, TX 78249, USA e-mail: [email protected]

Keywords Option pricing r Volatility smile r Nonparametric density

36.1 Introduction The Black-Scholes model is known to bias option prices for in-the-money and out-of-the-money options. Two unrealistic assumptions of the Black-Scholes model that may be responsible for this bias are continuous rebalancing and normality. The literature has dealt with these two issues separately. Some studies (e.g., Leland 1985; Constantinides 1998) have introduced transaction costs to the Black-Scholes model to relax the continuous rebalancing assumption, while other studies allow for stochastic volatility (e.g., Hull and White 1987; Heston 1993) and jump diffusions (e.g., Merton 1976; Cox and Ross 1976) to enrich the distributional assumption of the Black-Scholes model. In this study we consider both unrealistic assumptions. We find that eliminating the continuous rebalancing assumption eliminates some of the volatility smile, accounting for the actual underlying largely density eliminates most of it, and replacing both assumptions simultaneously eliminates the volatility smile. To examine the continuous trading assumption, we present a model in which returns are distributed normally, investors do not rebalance their portfolios,1 and asset prices are determined by a standard asset pricing model. We find that a model without continuous rebalancing is able to eliminate a relatively small fraction of the volatility smile. To examine the normality assumption, we avoid any parametric model but use the nonparametrically estimated density of the underlying asset. This design avoids the bias caused by any mispricing due to rigid parametric structure of the model and hence can truthfully gauge the impact of bias caused by normality.2 We first assume continuous rebalancing with the 1

Investors may not be able to rebalance due to transaction costs. The use of an empirically based distribution instead of a parametric model, such as a jump diffusion or stochastic volatility model, is motivated by repeated empirical findings that those models fail to explain the Black-Scholes biases (see, for example, Das and Sundaram 1999). 2

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_36, 

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Fig. 36.1 From normal to empirical density

From Normal to Empirical Density From Continuous to Discrete Trading

Black-Scholes III.A

Risk-neutral Empirical III.B

Static Lognormal III.C

Static Empirical III.D

empirical distribution by forcing all discount rates to be the risk free rate.3 We find that this model is more effective in eliminating the volatility smile. Then, we present a model based on an empirical distribution that does not permit rebalancing. Empirically, we find that once both the normality and continuous rebalancing assumptions are removed, the smile largely disappears. Our work differs from that of Rubinstein (1994) and AitSahalia (1996), who use option prices to back out the return distribution of the underlying asset. In their models, option prices by construction do not generate a smile because they are taken as an input. Rosenberg and Engle (2002) also estimate a monthly pricing kernel based on index option data. In contrast, our model uses the density estimated nonparametrically from the underlying asset’s historical returns. The assumption of a nonparametric distribution for the underlying asset is not new,4 but prior models have not directly used the historical density to operationalize their model. Figure 36.1 graphically illustrates the relationship between the models we consider, with continuous rebalancing models on top and the static, CAPM-based models below, and normal models on the left and nonparametrically estimated distributions on the right. After a brief review of the literature in Sect. 36.2, we discuss the basic BlackScholes model with normality and continuous rebalancing in Sect. 36.3.1. We then remove the continuous rebalancing assumption in Sect. 36.3.2 and present a static CAPM option pricing model instead. In Sect. 36.3.3 we replace the normality assumption of Black-Scholes with a density nonparametrically estimated from the actual returns

3

Note that continuous rebalancing is an important necessary condition for the use of the risk neutral pricing methodology. Typically, this linkage is provided by the Brownian motion mathematics. For the exposition here, we assume such a linkage exists for the empirical distribution without offering a proof and proceed to use the risk free rate as the discount rate. 4 See Stutzer (1996), Ait-Sahalia (1996), Ait-Sahalia and Lo (1998).

of the underlying asset, but assume costless rebalancing. In Sect. 36.3.4 we combine the static CAPM with our nonparametric density. Section 36.4 provides a graphical and econometric comparison of the implied volatilities from these models, and Sect. 36.5 concludes.

36.2 Literature Review The empirical literature has consistently demonstrated that the Black-Scholes model biases option values (or, equivalently, implied volatilities) along the moneyness and maturity dimensions. These biases are known as the volatility smile, which is the focus of this paper, and the volatility term structure, respectively. We begin with a brief review of the relevant literature.5 In a well-known study, Rubinstein (1994) demonstrates that the implied volatility for S&P 500 index option is a “sneer.”6 Other studies include Shimko (1993), who argues that the implied S&P 500 index distribution is negatively skewed and more leptokurtic than a lognormal distribution, and Jackwerth and Rubinstein (1996) who find that the S&P 500 index futures distribution before the crash of 1987 resembles the lognormal distribution, while the post-crash distribution exhibits leptokurtosis and negative skewness.7

5

We note that the empirical literature on the Black-Scholes model is voluminous and a comprehensive review is a paper itself. Black and Scholes (1972) provide the first empirical study and find option prices for high (low) variance stocks to be lower (higher) than predicted by the model. Subsequent studies include MacBeth and Merville (1979), Emanuel and MacBeth (1982), Whaley (1982), Geske et al. (1983), Geske and Roll (1984), Rubinstein (1985), and Whaley (1986). 6 Note that the Black-Scholes sneer found by Rubinstein (1994) is based upon data from 1 day. 7 For additional evidence, see Bates (1991 and 1996) and Dumas et al. (1998).

36 Are Tails Fat Enough to Explain Smile

The literature attributes the smile bias in the Black-Scholes model to two unrealistic assumptions: normality of stock returns and costless continuous rebalancing. The assumption that stock returns are distributed normally may underestimate option prices (particularly for deep inthe-money or far out-of-the-money options) because it implies that the tail probabilities of the return distribution are underestimated. To increase the tail probability, the theoretical research has used stochastic volatility, jumps, and GARCH to extend the Black-Scholes model. However, the empirical results have not been overly supportive. Scott (1987) and Bates (1996) find that the stochastic volatility model cannot explain the volatility smile.8 Bakshi et al. (1997) document that even a model with both jumps and stochastic volatility does not explain the volatility smile. Using S&P 500 index options with maturities less than 60 days from 1988 through 1991 they observe noticeable smiles for the three alternative models: stochastic volatility, stochastic volatility with jumps, and stochastic volatility with stochastic interest rates.9 Duan (1995, 1996) uses a GARCH model to price call options on the FTSE-100 index and demonstrates that the GARCH model can generate prices with a smile similar to the BlackScholes smile for most options. However, this result is based upon only a few days of data and no evidence is provided for whether the model can consistently remove the smile.10 More recently, Das and Sundaram (1999) point out that jump-diffusion and stochastic volatility models generate skew and excess kurtosis patterns that are inconsistent with reality. For example, the excess kurtosis generated by jump diffusion models (stochastic volatility models) declines with the holding horizon faster (more slowly) than in reality.11 In contrast, the literature includes a much smaller number of studies on the relationship between the cost of rebalancing and the smile. Assuming proportional transaction costs, Leland (1985, proposition 2) argues that transaction costs should depress the values of near-the-money options more

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than those of away-from-the-money options.12,13 Longstaff (1995), Jackwerth and Rubinstein (1996), Dumas et al. (1998), and Peña et al. (1999) report that transaction costs and liquidity contribute to, but do not completely explain, the volatility smile. In an empirical study, Dennis and Mayhew (2002) discover that the implied skewness of the stock return distribution (implied by option prices) is positively related with stock betas (the higher the beta the more negative the skewness). This finding indicates that risk premiums are important in pricing options. Our static models take this feature explicitly into consideration when computing option prices. In summary, the literature includes many attempts to explain the volatility smile.14 Most studies document that the curvature of the smile is negatively related to option maturity. However, to the best of our knowledge, none of the studies successfully and completely explains the phenomenon. In the rest of the paper, we examine to what degree eliminating the assumptions of continuous rebalancing (no transaction costs) and normality eliminates the Black-Scholes smile.

36.3 The Models 36.3.1 The Black-Scholes Model The Black-Scholes model assumes the following distribution for the underlying asset: ln ST D normal ln St C .r 

 t/;  2 .T  t/ (36.1)

where ST is the underlying asset price at time T given information at time t, r is the continuously compounded risk free rate and  is the annualized volatility of the underlying asset.15 Given that C is the call price, K is the strike price, 12

8 Scott (1987) uses selected stock options from July 1982 to June 1983, and Bates (1996) uses the Deutsche Mark options from 1984 through 1991. 9 See also Eberlein et al. (1998) and Peña et al. (1999). 10 Eberlein et al. (1998) use a hyperbolic function for the distribution of underlying returns and find that the smile and the time-to-maturity effects are reduced in comparison to the Black-Scholes model, but not completely eliminated. 11 For instance, in our sample of S&P 500 returns between January 3, 1950 and September 5, 2003, the excess kurtosis for the three-month period is 22% of the excess kurtosis for the 1-week holding period. In contrast, Das and Sundaram (1999) demonstrate that, using jump diffusion models, the corresponding number is less than 8%, while using stochastic volatility models, this number is more than 70%.

2 2 /.T

One intuitive explanation for this proposition is that at-the-money options have higher gamma, and thus would need to be dynamically hedged more frequently than in- and out-of-the-money options. 13 Boyle and Vorst (1992), and Cochrane and Saa–Requejo (2000) derive the upper and lower bounds for option prices if transaction costs are introduced but do not focus on the volatility smile. 14 In addition to the volatility smile, the literature also documents that the implied volatility depends on option maturity. For example, see Day and Lewis (1992), Canina and Figlewski (1993), Lamoureux and Lastrapes (1993), Heynen et al. (1994), and Xu and Taylor (1994), Campa and Chang (1995), Jorion (1995), and Amin and Ng (1997). However, this issue is largely beyond the scope of this paper. 15 The observed S&P 500 index (SPX) is an ex-dividend series. Usually, given the large number of stocks in the index, the ex-dividend process is assumed to be continuous and hence Equation (1) needs to be adjusted by the dividend yield (r must be replaced by r  d). This replacement could potentially have a large effect for American options, but the effect

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The definition of ˇt;T under the CAPM is:

the well-known Black-Scholes formula is:



CtBS D e r.T t / EO t ŒmaxfST  K; 0g D St N.d1 /  e r.T t / KN.d2 /

ST CT ; St Ct   ST var St



cov ˇt;T D

(36.2)

where EO t Πis the risk neutral expectation taken conditional on information given at time t and

2 .T  t/ ln S  ln K C r C 2 d1 D p  T t p d2 D d1   T  t

Solving (36.3) and (36.4) for the option price, we obtain the following: CtSL

( 1 covŒST ; CT Et ŒCT  D 1 C r.T  t/ varŒST )

.Et ŒST  Œ1 C r.T  t/ St /

Equation (36.2) is the standard no-arbitrage pricing theory that under the risk neutral measure, all assets must yield the risk-free return. A necessary condition for this pricing theory is continuous rebalancing, which under normality is equivalent to complete markets. In the empirical section, we calibrate the Black-Scholes model to the market price of the BS for a given option and solve for the implied volatility, t;T trade date and maturity date .t; T /.

36.3.2 The Static Lognormal Model

 

ST Et  1  r.T  t/ St (36.3)

2 .T  t/ ln S  ln K C  C 2 h1 D p  T t p h2 D h1   T  t p h3 D h1 C  T  t

for European options, like the ones we consider, is small. Further, note that our empirical distribution is formed by returns of ex-dividend levels, which already account for the dividend effect, and hence no additional adjustment is necessary. 16 It is possible for the risk neutral pricing methodology to co-exist with discrete trading but the option must be written on the only asset in the economy – usually assumed to be the aggregated consumption. See Brennan (1979) for a detailed discussion of this point. 17 In a recent working paper, Bondarenko (2003) uses CAPM to examine put option prices.

(36.6)

Also note that: 2

varŒST D St2 e 2.T t / e  .T t /  1

where Et Œ is the conditional expectation under the real measure, ˇt;T is the systematic risk, assumed fixed between t and T , and St is the value of the market portfolio at time t.

(36.5)

The large bracketed term in (36.5) can be regarded as a “certainty equivalent” expected cash flow. Since it carries no risk, it can be discounted at the risk free rate. The “certainty equivalent” expected cash flow is the original expected cash flow, Et ŒCT , minus the “dollar risk premium;” that is, the “market risk premium” adjusted by the “beta” of the option. In addition to the option price, we are also able to obtain the call’s beta and the risk-adjusted rate. In order to derive the closed form solution for the lognormal distribution (i.e., (36.5)), we first define:

In the absence of continuous rebalancing, the risk neutral expectation in Equation (36.2) is not obtainable, and thus we need to rely on an asset pricing model.16 Under the normality assumption the standard Capital Asset Pricing Model implies.17 For options written on the market portfolio: Et ŒCT  1 D r.T  t/ C ˇt;T Ct

(36.4)

covŒST ; CT D EŒST CT  EŒST EŒCT ;

(36.7)

where: EŒST CT D St2 e .2C

2 /.T t /

N.h3 /  KSt e .T t / N.h1 /

EŒST D St e .T t / EŒCT D St e .T t / N.h1 /  KN.h2 /:

(36.8)

These terms can then be substituted into (36.7), to provide a value for the lognormal call value, CtSL (see Appendix 36.A for further details). Alternatively, we can solve for the imSL D plied volatility in the S tatic Lognormal model, t;T p varŒST =St , by solving (36.5) given actual market prices.

36 Are Tails Fat Enough to Explain Smile

Note that the model presented here is static. It assumes no trading/hedging prior to the maturity date of the option. To demonstrate the relationship between our static model and a general model that allows for a finite number of times for trading/hedging between now and maturity, we derive a “two-period” model in which investors are allowed to trade/hedge once before maturity. In this case, at the only time for trading/hedging, symbolized by t1 , the option price should be exactly the same as (36.5) with the substitution of t1 for t. And today’s option price is the use of (36.5) again with the substitution of t1 for T . However, under this case, there is no closed form solution like (36.5) (and (36.5)) and an iterative solution process is necessary. In Appendix 36.B, we demonstrate that when continuous trading is allowed, the result converges to the Black-Scholes model. Note that (36.5) does not provide a solution for volatility when the market price is too low. In the empirical analysis of this model, we omit the approximately 8% of the observations where a solution does not exist.

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fitted to all the data. The differences in skewness and kurtosis are evident. The valuation equation under this model is: CtRE D e r.T t / EO t ŒmaxfST  K; 0g

To implement this empirical model, we normalize the variables as follows: 8 Ct ˆ ˆ Ct  ˆ ˆ St ˆ ˆ < ST Rt;T  (36.10) ˆ St ˆ ˆ ˆ ˆ ˆ : K  K St where St represents the current ex-dividend S&P 500 index (SPX) level. In order to impose the risk neutral pricing methodology, we define: RO t;T;i D Rt;T;i  RN t;T C e r.T t / i D 1; : : : ; N: where

36.3.3 The “Risk-Neutral" Empirical Model In this subsection, we present an empirically based option pricing model. We use a pre-specified moving window to nonparameterically estimate the density of the underlying asset’s return. However, we impose the assumption that continuous rebalancing is available and hence all assets should yield the risk free return. The nonparametric estimation techniques used are standard, and are briefly discussed in Appendix 36.C. Figure 36.2 provides a graph comparing the nonparametrically estimated density with the normal density

Fig. 36.2 Density function

(36.9)

1 XN RN t;T D Rt;T;i i D1 N

(36.11)

(36.12)

is the average return on the market portfolio, represented by the S&P 500 index, for the period Œt; T . This transformation shifts the mean of the distribution to the risk free rate. Thus, (36.9) turns into: CtRE D e r.T t / Et ŒmaxfRO t;T  K  ; 0g

(36.13)

In order to compute (36.13), the distribution of RO t;T needs to be estimated. We use past realized returns from a 5-year window where each return’s holding period length (i.e., T t)

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matches the time to expiration of the option.18 Then the distribution constructed from realized returns is smoothed using a Gaussian kernel, as described in Appendix 36.C. The valuation equation becomes: 

CT E St



36.3.4 The Static Empirical Model

D EŒmaxfRO t;T  K  ; 0g Z1 D .RO t;T  K  /'.RO t;T /dRO t;T K

Z1 D

RO t;T '.RO t;T /d RO t;T  K 

K

Z1

'.RO t;T /d RO t;T

Z1 xf .x/dx  K 

K

CtSE

f .x/dx

K

" ! 1 Xn RO t;T;i  K   O D .Rt;T;i  K /N i D1 n h 8 " #2 93 < 1 RO =  h t;T;i  K 5 (36.14) exp  Cp : 2 ; h 2 where f .x/ is a Gaussian kernel defined by:

f .x/ D

The empirical model described in this subsection assumes no rebalancing, and therefore employs a static asset pricing model for the option (as in Sect. 36.3.2), with a nonparametrically estimated stock return density (as in Sect. 36.3.3).20 Dividing (36.9) by St on both sides, using the notation in (36.10) we obtain:

K

Z1 D

RE and t;T is the implied volatility for the Risk-neutral Empirical model.

1 Xn x  Xi  i D1 nh h

.Et ŒRt;T  Œ1 C r.T  t/ /

covŒRt;T ; CT D EŒRt;T CT  EŒRt;T EŒCT



z2 1 .z/ D p e  2 2

EŒRt;T CT

(36.15)

t;T

C r.T  t/

t;T D

18

1 Xn .RO t;T;i  r.T  t//2 i D1 N 1

ST  K D E R max ;0 St



R.R  K  /'.R/dR

D K

Z1 x.x  K  /f .x/dx

D

(36.16)

K

Z1

(36.17)

D K

 K

Z1 xf .x/dx C

x 2 f .x/dx

K

(36.21) The variance term is more straightforward:  2   EŒRt;T 2 varŒRt;T D E Rt;T

where r



Z1

Plugging (36.15) back into (36.14) results in the call option pricing formula. We calibrate the return distribution to fit the model price generated by (36.13) to the market price of the call by scaling the density defined in (36.15) as follows:19 RE t;T

(36.20)

where

;

(36.19)

We then need to compute covŒRt;T ; CT and varŒRt;T . Note that:

where n is the total number of observations, h is the bandwidth, Xi is the i th observation in the data series, ./ is the kernel function that is chosen to be Gaussian:

RQ t;T D .RO t;T  r.T  t//

( covŒRt;T ; CT 1 D Et ŒCT  1 C r.T  t/ varŒRt;T )

(36.22)

where (36.18)

Our results are robust to other choices of window length from 2 to 30 years. 19 This transformation preserves the mean, skewness, and kurtosis of RE . the distribution while setting the standard deviation to t;T

EŒR D 20

1 Xn Ri i D1 n

This model is similar to the one in Chen and Palmon (2005). Here, we smooth the histogram with a Gaussian kernel to improve estimation realiability. Furthermore, with Gaussian kernel, the model is a closed form, as opposed to the numerical implementation in Chen and Palmon.

36 Are Tails Fat Enough to Explain Smile

EŒR2 D

537

1 Xn R2 i D1 i n

Substituting these results back to (36.19) leads to the final valuation equation. Implied volatility for the S tatic SE , is estimated from call prices as in the Empirical model, t;T risk-neutral empirical model by calibrating the distribution as follows: ^

Rt;T D .Rt;T  RN t;T /

SE t;T

t;T

C RN t;T

(36.23)

where RN t;T is defined in (36.12). Again, this transformation allows us to calibrate the standard deviation of the distribution without changing the mean, skewness, or kurtosis. As in the lognormal model without rebalancing, there is no guarantee that the model can always be solved for the implied volatility; when the option price is too low, the model may not lead to a positive volatility. However, we find that once a nonparametric density is used, all the implied volatilities are positive.

36.4 Data and Empirical Results We collect daily closing levels for the S&P 500 index for the period of January 3, 1950 to September 5, 2003, and we use subsets of these returns to form the nonparametric distributions. Table 36.1 provides summary statistics for the returns derived from these prices. For shorter return horizons, this historical return distribution presents large skewness and excess kurtosis that are substantially larger than the corresponding values for the normal distribution. Consistent with the literature (e.g., Das and Sundaram 1999), the skewness and excess kurtosis dissipate over longer time horizons.21 We use synchronized S&P 500 index call options for the period of June 1988 through December 1991 as in Bakshi et al. (1997).22 Table 36.2 presents the moneyness, defined 21 By excess kurtosis we mean the difference between the fourth centralized moment and the kurtosis for a normal distribution, which equals three. 22 We thank Charles Cao for kindly supplying us this data, which was originally produced by the CBOE.

Table 36.1 Distribution statistics of S&P 500 returns (annualized) for various holding horizons (in business days) from 1/3/1950 to 9/5/2003

as .S  K/=K and maturity distributions of the contracts in our sample. The sample contains 46,540 option price observations across various dates, strike prices, and maturities. We next estimate the implied volatilities for each of the models presented above. When estimating the underlying nonparametric returns density (models C and D), we use the actual returns on the S&P 500 from a 5-year moving window that ends just prior to the option pricing date.23 Figure 36.3 presents a smoothed graph of the implied volatility from the Black-Scholes model (IV_BS) on maturity (DTM) and moneyness (MNY).24 As expected, the graph shows a considerable smile (or smirk), where the implied volatilities are particularly high for shorter maturity in-themoney options. Figure 36.4 presents a similar graph of the implied volatilities from the Static Lognormal model (see III.B. above). Again, a similar smile is clearly visible. Figure 36.5 presents the implied volatility from the Risk-Neutral Empirical model (see III.C. above). Now the surface appears much smoother, and the smile is almost absent. Similarly, Fig. 36.6 presents the graph of implied volatility from the Static Empirical model (see III.D. above), and again the graph appears smoother. Thus, visually, it appears that much of the smile is related to the normality assumption, and using a nonparametric return density appears to remove much of the curvature. One possible concern is that the volatilities are muted relative to prices for some of our models, thus not providing a fair comparison. We therefore also examine the price impact of volatility for each of the four models and find them to be nearly identical. Thus the relationships we find when examining volatility are due to the impact of moneyness and not an ancillary artifact of the models. In order to quantitatively measure the magnitude and statistical significance of the smile under the various models, we regress implied volatility from each model on moneyness, moneyness squared, moneyness cubed, and moneyness raised to the forth power. In order to control for varying volatility across days and maturities, we include a separate

23

Recall that the choice of time horizon does not qualitatively affect the empirical results. 24 The smoothing algorithm used is the spline method provided in SAS.

1-day Mean Std. dev Skew Kurt

0:087 0:143 0:922 24:867

7-day

10-day

21-day

63-day

126-day

252-day

0:087 0:148 0:418 5:399

0:087 0:146 0:486 4:876

0:087 0:146 0:439 2:473

0:087 0:145 0:302 1:447

0:087 0:152 0:117 0:228

0:089 0:160 0:090 0:310

538

Table 36.2 Moneyness and maturity of the S&P 500 call options

R.-R. Chen et al.

Maturity moneyness

6 months

Total

>20% OTM 10 20% OTM 5 10% OTM 0: D Re .i /1 e i lnŒK f1 0 S

S 

 1 fQ1 D exp i .1 C i /V  C i lnŒS.t/ 2  

R ŒR  R .1  e R  /  2 2 ln 1 C ŒR  R  2R R  2i .1  e R  / C R.t/ ; (37.41) 2R  ŒR  R .1  e R  /

(37.43) V 

 Expressions for the gamma measures. The various secondorder partial derivatives of the call price in (37.10), which are commonly referred to as Gamma measures, are given below:

2

2

2

2

2

@ C.t; / @ ˘1 @ ˘2 D S.t/  KB.t; / 2 2 @V @V @V 2 (37.44)

R 

and,  1 fQ2 D exp i .i  1/V  C i lnŒS.t/  lnŒB.t; / 2 # " !  ŒR  R .1  e R  / R   2 2 ln 1 CŒR  R  2R R )  2.i  1/.1  e R  / C  R.t/ ; (37.42)  2R  ŒR  R .1  e R  /

(37.40)

@ C.t; / @ ˘1 D S.t/ 2 @R @R2   2 @ ˘2 @˘2 2 C % KB.t; /  2%./ ./˘ 2 : @R2 @R (37.45)

@˘1 @2 C.t; / D @S @V @V   Z 1 1 @f1 d : D Re .i /1 e i lnŒK 0 @V

S;V 

(37.46)

where for g D V; R and j D 1; 2 @2 ˘j 1 D @g 2 

Z

1 0

  @2 fj d : Re .i /1 e i lnŒK @g 2 (37.47)

Chapter 38

Application of the Characteristic Function in Financial Research H.W. Chuang, Y.L. Hsu, and C.F. Lee

Abstract In this chapter we introduce the application of the characteristic function in financial research. We consider the technique of the characteristic function useful for many option pricing models. Two option pricing models are derived in details based on the characteristic functions. Keywords Characteristic function r Constant elasticity of variance r Option pricing r Stochastic volatility

38.2 The Characteristic Functions In probability theory, the characteristic function of any random variable completely defines its probability distribution. On the real line it is given by the following formula, where X is any random variable with the distribution in question: 'X .t/ D E.e itX / D E.cos.tX// C iE.sin.tX//

38.1 Introduction The characteristic function in nonprobabilistic contexts is called the Fourier transform. The characteristic function was used widely in applied physics (signal process, quantum mechanics). The technique of the characteristic function is also useful for determining the option prices. In Heston (1993), the importance of the characteristic function was demonstrated to find a closed-form solution for options with stochastic volatility. It was subsequently considered by many authors, including Bates (1996), Bakshi and Chen (1997), Scott (1997), Carr and Madan (1999), among others. In addition, we consider the constant elasticity of variance (CEV) option pricing model (see Cox 1996; Schroder 1989; and Hsu et al. 2008) and options with stochastic volatility (see Heston 1993) based on the characteristic functions.

Y.L. Hsu () Department of Applied Mathematics and Institute of Statistics, National Chung Hsing University, Taichung, Taiwan e-mail: [email protected]

(38.1)

where t is a real number, I is the imaginary unit, and E denotes the expected value. The characteristic function is thus defined as the moment generating function but with the real argument s replaced by it; it has the advantage that it always exists because e itx is bounded. If FX is the cumulative distribution function, then the characteristic function is given by the Riemann-Stieltjes integral Z1 itX E.e / D e itx dFX .x/: (38.2) 1

If there is a probability density function, fX , this becomes Z1 E.e / D itX

e itx fX .x/dx:

(38.3)

1

If X is a vector-valued random variable, one takes the argument t to be a vector and t0 X to be an inner product. The characteristic functions for the common distributions are given in Table 38.1. Besides, if F is a one-dimensional distribution function and f is corresponding characteristic function, then the cumulative distribution function FX and its corresponding probability density function .x/ D F 0 .x/ can be retrieved via:

H.W. Chuang Department of Finance, National Taiwan University, Taipei, Taiwan C.F. Lee Department of Finance, Rutgers University, New Brunswick, NJ, USA

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_38, 

1

.x/ D 2

Z1 e itz f .t/dt

(38.4)

1

575

576

H.W. Chuang et al.

Table 38.1 The characteristic functions of the specific probability functions

Probability function

Distribution Normal Uniform Exponential Chi-Squared

Interval

Characteristic function

2 p1 e x =2 2

1 < x < 1

e t

1 e x

0<x m2 .K  S T / if ST < K and PL;T n and VRe;T LRe;T  m2 .K  S T / ;

The value dynamics for the reinsurer’s assets (VRe;T ) and liabilities (LRe;T ) are specifically governed by the following processes: dVRe;t D .r C VRe /VRe;t dt C VRe VRe;t d WVRe ;t ;

(48.23)

where Vi;t and Li;t represent the values of the insurer’s assets and liabilities at time t, respectively. The value dynamics for the insurer’s asset and liability are specified as follows: dV i;t D .r C Vi /Vi;t dt C Vi Vi;t dW Vi ;t ;

1 2 dLi;t D r C Li  P e yi C 2 yi Li;t dt CLi Li;t dW Li ;t C YPLi ;t Li;t dPL;t ;

(48.22)

otherwise

where PL;t is the loss number process, which is assumed to be driven by a Poisson process with intensity P ; Si;t denotes the insurer’s share price and can be shown as: Si;t D

759

(48.24)

(48.25)

where r is the risk-free interest rate; Vi is the risk premium associated with the insurer’s asset risk; Li denotes the risk premium for small shocks in the insurer’s liabilities; WVi ;t is a Weiner process denoting the asset risk; WLi ;t is a Weiner process summarizing all continuous shocks that are not related to the asset risk of the insurer; and YPLi ;t is a sequence of independent and identically-distributed positive random variables describing the percentage change in liabilities in the event of a jump. We assume that ln.YPLi ;t / has a normal distribution with mean yi and standard deviation yi . The 1 2 term P e yi C 2 yi offsets the drift arising from the compound Poisson component YPLi ;t Li;t dPL;t .

dLRe;t

1 2 D r C LRe  P e yRe C 2 yRe LRe;t dt

(48.26)

CLRe LRe;t d WLRe;t C YPLRe ;t LRe;t dPL;t ; (48.27) where VRe;t and LRe;t represent the values of the reinsurer’s assets and liabilities at time t, respectively; r is the riskfree interest rate; VRe is the risk premium associated with the reinsurer’s asset risk; LRe denotes the risk premium for continuous shocks in the insurer’s liabilities; WVRe ;t is a Weiner process denoting the asset risk; WLRe ;t is a Weiner process summarizing all continuous shocks that are not related to the asset risk of the reinsurer; and YPLRe ;t is a sequence of independent and identically-distributed positive random variables describing the percentage change in the reinsurer’s liabilities in the event of a jump. We assume that

6

Though CatPut issuer may adopt alternative credit instruments or derivative market vehicles to transfer its credit risk to the capital market as suggested in Saunders and Allen (2002). Watson (2008) also discussed the impact of transferring the credit risk by securitizing mortgage credit.

760

J.-P. Lee and M.-T. Yu

ln.YPLRe ;t / has a normal distribution with mean yRe and standard deviation yRe . In addition, assume that the correlation coefficient of ln.YPi ;t / and ln.YPLRe ;t / is equal to Y . The 1

2

term P e yRe C 2 yRe offsets the drift arising from the compound Poisson component YPLRe ;t LRe;t dPL;t . According to the payoff structures, the catastrophe loss number process, and the dynamics for the (re)insurer’s assets and liabilities specified above, the CatEPut can be valued as follows:   P LY D E  e rT PO LY :

(48.28)

Here, E  Œ denotes expectations taken on the issuing date under a risk-neutral pricing measure. The CAT put prices are estimated by the Monte Carlo simulation. Table 48.2 presents the numerical results. It shows that the possibility of a reinsurer’s vulnerability (credit risk) drives the put price down dramatically. We also observe that the higher the correlation coefficient of ln.YPi ;t / and

ln.YPLRe ;t / (i.e., Y ) is, the lower the value of the CatEPut will be. This implies that the reinsurer with efficient diversification in providing reinsurance coverage can increase the value of the CatEPut.

48.4 Catastrophe Derivatives Catastrophe risk for (re)insurers can be hedged by buying exchange-traded catastrophe derivatives such as catastrophe futures, catastrophe futures options, and catastrophe options. Exchange-traded catastrophe derivatives are standardized contracts based on specified catastrophe loss indices. The loss indices reflect the entire P&C insurance industry. The contracts entitle (re)insurers (the buyers of catastrophe derivatives) a cash payment from the seller if the catastrophes cause the index to rise above the trigger specified in the contract.

Table 48.2 Catastrophe put option prices with vs. without credit risk With credit risk y P

Without credit risk VRe Panel A: D1 Vi 2 0.12787 1 0.09077 0.5 0.05064 0.33 0.02878 0.1 0.00391

0.3

0.5

0.8

1

0.02241 0.01940 0.00872 0.00451 0.00048

0.02200 0.01913 0.00873 0.00448 0.00048

0.02140 0.01864 0.00861 0.00437 0.00043

0.02071 0.01843 0.00858 0.00433 0.00044

VRe D5 Vi 0.12787 0.09077 0.05064 0.02878 0.00391

0.03008 0.02637 0.01333 0.00693 0.00081

0.02918 0.02616 0.01319 0.00683 0.00081

0.02833 0.02576 0.01300 0.00664 0.00084

0.02778 0.02515 0.01280 0.00670 0.00085

VRe D 10 Vi 0.12787 0.09077 0.05064 0.02878 0.00391

0.03126 0.02730 0.01403 0.00733 0.00087

0.03029 0.02698 0.01389 0.00722 0.00086

0.02935 0.02654 0.01357 0.00707 0.00088

0.02860 0.02611 0.01347 0.00704 0.00091

Panel B: 2 1 0.5 0.33 0.1 Panel C: 2 1 0.5 0.33 0.1

Note. All values are calculated assuming option term T D 2, the number of catastrophe trigger n D 2; risk-free interest rate r D 5%, the mean of logarithmic jump magnitude yi D 2:3075651 (yRe D 2:3075651), the standard deviation of logarithmic jump magnitude yi D 0:5 (yRe D 0:5), the (re)insurer’s initial capital D 1:2), the volatility of (re)insurer’s assets Vi D 10% (VRe D 10%), and the position LVii D 1:2 ( LVRe Re volatility of (re)insurer’s pure liabilities Li D 10% (LRe D 10%). The catastrophe intensity P is set at 2, 1, 0.5, 0.33, and 0.1. All estimates are computed using 20,000 simulation runs

48 Catastrophic Losses and Alternative Risk Transfer Instruments

48.4.1 Catastrophe Derivatives Valuation Models A general formula for the catastrophe futures price can be developed as in Cox and Schwebach (1992) as follows: 1 .ALt C E ŒYt jJt / ; Q

Ft D

(48.29)

where Q is the aggregate premium paid for in the catastrophe insurance portfolio. Here, Yt denotes the losses of the catastrophe insurance portfolio which are reported after the current time t; but included in the settlement value, ALt is the current amount of catastrophe losses announced by the exchange, and Jt denotes the information available at time t. Cox and Schwebach (1992) further derived the catastrophe futures price by assuming Yt follows a compound Poisson distribution with a intensity parameter Y . The aggregate losses of a catastrophe insurance portfolio would be the sum of a random variable of individual catastrophe losses which are independent and identically distributed. In other words, Yt D X1 C X2 C    C XN , where X1 ; X2 ; :::; XN are mutually independent individual catastrophe losses. According to these assumptions, the futures price can be described as follows: Ft D

1 ALt C .T  t/ Y p1 ; Q

(48.30)

where p1 represents the first moment of the individual catastrophe loss distribution, i.e., p1 D E .Xi /. Assuming that the loss of a catastrophe insurance portfolio at maturity (i.e., YT ) T is lognormally distributed, that is, the logarithm of AL ALt is normally distributed with mean  .T  t/ and variance  2 .T  t/, the futures price can be described as:

ALt e Ft D Q

.T t /C 

2 .T t / 2



:

(48.31)

In the case where ALT is set to be lognormally distributed, Cox and Schwebach (1992) presented the value of a catastrophe futures call option with exercise price x; denoted as CCS , as follows: CCS D

e r.T t / Q

y1 D





2 C 2 .T t / ALt e ˆ .y1 /  xQˆ .y2 / ;

C  .T  t/ C p  T t p and y2 D y1   T  t :

log

ALt xQ

(48.32)  2 .T t / 2

Cummins and Geman (1995) used two different processes to describe the instantaneous claim processes during the

761

event quarter and the run-off quarter. They argued that the reporting claims by policyholders are continuous and take only a positive value, hence specifying the instantaneous claim to be a geometric Brownian motion during the run-off quarter. Moreover, they added a jump process to the process during the event quarter. Consequently, the two instantaneous claim  T  and run-off processes during the event quarter t 2 0; 2    quarter t 2 T2 ; T can be respectively specified as follows:   T ; for t 2 0; 2   T ;T ; for t 2 2

dct D ct  .c dt C c d Wc;t / C Jc dNc;t 0

0 dct D ct c dt C c d Wc;t

where ct denotes the instantaneous claim which means that the amount of claims reported during a small length of time 0 dt is equal to ct dt. Terms c and c represent the mean of the continuous part of the instantaneous claims during the event quarter and run-off quarter, respectively, while c and 0 c represent the standard deviation of the continuous part of the instantaneous claims during the event quarter and run-off quarter, respectively. Term Jc is a positive constant representing the severity of loss jump due to a catastrophe, Nc;t is a Poisson process with intensity c , and Wc;t is a standard Brownian motion. Cummins and Geman (1995) derived a formula to value the futures price at time t as follows: ! T exp˛. 2 t / 1 cs ds C ct Ft D ˛ 0 !   T exp˛. 2 t / ˛ T2  t  1 CJc c ˛2 ! 0 ˛ T2 0 1 ˛ . T2 t / exp Cc0 exp ˛0 ! ! 0 T T exp˛. 2 t / 1 exp˛ 2 1 CJc c ; (48.33) ˛ ˛0 Z

t

0

0

0

where ˛ D c  c and ˛ D c  c . Here,  represents the equilibrium market price of claim level risk and is assumed to be constant over period Œ0; T . Cummins and Geman (1995) also considered catastrophe call spreads written on the catastrophe loss ratio. The payoffs of European call spreads at maturity T , denoted as Cspread .S; k1 ; k2 /, can be written as follows: Cspread .c; k1 ; k2 / ( " D Min Max 100

# ) cs ds  k1 ; 0 ; k2  k1 ; (48.34) Qc

RT 0

762

J.-P. Lee and M.-T. Yu

Table 48.3 20/40 European catastrophe call spreads prices s Time to maturity

0.2

0.4

0.6

0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

3.234 3.192 3.155 3.122 3.095 3.071 3.052 3.035 3.022 3.010 3.002

3.842 3.798 3.760 3.727 3.698 3.674 3.653 3.635 3.620 3.608 3.598

4.421 4.376 4.336 4.301 4.270 4.244 4.221 4.202 4.185 4.171 4.160

Note. All values are calculated assuming the contract with an expected loss ratio of 20%, the risk-free rate r D 5%, c D 0:5; Jc D 0:8, and 0 the parameters ˛; ˛ and  to be set at 0.1, 0.1, and 0.15, respectively. Strike prices are also in points. Values are quoted in terms of loss ratio percentage points

Since the instantaneous mean and variance of calendartime futures return, Xt dt and X2 t dt, are linear to random information arrival, the information–time proportional factors, X and X2 , are constant. Substituting Equations (48.36) and (48.37) into Equation (48.35), the parent process in information time can be transferred into a lognormal diffusion process: dXn D X dt C X d WXn : Xn

According to the model, the value of the information-type European catastrophe call option with strike price k, denoted as c .X; n; k/, can be written as follows: c .X; n; k/ D

P1 mD0

ln d1 D

where k1 and k2 are the exercise prices of the catastrophe call spread and k2 > k1 , while Qc is the premiums earned for the event quarter. Since no close-form solution can be obtained, the catastrophe call spreads under alternative combinations of exercise prices can be estimated by Monte Carlo simulation. We report the values of 20/40 call spreads estimated by Cummins and Geman (1995) in Table 48.3 to present the effects of parameter values on the value of catastrophe call spreads. Chang et al. (1996) used the randomized operational time approach to transfer a compound Poisson process to a more tractable pure diffusion process and led to the parsimonious pricing formula of catastrophe call options as a riskneutral Poisson sum of Black’s call prices in information time. Chang et al. (1996) assumed catastrophe futures price changes follow jump subordinated processes in calendartime. The parent process is assumed to be a lognormal diffusion directed by a homogenous Poisson process as follows: dXt D Xt dt C Xt d WXt ; Xt

(48.35)

where Xt and Xt are the stochastic calendar-time instantaneous mean and variance, respectively, and: Xt dt D X d n .t/

(48.36)

X2 t dt D X2 d n .t/ ;

(48.37)

where dn.t/ D 1 dn.t/ D 0

if the jump occurs once in dt with probability jX dt; otherwise with probability 1  jX dt:

(48.38)

 .m; jX / B .Xˆ .d1 /  kˆ .d2 // ; (48.39)



1 X C X2 m p k 2 p ; d2 D d1  X m; X m jX .T t / Œj

/ m

t X where  .m; j / D e is the Poisson probamŠ bility mass function with intensity jx . Moreover, T  t is the option’s calendar-time maturity, r is the riskless interest rate, B D e r.T t / is the price of a riskless matching bond with maturity T  t, and m denotes the information time maturity index. Chang et al. (1996) followed Barone-Adsei and Whaley (1987) for an analytical approximation of the American extension of the Black formula to get the value of the information–time American catastrophe futures call option with strike price k, denoted as C .X; n; k/, as follows:

C .X; n; k/ D

1 X

.T

 .m; jX / CB .X; n; k/ ;

(48.40)

mD0

where 8 rm  q ŒXˆ .d1 /  kˆ .d2 / C A XX ; ˆ <e where X  X  ; CB .X; n; k/ D ˆ :  X  k; where X > X AD

 

X q

(48.41)   1  Bˆ d1 X  ; 

ln   d1 X  D

qD

1C

X X

C 12 X2 m p ; X m

p 2r 1 C 4h ; and h D 2 : 2 X .1  B/

48 Catastrophic Losses and Alternative Risk Transfer Instruments

Table 48.4 The Black .B/ and Information–time .I T / American spread values

763

Maturity (Years) Strike/Strike .k1 =k2 /

Model

0.025

0.05

0.1

0.25

0.5

20/40

B .I T1 / I T30 I T15 I T2 B .I T1 / I T30 I T15 I T2

0.756 0.549 0.424 0.154 0 0 0 0.013

1.067 0.943 0.774 0.303 0 0 0 0.024

1.505 1.503 1.322 0.580 0 0 0 0.053

2.231 2.583 2.369 1.288 0.030 0.056 0.064 0.146

3.048 3.706 3.348 2.140 0.228 0.288 0.281 0.321

40/60 y

Note. All option values are calculated assuming annual volatility X D 60%, annual risk-free interest rate r D 5%, and futures price X D 20. I T30 ; I T15 , and I T2 denote information–time values with annual jump arrival intensities jX D 30; 15; and 2, respectively. B denotes the Black value and is identical to I T1 , the information–time value when jump arrival intensity is infinity. The option is capped at 200

Here, CB .X; n; k/ represents the American extension of the Black formula based on MacMillan (1986) quadratic approximation of the American stock options. Moreover, X  is the critical futures price above where the American futures option should be exercised immediately and is determined by solving: X   k D e rm ŒXˆ .d1 /  kˆ .d2 / C A C

X q





X X

q

   1  Bˆ d1 X  :

Since a diffusion is a limiting case of a jump subordinated process when the jump arrival intensity approaches infinity and the jump size simultaneously approaches zero, the pricing model of Black (1976) is a special case of Equations (48.39) and (48.40). Table 48.4 reports the values of information–time American catastrophe call spreads estimated by Chang et al. (1996). It shows that the Black formula underprices the spread for the 40/60 case. However, for the 20/40 case, the Black formula overprices when the maturity is short. Chang et al. (1996) noted that the Black formula is a limiting case of information–time formula, so that the largest mispricing occurs when the jump intensity j is set at a low value.

48.5 Reinsurance with CAT-Linked Securities P&Cs traditionally diversify and transfer catastrophe risk through reinsurance arrangements. The objective of catastrophe reinsurance is to provide protection for catastrophe losses that exceed a specified trigger level. Dassios and Jang (2003) priced stop-loss catastrophe reinsurance contracts while using the Cox process to model the claim arrival

process for catastrophes. However, in the case of catastrophic events, reinsurers might not have sufficient capital to cover the losses. Recent studies of the catastrophe reinsurance market have found that these catastrophe events see limited availability of catastrophic reinsurance coverage in the market (Froot 1999, 2001; Harrington and Niehaus 2003). P&C reinsurers can strengthen their ability in providing catastrophe coverages by issuing CAT-linked instruments. For example, Lee and Yu (2007) developed a model to value the catastrophe reinsurance while considering the issuance of CAT bonds. The amount that can be forgiven by CAT bondholders when the trigger level has been pulled, ı, can be specified as follows: ı.C  / D FCAT  PCAT;T ;

(48.42)

where PCAT;T is the payoffs of the CAT bond at maturity and is specified as follows:  PCAT;T D

if C   KCAT bond FCAT : rp FCAT if C  > KCAT bond

(48.43)

Here, FCAT is the face value of CAT bonds, and C  can be the actual catastrophe loss facing the reinsurer (denoted as Ci;T , specified by Equation (48.3)) or a composite catastrophe index (denoted as Cindex;T , specified by Equation (48.12)) which depends on the provision set by the CAT bond. When the contingent debt forgiven by the CAT bond depends on the actual losses, there is no basis risk. When the basis risk exists, the payoffs of the reinsurance contract remain the same except that the contingent savings from the CAT bond, ı.C  /; depending on the catastrophic-loss index, become ı.Cindex;T /: Since the debt forgiven by the CAT bond does not depend on the actual loss, the realized losses and savings may not match and may therefore affect the insolvency of

764

J.-P. Lee and M.-T. Yu

the reinsurer and the value of the reinsurance contract in a way that differs from that without basis risk. Here, KCAT bond denotes the trigger level set in the CAT bond provision.

Pb;T

8 ˆ ˆM  A ˆ ˆ Ci;T  A ˆ ˆ ˆ ˆ < .M  A/.ARe;T C ı/ D DRe;T C M  A ˆ ˆ .C ˆ ˆ i;T  A/.ARe;T C ı/ ˆ ˆ ˆ ˆ DRe;T C Ci;T  A : 0

if Ci;T M and ARe;T C ı DRe;T C M  A if A  CiT < M and ARe;T C ı DRe;T C Ci;T  A if CiT M and ARe;T C ı < DRe;T C M  A

otherwise;

(48.45)

where ARe and ARe denote respectively the mean and standard deviation of the reinsure’s asset return; ARe is the instantaneous interest rate elasticity of the reinsurer’s assets; CRe;T is the catastrophe loss covered by the reinsurance contract; and M and A are respectively the cap and attachment level arranged in the reinsurance contract. In addition to the liability of providing catastrophe reinsurance coverage, the reinsurer also faces a liability that comes from providing reinsurance coverages for other lines. Since the liability represents the present value of future claims related to the noncatastrophic policies, the value of a reinsurer’s liability, denoted as DRe;t , can be modeled as follows: dDRe;t D .rt C DRe /DRe;t dt C DRe DRe;t drt CDRe DRe;t d WDRe ;t ;

(48.46)

where DRe is the instantaneous interest rate elasticity of the reinsurer’s liabilities. The continuous diffusion process reflects the effects of interest rate changes and other day-to-day small shocks. Term DRe denotes the risk premium for the small shock, and WDRe ;t denotes the day-to-day small shocks that pertain to idiosyncratic shocks to the capital market. In order to incorporate the effect of the interest rate risk on the reinsurer’s assets, the asset value of the reinsurance company is assumed to be governed by the same process as defined in Equation (48.10). Under the term structure assumption of Cox et al. (1985) the rate on line (ROL) or the fairly-priced premium rate can be calculated as follows: ROL D

h RT i 1

E0 e  0 rs ds Pb;T ; M A

(48.44)

if M > CiT A and ARe;T C ı < DRe;T C CiT  A

where ARe;T denotes the reinsurer’s asset value at time t, which is assumed to be governed by the following process: dARe;t D ARe dt C ARe drt C ARe d WARe ;t ; ARe;t

In the case where the reinsurer i issues a CAT bond to hedge the catastrophe risk, at maturity the payoffs of the reinsurance contract written by the reinsurer, denoted by Pb;T , can be described as follows:

(48.47)

where ROL is the premium rate per dollar covered by the catastrophe reinsurance; and E0 denotes the expectations taken on the issuing date under risk-neutral pricing measure. Table 48.5 reports ROLs with and without basis risk calculated by Lee and Yu (2007). When the coefficient of correlation between the individual reinsurer’s catastrophe loss and the composite loss index, X , equals 1, no basis risk exists. The lower the c is , the higher the basis risk the reinsurer has. The difference of ROLs for a contract with X D 1 and other alternative values is the basis risk premium. We note that the basis risk drives down the value of the reinsurance contract and the impact magnitude increases with the basis risk, catastrophe intensity, and loss volatility. We also note that the basis risk premium decreases with the trigger level and the reinsurer’s capital position, but increases with catastrophe occurrence intensity and loss volatility.

48.6 Conclusion This study investigates the valuation models for three types of CAT-linked securities: CAT bonds, CAT equity puts, and exchange-traded CAT futures and options. These three new types of securities are capital market innovations which securitize the reinsurance premiums into tradable securities and share the (re)insurers’ catastrophe risk with investors. The study demonstrates how prices of CAT-linked securities can by valued by using a contingent-claim framework and numerical methods via risk-neutral pricing techniques. It begins with introducing a structural model of the CAT bond that incorporates stochastic interest rates and allows for endogenous default risk and shows how its price can be estimated. The model can also evaluate the effect of moral hazard and basis risk related to the CAT bonds. This study then extends the literature by setting up a model for valuing CAT equity puts in which the issuer of the puts is vulnerable. The results show how the values of CAT equity puts change

48 Catastrophic Losses and Alternative Risk Transfer Instruments

765

Table 48.5 Values of reinsurance contracts (ROL) with CAT bonds and basis risk KCAT bond 80 100 c

0:3

0:5

1

0:3

0:5

(;  c ;  ci ndex )

120 1

0:3

0:5

1

ARe =DRe =1.1

(0.5,0.5,0.5) (0.5,1,1) (0.5,2,2) (1,0.5,0.5)

0:00283 0:01696 0:05335 0.01067

0:00283 0:01698 0:05354 0:01067

0:00283 0:01709 0:05424 0:01607

0:00283 0:01694 0:05327 0:01066

0:00283 0:01695 0:05343 0:01066

0:00283 0:01701 0:05403 0:01066

0:00283 0:01694 0:05326 0:01066

0:00283 0:01695 0:05340 0:01066

0:00283 0:01701 0:05392 0:01066

(1,1,1) (1,2,2) (2,0.5,0.5) (2,1,1) (2,2,2)

0:03989 0:10632 0:04331 0:09952 0:21774

0:03995 0:10663 0:04331 0:09964 0:21825

0:04018 0:10798 0:04337 0:10011 0:22703

0:03987 0:10615 0:04331 0:09931 0:21719

0:03990 0:10640 0:04331 0:09938 0:21767

0:04002 0:10749 0:04333 0:09967 0:21981

0:03978 0:10594 0:04331 0:09925 0:21687

0:03978 0:10620 0:04331 0:09930 0:21732

0:03986 0:10712 0:04331 0:09943 0:21910

0:00295 0:01908 0:06259 0:01123 0:04508 0:12488 0:04667 0:11292 0:24732

0:00295 0:01902 0:06175 0:01123 0:04473 0:12300 0:04663 0:11237 0:24506

0:00295 0:01902 0:06189 0:01123 0:04473 0:12326 0:04663 0:11242 0:24540

0:00295 0:01906 0:06246 0:01123 0:04482 0:12428 0:04663 0:11258 0:24677

0:00295 0:01906 0:06246 0:01123 0:04482 0:12428 0:04663 0:11258 0:24677

0:00296 0:02020 0:06887 0:01130 0:04763 0:13768 0:04734 0:12055 0:27038

0:00296 0:02020 0:06899 0:01130 0:04763 0:13793 0:04734 0:12058 0:27081

0:00296 0:02023 0:06962 0:01130 0:04773 0:13907 0:04734 0:12080 0:27268

ARe =DRe =1.3 (0.5,0.5,0.5) (0.5,1,1) (0.5,2,2) (1,0.5,0.5) (1,1,1) (1,2,2) (2,0.5,0.5) (2,1,1) (2,2,2)

0:00296 0:01904 0:06184 0:01123 0:04479 0:12335 0:04669 0:11265 0:24566

0:00296 0:01905 0:06203 0:01123 0:04484 0:12365 0:04669 0:11277 0:24603

0:00296 0:01918 0:06280 0:01123 0:04510 0:12512 0:04670 0:11337 0:24791

0:00295 0:01902 0:06177 0:01123 0:04492 0:12342 0:04666 0:11249 0:24531

0:00295 0:01902 0:06193 0:01123 0:04494 0:12367 0:04666 0:11257 0:24567 ARe =DRe =1.5

(0.5,0.5,0.5) (0.5,1,1) (0.5,2,2) (1,0.5,0.5) (1,1,1) (1,2,2) (2,0.5,0.5) (2,1,1) (2,2,2)

0:00296 0:02016 0:06887 0:01130 0:04767 0:13795 0:04735 0:12066 0:27106

0:00296 0:02018 0:06905 0:01130 0:04771 0:13824 0:04735 0:12076 0:27153

0:00296 0:02030 0:06988 0:01130 0:04800 0:13983 0:04735 0:12147 0:27417

0:00296 0:02017 0:06882 0:01130 0:04764 0:13796 0:04733 0:12051 0:27066

0:00296 0:02017 0:06897 0:01130 0:04766 0:13820 0:04733 0:12057 0:27110

Note. This table presents ROLs with CAT bond issuance and the payoffs to CAT bonds are linked to a catastrophe loss index. ROLs are calculated and report alternative sets of trigger values (KCAT bond ), catastrophe intensities (), catastrophe loss under volatilities (c ; ci ndex ) and the coefficient of correlation between the reinsurer’s catastrophe loss and the composite loss index (X ). ARe =DRe represents the initial asset-liability structure or capital position of the reinsurers. All estimates are computed using 20,000 simulation runs

with the issuer’s vulnerability and the correlation between the (re)insurer’s individual catastrophe risk and the catastrophe index. Both results indicate that the credit risk and the basis risk are important factors in determining CAT bonds and CAT equity puts. This study also compares several models in valuing CAT futures and options. Though differences exist in alternative models, model prices are within reasonable ranges and similar patterns are observed on price relations with the underlying elements. The hedging effect for a reinsurer issuing CAT bonds is also examined. As long as the threat that natural disasters pose to the financial viability of the P&C industry continues to exist, there

is a need for further innovations on better management of catastrophe risk. The analytical framework in this study in fact provides a platform for future research on catastrophic events with more sophisticated products and contracts in the insurance industry as well as other financial industries. For example, our analysis can be applied to the recent massive losses of derivatives associated with subprime mortgage loans and the vulnerability of the firm that has created these structured mortgage products.7

7

See Watson (2008).

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References Bantwal, V. J. and H. C. Kunreuther. 2000. “A CAT bond premium puzzle?” Journal of Psychology and Financial Markets 1(1), 76–91. Barone-Adesi, G. and R. E. Whaley. 1987. “Efficient analytic approximation of american option values, Journal of Finance 42(2), 301–320. Black, F. 1976. “The pricing of commodity contracts.” Journal of Financial Economics 3, 167–179. Chang, C., Chang, J. S., and M. -T. Yu. 1996. “Pricing catastrophe insurance futures call spreads: a randomized operational time approach.” Journal of Risk and Insurance 63, 599–617. Cox, S. H. and R. G. Schwebach. 1992. “Insurance futures and hedging insurance price risk.” Journal of Risk and Insurance 59, 628–644. Cox, J., J. Ingersoll, and S. Ross. 1985. “The term structure of interest rates.” Econometrica 53, 385–407. Cox, S. H., J. R. Fairchild, and H. W. Pederson. 2004. “Valuation of structured risk management products.” Insurance: Mathematics and Economics 34, 259–272. Cummins, J. D. and H. Geman. 1995. “Pricing catastrophe futures and call spreads: an arbitrage approach.” Journal of Fixed Income March, 46–57. Dassios, A. and J.-W. Jang. 2003. “Pricing of catastrophe reinsurance and derivatives using the cox process with shot noise intensity.” Finance and Stochastics 7(1), 73–95. Duan, J.-C. and M.-T. Yu. 2005. “Fair insurance guaranty premia in the presence of risk-based capital regulations, stochastic interest rat and catastrophe risk.” Journal of Banking and Finance 29(10), 2435–2454. Froot, K. A. (Ed.). 1999, The Financing of Catastrophe Risk, University of Chicago Press, Chicago. Froot, K. A. 2001. “The market for catastrophe risk: a clinical examination.” Journal of Financial Economics 60(2), 529–571. Harrington, S. E. and G. Niehaus. 2003. “Capital, corporate income taxes, and catastrophe insurance.” Journal of Financial Intermediation 12(4), 365–389.

J.-P. Lee and M.-T. Yu Jaimungal, S. and T. Wang. 2006. “Catastrophe options with stochastic interest rates and compound poisson losses.” Insurance: Mathematics and Economics 38, 469–483. Jarrow, R. and A. Rudd. 1982. “Approximate option valuation for arbitrary stochastic processes.” Journal of Financial Economics 10, 347–369. Lee, J.-P. and M.-T. Yu. 2002. “Pricing default-risky CAT bonds with moral hazard and basis risk.” Journal of Risk and Insurance69(1), 25–44. Lee, J.-P. and M.-T. Yu. 2007. “Valuation of catastrophe reinsurance with CAT bonds.” Insurance: Mathematics and Economics 41, 264–278. Litzenberger, R. H., D. R. Beaglehole, and C. E. Reynolds. 1996. “Assessing catastrophe reinsurance-linked securities as a new asset class.” Journal of Portfolio Management Special Issue , 76–86. Loubergé, H., E. Kellezi, and M. Gilli. 1999. “Using catastrophe-linked securities to diversify insurance risk: a financial analysis of cat bonds.” Journal of Insurance Issues 22, 125–146. MacMillan, L. W. 1986. “Analytic approximation for the American put options.” Advances in Futures and Options Research 1, 119–139. MMC Security. 2007. Market update: the catastrophe bond market at year-end 2006, Guy Carpenter & Company, Inc. Nielsen, J. and K. Sandmann. 1996. “The pricing of Asian options under stochastic interest rate.” Applied Mathematical Finance 3, 209–236. Saunders, A. and L. Allen. 2002. Credit risk measurement: new approaches to value at risk and other paradigms, Wiley, New York. Turnbull, S. and L. Wakeman. 1991. “A quick algorithm for pricing European average options.” Journal of Financial and Quantitative Analysis 26, 377–389. Watson, R. D. 2008. “Subprime mortgages, market impact, and safety nets: the good, the bad, and the ugly.” Review of Pacific Basin of Financial Markets and Policies 11, 465–492. Zajdenweber, D. 1998. The valuation of catastrophe-reinsurance-linked securties, American risk and insurance association meeting, Conference Paper.

Chapter 49

A Real Option Approach to the Comprehensive Analysis of Bank Consolidation Values Chuang-Chang Chang, Pei-Fang Hsieh, and Hung-Neng Lai

Abstract This study applies a modification of the Schwartz and Moon (Financial Analysts Journal 56:62–75, 2000) model to the evaluation of bank consolidation. From our examination of a bank merger case study (the first example of such a bank merger in Taiwan), we find that, from an ex-ante viewpoint, the consolidation value is, on average, about 30% of the original total value of the independent banks. We also find that the probability of bankruptcy was considerably lower following the merger than it would have been prior to the merger. Our case study therefore indicates that the merger was indeed a worthwhile venture for both banks involved. Furthermore, on completion of the merger, we are also able to determine that, in terms of the magnitude of the increased consolidation value, the most crucial roles are played by the resultant changes in the growth rates of the integrated loans and integrated deposits, as well as the cost-saving factors within the cost functions. Keywords A bank merger r The probability of bankruptcy r Consolidation values

49.1 Introduction Despite the growth in domestic financial institutions throughout the 1990s, there were significant declines in both return on equity (ROE) and return on assets (ROA) for Taiwanese domestic banks, which is, of course, one of the warning signs of an over-banking problem. In order to solve this problem, the Taiwanese government promulgated the Financial Institutions Merger Law in November 2000, which would subsequently allow banks to diversify the scope of their business, or to acquire other

C.-C. Chang (), P.-F. Hsieh, and H.-N. Lai Department of Finance, National Central University, Chung Li City, Taiwan, Republic of China e-mail: [email protected]

financial institutions. Thereafter, in June 2001, in an effort to accelerate the reformation of the island’s financial institutions, the Financial Holding Company Law was introduced as a means of simplifying the areas of cross-management and cross-selling through both horizontal and vertical integration. Following the passage of this Law, the island experienced a wave of financial institution consolidations, as the search began for capital efficiency and cost savings. Numerous studies have discussed the efficiency effects of bank mergers, with capital efficiency having been recognized as one of the issues of consolidation efficiency. In their review of over 250 references, Berger et al. (1999) found the evidence to be consistent with the increases in market power secured by certain other types of consolidations, through improvements in both profit efficiency and the diversification of risk. Rhoades (1998) summarized nine case studies of additional cost savings stemming from the elimination of redundant facilities, staff, or even departments, and found that in all nine cases the combined firms were able to successfully achieve their cost-cutting objectives. The literature on consolidation efficiency has focused on an examination of the consequences of consolidation for individual bank performance, including event studies of stock price responses and studies of post-merger performance based on income statement and balance sheet information; this essentially involves a kind of post-consolidation analysis. There have also been a few papers that explore the merger gains from an ex-ante viewpoint, and indeed, from a merger perspective, a discussion of the transaction values of participating banks prior to a merger does indeed play an important role. Moreover, an additional key determinant of a merger decision is the means of evaluating the merger gains in advance, since such means will help to determine whether the decision maker will actually decide to implement the project. In their analysis of a bank’s international expansion options, Panayi and Trigeorgis (1998) undertook an examination of multi-stage real option applications, dividing the expansion strategy into two stages. The first stage of this process involved investments that were being considered by the banks to take place in year 1, while the second stage involved the investment options that were being contemplated

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_49, 

767

768

for year 10. The traditional NPV method was used to estimate the investment value, including the two stages of NPV,1 whereas the Black and Scholes (1973) formula was used to measure the growth option value. In the Panayi and Trigeorgis (1998) study, the option model was used to apply a valuation, with the possible gains from bank consolidation then measured, ex ante. However, the use of the Black and Scholes formula still involves a number of restrictions, such as the limited number of parameters available for measuring bank consolidation value. This paper adopts a modified version of the Schwartz and Moon (2000) real-option concept, along with the least squared MonteCarlo simulation approach (which represents an easier and more general method of evaluating various types of investments) to the evaluation of an Internet company. We apply this method to the evaluation of financial institutions and formulate a model for ex-ante evaluation. Our model is implemented as a means of evaluating the first example of bank consolidation in Taiwan, involving the merger between Taishin International Bank and Dah An Commercial Bank. We calculate the individual premerger bank value, alongside the consolidated bank value, which might, theoretically, help to determine the conversion ratio in advance. Conversely, we examine the ratio of the increased value, resulting from the bank consolidation, which might also, theoretically, help to estimate the merger decision in advance. The remainder of this paper is presented as follows. In Sect. 49.2, we set up the model in continuous time from a discrete time approximation for the bank evaluation, followed, in Sect. 49.3, by a demonstration of the way in which our model is implemented. A number of simulations are carried out in Sect. 49.4 in an effort to investigate the key factors determining bank consolidation value. Finally, the conclusions drawn from this study are presented in Sect. 49.5.

49.2 The Model In developing a simple model for the evaluation of a bank’s value, we make a general assumption that the interest revenue earned from loans represents the bank’s major revenue source. We identify the credit scale as the bank’s total assets, and its deposits as the total liabilities. We also regard the cost function as a noninterest operating expense, combining both the fixed and variable components. We can determine the net post-tax cash flow rate from the revenue and cost functions, while also determining the cash available during each time 1

Panayi and Trigeorgis (1998) defined the expanded NPV as the sum of the base-case expanded NPV, with the second-stage opportunity NPV being valued as an expansion option.

C.-C. Chang et al.

interval. Thereafter, by discounting all of the cash available prior to the time horizon, we are able to determine the present value of the bank. We initially describe the model in continuous time, in an effort to develop it more precisely; within the process of implementation, we use the quarterly accounting data available from each of the banks and practice the use of the model in discrete time. In the following subsection, we focus on the development of a continuous, individual (premerger) bank model, along with a bank consolidation model.

49.2.1 The Premerger Bank Model The majority of a bank’s revenue comes from the interest spreads between loans and deposits. Let us assume that a bank’s loan at time t, is given by Lt , and that the dynamics of the bank’s loans are given by the stochastic differential process: dLt L D L (49.1) t dt C t d z1 Lt where L t (the drift) is the expected rate of growth in loans, which is assumed to follow a mean-reversion process with a long-term average drift, L I  L is the level of volatility in the loan growth rate, and z1 follows a standard Wiener process. We assume that the bank can earn an abnormal return within an observed period, and that it will converge stochastically to the more reasonable rate of growth for the banking industry as a whole. The mean-reversion process is given as:   L L dL N L  L t Dk t dt C t d z2 ;

(49.2)

where L t is the level of volatility in the expected loans growth rate. The mean-reversion coefficient k L describes the rate at which such growth is expected to converge to its longterm average. The term, z2 , also follows a standard Wiener process. We further assume that the unanticipated changes in loans converge to a normal level,  L , and that the unanticipated changes in the drift converge to zero. Similarly, the bank’s deposits at time t are given by Dt , and the dynamics of the bank’s loans are given by the stochastic differential process: dDt D D D t dt C t d z3 ; Dt

(49.3)

where D t is the expected rate of growth in deposits, which is also assumed to follow a mean-reversion process with a longterm average drift, D I  D represents the level of volatility in the growth rate of deposits, and z3 is a standard Wiener process.

49 A Real Option Approach to the Comprehensive Analysis of Bank Consolidation Values

The expected growth rate in deposits also converges to the long-term rate, as follows:   D D N D  D dD t D k t dt C t d z4

(49.4)

where D t is the level of volatility of the expected growth rate in deposits; the mean-reversion coefficient, k D , describes the rate at which the growth is expected to converge to its longterm average, and the term z4 follows a standard Wiener process. The unanticipated changes in deposits are also assumed to converge to  D , while the unanticipated changes in the expected growth rate in deposits are assumed to converge to zero. The interest spreads relate to the changes in the interest rates of both loans and deposits. Although there are various kinds of deposit rates, we use the average as a proxy. The loans and deposits rates follow two stochastic processes within which we define loan interest as a spread, St , above the average deposits rate, rt , with both of these following the square-root process of Cox, Ingersoll, and Ross (hereafter known as CIR) (1985): drt D a .b  rt / dt C t

p

rt d z5

p   dSt D aS b S  St dt C tS St d z6 ;

(49.5) (49.6)

where ˛ and ˛ s are the reversion speed parameters; b and b s are the values towards which the interest rates revert over time; t and ts are the standard deviation, and z5 and z6 follow standard Wiener processes. We define the net interest income, Rt , as the total amount of the interest spreads: Rt D Lt .rt C St /  Dt rt :

(49.7)

In defining all of the parameters for net interest income, there are six standard Wiener processes, z1 ; z2 ; z3 ; z4 ; z5 , and z6 , each of which is instantaneously and mutually correlated (see Appendix 49A). Furthermore, the noninterest operating expenses, Ot , are defined with both fixed and variable components. The fixed costs are the costs of buildings and operating equipment, which are assumed to remain unchanged within the estimation horizon; the variable costs are defined as the nonperforming loans (NPLs), which represent a proportion of the total amount of all bank loans: dOt D Ft C ˛Lt ; Ot where ˛ is the percentage of loans.

(49.8)

769

Given the total revenue and costs of the bank, we can define the net post-tax cash flow rate of Yt , which is given by: Yt D .Rt  Ot / .1  t / ;

(49.9)

where t is the corporate tax rate. Finally, the bank is assumed to have a certain amount of cash available, Xt , which evolves according to: dXt D Yt dt:

(49.10)

Schwartz and Moon (2000) assumed that a company goes into bankruptcy when its available cash level reaches zero; the Schwartz and Moon definition is adopted in this study as the base valuation. If a bank runs out of cash, we do not take into account the possibility of additional financing, even where the bank may be sufficiently attractive as to be able to raise cash. In order to simplify the model, we assume that all cash flow remains within the bank, where it earns a risk-free interest rate, and that it will be available for distribution to the shareholders at the time horizon. Finally, the objective of our model is to determine the total values of the individual banks, and indeed, Kaplan and Ruback (1995) provided empirical evidence to show that discounted cash flows provide a very reliable estimate of market value. Our estimation of the value of a bank is undertaken by calculating the discounted value at the risk-free measure:   V0 D EQ ŒXT C M .RT  OT /  e rT : (49.11) where the term EQ is the equivalent martingale measure, and e rT is the continuously compounded discount factor. We assume, in Equation (49.11), that bank value is similar to a European option. At the time horizon, the bank has two amounts of cash available; that is, the available cash accumulated to time horizon, and the terminal value of the bank. Schwartz and Moon (2000) set the terminal value at the time horizon as a multiple, M , (e.g., ten times) of the net post-tax cash flow rate. According to Brennan and Schwartz (1982), the measured variable should be converted so as to become risk-neutral; given that, in the real world, we use a risk-neutral measure (see Appendix 49B). In this model the value of a bank is a function of the state variables (loans, expected loan growth, deposits, expected deposit growth, interest rates, interest spreads, and cash balances) and time. This function can be written as follows:   V  V L; L ; D; D ; r; S; X; t :

(49.12)

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49.2.2 The Bank Consolidation Model Based on the study of Berger et al. (1999), we assume that the primary motive for consolidation is the maximization of shareholder value; all actions taken by the banks would therefore be aimed towards the maximization of the value of all shares owned by the existing shareholders. In this study, we design the consolidation model to focus on increasing efficiency. First of all, we discuss whether increasing the scale of loans and deposits could assist in both achieving economies of scale and improving the value of the existing bank. Thereafter, we go on to discuss the changes in the cost structure brought about as a result of consolidation. Assuming there is an existing bank after consolidation, the bank loans of bank M at time t are given by LM t , while the dynamics of the integrated loans are given by the stochastic differential equation, similar to that discussed above. However, the growth rate of the integrated loans processes needs to be redefined, since the loan sources of each of the banks may well overlap, or have some level of correlation. Let us assume, for example, that bank A merges with bank B; in order to consider the correlation between the two banks, we assume that there will be an instantaneous correlation, L , between the two Wiener processes. This is implemented by transformation to the new variables, tA and tB , as follows: d

A t

M LA LA LA D uLA t t dt C t t d z1

(49.13)

d

B t

M LB LB LB D uLB t t dt C t t d z2

(49.14)

M d zM 1 d z2 D L dt; M

(49.15)

M

where dz1 and dz2 are the standard Wiener processes of the new variables tA and tB , having a correlation, L . We define the loan growth rate of the consolidated bank t LM as the value-weighted average of the new growth rate processes of each of the banks: D ktA d dLM t ktA D

LA 0 B LA 0 C L0

A t

C ktB d

ktB D

B t

LB0 B LA 0 C L0

(49.16) ;

where ktA and ktB are the value-weighted initial loan amounts for each of the banks. The dynamics of the stochastic differential equation of the integrated loans is given as follows: dLM t D LM dt C tLM d zM t 3 ; LM t

(49.17)

where t LM represents the integrated loans growth rate, which is defined above, and tLM is the volatility of the integrated loans growth rate.

We define the integrated deposits as Dt M . We should also similarly define each of the bank’s expected deposit growth rates. The dynamics of the integrated deposits stochastic differential equation is as follows: dDM t D DM dt C tDM d zM t 4 ; DtM

(49.18)

where t DM is the growth rate of the integrated deposits, and tDM is the volatility of the growth rate of the integrated deposits. The interest rate models continue to follow the squareroot process of CIR (1985), and we use the existing bank’s average deposit rate and interest spread for the integrated bank’s average deposit rate, rtM , and interest spread, StM . The net interest income of the consolidated bank, RtM , is the difference between the interest earned from loans, and the interest paid to depositors:   M M  DtM rtM : RtM D LM t rt C St

(49.19)

Similarly, the six Wiener processes, the integrated bank’s loans, deposits, loan growth, deposit growth, deposit rate, and interest spread, may be mutually relative, as discussed earlier in the premerger model (see Appendix 49A). Rhoades (1998) found that in some cases of bank consolidation, there was some existence of cost efficiency, and in our study, we assume that there are cost reduction opportunities within the cost function of the consolidated bank:   MF

FtA C FtB C˛tMO LM OtM D FtM C˛tMO LM t D ˛ t ; (49.20) where ˛tMF are the fixed cost saving factors (or the economies of scale factor), and ˛tMO are the variable cost saving factors. Basically, there is a cost saving, or scale economy, motive for bank consolidation if the magnitude of ˛tMF is less than unity. The variable cost may produce a cost-saving opportunity either through increased loans scale, or a reduction in NPLs. We measure the effect of the variable cost savings by comparing the variable cost-saving factor with the rate of the value-weighted variable cost of each participating bank. The rate of post-tax net cash flow into the integrated bank, M Yt , the amount of cash available, XtM , and the estimation of the integrated bank, V0 M , are all the same as those in the premerger bank model. To reiterate, the integrated bank value is a function of the state variables (integrated loans, loan growth rate, integrated deposits, deposit growth rate, interest rate, interest spread and cash balances) and time. This can be shown as:   V M  V M LM ; LM ; D M ; DM ; r M ; S M ; X M ; t : (49.21)

49 A Real Option Approach to the Comprehensive Analysis of Bank Consolidation Values

The objectives of our study are to determine whether the effects of the bank merger are positive, and whether or not the merger achieves the goal of maximizing share value. If the consolidation value is greater than the summed-up value of the individual banks, then the bank merger does have synergy. We measure the increased value ratio ”, as: Increased bank value ratio . / D

V0M  V0A  V0B : V0A C V0B (49.22)

The model developed in the continuous time model, above, is path dependent, and such path dependencies can easily be taken into account by using the Monte Carlo simulation to solve the evaluation. In order to implement the simulation, we use the discrete version of the risk-adjusted process (see Appendix 49C).

49.3 Case Study In order to demonstrate the implementation of our model, we use, as an example, the first Taiwanese bank merger case, Taishin International Bank and Dah An Commercial Bank.

49.3.1 An Introduction to the Case Study The Financial Holding Company (FHC) Law, which was introduced in June 2001, brought about significant merger activity, beginning with the merger of Hua Nan Bank, Taiwan Bank, and First Bank on December 19, 2001. Thereafter, between 2002 and the first quarter of 2003, a total of 14 financial holding companies had been established in Taiwan, most of which comprised commercial banks and securities and insurance companies. No commercial bank mergers had taken place before the first quarter of 2002. This section introduces the first case of a commercial bank merger in Taiwan and discusses the value created by the merger, according to our model. Taishin International Bank and Dah An Bank held an extraordinary shareholder meeting in December 2001, at which they agreed to establish Taishin Financial Holding Company (FHC), which would comprise Dah An Bank, Taiwan Securities Company, and the Taishin Bills Finance Corporation. Within the planned FHC, Taishin International Bank offered shares, essentially for the takeover of Dah An Bank. Following the merger, Taishin International Bank was to be the surviving bank, under a share-exchange agreement of one share in Taishin Bank for two shares in Dah An Bank. As a result of the merger, the assets of Taishin FHC increased from NT$42 billion to NT$30 billion. In estimating

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the effects of the merger, it was expected that Taishin would be able to provide one-stop financial services to more than 3.5 million customers at its 133 branch offices (up from the previous 88) with 4,500 staff members. The company noted that, from 2002 to 2004, the consolidation was expected to lead to cost reductions estimated at between NT$600 and NT$700 million per year. The new company’s share of the loans market was predicted to reach 2.5%, while its nonperforming loan (NPL) ratio was expected to be reduced to 2.24% in 2002, giving the new company post-tax income of almost NT$5 billion. Taishin International Bank Chairman, Thomas Wu, estimated that the merger with Dah An Commercial Bank would create earnings of NT$1.60 per share by the end of 2002, an increase of 201% on 2001 earnings of NT$0.53 per share. The first stage in the formation of the FHC – the merger between Taishin International Bank and Dah An Commercial Bank – was presented to the Ministry of Finance, under plans to be completed in February 2002. The second stage was the plan to bring Taiwan Securities Company and Taishin Bills Finance Corporation into the FHC in the second quarter of 2002, once the share swap ratio had been negotiated at their June shareholder meeting. This merger, which was to be the first example of consolidation of domestic commercial banks in Taiwan, involved a very complex process of share exchanges, the elimination of the acquired bank, and the offering up of the existing bank. Hence, it was important to be able to accurately estimate the value of each bank in order to be able to handle the various problems. In this study, we are also keen to be able to determine the value created by the merger, from an ex-ante perspective.

49.3.2 Estimating the Parameters The model described above requires a considerable number of parameters to facilitate the calculation of overall bank value, and since Taishin International Bank applied for the merger in January 2001, with the expectation that it would complete it before February 2002, we set out in this study to estimate the bank value based on the third quarter of 2001, using the financial reports from the first quarter of 1997 to the third quarter of 2001 as the historical data.

49.3.2.1 The Premerger Parameters The implementation of the first part of the model requires more than 50 parameters, some of which are easily observed from each bank’s quarterly financial reports. However, some of the parameters, those which require the use

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Table 49.1 Taishin International Bank, quarterly sales and costs, March 1997–September 2001 (unit NT$  10 million)

Date

Loans

Deposits

Interest income

March 1997 June 1997 September 1997 December 1997 March 1998 June 1998 September 1998 December 1998 March 1999 June 1999 September 1999 December 1999 March 2000 June 2000 September 2000 December 2000 March 2001 June 2001 September 2001

92.453 103.432 115.105 130.221 145.511 161.354 161.134 163.441 161.889 162.930 165.610 176.137 175.961 176.166 180.174 185.694 183.183 182.630 180.272

95.657 96.726 112.576 148.140 166.797 175.910 186.273 197.166 198.380 200.010 195.101 197.314 210.578 214.840 219.554 224.619 231.224 232.996 229.595

2:508 4:750 8:217 10:841 4:043 7:648 12:996 16:245 4:820 8:756 14:465 17:722 5:008 9:372 15:507 19:226 5:420 9:792 15:941

Interest expenses 1:595 3:214 5:100 7:359 2:703 5:620 8:694 11:799 3:045 5:868 8:545 11:074 2:678 5:396 8:237 11:169 2:907 5:611 8:173

of judgment, can be determined either by the management’s estimated forecasts, or through investigation by various analysts. Finally, we determine those parameters that are difficult to either observe or estimate, by direct consultation with previous references. Tables 49.1 and 49.2 present the basic data on each bank, including quarterly loans, deposits, interest income, and interest expenses for the last 19 quarters. In Table 49.3, we use this basic data to determine the initial loans, deposits, fixed component of expenses, and the cash balance available for each bank. We take the average growth rate in loans and deposits over the last two quarters (March and June 2000) for the initial expected loan and deposit growth rates, while the expectations of Morgan Stanley analysts are used for the growth rates over the next four quarters. An average loans growth rate of 3.3% was expected for the Taiwanese financial industry in 2002. When estimating Taishin Bank’s loans growth rate, we can see that there was a high growth rate in nonmortgage consumer loans (24.5% of total loans), which was expected to continue growing in 2002. We forecast that its loans growth rate would be 7% per quarter. Dah An Bank, on the other hand, did not have any additional information on its loan growth rate, so this was estimated at 3%, based on analysts’ expectations. The growth rate in deposits has a high correlation with the currency rate and foreign interest rate, particularly with regard to the U.S. interest rate; hence, the currency remained under pressure during the continued decline of the U.S. interest rate in 2001. We forecast that the growth rate in deposits

Non-performing loan expenses

Operating expenses

Gross profit

Operating profit before taxes (EBITDA)

0.145 0.400 0.620 0.863 0.146 0.255 0.479 0.816 0.463 1.084 1.766 2.673 1.023 1.584 2.468 2.990 1.014 2.249 4.141

0.710 1.440 2.253 3.081 0.959 2.019 3.175 4.374 1.185 2.428 3.645 4.903 1.271 2.665 4.142 5.687 1.562 3.163 4.867

0.375 0.825 1.150 1.505 0.547 0.910 1.357 1.672 0.467 1.041 1.556 2.175 0.779 1.738 2.201 2.299 0.715 1.169 0.849

0.406 0.887 1.241 1.621 0.588 1.001 1.469 1.810 0.500 1.105 1.633 2.252 0.834 1.853 2.374 2.507 0.820 1.421 1.143

would be unchanged in 2002. Taishin Bank’s deposits growth rate was set at 6%, which was the same as its historical growth rate record, while the rate for Dah An Bank was set at 4%. The initial volatility of loans and deposits are the standard deviations of past changes in loans and deposits, with the initial volatility of the expected growth rate in loans and deposits being inferred from the observed stock price volatility. For the other parameters generated from the stochastic processes of loans and deposits, such as the long-term growth rates in loans and deposits, and the long-term volatility of loans, we use the industry average loans and deposits growth rate (2%) and the standard deviations (3%) as proxies. Finally, the mean-reversion coefficients are 7%.2 The cost parameters are classified as either fixed costs or variable costs, each of which is observed from the income statements of each bank. We find that the two main items of each banks’ expenses were financial activity expenses, and operating expenses. The financial activity expenses include interest expenses, NPL expenses and procedure fees; in our study we focus on a discussion of the net cash flow from the interest spread, so we do not take account of the procedure fees arising from other activities. The interest expense is considered as before. We also define two other items of cost, the first of which is the NPL expenses arising from bad loans, defining NPLs

2

All of the parameters are defined as in Schwartz and Moon (2000).

49 A Real Option Approach to the Comprehensive Analysis of Bank Consolidation Values

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Table 49.2 Dah An Commercial Bank, quarterly sales and costs, March 1997–September 2001 (unit NT$  10 million)

Date

Loans

Deposits

Interest income

March 1997 June 1997 September 1997 December 1997 March 1998 June 1998 September 1998 December 1998 March 1999 June 1999 September 1999 December 1999 March 2000 June 2000 September 2000 December 2000 March 2001 June 2001 September 2001

82:961 91:784 101:594 114:352 115:389 126:430 135:219 136:495 138:728 147:936 147:323 146:241 149:138 159:936 166:294 163:439 164:203 168:742 167:280

91:379 89:857 97:145 115:943 115:083 130:748 128:233 142:245 147:075 149:782 149:295 153:316 164:056 164:273 184:955 191:037 180:183 182:264 178:245

1:834 3:753 5:929 8:488 2:640 5:347 8:270 11:311 3:006 5:926 8:824 11:792 2:938 5:954 9:192 12:235 3:157 5:991 8:783

Interest expenses

Non-performing loan expenses

Operating expenses

1.270 2.583 4.012 5.764 1.904 3.877 5.986 8.150 2.159 4.273 6.324 8.270 2.040 4.196 6.565 8.717 2.374 4.562 6.567

0.101 0.116 0.324 0.623 0.065 0.113 0.228 0.909 0.284 0.645 1.001 5.235 0.319 0.398 0.493 0.920 0.593 1.008 1.703

0.434 0.891 1.366 1.917 0.527 1.098 1.685 2.286 0.604 1.218 1.719 2.204 0.536 1.095 1.668 2.261 0.560 1.137 1.703

Gross profit 0:344 0:755 1:169 1:480 0:473 0:889 1:336 1:102 0:324 0:552 0:813 2:572 0:571 1:224 1:732 0:400 0:343 0:677 0:441

Operating profit before taxes (EBITDA) 0:347 0:760 1:174 1:430 0:472 0:888 1:336 1:100 0:329 0:547 0:808 2:572 0:583 1:295 1:825 0:549 0:371 0:714 0:488

Table 49.3 Parameters used in the premerger base evaluation Initial estimations Parameters

Taishin International Bank

Dah An Commercial Bank

Loan parameters Initial loans L0 Initial expected loans growth rate (%) L 0 Initial volatility of loans (%) 0L Initial expected volatility of loans growth rate (%) L 0 Long-term loans growth rate (%)  NL Long-term volatility of loans growth rate (%) N L

NT$1,800 million/quarter 0.07/quarter 0.07/quarter 0.2082/quarter 0.02/quarter 0.03/quarter

NT$1,670 million/quarter 0.03/quarter 0.05/quarter 0.1882/quarter 0.02/quarter 0.03/quarter

Deposit parameters Initial Deposits D0 Initial expected deposits growth rate (%) D 0 Initial volatility of deposits (%) 0D Initial expected volatility of deposits growth rate (%) D 0 Long-term deposits growth rate (%)  ND Long-term volatility of deposits growth rate (%) N D

NT$2,290 million/quarter 0.06/quarter 0.08/quarter 0.2082/quarter 0.02/quarter 0.03/quarter

NT$1,780 million/quarter 0.04/quarter 0.07/quarter 0.1882/quarter 0.02/quarter 0.03/quarter

Cost parameters ˛ Variable component of expenses (as % of loans) F Fixed component of expenses

0.02/quarter NT$48 million/quarter

0.01/quarter NT$17 million/quarter

Other parameters Initial cash balance available X0

t Time increment for the discrete version of the model T Estimation horizon

NT$160 million/quarter one quarter 10 years

NT$220 million/quarter one quarter 10 years

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as a variable cost, which is a constant percentage of all loans. Taishin Bank’s NPL ratio was expected to fall to about 2% in 2002, so we assume its variable component of expenses to be 2% of all loans. Since the scale of loans at Dah An Bank was smaller, and the economic conditions looked to be better for the subsequent year, we assume its variable expenses component to be 1% of all loans. The second additional cost is operating expenses, which includes selling, general and administrative (SG&A) expenses, and other expenses, such as R&D expenses, training expenses and rental expenses and define these as a combined fixed expenses component. Net interest income is the difference between interest earned from loans and interest paid to depositors, and clearly, the change in interest rates on loans and deposits is a very important factor in estimating bank value. We use the average deposit rate as the initial interest rate for deposits and take the difference between the average lending rate and the deposit rate as the interest spread. These are acquired from the Taiwan Economic Journal (TEJ). As regards the setting up of the other interest rate parameters, we refer to Chen and Scott (1994). Other parameters that should be identified in the discrete model include the tax rate, which is the average rate of tax from the first quarter of 1997 to the third quarter of 2001;

these figures are available from the TEJ. Ten years was taken as the estimation horizon, and one quarter as the time increment. The correlation of each parameter was taken for the estimation of the parameters of each Wiener process, which was obtained from the historical data covering the period from the first quarter of 1997 to the third quarter of 2001.

49.3.2.2 The Consolidation Parameters A total of 39 parameters were required for the implementation of the second part of the model. As Table 49.4 shows, bank consolidation increases the scale of assets, and if we assume that all other things are equal, the initial integrated loans will be the sum of the two banks’ loans. In similar fashion, the initial deposits, following consolidation, will be equal to the sum of the two banks’ deposits. The drift and volatility terms of the loan and deposit stochastic processes are extremely important factors in the simulation of consolidation value; thus, we take the initial expected growth rate in loans and deposits as the value-weighted average of each bank’s initial expected loans and deposits growth rates. As in the previous section, in order to ensure that our model is more realistic, we consider the correlation between each bank, which involves the expected loans and deposits

Table 49.4 Parameters used in the consolidation base evaluation Parameters

Initial estimation

Loans parameters Initial Integrated Loans LM 0 Initial expected integrated loans growth rate (%) LM 0 Initial integrated loans volatility (%) 0LM Long-term integrated loans growth rate (%)  N LM Long-term volatility of integrated loans growth rate (%) N LM Correlation of expected loans growth rate of each bank (%) L

NT$3,470 million/quarter 0.05/quarter 0.07/quarter 0.02/quarter 0.03/quarter 0.58/quarter

Deposits parameters Initial Integrated Deposits D0M Initial expected growth rate in integrated deposits (%) DM 0 Initial volatility of integrated deposits (%) 0DM Long-term integrated deposits growth rate (%)  N DM Long-term volatility of integrated deposits growth rate (%) N DM Correlation of each bank’s expected rate of growth in deposits (%) D

NT$407 million/quarter 0.05/quarter 0.08/quarter 0.02/quarter 0.03/quarter 0.62/quarter

Cost parameters Variable expense cost saving factors ˛ MO Fixed expense cost saving factors ˛ MF Integrated fixed component of expenses FM

0.01/quarter 0.09/quarter NT$650 million/quarter

Other parameters Initial cash balance available X0M

t Time increment for the discrete version of the model T Estimation horizon Tax rate (%) c Risk-free interest rate (%) rf

NT$380 million/quarter one quarter 10 years 0.18/year 0.05/year

49 A Real Option Approach to the Comprehensive Analysis of Bank Consolidation Values

Table 49.5 Summary of the simulation results

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Pre-merger value Taishin Bank (NT$ bn)

Dah An Bank (NT$ bn)

Consolidation value (NT$ bn)

Ratio of value increase (%)

Schwartz and Moon (2000)

369.747

320.760

872.658

26.38

Longstaff and Schwartz (2001)

339.003

297.714

821.418

29.01

Method

growth rates. A correlation exists between each of the banks’ expected loans growth rate, with the initial value being assumed as the correlation between the two bank’s historical loans growth rates, in similar fashion to the correlation between each bank’s expected deposits growth rate. An additional major issue, with regard to the overall gains stemming from a merger, is the overall improvement in efficiency, with such efficiency gains coming from cost reductions. In our study, we assume that there are two distinct opportunities for cost savings: fixed cost saving factors and variable cost saving factors. The fixed cost saving factors are those arising from the reduction in the integrated bank’s fixed costs, vis-à-vis the sum of both banks prior to the merger. Analysts’ expectations, which were used for the initial fixed cost saving factors, estimated that the savings on operating expenses following the merger would be NT$3 billion to NT$4 billion; thus, the initial fixed cost saving factor was set at 0.09 per quarter. As was defined in the previous section, the variable cost saving factor is NPL expenses. Taishin FHC’s NPL ratio was expected to fall to 2.24% from 2.7% in 2002; hence, we assume the initial variable cost saving factor to be 0.01 per quarter. If we do not assume that an integrated bank that increases its market concentration may also increase its market power from setting prices on retail services,3 then we can assume that its initial deposit rate and interest spread would remain the same following the merger. Thus, the amount of cash available to the integrated bank would be NT$380 million, the sum of the initial available cash of the two banks.

49.4 Results After estimating the parameters, we use 100,000 Monte Carlo simulations to value the premerger and consolidated values, with the simulation results as follows:

49.4.1 The Fair Transaction Value of the Banks To simulate the value of each bank, we need to determine the bankruptcy assumption, first by using the Schwartz and Moon (2000) definition that a company will go into bankruptcy when the amount of available cash reaches zero. We then take the Longstaff and Schwartz (2001) “leastsquare method” (LSM) to determine the optimal stop point for each simulation path and to calculate the value of the bank. According to the LSM approach, it is more reasonable to determine the definition of a valuation stopping point as being that point where the conditional expected value is less than the cash available at T- t. We could regard the conditional expected value fitting under LSM as the expected continuing value of the bank at T- t., and then compare this to the actual cash available at the same time. Table 49.5 summarizes the simulation results of each method. Under the Schwartz and Moon base method, Taishin Bank’s value was NT$338.358 billion, while that of Dah An Bank was NT$315.542 billion. Since our intention here is to gain an accurate premerger perspective, our main concern is the increase in the value ratio; under this method, the consolidation value increase was 26.38%. Conversely, the LSM approach shows that Taishin Bank’s value was NT$339.003 billion, versus Dah An Bank’s NT$297.714 billion. Although each bank’s value is much the same under the LSM approach as it is under the base method, the increased value is, nevertheless, more significant. We relax the restriction that the cash available stochastic process may reach zero by simulation; therefore, the bank could not be seen as bankrupt if its expected continuing value was still greater than zero. Table 49.5 also clearly indicates that the ratio of the value increase is much more conspicuous under the LSM approach.

49.4.2 Bankruptcy and Stopping Points 3

Berger et al. (1999) found that mergers and acquisitions among institutions that have significant local market overlap, ex ante, may increase local market concentration and allow the consolidated firm to raise profits by setting less favorable prices to customers. This may affect rates and fees on retail deposits and small business loans.

Table 49.6 shows the base valuation method, following Schwartz and Moon (2000), in which the bankruptcy condition is defined as the point at which the cash available at each

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Table 49.6 Probability of bankruptcy per year, for base valuation (unit: %) Year Taishin Bank Dah An Bank Consolidation 1 2 3 4 5 6 7 8 9 10 Total

1.17 11.61 9.66 6.48 4.68 3.92 3.05 2.68 2.25 0.00 45.50

0.00 1.01 5.78 6.53 5.33 4.31 3.51 3.04 2.38 0.00 31.89

0.00 0.02 0.11 0.66 1.40 2.13 2.22 2.58 2.41 0.00 11.53

Table 49.7 Probability of stop points per year, by LSM approach (unit: %) Year Taishin Bank Dah An Bank Consolidation 1 2 3 4 5 6 7 8 9 10 Total

2.51 6.35 9.87 7.45 5.91 4.69 5.06 6.52 8.38 0.00 56.74

2.14 5.81 7.85 6.58 5.35 5.72 6.41 8.56 10.25 0.00 58.67

0.45 2.28 4.59 7.46 8.57 8.62 9.32 11.27 10.83 0.00 63.39

discrete time reaches zero. Note that in the beginning, the bankruptcy points per year for Taishin International Bank are greater than those for Dah An Commercial Bank, particularly in Year 2; however, the probability of bankruptcy for Taishin International Bank decreases rapidly after Year 4, when it becomes less than the probability of bankruptcy for Dah An Commercial Bank. The consolidated bankruptcy points start in Year 3, and by Year 8 the probability of bankruptcy for the consolidated bank is much lower than that of each individual bank. Table 49.7 provides details of the simulated distributions of the banks’ stopping points under the LSM approach, which is developed to determine the stop points for each simulation path, as discussed above. Note that the optimal stopping points for each bank, for each year, are closer than those under the base valuation. We also find that the total number of stopping probabilities for Dah An Commercial Bank was higher than that for Taishin International Bank in these simulation results. This result is more likely to accord with the real-world situation.

49.4.3 Sensitivity Analysis In this section we perform a sensitivity analysis to the most critical parameters for the total value of the consolidated bank – obtaining the bank values by use of a perturbation (a 10% higher value) for the indicated parameter, while all other parameters remain the same as in the base valuation of the consolidated bank value. The sensitivity analysis results, by base valuation, are reported in Table 49.8, where it is shown that three sets of parameters have a significant effect on the value of the bank. The first of these is the parameter for the fixed and variable cost components. Although the equations indicate that the two components have the same effect on the cost function, in the simulation results the variable component increases by 1%, while the consolidated bank value decreases by 1.73%, which was more significant than the 1% increase in the fixed component (Table 49.8). Second, an increase in the initial integrated loans and deposits growth rate, from 5% per quarter to 5.5% per quarter (a 10% increase) is also significant in the consolidated bank valuations. We find that the increase in the initial loans growth rate has a positive relationship with the valuation of the consolidated bank. Conversely, a negative relationship exists between the growth rate in deposits and the consolidated bank, because an increase in the deposits growth rate would lead to a rise in interest expenditure if the loans growth rate was unchanged. The distribution of loans and deposits growth rate variance is also important in the valuation because it determines the option value of growth, implying a higher probability of growth opportunities (Table 49.8). We find that such an increase in the standard deviation of both loans and deposits would lead to a rise in the value of the consolidated bank.

Table 49.8 Sensitivity of consolidation bank value meters Total Value of consolidation perturbed bank value parameter .NT$  1;000/ Parameter (per quarter)

to change para-

Base case LM 0 DM 0 0LM 0DM LM DM  LM  DM k1 k2 ˛ MO ˛ MF

–

– 0.055 0.055 0.077 0.088 0.022 0.022 0.033 0.033 0.077 0.077 0.011 0.099

868,031,438 913,308,472 846,838,917 875,041,757 868,454,744 873,199,259 865,663,870 868,370,641 868,218,622 867,598,615 867,760,803 853,037,257 866,134,239

Increase/ decrease on base case (%) 5:22 2:44 0:81 0:05 0:60 0:27 0:04 0:02 0:05 0:03 1:73 0:22

49 A Real Option Approach to the Comprehensive Analysis of Bank Consolidation Values

Table 49.9 Sensitivity of consolidation bank value to change parameters, by LSM Approach Total Value of consolidation perturbed Increase/decrease bank value parameter on base case (%) .NT$  1; 000/ (per quarter) Parameter Base case LM 0 DM 0 0LM 0DM LM DM  LM  DM k1 k2 ˛ MO ˛ MF

– 0.055 0.055 0.077 0.088 0.022 0.022 0.033 0.033 0.077 0.077 0.011 0.099

817,088,184 858,654,309 798,235,081 823,142,437 817,259,208 821,538,806 815,200,412 817,412,390 817,321,305 817,060,177 816,829,042 802,662,521 815,223,433

– 5:09 2:31 0:74 0:02 0:54 0:23 0:04 0:03 0:00 0:03 1:77 0:23

The third parameter involves an issue that is not so obvious, since the set of parameters that are generated from the mean-reversion process have an effect on the value of the consolidated bank. An increase in the mean-reversion coefficient in integrated loans and deposits, k1 and k2 (as in Table 49.8), would lead to a decline in the value of the consolidated bank, while an increase in the long-term growth rates of the integrated loans would lead to an increase in the overall value of the consolidated bank, which runs contrary to the increase in long-term growth rates of the integrated deposits. Table 49.9 shows the sensitivity of the total value of the consolidated bank under the LSM approach. We find that the results are similar to those in the earlier discussion.

49.5 Conclusions In this study, we apply a modification of the Schwartz and Moon (2000) model to the evaluation of bank consolidation. From our examination of a bank merger case study (the first example of such a bank merger in Taiwan), we find that, from an ex-ante viewpoint, the consolidation value is, on average, about 30% of the original total value of the independent

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banks. We also find that the probability of bankruptcy was considerably lower following the merger than it would have been prior to the merger. Our case study therefore indicates that the merger was indeed a worthwhile venture for both banks involved. Furthermore, on completion of the merger, we also find that, in terms of the magnitude of the increased consolidation value, the most critical roles are played by the resultant changes in the growth rates of the integrated loans and the integrated deposits, as well as the cost-saving factors within the cost functions. Acknowledgments The earlier version of this paper was presented at National Central University, National Taiwan University. We are especially grateful for Professor Pin-Hung Chou’s and San-Lin Chung’s helpful comments.

References Berger, A. N., R. S. Demsetz, and P. E. Strahan. 1999. “The consolidation of the financial services industry: causes, consequences and implications for the future.” Journal of Banking and Finance 23(2–4), 135–194. Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 81, 637–659. Boyle, P. 1997. “Options: a monte carlo approach.” Journal of Financial Economics 4, 323–338. Brennan, M. J. and E. S. Schwartz. 1982. “Consistent regulatory policy under uncertainty.” Bell Journal of Economics 13(2), 507–521. Cox, J. C., J. E. Ingersoll, and S. A. Ross. 1985. “A theory of term structure of interest rate.” Econometrica 53, 385–407. Longstaff, F. A. and E. Schwartz. 2001. “Valuing American options by simulation: a simple least-squares approach.” Review of Financial Studies 14(1), 113–147. Kaplan, S. and R. Ruback. 1995. “The valuation of cash flow forecasts: an empirical analysis.” Journal of Finance 50(4), 1059–1093. Panayi, S. and L. Trigeorgis. 1998. “Multi-stage real options: the cases of information technology infrastructure and international bank expansion.” Quarterly Review of Economics and Finance 38 (Special Issue), 675–692. Rhoades, S. A. 1993. “Efficiency effects of horizontal (in-market) bank mergers.” Journal of Banking and Finance 17, 411–422. Rhoades, S. A. 1998. “The efficiency effects of bank mergers: an overview of case studies of nine mergers.” Journal of Banking and Finance 22, 273–291. Ryngaert, M. D. and J. F. Houston. 1994. “The overall gains from large bank mergers.” Journal of Banking and Finance 18, 1155–1176. Schwartz, E. and M. Moon. 2000. “Rational pricing of internet companies.” Financial Analysts Journal 56, 62–75. Schwartz, E. and M. Moon. 2001. “Rational pricing of internet companies revisited.” University of California at Los Angeles, (Revised April 2001).

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Appendix 49A The Correlations Between the Standard Wiener Process Generated from a Bank’s Net Interest Income The relationships are as follows: dz1 dz2 dz1 dz5 dz2 dz4 dz3 dz4 dz4 dz5

D Lg Ldt D L dt D g Lg Ddt D Dg D dt D g D dt

dz1 dz3 dz1 dz6 dz2 dz5 dz3 dz5 dz4 dz6

D LD dt D LS dt D g L dt D D dt D g DS dt

dz1 dz4 dz2 dz3 dz2 dz6 dz3 dz6

D Lg Ddt D Dg Ldt D g LS dt D DS dt

where LgL is the correlation between loans and the loan growth rate; LD is the correlation between loans and deposits; LgD is the correlation between the loan growth rate and deposits; L is the correlation between loans and the deposit rate; LS is the correlation between loans and the interest spread; DgL is the correlation between deposits and the loan growth rate; gL gD is the correlation between the loan growth rate and the deposit growth rate; gL  is the correlation between the loan growth rate and the deposit rate; gL S is the correlation between the loan growth rate and the interest spread; DgD is the correlation between deposits and the deposit growth rate; D is the correlation between deposits and the deposit rate; DS is the correlation between deposits and the interest spread; gD  is the correlation between the deposit growth rate and the deposit rate; and gD S is the correlation between the deposit growth rate and the interest spread.

Appendix 49B The Risk-Adjusted Processes The model has six sources of uncertainty: (1) the uncertainty surrounding the changes in loans; (2) the uncertainty with regard to the expected loans growth rate; (3) the uncertainty surrounding the changes in deposits; (4) the uncertainty with regard to the expected deposit growth rate; (5) the uncertainty surrounding the changes in the average deposit rate; and (6) the uncertainty surrounding the changes in the interest spread. Following Brennan and Schwartz (1982), we have some simplifying assumptions: the risk-adjusted processes for the state variables can be obtained from the true processes, and the loans and deposits of the risk-adjusted processes are as follows:

  L L  dLt D L t  1 t Lt dt C  t Lt d z1     L L L  N L  L d—L t D k t  3 t dt C t d z2   D Dt dt C  D t Dt d z3 dDt D D t  2 t     D D  N D  D  4 D d—D t D k t t dt C t d z4 ; where the market prices of the factor risks, 1 ; 2 ; 3 and 4 are constants. Similarly, we have to adjust the interest rate to be riskneutral, as follows:  p  p drt D a.b  rt / C 5 t rt dt C t rt d z5 h p p i dSt D aS .b S  St / C 6 tS S t dt C tS St d z6 ; where 5 and 6 are the market prices of the factor risks.

Appendix 49C The Discrete Version of the Risk-Adjusted Process In order to implement the model developed in the previous section, we have to rewrite the model as a discrete-time version. For example, we use the loans and loan growth rate of each bank under the risk-adjusted process, as follows: ("

Lt C t D Lt  e L t C t

De

k L  t

L t 1 



L t

s

.tL / 2



2

#

p

t CtL  t"1

C 1e

k L  t

)



3 L t  N L  L k

1  e 2k L  t L p  t  t  "2 2k L

L L tL D 0L  e k1 t CN L 1  e k1 t C

L

L k2 t L : t D 0  e

where 0L and L 0 are the initial variance values in the loans and growth rate, and "1 and "2 are drawn from a standard normal distribution with correlation between the loans and growth rate. Under the risk-adjusted process, the deposits and deposit growth rate of each bank are similar to the loans process.

Chapter 50

Dynamic Econometric Loss Model: A Default Study of US Subprime Markets C.H. Ted Hong

Abstract The meltdown of the US subprime mortgage market in 2007 triggered a series of global credit events. Major financial institutions have written down approximately $120 billion of their assets to date and yet there does not seem to be an end to this credit crunch. With traditional mortgage research methods for estimating subprime losses clearly not working, revised modeling techniques and a fresh look at other macroeconomic variables are needed to help explain the crisis. During the subprime market rise/fall era, the levels of the house price index (HPI) and its annual house price appreciation (HPA) had been deemed the main blessing/curse by researchers. Unlike traditional models, our Dynamic Econometric Loss (DEL) model applies not only static loan and borrower variables, such as loan term, combined-loan-to-value ratio (CLTV), and Fair Isaac Credit Score (FICO), as well as dynamic macroeconomic variables such as HPA to project defaults and prepayments, but also includes the spectrum of delinquencies as an error correction term to add an additional 15% accuracy to our model projections. In addition to our delinquency attribute finding, we determine that cumulative HPA and the change of HPA contribute various dimensions that greatly influence defaults. Another interesting finding is a significant long-term correlation between HPI and disposable income level (DPI). Since DPI is more stable and easier to model for future projections, it suggests that HPI will eventually adjust to coincide with the DPI growth rate trend and that HPI could potentially experience as much as an additional 14% decline by the end of 2009. Keywords Dynamic econometric loss model r US subprime markets

50.1 Introduction Subprime mortgages are made to borrowers with impaired or limited credit histories. The market grew rapidly when C.H. Ted Hong () Beyondbond, Inc., USA e-mail: [email protected]

loan originators adopted a credit scoring technique like FICO to underwrite their mortgages. A subprime loan is typically characterized by a FICO score between 640 and 680 or less vs. the maximum rating of 850. In the first half of the decade, the real estate market boom and well-received securitization market for deals including subprime mortgages pushed the origination volume to a series of new highs. In addition, fierce competition among originators created various new mortgage products and a relentless easing of loan underwriting standards. Borrowers were attracted by new products such as “NO-DOC, ARM 2/28, IO” that provided a low initial teaser rate and flexible interest-only payments during the first 2 years, without documenting their income history. As the mortgage rates began to increase during the summer of 2005 and housing activity revealed some signs of a slowdown in 2006, the subprime market started to experience some cracks as delinquencies began to rise sharply. The distress in the securitization market backed by subprime mortgages and the resulting credit crisis had a ripple effect initiating a series of additional credit crunches. All this pushed the US economy to the edge of recession and is jeopardizing global financial markets. The rise and fall of the subprime mortgage market and its ripple effects raise a fundamental question. How can something as simple as subprime mortgages, which accounts for only 6–7% of all US mortgage loans, be so detrimental to the broader economy as well as to the global financial system? Before formulating an answer to such a large question, we need to understand the fundamental risks of subprime mortgages. Traditional valuation methods for subprime mortgages are obviously insufficient to measure the associated risks that triggered the current market turmoil. What is the missing link between traditional default models and reality? Since a mortgage’s value is highly dependent on its future cash flows, the projection of a borrower’s embedded options becomes essential to simulate its cash flows. Studying consumer behavior to help project prepayments and defaults (call/put options) of a mortgage is obviously the first link to understanding the current market conditions. This paper focuses on modeling the borrower’s behavior and resultant prepayment or default decision. A Dynamic

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_50, 

779

780

Econometric Loss (DEL) model is built to study subprime borrower behavior and project prepayment and default probabilities based on historical data from Loan Performance’s subprime database (over 17 million loans) and prevailing market conditions from 2000 to 2007. The paper is organized in the following manner. We start by constructing a general model framework in a robust functional form that is able to not only capture the impact of individual model determinants, but is also flexible enough to be changed to reflect any new macroeconomic variables. We then modeled default behavior through an individual factor fitting process. Prepayment modeling follows a similar process with consideration of the dynamic decision given prior prepayment and default history. The delinquency study builds the causality between default and delinquencies and the relationship within the spectrum of different delinquencies. We then utilized the delinquencies as a leading indicator and error correction term to enhance the predictability of the forecasted defaults by 15%. Our findings and forthcoming research are then drawn in the conclusion section (Fig. 50.1).

50.2 Model Framework When a lender issues a mortgage loan to its borrower, the loan is essentially written with two embedded American options with an expiration co terminus with the life of the loan. The lender will then receive payments as compensation for underwriting the loan. The payments will include interest, amortized principal and voluntary/involuntary prepayments along with any applicable associated penalties. The risk for lenders is that they might not receive the contractual payments and will need to go after the associated collateral to collect the salvage value of the loan. Additionally, the foreclosure procedure could be costly and time consuming. Unscheduled payments come in two forms. A voluntary prepayment is usually referred to simply as “prepayment” and an involuntary prepayment is known as “default” (with lags to potentially recover some portion of interest and principal proceeds). Prepayment is nothing but a call option on some or all of the loan balance plus any penalties at a strike price that a borrower has the right to exercise if the option is in-the-money. By the same token, default is a put option with the property’s market value as the strike price to the borrower. Understanding the essence of both options, we need to find the determining factors that trigger a borrower to prepay/default through filtering the performance history of the loan. A list of determinant factors regarding consumer behavior theory for modeling default and prepayment will be discussed in the next two sections. In order to construct a meaningful statistical model framework for empirical work, the availability of data and the data

C.H. Ted Hong

structure are essential. In other words, our model framework is designed to take full advantage of Loan Performance’s subprime mortgage historical information and market information. The model empirically fits to the historical default and prepayment information of US subprime loan performance from 2000 to 2007 (more than 17 million loans) (Fig. 50.2). Mathematically, our general framework constructs the default and prepayment rates as two separate functions of multiple-factors where the factors are categorized into two types – static and dynamic.1 The static factors are initially observable when a mortgage is originated such as borrower, characteristics and loan terms. Borrower characteristics include CLTV, FICO, and debt-to-income ratio (DTI). Loan terms include loan maturity, loan seasoning, original loan size, initial coupon reset period, interest only (IO) period, index margin, credit spread, lien position, documentation, occupancy, and loan purpose. The impact to the performance of a loan from the static factors provides the initial causality relationship, yet their influence may diminish or decay as the information ceases to be up to date. Dynamic factors include several macroeconomic variables such as HPA, prevailing mortgage interest rates, consumer confidence, gross disposable income, employment rate, and unemployment rate. These dynamic factors supply up-to-date market information and thus play an important role in dynamically capturing the market impact. The accuracy of capturing causality relationship due to the static factors and the predictability of the dynamic factors presents a constant challenge during the formulation of this model. For each individual factor, a non-linear function is formulated according to its own characteristics. For example, a “CLTV” factor for modeling default is formulated as the function of default rate over CLTV ratio. However, a DOC factor is formulated as the function of multiplier over discrete variables of “FULL” vs. “LIMITED” with percentages of respective groups. A general linear function of combined multifactor functions is then constructed as a basic model framework to fit the empirical data and to project forecasts for prepayments and defaults.2 In the following sections, we will discuss each factor in detail.

1

There is no industry standard measure for default rate, thus a different definition of default rate will give a very different number. As there is no set standard, we define our default rate to be based on the analysis in this paper, “Loss Severity Measurement and Analysis,” The MarketPulse, LoanPerformance, 2006, Issue 1, 2–19. Please refer to Appendix I for definition of default used throughout this paper. 2 See Appendix II for the details of model specification.

50 Dynamic Econometric Loss Model: A Default Study of US Subprime Markets 07/17 News Corp. reaches a tentative agreem ent to acquire Dow Jones at its original offer price of $60 a share.

01/18 A real-estate consortium unveils a $21.6 billion offer for Equity Office properties Trust.

07/20 The Dow industrials cross the 14000 m ilestone for the first time.

02/10 Fortress Investment Group LLC’s shares surge 68% in their debut to finish at $31.

14,000

06/13 U.S. bond yields hit a five-year high as inventors continue to sell Treasurys, with the yield on the benchm ark 10-year not rising to 5.249% .

03/09 New Century Financial Corp.’s creditors force the subprim e-m ortagage lender to stop m aking loans am id rising defaults.

10/25 Merrill posts a $2.24 billion loss as a larger-than-expected $8.4 billion write-down on m ortgage-related securities leaves the firm with its first quarterly deficit since 2001.

07/25 Countrywide Financial Corp. says profit slips 33%, dragged down by losses on certain types of prime mortgage loans.

06/22 Blackstone Group LP’s IPO is priced at $31 a share, raising as much as $4.6 billion.

04/26 ABN Am ro Holding NV receives a $98.58 billion takeover approach from a group led by Royal Bank of Scotland Group PLC.

14,500

781

11/27 Citigroup, seeking to restore investor confidence am id massive losses in the credit markets and a lack of perm anent leadership, receives a $7.5 billion capital infusion; 11/27 HSBC’s SIV bailout will move 2 SIV’s of $45 billion to its balance sheet.

08/11 Central banks pum p m oney into the financial system for a second day to ease liquidity strains.

13,500 10/27 Countrywide Financial Corp. posts its first quarterly loss in 25 years on about $1 billion in writedowns.

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12/21 Bear Stearns posts a loss of $854 million, the first in its 84-year history. The firm takes a $1.9 billion write down.

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08/14 Goldm an Sachs Group Inc. says three of its hedge funds have seen the net value of their assets fall about $4.7 billion this year.

06/23 Bear Stearns Cos. Agrees to lend as m uch as $3.2 billion to one of its own troubled hedge funds.

12/14 Citigroup bails out seven affiliated structuredinvestm ent vehicles, or SIVs, bring $49 billion in assets onto its balance sheet and further denting its capital base.

07

12,500

08/17 Blue chips fall more than 300 points at 845.78 after foreign m arkets tum ble on certain that U.S. credit problems could trigger a global slowdown; 08/17 Countrywide taps an $11.5 billion credit line in a bid to shore up its finances, but its stock falls 11% .

9/

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07/18 Bear Stearns Cos. Says two hedge funds it runs are worth nearly nothing.

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03/23 Blackstone files for an IPO to raise about $4 billion

What happened in 2008 7/31 The Senate pass the $300 billion Housing and Economic Recovery Act of 2008, which is designed to help struggling homeowners retain ownership. Signed by President Bush, the law goes into effect Oct. 1.

9/8 US rescue fires up markets. EUPHORIC 13000 investors are using the US Government’s historic bail-out 12500 of mortgage lenders Fannie Mae and Freddie Mac to 12000 justify a buying frenzy on share markets worldwide.

11500

11000

10500 Jan 2008 1/22 The largest rate cut since October 1984 by the U.S. Federal Reserve sends Wall Street lower today as Fed Chairman Ben Bernanke and other central bankers show they’re deeply worried about the immediate future of the U.S. economy.

Feb 2008

Mar 2008

3/14 Bear Stearns liquidity problem emerges, stock price goes down from $60 to $30 per share.

3/16 Shotgun wedding of Bear Stearns to J.P. Morgan for $2 a share, a bargain-basement $236.2 million. Federal Reserve bank to provide financing for the deal.

Apr 2008 7/11 FDIC takes over IndyMac Bank, the Second-largest bank failure in U.S. history. Iran missile launches send oil to $147 per barrel record.

May 2008

8/27 FDIC says “Problem bank list” in 2nd quarter grows to 117 with $78 billion in assets - up from 90 banks, $26 billion in assets in 1st quarter.

Fig. 50.1 What happened in 2007; What happened in 2008

Jun 2008

7/14 Fannie Mae & Freddie Mac post losses, stock prices both drop approximately 80%.

Jul 2008

9/3 Ospraie, the largest commodity hedge fund firm, closes to its biggest hedge fund after slumping 38% this year because of bad bets on commodity stocks. Oil plunges to 5month low at $105 per barrel.

9/8 South Korea’s financial regulator urges state-owned Korea Development Bank (KDB) to be cautious over any investment in Lehman Brothers.

Aug 2008

Sep 20

9/17 Fed agrees to take over the insurance giant AIG, an unprecedented $US85 billion bail-out. 9/15 Lehman expects to file for bankruptcy protection, the largest failure of an investment bank since the collapse of Drexel Burnham Lambert 18 years ago. Merrill agrees to sell itself to Bank of America Corp for $44 billion after more than $40 billion credit losses over the last year, Dow Jones Sinks 504 Points for Worst One Day Loss Since 9/11.

782

C.H. Ted Hong Type / Orig. Year ARM OTHER 2000 11,452 2001 11,389 2002 33,776 2003 51,548 2004 221,818 2005 496,697 2006 490,975 2007 99,946 Grand Total 1,417,601

ARM2/28 187,232 261,316 434,732 697,073 1,239,522 1,577,003 1,137,345 161,480 5,695,703

ARM3/27 68,430 67,018 100,939 164,228 413,366 393,020 234,344 36,795 1,478,140

ARM5/25 4,059 10,449 25,827 71,839 213,572 301,829 349,460 160,549 1,137,584

FIXED Grand Total 390,671 661,844 477,718 827,890 605,233 1,200,507 958,170 1,942,858 1,172,413 3,260,691 1,619,257 4,387,806 1,754,382 3,966,506 404,278 863,048 7,382,122 17,111,150

Fig. 50.2 Number of securitized Alt-A and subprime mortgage origination

50.3 Default Modeling

20 2003ACT 2003Prj 2004ACT 2004Prj 2005ACT 2005Prj

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Default Modeling Factor Components

Refinance Cashout

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Occupancy Owner Second home Investor Property Type Single-Family Multi-Family Condo Loan Documentation Full Limited House Price Appreciation (HPA) State Level CBSA Level

14 CDR (%)

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12 10 8 6 4 2 0 02/03 08/03 02/04 08/04 02/05 08/05 02/06 08/06 02/07 08/07 Date

Fig. 50.3 Seasoning: CDRs by date and vintages of ARM 2/28 20 18

2003ACT 2003Prj 2004ACT 2004Prj 2005ACT 2005Prj

16

50.3.1 Seasoning

CDR (%)

14 12 10 8

Loan information regarding borrower’s affordability is usually determined at origination. As a loan seasons, its original information decays, and its default probability starts to surge. A seasoning baseline curve with annualized Constant Default Rate (CDR) against its seasoning age would post a positive slope curve for the first 3 years. Figure 50.3 shows actual CDR curves and their fitted result for different vintages of ARM 2/28 mortgage pools. They roughly follow a shape similar to the Standard Default Assumption (SDA) curve.3 However, as shown in Fig. 50.4, the ramp-up curve can be very different for different vintages.

3

SDA is based on Federal Housing Administration (FHA)’s historical default rate and was developed by Bond Market Association (BMA), now known as Securities Industry and Financial Markets Association (SIFMA).

6 4 2 0 0

6

12

18 24 Age (month)

30

36

Fig. 50.4 Seasoning: CDRs by age and vintages of ARM 2/28 (Source: Beyondbond Inc, LoanPerformance)

50.3.1.1 Why Is the 2005 Seasoning Pattern Faster Than Prior Vintages? Since the seasoning baseline curve is not independent of dynamic factors, a dynamic factor such as HPA could tune vintage seasoning curves up and down. In Fig. 50.4, the 2005 seasoning pattern is significantly steeper than its prior

50 Dynamic Econometric Loss Model: A Default Study of US Subprime Markets 25

vintages. Looser underwriting standards and deteriorating credit fundamentals can be important reasons. Negative HPA obviously starts to adversely impact all vintages after 2005.

20 CDR (%)

50.3.2 Payment Shock – Interest Only (IO)

783

ACT_NIO Prj_NIO ACT_IO Prj_IO

15 10

The boom in the subprime market introduced new features to the traditional mortgage market. An ARM 2/28 loan with a 2-year interest-only feature has a low fixed initial mortgage rate and pays no principal for the first 2 years prior to the coupon reset.4 When the IO period ends, the borrower typically faces a much higher payment based on its amortized principal plus the fully indexed interest amounts. This sudden rise in payments can produce a “Payment Shock” and test the affordability to borrowers. Without the ability to refinance, borrowers who are either under a negative equity situation or not able to afford the new rising payment will have a higher propensity to default. Consequently, we see a rapid surge of default rates after the IO period. The ending of the IO period triggers payment shock and will manifest itself with a spike in delinquency.5 Delinquent loans eventually work themselves into the defaulted category within a few months after the IO period ends. Figure 50.5 shows the different patterns and the default lagging between IO and Non-IO ARM 2/28 pools.

50.3.3 Combined Loan-to-Value (CLTV) LTV measures the ratio of mortgage indebtedness to the property’s value. When multiple loans have liens added to the indebtedness of the property, the resulting ratio of CLTV becomes a more meaningful measure of the borrower’s true equity position. However, the property value might not be available if a “market” property transaction does not exist. A refinanced mortgage will refer to an “appraisal value” as its property value. Note that “appraisal value” could be manipulated during ferocious competition among lenders in a housing boom market and undermine the accuracy of CLTV. 4

The reset is periodical, and the interest rate is set as Index C Margin. The delinquency rate is measured by OTS (Office of Thrift Supervision) or MBA (Mortgage Bankers Association) convention. The difference between these two measures is how they count missed payments. MBA delinquency rate counts the missed payment at the end of the missing payment month while OTS delinquency rate counts the missed payment at the beginning of the following month after missing payment. This difference will pose a 1–30 days delay of record. OTS delinquency rate is the prevailing delinquency measure in subprime market.

5

5 0 02/04 08/04 02/05 08/05 02/06 08/06 02/07 08/07 Date

Fig. 50.5 IO payment shock: CDRs by date of ARM 2/28

As we know, default is essentially a put option embedded in the mortgage for a borrower. In a risk neutral world, a borrower should exercise the put if the option is in-the-money. In other words, a rational borrower should default if the CLTV is greater than one or if the borrower has negative equity. At higher CLTVs, it becomes easier to reach a negative equity level as the loan seasons and its default probability increases. Figure 50.6 provides the actual stratification result of CDR over various CLTV ranges. Obviously, CDR and CLTV are positively correlated. In addition, lower CDR values are observed for higher subprime tiered FICO ranges. This shows that the FICO tier granularity is another important factor in modeling. However, since CLTV is obtained at the loan’s origination date, it does not dynamically reflect housing market momentum. We introduce a dynamic CLTV that includes housing price appreciation from loan origination in order to estimate more precisely the actual CLTV. This dynamic CLTV allows us to better capture the relationship between CLTV and default. Figure 50.7 clearly illustrates that different CLTV groups show a different layer of risk level.

50.3.4 FICO FICO score is an indicator of a borrower’s credit history. Borrowers with high FICO scores maintain a good track record of paying their debts on time with a sufficiently long credit history.6 6 According to Fair Isaac Corporation’s (The Corporation issued FICO score measurement model) disclosure to consumers, 35% of this score is made up of punctuality of payment in the past (only includes payments later than 30 days past due), 30% is made up of the amount of debt, expressed as the ratio of current revolving debt (credit card balances,

784

C.H. Ted Hong CDR vs. CLTV of ARM 2/28 Non-IO with age>24

CDR vs. CLTV of ARM 2/28 Non-IO with age >24 and FICO between 641 and 680

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Source: Beyondbond Inc, LoanPerformance

Fig. 50.6 Stratified seasoned CDR over CLTV ranges (Source: Beyondbond Inc, LoanPerformance)

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Source: Beyondbond Inc, LoanPerformance

Fig. 50.7 CDRs by date and CLTVs of ARM 2/28 (Source: Beyondbond Inc, LoanPerformance)

In recent years, people have come to believe that FICO is no longer an accurate indicator due to the boom in hybrid ARM loans and fraudulent reporting to the credit bureaus. With refinancing becoming much easier to obtain, issuers have been giving out tender offers to borrowers in order to survive the severe competition among lenders. CLTV and FICO scores are two common indicators that the industry uses to predict default behavior.7 We examine etc.) to total available revolving credit (credit limits), and 15% is made up of length of credit history. Severe delinquency (30 plus) and credit history length make up 50% of the FICO score. This score reflects people’s willingness to repay. It’s essentially the probability distribution for people’s default activity on other debts such as credit card and/or utility bills, etc. Statistically speaking, people with higher FICO scores will have lower probability to default. 7 Debt-to-Income ratio is also an important borrower characteristic, but in recent years, more Limited-Doc or/and No-Doc loans have been

the combined CLTV and FICO effects on CDR as shown in Fig. 50.8. The figure presents a 3-D surface of stratified CDR rates over CLTV and FICO ranges from two different angles for seasoned ARM 2/28 pools. The relationship between CLTV and CDR is positively correlated across various FICO ranges. On the other hand, the relationship between FICO and CDR is somewhat negatively correlated across various CLTV ranges. However, the case is not as significant. FICO’s impact is obviously not as important as we originally expected. In our analysis, CLTV D 75 and FICO D 640 serve as the base case, and then we adjust the CDR according to movements of other default factors. Figure 50.9 gives an example of fitting results based on ARM 2/28 2004 vintage pools. The difference between 600– 640 and 680–700 FICO ranges makes only a small difference of 1% in CDR for a seasoned pool.

50.3.5 Debt-to-Income Ratio (DTI) and Loan Documentation (DOC) The DTI in this paper is defined as the back-end DTI, which means the debt portion for calculating the DTI ratio includes not only PITI (Principal C Interest C Tax C Insurance) but also other monthly debts such as credit card payments, auto loan payments and other personal obligations.8 The DTI

issued. For these loans, many of them do not have DTI ratio report, so we consider DTI separately for different DOC type. 8 There are two major measures of DTI in the industry: Front-End DTI ratio D PITI/Gross Monthly Income, and Back-End-DTI ratio D PITI C Monthly Debt/Gross Monthly Income. PITI D Principle C Interest C Tax C Insurance.

50 Dynamic Econometric Loss Model: A Default Study of US Subprime Markets CDR vs. FICO and CLTV of Seasoned ARM 2/28

CDR vs. FICO and CLTV of Seasoned ARM 2/28

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Fig. 50.33 Wealth: CPR over various CLTV of ARM2/28 (Sources: Beyondbond Inc, LoanPerformance)

We can see a combined effect of FICO and CLTV on CPR. Those people who have a low CLTV and a high FICO score can easily refinance and will have the highest prepayment rate; while people who have high CLTV and low FICO score will be on the other side of the pendulum with the lowest prepayment rate. Figure 50.36 gives a sample CPR fitting result based on ARM 2/28, 2004 vintage pools.

796

C.H. Ted Hong 70

FICO 641-680 of ARM2/28, 2004 vintage LTV81ACT

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Fig. 50.34 Fitted CPR over CLTV 81–90 of ARM2/28, 2004 vintage (Sources: Beyondbond Inc, LoanPerformance)

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Fig. 50.36 Credit: fitted CPR (Sources: Beyondbond Inc, LoanPerformance)

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Fig. 50.37 CPR over various vintages of ARM2/28 (Sources: Beyondbond Inc, LoanPerformance)

50.4.9 Prepayment Penalty

50.4.10 Interaction Between Prepayment and Default

A prepayment penalty fee in the loan structure is a negative incentive to refinance and deters prepayment. Prepayment is essentially an embedded call option with the remaining balance as its strike. The penalty simply adds to that strike price as an additional cost when borrowers exercise the option. That additional cost will be reduced to zero when the penalty period ends. Figure 50.37 shows the prepayment difference when a penalty clause is in place. Before the 2-year penalty period ends, prepayment is consistently slower than no-penalty loans. As soon as the penalty period ends, prepayments surge dramatically and surpass the no-penalty loans within 3 months and consistently maintain a faster prepayment speed thereafter.

As we stated in the beginning of the model framework, prepayment and default can be viewed as embedded call and put options, respectively, on the mortgage. A borrower will continuously find incentives to exercise the option if it is inthe-money (Fig. 50.38). When we estimate prepayment and default for a pool of mortgages, the remaining principal factor encompasses the entire history of the pool’s prepayment and default rates. Since estimating losses is a main focus of modeling default and prepayment, it is of particular importance in a slow prepayment environment. Given the same default probability, the tail risk to the loss curve will still increase

50 Dynamic Econometric Loss Model: A Default Study of US Subprime Markets

80

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CDR (%)

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Fig. 50.39 Loss projection of ARM2/28, 2004 vintage (Sources: Beyondbond Inc, LoanPerformance)

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Fig. 50.38 CDR and CPR of ARM2/28, 2004 vintage (Sources: Beyondbond Inc, LoanPerformance)

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substantially. Figure 50.39 presents a tail risk example. When the prepayment speeds double, the total loss decreases to 21% from 29% given the same default speeds. Because the history of prepayment and default rates can seriously affect the remaining principal factor for any given pool of loans, tracking and rolling the principal factor for a loan pool is one of the most important elements in the model projections and future forecasts. Prepayments are removed from the outstanding balance and, as a result, are not available to default in the future.

0 40

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delinquencies should be leading indicators of future defaults. We should be able to simply roll delinquency numbers month to month into actual defaults. The question is whether there is a constant relationship that can be parameterized or not. The time series plots of defaults and the spectrum of delinquencies for the 2003 vintage are shown in Fig. 50.40. The cross correlations indicate an approximately 6-month period for a 30-day delinquency to manifest into default as shown in Fig. 50.41.

50.5.2 Analysis Among Delinquency Spectrum 50.5 Delinquency Study 50.5.1 Delinquency, the Leading Indicator Is delinquency a good leading indicator for default? When a borrower is late for a payment for more than 30 days, a 30-day delinquency is reported. If the payment is late for more than 2 months, a 60-day delinquency is reported. After a 90-day delinquency, the loan is considered to be in default, and the bank holding the mortgage will likely initiate its foreclosure process depending on the judicial status of each state. Since a default is a consequence of delinquency,

The results among delinquency spectrums show a very significant cross correlation between delinquency and its lagged earlier tenor (Fig. 50.42).

50.5.3 A Delinquency Error Correction Default Model Based on the results shown previously, the spectrum of various delinquencies can be parameterized for near-term projections. The benefit of including delinquency to project

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C.H. Ted Hong

Fig. 50.40 Source: Default and delinquency over time for 2003 vintage

2.8

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Fig. 50.42 Source: Correlations between various delinquencies

defaults is that it does not require specific consumer behavior theory to be applied. By simply looking at a delinquency

report, we are able to project the likelihood of defaults. It, however, suffers from the long-term view that if a loan fundamentally carries lower credit-worthy characteristics such as a high CLTV, it has a greater propensity to default. Because we are impressed with the short-term forecast ability, and in order to provided by delinquency and the econometric model based on consumer behavior theory, we have integrated both and created a delinquency error correction model.

50 Dynamic Econometric Loss Model: A Default Study of US Subprime Markets

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Fig. 50.43 Source: Delinquency error model: actual vs. fitting

The fundamental idea is that not only can the long-term view and various scenarios based on changing views of macroeconomic variables be adopted, but also the immediate/early warning signs from delinquency can be observed and utilized (Fig. 50.43). In our error correction model, we start by projecting default rates using the default function with fitted parameters.

We then layer on a 6-month lagged 30-day delinquency as an additional exogenous variable to regress the fitted errors. The process is then repeated sequentially by adding 5-month lagged 60-day and then 4-month 90-day delinquency rates as new regressors. The results are very encouraging when compared to the base model without error correction. The additional R2 pickup is about 15% (Fig. 50.44).

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Fig. 50.44 Comparison of model explanation power for 2000–2007 vintages (Sources: Beyondbond Inc, LoanPerformance)

5.

50.6 Conclusion

6.

50.6.1 Traditional Models As a result of the credit crisis, we now know we must have missed something in the traditional models. It requires us to take a hard look at the models and methodologies employed previously and see what is needed to provide a better interpretation of the current market data and conditions. Traditionally, practitioners have observed consumer behavior through historical defaults and prepayments while building an econometric model with several quantifiable factors. These factors include seasoning patterns, underlying loan characteristics, such as mortgage coupon, FICO score, loan-to-value, and debt-to-income ratio, and macroeconomic variables, such as prevailing mortgage rate and housing price appreciation. In order to fit the historical data, non-linear functions are usually constructed with parameters around the factors to explain default and/or prepayment probabilities. During the process of historical sample fitting to the econometric model, the traditional modelers usually miss the following: 1. Traditional models focus on fitting in-sample data with a unique parameter set by vintage. Although the in-sample data fitting provides a much easier fit of the parameter set, it assumes that borrower’s behavior varies given the same loan characteristics and loan age. It creates a disconnection among vintages and cannot be applied to new loans. 2. Borrower behaviors underlying LTV, FICO, and DTI were implicit but not fully quantified in a dynamic form by traditional models. Since the borrower and loan information such as LTV, FICO, and DTI levels are not periodically

updated after the loan origination date, the accuracy of the projected performance of seasoned loans diminishes as the original data becomes aged and less relevant. Out-of-sample projections may produce counterintuitive results. Macroeconomic variables, such as HPA, unemployment level, personal gross income, and so on can be very important factors for in-sample fitting. However, they do not provide insight for new scenarios. If a new scenario has not occurred historically, a stress test for the new scenario should be thoroughly pre-examined. Traditional models focus on the national level rather than the local housing markets. Since house prices are highly dependent on location, a model with more detailed housing information can make a dramatic difference in the accuracy of its forecasts. Traditional models treat prepayments and defaults independently and ignore the complexity and interaction between these embedded call and put options. Traditional models do not dynamically quantify feedback from other leading indicators such as delinquency rates.

50.6.2 Innovation Having addressed the pitfalls of the traditional models, we have built a Dynamic Econometric Loss (DEL) model framework with the following innovations: Consistent parameter sets for all vintages via the addition of consumer behavior factors. 1. Dynamic consumer behavior factors (a) CLTV ratio (via cumulative HPA since origination) that reflects housing market wealth effects during housing boom/bust periods. (b) DTI ratio (via unemployment rate forecasts) that addresses housing affordability. 2. Complete study of HPA index prior to model-fitting (a) HPCUM as the cumulative HPA since origination to capture the wealth effect. (b) HPA to capture the pulse of the housing market. (c) HPA2D as the change of HPA to capture the trend of the housing market. HPA2D successfully captures the timing of defaults for 2005 to 2006 vintages. (d) In-sample and out-of-sample HPA fit testing to ensure the model’s robustness. 3. A detailed CBSA-level HPA model allows us to understand local housing markets better and to generate more precise projections. 4. Recursive calculations along seasoning paths while estimating/projecting prepayments and defaults.

50 Dynamic Econometric Loss Model: A Default Study of US Subprime Markets

5. An error correction model that systematically builds the linkage between delinquency and default to enhance default forecast accuracy.

50.6.3 Advantages The implementation based on our model framework will capture the default and loss patterns exhibited during the recent period and use the information contained in them to forecast future prepayments, defaults and losses based on various macroeconomic market scenarios. The implementation advantages are as follows: 1. Multiplicative and additive factors for each nonlinear function (boot-strapping Maximum Likelihood Estimation) 2. Comprehensive consumer behavioral economic theory applied in practice (a) Develop a consumer behavior-based economic theory. (b) Estimate consumer behavior via an econometric model. (c) Apply the econometric model to prepayment and default. 3. Fully utilize HPA time-series information (a) A built-in time-series fitting model that dynamically estimates parameters and generates forecasts on the fly. For example, HPCUM #(below 5%) ) CLTV") MDR", SMM# HPA # (below 2%) ) MDR", SMM# HPA2D #(below 5%) ) MDR", SMM # 4. Multiple built-in time-series fitting models at the national, state, and CBSA level that dynamically estimate parameters and generate forecasts on the fly. 5. Built-in recursive calculator along seasoning paths for projecting prepayments and defaults. 6. A set of error correction fitting models that estimate parameters within the spectrum of delinquencies and defaults.

50.6.4 Findings In order to understand how a loan prepays or defaults, we have investigated consumer behavior via loan characteristics utilizing static factors and relevant macroeconomic variables as dynamic factors. For each factor, we have constructed a non-linear function with respect to the magnitude of the factor. We build the default/prepayment function as a linear combination of these factors to justify the impact of each

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factor accordingly. Since a loan can either prepay or default over time, we continue to ensure that the principal factors are rolled properly for prepayment and default forecasts. While the level of HPA is considered the main blessing/curse for the rise and fall of the subprime market, we find that cumulative HPA and the change of HPA contribute to effect prepayment and defaults. 1. HPI is significantly correlated with DPI over a long-term period. Since DPI is a more stable time series, it suggests that HPI will eventually adjust to coincide with DPI growth rate. 2. Default is strongly correlated with the spectrum of delinquency rates. By applying the fitted parameters between default and delinquency rates to an error correction model, we are able to effectively improve default predictability.

50.6.5 Future Improvements Modeling mortgage defaults and prepayments as embedded options is an ongoing learning process. While we are encouraged by our findings, there is a myriad of new questions for us to address with an aim to continuously improve and finetune the model. Some areas for further investigation are briefly described below.

50.6.5.1 Business Cycle – Low Frequency of Credit Spread While studying the dynamic factors in the Default Modeling section, we have focused mainly on the HPI impact on consumer behavior and have introduced the DPI as another macroeconomic variable to determine the long-term growth of the economy. At the beginning of this paper, we wondered how a relatively small volume of loans could result in a subprime crisis that proved to be so detrimental to the entire US financial market and global financial system. We believe that the subprime crisis was merely the tipping point of unprecedented credit market easing that has existed since early this century. During this era of extremely easy credit, yield hungry investors sought to enhance their returns through investment in either highly leveraged securities or traditionally highly risky assets such as subprime loans. Through the rapid growth of the credit default swap in derivative markets and RMBS, ABS, and CDOs in the securitization markets, subprime mortgage origination volume reached record highs after 2003. The credit ease impacted not just the subprime market. All credit-based lending, from credit cards to auto loans and leveraged buy-out loans, were enjoying a borrower friendly environment as lenders went on a lending spree. While the credit default rates reached their historical low

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Fig. 50.45 Historical TED spread and histogram (Source: Beyondbond Inc.)

last decade and resulted in extremely tight spreads among credit products, a longer view of the history of business cycles started to reveal warning signs of the potential downside risk. For example, the TED Spread dramatically widened after August 2007, which was a re-occurrence of the late eighties market environment (Fig. 50.45). Over the past 20 years, traditional calibration models that only focused on shorter time frames missed the downside “fat tail.” The improbable is indeed plausible. Is there a better method to mix the long-term low frequency data with the short-term high frequency data and provide a better valuation model?

50.6.5.2 Dynamic Loss Severity It is a usual practice, when using prepayment and default rates to forecast mortgage and mortgage-derived securities performance, to treat the lagged timing of loan loss/recovery and the loan loss/recovery level as given assumptions. The detailed HPA information provided at the CBSA-level and better detailed information from the loan servicers in recent years have allowed us to begin to model these variables to create dynamic loss severity percentages. Greater cooperation with the servicers will lead to more robust estimations.

Hayre, L. S., M. Saraf, R. Young, and J. D. Chen. 2008. Modeling of mortgage defaults, Citigroup Global Markets, Inc., U.S. Fixed Income Strategy & Analysis – Mortgages, January 22. Laderman, E. 2001. Subprime mortgage lending and the capital markets, Federal Reserve Bank of San Francisco, FRBSF Economic Letter (Number 2001–38), December 28. Mago, A. 2007. Subprime MBS: grappling with credit uncertainties, Lehman Brothers Inc., March 22. Mason, J. R. and J. Rosner. 2007. Where did the risk go? How misapplied bond ratings cause mortgage backed securities and collateralized debt obligation market disruptions, Mortgage-Backed Security Ratings, May 3. Nera Economic Consulting. 2007. At a glance: the chilling effects of the subprime meltdown, Marsh & McLennan Companies, September. Parulekar, R., U. Bishnoi, and T. Gang. 2008. ABS & mortgage credit strategy cross sector relative value snapshot, Citigroup Global Markets Inc., June 13. Peterson, C. L. 2007. Subprime mortgage market turmoil: examining the role of securitization – a hearing before the U.S. Senate Committee on Banking, Housing, and Urban Affairs Subcommittee on Securities, Insurance, and Investment, University of Florida, April 17. Ramsden, R., L. B. Appelbaum, R. Ramos, and L. Pitt. 2007. The subprime issue: a global assessment of losses, contagion and strategic implications, Goldman Sachs Group, Inc. – Global: Banks, November 20. Risa, S. 2004. The New Lehman HEL OAS model, Lehman Brothers Inc., December 8. Ted Hong, C. H. 2005. Modeling fixed rate MBS prepayments, Beyondbond Inc, October. Ted Hong, C. H. and M. Chang. 2006. Non-agency hybrid ARM prepayment model, Beyondbond Inc, July, 6–17. Wang, W. 2006. “Loss severity measurement and analysis.” The MarketPulse, LoanPerformance, Issue 1, 2–19.

References Bergantino, S. and G. Sinha. 2005. Special report: subprime model update, Bear Stearns & Co. Inc., May 26. Dubitsky, R., J. Guo, L. Yang, R. Bhu, and S. Ivanov. 2006.“Subprime prepayment, default and severity models.” Credit Suisse, Fixed Income Research, May 17. Flanagan, C. 2008. Subprime mortgage prepayment and credit modeling, J.P. Morgan Chase & Co., April 2. Hayre, L. S. and M. Saraf. 2008. A loss severity model for residential mortgages, Citigroup Global Markets, Inc., U.S. Fixed Income Strategy & Analysis – Mortgages, January 22.

Appendix 50A Default and Prepayment Definition We consider a loan to be in default if it meets both of the following criteria: 1. The loan is not able to generate any future investor cash flow

50 Dynamic Econometric Loss Model: A Default Study of US Subprime Markets

2. The loan has been in foreclosure, REO, or reporting loss in the prior reporting period The Monthly Default Rate (MDR) is defined as the percentage of defaulted amount as a sum of all default loan balance compared with the aggregate loan balance of that period. SMM (Single Month Mortality) is calculated by formula: SMM D

Scheduled Balance  Current Balance Scheduled Balance

If we have MDR and SMM, then we can derive CDR and CPR from them by using the formula: CDR D 1  .1  MDR/ CPR D 1  .1  SMM/12

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I is the number of multiplicative ! spline functions .j / M P .j / .j / Xt;m ˇm and is a linear combij is equal to 1 C m

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Appendix 50C Default Specification

12

Appendix 50B General Model Framework

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.i / .i / ˛m ; ˇm knots .k/

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i is a spline .k/ .k/ ˛m ; ˇm knots .i /

Xt

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A whole loan mortgage starts at t0 and matures by tn , its MDR by time t can be driven by two types of variables – static and dynamic. Collateral characteristics such as mortgage rate, loan size, IO period, teaser period, loan structure, term to maturity, geographic location, FICO, and CLTV are static factors since their impact diminish over time while the loan is getting seasoned. Macroeconomic variables over time such as Housing Price Index, mortgage interest rate, unemployment rates, Gross Disposable Income, and inflation rates are dynamic. They are publicly observable and will adjust the default rate forecasts based on the scenario assumption. We formulate our default function MDR as follows:  ˇ  Dt D LTV vt ˇLTV j ; ht C FICO .cj /    rate .rt jWACt /  age .ai ja0 /  DTI dj jDTIj ; DOCj    IO gt jIOj ; ai  size .s/  HPA .HPA/  H2D .H2D/  DOC .Docm /  LIEN .LIEN m /  PURPOSE .PURPOSEm /  where ®’s are spline functions in MDR % and are additive to form a base value œ’s are spline functions as multipliers for the MDR adjustments vt : CLTV by time t where initial CLTV is assumed at time t0 rt : Ratio spread of WACt over original WAC rate cj : FICO score of loan j ai : Age of loan j dt : DTI gi : Remaining IO period if IO exists and is positive lj : Size of loan j ®LTV : Original LTV level & HPAt   vt D vt v0 ; ht ; zj

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Hti : HPI at time ti since origination date t0 zt : Geographic zip code j, e.g., z1 D z .CA/ D 1:3 z2 D z .OH/ D 1:1 z3 D z .MI/ D 1:01 z0 D z .Other/ D 1

rt D .WACt  MTGt / ®rate .rt / is a spline function of rt WACt is gross coupon that is either observable or can be

simulated from index rates and loan characteristics

the function form of vt vt D

RATE Effect

Index rate forecasting will be a spread

v0  Ht.i lag/ Ht.0lag/

:zj

yt0 s D “0 C “0 yt1 C “1 Swp2Yt C “2 Swp5Yt C“3 Swp10Yt C “4 LIBOR1Mt C ©t

ht : the functional form of ht as simple AR(2) model ht D “h0 C “h1 ht1 C “h2 ht2 C ©t Where all the parameters can be independently regressed by ht ’s time series data zj : the functional form of zj is setup as a dummy variables zj D “zj  z.j/ if j D “CA” and parameter “zj can be calibrated by default data by bootstrapping the value ft : is the actual principal factor and will be either observed for in-sample filtering or simulated for out-of-sample forecast FICO: Checks if credit scores (original) are a good measure of default cj : the functional form of cj will be a spline (natural, Linear, tension spline) function with fixed FICO locators, j’s (suggested only) [250, 350, 450, 500, 525, 550, 550, 580, 600, 625, 650, 680, 700, 720, 750, 800, 820] and parameters can be calibrated for default data base & fine-tuned AGE: Default probability increases as loan get seasoned but eventually reach a plateau given other constants at : we will sample linear spline function from 0 to 1 to apply age locators [0, 1, 5, 10, 15, 20, 30, 45, 60, 120] DTI Effect: Income level will affect default under assumption of DOC if it’s fully available ut D u0

GDPt  GDP0



UMt UM0

“.UM/

the functional form œu .ut / is a linear spline function of ut   œDTI ut ; wj D .œu .ut //œw .wj / where œw .w0 / D 1 ! Full D w0 œw .w1 / D 0:1 ! Low D w1 œw .w2 / D 0 ! No D w2

for corresponding index rate LIBOR6M, 1Y-CMT, COFI, 5YY – CMT, : : :etc. IO-Payment-Shock: Increased payments at the end of IO period will increase defaults. gt D IO0  at œIO .gt / D is a linear spline function of locators [30; 20, 10; 5; 2, 0, 2, 0, 2, 5, 10, 20] Crowding Out: Measures if the underwriting standard has deteriorated œvolume is a spline function vmt is whole loan issue amount ratio .FICO  580, 580 < FICO  700/  Note: 30-day Delinquency rate for the (12-month) ratio if delinquency report is available œsize is a simple step-spline function to certain loan size after default with locators [ 50k;  100k;  150k;  250k, 500k, 800k, 1million]

Occupancy ocp has 3 kinds of occupancy (Owner, Second Home, Investor)

Loan Purpose prs has 3 kinds of purpose (Purchase, Refi, Cash Out)

Lien lien has 2 lien positions (First lien, Second lien)

Loan Document doc has 3 kinds of documentation type (Full, Limit, and No Document)

50 Dynamic Econometric Loss Model: A Default Study of US Subprime Markets

Appendix 50D Prepayment Specification Single Monthly Mortality (SMM) Rate Function St D rate .rt / turnoverrate ./ teaser .tst / seasonality ./ cashout ./ age .at / burnout .ft / yieldcurve ./ equity ./ credit ./ IO .gt / credit .Vt / issuer .IY j 0 s / size .lj 0 s / penality .Nyes=no /

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mer, decreases through the fall, and slows down even more in the winter. The pattern may be different geographically and demographically.

Cash-Out Prepayment is driven by general housing price appreciation. Rate Factor ®rate .rt / (to grab REFI-incentive) ®rate : a natural spline function 20 locators [10; 5; 2; 1, 0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 5, 6, 7, 9, 10, 15, 20]  rt D

WAC  mt .Fixed/ WACD  mt .ARM=Hybrid/

mt : FH 30-yr/10 day commitment rate (FHR3010) as prevailing mortgage rate to measure SATO effect  Age Factor: PPY has less incentive due to the consideration of initial financing sunk cost but the probability increases as 3-year costs average out over time.

Housing Turnover Rate Age Prepayment based on long-term housing turnover rate and composed of existing sales over single-family owner housing stock.

Mortgages generally display an age pattern.

Seasonality

Burnout Effect

Monthly seasonality is generally believed to affect prepayments. The belief stems from the mobility of mortgagors, time of housing construction, school year, and weather considerations. For a specific month of the year and ceteris paribus, prepayment rates are directly affected by the related month-of-year’s coefficient. Usually, the seasonality pattern shows greater active in the spring, rises to a peak in the sum-

Borrowers don’t behave homogeneously when they encounter the same refinancing opportunities. Some are more sensitive than others. If the borrowers are heterogeneous with respect to refinancing incentives, those who are more interest sensitive will refinance sooner. The remainder will be composed of less interest sensitive borrowers.

Chapter 51

The Effect of Default Risk on Equity Liquidity: Evidence Based on the Panel Threshold Model Huimin Chung, Wei-Peng Chen, and Yu-Dan Chen

Abstract This research sets out to investigate the relationship between credit risk and equity liquidity. We posit that as the firm’s default risk increases, informed trading increases in the firm’s stock and uninformed traders exit the market. Market-makers widen spreads in response to the increased probability of trades against informed investors. Using the default likelihood measure calculated by Merton’s method, this paper investigates whether financially ailing firms do indeed have higher bid-ask spreads. The panel threshold regression model is employed to examine the possible non– linear relationship between credit risk and equity liquidity. Since high default probability and worse economic prospects lead to greater expropriation by managers, and thus greater asymmetric information costs, liquidity providers will incur relatively higher costs and will therefore offer higher bid-ask spreads. This issue is further analyzed by investigating whether there is any evidence of increased vulnerability in the equity liquidity of firms with high credit risk. Our results show that the effects caused by increased default likelihood might precipitate investors’ loss of confidence in equities trading and thus a decrease in liquidity as evident during the Enron crisis period. Keywords Default risk r Equity liquidity r Financial distress costs r Panel data r Threshold regression

W.-P. Chen () Department of Finance, Shih Hsin University, Taiwan e-mail: [email protected] H. Chung Graduate Institute of Finance, National Chiao Tung University, Hsinchu 30050, Taiwan Y.-D. Chen Graduate Institute of Finance, National Chiao Tung University, Taiwan

51.1 Introduction A lot of researchers have studied the costs of financial distress, including direct costs and indirect costs.1 Nevertheless, except for direct costs and indirect costs, recent research has revealed a new notion of financial distress costs. Previous studies that examine the effect of default risk on equities focus on the ability of default spreads to explain or predict returns. Vassalou and Xing (2004) uses Merton’s (1974) options pricing model to compute default measures for individual firms and access the effect of default risk on equity returns. Although considerable research effort has been put toward modeling default risk for the purpose of valuing corporate debt ad derivative products written on it, little attention has been paid to the effects of default risk on equity liquidity. This paper considers the effects of default risk on equity liquidity. Agrawal et al. (2004) demonstrate that firms with financial distress suffer from reduced stock liquidity by increasing bid-ask spread. In other words, as a firm’s performance deteriorates and likelihood of financial distress increases, the stock liquidity of the firm will be worse. In the literature on market microstructure, the bid-ask spread is modeled as arising from three sources: adverse selection, order processing costs, and inventory holding costs. The adverse-selection component compensates the market-maker for losses incurred on trades against informed traders. Market makers will normally want to profit in their transactions with noise traders who do not obtain any private information. Therefore, market makers set the spreads wide enough to guarantee that their profits in trades to uninformed traders will cover the expected losses from the trades with informed traders. In the period of expected high financial distress costs, default risk might affect equity liquidity in a more severe pattern. This is because the managerial agency costs are 1 The direct costs comprise legal and administrative expenses of bankruptcy proceedings (Warner 1977; Weiss 1990); the indirect costs consist of management resources devoted to resolving financial distress, loss of suppliers and customers, and constraints on the firm’s financing and investment opportunities (Altman 1984; Titman 1984; Wruck 1990).

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particularly severe in stressful market conditions and managers’ incentive to pursue their private benefits increases. Market makers might have to widen the bid ask spread in response to the perceived increase in information asymmetry costs. There are also several other reasons why uninformed investors are less likely to trade in the securities of financially troubled firms. Information asymmetry costs increase with firms’ financial distress conditions. When a firm’s performance and financial condition deteriorate, trading in the firm’s security may be increasingly dominated by informed investors. If the liquidity provider believes that the probability of some traders possessing superior information has increased for equities with high default risk, the liquidity provider could protect himself by quoting a higher bid-ask spread. This paper explores the default risk impacts on liquidity for the Enron crisis period when stock investors’ strong lack of confidence was observed and the expected financial distress costs were extreme.2 The bankruptcy of Enron Corp. was infectious. Investors became more and more sensitive to financial statement and credit risk of companies. In this paper, we will investigate the cross-sectional relationship between firms’ default risk and their equity liquidity and whether the results are any more significant in the period with high financial distress costs. We posit that the recent corporate accounting scandal of Enron and other listed companies’ bankruptcy be a precipitating factor to the investors’ loss of confidence in equity trading, particularly for those stocks with high default risk. To account for the potential non-linear relationship between default risk and equity liquidity, the analysis is conducted by using the panel threshold regression model of Hansen (1999). We apply Merton’s (1974) option pricing model to estimate default risk for individual firms. The default risk measure is called the “Default Likelihood Indicator (DLI).” Our empirical results provide interesting factors on the influence of default risk on liquidity. The results show that the perceived default risk at the beginning of the month has predictive contents on the liquidity (average bid-ask spread) of the month. DLI reveals an extremely significant positive relation to percentage bid-ask spread, showing that DLI is one of the key determinants to the liquidity cost, indicating that financially troubled firms suffer from the cost of reduced liquidity. More importantly, in the period of high expected 2 During the fourth quarter in 2001, most of the global finance markets were sluggish, especially the U.S. stock market, owing to explosion of a series of corporate scandals. Most famous of all, on December 2, 2001, the energy giant “Enron Corp.” announced that it and 14 of its subsidiaries have filed voluntary petitions for Chap. 14 reorganizations with the U.S. Bankruptcy Court for the Southern District of New York. Enron listed $49.8 billion in assets and $31.2 billion in debts, making it the biggest corporate bankruptcy in U.S. history.

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financial distress costs the economic costs of liquidity appear to be higher for firms with high default probabilities. The remainder of this paper is organized as follows. The next section discusses data and methodologies, such as default likelihood indicators, panel data regression model, and threshold regression model. Empirical tests and results are undertaken in Sect. 51.3. Section 51.4 outlines our conclusion

51.2 Data and Methodology 51.2.1 Data The sample data include the component stocks of S&P 500. To avoid the effect of different market mechanisms on equity liquidity, only the component stocks listed on the New York Stock Exchange (NYSE) are included, as Stoll (2000) concludes that market structure has a clear effect on bid-ask spreads. The sample period is from February 1, 2001 to May 31, 2002.3 Intraday trade and quoted data are taken from the NYSE Trade and Quoted (TAQ) database. Following Stoll (2000), the averages of each of the underlying variable are taken across all of the days in each month in order to reduce the errors associated with a single day. The firms that have missing values of variables used in this study are deleted from our sample. Consequently, our sample size reduces to 276 firms and the number of firm-months is 4,416. We delete all trades and quotes that were out of time sequence, as well as those that involved any errors. Following Huang and Stoll (1996) and many previous studies, we attempt to further minimize data errors by eliminating quotes with the following characteristics: (1) where either the bid or the ask price is equal to, or less than, zero; (2) where either the bid or the ask depth is equal to, or less than, zero; (3) where either the price or volume is equal to, or less than, zero; (4) all quotes with a negative bid-ask spread, or a bidask spread of greater than US$4; (5) all trades and quotes that are either “before-the-open” or “after-the-close”; (6) all Pt trade prices, where: j.Pt  Pt 1 /=Pt 1 j > 0:1; and (7) all at ask quotes, where j.at  at 1 /=at 1 j > 0:1 and all bt bid quotes, where j.bt  bt 1 /=bt 1 j > 0:1. In our cross-sectional model, the depend variable is percentage bid-ask spread (PSP) and explanatory variables are close price (P), the number of trade (NT), volatility (SIG), market value (MV) and default likelihood indicators (DLI). PSP is calculated as the average of percentage spread in final 5 min for each trading day. The average of the daily observation of PSP, closing price and number of trades are 3

NYSE switched to the decimal pricing system on January 29, 2001. Thus the period of our research starts from February 1, 2001.

51 The Effect of Default Risk on Equity Liquidity: Evidence Based on the Panel Threshold Model

calculated for each stock in each month. We compute the standard deviation of daily stock returns for a given month (SIG) as the volatility of these daily returns over that 1-month period. MV is the stock’s market value. The book value of total debt includes both current and long term. DLI are nonlinear functions of the default probabilities of the individual firms. We estimate DLI by using the contingent claims methodology of Black and Scholes (1973) and Merton (1974). This paper employs the COMPUSTAT database to retrieve the firm’s “Market Value,” “Debt in One Year,” “Long-Term Debt,” and “Total Liabilities” series for all companies. We acquire information on daily stock returns from the Center for Research in Securities Prices (CRSP) daily return files. To calculate DLI, we observe monthly 1-year T-bill rate r from The Federal Reserve’s website.

809

Let xt be the book value of the debt at the time t, that has maturity equal to T. xt is used as the strike price of a option, since the market value of equity, VE , can be regarded as a call option on VA with time to expiration equal to T. Using the Black and Scholes formula for call options, the market value of equity will be estimated: VE D VA N.d1 /  Xe rT N.d2 /;

(51.3)

and   p log.VA =X / C r C 12 A2 T ; d2 D d1  A T ; p d1 D A T (51.4)

51.2.2 Methodology

r is the risk-free rate, and N./ is the cumulative density function of the standard normal distribution. An iterative procedure is adopted to calculate A and then to estimate daily value of VA for each month. See the following steps:

51.2.2.1 Bid-Ask Spread

Step 1.

The percentage spread is used as a measure of trading cost. They are then averaged for each security for each month of the sample. The nominal spread of security i at time t, Traded Spreadit , is calculated as Askit Bidit , where Askit and Bidit are the respective average intraday ask and bid prices at time t for security i . The percentage spread is calculated as: Askit Bidit : Percentage Spreadit D .Askit C Bidit /=2

Step 2.

Step 3.

(51.1) Step 4.

51.2.2.2 Measure of Default Risk Step 5. Vassalou and Xing (2004) is the first research that uses Merton’s (1974) option pricing model to calculate default measures for individual firms and appraise the effect of default risk on equity returns. The basic concept of Merton (1974) is that the equity of a firm is thought of as a call option on the firm’s assets and then one can estimate the value of equity by using Black and Scholes (1973) model. Our procedure in calculating default risk measures using Merton’s option pricing model is similar to the one used by KMV. We assume the capital structure of the firm including equity and debt. The market value of a firm’s underlying assets follows a Geometric Brownian Motion (GBM) of the form: dV A D VA dt C A VA dW

The above procedure is repeated at the beginning of every month, resulting in the estimation of monthly values of A . The estimation window is always kept equal to 12 months. The risk-free rate is the 1-year T-bill rate. Once daily values of VA for month t are estimated, we can compute the drift , by calculating the annul compound return rate of asset value, log.VA;t 1 =VA;t q /. The default probability is the probability that the firm’s assets will be less than the book value of the firm’s liabilities as maturity T4 . In other words, Pdef ;t D Pr .VA;t CT 6 Xt jVA;t / D Pr .log .VA;t CT /

(51.2)

where VA is the firm’s value with an instantaneous drift  and an instantaneous volatility A . A standard Wiener process is W .

Daily data from the past 12 months are used to estimate the volatility of equity .E /, which is then viewed as an initial value for the estimation of A . Using the Black and Scholes formula given in Equation (51.3), we obtain VA using VE as the market value of equity of that day for each trading day of the past 12 months. In this mode, we capture daily values of VA for the past 12 months. We then compute the standard deviation of those VA from the previous step, which is used as the value of A , for the next iteration. Steps 2 and 3 are repeated until the values of A from two consecutive iterations converge. Our tolerance level for convergence is 10E-4. Once the converged value of A is obtained, we use it to back out VA through Equation (51.3).

6 log .Xt / jVA;t / 4

(51.5)

We use the “Debt in One Year” plus half the “Long-Term Debt” as book value of debt.

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Because the value of the assets follows the GBM of Equation (51.2), the value of the assets at any time t is given by:

p A2 T C A T "tCT ; log.VA;t CT / D log.VA;t / C   2 "tCT D

W .t C T /  W .t/ p ; and "tCT  N.0; 1/ T

(51.6)

Hence, we can get the default probability as follows:   Pdef ;t D Prob. log.VA;t /  log.Xt / C   0:5A2 T p CA T "t CT < 0/ Pdef ;t D Prob."t CT 6 Πlog.VA;t =Xt / p   C   0:5A2 T =A T /

(51.7)

Distance to default (DD) could be defined as follows: p   DDt D . log.VA;t =Xt / C   0:5A2 T /=A T

(51.8)

When the ratio of the value of assets to debt is less than 1 or its log is negative, default occurs. The DD tells us by how many standard deviations the log of this ratio needs to deviate from its mean in order for default to occur. Although the value of the call option in Equation (51.3) does not depend on , DD does, as it depends on the future value of assets that is given in Equation (51.4). Using the normal distribution implied by Merton’s model, the theoretical probability of default, called Default Likelihood Indicators (DLI), will be given by5 :

51.2.2.3 Panel Data Analysis of Default Risk on Equity Liquidity The important factors of effect percentage bid-ask spreads have been approved by numerous earlier papers; see Benston and Hagerman (1974), Tinic and West (1972), and Stoll (1978a, 1978b). Besides stock price, the spread is influenced by other factors such as trading volume, variance of stock returns, market value of equity, and even the structure of exchange market. A word of caution is necessary here because some of these empirical factors will also be affected, in turn, affecting spreads when a firm’s performance deteriorates. For example, the declining stock prices of poorly performing firms will make percentage bid-ask spreads rise. In particular, variance of stock returns, which proxies for informed trading (Black 1986), may rise and increase in spreads. It is also possible that although trading by informed investors rises, trading volume in general will decline because of fewer investors in the market, hence higher spreads will ensue. For the purpose of examining the influence of the firm’s default risk on the bid-ask spreads clearly, we recommend controlling for these determinants. Using controlling variables in our regressions we find that the increase in spreads is still directly linked to the firm’s financial condition. Following Stoll (2000), the averages of each of the underlying variables are taken across all of the days in each month in order to reduce the errors associated with a single day. Every variable is the monthly average except for DLI. DLI is defined as the default probability of the first trading day for each month. That implies that DLI forecasts PSP in an ex ante base. The following regression model is analyzed: PSPit D a C b1 log Pit C b2 log NT it C b3 SIGit

Pdef D N.DD/

p   D N .  Πlog.VA;t =Xt / C   0:5A2 T =A T /

C b4 log MV it C b5 log PDLI it C "1

(51.9)

(51.10)

DLI increases as (1) the value of debt raises, (2) the market value of equity as well as assets goes down, and (3) the assets’ volatility increases. Comparatively, DLI is superior to other approaches using accounting financial information to measure default risk. DLI has a range of strengths as follows: (1) it has strong theoretical underpinnings; (2) it takes into account the volatility of a firm’s assets; (3) it is forward looking based on stock market data rather than historic book value accounting data.

Based on the results of past literature, we expect that b1 ; b2 and b4 should be negative, and b3 be positive. Because higher default risk implies financial condition exacerbation, b5 is expected to be positive. To account for the potential heteroscedasticity, we use generalized least squares estimation (GLS) to estimate the coefficients of these explanatory variables in Equation (51.10). The important motivation for using panel data is to solve the omitted variables problem. If the omitted variable is correlated with explanatory variables, we cannot consistently estimate parameters without additional information. The use of fixed effect panel regression might mitigate the omitted variable problem.

5

Strictly speaking, Pdef is not a default probability because it does not correspond to the true probability of default in large samples.

51 The Effect of Default Risk on Equity Liquidity: Evidence Based on the Panel Threshold Model

51.2.2.4 Panel Threshold Regression Analysis of Default Risk on Liquidity The previous model considered in this paper assumes a linear relationship between credit risk and equity liquidity. The relationship could be non-linear as perceived high default risk might signal increasing information asymmetry costs. This issue may be addressed by using threshold regression techniques. Threshold regression models express that individual observations can be divided into classes by the value of an observed variable. This paper treats DLI as threshold variables and our single threshold regression equation has the following form: PSPit D a C b1 log Pit C b2 log NT it C b3 SIGit

(51.11)

where I() is the indicator function and 1 is threshold. PDLI is equal to DLI times 100. Alternatively, the double threshold regression model for the credit risk and liquidity costs can be expressed as: PSPit D a C b1 log Pit C b2 log NT it C b3 SIGit C b4 log MV it C b5 log PDLI it I.DLI it < 1 / C b6 log PDLI it I.1 < DLI it < 2 / Cb7 log PDLI it I.DLI it > 2 / C "it

(51.12)

where 1 and 2 are the threshold values for DLI. The observations are divided into two and three regimes. Both models have only the slope coefficient on log PDLI switch between regimes, because we can focus attention on this key variable of interest. Based on the results of past literature, b1 ; b2 and b4 should be negative, and on the other hand, b3 ; b5 ; b6 and b7 should be positive. In contrast to deciding threshold levels arbitrarily, the above panel threshold regression model chooses the threshold points statistically. Hansen’s (1999) threshold regression methods are suitable for non-dynamic panels with individual specific fixed effects and show that the model is rather straightforward using a fixed-effects transformation. An asymptotic distribution theory is derived, which is used to construct confidence intervals for the parameters. A bootstrap method to assess the statistical significance of the threshold effect is also described. Our sample is designed for balanced panels, so we took the subset of 276 stocks, which are observed for the period 2001/02–2002/05. GAUSS programs are modified to fit our sample and regression model. Let yit be the percentage spread for stock i at month t, the threshold model in Equation (51.11) can be rewritten as: yit D xit . /ˇ C ci C uit

where xit . / and ˇ corresponds to the explanatory variables and parameters in Equation (51.13), respectively. Similar to the procedure of fixed effects transformation introduced in previous section, transformation is used to eliminate the individual effect ci and now let Y ; X and u denote data stacked over all individuals. The threshold equation with fixed effects transformation could be written as Y  D X . /ˇ C u

(51.13)

(51.14)

Hansen (1999) suggests that the parameter ˇ can be esO / D .X . /0 timated by OLS for any given  , i.e., ˇ.  1  0  X . // X . / Y . The regression residuals is O / uO  . / D Y  X . /ˇ.

C b4 log MV it C b5 log PDLI it I.DLI it < 1 / C b6 log PDLI it I.DLI it > 1 / C "it

811

(51.15)

Then, the sum of squared errors (SSE) is S1 . / D uO  . /0 uO  . / D Y0 .I  X . /0 .X . /0 X . //1 X . /0 /Y (51.16) Using least square to estimate  is suggested by Chan (1993) and Hansen (1999). It is easiest to achieve by minimizing the sum of squared errors. Thus, the least squares estimator of  is (51.17) O D arg min S1 . / 

when the threshold O is available, the coefficient of the reO and the gressor is ˇO D ˇO .O /, the residual is uO  D uO  .”/, S1 .O / 2 . residual variance is O D n .T  1/ It is very important to identify whether the threshold effect has observable influence on coefficient estimator. In other words, it is necessary to test the hypothesis of no threshold effect in Equation (51.12). The null hypothesis, H0 W b5 D b6 , can be tested by using the likelihood ratio test. If the null hypothesis holds, the model is reduced to Equation (51.10). Let S0 be the sum of squared error of Equation (51.10). The likelihood ratio test of no threshold effect is F1 D .S0  S1 .O // = O 2

(51.18)

Because the asymptotic distribution of F1 is non-standard, Hansen (1999) suggests a bootstrap procedure to simulate the asymptotic distribution of likelihood ratio test. The bootstrap method estimates the asymptotic p-value for F1 under H0 . If p-value is less than the desired critical value, the null hypothesis of no threshold effect is rejected. For the double threshold panel model in Equation (51.12), a sequential OLS method described in Hansen (1999) can be directly applied to estimate the threshold parameters.

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51.3 Empirical Results 51.3.1 Descriptive Statistics Our sample period extends over the 351 trading days from February 2001 to June 2002. The sample is divided into two subperiods: the first from February 2001 to September 2001, and the second from October 2001 to June 2002. In this paper, the period 2 from October 2001 to June 2002 will be elected the period of high financial distress costs. Table 51.1 presents descriptive statistics, such as the mean, median, standard deviation, and minimum and maximum of selected variable for three groups of pooled data. Pearson correlations coefficients of selected variables for three groups of pooled data are given in Table 51.2. For price (P), number of trades (NT), and market values (MV), there is negative correlation to percentage bid-ask spread (PSP). In the other hand, volatility (SIG) and DLI have positive relation to percentage bid-ask spread (PSP). We first provide some preliminary analysis on the relationship between percentage spread and trading characteristic variables, such as price (P), number of trades (NT),

volatility (SIG), and market values (MV). The monthly regression results using ordinary least squares estimation is shown in Appendix Table 51A.1. For most months, the coefficients of log P and SIG are significant at the 1% level. As expected, the signs of the parameters of the two explanatory variables are both consistent with our prediction. There is a negative cross-sectional relationship between price and percentage spread. While volatility is positively related to percentage bid-ask spread for every month, the coefficients of log NT and log MV are significant only for fewer months.

51.3.2 Results of Panel Data Regression If the structure model for the cross-sectional determinant of percentage spread equation has the potential omitted variable problem, pooled ordinary least squares estimator is not consistent. Unobserved effects panel data model can be used to address this problem. We thus use the fixed effect model to estimate the panel data regression model. The data is also

Table 51.1 Descriptive statistics of selected variable for three groups of pooled data N Mean Median S.D Minimum

Maximum

Panel A: Whole period (February 2001–May 2002) PSP (%) 4,833 0.1285 0.1054 CLP 4,833 40.0159 38.1913 NT 4,833 1,068 859 SIG 4,833 2.0782 1.8841 MV(million’s) 4,833 19,612 7,855 DLI (%) 4,833 4.3084 0 Panel B: First sub-period (February 2001–September 2001)

0.0837 19.7966 681 0.8603 42,885 14.8112

0.0195 2.8175 162 0.4489 506 0

0.8140 159.9965 5,005 6.0690 486,720 100

PSP (%) 2,267 0.1455 0.1202 P 2,267 40.5592 38.9771 NT 2,267 938 741 SIG 2,267 2.1030 1.9298 MV(million’s) 2,267 20,476 7,865 DLI (%) 2,267 4.7494 0 Panel C: Second sub-period (October 2001–May 2002)

0.0880 19.3324 632 0.8370 45,095 15.8597

0.0316 5.8313 162 0.6046 506 0

0.8140 133.1581 4,098 6.0690 486,720 100

PSP(%) P NT SIG MV(million’s) DLI (%)

0.0767 20.1895 701 0.8799 40,826 13.8103

0.0195 2.8175 196 0.4489 510 0

0.7745 159.9965 5,004 6.0177 398,105 100

2,566 2,566 2,566 2,566 2,566 2,566

0.1134 39.5359 1,182 2.0563 18,849 3.9188

0.0913 37.3880 982 1.8439 7,838 0

This table presents descriptive statistics of all selected variables during our sample period. Since the firms that have missing values of variables are deleted, our sample size reduces to 276 firms that were listed on the NYSE Notes: PSP the monthly average percentage bid-ask spreads for company i ; P the monthly average close price for company i ; NT the monthly average of the number of trades for company i ; SIG the standard deviation of daily stock returns for a month for company i ; MV the market value of company i ; DLI default likelihood indicators of company i ; N the number of observation

51 The Effect of Default Risk on Equity Liquidity: Evidence Based on the Panel Threshold Model

813

Table 51.2 Pearson correlations coefficients of selected variable for three groups of pooled data P NT SIG MV

DLI

Panel A: The whole period (February 2001–May 2002) PSP 0:5991 . jtj 0.1807

Notes: N is the number of the sample firms. Period 1 corresponds to February 2001–September 2001. Period 2 corresponds to October 2001–June 2002. “Diff” is the difference of average DLI between Period 2 and Period 1 for each firm. H0 : the difference of DLI for company i between Period 1 and Period 2 equal zero

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Table 51A.3 Percentages of firms in each regime by month: double threshold model

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Firms class

Month

DLI 5 0:0843 APSP D 0:1188

0:0843 < DLI 5 0:6735 APSP D 0:2036

DLI > 0:6735 APSP D 0:2281

2001 Feb 2001 Mar 2001 Apr 2001 May 2001 June 2001 July 2001 Aug 2001 Sep 2001 Oct 2001 Nov 2001 Dec 2002 Jan 2002 Feb 2002 Mar 2002 Apr 2002 May 2002 June

74.6% 91.7% 88.0% 90.6% 92.8% 91.3% 91.3% 89.5% 84.4% 86.6% 89.9% 89.9% 90.2% 89.9% 93.1% 92.4% 91.3%

18.5% 8.0% 9.8% 6.5% 4.7% 6.2% 6.5% 8.0% 13.0% 11.2% 8.0% 8.7% 8.0% 8.7% 6.2% 6.5% 6.9%

6.9% 0.4% 2.2% 2.9% 2.5% 2.5% 2.2% 2.5% 2.5% 2.2% 2.2% 1.4% 1.8% 1.4% 0.7% 1.1% 1.8%

We find that the percentage of companies in the “very high default risk” category ranges from 0.4 to 6.9% of the sample in each month. The “low default risk” firms range from 74.6 to 91.7% of the sample over the months. The average of percentage bid-ask spreads for each regime are 0.1188, 0.2036 and 0.2281 in that order. The “very high default risk” class has the highest average of percentage bid-ask spreads, 0.2281. This proves that the firms with higher credit risk will suffer higher liquidity cost Note: APSP D the average percentage bid-ask spreads for each regime

Chapter 52

Put Option Approach to Determine Bank Risk Premium Dar Yeh Hwang, Fu-Shuen Shie, and Wei-Hsiung Wu

Abstract In this paper, we briefly review the pricing of deposit insurance under the option approach. First, we review the theoretical works of deposit insurance pricing model. And second, we summarize the applications of deposit insurance pricing model, focusing on the issue of whether fees charged by the insuring agency are excessive. Keywords Deposit insurance r Capital standard r Options pricing

52.1 Introduction A main function of a bank is to lend money to borrowers and receive funds from depositors. Because of the information asymmetry and the transaction costs, depositors lend funds to banks instead. To insure that the banks function well, banks should be risk-free. Hence, there is a need for the third-party guarantee on the deposits. But the guarantee incurs costs on the insurer, and how the costs are priced is an important issue. There is an isomorphic relationship between deposit insurance and put options. Merton (1977) first presents this insightful relationship. In short, with deposit insurance, the bank receives nothing as it is solvent. But the insurer pays the deposits that cannot be met by the bank assets when insolvency occurs. Therefore, the deposit insurance contract corresponds to the put option written on the bank assets with strike price equal to the insured deposits. There are lots of studies that follow the work of Merton (1977). First, Merton (1978) considers the auditing costs of insurers and finds that deposit insurance costs per dollar

D.Y. Hwang () and W.-H. Wu Department of Finance, College of Management, National Taiwan University, 85, Roosevelt Road, Section 4, Taipei Taiwan, R.O.C. e-mail: [email protected] F.-S. Shie Department of Finance, College of Management, Asia University, 500, Lioufeng Road, Wufeng, Taichung County, Taiwan, R.O.C.

are no longer monotonically decreasing in the asset-todeposit ratio. Mullins and Pyle (1994) show that the riskbased capital ratio increases with the auditing costs. Allen and Saunders (1993) take into account the capital forbearance. However, their works are modified by Hwang et al. (2009), who introduce the variable of bankruptcy costs. Lee et al. (2005) derive a formula of fair deposit insurance premium, which explicitly incorporates the forbearance and the capital requirement of Basel Accord. To relax the assumptions of constant risk-free rate and asset volatility, Kerfriden and Rochet (1993) discuss how the deposit insurance is priced under stochastic interest rates economy, and Duan and Yu (1999) use the CARCH option technique to fairly price the deposit insurance. One of the main applications of Merton (1977) is to examine whether the deposit insurance provided by the FDIC is excessive. Markus and Shaked (1984) first proposed a method that can be implemented to price the risk-based deposit insurance premium of each bank. Their empirical results show that current fees charged by the FDIC are overpriced. Ronn and Verma (1986) present an empirical approach as well, but they point out that considering forbearance, the deposit insurance premia are underpriced. Pennacchi (1987) considers the power of the insurer, and he indicates that whether the deposit insurance premia are under- (or over-) priced depends on the level of insurer’s control power over the banks. His empirical results show that current premium lies within the upper and lower bound he presents, which shows that the FDIC has mild control over the banks. Flannery (1991) states that the insurer can observe the financial data of banks only with errors, which tends to underestimate the true value of deposit insurance. Since measurement errors exist, it is socially optimal to both adopt risk-based deposit insurance scheme and risk-based capital requirements. This paper is organized as follows. Section 52.2 reviews the pioneer work of deposit insurance pricing in Merton (1977). Section 52.3 discusses some of the major extensions of Merton (1977). Section 52.4 presents the applications for option pricing deposit insurance. The major issue is whether the deposit insurance premium is over- or undercharged. Section 52.5 concludes.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_52, 

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52.2 Evaluating Insurer’s Liability by Option Pricing Model: Merton (1977) One of the main functions of a bank is to lend money to borrowers and receive deposits from firms and individuals. People would like to deposit funds in a bank because of lower transactions costs, liquidity, and convenience. But a necessary condition for a depositor to lend funds to banks is that the deposits are risk-free. Otherwise, depositors would have to collect information on the banking industry, analyze the balance sheets of banks, and diversify their funds across banks. This is costly to them. Risky deposits will jeopardize the main function of a bank, which is serving as an intermediary. Introducing a third-party guarantee on the deposits that a bank takes is a solution. However, considering the scale of the banking industry, the third-party guarantor can almost only be the government or one of its agencies, such as the Federal Deposit Insurance Corporation (FDIC). It is publicly believed that even in countries where explicit deposit insurance systems are not adopted, the government guarantees the deposits implicitly. Hence the deposit insurance imposes costs on the guarantor. There is an isomorphic relationship between the deposit insurance and the common stock put options. First we discuss the payoff of depositors without deposit insurance, and then we consider the effects of deposit insurance provided by the insurer. Let V be denoted as the value of a bank’s assets at maturity date, and D the total value of deposits at maturity date. For deposits, if the value of the bank’s assets exceeds total deposits, the payoff of deposits is D. But if V is so low (V < D) that the bank cannot meet its obligations, deposits of the bank can only receive V in return. In sum, at the maturity date, the depositors receive Min (V; D). Similarly, for the equityholders of the bank, their payoff is Max (0; V D). Second, consider the impact of a third-party guarantee on the obligations that the bank should meet at maturity date. If the bank assets value V is greater than the insured deposits D, then the deposit insurance has no value and the bank continues to operate. But if at the end of the contract period, V is smaller than D, then the insurer pays the deficits D  V and closes the bank. The costs of an insurer for providing the deposit insurance contract are thus Max .0; D  V /. Put it another way, without deposit insurance, the original payoff of a depositor is Min (V; D). But Min .V; D/ D D CMin .0; V D/ D D Max .0; D V /. Since Max (0; D  V ) is nonnegative, it is the guarantor’s task to make payment of Max (0; D  V ) to depositors to insure that the deposits are risk-free. Hence, pricing the deposit insurance corresponds to value a put option written on the bank assets with strike price equal to the insured deposits. If a guarantor (for example, the FDIC) guarantees that a bank can meet the obligation of its deposits when they are

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due, it is essentially the same that the guarantor sells the bank a put written on its total asset. Hence the deposit insurance premium can be valuing by the put option model. The deposit insurance premium that Merton (1977) has originally derived is of the form: g.d;  / D ˆ.h2 / 

1 ˆ.h1 /; d

(52.1)

where n  o .p ; h1  log.d /  2 p h2  h1 C ; and g is the cost of insured deposit per dollar, d  D=V is the current deposit-to-asset value ratio,    2 T is the variance of the logarithmic change in the value of the assets during the term of the deposits, and ˆ./ denotes the CDF of the normal distribution. The proof of Equation (52.1) is shown in the Appendix 52A. From Equation (52.1), it is shown that the change in the cost with respect to an increase in the deposit-to-asset value ratio (d ) is positive; that is, ˆ.h1 / @g D : @d d2

(52.2)

The change in the cost with respect to an increase in the variance of the logarithmic change in the value of the assets (£) is also positive; that is, ˆ0 .h1 / @g D p : @ 2d 

(52.3)

Equations (52.2) and (52.3) are easily understood because the increase of d and £ will both increase the probability of insolvency. The proof of Equations (52.2) and (52.3) are shown in the Appendix 52B.

52.3 Extensions of Merton (1977) There are lots of researchers that follow the work of Merton (1977). Merton (1978) considers the surveillance costs of insurer and finds that deposit insurance costs are no longer monotonically decreasing in the asset-to-deposit ratio. Mullins and Pyle (1994) evaluate the impact of auditing costs on the risk-based capital ratio and the deposit rate spread. They show that the risk-based capital ratio and the deposit rate spread increase with auditing costs. Allen and Saunders (1993) take into account the capital forbearance.

52 Put Option Approach to Determine Bank Risk Premium

821

Hwang et al. (2009) introduce the variable of bankruptcy costs to modify the work of Allen and Saunders (1993). Lee et al. (2005) derive a formula of fair deposit insurance premium, which explicitly incorporates the forbearance and the capital requirement of Basel Accord.

52.3.1 The Consideration of Surveillance Costs Merton (1978) derives the explicit form of deposit insurance premium when the surveillance costs of the guarantor are taken into consideration. In this model, random auditing times are also allowed. Under this setting, Merton shows that costs of deposit insurance per dollar are no longer monotonically decreasing in the asset-to-deposit ratio, contrary to the findings in Merton (1977). The reason is that although the costs of deposits guarantee are monotonically decreasing, the audit costs are not. A bank is audited more frequently if it is likely to be insolvent. Another finding is that, in equilibrium, the market interest rate is higher than the total rate of return on deposits. The difference is just the expected auditing costs per dollar per unit time. So, the depositors pay for the auditing costs, and the bank’s shareholders pay for the put option component. Mullins and Pyle (1994) find that the deposit rate spread (risk-free rate minus the deposit rate) increases with the volatility of bank asset value. In Mullins and Pyle (1994), they assume that the deposit insurance guarantor can measure the risks of bank assets without error, and auditing times are known beforehand. They use the isomorphic relationships between the deposit insurance premium and put options and the shareholder value and the call options to determine the risk-based capital ratio under current deposit insurance scheme. That is, p0 D e D N.y C V /  .1 C k0  p0 /N.y/ Ca.1 C k0  p0 /; k0 D .1 C k0  p0 /N.y/  e D N.y  V /:

(52.4) (52.5)

where p0  P0 =D0 and k0  K0 =D0 , denote the insurance premium per dollar of deposits and capital-to-deposit ratio in the beginning of the period, respectively. And a denotes the auditing cost per dollar of asset value, D  rf  rD , which is the spread of deposit rate, N./ is the CDF of the standard Normal distribution, and y D .ln.1 C k0  p0 / C D =V / C V =2. Note that the guarantor incurs the auditing costs, and the auditing costs per dollar of deposits, a.1 C k0  p0 /, is added to the Black-Scholes term in Equation (52.4). Since there are two equations, two known variables, k0 .V / and

D .V /, which represent the risk-based capital-to-deposit ratio and the deposit rate spread, can be solved. And p0 is set to be 0.0023, the deposit insurance premium charged by FDIC before 1993. The results show that both the risk-based capital requirements and the deposit rate spread increase with the asset volatility and the auditing costs per dollar of deposits. Using two isomorphic correspondences between options and deposit insurance premium or equityholders’ value can also be seen in Ronn and Verma (1986), see Sect. 52.4.2.

52.3.2 Incorporating Forbearance 52.3.2.1 Allen and Saunders (1993) Allen and Saunders (1993) evaluate the deposit insurance with the forbearance policy exercised by the insurer. Forbearance is the delay in implementing or enforcing specific regulations. With regard to closure policy, forbearance is granting the institution time to return to solvency before it is closed. Previous researches ignore the call provision feature of deposit insurance; that is, the insurer (FDIC) has the right to exercise the option before the insolvent banks choose to exercise it. The call provision has value if insurer-induced closure policy is more stringent than the self-closure policy. Allen and Saunders (1993) hence utilize the callable American put option to price the deposit insurance. Allen and Saunders (1993) show that the value of the deposit insurance can be calculated as the value of the noncallable American put option minus the value from the call provision. The value of the noncallable American put option is:   .1 C  /a  1 ; (52.6) p.a; 1I 1/ D 1C  where a is the asset-to-deposit ratio;  D 2r= 2 , r is the risk-free rate. And value from the call provision is:  c.a; 1I 1/ D a aN  C1  aN  C

  ; . C 1/ C1

(52.7)

where aN denotes the asset-to-deposit ratio that the FDIC will call the put option. So the net value of deposit insurance, evaluated as a callable perpetual American put option is Equations (52.6)–(52.7):  aN : i.a; 1I 1/ D .1  a/ N a

(52.8)

Allen and Saunders (1993) indicate that, after deducting the call provision value, the deposit insurance premium has three

822

D.Y. Hwang et al.

features. First, the simulations show that the looser the closure policy, the more valuable the deposit insurance premium. Second, the value of the deposit insurance subsidy from the insurer increases monotonically with increases in the volatility of bank assets. And third, if the insurer rewards the safer banks with looser closure policy, then the deposit insurance value attains maximum at some level of assets’ volatility, not a monotonic function. Let aN  f ./, where f 0 ./ > 0, then:

risk. By differentiating Equation (52.12) with respect to , one can obtain

  @i f 0 ./ 4r 2r f 0 ./ a Di C 3 ln C 2 : @ 1  f ./  f ./  f ./ (52.9)

That is, the value of deposit insurance with bankruptcy cost increases with the bank asset risk, which is a desirable result.

Note that the sign of Equation (52.9) is undetermined, because although the value of deposit insurance increases with assets’ risk, the likelihood of closure increases as well, which reduces the value of deposit insurance.

52.3.2.2 Hwang et al. (2009) Based on the framework of Allen and Saunders (1993), Hwang et al. (2009) introduce the bankruptcy costs into the model and derive a closed-form solution of fair deposit insurance premium, in which they set the bankruptcy cost as a function of asset return volatility. The comparative static analysis and simulation results show that the value of deposit insurance increases with the increase of asset volatility. Let kx denote discount rate at the self-closure asset-todeposit ratiox, and kaN denote the discount rate at the regulatory closure point, a, N where 0  kx ; kaN  1. Then the bankruptcy cost at the self-closure point is 1kx x and 1kaN aN at the regulatory closure point. Under this setting, the value of the noncallable perpetual put with bankruptcy cost is pbc .a; 1I 1/ D

1 1C



.1 C  /kx a 

 ;

(52.10)

and the call provision of the perpetual put with bankruptcy cost becomes      C1  cbc .a; 1I 1/ D a  aN C  kaN aN : kx . C 1/ C1 (52.11) Similarly, subtracting the value of call provision from the value of the noncallable perpetual put yields the value of the callable deposit insurance with bankruptcy costs; that is, ibc .a; 1I 1/ D .1  kaN a/ N

a  aN

:

(52.12)

Define kaN  kaN ./ where ka0N ./ < 0. That is, the bankruptcy cost is a function of bank asset risk and increases with asset

a  @ibc D .ka0N ./a/ N @ aN a  a  4r a C .1  kaN a/ N 0: ln aN aN 3 aN (52.13)

52.3.2.3 Lee et al. (2005) With the implementation of the Basel Accord, the pricing of fair deposit insurance premium becomes more complicated, since the threshold of minimum capital requirement must be considered with regard to the forbearance policy. Explicitly considering the forbearance policy taken by the regulator and the capital requirement from Basel Accord, Lee et al. (2005) derive the formula for fair deposit insurance price under this setting. They assume that banks’ auditing times are known in advance. At audit time, if the required capital ratio is maintained, the value of deposit insurance is zero. If the required capital ratio is slightly violated, but is still forborne by the regulator, the bank is allowed to operate, and will be audited later. But if the required capital ratio is severely violated, the bank is closed and the regulator assumes the capital deficits. The derived formula can be decomposed into parts – the capital forbearance component and the time forbearance component. And they have shown that a lower threshold of forbearance however increases the value of capital forbearance part because fewer insolvent banks will be closed, the value of time forbearance decreases accordingly because more banks will be closed later. In sum, lowering forbearance decreases the fair value of deposit insurance.

52.3.3 The Consideration of Interest Rate Risk Previous researches (e.g., Markus and Shaked 1984; Ronn and Verma 1986) assume that interest rates and the volatility of bank assets are constant, which are unrealistic. Using option pricing formula on coupon bond, Kerfriden and Rochet (1993) derive a formula for fair deposit insurance premium. Kerfriden and Rochet (1993) assume the interest rate follows a diffusion process, and at the same time they consider the cash flows of assets come from multi-periods. The formula that they derive is not explicit, but can be computed

52 Put Option Approach to Determine Bank Risk Premium

823

numerically. The sensitivity analysis shows that the deposit insurance premium decreases with capital-to-assets ratio, and increases with short or long interest rates.

52.3.4 Utilization of GARCH Option Pricing In Sect. 52.3.3, it was mentioned that previous studies assume that interest rates and the volatility of bank assets are constant. In empirical studies, it is often found that financial asset returns have features such as fat-tailed distribution, volatility clustering and leverage effects. Duan and Yu (1999) employ the GARCH option pricing technique to fairly value the deposit insurance. Under their approach, they consider the capital standard and the forbearance that the regulator exercises, and they use the NGARCH model to describe the bank assets return process. In their framework, deposit insurance premium can be numerically priced fairly. The remarkable feature of GARCH option pricing method is that, the fair deposit insurance premium is a function of asset risk premium. Consider an n-period model in which banks are audited at the end of each period. The asset value adjustment mechanism at the auditing time ti is 8 rti ˆ A .ti / qu Derti A .ti / Derti : ˆ : ql Derti ; otherwise (52.14) where A .ti / represents the bank asset value prior to adjustment. qu is a threshold level of asset-to-deposit ratio above which the bank will pay the cash dividends. D denotes the face value of insured deposits. The parameter  captures the level of capital forbearance, where 0 <  < 1. The bank is still allowed to operate even though the capital requirement is violated if it is forborne by the regulator. But if the capital requirement is severely violated, the insurer infuses funds to meet the capital requirement. The parameter ql D 1:087 corresponds to the 8% capital requirement agreed by the Basel Accord. The deposit insurance payoff at the auditing time ti , denoted by P .ti /, is described as follows  P .ti / D

0; A .ti / min.; 1/Derti : rti  De  A .ti /; otherwise (52.15)

The NGARCH model of bank asset return Rt C1 D ln.At C1 =At / under the risk-neutral probability measure is described as 1 Rt C1 D r  t2C1 C t C1 t C1 ; 2

(52.16)

t2C1 D ˇ0 C ˇ1 t2 C ˇ2 t2 .t    /2 ; (52.17) where t C1  "t C1 C is a standard Normal random variable w.r.t. the risk-neutral probability measure. Finally, taking expectations at time 0 w.r.t. the dynamic process in Equations (52.16) and (52.17), one can obtain the fair deposit insurance premium per dollar of deposits in each period as ın D

n 1 X rti  e E0 ŒP .ti / : nD i D1

(52.18)

52.4 Applications for Merton (1977) One of the main applications of Merton (1977) is to examine whether the deposit insurance provided by the FDIC is overpriced, underpriced, or fairly priced. Markus and Shaked (1984) propose a method that can be implemented to price the risk-based deposit insurance premium of each bank. The empirical results show that current fees charged by the FDIC are excessive. Ronn and Verma (1986) also present an empirical approach, but they point out that after taking the forbearance policy into consideration, the deposit insurance premia are underpriced rather than overpriced. Pennacchi (1987) indicates that whether the deposit insurance premia are under- (or over-) priced depends on the level of insurer’s control power over the banks. And the empirical results show that current premium lies within the upper and lower bound he presents, which shows that the FDIC has mild control over the banks. Flannery (1991) argues that the insurer can observe the financial data of banks only with measurement errors. This tends to underestimate the true value of deposit insurance that they provide. Without measurement errors, it is equivalent to adopt the risk-based deposit insurance or risk-based capital requirement. But since measurement errors exist, it is socially optimal to both adopt risk-based deposit insurance scheme and risk-based capital requirements.

52.4.1 Marcus and Shaked (1984) Utilizing the analogue between deposit insurance and put option proposed by Merton (1977), and Markus and Shaked (1984) provide a way to evaluate the risk-based deposit insurance premium of publicly traded banks. If the value of bank assets follows a stochastic diffusion process, then the fair value of deposit insurance (I / at time 0 is: I D DerT Œ1  ˆ.x2 /  e ıT V0 Œ1  ˆ.x1 / ;

(52.19)

824

D.Y. Hwang et al.

where • denotes dividend rate per dollar of bank assets, V0 denotes value of bank assets at time 0, x1 D  p p log.V0 =D/ C .r C  2 =2  ı/T = , and x2 D x1  . To calculate the volatility of bank assets value, the following equation is used:   DerT ˆ.x2 / ;  D E 1  ıT e V0 ˆ.x1 /

(52.20)

where E denotes the standard deviation of the rate of return on bank equity. Using recursive substitutions, ¢ and I can be derived from Equations (52.19) and (52.20). Two main findings from Markus and Shaked (1984) are that deposit insurance premium charged by FDIC is excessive, the same is true after taking account of the administrative costs. And the estimated risk-based premia of each bank are skewed, indicating that many relative safe banks subside the fewer but much riskier banks.

52.4.2 Ronn and Verma (1986) Contrary to Markus and Shaked (1984), Ronn and Verma (1986) point out that the deposit insurance premium charged by the FDIC is underpriced rather than overpriced. The main difference is that Ronn and Verma (1986) explicitly consider the direct assistance from the insuring agency in their model, while Markus and Shaked (1984) do not. That is, at the time of audit, if the market value of bank assets V is below the deposit liability D, the bank is not liquidated immediately. Instead, the insurer infuses funds to the bank when D  V  D .  1/. The bank is closed when V < D. The insurer exercises a kind of forbearance policy. It is the forbearance that increases the value of deposit insurance. To calculate the current market value V0 and the return volatility  of bank assets, Equation (52.20) is not used in Ronn and Verma (1986). They use the isomorphic relationship between equity value of a bank and the call option: E D V0 ˆ.d1 /  DerT ˆ.d2 /; and D

E E ; V0 ˆ.d1 /

(52.21)

(52.22)

where E denotes the equity market value of a bank, d1 D p p .log.V0 =D/ C =2/= , d2 D d1  . Implementing recursive substitutions in Equations (52.21) and (52.22) one can obtain V0 and . Inserting V0 and  into Equation (52.19), the risk-based deposit insurance premium of each bank is obtained. One caution is that, taking forbearance policy into consideration, Ronn and Verma (1986) indicate that current deposit insurance premium is underestimated, but Allen and

Saunders (1993) point out that the subsidies from FDIC tend to be overestimated. The reason is that Ronn and Verma (1986) consider the direct assistance from the insurer, so the extra costs must be added into the premium. While Allen and Saunders (1993) regard the forbearance policy as the FDIC’s right to exercise the put option before the bank chooses to self-close, hence the value of call provision has to be deducted from the premium. One can compare the methods implemented in Ronn and Verma (1986) with Mullins and Pyle (1994). Although they both use the isomorphic correspondences between options and deposit insurance or equity value, the purposes are different. In Ronn and Verma (1986), they try to solve the fair riskadjusted deposit insurance premium of listed banks. So they use the two kinds of isomorphic relationships in Equations (52.19) and (52.21). They use the publicly traded data (stock prices of listed banks) to see if the current premium is fairly priced. Mullins and Pyle (1994) use similar isomorphic relationship in Equations (52.4) and (52.5). Their purpose is to calculate the risk-based capital ratio of each bank, taking the current deposit insurance premium as given. Simply speaking, in Ronn and Verma (1986), bank’s equity value is used as inputs and deposit insurance premium as outputs. In Mullins and Pyle (1994), current deposit insurance premium is used as inputs and the equity value of banks (and hence the riskbased capital ratio) are derived as outputs.

52.4.3 Pennacchi (1987) Marcus and Shaked (1984) have reached the conclusion that the fees charged by FDIC are overpriced under current deposit insurance scheme. Pennacchi (1987) uses a different argument to explain this phenomenon. He states that the judgment of whether most banks are currently under- or overcharged by the FDIC must hinge on the degree of FDIC’s regulatory control over banks. If the FDIC has full control, then FDIC can force the bank to infuse funds immediately as its minimum capital requirements are not met. In this case, the deposit insurance premium is overcharged because the insurer’s liability is lower. On the contrary, if the FDIC has no control over the banks; that is, it has no power to force the bank to increase capital ratio, then the premium is undercharged because the liability the FDIC assumes increases. In Marcus and Shaked (1984), it is assumed that a bank is closed by the insurer when it is insolvent. So it is implicitly assumed that the FDIC has full control over the banks in Marcus and Shaked (1984). The deposit insurance premium tends to be underpriced under this assumption. Pennacchi (1987) assumes that the bank assets market value and the deposit liability both follow diffusion processes. Merton (1978), also assumes that the insuring agency incurs surveillance costs and the audit times are

52 Put Option Approach to Determine Bank Risk Premium

Poisson-distributed. Under this setting, he derives the fair deposit insurance premium for both scenarios – the FDIC has full control and the FDIC has no control the banks. Because in fact the FDIC has some power over the banks, these two premia are the upper and lower bound of fair premia. And the empirical results show that the current premium lies between the upper and lower bound.

52.4.4 Flannery (1991) To price the deposit insurance premium fairly, one necessary condition is that the  and D are accurately estimated in Equation (52.4). That is, the true financial data of a bank, the risk of bank assets value, and the liability level of a bank must be known to the insurer without errors. Flannery (1991) indicates that there are reasons that the regulator measures the financial data of banks with errors. First, bank assets are often opaque and hard to monitor. Evaluating the value of bank assets is costly. Second, even if the insurer knows the true value in advance, bankers have strong incentives to change the financial condition ex post, because increasing assets volatility and liability level raises the value of deposit insurance that they receive. If there are no measurement errors, it is equally well to adopt the risk-based capital requirement or risk-based premia. But since measurement errors exist, it is socially optimal to maintain risk-based capital requirement while charge riskbased deposit insurance premia. The argument is as follows. The value of deposit insurance increases with the leverage level of a bank. Although the measurement errors increases with leverage level and are socially costly, leverage levels are also socially beneficial. Hence for the FDIC to maximize the social welfare, there exists an optimal level of leverage. So to maximize the social welfare, both risk-based capital ratio and risk-based deposit insurance scheme should maintain.

52.5 Conclusion A remarkable feature of banking industry is to serve as an intermediary. That is, to lend money to borrowers and receive funds from depositors. To function well, from the perspective of depositors the banks should be risk-free. Otherwise, depositors have to collect information from banks, or to diversify their deposits across banks, which is costly. One way to solve the problem is to introduce the third-party liability guarantee. Considering the size of the whole banking industry, only the government or its agencies can be the liability guarantee guarantor. Such guarantee incurs costs to the insurer. To measure the costs of the guarantor, Merton (1977)

825

first proposes an isomorphic correspondence between a put option and the deposit insurance guarantee, and then uses the Black-Scholes formula to price the fair deposit insurance premium. The reason behind the isomorphic correspondence is that at the end of the period, if the bank is solvent, then the deposit insurance is worthless. But if the bank is insolvent, the insurer must infuse funds to meet the liabilities. Hence the payoff of the deposit insurance corresponds to the put option written on the bank assets with strike price equal to the insured deposits. The simplest version of deposit insurance pricing shows that the deposit insurance premium decreases with the asset-to-deposit ratio, and increases with the volatility of bank asset value. Since the pioneering work of Merton (1977), many studies follow on. Merton (1978) considers the surveillance costs with random auditing times and shows that the fair deposit insurance premium does not decrease in the asset-to-deposit ratio monotonically. Mullins and Pyle (1994) also consider the auditing costs, but they calculate the risk-based capital ratio given the deposit insurance premium set by the regulator. They show that both the risk-based capital ratio and the deposit rate spread increases with the auditing costs. Allen and Saunders (1993), Hwang et al. (2009), and Lee et al. (2005) all value the deposit insurance premium with the forbearance policy taken into account explicitly. Allen and Saunders (1993) show that if the regulator rewards the safer banks with looser closure policy, the value of deposit insurance will not increase with the bank asset risks monotonically. Inspired by Allen and Saunders (1993), Hwang et al. (2009) show how the bankruptcy costs can be incorporated into the deposit insurance pricing model. Besides, Lee et al. (2005) derive a formula of fair deposit insurance premium, which explicitly considers the capital requirements by Basel Accord. Previous studies follow the framework of Black-Scholes and assume that the risk-free rate and the volatility of bank asset value are constant. Kerfirden and Rochet (1993) provide a way of pricing deposit insurance premium under stochastic interest rate. Duan and Yu (1999) show how the CARCH option pricing model can relax the constant volatility assumption. An important application of Merton (1977) is to test whether the deposit insurance is fairly priced. Markus and Shaked (1984) point out that the deposit insurance charged at that time are excessive even after considering the administration costs of the FDIC. However, Ronn and Verma (1986) claim that the deposit insurance is underpriced. The main reason is that the forbearance policy exercised by the regulator is neglected in Markus and Shaked (1984). Pennacchi (1987) argues that whether the deposit insurance is under- (over-) charged depends on the level of insurer’s control power over the banking industry. Flannery (1991) indicates that the deposit insurance premium tends to be underpriced because the

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D.Y. Hwang et al.

insurer can only measure the bank asset risks with errors (underestimate the risks ex ante). Regarding the deposit insurance scheme, there is still much room for improvement. For example, a mechanism is needed that minimizes the effects of moral hazard, or modifies the deposit insurance scheme, which will comply with the supervision of capital requirement.

(1) Cox and Ross (1976) and Harrison and Kreps (1979) propose the martingale pricing method to price the value of an option. This method is briefly introduced as follows. Assume that the price of the underlying asset of a put option follows a diffusion process:

References

where W P denotes a Wiener process under the probability measure (P-measure).

Allen, L. and A. Saunders. 1993. “Forbearance and valuation of deposit insurance as a callable put.” Journal of Banking and Finance 17, 629–643. Bjork, T. 1998. Arbitrage theory in continuous time, Oxford University Press, Oxford. Cox, J. C. and S. Ross. 1976. “The valuation of options for alternative stochastic process.” Journal of Stochastic Process 3, 145–166. Duan, J. C. and M. T. Yu. 1999. “Capital standard, forbearance and deposit insurance pricing under GARCH.” Journal of Banking and Finance 23, 1691–1706. Epps, T. W. 2000. Pricing derivatives securities, World Scientific, Singapore. Flannery, M. J. 1991. “Pricing deposit insurance when the insurer measures bank risk with an error.” Journal of Banking and Finance 15, 975–998. Harrison, M. and D. Kreps. 1979. “Martingales and multiperiod securities markets.” Journal of Economic Theory 20, 381–408. Hwang, D.-Y., F.-S. Shie, K. Wang and J.-C. Lin. 2009. “The pricing of deposit insurance considering bankruptcy costs and closure policies.” Journal of Banking and Finance 33(10), 1909–1919. Kerfriden, C. and J. C. Rochet. 1993. “Actuarial pricing of deposit insurance.” Geneva Papers on Risk and Insurance Theory 18, 111–30. Lee, S. C., J. P. Lee, and M. T. Yu. 2005. “Bank capital forbearance and valuation of deposit insurance.” Canadian Journal of Administrative Sciences 22, 220–229. Marcus, A. J. and I. Shaked. 1984. “The valuation of FDIC deposit insurance using option-pricing estimates.” Journal of Money, Credit and Banking 16, 446–460. Merton, R. C. 1977. “An analytic derivation of the cost of deposit insurance and loan guarantees.” Journal of Banking and Finance 1, 3–11. Merton, R. C. 1978. “On the cost of deposit insurance when there are surveillance costs.” Journal of Business 51, 439–452. Mullins, H. M. and D. H. Pyle. 1994. “Liquidation costs and risk-based bank capital.” Journal of Banking and Finance 18, 113–138. Pennacchi, G. G. 1987. “A reexamination of the over- (or under-) pricing of deposit insurance.” Journal of Money, Credit and Banking 19, 340–360. Ronn, E. I. and A. K. Verma. 1986. “Pricing risk-adjusted deposit insurance: an option-based model.” Journal of Finance 41, 871–895.

dS D dt C dW P ; S

Let dW P D dW Q 

  r 

Proof of Equation (52.1) In this appendix, we first derive the European put option pricing formula, and then apply this formula to the deposit insurance premium in Merton (1977).

dt;

(52.24)

where W Q is the Wiener process under the equivalent martingale measure (Q-measure) of the original measure (P-measure), and the asset return process under the Q-measure is dS D rdt C dW Q : (52.25) S By Ito’s Lemma,

2 Q dt C  Wt ; d ln St D r  2

(52.26)

2 Q T C  WT ; ST D S exp r  2

(52.27)

and

Q

where WT  N Q .0; T /. Furthermore, let d W Q D d W R C dt, then under Rmeasure

2 T C  WTR ; ST D S exp r C 2

(52.28)

where WTR  N R .0; T /. Rt 2 2 Q Let t  exp. 2 t C  Wnt / D exp. 1 2 0  ds

o Rt Rt and since E exp 12 0  2 ds D C 0 dW Q s /, 1 2  exp 2  t < 1; by the Girsanov Theorem, t is a martingale process adapted to a filtration Ft and E Q .t IA / D E R .IA /;

Appendix 52A

(52.23)

(52.29)

where IA is an indicator function; that is, IA D 1 if event A occurs and IA D 0 otherwise. Interested readers not familiar with these tools, please refer to Pricing Derivatives Securities (Epps 2000) and Arbitrage Theory in Continuous Time (Bjork 1998) for more details. (2) The payoff of a European put option holder at time T is Max .K  ST /. Where K is the strike price and ST

52 Put Option Approach to Determine Bank Risk Premium

827

denotes the price of the underlying asset at T . The fair price of the put P in the beginning is thus P D e rT E Q Œmax.K  ST ; 0/ D e rT E Q Œ.K  ST /IA ; (52.30) where A D fST jST < Kg. Hence P D KerT E Q .IA /  e rT E Q .ST IA /:

(3) Return to Merton (1977). To derive Equation (52.1), one may simply substitute V D D, KerT D D (the value of insured deposits in the beginning) and  2 =2 D  into Equation (52.35). Let d  D=V and divide both sides of (A13) by D, Equation (52.1) can be easily obtained. Note that  p  x D ln.K=S/  .r   2 =2/T = T   p D ln.K=S/ C ln e rT C  2 =2 = ;   p D ln.Ke rT =S/ C  2 =2 = 

(52.31)

We deal with the term E Q .IA / first and then calculate E Q .ST IA /. By Equation (52.27), E Q .IA / D PrQ .ST < K/ D PrQ .ln ST < ln K/ h i Q D PrQ ln S C .r   2 =2/T C  WT < ln K : " # Q ln.K=S/  .r   2 =2/T Q WT < p p D Pr T  T D ˆ.x/

D h2

and h1 D x 

p .

Appendix 52B

(52.32) Proof of Equations (52.2) and (52.3)

Q

WT

 N .0; T /, ˆ./ denotes the Recall that CDF of the standard Normal   p distribution, and x D ln.K=S/  .r   2 =2/T = T . Under Q-measure, Q

h i Q ST D S exp .r   2 =2/T C  WT

2  Q D Se rT exp  T C  WT : 2 D SerT T

Pf:

Q

(52.33)

R

D SerT E R .IA / D SerT PrR .ST < K/ D Se

rT

and

@h2 @d

D

@h1 , @d

then

< ln K/:  D SerT PrR ln S C .r C  2 =2/T  C WTR < ln K  

W R ln.K=S/  .r C  2 =2/T D SerT PrR p T < p T  T

p D Se rT ˆ x   T (52.34)

Substituting Equations (52.33) and (52.34) into Equation (52.31), one may obtain the premium of a European put (52.35)

ˆ0 .h1 / d

D

ˆ0 .h1 / p . 2d 

then

ˆ0 .h1 / @g D @ d We know that

@g @

@g @h2 ˆ0 .h1 / @h1 D ˆ0 .h2 /  : @ @ d @

Since ˆ0 .h2 / D

PrR .ln ST

p P D KerT ˆ.x/  Sˆ x   T :

ˆ0 .h1 / d

(2) Proof of Equation (52.3), Pf:

rT

ˆ.h1 / . d2

ˆ.h1 / @g D @d d2

E .ST IA / D Se E .T IA / D Se E .IA / rT

D

ˆ.h1 / ˆ0 .h1 / @h1 @g @h2 D ˆ0 .h2 / C :  @d @d d2 d @d

Since ˆ0 .h2 / D

By Equations (52.28) and (52.29), Q

@g @d

(1) Proof of Equation (52.2),

@h2 @



@h1 @

D



@h2 @h1  @ @

@.h2 h1 / @

ˆ0 .h1 / @g D p : @ 2d 

D

:

1 p , 2 

we have

Chapter 53

Keiretsu Style Main Bank Relationships, R&D Investment, Leverage, and Firm Value: Quantile Regression Approach Hai-Chin Yu, Chih-Sean Chen, and Der-Tzon Hsieh

Abstract Using quantile regression, our results provide explanations for the inconsistent findings that use conventional OLS regression in the extant literature. While the direct effects of R&D investments, leverage, and main bank relationship on Tobin’s Q are insignificant in OLS regression, these effects do show significance in quantile regression. We find that firms’ advantages with high R&D investment over low R&D monotonically increase with firm value, appearing only for high Q firms; while firms’ advantages with low R&D over high R&D monotonically increase with firm value for low Q firms. Tobin’s Q is monotonically increasing with leverage for low Q firms; whereas it is decreasing in high Q firms. Main banks add value for low to median Q firms, while value is destroyed for high Q firms. Meanwhile, we find the interacted effect of main bank and R&D investment which increases with firm value, only appears in medium quantiles, instead of low or high quantiles. Results of this work provide relevant implications for policy makers. Finally, we document that industry quantile effect is larger than the industry effect itself, given that most of the firms in higher quantiles gain from industry effects while lower quantile firms suffer negative effects. We also find the results of OLS are seriously influenced by outliers. In stark contrast, quantile regression results are impervious to either inclusion or exclusion outliers. Keywords R&D investment r Quantile regression r Keiretsu r Main bank r Leverage

H.-C. Yu () Department of International Business, Chung Yuan University & Rutger University Chung-Li, 32023, Taiwan e-mail: [email protected]; [email protected] C.-S. Chen Institute of Management, Chung Yuan University, Chung-Li, 32023, Taiwan e-mail: [email protected] D.-T. Hsieh Department of Economics, National Taiwan University, Taipei, 100, Taiwan e-mail: [email protected]

53.1 Introduction Research and development (R&D) investment is considered a crucial input for firms’ growth and performance, especially in technology and science intensive industries. Among developed countries, Japan is thought to be one of the most successful countries in manufacturing. Why is manufacturing so successful in Japan but not in the U.S. or other countries? Our data shows that more than 50% of Japanese listed firms belong to the technology-based industry underscoring the importance of R&D in boosting long-term economic growth. Owing to the fact that R&D investments is intangible, the asymmetric information problem with respect to R&D investment is expected to be more serious than physical investment, since managers may be inclined to keep confidential information to maintain competitive advantage. This, in turn, makes it difficult for investors to measure the effect of R&D investment on firm value. Extant literature shows divergent findings regarding the impacts of R&D on firms’ value rendering these results inconclusive. For example, Woolridge (1988) finds a significant market response to the increase in R&D expenditure; however, the reaction differs based on different industries. Chan et al. (1990) and Zantout and Tsetsekos, (1994) report a positive market response to R&D increases for firms in high-tech industries vs. a negative market response for low-tech industries. Acs and Isberg (1996) suggest the final impact of financial leverage and R&D investment on Tobin’s Q depends on firm size. Ho et al. (2006) also indicates that a firm’s growth opportunity from R&D depends on its size, leverage, and industry concentration. Conversely, some researchers such as Doukas and Switzer (1992) and Sundaram et al. (1996) find an insignificant effect. When combined, the above evidence suggests that due to an asymmetry of information outside investors may not completely understand the value-adding characteristic of R&D. More to the point, if a significant asymmetric information problem exists in intangible R&D investment, then firms with main bank relationships may help reduce this uncertainty and convey a positive signal. If this logic is correct, then firms with a main bank relationship facing high R&D

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_53, 

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investment should provide more credible information in order for investors to accurately evaluate firm value than those firms without it. In turn, this superior level of information should be reflected on firm value. Additionally, this signal is expected to be stronger when the asymmetric information problem is more serious between managers and investors. Another point that makes this issue valuable is the impact of R&D investment on long-term performance. This characteristic conforms with the Japanese sample where the main bank dominates almost 71% of outstanding shares (Sheard 1994). This concentrated and stable ownership structure promotes efficient monitoring and long-term performance without regard to short-term pressures. Given that the main bank helps to reduce the conflict between managers and creditors and positively aligns their relationship, using the main bank as a moderator variable is expected to produce different findings vs. to those countries that utilize highly individual ownership data (Aoki 1990; McGuire and Dow 2002). In Japan, banks traditionally maintain multidimensional ties with their clients, providing security, insurance, or banking services. This approach of banking has come to be commonly known as Keiretsu (or main bank) style of banking. From a positive viewpoint, a main bank gathers proprietary information through the financing process. This inside information (and hence the quality of the firm) is then signaled to the market by way of a main bank financing decision. Specifically, if the main bank is willing to continue financing, a signal of firm quality is conveyed to outside investors. It is through this mechanism that affiliation with main banks creates value. Moreover, due to lower agency cost induced by significant equity holdings of main banks, a higher level of performance is expected. From a negative point of view, some high-tech intensive firms may not prefer to interact with a main bank because potential activity on the part of the main bank may inadvertently reveal a firm’s prospects to its competitors. Additionally, in the case of financial distress, managers may expect the main bank to solve their financial shortcomings, leading to moral hazard problems and an increased propensity for bankruptcy. Therefore, main bank relationships may exacerbate a firm’s problems and reduce its performance. Extant literature has not revealed a consistent finding regarding whether main banks help increase or decrease firm value. Given that main banks serve simultaneously as the internal governance (board) and external monitoring (principal creditor), more research designed to clarify the relationship of main banks and their influence on a firm’s values is warranted. Our paper differs from the existing literature in a number of ways. First, we use quantile regression instead of conventional OLS to add additional evidence that helps explain the empirical puzzle concerning the effects of R&D expenditure,

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leverage, and main bank relationships on firms’ value. Since there were a great deal of bubble Internet stocks in the hightech industry during 1990s in Japan, the average Tobin’s Q would be overestimated by using conventional OLS. To avoid this upward bias resulting from including extremely high Tobin’s Q firms, a more realistic view may be gained by using lower quantiles as the target point. Our data from 1994 to 2004 covers the whole Internet evolution period, illustrating again the advantage of using quantile approach. Second, we include the presence of main bank relationships in model of firm value in order to capture the impact of this relationship and its ability to reduce asymmetric information and enhance firm value. Third, besides testing the independent effects of R&D and main bank influence on firm value, the interaction effects of R&D with leverage and with main bank relationship are also included in the model. Prior research treats these characteristics as independent effects, thus failing to test for possible interaction effects of these variables. Fourth, industry dummies are being controlled in order to avoid any biases resulting from industry characteristics. Fifth, our data set is unique in that we are able to merge a number of sources of data (such as COMPUSTAT Global and Emerging Markets database and PACAP database), allowing for a through examination (by way of quantile estimators) of the impact of Keiretsu style banking on firm value. We find that R&D investment, financial leverage, and Keiretsu style main banking all have quantile effects on firm value. The impact of R&D investment, financial leverage, and Keiretsu style banking on firm value varies with the various Tobin’s Q quantile of the firms rather than firm size as suggested by Acs and Isberg (1996). These results are robust after controlling for industry types. Additional findings include, but are not limited to the following: (a) Firms’ advantages with high (low) R&D investment over low (high) R&D monotonically increase with firm value for high (low) Q firms; (b) Tobin’s Q is monotonically increasing (decreasing) with leverage for low (high) Q firms; (c) for low Q firms, R&D investment and leverage complement each other to create value; for high Q firms, R&D investment and leverage substitute each other; (d) main bank relationships play a significant role in adding value only for low to median Q firms whereas, in contrast, the main bank effect turns out to be negative with firms staying at extremely high Q; and (e) finally, we find that R&D efforts from the industry itself do not have a systematic positive or negative impact on firms’ value. Instead, most of the firms in higher quantiles gain from industry effects while lower quantile firms suffer negative effects. The paper is organized as follows. A review of the extant literature is provided in Sect. 53.2. Section 53.3 presents the data and sample. Section 53.4 reports the methodology and results, Section 53.5 concludes.

53 Keiretsu Style Main Bank Relationships, R&D Investment, Leverage, and Firm Value: Quantile Regression Approach

53.2 Literature Review The effect of R&D investments on Tobin’s Q and the relationship between R&D investments and debt financing is explored in numerous studies (Szewezyk et al. 1996; Zantout 1997; Vincente-Lorente 2001; O’Brien 2003). However, their findings are inconsistent. Some findings document that debt has a positive impact on R&D investment; others document a negative effect, while still others find no effect at all. These inconsistencies are not surprising. From one point, debt can act as a positive influence on managerial behavior by motivating managers to invest in projects with positive net present value. Thus, R&D investments undertaken in a high debt ratio regime are expected to positively enhance value. This helps explain why the relationships between stock returns and R&D investments announcements is positive only in firms with relatively high debt ratio (Zantout and Tsetsekos 1994). From another point, R&D investments have been viewed as high risk and intangible such that managers may not be interested in firms with higher debt ratios. In such cases, underinvestment (or asset substitution) may occur more easily than other physical investments leading to negative impacts reflected in debt ratios. Hence, Bhagat and Welch (1995) suggest a negative relationship between R&D investments (witnessed through debt ratios) and firm value – Bradley et al. (1984) and Long and Malitz (1985) originally link leverage and R&D expenditures. Thereby, highly leveraged firms are supposed to invest less in R&D than other physical assets. Acs and Isberg (1996) address the additional issue of growth opportunities, finding that financial leverage’s and R&D investment’s influence on growth opportunities depends on firm size and industry structures. As we know, agency cost and information asymmetry problems are likely to negatively influence the benefits of a highly leveraged firm from its R&D investments. Daniel and Titman (2006) suggest asymmetric information reduces the attractiveness of R&D projects to outside investors (debt holders) with managers keeping “know-how” for competitive reasons. This problem renders debt financing of R&D projects more expensive and less preferred. Daniel and Titman (2006) document that stock returns are positively related to price-scaled variables such as the book-to-market ratio (BM) and suggest the book-to-market ratio can forecasts stock returns because it is a good proxy for the intangible return. Daniel and Titman (2006) provide an argument for why high book-to-market value firms realize high future returns and find that future returns are unrelated to the accounting measures of past performance (tangible information), but are negatively related to future performance (intangible information). Among different types of intangible information, R&D is the most crucial variable when it comes assessing future performance. We measure firms’ future performance

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by Tobin’s Q ratio as reported in the extant studies (Lang et al. 1989, 1991; Doukas 1995; Szewczyk et al. 1996, etc.).1 Chan et al. (1990) and Zantout and Tsetsekos (1994) suggest that R&D investment by firms with promising growth opportunities are basically worthy of inclusion into any model of firm value. Due to the potential uncertainty caused by the presence of asymmetry of information between firms and investors and the unobservable nature of management, R&D is likely to be discounted in high-leverage firms. Said another way, asymmetric information problems and agency costs may render debt financing of R&D investments more expensive than other options. Support for this argument is found in Bhagat and Welch (1995), which find a negative relationship between debt ratios and R&D investments on firm value; and Kamien and Schwartz, 1982), which find that investors value a firm’s investments in R&D when they use their own funds. In addition to asymmetric information and agency, transaction cost economics may explain part of the evidence about the types of investments made by managers. Williamson (1988) points out that the choice of project financing depends on the characteristics of the assets. Titman and Wessels (1988) also postulate that “debt level is negatively related to the uniqueness of a firm’s line of business.” Therefore, firms with higher degrees of firm-specific assets are found to have less debt (Williamson 1988; Balakrishnan and Fox 1993; Vincente-Lorente 2001; O’Brien 2003). Our results compensate the lack of literature in addressing the relationship between R&D and firm value, R&D and leverage, and main bank and R&D. The results are robust and consistent after making some relative tests.

53.3 Data and Sample 53.3.1 Data Source and Sample Description To construct the sample for this research, two different databanks are employed: (a) COMPUSTAT Global and Emerging Markets database and (b) PACAP (Pacific-Basin Capital Markets) database. The financial data for Japanese-listed companies is obtained from PACAP database. Starting with an original sample of 3,499 non-financial firms listed on the Tokyo Stock Exchange and, after eliminating the firms with reported data that are not creditable,2 an effective sample 1

The theoretical Tobin’s Q is the ratio of the market value to the replacement cost of the assets. Because some data is not available, it has been often estimated by a simple measure of Q (the “pseudo Q”): market value to book value of the total assets. 2 For example, some firms are characterized by negative debt or negative sales without proper explanations in footnotes.

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of 40,575 year-firm observations is derived from an original sample of 41,470 observations.3 Additional data, such as firm R&D expenditure, is taken from the Toyo Keizai’s Tokei Geppo Statistics Monthly. The sample is next classified into eight industry types based on industry classifications proposed by Chan et al. (2001). Specifically, our samples is comprised of transportation (SIC D 37), communications (SIC D 48), electronic equipments (SIC D 36), measuring instruments (SIC D 38), computer and office equipment (SIC D 357), drugs and pharmaceuticals (SIC D 283), and computer programming and software (SIC D 737) eight industries. Panel A of Table 53.1 reports the summary statistics of the full sample, while Panel B, C, and D presents the statistics for the main bank clients, unaffiliated firms, and selected industries, respectively. It is worth noting that in order to avoid biases resulting from extreme values in descriptive statistics, we drop the sample outliers based on if their Tobin’s Q is less than 1% or larger than 99%.4 Panel A reveals that the mean (median) of RD_TA in Japan is 2.0% (1.2%), which is low compared with the current American firms that have an average of 10.2–16.0% (Ho et al. 2006; Cui and Mak 2002). The median of RD_TA is 1.2% (smaller than the mean of 2.0%), implying the distribution of RD_TA is dominated by firms with higher RD_TA. Figure 53.1a shows the trend of average RD_TA rose from 1.915% in 1994 to 2.15% in 1998, then dropped to 1.85% in 1999, and then increased to 2.003% in 2004, an increase of only 0.105% during these 10 years. This trend reveals there was no significant change on R&D investment in Japan during our sample period. Figure 53.1a also shows main bank clients do not always have significantly higher or lower R&D expenditure than their counterparts of unaffiliated firms. Besides, main bank clients manifest a similar trend on R&D expenditure as the full sample, which shows the highest peak during 1997–1998. This peak may have been caused by the shrinkage of total assets during Asian financial crisis. The leverage ratio (LEV) is 0.591, which is extremely high compared to the American firms that have an average of 0.158–0.25 only (Ho et al. 2006; Johnson 1997). This figure implies Japanese firms use higher financial leverage in their capital structure, which may related to their main bank system. Furthermore, among these leverages, the long3

The financial firms were excluded from the overall sample as the financial firms exhibit different balance sheet items from those of the non-financial firms. 4 Our extreme value definition is different from Ho et al. (2006) who deleted the extreme value below the first quantile less 1.5 times the interquantile range or any value greater than the third quantile plus 1.5 times the interquantile range. The reason why we need to delete the outliers is that we find some firms with Tobin’s Q extremely high, which is beyond our understanding. We think it may be a result of the bubble Internet stock collapse before Asian Financial Crisis.

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term debt ratio is only 11.6%, which is lower than those of U.S. firms averaging 18% (Aivazian et al. 2005), revealing another unhealthy debt structure, since most of the leverages are short-term debts. Figure 53.1b shows the time trend of average leverage. The average leverage falls from 61.4% in 1994 to 53.8% in 2004. Figure 53.1b also shows main bank clients have significantly higher leverages (with an average of 0.652) compared to independent firms (with an average of 0.576) during 1994–2004. The average Tobin’s Q is 8.012 (with a median of 1.466), which is high compared to U.S. listed firms that have an average of 2.89 (Cui and Mak 2002). This high Tobin’s Q may be attributable to the Asian Financial Crisis during 1997–1998 when the economy was bubbling and the dramatic volatility of high tech stocks in 1999. Figure 53.1c displays the trend of Tobin’s Q over 1994–2004. It shows the average Tobin’s Q rose to 66.0 in 2004 from 10.0 in 1994, an increase of six times the value of 1994 levels5 It also shows main bank clients have the significantly lower Tobin’s Q (5.137) vis-àvis independent firms (8.78) over 1994–2004, indicating that main bank clients do not perform better. Keiretsu-style main bank ratio (MB_D) is 19.5%, meaning only one fifth of listed firms have affiliated relationship with Keiretsu-style banking. The average log size (LOG_TA) is 4.3, which is small in comparison to American firms that average 17.89 (Cui and Mak 2002), implying a potential difference in size effect may exist. The average (median) capital expenditure rate (CAPEX_TA) is 2.35% (0.00%), which is negative, and the median of zero% revealed that half of the Japanese firms did not have capital expenditures during the sample period. The net cash flow ratio on average is 7.24% with a negative skewness, implying a high likelihood of negative cash flow among the listed firms. Earnings volatility (EBIT_VOL) is 5.316%, similar to American firms of 4.79% (Jonhson 1998) Dividend payout ratio (DIV_TA) on average (median) is only 0.5% (0.4%), which is low when compared to U.S. firms (Jonhson 1998). Panels B and C then compare the characteristics of Keiretsu-style main bank clients and unaffiliated firms. Not surprisingly, main bank clients show higher leverage (0.65 vs. 0.58), larger size (4.47 vs. 4.25), and lower earnings volatility (3.21 vs. 5.84) than unaffiliated firms. However, it shows a lower median capital expenditure ratio (3.89 vs. 1.97) and cash flow ratio (1.82 vs. 0.46) in comparison to independent firms. This implies main bank clients are larger, have higher leverage, and lower earning volatility. However, these firms paid lower dividends and have lower cash flows ratios and capital expenditure.

5

The time trend utilizes the original sample without deleting the outliers. The summary statistics in Table 53.1 use the sample after deleting Tobin’s Q, which is less than 1% or larger than 99% in the whole sample

53 Keiretsu Style Main Bank Relationships, R&D Investment, Leverage, and Firm Value: Quantile Regression Approach

Surprisingly, main bank clients show a lower R&D expenditure on average (0.019 vs. 0.020), but a higher R&D expenditure on median (0.012 vs. 0.011), suggesting there is no significant difference on investing R&D expenditures. Furthermore, main bank clients show a lower Tobin’s Q on average (5.14 vs. 8.71), but a higher Tobin’s Q on median (1.73 vs. 1.36), suggesting main bank clients perform poorly and are dominated by lower Q firms. Panel D reports the selective statistics on different industries. The drug and pharmaceutical industry shows the highest R&D expenditure of 6% associated with the lowest leverage of 0.38, followed by the computer and office equipment industry with R&D expenditure of 4% associated with the leverage of 0.51, and the measuring instruments industry with R&D inputs of 3.6% and leverage of 0.54. Communication industry shows the highest Tobin’s Q of 58.72, followed by the computer programming and soft-

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ware (26.79) and computer and office equipments (10.60). The extremely high Tobin’s Q reflects the bubble stocks of high tech during the sample period and underscores the reason for employing quantile regression in this work. The transportation equipment industry shows the lowest Tobin’s Q of 3.23 with the second lowest R&D inputs of 2.4% and highest leverage of 0.61, reflecting the general traditional industry characteristic: low R&D inputs, highly leveraged and low performance of Q.

53.3.2 Keiretsu and Main Bank Sample The Keiretsu data comes from Industrial Groupings in Japan. This handbook provides the data for each company belonging to specific Keiretsu and its relationship strength with the

Table 53.1 Descriptive statistics and industry distribution Panel A: Full sample statistics Variables

Obs

Mean

25%

TobinQ RD_TA LEV MB_D LOG_TA CAPEX_TA CF_TA EBIT_VOLA DIV_TA

35990 16694 36725 36725 36725 33700 36080 25055 36725

8:012 0:020 0:591 0:195 4:293 2:335 0:724 5:316 0:005

0:768 0:004 0:429 0:000 3:935 0:008 0:000 0:278 0:000

Median 1:466 0:012 0:602 0:000 4:342 0:000 0:026 0:735 0:004

75%

Skewness

Kurtosis

3:546 0:026 0:758 0:000 4:752 0:015 0:052 1:952 0:007

12:420 2:707 5:774 1:564 0:657 19:437 94:142 104:394 8:201

171:485 14:896 220:649 3:447 3:945 495:726 11695:460 11331:750 186:577

75%

Skewness

Kurtosis

Panel B: Main bank clients statistics Variables

Obs

Mean

25%

TobinQ RD_TA LEV MB_D LOG_TA CAPEX_TA CF_TA EBIT_VOLA DIV_TA

7018 4132 7049 7049 7049 6386 6964 5000 7049

5:137 0:019 0:652 1:000 4:467 3:891 1:817 3:219 0:004

1:100 0:005 0:521 1:000 4:064 0:011 0:000 0:305 0:000

Median 1:726 0:012 0:668 1:000 4:577 0:000 0:019 0:913 0:004

3:114 0:025 0:800 1:000 5:023 0:009 0:041 2:296 0:007

18:995 1:979 4:004

403:701 8:346 135:686

0:743 12:425 35:304 12:173 5:415

3:165 172:491 1982:061 221:677 94:367

75%

Skewness

Panel C: Unaffiliated firms statistics Variables

Obs

Mean

25%

TobinQ RD_TA LEV MB_D LOG_TA CAPEX_TA CF_TA EBIT_VOLA DIV_TA

28972 12562 29676 29676 29676 27314 29116 20055 29676

8:708 0:020 0:576 0:000 4:252 1:971 0:462 5:838 0:005

0:700 0:004 0:410 0:000 3:913 0:008 0:001 0:272 0:000

Median 1:362 0:011 0:583 0:000 4:301 0:000 0:027 0:693 0:004

3:674 0:027 0:745 0:000 4:680 0:018 0:055 1:869 0:008

Kurtosis

11:629 2:776 6:170

150:366 15:159 234:627

0:711 22:455 101:212 93:482 8:173

4:321 667:718 12523:120 9081:347 179:844 (continued)

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Table 53.1 (continued) Panel D: Selected industries statistics Industries

TobinQ

RD_TA

IND_D1(Transportation equipment)

37

1146

3:203

0:024

0:609

IND_D2(Communications)

48

150

58:720

0:007

0:532

IND_D3(Electrical equipment excluding)

36

2336

9:280

0:032

0:558

IND_D4(Measuring instruments)

38

754

7:442

0:036

0:537

IND_D5(Computers and office equipment)

357

335

10:599

0:039

0:513

IND_D6(Drugs and pharmaceuticals)

283

498

8:358

0:060

0:385

IND_D7(Computer programming, and software)

737

1199

26:789

0:022

0:419



29572

7:058

0:014

0:607

IND_D8(Others)

SIC

Obs

LEV

For sample period from 1994 to 2004, Panel A presents the basic statistics in the full sample, Panel B, C and D present the statistics for the main bank clients, unaffiliated firms, and selected industries, respectively. To avoid the bias from extreme value, we drop the sample outliers based on if their Tobin’s Q is less than 1% or larger than 99%. TOBINQ is the sum of market value of firms’ equity and book value of total liabilities divided by total assets; RD_TA is R&D expenditure divided by total assets; LEV is total debt divided by total assets; MB_D is the Keiretsu style main bank relationship which is given a 1 or 0; LOG_TA is the log of firm’s total assets; CAPEX_TA is capital expenditure divided by total assets; CF_TA is the ratio of net cash flow to total assets; EBIT_VOLA, the volatility of earnings before interest and taxes, is the standard deviation of first differences of earnings before interest and tax (EBIT) divided by total assets; DIV_TA is the cash dividend payout ratio. IND_D is the dummy variable of industries, IND_D1 denotes transportation equipment, IND_D2 denotes communications industry, IND_D3 denotes electrical equipment excluding computers, IND_D4 denotes measuring instruments, IND_D5 denotes computers and office equipment, IND_D6 denotes drugs and pharmaceuticals industries, IND_D7 denotes computer programming, software, and services industries, IND_D8 denotes the others industries

Keiretsu. There are eight commonly recognized horizontal keiretsu: Mitsubishi, Mitsui, Sumitomo, Fuyo, DKB, Sanwa, Tokai, and IBJ. Our sample shows that these eight commonly recognized horizontal keiretsu – Mitsubishi, Mitsui, Sumitomo, Fuyo, DKB, Sanwa, Tokai, and IBJ – have 269, 209, 247, 222, 207, 191, 60, and 47 affiliated firms, respectively. Of those, only 127, 85, 109, 113, 91, 94, 37, and 26 are publicly listed. Hence, our data shows only 19.5% of Japanese listed firms having strong ties with one of the eight keiretsu groups; this ratio is lower than that of 89 out of 200 reported by Hoshi et al. (1991). Our analysis of main banks is operationally defined as each firm’s top bank on its trading bank list in Kaisya Shikiho (the Japan Company Handbook). Thus, a firm’s main bank is its largest lender as well as the top shareholder among banks with the firm. This data is collected from Toyo Keizai’s Kigyou Keiretsu Soran (the Japanese Keiretsu Handbook). Industrial groupings in Japan provides information about the amount of loans provided by each bank for Keiretsu members. Finally, data and names of lenders are obtained from the Japan Company Handbook. The handbook discloses the major sources of funding for each Japanese listed company. The names of principal banks were obtained from the reference list.

53.3.3 Model Specification Our empirical quantile model is specified as Equation (53.1): Tobin0 sQit D ˇ0 C ˇ1 RD_TAit C ˇ2 LEV it C ˇ3 MB_Dit Cˇ4 LOG_TAit C ˇ5 CAPEX_TAit Cˇ6 CF_TAit Cˇ7 EBIT_VOLAit Cˇ8 DIV_TAit Cˇ9 LEV it RD_TAit Cˇ10 MBit RD_TAit C ˇ1117 INDit _D17 C "i t (53.1) We measure firm value by Tobin’s Q similar to the prior studies (Lang et al. 1989; Doukas 1995; Szewczyk et al. 1996; Claessens et al. 2002, etc.) Tobin’s Q is measured by marketto-book ratio. Market value is defined as the sum of the market value of common stock and the book value of debt and preferred stock. Following the approach of Chan et al. 1990), R&D expenditure ratio (RD_TA) is measured by R&D expenditure divided by total assets. R&D expenditure has been consistently found to be significantly positively related to a firm’s Tobin’s Q in at least two studies (Lang and Stulz 1994;

53 Keiretsu Style Main Bank Relationships, R&D Investment, Leverage, and Firm Value: Quantile Regression Approach

a

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The R&D Expenditure Ratio of Japanese Firms, 1994-2004 0.021 0.0205 RD_TA

0.02

Unaffiliated Firms

0.0195

Main Bank Clients Full Sample

0.019 0.0185

2004

2003

2002

2001

2000

1999

1998

1997

1996

1995

1994

0.018

Year

b

The Leverage of Japanese Firms, 1994-2004 0.750

Leverage

0.700 Unaffiliated Firms

0.650

Main Bank Clients 0.600

Full Sample

0.550

2004

2003

2002

2001

2000

1999

1998

1997

1996

1995

1994

0.500

Year

c

The Tobin's Q of Japanese Firms, 1994-2004

Tobin's Q

14 12 10

Unaffiliated Firms

8 6

Main Bank Clients Full Sample

4 2 2003

2002

2001

2000

1999

1998

1997

1996

1995

1994

Year

0

Year

Fig. 53.1 (a) The R&D expenditure ratio (RD_TA) of Japanese firms, 1994–2004. (b) The leverage ratio of Japanese firms, 1994–2004. (c) The Tobin’s Q of Japanese firms, 1994–2004

Chen and Steiner 2000). The main bank dummy (MB_D) is used to measure the impacts of bank monitoring on performance (Jensen and Meckling 1976), reducing information asymmetry and enhancing information signaling (Leland and Pyle 1977). If firms with Keiretsu style main bank relationships are expected to have lesser problems with asymmetric

information or more efficient monitoring, then a lower cost and higher performance may appear. MB_D is given a “one” if firms affiliate with any of Keiretsu main bank groups, otherwise a “zero” will be given. Firm size (LOG_TA) is included because large firms may be more successful in developing new technology (Chauvin

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and Hirschey 1993) resulting in different Q scores. The capital expenditure ratio (CAPEX_TA) is the ratio of capital investment expenditure to total assets. Houston and James (1996) document that market value (Tobin’s Q) obviously relies on future growth opportunities, which are affected by capital expenditure. We also control free cash flow (CF_TA) for cross sectional differences among firms since R&D investment for low-free-cash-flow firms may increase the probability for seeking external financing. If main bank clients are expected to be less sensitive to cash flow problems, then the expenditure on R&D should have less aversive credit rationing from their main banks as well. Leverage ratio (LEV) and dividend yields (DIV_TA) are included as alternative measures of free cash flow (Jensen 1986) and investment opportunity (Smith and Watts 1992), respectively. Gugler (2003) finds an inverse relationship between payout ratios and R&D investments. Although a tremendous amount of extant literature demonstrates a negative relationship between leverage and Tobin’s Q, the impact of leverage on firms’ value from R&D investments is ambiguous in our research. Bradley et al. (1984) and Long and Malitz (1985) find that the use of leverage decreases with R&D expenditures. However, these authors do not provide any evidence of the impact of leverage on firm value while R&D investment is high, or the impact of R&D expenditure on firm value while leverage is high (or low). To clarify this point, the interaction effects of leverage and R&D expenditure (LEV*RD_TA) is included. Moreover, the interaction effect of main bank dummy and R&D expenditure (MB*RD_TA) is also included to further elucidate our findings in supplementing the extent literature. Earnings growth volatility (EBIT_VOLA) as a proxy for controlling observable credit risk (Johnson 1997) is included in the model. Finally, seven industry dummies (IND_D1IND_D7) are controlled in this model.

53.4 Empirical Results and Analysis 53.4.1 Quantile Regression and Bootstrapping Analysis Quantile regression is originally proposed by Koenker and Bassett (1978). The general form of the equation is as follows: yi D ˇq0 xi C uqi (Koenker and Hallock 2001). This equation implies that the coefficients differ by quantile. Mean-based procedures such as ordinary least squares are more sensitive to outliers than median-based quantile estimators. The quantile estimator places less weight on outliers than do OLS estimators. Therefore, the bias should be smaller using a median-based quantile estimator.

H.-C. Yu et al.

The target for quantile regression estimates is a parameter specified before estimation. Letting eit be the residuals and q represent the target quantile from the distribution of the residuals. Quantile parameter estimates are the coefficients that minimize the following objective function in Equation (53.2): X eit>0

2qjeit j C

X

2.1  q/jeit j

(53.2)

eit P0 .

848

J. Ronen and B. Sarath

some period of time (perhaps until price support is no longer needed). This is part of the initial tie-in agreement with the underwriter. In the interim period, traders may update their beliefs as to whether momentum has been caused by fundamentals or because of laddering. Traders who decide that laddering is present revise their beliefs downwards and sell (or short) the security (at t D 3). In turn, this affects the beliefs of other traders about whether the price momentum is upwards or downwards. Finally, the insiders dump their shares at t D 4. The time line is summarized below.

At time t D 3, the outsiders trade with each other where some might have acquired inside information about the possibility of laddering. So P3 solves: S2  K0  K1 D

M X

Qm .P3  .P2  P1 /; x.P0 //

mD1

C

N X

Œ.1  p/Qn .P3 ; x.P3 // C pQn .P3 ; x.P0 //

nD1

(54.9)

Time Line tD0 tD1 tD2 tD3

tD4

IPO issued consisting of S shares at a price P0 . K0 shares overallotted to insiders. The outside investors trade among themselves to reach a market equilibrium price P1 . Insiders step in and buy K1 additional shares driving the price up to P2 . Outside investors update beliefs and trade with each other at a price P3 . If there are no traders that suspect laddering, no trade takes place at time t D 3. The price support period ends and Insiders sell their shares at a market clearing price P4 .

At time 1, the insiders do not trade. The momentum traders base their beliefs about fundamental value based on past price P0 whereas rational expectations traders use the current price P1 . As a consequence, the market equilibrium price P1 is given as the price that solves: S2  K0 D

M X

Qm .P1 ; x.P0 // C

mD1

N X

Qn .P1 ; x.P1 //

nD1

(54.6) Note that we are fixing the momentum traders beliefs about fundamental values to be consistent with the IPO price. For simplicity, we shall assume that: x.P0 / D x0

(54.7)

that is, momentum traders start with the correct initial beliefs but subsequently adjust their demand based on momentum. At time t D 2, insiders step in and buy an additional K1 shares. Thus the equilibrium price P2 solves: S2  K0  K1 D

M X

Qm .P2  .P1  P0 /; x.P0 //

mD1

C

N X nD1

Qn .P2 ; x.P2 /

(54.8)

Finally, at time t D 4, the insiders sell all their excess shares and revert to the (unmanipulated) optimal holdings – that is S2 D

M X

Qm .P4  .P3  P2 /; x.P0 //

mD1

C

N X

Œ.1  p/Qn .P4 ; x.P4 // C pQn .P4 ; x.P0 //

nD1

(54.10) The question we address is whether this process could possibly lead to excess profits for the insiders. The excess profits (to all insiders) are determined as follows: K0 .P4  P0 / C K1 .P4  P2 / On a per-share basis, this becomes: K0 .P4  P0 / C K1 .P4  P2 / K1 C K0     K0 K1 D .P4  P0 / C .P4  P2 / K1 C K0 K1 C K0 (54.11) The basic question we shall address is whether the quantity in Equation (54.11) can be positive, for then, every insider gains more than their normal IPO return by becoming a part of the laddering cabal. The excess profit measure in Equation (54.11) is structured to be consistent with our assumptions that the IPO offer price is set competitively and the issue is whether manipulation is possible in the after market. If IPO prices are set competitively, there is no reason to add the returns on the optimal (unmanipulated) holdings to the left-hand-side of Equation (54.11). Note that we are not assuming a normal return on all the “excess” shares allocated and are only assuming a normal return with respect to their unmanipulated holdings. Similarly, we are not examining potential costs that may be

54 On the Feasibility of Laddering

imposed on the ladderers as a result of refusing tie-in agreements (such as an exclusion from future IPO’s conducted by the underwriter). If issuers can be forced to a low initial offer price, ladderers benefit from the initial overallocation; under such a circumstance, they would be willing to enter into laddering schemes even though they may lose money on their aftermarket purchases. But again, this need not give rise to inflated prices; the existence of information seeking traders will bring back prices to their equilibrium value levels. Again, these types of strategic collusion between the underwriter and ladderers lie outside our model. Our focus is to examine whether ladderers, acting on their own or in concert with the underwriter, can make abnormal profits under a maintained assumption that there is no price manipulation at the offer.

849

function Q.P C P2  P1 ; x.P0 // at time t D 4. So the period t D 4 market clearing price (with N D 0) solves: S2 D

M X

Qm .P4  .P2  P1 /; x.P0 //

mD1

H) P4  .P2  P1 / D P0

(54.12)

where the last implication follows from Equation (54.4). Let 0 1 ˛ D K1KCK and 1  ˛ D K1KCK . Then (54.11) transforms to 0 0 showing that: ˛.P4  P0 / C .1  ˛/.P4  P2 / 0

(54.13)

Equivalently, substituting for .P4  P0 / and .P4  P2 / from Equation (54.12), it suffices to show: ˛.P2  P1 / C .1  ˛/.P0  P1 / D ˛P2 C .1  ˛/P0 0 (54.14)

54.3 Results The key theme in our formal analysis is to examine conditions when the excess profit determined in (54.11) has a determinate sign; that is, we examine conditions under which excess profits are positive or have to be negative. If ladderers can successfully manipulate the after market prices, then laddering is a rational equilibrium phenomenon. On the other hand, if the conditions are such that the excess profits have to be negative, then laddering is unsustainable. The main result that we establish (Proposition 3) is that the presence of traders who indulge in private information search can result in laddering becoming unsustainable. The first result that we examine considers the benchmark situation where no traders suspect that laddering is taking place and assume instead that the manipulated prices reflect information about fundamental values. We do not believe this situation is realistic – in a Bayesian-rational market with laddering, there will be some prior beliefs that laddering is taking place. However, we start with this benchmark of no priors on laddering both because it is the situation analyzed in prior literature (Qing Hao 2007) and because it provides a contrast with rational and efficient market behavior. Under this restrictive hypothesis, it turns out that laddering is a positive payoff strategy. Proposition 1 (Momentum traders and the profitability of laddering). Suppose that N D 0, that is, that there are no outsider traders who suspect that laddering is taking place. Then ladderers earn abnormal positive profits by purchasing shares in the aftermarket. Proof. In this case, there is no trade at t D 3 so we only have an observed price sequence P0 ; P1 ; P2 ; P4 , (or alternatively, set P3 D P2 ). Therefore, momentum traders have a demand

Adding ˛ times the demand at t D 2 (Equation (54.8)) to .1  ˛/ times the demand in (54.4), we get: ˛

M X

Qm .P2  .P1  P0 /; x.P0 //

mD1

C .1  ˛/

M X

Qm .P0 ; x.P0 //

mD1

D ˛.S2  .K0 C K1 // C .1  ˛/S2 D S2  K0 D

M X

Qm .P1 ; x.P0 //

(54.15)

mD1

where the last equality follows from Equation (54.6). From Equation (54.15), the fact that P1 > P0 and the convexity assumption on Q, we get the following chain of relations: M X

Qm .P1 ; x.P0 //

mD1

D˛

M X

Qm .P2  .P1  P0 /; x.P0 //

mD1

C .1  ˛/

M X

Qm .P0 ; x.P0 //

mD1

˛

M X

Qm .P2 ; x.P0 // C .1  ˛/

mD1



M X

M X

Qm .P0 ; x.P0 //

mD1

Qm .˛P2 C .1  ˛/P0 ; x.P0 //

mD1

H) P1  ˛P2 C .1  ˛/P0 Rearranging, we obtain Equation (54.14).

(54.16) t u

850

J. Ronen and B. Sarath

Proposition 1 shows that with momentum traders, laddering can be a viable economic strategy. Before clarifying the intuition, we note that the price at which ladderers exit their excess holdings would, in general, lie between the inflated price P2 and the IPO price P0 . For this reason, the sign of the expression in Equation (54.11) is indeterminate. However, momentum traders intensify their demand when the price rises (through laddering). Given the greater demand intensity, the price falls less when ladderers sell than it rose at the time they bought the shares. It is precisely this asymmetry, caused by a manipulation of expectations, that results in abnormal profits to laddering. In contrast, the next result shows that if demand intensity cannot be manipulated, then laddering ceases to be profitable. The next proposition deals with the opposite benchmark to Proposition 1 in that we assume there are no momentum traders and that all traders are rational expectations traders. The beliefs of these traders is such that they assign some probability to the fact that ladderers are present and they condition their demands on the spot price rather than past prices. Obviously, if they discover that laddering has taken place, they reduce their demands driving the price downwards before the ladderers can exit the market resulting in losses. Less obviously, the fact that they respond instantaneously to selling pressure (observed through the spot price) makes them immune to laddering schemes. We summarize this basic economic point in the next lemma where we show that the demand of rational expectations traders is determined by the initial (fair) offer price. Lemma 1 (Rational expectation traders demands). For a rational expectations trader n, the demand at time t, Qtn .Pt jPt ;    ; P0 / is either Q0n .Pt ; x.P0 // (if the true value parameter x0 D x.P0 / is discovered) or Q0n .P0 ; x.P0 // D Qtn .Pt ; x.Pt // (if the parameter is not discovered). Proof. The assertion is obvious if trader n discovers that laddering has taken place. In contrast, if laddering is not detected, the inferred parameter x.Pt / is such that the demand at Pt with this belief x.Pt / is the same as the demand at P0 with belief x0 (see Assumption 2 Equation 54.1). The lemma follows. As might be expected based on Lemma 1, if all traders are rational expectations traders, laddering schemes fail. Proposition 2 (Rational expectations and the unprofitability of laddering). Suppose that M D 0, that is, all outside traders are rational expectations traders. Then Laddering is unprofitable. Proof. We follow the same notation as Proposition 1 and will show that the following holds: ˛.P4  P0 / C .1  ˛/.P4  P2 / < 0

0 1 and 1  ˛ D K1KCK ). When M D 0, the (where ˛ D K1KCK 0 0 t D 0 and t D 4 equilibrium conditions become:

S2 D

N X

Qn .P0 ; x0 /

nD1

S2 D

N X

Œ.1  p/Qn .P4 ; x.P4 // C pQn .P4 ; x.P0 //

nD1

(54.17) It follows that P4  P0 ; for if P4 > P0 , x.P4 / > x.P0 / and: Qn .P4 ; x.P4 // D Qn .P0 ; x.P0 //I Qn .P4 ; x.P0 // < Qn .P0 ; x.P0 // where the first equality follows from Equation (54.1); this contradicts Equation (54.17). In fact, the same argument shows that P4 P0 , that is, P4 D P0 . As P2 > P1 > P0 D P4 (ladderers buy shares at time t D 2), we conclude t u that ˛.P4  P0 / C .1  ˛/.P4  P2 / < 0 As we had observed earlier, the key issue is whether the price falls by the same level when ladderers sell as it rises when they buy. In the absence of momentum traders, this is precisely what happens, that is, when the ladderers sell their excess holdings, the price reverts to the IPO level. Hence, Laddering is not a profitable equilibrium strategy. Our last result combines the two benchmark cases and examines what happens when there are both momentum traders and rational expectations traders. The interest in the proposition stems from the fact that the presence of some information traders is enough to create a spiraling effect that ends in laddering being unprofitable. Proposition 3 (Laddering is unsustainable in equilibrium). Let N , p be such as to ensure that P3  P2 , that is, rational information traders who suspect that laddering has taken place, investigate the fundamental value of the security, and sell their holdings driving the t D 3 price below the inflated price at t D 2. Then the ladderers lose money in equilibrium and laddering is unsustainable. Proof. Under the hypothesis of this proposition, some proportion pN of traders act on their rational suspicions of laddering and discover the true fundamental value of the security, x0 . This is below the inferred value based on the inflated price, P2 . Under these circumstances, this subset of traders reduce their holdings creating a downward pressure on prices; if this pressure is sufficient, P3 is less than P2 . Then the market clearing price at time t D 4 is also lower both due to the downgrading by informed traders and due to the reduced demand of momentum traders. Specifically,

54 On the Feasibility of Laddering

if P3 < P2 , then it follows that P4 D P0 C .P3  P2 / < P0 . The proof is now similar to Proposition 2. Because P4 < P0 < P2 , the net payoff in Equation (54.13) will be negative, resulting in a loss to ladderers. t u The intuition behind Proposition 3 is a development of the ideas in Propositions 1 and 2. Specifically, the t D 3 price fall forces the t D 4 price to be lower than the IPO price. The reason is that momentum traders now shift their demand curves downwards resulting in lower demand intensity than at the time of the IPO. We have modeled the price fall at t D 3 to be a consequence of Bayesian rationality where at least some traders suspect that laddering might have taken place and therefore investigate the fundamentals underlying the valuation. The same result will hold as long as some traders (rational or otherwise) spend resources to gather information about the security. If the price of the security has been inflated through laddering, private information search leads to a downward trend in prices. This trend is magnified by momentum traders and the price drops significantly if the ladderers try to dispose of their excess holdings.

54.4 Conclusion We investigate the feasibility of laddering in an IPO market. Earlier studies have assumed that laddering is sustainable in equilibrium and have investigated its effects on market prices. In contrast, we study whether laddering itself is economically rational. Our model features three distinct sets of traders: insiders who ladder, rational traders who are generally suspicious of price manipulation and willing to invest effort and resources to acquire information about fundamental values, and, finally, momentum traders whose beliefs are conditioned on past prices. Our findings are that laddering cannot be sustained in equilibrium unless there is a significant proportion of traders whose beliefs about fundamental value can be manipulated through strategic purchases – the momentum traders – and there do not exist information traders who have a significant effect on prices. In contrast, if there are traders that behave in a way consistent with rational expectations and remove price inflation, laddering cannot be a profitable strategy. Our conclusions are dependent on an assumption that offer prices at IPO time are set rationally and that monopolistic (or oligopolistic) setting of issue prices is not feasible. If offer prices can be artificially lowered, then laddering becomes feasible in the limited sense that the profits made through the initial overallocation outweigh the losses made in the aftermarket. With the existence of information traders that include short-sellers, any inflation induced by laddering activities can dissipate quickly and ladderers are saddled with

851

losses in the aftermarket. In particular, if the market for underwriting is competitive with regard to issuers, there are no abnormal returns associated with participating in the IPO. In such a setting, attempts by ladderers to profit from their ability to exploit market sentiment will fail as long as some traders in the market function in a manner consistent with the economics of rational expectations. These results contribute to a better understanding of markets and shed some light on the recent controversy as to whether laddering takes place, and whether such laddering can be successful in yielding profits to the ladderers and underwriters such as to make it sustainable in equilibrium. Our conclusion is that if underwriters do not act as monopolists, laddering is not a sustainable activity in a semistrong efficient market in which price – but not the history of prices – is seen as reflecting public information unbiasedly and instantaneously.

References Aggarwal, R. 2000. “Stabilization activities of underwriters after initial public offerings.” Journal of Finance 55, 1075–1103. Aggarwal, R. and P. Conroy. 2003. “Price discovery in initial public offerings.” Journal of Finance 55, 2903–2922. Aggarwal, R., K. Purnanandam, and G. Wu. 2006. “Underwriter manipulation in initial public offerings.” Working Paper, University of Minnesota. Benveniste, L. M. and P. A. Spindt. 1989. “How investment bankers determine the offer price and allocation of new issues.” Journal of Finance Economics 25, 343–361. Choi, S. and A. C. Pritchard. 2004. “Should issuers be on the hook for laddering?.” Working Paper, Law and Economics, Michigan University, No. 04-009/CLEO USC No. C04-3. Derrien, F. 2005. “IPO pricing in ‘hot’ market conditions: who leaves money on the table?” Journal of Finance 60, 487–521. Edwards, A. K. and K. W. Hanley. 2007. “Short selling and initial public offerings.” Working Paper, Office of Economic Analysis, U.S. Securities and Exchange Commission, April 2007; http://ssrn.com/abstract=981242. Griffin, J. M., J. H. Harris, and S. Topaloglu. 2007. “Why are investors net buyers through lead underwriters?” Journal of Finance Economics 85, 518–551. Hanley K. W. and W. J. Wilhelm. 1995. “Evidence on the strategic allocation of initial public offerings.” Journal of Finance Economics 37, 239–257. Hao, Q. 2007. “Laddering in initial public offerings.” Journal of Finance Economics 85, 102–122. Loughran, T. and J. R. Ritter. 2002. “Why don’t issuers get upset about leaving money on the table.” Review of Financial Studies 15, 413–443. Loughran, T. and J. R. Ritter. 2004. “Why has IPO underpricing changed over time?” Financial Management, Autumn 33, 5–37. Smith R. and R. Sidel 2006. “J.P. Morgan agrees to settle IPO cases for $425 million.” Wall Street Journal April 21, C4. Wurgler, J. and E. Zhuravskaya. 2002. “Does arbitrage flatten demand curves for stock?” The Journal of Business 75, 583–609. Zhang, D. 2004. “Why do IPO underwriters allocate extra shares when they expect to buy them back?” Journal of Financial and Quantitative Analysis 39, 571–594.

Chapter 55

Stock Returns, Extreme Values, and Conditional Skewed Distribution Thomas C. Chiang and Jiandong Li

Abstract This paper investigates stock returns presenting fat tails, peakedness (leptokurtosis), skewness, clustered conditional variance, and leverage effects. We apply the exponential generalized beta distribution of the second kind (EGB2) to model stock returns as measured by six AMEX industry indices. The evidence suggests that the error assumption based on the EGB2 distribution is capable of accounting for skewness and kurtosis and therefore of making good predictions about extreme values. The goodness-of-fit statistic provides supporting evidence in favor of an EGB2 distribution in modeling stock returns. Keywords EGB2 distribution r Stock return r Fat tails r Risk management r VaR

55.1 Introduction In analyzing financial time series, the conventional approach usually assumes a Gaussian distribution. The advantage of using a Gaussian assumption is that the statistical analysis of asset returns can be simplified: the main focus is on the first two moments. This simplification is appealing to practitioners, since it can directly link statistical tools to the meanvariance framework in making daily portfolio decisions. This advantage, however, entails a cost. Recent financial disturbances suggest that significant daily loss occurs more frequently and the behavior of volatility cannot reasonably be predicted based on a normal distribution. To address this issue, practitioners have developed the Value at Risk (VaR) and Expected Tail Loss methods to T.C. Chiang () Department of Finance, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA e-mail: [email protected] J. Li Chinese Academy of Finance and Development (CAFD), Central University of Finance and Economics (CUFE), 39 South College Road, Beijing 100081, China e-mail: [email protected]

deal with extreme values of asset price movements. Yet, using VaR to measure a portfolio’s maximum loss over a target horizon within a given confidence interval is essentially based on a normal distribution. This scheme may work well at the 95% confidence interval. However, using the normal model, VaR estimates in general lead to under-forecasts of losses at the 99% level. The 1987 market crash with its 17¢ attests to the failure of the VaR method in estimating variance by employing the normal distribution. Recognizing the frequent occurrences of market crashes, major defaults, financial turmoil, and the collapse of asset prices, financial institutions have begun to treat extreme values as a kind of common risk when managing their portfolios. From a practical perspective, if portfolio distributions depend on more than two parameters, optimal choice cannot be satisfied by the mean-variance approach. To account for the distributional anomalies of stock returns, we must incorporate skewness and kurtosis into portfolio decisions and risk management. From an academic point of view, although it is generally recognized that the GARCH type model is capable of dealing with the volatility cluster phenomenon, it nonetheless fails to account for the fat tails and skewness. In fact, as argued by Brooks et al. (2005), “If asset returns are fat-tailed, it will lead to a systematic underestimate of the true riskiness of a portfolio, where risk is measured as the likelihood of achieving a loss greater than some threshold” (p. 400). From an econometric point of view, the exclusion of fat-tailed and skewed information in the asset return model is bound to result in missing variables and misspecification problems. Several non-Gaussian distributions have been proposed in the finance literature. For example, Bollerslev (1987) suggests a Student’s t-distribution; Nelson (1991) proposes a general error distribution (GED, also known as an exponential power distribution); and Fama (1963), Liu and Brorsen (1995), Rachev and Mittnik (2000), and Khindanova et al. (2001) use Paretian distributions (or stable distributions). Despite the large volume of research into the distribution of stock returns, attention has been directed to a particular feature of asset return behavior. The research lacks a more general treatment of stock return characteristics such as fat tails, peakedness (leptokurtosis), skewness, clustered

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_55, 

853

854

conditional variance, and leverage effects. This paper addresses these non-normality issues and provides empirical evidence based on six industry indices from the American Stock Exchange (AMEX): biotechnology, computers, natural gas, telecommunications, oil, and pharmaceuticals. In this study, we employ a distribution called the exponential generalized beta distribution of the second kind (EGB2).1 The distribution proposed by McDonald and Xu (1995) and Wang et al. (2001) has several advantages compared with alternative distributions. First, as a four-parameter distribution, it allows diverse levels of skewness and kurtosis. Its parameters are estimated simultaneously with the model structure estimation process so that it is able to accommodate a wider range of data characteristics. The skewness and excess kurtosis of EGB2 are in the range of (2, 2) and (0, 6), respectively. Second, a number of distributions used in statistical analysis, such as logistics distribution and log-normal distribution, are nested in the EGB2 distribution.2 Third, unlike a Hansen-type (1994) skewed generalized t-distribution requiring the imposition of several restrictions on the parameter values to permit estimation (Brooks et al. 2005),3 the EGB2 model is simple and has a closed-form density function for the distribution; its higher order moments are finite and explicitly expressed by its parameters. Although the EGB2 is capable of accounting for fat tails and the higher order moments of returns, previous studies using EGB2 distribution fail to incorporate a recent advancement in asset return behavior: the asymmetric response of asset return volatility to news (McDonald and Xu 1995). In addition, their analysis has been focused on aggregate stock market returns (Hueng and McDonald 2005). In this study, we introduce this asymmetric behavior into an EGB2 distribution and examine stock indices for six industries. Thus, our model is characterized as an asymmetric GARCH(1,1) cum EGB2 distribution model and provides evidence for industrial stock return analysis.4 1 There are other names for the EGB2 distribution used in non-financial fields or in non-American journals; for example, generalized logistic distribution is used in Wu et al. (2000), z-distribution in BarndorffNielsen et al. (1982), the Burr type distribution in actuarial science (Hogg and Klugman 1983), and four-parameter Kappa distribution in geology (Hosking 1994). 2 Wang et al. (2001) show that the EGB2 distribution is very powerful in modeling exchange rates that have fat tails and leptokurtic features. 3 In following Hansen (1994), Theodossiou (1998) also develops the skewed generalized t distribution (SGT) to examine stock market indices. 4 It is not our intention to exhaust all the non-Gaussian models in modeling stock returns, which is infeasible. Rather, our strategy is to adopt a distribution that is rich enough to accommodate the features of financial data. To our knowledge, there are different types of flexible parametric distributions parallel to the EGB2 distribution, such as a skewed generalized t -distribution (SGT) (Hueng and Brooks 2002) and an inverse hyperbolic sine distribution (IHS) (Johnson et al. 1994), among others.

T.C. Chiang and J. Li

The remainder of the paper is organized as follows. Section 2 describes the methodology of the AGARCH-EGB2 model. Section 3 discusses the data used in this study. Section 4 presents the empirical results for U.S. industrial stock returns. Section 5 reports the goodness-of-fit tests to assess alternative distribution. Section 6 contains the probability evaluation via non-Gaussian distributions. Section 7 contains concluding remarks and a summary.

55.2 The AGARCH Model Based on the EGB2 Distribution The AGARCH(1,1)-EGB2 stock return model can be represented by: 8 Rt D t C "t ˆ ˆ m n ˆ X X ˆ ˆ ˆ ˆ  D

C

R  'j "t j t 0 i t i ˆ ˆ < i D1 j D1 p "t D ht zt ˆ ˆ ˆ ˆ ht D vc C va "2t1 C vb ht 1 C vd "2t1 I."t 1 < 0/ ˆ ˆ ˆ ˆ ˆ : "t jˆt 1  EGB2.0; ht ; p; q/ (55.1) where Rt is the portfolio’s return at time t; and t is the conditional mean that follows an ARMA(m; n) process.5 The error term "t is assumed to be independent. The conditional variance, ht , is assumed to follow GARCH(1,1); c ; a , and b > 0 ensure a strictly positive conditional variance; I is an indicator variable that takes the value of unity only when the error term is negative; the negative error term component in the variance equation captures the asymmetric effect of an extraordinary shock to the variance: bad news usually has a more profound effect than good news (Glosten et al. 1993); zt is assumed to have a zero mean and unit variance and is independent and identically distributed .i:i:d / if the model is correctly specified. The conditional distribution is assumed to be the exponential generalized beta distribution of the second kind (EGB2). The EGB2 distribution has the probability density function (pdf) given by: ip xı e.  / EGB2.xI ı; ; p; q/ D ipCq h xı jjB.p; q/ 1 C e .  / (55.2) where x is a random variable; ı is a location parameter that affects the mean of the distribution;  reflects the scale of h

5

The GARCH-M specification was explored. The conditional variance variable is insignificant for all six indices. Other macro variables and industry factors may be added to the mean equation or variance equation. Here, we focus on the time series property.

55 Stock Returns, Extreme Values, and Conditional Skewed Distribution

the density function; p and q (p > 0 and q > 0) are shape parameters that together determine the skewness and kurtosis of the distribution of the asset return series; and B.p; q/ is the beta function.6 As suggested by McDonald (1991), the EGB2 is suitable for coefficient of skewness values between 2 and 2 and coefficient of excess kurtosis values of up to 6. Thus, it is capable of accommodating fat-tailed and skewed error distributions pertinent to stock return modeling. Different from the Student’s t-distribution, which has a fat-tail feature, the EGB2 distribution allows both fat fail and peakedness features. In other words, the pdf curve of Student’s t-distribution is flat-topped; the pdf curve of the EGB2 distribution has a peak around the mean. This feature makes the EGB2 distribution superior to the Student’s t-distribution, since a histogram of actual equity returns shows fat tails and peakedness. The disadvantage is that the EGB2 distribution has four parameters to be estimated, while the Student’s t-distribution has only three parameters. Compared with other four-parameter distributions, such as stable distributions and skewed t-distributions, the EGB2 distribution is plausible in that it has a closed-form pdf and contains higher-order moments. It follows that the parameters can be estimated by using a maximum likelihood function and the model can be evaluated based on skewness and kurtosis coefficients. The first four central moments are: mean D ı C . .p/  .q//   Var D  2 0 .p/ C 0 .q/  The 3rd Moment D  3 00 .p/   The 4th Moment D  4 000 .p/ C

00

.q/

000



.q/



(55.3)

0 00 000 ./; ./ and ./ denote digamma, where ./; trigamma, tetragamma, and pentagamma functions, respectively.7 Since the error term "t is standardized in the model, we can express ı and  by using the above polygamma functions as: q q 1 1 D 0 .p/C 0 .q/ D  q ı D .. .p/  .q// D  1 (55.4)

where

6

D D

.p/  .q/ 0 .p/ C 0 .q/

It should be noted that beta function here has nothing to do with the stock’s beta. 7 The digamma function is the logarithmic derivative of the gamma function; the trigamma function is the derivative of the digamma function.

855

With some algebraic manipulation, we can write the univariate GARCH-EGB2 log-likelihood function as follows8 : i h p log L D T log. /  log.B.p; q// C p

" p X "t p. p /  0:5 log.ht /  .p C q/ C ht # p "t C // (55.5) log.1 C exp. p ht where T is the number of periods, B.p; q/ is the beta function, ht is the conditional variance, and "t is the standardized error term. Various analysis packages can be used to conduct the maximum likelihood estimation. This paper uses the R BFGS algorithm in a RATS program. The skewness and excess kurtosis for the EGB2 distribution are given, respectively, by: Skewness D Kurtosis D

.p/  00 .q/ . 0 .p/ C 0 .q//1:5 00

.p/ C 0 . .p/ C 000

000

.q/

0 .q//2

(55.6)

The standard deviation of skewness and kurtosis coefficients can be drawn by using a standard delta method.9 This model is appealing, since it contains richer features pertinent to asset return behavior. First, the asset return in the mean equation embodies time series patterns, which can facilitate forecasting. Second, the variance equation allows for the evolution of volatility and treats the impacts of bad news and goods news on variance asymmetrically. Third, the distribution is flexible enough to nest alterative distributions. For instance, the EGB2 converges to normal distribution as p D q approaches infinity, to log-normal distribution when only p approaches infinity, to the Weibull distribution when p D 1 and q approaches infinity, and to the standard logistic distribution when p D q D 1. It is symmetric if p D q. The EGB2 is positively (negatively) skewed as p > q .p < q/ for  > 0.

55.3 Data The use of industrial indices by institutional and individual investors in making portfolio decisions has become a common strategy in pursuing higher returns, liquidity, investment 8 9

See the appendix in Wang et al. (2001). The derivation is available upon request.

856

T.C. Chiang and J. Li

style, and diversification. For this reason, in this paper we explore indices for six industries traded on the American Stock Exchange (AMEX): biotechnology, computers, natural gas, telecommunications, oil, and pharmaceuticals. The sample is non-balanced weekly data covering the period from 8/26/1983 through 9/17/2006 The earliest data start on 8/26/1983 for the computer and oil indices; the latest begin on 12/9/1994 for biotechnology. The data source is Trade Tool, Inc. We use weekly data in order to be consistent with industry practices. For example, Value Line, Bloomberg, and Baseline all use weekly stock data to calculate a stock’s beta. In addition, weekly data are free of the Monday effect and other calendar effects. Figure 55.1 shows the time series plots for these six indices. We can easily see the boom period in 2001 for the computer, telecom, and biotech industries and the rise in oil and natural gas in recent years. The weekly returns along with other statistics for these six indices are reported in Table 55.1. Looking at the skewness coefficient, we find that all six indices have negative values and four of them are significant at the 1% level. A negative skewness coefficient means that there are more negative extreme values than positive extreme values in the sample period. With respect to the excess kurtosis (the kurtosis coefficient minus 3), all of the estimated values are statistically significant at the 1% level, suggesting that the presence of fat tails is confirmed. The range of the excess kurtosis coefficient is between 1.06 and 4.85. If we check the range of peakedness, the values are between 1.08 and 1.20.10 This range is much lower than the referenced figure, 1.35, indicating the presence of a high peak in the probability density func10

tion for all of the indices under investigation. Further, by inspecting the Jarque–Bera statistics, the normality assumption is uniformly rejected for all six indices. The Ljung–Box Q statistics show that we cannot reject the null hypothesis of the absence of serial correlation at the 5% level for all of the series, indicating no evidence of serial correlation for the stock return series under investigation. However, the Ljung– Box Q2 tests indicate that all six statistics of return squares are significant at the 1% level, suggesting volatility clustering and that this finding would be consistent with a GARCH-type specification. In sum, the above statistics clearly indicate that the popular normality assumption does not match up well with the weekly returns under investigation. Stock index returns often show positive excess kurtosis (fat tails), accompanied by negative skewness. The peakedness does not conform to that of the normal distribution either. Besides the nonGaussian features, some weekly returns show autocorrelation, and all of the return squared series display significant volatility clustering.

55.4 Empirical Evidence 55.4.1 GARCH(1,1) Model: The Normal Distribution It is convenient to start with a GARCH(1,1) model based on the normal distribution, which sets a basis for comparison. With a general model in hand, which is given by the set

See the notes to Table 55.1.

6 index level 1800

AMEXBI OT

1700

AMEXCPT

AMEXNGAS

AMEXNTEL

AMEXPHAR

AMEXOIL

1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100

date

JAN07

JAN06

JAN05

JAN04

JAN03

JAN02

JAN01

JAN00

JAN99

JAN98

JAN97

JAN96

JAN95

JAN94

JAN93

JAN92

JAN90

JAN91

JAN89

JAN87

JAN88

JAN85

JAN86

JAN84

0 JAN83

Fig. 55.1 Level of six indices Six industries traded on the American Stock Exchange (AMEX) are: biotechnology (AMEXBIOT), computers (AMEXCPT), natural gas (AMEXNGAS), telecommunications (AMEXNTEL), oil (AMEXOIL), and pharmaceuticals (AMEXPHAR). Data source: Trade Tool, Inc.

55 Stock Returns, Extreme Values, and Conditional Skewed Distribution

857

Table 55.1 Descriptive statistics for six industry indices’ weekly returns Jarque Ticker AMEXBIOT AMEXCPT AMEXNGAS AMEXNTEL AMEXOIL AMEXPHAR

Nobs 614 1; 203 652 671 1; 203 655

Mean

Variance

Skew

Kurtosis

Peakedness

–Bera

Q(13)

Q.13/2

0:0034

0.0025

0:205

3.187

1.087

264:328

21:354

412:762

Œ2:08

[16.12]

0:000

0:070

0:000

0:581

3.390

643:996

19:555

210:064

Œ8:24

[24.00]

0:000

0:110

0:000

0:309

1.060

40:927

14:0289

86:444

Œ3:22

[5.53]

0:000

0:370

0:000

0:760

4.858

724:532

18:083

197:119

Œ8:04

[25.69]

0:000

0:150

0:000

0:335

1.267

103:157

18:577

74:132

Œ4:75

[8.98]

0:000

0:140

0:000

0:060

1.494

61:357

11:957

83:497

Œ0:63

[7.81]

0:000

0:530

0:000

0:0016 0:0017 0:0014 0:0019 0:0022

0.0012 0.0012 9.93E-04 6.72E-04 6.67E-04

1.108 1.191 1.083 1.202 1.187

Notes: The six indices are biotechnology, computers, natural gas, telecommunications, oil, and pharmaceuticals. The numbers in brackets below the estimated values are t-values. The critical t-value at the 1% level is 2.57. The standard deviations of skewness and excess kurtosis coefficients are approximately given by .6=T/0:5 and .24=T/0:5 , respectively. Peakedness is measured by f0:75  f0:25 , the distance between the value of the standardized variable at which the cumulative distribution function equals 0.75 and the value at which the cumulative distribution function equals 0.25. The reference value of the standard normal distribution is 1.35. Peakedness of less than 1.35 means there is a high peak in the probability density function. A normality test is conducted using the Jarque–Bera statistics. Q and Q2 are Ljung–Box tests for examining independence up to the order of 13. The numbers below these statistics are p-values

of equations in (52.1), we proceed with our estimation by setting the error distribution to be normal; that is, "t jˆt 1  N.0; ht /, where N stands for the normal distribution. Before estimating the AGARCH-EGB2 model, we apply the Box–Jenkins method to filter out the time series pattern.11 The residuals from the mean equation are normalized by dividing by the estimated standard deviation obtained from the conditional variance equation. Using standardized residuals to fit the model allows us to remove the heteroscedasticity due to stochastic variance. The statistics are reported in Table 55.2. Checking the variance equation, we find that most of the coefficients in the GARCH(1,1) equations are statistically significant, indicating that stock-return volatilities are characterized by a clustering phenomenon. Further, the average variances for various stock returns are rather high, indicating that variances display a higher degree of persistence. The information from the asymmetric coefficient suggests that only half of the six cases show a significant asymmetric effect, which seems to be less apparent than those found in the daily data in the literature. This may be the result of using weekly data, which moderates the effect. 11 According to the correlogram of each stock index, the following ARMA processes are revealed. For biotechnology, AR(12) is detected; AR(11) for computers; AR(10) for natural gas; AR(11,15) for telecommunications; AR(1) for oil; and AR(25) for pharmaceuticals.

Looking at the statistics for skewness, the evidence shows that the null hypothesis of the absence of skewness is rejected at the 1% level in three out of six cases, while the null hypothesis of the absence of kurtosis is rejected in five cases. The significant excess kurtosis coefficients are in the range of 0.52–1.26. A joint test of the Jarque–Bera statistics shows that all of the return residuals are rejected by assuming a Gaussian distribution. The rejection occurs because the stock-index changes are either not independent, not normally distributed, or both. Further looking into the measure of peakedness, the estimate values are around 1.25, lower than the reference point of the standard normal distribution of peakedness, 1.35, indicating that all of the returns are leptokurtic. It is apparent that the normality assumption of the residuals is problematic. We then examine the independence of stock returns up to the 13th order, which is one-quarter for weekly data. On the basis of the Ljung–Box statistics, none of Q(13) is significant, indicating the absence of autocorrelation in the weekly data. Further checking the Q.13/2 statistics for examining the null hypothesis of dependency on the squared returns also finds no case that is significant, suggesting that the volatility clustering phenomenon has been reduced significantly as evidence of implementing an AGARCH(1,1) specification. From the test results then, it can be argued that stock returns departing from normality may be mainly attributed to leptokurtosis.

1; 191

641

655

1201

629

AMEXCPT

AMEXNGAS

AMEXNTEL

AMEXOIL

AMEXPHAR

0:0247 Œ0:25 0:3649 Œ5:14 0:3113 Œ3:22 0:2189 Œ2:29 0:3199 Œ4:53 0:1645 Œ1:68 0.6825 [3.42] 1.2623 [8.89] 0.3276 [1.69] 0.5206 [2.72] 0.9927 [7.02] 0.5661 [2.90] 1.2777

1.2568

1.2975

1.281

1.2203

1.2479

11:72666 0:00 105:4992 0:00 13:21887 0:00 12:62975 0:00 69:80144 0:00 11:23468 0:00

6:8348 0:91 7:4283 0:88 9:2334 0:76 5:2553 0:97 8:9519 0:78 11:3585 0:58

19:0509 0:12 17:189 0:19 13:0559 0:44 5:5316 0:96 9:156 0:76 19:2286 0:12 0.00002 [1.09] 0.000021 [2.08] 0.000034 [2.26] 0.00001 [1.94] 0.000007 [1.21] 0.000007 [1.28]

vc 0.0629 [3.35] 0.0727 [3.84] 0.0089 [0.48] 0.0537 [2.81] 0.0424 [3.51] 0.0248 [1.65]

va

0.9214 [37.00] 0.9052 [40.06] 0.9218 [38.95] 0.9080 [52.25] 0.9479 [52.09] 0.9424 [64.87]

vb

0.0175 [0.53] 0.0127 [0.53] 0.0859 [2.77] 0.0612 [1.99] 0.0014 [0.07] 0.0461 [1.75]

vd

2; 012:15

3; 833:33

2; 037:75

1; 847:77

3; 485:60

1; 573:1

ln L

Notes: The six indices are weekly data for biotechnology, computers, natural gas, telecommunications, oil, and pharmaceuticals. The numbers in brackets below the estimated values are t-values. The critical t-value at the 1% level is 2.57. The standard deviations of skewness and excess kurtosis coefficients are approximately given by .6=T/0:5 and .24=T/0:5 , respectively. Peakedness is measured by f0:75  f0:25 , the distance between the value of the standardized variable at which the cumulative distribution function equals 0.75 and the value at which the cumulative distribution function equals 0.25. The reference value of the standard normal distribution is 1.35. Peakedness of less than 1.35 means there is a high peak in the probability density function. A normality test is conducted using the Jarque–Bera statistics. Q and Q2 are Ljung–Box tests for examining independence up to the order of 30. The numbers below these statistics are p-values. vc ; va ; vb ; vd are the coefficients of the variance equation. lnL is the log-likelihood function value. The numbers in brackets below the estimated statistics are the t-values

601

AMEXBIOT

Table 55.2 Statistics for higher moments of the standardized errors on the GARCH(1,1)-normal distribution Ticker Nobs Skew Kurtosis Peakedness Jarque–Bera Q(13) Q.13/2

858 T.C. Chiang and J. Li

55 Stock Returns, Extreme Values, and Conditional Skewed Distribution

55.4.2 AGARCH(1,1): The EGB2 Distribution Model In this section, we estimate the parameters represented by (55.1) that assumes an EGB2 distribution. Table 55.3 reports the comparable statistics based on the standardized return residuals from an AGARCH(1,1) cum EGB2 distribution. The statistics, including the estimated coefficients and Q(13) and Q.13/2 , turn out to have results similar to those reported in Table 55.2, which is based on a normal distribution. However, by using the EGB2 distribution, the skewness problem has been removed for all six indices, and no coefficient is rejected at the conventional significance levels. Turning to the statistics of kurtosis, we find that none of the cases are significant at the 1% level. This suggests that the EGB2 distribution works well on the kurtosis. By inspecting the estimates of p and q, we find that p ranges from 1.25 to 2.28 and q ranges from 1.60 to 4.53. These pair-wise figures are nowhere near being equal and are far from the normal distribution that requires that both p and q approach infinity. Finally, we check the peakedness, which ranges from 1.22 to 1.28, conforming to the existence of a high peak. In sum, the testing results suggest that the EGB2 distribution produces a much more satisfactory result for modeling skewness and tailed information than can be achieved by assuming a normal distribution.

55.5 Distributional Fit Test While highlighting the model’s ability to account for the skewness and kurtosis of the residuals may not be adequate to justify the performance of the distribution, in this section we shall conduct the log-likelihood ratio test and goodness-offit test for different distributions. The log-likelihood function values in Tables 55.2 and 55.3 already clearly show that the EGB2 distribution has outperformed the normal distribution. We conduct a log-likelihood ratio test, which is expressed as: LR D 2.In L0  In L1 /

859

standardized residuals and theoretical distributions based on estimated shape parameters (Snedecor and Cochran 1989).12 The null hypothesis tested by the GoF statistic is that the observed and predicted distribution functions are identical. The test statistic is given by: GoF D

k X .fi  Fi /2 Fi i D1

(55.8)

where fi is the observed count of actual standardized residuals in the i th data class (interval), Fi is the predicted count derived from the estimated values for the distribution parameters, and k is the number of data intervals used in distributional comparisons. GoF has an asymptotic 2 distribution with degrees of freedom equal to the number of intervals minus the number of estimated distribution parameters minus one. For the EGB2 distribution, two parameters are estimated; for the normal distribution, no parameter is required, since the error term has been standardized. Table 55.4 reports the results of the 2 test for the standardized residuals generated by the AGARCH(1,1) – the EGB2 distribution model. The test power is maximized by choosing a data class equiprobably (equal probability). The rule of thumb of a Chi-squared test is to choose the number of groups starting at 2 T 0:4 . Our sample contains either 660 or 1,200 observations with 40 intervals being used. The degree of freedom is 37 for the EGB2 distribution and 39 for the normal distribution. The critical values for the chi-squared distribution are provided in the notes to the table. The testing results show that the null is rejected for the computer index on the normal distribution at the 5% level; however, none is rejected on the EGB2 distribution. Although both distributions in general are acceptable for the sample, the EGB2 distribution yields lower values of the 2 statistics. Putting the goodness-of-fit test, LR test, and previous skewness and kurtosis coefficient tests together, plus the peakedness analysis, it is clear that the EGB2 distribution is superior to the normal distribution in our empirical analysis.

(55.7)

55.6 The Implication of the EGB2 Distribution where ln L1 is the maximum log-likelihood of the unrestricted distribution and ln L0 is the maximum log-likelihood of the restricted distribution. The likelihood ratio statistic, LR, follows an asymptotic 2 .k/ distribution with k degrees of freedom under the null hypothesis that the restrictions are valid, where k is the number of restrictions imposed. We find that in Table 55.4 all of the LR statistics are highly significant at the 1% level. The evidence is firmly in favor of the EGB2 distribution. We further use the 2 goodness-of-fit (GoF) statistic to compare differences between observed distributions of

In addition to the superior performance justified by statistical criteria, the EGB2 distribution offers a useful tool for evaluating the probability of those extreme values. According 12 The chi-square test is an alternative to the Anderson-Darling and Kolmogorov-Smirnov goodness-of-fit tests. The chi-square test and Anderson-Darling test make use of the specific distribution in calculating critical values. This has the advantage of allowing a more sensitive test and the disadvantage that critical values must be calculated for each distribution.

va 0.0664 [2.73] 0.0617 [3.46] 0.0088 [0.42] 0.0541 [2.71] 0.0423 [2.85] 0.0182 [0.95]

vb 0.9099 [27.91] 0.9317 [50.52] 0.9263 [43.29] 0.9026 [46.42] 0.9385 [48.42] 0.9306 [39.90]

vd 0.0299 [0.70] 0:006 Œ0:30 0.0826 [2.45] 0.0607 [1.98] 0.0095 [0.38] 0.0642 [1.65]

q 1.604 2.017 4.536 3.721 2.324 2.389

p 1.365 1.253 2.080 2.281 1.593 2.056

1,437.16

2,742.79

1,439.59

1,264.41

2,409.75

ln L 1,024.92

Notes: The six indices are weekly data for biotechnology, computers, natural gas, telecommunications, oil, and pharmaceuticals. The numbers in brackets below the estimated values are t-values. The critical t-value at the 1% level is 2.57. Peakedness is measured by f0:75  f0:25 , the distance between the value of the standardized variable at which the cumulative distribution function equals 0.75 and the value at which the cumulative distribution function equals 0.25. The reference value of the standard normal distribution is 1.35. Peakedness of less than 1.35 means there is a high peak in the probability density function. A normality test is omitted in the table, since the assumed distribution is the EGB2 distribution. An independence test is conducted using a Ljung–Box Q test up to the order of 13. The Q2 is the Ljung–Box statistics for testing returns squared up to the order of 13. The numbers below these statistics are p-values. vc ; va ; vb ; vd are the coefficients of the variance equation. The numbers in brackets below the estimated statistics are t-values. New columns show two distribution parameters: p and q The lnL is the log-likelihood function value

Table 55.3 Statistics for higher moments of the standardized errors on the AGARCH(1,1)-EGB2 distribution Ticker Nobs Skew Kurtosis Peakedness Q(13) Q.13/2 vc AMEXBIOT 601 0:0268 0.7078 1:2376 6:888 18:416 0.000029 [0.69] Œ0:31 0:91 0:14 [1.16] AMEXCPT 1; 191 0:4084 1.5679 1:2198 7:186 19:408 0.000013 Œ0:34 [1.92] 0:89 0:11 [1.75] AMEXNGAS 641 0:3138 0.3286 1:2634 9:431 13:056 0.000032 [0.51] Œ0:68 0:74 0:44 [2.28] AMEXNTEL 655 0:238 0.5981 1:2834 5:313 5:567 0.000014 [0.12] [0.30] 0:97 0:96 [2.12] AMEXOIL 1; 201 0:3333 1.0343 1:2672 8:709 8:396 0.00001 Œ0:71 [1.11] 0:79 0:82 [1.59] AMEXPHAR 629 0:1879 0.6952 1:2739 10:96 19:463 0.000013 Œ0:76 [0.47] 0:61 0:11 [1.39]

860 T.C. Chiang and J. Li

55 Stock Returns, Extreme Values, and Conditional Skewed Distribution

Table 55.4 Goodness-of-fit (Chi square statistics) and log-likelihood ratio test between Gaussian distribution and the EGB2 distribution

861

Ticker

Nobs

Normal

EGB2

LR

AMEXBIOT AMEXCPT AMEXNGAS AMEXNTEL AMEXOIL AMEXPHAR

601 1; 191 641 655 1; 201 629

37.867 54:83 40.875 29.000 35.300 35.313

37.533 39.767 27.312 24.412 18.567 28.312

1; 096:372 2; 151:709 1; 166:723 1; 196:314 2; 181:069 1; 149:995

Notes: Normal is the standardized residual from GARCH(1,1)-normal distribution; EGB2 is the standardized residual from GARCH(1,1)-EGB2; The two columns are based on the same model structure. The quantiles are computed via 40 intervals. For the EGB2 distribution, the degree of freedom (d.f.) is 37, and the critical values at the 1%, 5%, and 10% levels are 59.89, 52.19 and 48.36, respectively. For the normal distribution, the degree of freedom is 39, and the critical values at the 1%, 5% and 10% levels are 62.43, 54.57 and 50.66, respectively. LR is log-likelihood ratio. and  indicate significance at the 1% and 5% levels, respectively Table 55.5 The probability of negative extreme shocks in the error term between EGB2 and normal distributions Shocks Index 9¢ 8¢ 7¢ 6¢ 5¢ 4¢ 3¢ AMEXBIOT AMEXCPT AMEXNGAS AMEXNTEL AMEXOIL AMEXPHAR

2¢

1¢

EGB2

4.58E-08 1.91E-07 4.82E-08 1.40E-08 5.34E-08 6.49E-09

3.49E-07 1.23E-06 3.80E-07 1.29E-07 4.05E-07 6.49E-08

2.36E-06 6.92E-06 2.66E-06 1.08E-06 2.72E-06 5.89E-07

1.56E-05 3.79E-05 1.83E-05 8.86E-06 1.80E-05 5.29E-06

0.000103 0.000207 0.000125 7.22E-05 0.000118 4.73E-05

0.000681 0.001125 0.000837 0.000575 0.00077 0.000416

0:004435 0:006063 0:005404 0:004362 0:004927 0:003505

0:02758 0:031589 0:031757 0:029306 0:029597 0:026360

0:145553 0:146058 0:150973 0:151308 0:148128 0:149579

Normal

1.13E-19

6.22E-16

1.28E-12

9.87E-10

2.87E-07

3.17E-05

0:00135

0:02275

0:158655

Notes: The probability values are calculated based on estimated parameter values of the EGB2 distribution. The probability values based on the normal distribution are the same for all six indices

to the normal distribution, the 1987 market crash, which is beyond 17¢ (daily data), would have never happened. However, recent market crashes indicate that big market swings or significant declines in asset prices happen more frequently than we expect. Although VaR is one of the most prevalent risk measures under normal conditions, it cannot deal with the involvement of those extreme values, since extreme values are not in the state of normal. From this perspective, the EGB2 distribution provides a management tool for calculating risk. Table 55.5 reports the probability of the semivolatility of shocks. We find that the predicted probabilities for extreme values for the EGB2 distribution become greater than that of the normal distribution, especially in the horizon beyond 2¢. For instance, the probability of a 5¢ and 7¢ shock in the biotechnology index for the EGB2 distribution is 1.03E-4 and 2.36E-6. These values are much greater than the 2.87E-7 and 1.28E-12 values, respectively, based on the normal distribution. Yet, the probabilities for the EGB2 distribution under a moderate regime (within ˙2¢) are less than that of the normal distribution. This is another way to tell the peakedness and fat tails of portfolio returns. Notice that the dividing point between the EGB2 distribution and the normal distribution is

in the neighborhood of ˙2¢, where the probabilities of both distributions are equivalent. This feature implies that VaR at the 95% confidence level based on the normal distribution is by chance consistent with reality. However, beyond this critical level, the VaR method based on the normal distribution leads to overly optimistic forecasts of losses.

55.7 Conclusion This study provides a time series model that incorporates the asymmetric GARCH feature into a non-Gaussian distribution: an exponential generalized beta distribution of the second kind (EGB2). When we apply data for six stock indices to the EGB2 distribution, the evidence consistently shows that the model is capable of dealing with skewness and kurtosis of stock returns. Testing the model by using goodness-of-fit and likelihood ratio statistics, the EBG2 distribution outperforms the model by assuming normal distribution. Because of its capacity for modeling skewness and fat tails, the EGB2 model provides a useful tool for forecasting variances involving extreme values. As a result, this model can be practically used for risk management.

862 Acknowledgements We are grateful to Michael J. Gombola, Daniel Dorn, Hazem Maragah, and C.F. Lee for their helpful comments in an earlier draft of this paper. We alone are responsible for any errors. Thomas C. Chiang would like to acknowledge the research support received from the Marshall M. Austin Endowed Chair, LeBow Business College, Drexel University.

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T.C. Chiang and J. Li conditional densities, working paper, WP02–08–01, The University of Alabama. Hueng, C. J. and J. B. McDonald. 2005. “Forecasting asymmetries in aggregate stock market returns: evidence from conditional skewness.” Journal of Empirical Finance 12, 666–685. Johnson, N. L., S. Kotz and N. Balakrishnan. 1994. Continuous univariate distributions, vol. 1, 2nd ed., Wiley, New York. Khindanova, I., S. T. Rachev, and E. Schwartz. 2001. “Stable modeling of value at risk.” Mathematical and Computer Modeling 34, 1223–1259. Liu, S. M. and B. Brorsen. 1995. “Maximum likelihood estimation of a GARCH – stable model.” Journal of Applied Econometrics 2, 273–285. McDonald, J. B. and Y. J. Xu. 1995. “A generalization of the beta distribution with applications.” Journal of Econometrics 66, 133–152. McDonald, J. B. 1991. “Parametric models for partially adaptive estimation with skewed and leptokurtic residuals.” Economics Letters 37, 273–278. Nelson, D. B. 1991. “Conditional heteroskedasticity in asset return: a new approach.” Econometrica 59, 347–370. Rachev, S. T. and S. Mittnik. 2000. Stable paretian models in finance, Wiley, Chichester. Snedecor, G. W. and W. G. Cochran. 1989. Statistical methods, 8th ed., Iowa State Press, Ames, Iowa. Theodossiou, P. 1998. “Financial data and the skewed generalized t distribution.” Management Science 44, 1650–1661. Wang, K.-L., C. Fawson, C. B. Barrett and J. B. McDonald. 2001. “A flexible parametric GARCH model with an application to exchange rates.” Journal of Applied Econometrics 16, 521–536. Wu, J.-W., W.-L. Hung, and H.-M. Lee. 2000. “Some moments and limit behaviors of the generalized logistic distribution with applications.” Proceedings of the National Science Council, ROC 24, 7–14.

Chapter 56

Capital Structure in Asia and CEO Entrenchment Kin Wai Lee and Gillian Hian Heng Yeo

Abstract We examine the association between CEO entrenchment and capital structure decisions of Asian firms. We find that firms with higher CEO entrenchment have a lower level of leverage. Specifically, firms with CEOs who chair the board, have a higher CEO tenure, and have a lower proportion of outside directors, have lower leverage. The negative association between CEO entrenchment and corporate leverage is more pronounced in firms with higher agency costs of free cash flow. In addition, for firms with entrenched CEOs, those firms with greater institutional investor equity ownership have higher leverage. This result suggests that active monitoring by large shareholders mitigate sentrenched CEOs’ incentives to avoid debt. Keywords Capital structure entrenchment

r

Agency costs

r

Managerial

56.1 Introduction A considerable body of research has focused on managers deviating from optimal level of capital structure due to conflict of interest between managers and shareholders. One stream of research suggests that leverage reduces managerial discretion over corporate resources because higher debt financing increases the commitment and pressure to distribute surplus cash as repayment of debt obligations (Jensen 1986). Thus, entrenched managers prefer capital structures with low leverage. Another stream of research suggests that entrenched managers have greater incentives to increase leverage beyond the optimal level to reduce the probability of

K.W. Lee () S3-B2A-19 Nanyang Business School, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore e-mail: [email protected] G.H.H. Yeo S3-1A-21 Nanyang Business School, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore

successful takeovers by increasing the concentration of their shareholdings, which enables them to have greater control of their firms (Harris and Raviv (1988), and Stulz (1988). Prior studies on U.S. listed firms provide some evidence that entrenched managers prefer low corporate leverage. Friend and Lang (1988) and Mehran (1992) find that firms with high agency costs of managerial discretion have low leverage levels. Using a sample of large U.S. industrial firms, Berger et al. (1997) find that leverage levels are lower when CEOs do not face pressure from ownership and compensation incentives or active monitoring. They also document that leverage increases significantly in the aftermath of events that represent entrenchment-reducing shocks to managerial security, including unsuccessful tender offers, involuntary CEO replacements, and the addition to the board of major shareholders. However, given the opposing theories on the association between the capital structure decisions and managerial entrenchment, the empirical evidence supporting these views are hard to generalize, especially in other economies. For example, the market for corporate control in Asia is relatively less hostile and involuntary CEO turnover is quite infrequent. In addition, most firms in Asia are closely held and controlled by a large ultimate shareholder (Claessens et al. 2000). These ownership structures allow controlling owners to commit low equity investment while maintaining tight control of the firm, creating a separation of cash flow rights and voting rights. As the separation of cash flow rights from voting rights increases, the controlling owner becomes more entrenched with levels of control, and therefore entrenched managers in Asia may have less incentive to increase corporate leverage so that they can increase their voting rights. This paper examines the association between CEO entrenchment and capital structure decisions of Asian firms. We find that firms with higher CEO entrenchment have lower levels of leverage. Specifically, our results indicate that firms with CEOs who chair the board, and have a lower proportion of outside directors, have a higher CEO tenure, tend to have lower leverage. We then examine factors that may affect the association between leverage and CEO entrenchment. We find that the

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_56, 

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negative association between CEO entrenchment and corporate leverage is more pronounced in firms with higher agency costs of free cash flow. Thus, in firms with entrenched CEOs, those with greater managerial discretion associated with free cash flow have lower leverage. We interpret our result as suggesting that high agency costs of free cash flow exacerbates the agency costs associated with CEO entrenchment, resulting in lower levels of leverage. We also find that for firms with entrenched CEOs, those with greater institutional equity ownership have higher leverage. This result suggests that active monitoring by large institutional shareholders mitigate entrenched CEOs’ incentives to avoid debt. Our result contributes to stream of research on the corporate governance role of institutional shareholders in curtailing the managerial opportunism or self-serving behavior (Gillan and Starks 2000; Hartzell and Starks 2003; Parrino et al. 2003). Debt-to-asset ratios represent the cumulative result of years of separate decisions. Thus, cross-sectional tests based on a single aggregate of different decisions are likely to have low power (Jung et al. 1996). To increase the power of our test, we also examine the decisions to change leverage. Further analysis indicates that year-to-year change in leverage is negatively associated with CEO entrenchment. Specifically, net debt issued during the year is lower when a CEO also chairs the board, and when the CEO has longer tenure. We also find significant leverage decreases in firms with low board independence. As a robustness test, we also examine how leverage changes as a function of the firm’s financing deficit at the start of the year. The financing deficit variable is computed as cash dividends paid plus investments plus capital expenditure plus change in working capital less cash flow. We find that firms with entrenched CEOs tend to fund their financing deficit by issuing less debt. We interpret our result to show that entrenched CEOs avoid increases in debt to mitigate the disciplinary role of debt. This result suggests that in firms with entrenched CEOs, those firms with high free cash flow are even less likely to fund their financing deficit with debt. In other words, the entrenchment effect of CEOs on net debt issued is more pronounced in firms with high agency costs of free cash flow. In firms with entrenched CEOs, those firms with high institutional equity ownership are more likely to fund their financing deficit with debt. Hence, the entrenchment effect of CEOs on net debt issued is mitigated by large institutional equity ownership, suggesting institutional shareholders play an effective monitoring role. The rest of the paper proceeds as follows. Section 56.2 develops the hypotheses and places our paper in the context of related research. Section 56.3 describes the sample. Section 56.4 presents our results. We conclude the paper in Sect. 56.5.

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56.2 Prior Research and Hypothesis 56.2.1 CEO Entrenchment and Leverage Conflicts of interest over capital structure decisions arise between managers and shareholders because managers may deviate from value-maximizing levels of debt. Managers may prefer less leverage than optimal because of their preference for lower firm risk to protect their undiversified human capital (Fama 1980) and their preference to avoid the performance pressures associated with fixed debt payments (Jensen 1986)1. Jung et al. (1996) provide evidence supporting the entrenchment affecting capital structure choices. They find that equity issuers are low-leverage firms with limited investment opportunities. These firms also invest more than similar firms issuing debt. These results suggest that agency costs of managerial discretion lead certain firms to issue equity when debt issuance would be the firm-value enhancing alternative. Using a sample of 452 industrial firms in the United States, Berger et al. (1997) find that firm leverage is negatively associated with the degree of entrenchment of managers. Specifically, they find that leverage is lower when the CEO has a long tenure in office, has weak compensation incentives, and does not face strong monitoring from the board of directors. In further analysis of leverage changes, they find that leverage increases significantly in the aftermath of events that represent entrenchment-reducing shocks to managerial security including unsuccessful tender offers, involuntary CEO replacements, and the addition to the board of major shareholders. Another stream of research suggests that entrenched managers have greater incentives to increase leverage beyond the optimal level so that they can increase the voting power of their equity ownership and reduce the probability of successful takeovers (Harris and Raviv 1988; Stulz 1988). In a similar vein, Israel (1992) argues that by issuing debt, the management of the target firm transfers some of the value from equity holders to debt-holders in exchange for private benefits of control, which lowers the acquirer’s premium. Unlike U.S., the market for corporate control is less hostile in Asia and proxy fights are rare. In addition, involuntary CEO turnover is relatively infrequent in Asia. Most firms in Asia are closely held and controlled by a large ultimate shareholder (Claessens et al. 2000). Corporate ownership in Asia is complicated by pyramidal and cross-holding structures. Specifically, these ownership structures allow controlling owners to commit low equity investment while maintaining

1

See Chen et al. (2009) for a review of important and representative capital structure models.

56 Capital Structure in Asia and CEO Entrenchment

tight control of the firm, creating a separation of cash flow rights and voting rights. As the separation of cash flow rights from voting rights increases, the controlling owner becomes more entrenched with levels of control, while the low cashflow level provides a low degree of alignment of interest between the controlling owner and minority shareholders. Given the relatively inactive market for corporate control and high voting rights concentration in the controlling shareholder, we conjecture that entrenched managers in Asia have relatively weak incentives to increase corporate leverage with the aim of increasing their voting rights. Thus, on balance, entrenched CEOs of Asian firms are likely to prefer lower corporate leverage to avoid the monitoring associated with debt financing. If CEOs in Asian firms face less entrenchment-reducing shocks to managerial security such as unsuccessful tender offers and involuntary CEO replacements, the degree of CEO entrenchment will be higher. Our first hypothesis is as follows: H1: In Asian firms, leverage is negatively associated with the CEO entrenchment.

56.2.2 Free Cash Flow, CEO Entrenchment, and Leverage Firms with high free cash flow may overinvest in negative net present value projects (Jensen 1986; Stulz 1988; Zwiebel 1996). Leverage reduces management’s discretion and mitigates the agency costs of managerial discretion. Since debt financing without retention of proceeds of issue commits the free cash flow to pay creditors, management has less control over the firm’s cash flow. Management will be monitored by creditors who want to ensure that they will be repaid. We posit that entrenched CEOs have incentives to avoid the monitoring associated higher leverage so that they have more discretion over corporate resources. We extend this notion to suggest that entrenched CEOs’ propensity to avoid leverage is exacerbated in firms with high free cash flow. Thus, the combination of higher free cash flow and higher CEO entrenchment amplifies the agency costs of managerial discretion and results in lower leverage. H2: The negative association between CEO entrenchment and corporate leverage is more pronounced in firms with higher free cash flow.

56.2.3 Institutional Ownership, CEO Entrenchment, and Leverage Past studies document the role of large shareholders in corporate governance (Shleifer and Vishny 1986). Due to the

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high cost of monitoring, large shareholders such as institutional investors can achieve sufficient benefits by having an incentive to monitor (Grossman and Hart 1980). Specifically, large institutional investors have the opportunity, resources, and ability to monitor, discipline, and influence managers. McConnell and Servaes (1990), Nesbitt (1994), Smith (1996), and Del Guercio and Hawkins (1999) find that firm performance (as measured by Tobin’s Q) is positively related to institutional investor ownership. These studies prove consistent with the hypothesis that corporate monitoring by institutional investors can result in managers focusing more on corporate performance and less on opportunistic or selfserving behavior. Other studies find that institutional investors have increasingly used their ownership rights to pressure managers to act in the interest of shareholders. Gillan and Starks (2000) find that corporate governance proposals sponsored by institutional investors receive more favorable votes than those sponsored by independent individuals or religious organizations. Hartzell and Starks (2003) find that institutional ownership is negatively associated with the level of executive compensation and positively associated with pay-for-performance sensitivity. Finally, Parrino et al. (2003) show that institutional selling is associated with forced CEO turnover and that these CEOs are more likely to be replaced with an outsider. In this study, agency costs arise because entrenched CEOs underlever their firms to avoid the tighter monitoring associated with higher debt financing. If institutional shareholders have the incentives and ability to perform an effective monitoring role, we predict that large institutional shareholders mitigate entrenched CEOs’ incentives to reduce corporate leverage. Thus, for firms with entrenched CEOs, those firms with higher institutional ownership have higher leverage. Our third hypothesis is as follows: H3: The negative association between CEO entrenchment and corporate leverage is mitigated in firms with higher institutional shareholders.

56.3 Data and Method 56.3.1 Sample Construction We begin with the Worldscope database to identify listed firms in eight East Asian countries comprising Hong Kong, Indonesia, Japan, South Korea, Malaysia, Philippines, Singapore and Thailand for the period 2000–2005. We exclude financial institutions because of their unique financial structure and regulatory requirements. We eliminated observations with extreme values of financial statement variables such as leverage and return-on-assets (discussed in

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Sect. 56.3.2 below). This procedure yielded an initial sample of 4,720 firms. In view of the costs of manually collecting CEO entrenchment and ownership variables from the annual reports, we randomly selected 834 firms to obtain 18% of the firms in the initial sample. We obtained annual reports for fiscal year 2000–2005 from the Global Report database and company websites. Our final sample consisted of 834 firms for 4,301 firm-year observations during the period 2000–2005 in eight East Asian countries (Hong Kong, Indonesia, Japan, South Korea, Malaysia, Philippines, Singapore, and Thailand). On average, our final sample accounts for 62% of the market capitalization of all the listed firms in these countries.

56.3.2 Empirical Model We use the following regression model below to test the association between the corporate leverage and the CEO entrenchment: LTD D ˇ0 C ˇ1  CEODUAL C ˇ2  TENURE C ˇ3  OUTDIR C ˇ4 INSTIOWN C ˇ5 ROA C ˇ6  SIZE C ˇ7  MB C ˇ8  TANGIBLE C ˇ9  FCF C Country Controls C Year Controls C Industry Controls C e

(56.1)

where: LTD D long term debt divided by total assets. CEODUAL D A dummy variable that equals 1 if the CEO is the chairman of the board of directors, and 0 otherwise. TENURE D The number of years the CEO is in office. OUTDIR D The number of outside directors divided by board size. INSTIOWN D percentage of common stock outstanding owned by institutional shareholders ROA D net profit after tax divided by total assets. SIZE D natural logarithm of total assets. MB D market value of equity divided by book value of equity. TANGIBLE D Net Property, Plant and Equipment divided by total assets. FCF D (Cash flow from operations less capital expenditure less common dividends paid) divided by total assets. Country Controls D a set of country dummy variables Year Controls D a set of year dummy variables Industry Controls D a set of industry dummy variables e is the error term.

56.3.2.1 Test Variables We have three proxies for CEO entrenchment: (1) CEO duality, (2) CEO tenure, and (3) board independence. The board of directors cannot objectively monitor a CEO who also chairs the board. Consistent with this hypothesis, Dahya et al. (2002) find higher CEO turnover following poor firm performance for firms that separate CEOs and board chairs. They also find that operating performance improves for these firms. We measure CEO duality with a dummy variable (CEODUAL) that equals 1 if the CEO is the chairman of the board of directors, and 0 otherwise. Following Hermalin and Weisbach (1998), we argue that independent boards are more willing to monitor the CEO, whose ability to impose costs on them declines with their independence. We measure board independence as the proportion of outside directors on the board (OUTDIR). Outside directors are defined as those who are not current or former employees of the company and those who do not have any related party transactions with the company. The CEO’s control over the internal monitoring mechanisms increases as his tenure increases. An entrenched CEO who is insulated from the threat of disciplinary action from the managerial labor market and the market for corporate control is likely to have a larger number of years in the office. We compute the CEO tenure as the number of years in office until the start of the current year (TENURE). Hypothesis H1 predicts that corporate leverage is negatively associated with CEO entrenchment. Thus, we expect firms with CEOs who chair the board to have a higher CEO tenure and have a lower proportion of outside directors and have lower leverage. Hence, we predict the coefficients “1 and “2 to be negative and the coefficients “3 to be positive. Next, we construct a composite entrenchment variable using principal component analysis that summarizes the information contained in the individual entrenchment variables by detecting linear relationships among these variables. The advantage of this method is that it reduces the dimension of the explanatory variables and the potential multi-collinearity problem. To test hypothesis H2 and H3, we modify model (1) as follows: LTD D ˇ0 C ˇ1  ENTRENCH C ˇ2  ENTRENCH FCF C ˇ3 ENTRENCH INSTIOWN C ˇ4  INSTIOWN C ˇ5  ROA C ˇ6  SIZE C ˇ7  MB Cˇ8  TANGIBLE C ˇ9  FCF C Country Controls C Year Controls C Industry Controls C e

(56.2)

56 Capital Structure in Asia and CEO Entrenchment

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where: ENTRENCH D a composite CEO entrenchment index based on the principal component analysis of CEODUAL, TENURE and OUTDIR with higher values of ENTRENCH denoting higher CEO entrenchment. Other variables are previously defined. From Hypothesis H1, we predict the coefficient for ENTRENCH to be negative. Hypothesis H2 predicts that the negative association between leverage and CEO entrenchment is more pronounced in firms with higher agency costs of free cash flow. We measure free cash flow (FCF) as cash flow from operations less capital expenditure less common dividends paid divided by total assets. Thus, we expect the coefficient on the interaction term, ENTRENCH FCF, to be negative. Hypothesis H3 predicts that the negative association between leverage and CEO entrenchment is more mitigated in firms with higher institutional equity ownership. Thus, we expect the coefficient on the interaction term, ENTRENCH INSTIOWN, to be positive.

dummies to control for time-series effects and country dummies to control for country-specific effects. We include industry dummies to control for industry-level determinants of capital structure.

56.4 Results 56.4.1 Descriptive Statistics Table 56.1 reports the descriptive statistics. Mean leverage, computed as long-term debt divided by total assets, is 0.1178. On average, the firms are profitable (mean ROA D 0:037), are relatively low growth firms (mean MB D 1:096), and have high proportion of assets comprising of tangible assets (mean TANGIBLE D 0:406). About 26% of the firms have a CEO who also chairs the board. The mean CEO tenure is 6.1 years. The mean proportion of outside directors on the board is 0.4002. The mean of INSTIOWN is 0.19, suggesting that 19% of the firms have large institutional shareholders.

56.3.2.2 Control Variables We also include in our model standard control variables for other firm characteristics expected to influence leverage. These variables were considered in prior studies such as Titman and Wessels (1988) and Rajan and Zingales (1995). To control for firm size, we use the natural logarithm of the book value of total assets (SIZE). To control for profitability, we include a return on assets variable (ROA) defined as net income after tax divided by total assets. We also control for the collateral value of tangible assets by including the net property, plant, and equipment divided by total assets (TANGIBLE). To control for growth options in the firm’s investment opportunity set, we include the ratio of market value of equity to book value of equity (MB). We also include year

56.4.2 Leverage Levels Table 56.2 presents the results of regressions of corporate leverage levels. The dependent variable is the level of leverage defined as long-term debt divided by total assets. In column (1), the results indicate that firms with entrenched CEOs have lower leverage. Specifically, the estimated coefficient on CEODUAL is negative and significant at the 1% level, suggesting that firms with CEOs who chair the board have lower leverage. Furthermore, the estimated coefficient on TENURE is negative and significant at the 1% level, suggesting that the longer the CEO tenure, the lower the corporate leverage. This result is consistent with entrenched CEOs

Table 56.1 Descriptive statistics LTD ROA SIZE MB TANGIBLE FCF CEODUAL TENURE OUTDIR INSTIOWN

Mean

25th percentile

Median

75th percentile

Standard deviation

0:1178 0:0373 12:4491 1:0969 0:4067 0:0026 0:2600 6:149 0:4002 0:1908

0:0023 0:0038 11:4710 0:4673 0:2464 0:0331 0 4:662 0:2864 0:1129

0:0734 0:0384 12:3871 0:7771 0:4030 0:0062 0 5:223 0:3247 0:1912

0:1972 0:0849 13:406 1:3248 0:5558 0:0643 1 6:0379 0:5565 0:2653

0.1299 0.1451 1.3999 1.0778 0.2048 0.0781 0.4382 2.4584 0.1514 0.1037

The sample consists of 4,301 firm-year observations in the period 2000–2005. All variables are defined in Appendix 56.1

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Table 56.2 Regressions of capital structure levels

1 0:0185 .4:26/ 0:0026 .3:17/ 0.0413 (3.08) 0.0853 (5.23)

CEODUAL TENURE OUTDIR INSTIOWN ENTRENCH

2

0.0653 (3.41) 0:0249 .6:84/

ENTRENCH FCF ENTRENCH INSTIOWN ROA SIZE MB TANGIBLE FCF Year indicator variables Country indicator variables Industry indicator variables N Adjusted R2

0:1052 .8:34/ 0.0322 (14.16) 0.0045 (2.11) 0.1523 (7.62) 0:0147 (2.10) Yes Yes Yes 4,301 29.1%

0:2172 .9:02/ 0.0324 (16.62) 0.0035 (2.15) 0.1516 (7.49) 0:0285 .2:21/ Yes Yes Yes 4,301 28.6%

3

0:01708 .3:95/ 0:0147 .2:23/ 0.0341 (4.51) 0:1467 .6:53/ 0.0317 (14.28) 0.0071 (1.09) 0.1495 (7.31) 0:0382 .1:71/ Yes Yes Yes 4,301 30.7%

The sample consists of 4,301 firm-year observations in the period 2000–2005. The dependent variable is long-term debt divided by total assets (LTD). ENTRENCH is a composite entrenchment index based on the principal component analysis of CEODUAL, TENURE, and OUTDIR with higher values of ENTRENCH denoting higher CEO entrenchment. The clustered t-statistics, adjusted for heteroscedasticity and serial correlation, are reported in parenthesis. All other variables are defined in Appendix 56.1.  ,  , and  indicate two-tailed statistical significance at the 10%, 5%, and 1% levels, respectively

taking on less debt, under the assumption that entrenched CEOs have longer job tenure in office. Variables associated with stronger monitoring have positive associations with leverage. When the CEO’s power over the board decreases, corporate leverage increases. Specifically, the positive and significant coefficient on OUTDIR suggests that firms with higher proportion of outside directors on the board have higher leverage. Thus, independent boards have greater influence over entrenched CEOs’ incentives to avoid leverage. In Table 56.2 column (2), we include a composite CEO entrenchment variable (ENTRENCH) based on the principal component analysis of CEODUAL, TENURE, and OUTDIR. The composite CEO entrenchment proxy is an increasing function of managerial entrenchment given the relative importance of the overall factor loadings of each individual entrenchment proxy. Thus, higher values of ENTRENCH denote higher CEO entrenchment. The estimated coefficient on

ENTRENCH is negative and significant at the 1% level. This result supports the hypothesis that entrenched CEOs prefer less debt. In column (3), the coefficient on the interaction term between CEO entrenchment and free cash flow (ENTRENCH FCF) is negative and significant at the 5% level. This finding suggests that the negative association between leverage and CEO entrenchment is more pronounced in firms with higher agency costs of free cash flow. Thus, consistent with hypothesis H2, in firms with entrenched CEOs, those with greater managerial discretion associated with free cash flow have lower leverage. In other words, higher free cash flow exacerbates the agency costs associated with CEO entrenchment, resulting in lower level of leverage. Furthermore, our results indicate that leverage increases with the proportion of equity owned by institutional investors (INSTIOWN). This finding suggests CEOs are more

56 Capital Structure in Asia and CEO Entrenchment

likely to take on more debt when there is active monitoring by large external shareholders. More generally, large shareholders appear to act as monitoring complements to debt-holders. We also document that the coefficient on the interaction term between CEO entrenchment and free cash flow (ENTRENCH INSTIOWN) is positive and significant at the 1% level. Thus, consistent with hypothesis H3, this finding suggests that the negative association between leverage and CEO entrenchment is mitigated in firms with higher institutional equity ownership. By demonstrating that in firms with entrenched CEOs, those with larger institutional equity ownership have higher leverage, our result provides evidence supporting the notion that large institutional shareholders play an effective monitoring role in reducing managerial discretion. In general, the control variables are in their predicted direction. Firms with larger profitability and those with lower free cash flow have lower leverage. Larger firms and firms with higher proportion of tangible assets have higher leverage. As

DEBTISSUE D

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robustness tests, we repeat our analysis by country. Regression results by country (not tabulated) yield qualitatively similar results. Similarly, when we repeat our tests using annual regressions, our inferences are not changed. Thus, there is no clustering of results by economy and by year.

56.4.3 Changes in Leverage Prior studies (Jung et al. 1996; Berger et al. 1997) suggest that debt-to-asset ratios represent the cumulative result of years of separate decisions. Thus, cross-sectional tests based on a single aggregate of different decisions are likely to have low power. Following Berger et al. (1997), as an additional test, we analyze the decisions to change leverage rather than use the cross-sectional variation in debt-to-asset ratios. To examine the net change in leverage during the year, we compute the net debt issued variable (DEBTISSUE) as

.debt issued less debt retired/ minus .equity issued less equity retired/ Total assets at the start of the year

Table 56.3 presents the regression results for our model of year-to-year net change in leverage. The dependent variable is net debt issued during the year. All independent variables are measured at the start of the year. In column (1), we find that firms with a CEO who also chairs the board have significant decrease in leverage. The results also indicate significant increases in leverage in firms with higher CEO tenure. We also document that firms with higher proportion of outside directors on the board have larger leverage increases. Furthermore, net debt issued by firms is positively associated with the proportion of institutional equity ownership. Collectively, these results suggest that entrenched CEOs are less likely to issue net debt. In column (2), we replace three of the proxies for CEO entrenchment (CEODUAL, TENURE, and OUTDIR) with a composite CEO entrenchment variable (ENTRENCH). Recall that the variable ENTRENCH is computed based on the principal component analysis of CEODUAL, TENURE, and OUTDIR with higher values of ENTRENCH denoting greater CEO entrenchment. The estimated coefficient on ENTRENCH is negative and significant at the 1% level. This finding implies that entrenched CEOs issue less net debt. In column (3), we test whether agency costs of free cash flow exacerbate entrenched CEOs’ incentives to issue less debt. The coefficient on the interaction term between CEO entrenchment and free cash flow (ENTRENCH FCF) is negative and significant at the 5% level. This finding suggests

that in firms with higher agency costs of free cash flow, those with more entrenched CEOs have lower propensity to issue net debt. Hence, the combination of higher agency costs of free cash flow and greater CEO entrenchment is associated with greater decline in leverage. We also document that the coefficient on the interaction term between CEO entrenchment and free cash flow (ENTRENCH INSTIOWN) is positive and significant at the 1% level. Consistent with hypothesis H3, this finding suggests that in firms with entrenched CEOs, those with higher institutional equity ownership have higher increases in leverage.

56.4.4 Financing Deficit Our main results indicate that less entrenched CEOs issue more debt. However, as Berger et al. (1997) note, more leverage may not represent a value-increasing strategy, and it is possible that CEOs increase their firms’ leverage beyond optimal levels as a defensive measure when their security is threatened. For example, increasing corporate leverage increases the concentration of equity, which may reduce the likelihood of takeover. To address this issue, we examine how leverage changes as a function of the firm’s financing deficit at the start of the year. Table 56.4 presents the results of regressions of

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Table 56.3 Regressions of net debt issued

1 0:0542 .2:75/ 0:0021 .3:18/ 0.0487 (2.03) 0.0457 (2.17)

CEODUAL TENURE OUTDIR INSTIOWN ENTRENCH

2

0.0283 (2.09) 0:0611 .4:86/

ENTRENCH FCF ENTRENCH INSTIOWN ROA SIZE MB TANGIBLE LTD FCF Year indicator variables Country indicator variables Industry indicator variables N Adjusted R2

0.1819 (3.95) 0.0178 (5.65) 0:0141 .1:87/ 0:0135 .0:65/ 0:1838 .5:85/ 0:2193 .5:52/ Yes Yes Yes 1,198 15.8%

0.1832 (3.98) 0.0177 (1.89) 0:0139 .1:86/ 0:0123 .0:60/ 0:1838 .5:63/ 0:2154 .5:44/ Yes Yes Yes 1,198 15.7%

3

0:0484 .5:17/ 0:0122 .2:14/ 0.0155 (1.99) 0.1622 (3.49) 0.0058 (0.94) 0:0146 .1:96/ 0:0041 .0:19/ 0:1792 .5:72/ 0:2250 .5:64/ Yes Yes Yes 1,198 16.3%

The sample consists of 1,198 firm-year observations in the period 2000–2005. The dependent variable is net debt issued divided by total assets at the start of the year (DEBTISSUE). Net debt issued is (debt issued less debt repurchased) minus (equity issued less equity repurchased) during the year. All the independent variables are measured at the start of the year. ENTRENCH is a composite entrenchment index based on the principal component analysis of CEODUAL, TENURE, and OUTDIR with higher values of ENTRENCH denoting higher CEO entrenchment. The clustered t-statistics, adjusted for heteroscedasticity and serial correlation, are reported in parenthesis. All other variables are defined in Appendix 56.1.  ,  , and  indicate two-tailed statistical significance at the 10%, 5%, and 1% levels, respectively

changes in leverage on financing deficit and CEO entrenchment variables. The dependent variable is net debt issued during the year (defined in Sect. 56.4.3). The financing deficit variable (DEFICIT) is computed as cash dividends paid plus investments plus capital expenditure plus change in working capital less cash flow. A positive coefficient on DEFICIT indicates that the firm funds its financing deficit with debt. In column (1), the financing deficit variable is positive and significant, indicating that firms fund their financing deficit mainly with debt issuance. The interaction term between financing deficit and composite entrenchment index (ENTRENCH DEFICIT) is negative and significant at 5% level. This result suggests that firms with entrenched CEOs

tend to fund their financing deficit by issuing less debt. This appears to indicate that entrenched CEOs avoid increases in debt to fund their firm-level financing deficit to mitigate the disciplinary role of debt in mitigating managerial discretion. Hence, this result casts doubt that the changes in leverage are likely to be firm value-enhancing. In column (2), we examine how agency costs of free cash flow affects the association between changes in leverage and the financing deficit in firms with entrenched CEOs. The three-way interaction term (ENTRENCH DEFICIT FCF) is negative and significant at the 5% level. This result suggests that in firms with entrenched CEOs, those firms with high free cash flow are even less likely to fund their financing

56 Capital Structure in Asia and CEO Entrenchment

871

Table 56.4 Regressions of net debt issued on financing deficit 1 2 DEFICIT ENTRENCH ENTRENCH DEFICIT

0.0745 (2.84) 0:0197 (2.25) 0:0116 .2:29/

ENTRENCH DEFICIT FCF ENTRENCH DEFICIT INSTIOWN CHGROA CHGSIZE CHGMB CHGTANGIBLE Year indicator variables Country indicator variables Industry indicator variables N Adjusted R2

0:1877 .4:06/ 0.1159 (8.12) 0.0026 (0.61) 0.0745 (1.67) Yes Yes Yes 1,162 10.5%

0.8503 (2.19) 0:0211 .1:81/ 0:0134 .2:01/ 0:1006 .6:41/ 0.0345 (1.93) 0:1828 .4:02/ 0.1041 (7.36) 0.0041 (0.98) 0.0545 (1.24) Yes Yes Yes 1,162 13.7%

The sample consists of 1,162 firm-year observations in the period 2000–2005. The dependent variable is net debt issued divided by total assets at the start of the year (DEBTISSUE). Net debt issued is (debt issued less debt repurchased) minus (equity issued less equity repurchased) during the year. All the independent variables are measured at the start of the year. ENTRENCH is a composite CEO entrenchment index based on the principal component analysis of CEODUAL, TENURE, and OUTDIR with higher values of ENTRENCH denoting higher CEO entrenchment. DEFICIT is financing deficit for the year computed as the sum of cash dividend plus capital expenditure plus investments plus change in working capital less cash flow from operations divided by total assets. CHGROA is the change in return on assets. CHGMB is the change in market-to-book equity. CHGSIZE is the change in firm size as measured by logarithm of total assets. CHGTANGIBLE is the change in tangible assets. The clustered t-statistics, adjusted for heteroscedasticity and serial correlation, are reported in parenthesis.  ,  , and  indicate two-tailed statistical significance at the 10%, 5%, and 1% levels, respectively

deficit with debt. In other words, the entrenchment effect of CEOs on net debt issued is concentrated in firms with high agency costs of free cash flow. In column (3), we explore how institutional equity ownership affects the changes in leverage and the financing deficit in firms with entrenched CEOs. The three-way interaction term (ENTRENCH DEFICIT INSTIOWN) is positive and significant at the 10% level. Thus, there is some evidence supporting the notion that in firms with entrenched CEOs, those with high institutional equity ownership are more likely to fund their financing deficit with debt. In other words, the entrenchment effect of CEOs on net debt issued is reduced in the presence of large institutional equity ownership.

56.5 Conclusion Entrenched CEOs have discretion over their firms’ leverage choices. Following Jensen (1986), leverage reduces the agency costs of managerial discretion. Since debt issuance commits the free cash flow to pay creditors, management has less control over the firm’s cash flow. Thus, we posit that entrenched CEOs have incentives to avoid the monitoring associated higher leverage so that they have more discretion over corporate resources. We find that firms with higher CEO entrenchment have lower level of leverage. Specifically, firms with CEOs who chair the board, have a higher CEO tenure and have a lower proportion of outside directors and have lower leverage. We also predict that the combination of higher free cash flow and higher CEO entrenchment exacerbates the agency costs of managerial discretion and results in lower leverage. Consistent with our hypothesis, we find that the negative association between CEO entrenchment and corporate leverage is more pronounced in firms with higher agency costs of free cash flow. Finally, we document that for the sub-sample of firms with entrenched CEOs, those firms with greater institutional equity ownership have higher leverage. This result suggests large institutional shareholders play an important corporate governance role in mitigating entrenched CEOs’ incentives to avoid debt.

References Berger, P. G., E. Ofek, and D. L. Yermack. 1997. “Managerial entrenchment and capital structure decisions.” Journal of Finance, 52(4), 1411–1438. Chen, S.S., C.F. Lee, and H.H. Lee. (2010). “Alternative methods to determine optimal capital structure: theory and application.” Handbook of Quantitative Finance and Risk Management, (forthcoming). Claessens, S., S. Djankov, and L. Lang. 2000. “The separation of ownership and control in East Asian corporations.” Journal of Financial Economics 58, 81–112. Dahya, J., J. J. McConnell, and N. G. Travlos. 2002. “The cadbury committee, corporate performance, and top management turnover.” Journal of Finance 57(1), 461–483. Del Guercio, D. and J. Hawkins. 1999. “The motivation and impact of pension fund activism.” Journal of Financial Economics 52, 293–340. Fama, E. 1980. “Agency problems and the theory of the firm.” Journal of Political Economy 88, 288–307. Friend, I. and L. H. P. Lang. 1988. “An empirical test of the impact of managerial self-interest on corporate capital structure.” Journal of Finance 47, 271–281. Gillan, S. and L. Starks. 2000. “Corporate governance proposals and shareholder activism: the role of institutional investors.” Journal of Financial Economics 57, 275–305. Grossman, S. and O. Hart. 1980. “Takeover bids, the free rider problem, and the theory of the corporation.” Bell Journal of Economics 11, 42–64.

872 Harris, M. and A. Raviv. 1988. “Corporate control contests and capital structure.” Journal of Financial Economics 20, 55–86. Hartzell, J.C. and L. T. Starks. 2003. “Institutional investors and executive compensation.” Journal of Finance 58(6), 2351–2375. Hermalin, B. E. and M. S. Weisbach. 1998. “Endogenously chosen board of directors and their monitoring of the CEO.” American Economic Review 88(1), 96–188. Jensen, M. C. 1986. “Agency costs of free cash flow, corporate finance and takeovers.” American Economic Review 76, 323–329. Jung, K., Y. C. Kim, and R. M. Stulz. 1996. “Timing, investment opportunities, managerial discretion, and security issue decision.” Journal of Financial Economics, 42, 159–185. Israel, R.. 1992. “Capital and ownership structures, and the market for corporate control.” Review of Financial Studies 5(2), 181–198. McConnell, J. J. and H. Servaes. 1990. “Additional evidence on equity ownership and corporate value.” Journal of Financial Economics 27(2), 595–612. Mehran, H. 1992. “Executive incentive plans, corporate control, and capital structure.” Journal of Financial and Quantitative Analysis 27, 539–560. Nesbitt, S. L. 1994. “Long-term rewards from shareholder activism: a study of the CalPERS effect.” Journal of Applied Corporate Finance 6, 75–80. Parrino, R., R. W. Sias, and L. T. Starks. 2003. “Voting with their feet: institutional ownership changes around forced CEO turnover.” Journal of Financial Economics 68, 3–76. Rajan, R. G. and L. Zingales. 1995. “What do we know about capital structure? Some evidence from international data.” Journal of Finance 50, 1421–1460. Shleifer, A. and R. Vishny. 1986. “Large shareholders and corporate control.” Journal of Political Economy 94, 448–461. Smith, M. 1996. “Shareholder activism by institutional investors: evidence from CalPERS.” Journal of Finance 51, 227–252.

K.W. Lee and G.H.H. Yeo Stulz, R. 1988. “Managerial control of voting rights: financing policies and the market for corporate control.” Journal of Financial Economics 20, 25–54. Titman, S. and R. Wessels. 1988. “The determinants of capital structure.” Journal of Finance 43(1), 1–19. Zwiebel, J. 1996. “Dynamic capital structure under managerial entrenchment.” American Economic Review 86(5), 1197–1215.

Appendix 56A Variables Definition

Variables LTD ROA SIZE MB TANGIBLE

Definition

Long-term debt divided by total assets. Net income after tax divided by total assets. Natural logarithm of book value of total assets. Market value of equity divided by book value of equity Net Property, Plant, and Equipment divided by total assets. FCF (Cash flow from operations less capital expenditure less common dividends paid) divided by total assets. CEODUAL A dummy variable that equals 1 if the CEO is the chairman of the board of directors, and 0 otherwise. TENURE The number of years the CEO is in office. OUTDIR The number of outside directors divided by board size. INSTIOWN The proportion of common equity held by institutional investors.

Chapter 57

A Generalized Model for Optimum Futures Hedge Ratio Cheng-Few Lee, Jang-Yi Lee, Kehluh Wang, and Yuan-Chung Sheu

Abstract Under martingale and joint-normality assumptions, various optimal hedge ratios are identical to the minimum variance hedge ratio. As empirical studies usually reject the joint-normality assumption, we propose the generalized hyperbolic distribution as the joint log-return distribution of the spot and futures. Using the parameters in this distribution, we derive several widely used optimal hedge ratios: minimum variance, maximum Sharpe measure, and minimum generalized semivariance. Under mild assumptions on the parameters, we find that these hedge ratios are identical. Regarding the equivalence of these optimal hedge ratios, our analysis suggests that the martingale property plays a much important role than the joint distribution assumption. Keywords Optimal hedge ratio distribution r Martingale property

r

Generalized hyperbolic

57.1 Introduction Because of their low transaction cost, high liquidity, high leverage, and ease of short position, stock index futures are among the most successful innovations in the financial markets. Besides speculative trading, they are widely used to hedge against the market risk of the spot position. One of the most important issues for investors and portfolio managers is to calculate the optimal futures hedge ratio, the proportion of the position taken in futures to the size of the spot so that the risk exposure can be minimized. The optimal hedge ratios typically depend on the objective functions under consideration. In literature on futures K. Wang () and Y.-C. Sheu National Chiao-Tung University, Hsinchu City, Taiwan, ROC e-mail: [email protected]; [email protected] C.-F. Lee Rutgers University, New Brunswick, NJ, USA e-mail: [email protected] J.-Y. Lee Tunghai University, Taichung, Taiwan, ROC e-mail: [email protected]

hedging, there are two different types of objective functions: the risk function to be minimized, and the utility function to be maximized. Johnson (1960) obtains the minimum variance hedge ratio by minimizing the variance of the change in the value of the hedged portfolios. On the other hand, as Adams and Montesi (1995) indicate, corporate managers are more concerned with the downside risk rather than the upside variation. A measure of the downside risk is the generalized semivariance (GSV) where the risk is computed from the expectation of a power function of shortfalls from the target return (Bawa 1975, 1978; Fishburn 1977). De Jong et al. (1997) and Lien and Tse (1998, 2000, 2001) have calculated several GSV-minimizing hedge ratios. Regarding the utility function approach, we consider the Sharp measure (SM) criteria; that is, the ratio of the portfolio’s excess return to its volatility. Howard and D’Antonio (1984) formulate the optimal hedge ratio by maximizing the Sharp measure. Normally, these optimal hedge ratios under different approaches are not the same. However, with the jointnormality and martingale assumptions, they are identical to the minimum variance hedge ratio. Unfortunately, many empirical studies indicate that major markets typically reject the joint-normality assumption (Fig. 57.1) (Chen, Lee and Shrestha 2001; Lien and Tse 1998). In particular, the fat-tail property of the return distribution affects the hedging effectiveness substantially (Fig. 57.2). It will be useful to find out the nature of the optimal hedge ratios under more realistic assumptions. In this paper we introduce the bivariate generalized hyperbolic distributions as alternative joint distributions for returns in the spot and futures markets (Fig. 57.3). Barndorff-Nielsen (1977, 1978) develops the generalized hyperbolic (GH) distributions as a mixture of the normal distribution and the generalized inverse Gaussian (GIG) distribution first proposed in 1946 by Etienne Halphen. The class of the generalized hyperbolic distributions includes the hyperbolic distributions, the normal inverse Gaussian distributions, and the variance-Gamma distributions, while the normal distribution is a limiting case of the generalized hyperbolic distributions. Use of the generalized hyperbolic distributions have been increasing in finance literature. To model the log returns of some financial assets,

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_57, 

873

874

C.-F. Lee et al.

Fig. 57.1 Normal density and Gaussian kernel density estimators

Fig. 57.2 Log-densities of daily log returns of major indices and futures (2000–2004)

Eberleinand Keller (1995) consider the hyperbolic distribution and Barndorff-Nielsen (1995) proposes the normal inverse Gaussian distribution. For more recent applications of the generalized hyperbolic distributions in finance, see Bibby and Sørensen (2003), Eberlein, Keller, and Prause (1998), Rydberg (1997, 1999), Kücher et al. (1999), and Bingham and Kiesel (2001). In terms of the parameters for the bivariate hyperbolic distributions, we have developed in this paper the minimum variance hedge ratio, GSV-minimizing hedge ratio, and the SM-maximizing hedge ratio. Moreover, the relationships between these hedge ratios are explored. In particular, under the martingale assumption, we can still obtain the result

that these hedge ratios are the same as the minimum variance hedge ratio. Based on the maximum likelihood estimation of the parameters and the numerical methods, we calculate and compare the different hedge ratios for TAIEX futures and S&P 500 futures (Fig. 57.4). The paper is divided into five sections. Section 57.2 first introduces the definitions and some basic properties for GIG and GH distributions. In Sect. 57.3, we study the optimal hedge ratios under different approaches and estimate these ratios in terms of the parameters for GH distributions. In Sect. 57.4, we discuss the kernel density estimators and MLE method for parameter estimation problem. The last section provides the concluding remarks. Proofs are relegated to the Appendix.

57 A Generalized Model for Optimum Futures Hedge Ratio Fig. 57.3 Estimated symmetric H2 distributions

Fig. 57.4 Estimated hedge ratios

875

876

C.-F. Lee et al.

Z

57.2 GIG and GH Distributions

1

MGIG.;ı; / .u/ D

e ux dGIG.;ı; / .x/dx 0

57.2.1 The Generalized Hyperbolic Distributions

! D

To introduce the generalized hyperbolic distribution, we first recall some basic properties of generalized inverse Gaussian (GIG) distributions. Note that for any ı; > 0 and  2 R, the function dGIG.;ı; / .x/ D

. =ı/ 1  1 .ı2 x 1 C x e 2 2K .ı /

2 x/

with the restriction 2u
0, and then U f .Dt I S0 / C S Ct;T < Ft;T

(59.4)

Given that the futures price cannot completely explain the cost-of-carry model, the spot price will be higher than the futures price when the market faces excess demand, and the convenience yield from holding the commodity over the period from t to T may be expressed as: CY t;T D Pt C S Ct;T  Ft;T :

OCY t;T D Max.Pt  Ft;T ; 0/:

(59.5)

When Equation (59.3) holds, we have C Yt;T D 0. When Equation (59.5) holds, we obtain C Yt;T > 0. Therefore, a temporary shock in demand/supply during the cycle will cause the storage to drop and the spot price to rise, which gives rise to a risk premium from holding the commodity and results in a positive convenience yield. Given that the storage cost in Equation (59.5) is difficult to estimate, observing the convenience yield becomes infeasible. Fama and French (1988) considers the behavior of the convenience yield on an interest-adjusted basis. Such an approach avoids directly estimating the convenience yield but is unable to give the whole picture of the convenience yield from holding the commodity. Milonas and

(59.6)

Instead of using spot price as an underlying variable, as in Equation (59.6), we set the price of a nearby futures contract as the underlying variable, and the price of a distant contract as the exercise price. Under a cost-of-carry framework with zero storage cost, the convenience yield is the difference between the net cost of carrying a nearby and a distant futures contract observed at time 0, OCY 0t;T D Max.F0;t  Ft;T ; 0/:

(59.7)

From the above equation the convenience yield from t to T is observed at date 0 in the beginning month of business cycle. However, it ignores storage cost, which might result in a negative convenience yield. The storage cost (S Ct;T ) is incurred from date t when the nearby contract matures, to date T when the commodity is delivered and the distant contract matures. Equation (59.7) is then revised as:     Ft;T ; 0 ; OCY 0t;T D Max F0;t

(59.3)

U , then When there is excess demand, St D 0; Ft;T D Ft;T U : f .Dt I S0 / C S Ct;T > Ft;T

Thomadakis (1997) treats the convenience yield as a call option. With the assumption of zero storage cost and the spot price as the underlying variable, the payoff function from the convenience yield (OCY) at maturity, or the boundary condition of the option pricing formula, can be written as

(59.8)

where   Ft;T C S Ct;T : F0;t

Both the underlying variable and the exercise price in Equation (59.8) are uncertain variables, which follow standard diffusion processes. We assume that the nearby futures n contract price (F0;t ) with a maturity date at time t and the disd ) with a maturity date at time tant futures contract price (Ft;T T , and they both follow geometric Brownian motion (GBM): n n n D n F0;t dt C n F0;t d zn dF0;t d d d dFt;T D d Ft;T dt C d Ft;T d zd :

(59.9) (59.10)

Based on the deduction of Fisher (1978) that utilizes a nontraded asset as strike price, the expected rate of return on the hypothetical security may be solved by the capital asset pricing model (CAPM). Unlike Fisher where the hedge against the uncertainty of strike price is through a non-traded asset, the strike price in our model is the price of a traded asset such as crude oil futures. The expected rate of return will be n d zero. Simplifying F0;t  as F0;t and Ft;T as Ft;T , referring to Equation (59.8), the boundary condition can be rewritten as: F n Max.XT  1; 0/;

(59.11)

59 Raw Material Convenience Yields and Business Cycle

921

where XT D F n =F d Applying Itô’s Lemma on the transformed variables, we obtain a closed-form solution to the call option (OCY): 0 D Fn N.d1 /  Fd N.d2 /; OC Yt;T

(59.12)

where d1 D

ln.Xt / C .p2 =2/  p ; p 

d2 D

ln.Xt /  .p2 =2/  p ; p 

and p2 D F2 n C 2F n F d F n F d C F2 d :

(59.13)

,  and £ are the volatility, correlation coefficient, and the period between the nearby and distant contracts, respectively. We use the variance of daily logarithmic futures price changes from all trading days during the beginning of the first month of the business cycle as an estimate of volatility of the futures contract in the model. The correlation coefficient is estimated using the nearby and distant daily logarithmic futures price changes from all trading days during the beginning month.

59.4 Data We now turn to the estimation of convenience yield, as well as the design of the business cycle for four different raw materials. The applied dataset consists of daily observation of futures prices on WTI crude oil, Brent crude oil, corn and soybean in the period January 1995 to July 2006. The futures for crude oils have 12 expiration months in 1 year. CBOT corn futures have five expiration months: March, May, July, September, and December. The futures of soybean have 7 expiration months: January, March, May, July, August, September, and November. For each commodity in our sample, we need to estimate the convenience yield and then determine how these values relate to inventory and interest rate by cross-sectional regression. Our analyses test the consistency and validity of the theory of storage. Finally, we also test Samuelson (1965) hypothesis for our estimation of convenience yield with the same approach as that in Fama and French (1988). The issue of whether commodity prices and convenience yields have seasonal cycles is closely related to whether they are subject to supply/demand shocks. We employ a sample of WTI and Brent crude oil prices that are subject to the effect

of supply/demand shocks as well as a sample of corn and soybean prices that are affected by seasonality. The samples contain daily data during the years 1995 and 2006. The commodity price data are from the Commodity Research Bureau database, the Wall Street Journal, the NYMEX, CBOT and ICE. Inventory data are obtained from the U.S. Energy Information Administration (EIA), U.S. Department of Agriculture (USDA), and the Economagic database. Because the study involves many years of data with varying levels of inflation, crude oil prices are translated using the producer price index (PPI) with the base adjusted to January 1996 and agricultural prices are translated using the consumer price index (CPI). We take the 3-month U.S. Treasury bill rate as the risk-free rate, and discount all commodity futures prices at the risk-free rate to the beginning month of the business cycle using the following method: Ft;T D ft;T e rf .T t /=365 ;

(59.14)

where f is the pre-discounted futures price. In this way, all futures prices become free of carrying charges. Brennan (1986, 1991) and Milonas and Thomadakis (1997) also adopted the same discounting procedure. According to the EIA report,4 the storage cost of crude oil is approximately 30 cents per barrel/month during this period, which accounts for about 1% of the crude oil spot price, and is a fixed cost for holders of storable commodities. Moreover, according to the report of the research center of Consumer and Environmental Sciences, College of Agricultural, University of Illinois, the 7-months commercial storage costs5 of corn during the harvest period of May in spring of each year are between US$0.25 and US$0.45 per bushel, accounting for about 15 to 30% of the spot price; those of soybean are close to this figure. Thus, we set the monthly storage cost for crude oil at a fixed percentage of 1% of the spot price and those for corn and soybean at an average percentage of 3.2%6 of spot prices to estimate convenience yields. Finally, to estimate the means and volatilities of these variables, we apply the observed spot and futures prices for all trading days during the beginning month. The convenience yield is calculated on a daily basis throughout the observation month and then averaged over the observation month. In addition to calculating the monthly convenience yields from March to November, we also use September for crude oil, when spot prices peak as the shock month in order to estimate convenience yields. We observe the convenience yields of May to December, and for corn we observe those of May to November for soybean. 4 Energy Information Administration, Petroleum Supply Monthly. DOE/EIA-0109, Washington DC, March 1996. 5 Drying and shrinking costs are included. 6 3.2% D [(15% C 30%)/2] 7 D 3.2% (one-month).

922

Table 59.2 Estimated sample means for crude oil convenience yields

C.-W. Duan and W.T. Lin

WTI crude oil (US$/spot price)

Brent crude oil (US$/spot price)

Period

OCY.1/

CCY.2/

(1)–(2)

OCY.3/

CCY.4/

(3)–(4)

Mar–Apr

0.0707 (12.98) 0.1062 (13.45) 0.1350 (13.47) 0.1619 (13.33) 0.1970 (13.31) 0.2242 (13.26) 0.2529 (12.82) 0.2783 (13.79) 0.3088 (13.73) 0.0677 (12.08) 0.0633 (11.67) 0.0616 (11.31) 0.0637 (11.43) 0.0671 (11.93) 0.0697 (12.66) 0.0699 (12.42) 0.0740 (11.35)

0.0242 (4.42) 0.0476 (4.68) 0.0698 (4.81) 0.0918 0.0700 (5.00) 0.1126 (5.19) 0.1316 (5.37) 0.1502 (5.56) 0.1675 (5.71) 0.1844 (5.86) 0.0232 (4.93) 0.0219 (5.09) 0.0216 (5.64) 0.0202 (6.32) 0.0183 (6.77) 0.0178 (7.58) 0.0165 (7.55) 0.0159 (8.22)

0.0464 (5.98) 0.0586 (5.08) 0.0652 (4.56) 0.1624 (4.27) 0.0844 (4.59) 0.0926 (4.82)  0.1027 (5.23) 0.1108 (4.86) 0.1244 (4.85) 0.0445 (6.08) 0.0414 (6.03) 0.0400 (6.08) 0.0435 (6.72) 0.0487 (7.83) 0.0518 (8.94) 0.0533 (8.79) 0.0581 (8.14)

0.0715 (12.17) 0.1071 (12.67) 0.1355 (12.61) 0.0866 (12.73) 0.1948 (12.21) 0.2226 (11.98) 0.2539 (11.69) 0.2772 (12.91) 0.3007 (11.80) 0.0684 (11.93) 0.0644 (11.26) 0.0630 (11.08) 0.0644 (10.58) 0.0671 (9.98) 0.0720 (10.49) 0.0728 (11.04) 0.0744 (10.86)

0.0236 (4.50) 0.0455 (4.60) 0.0663 (4.69) 0.0757 (4.80) 0.1055 (4.91) 0.1234 (5.00) 0.1408 (5.13) 0.1573 (5.24) 0.1740 (5.35) 0.0217 (4.60) 0.0205 (4.84) 0.0199 (5.16) 0.0184 (5.49) 0.0173 (5.72) 0.0167 (6.36) 0.0157 (6.55) 0.0158 (6.84)

0.0479 (6.38)  0.0615 (4.60)  0.0692 (4.96) 

Mar–May Mar–Jun Mar–Jul Mar–Aug Mar–Sep Mar–Oct Mar–Nov Mar–Dec Apr–May May–Jun Jun–Jul Jul–Aug Aug–Sep Sep–Oct Oct–Nov Nov–Dec

(4.69)  0.0892 (4.88)  0.0991 (4.86)  0.1130 (5.42)  0.1198 (5.26)  0.1267 (5.26)  0.0466 (6.36)  0.0438 (6.34)  0.0430 (6.37)  0.0459 (6.74)  0.0498 (6.98)  0.0552 (7.79)  0.0570 (8.40)  0.0585 (8.35) 

Notes: “OCY” indicates estimation of convenience yield by option model, and “CCY” is by the cost-ofcarry model  Significant at the 0.01 levels. T-statistic is in parentheses

59.5 Empirical Results Table 59.2 presents the convenience yields calculated based on the call option and cost-of-carry models. The results show that the values of the convenience yields estimated from the options model are higher than those from the cost-of-carry model, implying the strategic and management flexibility of the options approach. Moreover, the cost-of-carry model could produce results that contradict the options theory and yield negative estimates. Figures 59.2 and 59.3 draws the term structure of the convenience yield. We find that the behavior of crude oil and agricultural commodity convenience yields exhibits seasonality (Table 59.3). The convenience yields of crude

oils are the highest during the November–December holding period, and then the yields gradually fall to reach the lowest point in June–July. Based on this seasonal behavior, the convenience yields of Brent are higher than those of WTI in the winter, while the same is the case in the summer, suggesting that greater benefits accrue to the holder of Brent crude in the winter when it is subject to supply shocks. Moreover, the convenience yields for Brent crude are more volatile than those for WTI. The convenience yields of corn are at their highest during the September–December holding period, and then they decline gradually to reach their lowest point in March–May. The convenience yields relating to soybean are at their highest during the September–November holding period, and then they decline gradually to reach their lowest

59 Raw Material Convenience Yields and Business Cycle

923

Fig. 59.2 Term structure of convenience yields for crude oil

WTI

Brent

Convenience Yield/Spot price

0.075

0.07

0.065

0.06 Mar-Apr Apr-May May-Jun Jun-Jul Jul-Aug Aug-Sep Sep-Oct Oct-Nov Nov-Dec Holding Periods

Convenience Yield/Spot Price

0.16 Convenience Yield/Spot Price

Fig. 59.3 Term structure of convenience yields for agricultural commodities

0.14

0.12

0.1

0.1

0.08

0.06

0.08 Corn Mar-May May-Jul Jul-Sep Holding Period

point in July–August, which indicates that in the initial stage of crop years, corn and soybean have highest convenience yields showing that convenience yields are affected by seasonality. We find that the inventory of a commodity tends to decrease as the end of the cycle year draws near and the probability of a stock-out increases, which means that the benefit of holding inventory also rises, resulting in higher convenience yields. Thus the term structure of the convenience yields is upward sloping. Figures 59.4 and 59.5 support this conclusion. The empirical results above indicate that WTI possesses a price advantage over Brent in the spot market, but its convenience yields are not necessarily higher than those of Brent. According to our estimates for convenience yields, although the price of soybean is naturally higher than that of corn, the convenience yield of soybean is not necessarily greater than but less than that of corn. The theory of storage also suggests that holding inventory becomes more costly in periods of high interest rates. Furthermore, the convenience yield can be treated as a call

0.12

Sep-Dec

Mar-May May-Jul Jul-Aug Aug-Sep Sep-Nov Soybean Holding Period

option; in theory it is positively correlated with the covariance (p2 ) of two futures contracts and the risk-free rate (rf ). We use a multiple regression to test the relationship between the convenience yield estimated on the basis of the option model (OCY) and the independent variables: 2 C ˇ3 rf t C "t ; OCY t;T D ˇ0 C ˇ1 log.It 1 / C ˇ2 pt (59.15)

where “ and © are the regression coefficient and the error term, respectively. The independent variable log(It 1 ) is log inventories at term t  1. The relationships between convenience yields and inventories are suggested negative by the regression results in Tables 59.4–59.7. In Tables 59.4 and 59.5, we find that the sign of the coefficient is consistent with the theory. “1 is, however, statistically insignificant for crude oils. Tables 59.6 and 59.7 are regression results for corns and soybeans respectively. “1 are mostly negative and statistically significant, strongly suggesting that the negative relationship between estimated convenience yield and inventory.

924

Table 59.3 Estimated means for agricultural commodity convenience yields

C.-W. Duan and W.T. Lin

Corn (US cents/spot price)

Soybean (US cents/spot price)

Perioda

OCY.1/

CCY.2/

(1)–(2)

OCY.3/

CCY.4/

(3)–(4)

Mar–May

0.0909 (22.91) 0.1621 (19.40)

0.0483 (11.63) 0.0999 (11.34)

0.0426 (10.40) 0.0621 (8.56)

0.2409 (17.63)

0.1735 (7.17)

0.0673 (5.95)

0.1047 (27.54) 0.1830 (22.84) 0.2217 (31.47) 0.2727 (26.12) 0.3642 (19.80)

0.0672 (36.36) 0.1343 (33.25) 0.1772 (26.06) 0.2309 (16.84) 0.3136 (14.05)

0.0375 (8.33) 0.0487 (5.57) 0.0445 (5.42) 0.0417 (4.34) 0.0505 (4.73)

0.3483 (15.87) 0.1003 (19.90)

0.2654 (8.08) 0.0527 (11.57)

0.0829 (6.44) 0.0475 (8.75)

0.1075 (18.50) 0.0749 (16.02)

0.0671 (28.64) 0.0427 (11.92)

0.0404 (6.46) 0.0321 (5.99)

0.1218 (18.08)

0.0755 (4.69)

0.0463 (4.30) 0.0778 (19.23) 0.1181 (19.31)

0.0534 (7.84) 0.0806 (10.30)

0.0244 (4.38) 0.0375 (5.51)

Mar–Jul Mar–Aug Mar–Sep Mar–Nov Mar–Dec May–Jul Jul–Aug Jul–Sep Aug–Sep Sep–Nov Sep–Dec

0.1551 (36.39)

0.0938 (14.43)

0.0612 (6.58)

Notes: “OCY” indicates estimation of convenience yield by option model, and “CCY” is by the cost-ofcarry model a The five expiration months for CBOT corn futures contracts are March, May, July, September, and December. CBOT soybean has seven expiration months: January, March, May, July, August, September, and November  Significant at the 0.01 levels. T-statistic is in parentheses

Fig. 59.4 Plot convenience yields by March as shock month for crude oil Convenience Yield/Spot Price

WTI

Brent

0.25

0.15

0.05 Apr

May

Jun

Jul

Aug Event Months

Sep

Oct

Nov

Dec

925

Convenience Yield/Spot Price

Fig. 59.5 Plot convenience yields by March as shock month for agricultural commodities

Convenience Yield/Spot Price

59 Raw Material Convenience Yields and Business Cycle

0.35

0.25

0.15

0.25

0.15

0.05

0.05 May

Jul

Corn

Table 59.4 WTI crude oil convenience yields regressed on inventory, volatility and risk-free rate

0.35

Sep

May

Dec

Jul

Soybean

Event Months

Aug

Sep

Nov

Event Month

Period

ˇ0

ˇ1

ˇ2

ˇ3

R2

F

Mar–Apr

0.6206 (1.04) 1.1620 (1.07) 1.6936 (1.08) 2.2397 (1.10) 2.6897 (1.08) 3.0435 (1.04) 3.1141 (0.92) 3.7660 (1.06) 4.1770 (1.08)

0.0841 (0.96) 0.1589 (1.00) 0.2327 (1.01) 0.3088 (1.03) 0.3714 (1.02) 0.4204 (0.98) 0.4292 (0.87) 0.5196 (1.00) 0.5758 (1.02)

20.7896 (4.09) 27.4722 (2.87) 31.4704 (2.18) 34.6799 (1.79) 40.9165 (1.87) 45.8778 (1.73) 55.3662 (1.95) 48.4808 (1.65) 46.3762 (1.76)

0.1092 (0.51) 0.2476 (0.64) 0.4033 (0.72) 0.5691 (0.78) 0.6518 (0.75) 0.7170 (0.69) 0.6886 (0.61) 0.8587 (0.70) 0.954 (0.72)

0.78

7.43

0.66

4.00

0.56

2.60

0.49

1.96

0.49

1.99

0.47

1.81

0.51

2.15

0.46

1.74

0.51

2.17

0.5072 (1.03) 0.4237 (1.03) 0.3708 (1.16) 0.2323 (1.01) 0.0701 (0.43) 0.1632 (0.79) 0.2759 (0.96) 0.4398 (1.33)

0.0678 (0.94) 0.0561 (0.93) 0.0488 (1.04) 0.0288 (0.86) 0.0052 (0.22) 0.0292 (0.97) 0.0458 (1.10) 0.0696 (1.44)

24.3732 (5.35) 26.2683 (6.40) 29.2983 (8.60) 30.0085 (12.98) 27.9118 (18.80) 26.7628 (15.12) 24.8389 (11.30) 24.9073 (11.32)

0.1449 (0.82) 0.1463 (1.00) 0.1350 (1.20) 0.1186 (1.50) 0.0931 (1.66) 0.0626 (0.92) 0.0638 (0.70) 0.0110 (0.10)

0.86

12.56

0.89

17.51

0.93

30.46

0.96

63.85

0.98

131.24

0.97

84.76

0.95

46.80

0.95

47.82

Mar–May Mar–Jun Mar–Jul Mar–Aug Mar–Sep Mar–Oct Mar–Nov Mar–Dec

Apr–May May–Jun Jun–Jul Jul–Aug Aug–Sep Sep–Oct Oct–Nov Nov–Dec

2 Notes: Model: OC Yt;T D ˇ0 C ˇ1 log.It1 / C ˇ2 pt C ˇ3 rt C "t Standard errors are analyzed with the correction of heteroscedasticity and results differ only marginally ; ;  Significant at the 0.01, 0.05, and 0.1 levels, respectively. T-statistic is in parentheses

926

Table 59.5 Brent crude oil convenience yields regressed on inventory, volatility and risk-free rate

C.-W. Duan and W.T. Lin

Period

ˇ0

ˇ1

ˇ2

ˇ3

R2

F

Mar–Apr

0.9170 (1.45) 1.5182 (1.34) 2.0842 (1.28) 2.6232 (1.25) 3.1276 (1.21) 3.5752 (1.16) 3.6817 (1.08) 4.3115 (1.15) 4.3029 (1.02)

0.1292 (1.39) 0.2137 (1.28) 0.2934 (1.23) 0.3692 (1.20) 0.4406 (1.16) 0.5034 (1.12) 0.5177 (1.04) 0.6056 (1.10) 0.6038 (0.98)

24.6410 (4.37) 32.4569 (3.12) 38.2187 (2.49) 41.8079 (2.06) 48.5656 (2.13) 52.5074 (2.06) 63.0646 (2.40) 57.1504 (1.75) 67.8829 (2.12)

0.0559 (0.25) 0.0306 (0.08) 0.0167 (0.03) 0.0601 (0.08) 0.0853 (0.10) 0.1289 (0.12) 0.1441 (0.13) 0.1910 (0.15) 0.0730 (2.12)

0.79

7.70

0.67

4.21

0.58

2.87

0.51

2.12

0.54

2.35

0.52

2.25

0.57

2.68

0.47

1.79

0.56

2.57

0.7084 (1.23) 0.6273 (1.29) 0.5241 (1.24) 0.3817 (1.08) 0.2134 (0.75) 0.1551 (0.36) 0.1495 (0.28) 0.1926 (0.55)

0.0985 (1.17) 0.0871 (1.22) 0.0723 (1.17) 0.0515 (1.00) 0.0269 (0.64) 0.0177 (0.29) 0.0165 (0.21) 0.0324 (0.55)

26.6232 (4.85) 28.7516 (5.91) 31.0707 (7.00) 31.6910 (9.31) 30.2982 (13.19) 27.4162 (8.92) 23.8895 (6.72) 28.3505 (11.20)

0.0050 (0.03) 28.7516 (0.13) 0.0070 (0.05) 0.0264 (0.22) 0.0416 (0.44) 0.0722 (0.52) 0.0635 (0.36) 0.0519 (0.46)

0.82

9.15

0.86

13.35

0.90

18.14

0.94

31.80

0.96

64.15

0.93

28.74

0.88

16.07

0.95

47.38

Mar–May Mar–Jun Mar-Jul Mar–Aug Mar–Sep Mar–Oct Mar–Nov Mar–Dec

Apr–May May–Jun Jun–Jul Jul–Aug Aug–Sep Sep–Oct Oct–Nov Nov–Dec

2 Notes: Model: OC Yt;T D ˇ0 C ˇ1 log.It1 / C ˇ2 pt C ˇ3 rt C "t Standard errors are analyzed with the correction of heteroscedasticity and results differ only marginally ; ;  Significant at the 0.01, 0.05, and 0.1 levels, respectively. T-statistic is in parentheses

The theory of storage points out that the convenience yield has a negative relation with inventory which could be verified through those products affected by seasonality, but could not be observed by products affected by demand/supply. Furthermore, “2 are consistently positive and mostly statistically significant, suggesting that the more uncertain the prices, the higher the convenience yields, which represents the management flexibility for the value of holding spot. Following the theory of storage, we may expect a significant non-zero coefficient of “3 . In fact, to the extent that holding inventory becomes more costly in period of high interest rates, we may expect negative correlation between interest rates and inventory, and positive correlation between interest rates and convenience yields, and hence a positive “3 . In testing the risk-

free rate coefficient, “3 of crude oil and corn are negative statistically insignificant, while that of soybean is negative statistically significant for some holding periods. This negative value implies that the increase in the carrying cost of commodity – namely the interest rate – would cause the yield of holding spot to decline. To observe the effect of a variable explaining the convenience yield, Lin and Duan (2007) use the spot price spread and futures price spread in the two crude oil markets as explanatory variables and perform regression on the following equation, -Brent C " ; OCY t;T D ˛0 C ˛1 SPFnt;T C ˛2 SPFdt;T C ˛3 SPSWTI t t;T (59.16)

59 Raw Material Convenience Yields and Business Cycle

Table 59.6 Corn convenience yields regressed on inventory, volatility and risk-free rate

927

Perioda

ˇ0

ˇ1

ˇ2

ˇ3

R2

F

Mar–May

0.7675 (3.73) 1.7897 (7.89) 5.1443 (6.09) 9.2369 (5.69) 0.6959 (5.32) 2.2816 (3.94) 0.5170 (1.83)

0.0445 (3.46) 0.1058 (7.45) 0.3073 (5.73) 0.5602 (5.46) 0.0396 (4.84) 0.1364 (3.74) 0.0259 (1.44)

51.5441 (7.31) 69.2155 (13.87) 34.5940 (1.27) 18.8752 (0.35) 45.9741 (15.83) 5.2640 (0.44) 52.9910 (7.17)

0.0443 (0.51) 0.0968 (1.04) 0.0037 (0.01) 0.4290 (0.67) 0.0381 (0.70) 0.2676 (1.09) 0.0451 (0.46)

0.92

23.28

0.97

89.63

0.91

22.04

0.85

11.40

0.98

98.01

0.80

8.41

0.93

27.43

Mar–Jul Mar–Sep Mar–Dec May–Jul Jul–Sep Sep–Nov

Notes: Model:OCY t;T D ˇ0 C ˇ1 log.It1 / C ˇ2 pt2 C ˇ3 rt C "t Standard errors are analyzed with the correction of heteroscedasticity and results differ only marginally a The five expiration months for CBOT corn futures contracts are March, May, July, September, and December  Significant at the 0.01 level. T-statistic is in parentheses

Table 59.7 Soybean convenience yields regressed on inventory, volatility and risk-free rate

Perioda

ˇ0

ˇ1

ˇ2

ˇ3

R2

F

Mar–May

0.2828 (2.48) 0.9490 (3.38) 1.6586 (4.87) 3.6699 (7.82) 6.0820 (8.25) 0.5410 (2.70) 0.5664 (1.83) 1.3306 (9.19) 1.0458 (5.26)

0.0141 (1.80) 0.0556 (2.87) 0.1016 (4.31) 0.2359 (7.28) 0.3961 (7.82) 0.0319 (2.31) 0.0352 (1.65) 0.0873 (8.72) 0.0663 (4.85)

43.6339 (14.06) 58.1059 (11.93) 67.5908 (8.36) 67.5973 (6.09) 76.4794 (4.65) 44.0575 (12.20) 28.0626 (6.07) 29.3109 (9.41) 43.6297 (10.60)

0.0600 (1.34) 0.1203 (1.08) 0.2589 (1.98) 0.5214 (2.80) 0.6861 (2.12) 0.0629 (0.80) 0.0818 (0.68) 0.1890 (3.34) 0.0752 (0.89)

0.97

68.11

0.96

49.21

0.92

26.18

0.93

28.63

0.94

35.50

0.96

50.83

0.86

12.74

0.95

47.89

0.96

55.06

Mar–Jul Mar–Aug Mar–Sep Mar–Nov May–Jul Jul–Aug Aug–Sep Sep–Nov

Notes: Model: OCY t;T D ˇ0 C ˇ1 log.It1 / C ˇ2 pt2 C ˇ3 rt C "t Standard errors are analyzed with the correction of heteroscedasticity and results differ only marginally a The seven expiration months for CBOT soybean futures contracts are January, March, May, July, August, September, and November ; ;  Significant at the 0.01, 0.05, and 0.1 levels, respectively. T-statistic is in parentheses

where ˛˛ and " are the regression coefficient and the error term, respectively, and SPFn , PFd , and SPSWTI -Brent are the price spreads of a nearby contract, the price spreads of

a distant contract, and the spot price spreads, respectively. However, their testing results for crude oil exhibit inconsistent coefficient signs.

928

Table 59.8 Regression of spot price spread on WTI and Brent convenience yields

C.-W. Duan and W.T. Lin

Period

0

1

2

R2

F

Mar–Apr

0.3330 (0.91) 0.0929 (0.24) 0.2381 (0.59) 0.4485 (1.10) 0.6147 (1.55) 0.0876 (0.25) 0.9046 (2.18) 0.2234 (0.62) 0.1707 (0.47)

126.0478 (7.52) 70.5103 (5.95) 43.4686 (4.65) 30.4990 (4.12) 19.7444 (3.31) 32.4723 (8.11) 13.4769 (3.04) 44.1757 (8.75) 27.2599 (7.96)

150.1493 (9.72) 90.9732 (8.23) 60.9317 (0.99) 46.4001 (6.59) 34.1160 (6.15) 42.7813 (11.70) 1.4364 (0.36) 52.9324 (11.11) 34.5456 (11.38)

0.39

64.16

0.34

53.34

0.30

44.46

0.29

42.01

0.32

48.00

0.49

95.71

0.22

28.66

0.48

94.11

0.50

102.52

0.6735 (1.95) 1.3067 (3.80) 1.2211 (3.83) 1.4472 (4.51) 2.5587 (10.07) 1.8793 (4.52) 0.8636 (1.98) 1.1570 (6.21)

187.2975 (9.31) 195.4551 (8.98) 191.1563 (10.39) 178.1770 (9.98) 187.1712 (17.72) 105.3250 (6.34) 20.4706 (1.23) 171.8386 (24.62)

207.2445 (10.56) 205.5767 (9.97) 201.8633 (11.49) 187.0148 (11.47) 180.7370 (20.58) 105.8573 (7.94) 37.3382 (2.62) 184.1415 (27.82)

0.38

62.78

0.35

53.89

0.40

69.12

0.42

73.69

0.69

225.08

0.27

37.78

0.08

9.20

0.79

395.87

Mar–May Mar–Jun Mar–Jul Mar–Aug Mar–Sep Mar–Oct Mar–Nov Mar–Dec

Apr–May May–Jun Jun–Jul Jul–Aug Aug–Sep Sep–Oct Oct–Nov Nov–Dec

-Brent D  C  OCY WTI C  OCY Brent C " Note: Model: SPSWTI 0 1 2 t t;T t;T t;T  and Significant at the 0.01 and 0.05 levels. T-statistic is in parentheses



Table 59.9 Regression of spot price ratio between soybean and corn on corn and soybean convenience yields

Period

0

1

2

R2

F

Mar–May

0.9449 (5.10) 0.7951 (4.70) 0.8665 (7.45) 1.1356 (7.38)

8.1467 (4.97) 3.1352 (4.62) 2.5103 (8.53) 4.8641 (4.13)

6.5683 (3.79) 5.8483 (8.08) 7.7554 (19.50) 6.9854 (6.70)

0.23

31.69

0.30

45.15

0.66

199.35

0.25

34.72

Mar–Jul Mar–Sep May–Jul

Psoybean =Pcorn

Soybean

Corn Note: Model: Ratiot;T D 0 C 1 OCYt;T C 2 OCYt;T C "t The five expiration months for CBOT corn futures contracts are March, May, July, September, and December. CBOT soybean has seven expiration months: January, March, May, July, August, September, and November ; ;  Significant at the 0.01, 0.05, and 0.1 levels, respectively. T-statistic is in parentheses

59 Raw Material Convenience Yields and Business Cycle

929

Since Equation (59.16) could lead to multicollinearity among explanatory variables, which in turn might lead to insignificant coefficient test results or inconsistency in the signs of the coefficients, we then use the convenience yields of WTI, Brent, corn, and soybean as explanatory variables and respectively run the following equations: -Brent D  C  OCY WTI C  OCY Brent C " : SPSWTI 0 1 2 t t;T t;T t;T (59.17) P

Ratiot;Tsoybean

=Pcorn

Soybean

D 0 C 1 OCY Corn t;T C 2 OCY t;T

C "t : (59.18)

where ” and " are the regression coefficient and the error term, respectively, and Ratio is the ratio on spot prices of Table 59.10 Convenience yields regression results: Holding period from the shock month to the final month of business cycle

soybean to corn. The results presented in Table 59.8 show that 1 is negative statistically significant excluding March– October holding period, while 2 is significantly positive, indicating that the lower the convenience yield for WTI, the higher the convenience yield for Brent and the greater the spot price spread between WTI and Brent. The results for agricultural commodities illustrated in Table 59.9 show that 1 and 2 are positive statistically significant excluding March-September holding period, indicating that the higher the convenience yields for corn and soybean and the greater the ratio of spot price between soybean and corn. Table 59.10 illustrates the convenience yields from the shock month till the final month of the cycle, where the shock month is the futures contract month in which the peak of spot prices for underlying occurs during the cycle year, and the fi-

Commodity/Period

ˇ0

ˇ1

ˇ2

ˇ3

R2

F

WTI/ Sep–Dec Brent/ Sep–Dec Corn/ May–Dec Soybean/ May–Nov

0.1965 (0.64) 0.0802 (0.17) 7.6373 (4.42) 5.3587 (8.04)

0.0394 (0.87) 0.0027 (0.04) 0.4638 (4.24) 0.3518 (7.67)

44.09 (19.33) 51.12 (14.82) 29.03 (0.49) 76.64 (4.89)

0.0586 (0.57) 0.0041 (0.03) 0.3410 (0.50) 0.5760 (1.99)

0.99

143.55

0.98

84.30

0.77

6.77

0.95

34.54

Notes: Model: OCY t;T D ˇ0 C ˇ1 log.It1 / C ˇ2 pt2 C ˇ3 rt C "t Standard errors are analyzed with the correction of heteroscedasticity and results differ only marginally ; ;  Significant at the 0.01, 0.05, and 0.1 levels, respectively. T-statistic is in parentheses

Table 59.11 Testing the hypothesis of Samuelson for crude oil WTI

Brent

Sample

OCY level

No

OC Y

1

No

OC Y

1

Full

High

67 103

High

43

Low

47

High

13

0.3043 (10.49) 0.6262 (13.40) 0.3027 (9.22) 0.3619 (8.29) 0.3912

60

Low

0.2559 (30.37) 0.0772 (29.11) 0.2972 (39.89) 0.1479 (24.48) 0.2606

0.2397 (26.38) 0.0758 (32.02) 0.2763 (30.65) 0.1241 (20.96) 0.0830

0.1709 (4.58) 0.2390 (6.03) 0.1526 (3.30) 0.1974 (4.80) 0.1340

Low

77

(8.50) 0.0659 (34.81)

(4.66) 0.6818 (11.83)

(67.92) 0.0508 (24.95)

(2.64) 0.3858 (4.95)

High

130

Low

210

0.2459 (39.60) 0.0755 (44.33)

0.2420 (10.22) 0.4293 (12.77)

March as shock

March to December as shock

Total sample WTI and Brent

Notes: The estimated model is ln.Ft;T =Ft1;T 1 / D 0 C 1 ln.St =St1 / C "t Significant at the 0.01 level. T-statistic is in parentheses



110 40 50 50

40

930

C.-W. Duan and W.T. Lin

Table 59.12 Testing the hypothesis of Samuelson for agricultural commodities Corn Soybean Sample

OCY level

No

OC Y

1

No

OC Y

1

Full

High

24 46

High

18

Low

22

High

17

0.7292 (16.89) 0.7829 (24.37) 0.7150 (14.31) 0.7853 (20.81) 0.6775

36

Low

0.2773 (16.57) 0.1204 (28.45) 0.3058 (17.33) 0.1326 (14.08) 0.1457

0.2717 (22.23) 0.1012 (27.01) 0.3029 (22.72) 0.1556 (16.74) 0.1153

0.8087 (28.32) 0.8855 (42.16) 0.7664 (21.57) 0.9092 (44.64) 0.8680

(31.16) 0.0958 (36.29) 0.2723 (27.71) 0.1094 (37.53)

(10.46) 0.8714 (29.27) 0.7770 (32.08) 0.8388 (43.21)

(36.45) 0.0793 (31.43)

(24.44) 0.8971 (31.55)

March as shock

March to December as shock

Corn and soybean

Low

23

High

61

Low

99

54 25 25 24

26

Notes: The estimated model is ln.Ft;T =Ft1;T 1 / D 0 C 1 ln.St =St1 / C "t Significant at the 0.01 level. T-statistic is in parentheses



nal month is the last futures contract month of business cycle. The negative correlation between these commodities’ convenience yield and the inventory level suggests that it is closely linked to business cycle. In the case of agricultural commodities, we find the convenience yield with the inventory level has strong negative correlation in the regression analysis, which is identical to the previous results of this article. Samuelson (1965) argues that futures prices are less variable than spot prices. The theory of storage also predicts that, at a low inventory level, futures prices vary less than spot prices; at a high inventory level, spot prices and futures prices have similar variability. Fama and French (1988) supported Samuelson’s hypothesis by examining the interest-adjusted basis of industrial metals. Thus it is believed that the convenience yield declines at higher inventory levels, and that spot and futures prices have roughly the same variability. Conversely, the convenience yield rises at a low inventory level, and spot prices are more variable than futures prices. To test the Samuelson (1965) hypothesis, we adopt the same approach as Fama and French (1988) and perform the following regression: ln

Ft;T Ft 1;T 1





St D 0 C 1 ln St 1

C "t :

(59.19)

We apply the regression analysis in the event months, and divide the estimated values of the coefficients 1 into high convenience yields and low convenience yields based on the means of the convenience yields. We find that high conve-

nience yields have smaller average values for coefficients, while low convenience yields have average coefficient values close to one. This implies that at a low inventory level, the spot prices of crude oil vary more than the futures prices and that the convenience yields derived from the options model are higher; at a high inventory level, the spot and futures prices of these commodities have similar variability, while the convenience yields are lower. These results are consistent with the hypothesis of Samuelson (1965) and the findings of Fama and French (1988) (Tables 59.11 and 59.12).

59.6 Conclusion The commodity convenience yield calculated by the methodology of Milonas and Thomadakis (1997) exhibits seasonality in the presence of the business cycle. The paper has provided a framework for estimating convenience yields by option approach. The empirical results show that the values of the convenience yields estimated from the options model are higher than those from the cost-of-carry model, implying the strategic and management flexibility of the options approach. Our results show that the negative correlation between the convenience yields for crude oil and the inventory level is weak, yet those for agricultural commodities are strong. This demonstrates that when using our option model the choice of the timing of the business cycle is critical to the calculation of the commodity convenience yield.

59 Raw Material Convenience Yields and Business Cycle

We find that the convenience yields of crude oil can explain the spot price spreads between WTI and Brent. It is thus implied that the higher the convenience yield in relation to WTI crude oil the lower the convenience yield in terms of Brent crude oil, and the greater/lower the spot price spread between WTI and Brent crude oils. In the evidence for agricultural commodities, convenience yields could explain the ratio between soybean and corn. Samuelson (1965) proposed that spot and futures price variations will be similar when a supply/demand shock occurs during higher inventory levels but that spot prices will be more variable than the futures prices at lower inventory levels. Using a regression analysis method for both futures prices against spot prices and their volatilities as in Fama and French (1988), to verify the Samuelson hypothesis, we find that at a low inventory level spot prices vary more than futures prices. Thus our estimated convenience yields are shown to be higher at a lower inventory level than at a higher inventory level. At a high inventory level, spot and futures prices share roughly the same variability, which leads to a lower convenience yield at a higher inventory level, verifying the Samuelson hypothesis.

References Brennan, M. J. 1958. “The Supply of Storage,” American Economic Review 48, 50–72. Brennan, M. J. 1986 The cost of convenience and the pricing of commodity contingent claims, Working paper, University of British Columbia.

931 Brennan, M. J. 1991 “The price of convenience and the valuation of commodity contingent claims” in Stochastic models and options values, D. Lund and B. Oksendal (Eds.). Elsever North Holland. Casassus, J. and P. Collin-Dufresne, 2005. “Stochastic convenience yield implied from commodity futures and interest rates.” Journal of Finance 60, 2283–2331. Fama, E. F. and K. R. French 1987 “Commodity futures prices: some evidence of forecast power premiums, and the theory of storage” Journal of Business 60, 55–73. Fama, E. F. and K. R. French 1988 “Business cycles and the behavior of metals prices” Journal of Finance 43, 1075–1093. Fisher, S. 1978 “Call option pricing when the exercise price is uncertain and the valuation of index bonds” Journal of Finance 33, 169–176. Gibson, R. and E. S. Schwartz 1990 “Stochastic convenience yield and the pricing of oil contingent claims” Journal of Finance 45, 959– 976. Heinkel, R., M. Howe, and J. S. Hughes 1990 “Commodity convenience yields as an option profit” Journal of Futures Markets 10, 519–533. Kaldor, N. 1939 “Speculation and economic stability” Review of Economic Studies 7, 1–27. Keynes, J. M. 1930. A treatise on money, Harcourt Brace, New York. Lin, T. and C. W. Duan 2007 “Oil convenience yields estimated under demand/supply shock.” Review of Quantitative Finance and Accounting 28, 203–225. Milonas, N. T and S. B. Thomadakis 1997 “Convenience yields as call options: an empirical analysis” Journal of Futures Markets 17, 1–15. Samuelson, P. A. 1965. “Proof that properly anticipated prices fluctuate randomly.” Industrial Management Review 6, 41–49. Schwartz, E. S. 1997. “The stochastic behavior of commodity prices: implication for valuation and hedging.” Journal of Finance 52, 923–973. Telser, L. G. 1958 “Futures treading and the storage of cotton and wheat” Journal of Political Economy 66, 233–255. Working, H. 1948 “Theory of the inverse carrying charge in futures markets” Journal of Farm Economics 30, 1–28. Working, H. 1949 “The theory of the price of storage” American Economic Review 39, 1254–1262.

Chapter 60

Alternative Methods to Determine Optimal Capital Structure: Theory and Application Sheng-Syan Chen, Cheng-Few Lee, and Han-Hsing Lee

Abstract In this paper, we review the most important and representative capital structure models. The capital structure models incorporate contingent claim valuation theory to quantitatively analyze prevailing determinants of capital structure in corporate finance literature. In capital structure models, the valuation of corporate securities and financial decisions are jointly determined. Most of the capital structure models provide closed-form expressions of corporate debt as well as the endogenously determined bankruptcy level, which are explicitly linked to taxes, firm risk, bankruptcy costs, risk-free interest rate, payout rates, and other important variables. The behavior of how debt values (and therefore yield spreads) and optimal leverage ratios change with these variables can thus be investigated in detail. Keywords Optimal capital structure Capital structure model r Credit risk

r

Structural model

r

60.1 Introduction The traditional capital structure theories investigate how the capital structure of a firm can be affected by some crucial determinants, such as the financial distress costs and agency costs. However, it is a difficult task to measure their effects on the capital structure decisions and security pricing. It is of great interest for corporate managers and researchers to quantitatively examine how these effects influence a company and its financial decisions under various conditions. The capital structure models employ contingent claim valuaS.-S. Chen National Taiwan University, Taiwan e-mail: [email protected] C.-F. Lee Rutgers University, USA e-mail: [email protected] H.-H. Lee () National Chiao Tung University, Taiwan e-mail: [email protected]

tion (option pricing) theory for analyzing classical corporate finance issues and can deepen the understanding of the firm’s financial decisions. Furthermore, the contingent claim valuation enables risky debt pricing using marketable securities of the firm, such as common stock, bond or credit default swaps (CDS) prices, to estimate the firm value parameters and in turn the default risk. This approach avoids the use of bond rating and risk premium (markup), which are difficult to pin down and often are stale in contrast to the trading data of marketable securities.1 The structural credit risk modeling or defaultable claim modeling is pioneered by the seminal paper by Black and Scholes (1973) in which corporate liabilities can be viewed as a covered call – own the asset but short a call option. Black and Scholes (1973) assume that a company issues only one zero-coupon bond and equity. At the maturity, debt holders either receive the face value of debt if the company is solvent, or take control of the company otherwise. This option analogy allows the closed-form expression for the corporate securities. Later on, Merton (1974) rigorously elaborates the pricing of corporate debt. In addition, the valuation of any corporate security can be modeled as a contingent claim on the underlying value of firm. This Black-Scholes-Merton approach of modeling default claims is named structural approach since the model explicitly ties default risk to the firm value process and its capital structure. In modeling the cause of default-trigger mechanism, the Black-Scholes-Merton approach implicitly assumes that a company can only default at a certain time point when its bond principal is due. While this default trigger mechanism is apparently unrealistic, Black and Cox (1976) first propose a model in that default can occur during the life of the bond – a company defaults on its debt if the asset value of the company falls below a certain exogenous default boundary due to bond covenant. In the same paper, Black and Cox (1976) also point out that default decision can be endogenized by 1

Another advantage of the structural approach is that macro- and microconditions have been implicitly incorporated into the evolution (change) of firm values and, in turn, security prices, and need not be modeled separately.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_60, 

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shareholders’ discretion to maximize equity value. These two types of default mechanisms, exogenously specified by bond covenant and endogenously determined by management, have been widely adopted in most of the succeeding structural models.2 Subsequent researchers extend the structural approach further and incorporate more realistic assumptions. Based on the key objectives of these models, one can classify them as corporate bond pricing model and capital structure model. Corporate bond pricing models, typically assuming stochastic interest rate, are primarily developed to price defaultable corporate debt. In order to accommodate a more realistic and complex debt structure, these models typically assume an exogenous default barrier and bondholders receive a recovery rate times the face value of debt at maturity if a firm defaults.3 For example, the Longstaff and Schwartz model (1995) is the most well-known corporate bond pricing model. By contrast, the capital structure models incorporate corporate finance theory and put the emphasis on the default event itself. The valuation of corporate securities and financial decisions are jointly determined, and the analyses focus on the complex relations between credit risk, risk premiums, and the firm’s financing decisions rather than on the valuation of complex credit securities. In contrast to corporate bond pricing models, the simple stationary debt structure assumption, debt structure with time-independent payouts, is widely adopted in order to obtain closed-form expression of debt, equity, and firm value. In this paper, we restrict ourselves to capital structure models and exclude those models focusing on risky debt pricing.4 In the capital structure models, the most notable developments are the works by Leland (1994) and Leland and Toft (1996). They extend the endogenous default approach by Black and Cox (1976) to include the tax shield of debt and bankruptcy costs, and explicitly analyze the static tradeoff theory of capital structure. Various important issues including the endogenous default boundary, leverage ratio, debt value, debt capacity, yield spread, and even potential agency costs are discussed in detail. Note that apart from the early structural models, Leland and Toft (1996) point out that equity in a capital structure model is not precisely analogous to an ordinary call option or vanilla barrier option. The bankruptcy boundary varies with the risk of the firm’s activities. Furthermore, the tax benefits and the potential loss in

2 Geske (1977) proposes another way of modeling corporate debts as compound options. 3 There are some exceptions. For example, Briys and de Varenne (1997) assume recovery of firm value when a firm defaults. However, the tradeoff is its simpler debt structure, a zero-coupon bond. 4 Readers who are interested in other structural models can refer to the survey paper by Elizalde (2005).

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bankruptcy imply that debt and equity holders do not split a claim depending only on the underlying asset value. While the Leland (1994) and the Leland and Toft (1996) models provide some insights of the capital structure issues, their predicted optimal leverage and yield spreads seem not to be consistent with historical average. Consequently, the more recent studies introduce additional realistic features in order to meliorate the model-predicted leverage and yield spreads. We will also review the some of these extensions. This paper is organized as follows: In Sect. 60.2, we will briefly review traditional capital structure theory. In Sect. 60.3, we will present in detail two most representative capital structure models – Leland (1994), and Leland and Toft (1996) models – and their implications. In Sect. 60.4, more recent studies including debt renegotiation, dynamic capital structure, as well as other developments are presented. In Sect. 60.5, the applications and empirical performance of capital structure models are discussed. In Sect. 60.6, conclusion is given.

60.2 The Traditional Theory of Optimal Capital Structure5 Durand (1952) proposes an approach to explain why there is an optimal capital structure for a firm. Called the traditional approach, it suggests that “moderate” amounts of debt do not noticeably increase the risks to either the debtholders or the equityholders. Hence, both cost of debt (i ) and cost of equity (rE ) are relatively constant up to some point. However, beyond some critical amount of debt used, both i and rE increase. These increases in both the cost of debt and equity begin to offset the decrease in the average cost of capital because of cheaper debt. Therefore, the average cost of capital (rA ), initially falls and then increases. This allows for a minimum value for the average cost of capital function. If the average cost of capital is at the minimum, then the firm’s value (V ) is maximized V D

.1  TC /EBIT rA

(60.1)

where TC D corporate tax rate and EBIT D earnings before interest and tax. This implies that there is an optimum capital structure for the firm.

60.2.1 Bankruptcy Costs The theoretical analysis of Modigliani and Miller (1958) indicates that in the presence of corporate taxes (TC ), the 5

This chapter is an excerpt from the book by Lee et al. (2009).

60 Alternative Methods to Determine Optimal Capital Structure: Theory and Application

value of the levered firm (VL ) exceeds the value of the otherwise comparable unlevered firm (VU ). The difference can be viewed as an increment to present value derived from the tax shelter (TC B). Therefore, we can write VL D VU C T C B

(60.2)

where B D total amount of debt. One issue not considered in deriving this result is the possibility of corporate bankruptcy. Obligations to bondholders constitute a first claim on a company’s assets. However, under US law, stockholders, who are the owners of public companies, have only limited liability. Their personal assets, other than holdings in the firm, are protected from the claims of bondholders. If operating performance is sufficiently poor, stockholders can avoid obligations to bondholders through the firm’s declaring bankruptcy. Essentially, bankruptcy turns over the firm to its debtholders. Bankruptcy proceedings can be very costly. Heavy legal fees must be paid, and assets must often be sold at “bargain basement” prices, frequently following lengthy and expensive disputes among competing claimants. Taken together, these bankruptcy costs may be far from negligible, and the possibility of their being incurred must be considered in assessing a firm’s market value. After bankruptcy, the firm belongs to its creditors, who must therefore bear these costs. However, purchasers of corporate bonds will be aware of this and will charge a higher price – that is, demand a higher interest rate – for debt. This, in turn, reduces the amount of earnings available for distribution to stockholders, reducing the value of shares. Hence, before bankruptcy, costs are carried by stockholders as reductions in the value of their equity holdings. How much does the possibility of bankruptcy lower the firm’s value? Two factors must be considered: (1) the costs

Fig. 60.1 Possibility of optimal debt ratio when allowing for present value of bankruptcy costs

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incurred in the event of bankruptcy and (2) the likelihood, or probability, of bankruptcy occurring. The greater the amount of debt a firm has, the more likely it will fail to meet its obligations and be driven to bankruptcy. Therefore, seeing this risk, bondholders will demand higher rates of interest the larger the debt. Thus, expected bankruptcy cost is an increasing function of the proportion of debt in a firm’s capital structure. Taking into account the possibility of bankruptcy and its effect on firm value, we can augment Equation (60.2) as follows: VL D VU C TC B  present value of bankruptcy costs (60.3) The higher the level of debt, the greater the present value of the tax shield. As we saw in Equation (60.2), the effect is linear, in the sense that the value of the tax shield is a constant multiple TC (the corporate tax rate) of the debt value. However, as Equation (60.3) indicates, one must now subtract the present value of bankruptcy cost. This, too, is an increasing function of the debt level, reflecting the increase in the probability of bankruptcy that attaches to increasing debt levels. If these factors operate in conjunction, as illustrated in Fig. 60.1, then an optimal capital structure results. One sees for the unlevered firm, zero values for the tax shield and bankruptcy costs. Each increases with the level of debt issued, the net effect being a relationship between firm value and debt level that shows a maximum value for a debt level well below 100% of total value. Myers (1984) called the optimal capital structure discussed in this section the static tradeoff theory.6 6

Myers (1984) also proposes a pecking-order theory to contrast to the static trade-off theory.

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Fig. 60.2 Possibility of optimal debt ratio when allowing for present value of bankruptcy and agency costs

60.2.2 Agency Costs Agency costs are legal and contract costs that protect the rights of stockholders, bondholders, and management, and arise when the interests of these parties come into conflict. Such organizational conflict may lead to capital structure decisions that affect the firm’s value. Asset substitution problem arises when a firm invests in assets that are riskier than those that the debtholders expect. For example, having sold debt, stockholders may gain if the corporation abandons its low-risk projects in favor of more risky projects. If the higher-risk projects are successful, the stockholders will benefit from the high returns. On the other hand, this exposes bondholders to a higher degree of risk than they anticipate when the bonds were purchased, and thus they face a higherthan-expected chance of corporate bankruptcy. Bondholders anticipate such possibilities and require that bonds be protected by restrictive covenants. Such covenants involve two types of costs to the firm. First, there are costs in monitoring the covenants, and second, these covenants restrict management flexibility, possibly precluding courses of action that would raise the firm’s market value. These costs are borne by stockholders, resulting in a lower market value for equity. Jensen and Meckling (1976) use agency costs of debt and equity to suggest that there is an optimum combination of outside debt and equity that will be chosen to minimize total agency costs. Therefore, it is possible for the existence of an optimal capital structure even in the absence of taxes and bankruptcy costs. In general, one might expect that agency costs of this sort will be greater the higher the debt. If there is little debt, bondholders will consider their investments relatively secure and demand only modest, fairly inexpensive protection of their interest. At higher debt levels, more stringent assurances will

be required. Thus, agency costs affect firm value in much the same way as bankruptcy costs. The greater the debt, the greater the agency cost against firm value. This is illustrated in Fig. 60.2, which extends Fig. 60.1 by allowing for agency cost. At any level of debt above zero, agency cost causes a decrease in market value; the greater the debt level, the larger the agency cost factor. Hence, agency cost yields a lower optimal amount of debt in the capital structure than would otherwise prevail. If there is agency cost, then Equation (60.3) can be rewritten as VL D VU C TC B  present value of bankruptcy costs Agency cost

(60.3a)

In a review paper, Harris and Raviv (1991) survey capital structure theories based on agency costs, asymmetric information, product/input market interactions, and corporate control considerations (but excluding tax-based theories). Therefore, this paper gives us further detailed information about the result discussed in this section.

60.3 Optimal Capital Structure in the Contingent Claims Framework The traditional capital structure theory identifies some prime determinants of optimal capital structure. Nevertheneses, the theory is less useful in practice since it provides only qualitative guidelines. The structural approach enables quantitative analysis and thus has become very popular in the corporate finance literature. We will review in detail how the determinants mentioned in Sect. 60.2 affect capital structure decisions under the contingent claim framework in this section.

60 Alternative Methods to Determine Optimal Capital Structure: Theory and Application

In the analysis of optimal capital structure, Brennan and Schwartz (1978) are the first to provide the quantitative examination of optimal leverage using contingent claims approach. However, their analysis employs numerical techniques and lacks a close-form solution. As a result, their numerical-based study cannot provide general comparative statics of financial variables and further discussion. In this section, we review in detail two most representative capital structure models – Leland (1994) and Leland and Toft (1996) models – and the implication of these models. These two studies provide closed-form expressions of corporate securities as well as comparative statics. Their methodology and stationary debt structures are widely adopted in the literature. In addition, these two models have served as benchmark models in the optimal capital structure literature.

60.3.1 The Leland (1994) Model Leland (1994) extends the endogenous default approach by Black and Cox (1976) to include the tax shield of debt and bankruptcy costs, which is also termed the Static tradeoff model in the corporate theory. In the Leland (1994) model, financial distress is triggered when shareholders no longer find that running a company is profitable, given the revenue produced by the assets to continue servicing debt. Bankruptcy is determined endogenously rather than by the imposition of a positive net worth condition or by a cash flow constraint. In this model, two major assumptions are as follows: (1) The activities of the firm are unchanged by financial structure, and the capital structure decisions, once made, are not subsequently changed; (2) The face value of debt, once issued, remains static through time. Consider a firm with asset value V , which follows a diffusion process with constant volatility of rate of return: dV D .V; t/dt C dW V

(60.4)

where W is a standard Brownian motion. The stochastic process of V is assumed to be unaffected by the financial structure of the firm. Therefore, any net cash outflows associated with the choice of leverage such as coupons after tax benefits must be financed by selling additional equity. In addition, there exists a riskless asset that pays a constant rate of interest r. Denote any claim F .V; t/ on the firm that continuously pays a nonnegative coupon, C , per instant of time when the firm is solvent. When the firm finances the net cost of the coupon by issuing additional equity, any such asset’s value must satisfy the PDE: 1 2 2  V FVV .V; t/CrVFV .V; t/rF.V; t/CFt .V; t/CC D 0 2

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with boundary conditions determined by payments at maturity, and by payments in bankruptcy happens prior to maturity. When securities have no explicit time dependence as this case, the term Ft .V; t/ D 0, and the above equation becomes an ordinary differential equation with F .V / satisfying 1 2 2  V FVV .V; t/ C rVFV .V; t/  rF.V; t/ C C D 0: 2 (60.5) The equation has the general solution F .V / D A0 C A1 V C A2 V X ;

(60.6)

where X D 2r= 2 and the constants A0 , A1 , and A2 are determined by boundary conditions. Any time-independent claim with an equity-financed constant payout C 0 must have this functional form. First, Leland (1994) provides the solution of the perpetual debt. Debt promises a perpetual coupon payment, C , whose level remains constant unless the firm declares bankruptcy. Let VB denote the constant level of asset value at which bankruptcy is declared. If bankruptcy occurs, a fraction 0  ˛  1 of asset value will be lost to bankruptcy cost. Accordingly, debtholders receive .1˛/VB and shareholders receive nothing.7 Boundary conditions are: At V D VB ;

D.V / D .1  ˛/VB

As V ! 1, D.V / D C =r D.V / D

  X C C V C .1  ˛/VB  r r VB

(60.7)

The above equation can be rewritten as D.V / D Œ1  pB .C =r/ C pB Œ.1  ˛/VB where pB  .V =VB /X has the interpretation of the present value of $1 contingent on future bankruptcy; that is, V falls to VB . Next, Leland (1994) derives the total value of the firm, v.V /, which reflects three terms: the firm’s asset value, plus the value of the tax deduction of coupon payments, less the value of bankruptcy costs. Consider a security pays no coupon, but has a value equal to the bankruptcy costs. Because its returns are time independent, it must satisfy Equation (60.6) with boundary conditions At V D VB ; BC.V / D ˛VB As V ! 1; BC.V / D 0:

7

Leland (1994) also has a discussion (in Sect. VI.C) under the condition of deviation of absolute priority, in which bondholders do not receive all of the remaining asset value and stockholders also receive a fraction of the remaining asset value.

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The solution in this case is ˛VB .V =VB /X ;

(60.8)

and BC.V / is a decreasing convex function of V . Next consider the value of tax benefits associated with debt financing. Denote the corporate tax rate as TC . These benefits resemble a security that pays a constant coupon equal to the tax-sheltering value of interest payments, TC C , as long as the firm is solvent. In other words, TB.V / equals the value of the tax benefit of debt. Since it is also time independent and must satisfy Equation (60.6) with boundary conditions At V D VB ; TB.V / D 0 As V ! 1; TB.V / D TC C =r: The solution with these boundary conditions is TC C =r  .TC C =r/.V =VB /X

Note that Equation (60.10) is similar to Equation (60.3). If .V =VB /X approaches to zero, then Equation (60.10) reduces to original M&M proposition I as defined in Equation (60.2). For this condition to hold, either (i) .V =VB / ! 1 or (ii)  2 ! 0 and V > VB . In both cases, the firm is unlikely to go bankrupt, and therefore tax saving per period on the interest payments becomes a sure stream. As a result, the tax benefit claim in Equation (60.9) will be TC C =r, and Equation (60.10) will coincide with the original M&M proposition I in Equation (60.2), even with the existence of bankruptcy cost. On the other hand, without the bankruptcy cost (˛ D 0), the tax benefit claim under Leland’s specification is still different from that by M&M. The difference, a minus term (.TC C =r/.V =VB /X ), is coming from the possibility of future bankruptcy, and thus lowers the tax shield of debt. The value of equity is the total value of the firm less the value of debt as follows: E.V / D v.V /  D.V / D V  .1  TC /.C =r/

(60.9)

CŒ.1  TC /.C =r/  VB .V =VB /X: 8

The TB.V / is an increasing, strictly concave function of V . Note that the value of tax benefits above presumes that the firm always benefits fully in amount TC C from the tax deductibility of coupon payments when it is solvent. However, under U.S. tax codes, to benefit fully, the firm must have earnings before interest and taxes (EBIT) that is at least large as the coupon payments C . Leland (1994) also provides the solution, in which EBIT is related to asset value,V , and tax benefits may be partially lost when the firm is still solvent. In sum, the total value of the firm, v.V /, consisting of the above three terms, the firm’s asset value, the value of the tax deduction of coupon payments, and the value of bankruptcy costs, can be expressed as: v.V / D V C TB.V /  BC.V / D V C .TC C =r/Œ1  .V =VB /X  ˛VB .V =VB /X (60.10) v.V / is strictly concave in asset value, V , when C > 0 and either ˛ > 0 or TC > 0. Note that if ˛ > 0 or TC > 0, then v.V / < V as V ! VB , and v.V / > V as V ! 1. This coupled with concavity implies that v.V / is proportionally more volatile than V at low values of V , and is less volatile at high value of V .

8 Note that this equation represents that the firm always benefitfully from the tax deductibility of coupon payment when it is solvent. However, to benefit fully the firm must have EBIT over the coupon payment, C . Leland (1994) provides an alternative approach in Appendix A in which tax benefits may lost when the firm is still solvent.

(60.11) Leland next provides the bankruptcy-triggering asset level VB chosen by the firm. To maximize the value of equity at any level of V , the equilibrium bankruptcy-triggering asset value VB is determined ˇ endogenously by the smooth-pasting @E.V IVB ;T / ˇ D 0. condition ˇ @V V DVB

VB D Œ.1  TC /C =r ŒX=.1 C X / D .1  TC /C =.r C 0:5 2 / (60.12) Since VB < .1 TC /C =r, equity is indeed convex in V . Several observations of the asset value, VB , at which bankruptcy occurs are: (1) (2) (3) (4) (5)

It is proportional to the coupon, C . It is independent of the current asset value, V . It decreases as the corporate tax rate, TC , increases. It decreases as the risk-free interest rate, r, rises. It decreases with increases in the riskiness of the firm,  2 .

Finally, VB in Equation (60.12) is independent of time and it confirms the assumption of the constant bankruptcytriggering asset level VB . In addition to the determinants of bankruptcy level, with the closed-form expressions, corporate debt values and optimal leverage are explicitly linked to taxes, firm risk, bankruptcy costs, risk-free interest rate, and bond covenant. The behavior of how debt values (and therefore yield spreads) and optimal leverage ratios changes with these variables is discussed in detail. In addition, Leland shows that a

60 Alternative Methods to Determine Optimal Capital Structure: Theory and Application

positive net worth covenant9 can eliminate the firm’s incentive to increase risk and lower the agency problem. However, optimal capital structure refers not only to leverage ratio, but also to the maturity of debt. The consol bond assumption of the Leland model confines the analysis in this dimension. As a result, Leland and Toft (1996)10 propose a finite maturity debt structure for further discussion of this issue.

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of bankruptcy. Let f .sI V; VB / denote the density of the first passage time s from V to VB , and F .sI V; VB / is the cumulative distribution function of the first passage time to bankruptcy; that is, the default probability at time s. Under risk-neutral valuation with the drift rate .r  ı/, the bond with maturity t has the value Z

t

d.V I VB ; t/ D

e rs c.t/Œ1  F .sI V; VB / ds

0

Ce rt p.t/Œ1  F .sI V; VB / Z t C e rs .t/VB f .sI V; VB /ds

60.3.2 The Leland and Toft (1996) Model

0

In the Leland and Toft (1996) model, financial distress is triggered, as assumed in Leland (1994), when shareholders no longer find that running a company is profitable. In this model, they further examine the optimal capital structure of a firm that can choose both the amount and maturity of its debt. They again setup a stationary debt structure by issuing finite maturity debt with continuous coupon and retiring the same constant amount (principal) of debt issued in the past simultaneously. As a result, at any given time, the firm has many overlapping debt outstanding, while the total cash flow to debt holders is constant. The results extend Leland’s (1994) closed-form results to a much richer class of possible debt structures and dividends payment and permit study of the optimal maturity of debt as well as the optimal amount of debt. The firm has productive assets whose unleveraged value V follows a continuous diffusion with constant proportional volatility  dV D Œ.V; t/  ı dt C dZ V

The first term represents the discounted expected value of the coupon flow; the second term represents the expected discounted value of repayment of principal; and the third term represents the expected discounted value of the fraction of the assets, which will go to debt with maturity t, if bankruptcy occurs. Integrating the first term by parts yields d.V I VB ; t/ D

  c.t/ c.t/ Œ1  F .t/ C e rt p.t/  r r   c.t/ C .t/VB  G.t/ (60.14) r Z

where

t

G.t/ D 0

Using the result of F .t/ from Harrison (1985) and G.t/ from Rubinstein and Reiner (1991):

(60.13)

where .V; t/ is the total expected rate of return on asset value V ; ı is the constant fraction of value paid out to security holders; and dZ is the increment of a standard Brownian motion. The process continues without time limit unless V falls to a default-triggering valueVB . VB will be determined endogenously and shown to be constant in a rational expectations equilibrium. Finally, a default-free asset exists and pays a continuous constant interest rate r. Consider a bond issue with maturity t periods from the present, which continuously pays a constant coupon flow c.t/ and has principal p.t/. Let .t/ be the fraction of asset value VB , which debt of maturity t receives in the event 9 The positive net-worth covenant is modeled by setting default boundary equal to the principal value of debt. 10 Leland (1994b) proposes an exponential model similar to sinking fund that firms retire debt at a proportional rate m through time. In the absence of default, the average maturity of debt is finite as the timeR1 weighted average: T D 0 t .memt /dt D 1=m.

e rs f .sI V; VB /ds

F .t/ D N Œh1 .t/ C G.t/ D

V VB

V VB

2a N Œh2 .t/

aCz N Œq1 .t/ C

V VB

(60.15)

az N Œq2 .t/ (60.16)

where q1 .t/ D

.b  z 2 t/ .b C z 2 t/ I q2 .t/ D p p  t  t

.b  a 2 t/ .b C a 2 t/ p I h2 .t/ D p  t  t

.r  ı  . 2 =2// V I aD I b D ln 2 VB

h1 .t/ D

zD

Œ.a 2 /2 C 2r 2 1=2 2

and N./ is the cumulative standard normal distribution.

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A stationary debt structure with time-independent debt service payments is assumed by Leland and Toft (1996) as the following. Consider an environment where the firm continuously sells a constant (principal) amount of new debt with maturity of T years from issuance, which (if solvent) it will redeem at par upon maturity. New bond principal is issued at a rate p D .P =T / per year, where P is the total principal value of all outstanding bonds. The same amount of principal will be retired when the previously issued bonds mature. Therefore, as long as the firm remains solvent, at any time s the total outstanding debt principal will be P , and have a uniform distribution of principal over maturities in the interval .s; s C t/. Bonds with principal p pay a constant coupon rate c D .C =T / per year, implying the total coupon paid by all outstanding bonds is C per year. Total debt service payments are therefore time-independent and equal to .C C P =T / per year. Accordingly, this environment assumed by Leland and Toft (1996) avoids the time-dependent debt service requirements such as the single issue of debt outstanding assumed by Merton (1974). Assuming the fraction of firm asset value lost in bankruptcy is ˛ and .t/ D =T per year for all t, this implies  D .1  ˛/. Therefore, writedown for debt is the fixed fraction of the principal value of debt lost in bankruptcy, which is .1  .1  ˛/VB =P /. Leland and Toft derive the closed-form solution of the value of all outstanding bonds, with debt of maturity T years, by integrating all the outstanding debt during this period: Z T d.V I VB ; t/dt D.V I VB ; T / D 0





C C 1  e rT D C P  I.T / r r rT

C J.T /; (60.17) C .1  ˛/VB  r where 1 T

Z

T

1 .G.T /  e rT F .T //; rT 0 aCz Z 1 T rt 1 V J.T / D e G.t/dt D p  T 0 V z T B ! az V

N Œq1 .T / q1 .T / C N Œq2 .T / q2 .T / : VB I.T / D

e rt F .t/dt D

Following Equation (60.10) by Leland (1994), total firm value is  az  az V V TC C 1  ˛VB v.V I VB / D V C r VB VB (60.18) where a and z have been defined in Equation (60.16).

The equilibrium bankruptcy-triggering asset value VB is determinedˇ endogenously by the smooth-pasting condition @E.V IVB ;T / ˇ D 0. ˇ @V V DVB

VB D

.C =r/ .A=.rT/  B/  AP=.rT/  TC Cx=r 1 C ˛x  .1  ˛/B (60.19)

p p where A D 2aerT N.a T /  2zN.z T /  p p rT p n.z T / C .z  a/ .z T / C 2e  T B D  2z C

2 z 2 T

C.z  a/ C





2 p T

n

p p 2 N.a T /  p n.z T /  T

1 z 2 T

x D a C z11 and n./ denotes the standard normal density function. The solution is independent of time, indicating that the earlier assumption of a constant VB is warranted: it represents a rational expectations equilibrium. As T ! 1, it can be shown that VB ! .1  TC /.Cx=r/=.1 C x/ as in Equation (60.12). Note that VB will depend on the maturity T of debt chosen, for any given values of total bond principal P and coupon rate C . This is an important difference between Leland and Toft model and the flow-based bankruptcy model (e.g., Kim et al. (1993) or with a positive net worth covenant model (e.g., Longstaff and Schwartz (1995), in which they imply that VB is independent of debt maturity. And finally, the value of equity is given by E.V I VB ; T / D v.V I VB /  D.V I VB ; T /

(60.20)

In contrast to the Leland (1994) model, the Leland and Toft (1996) model has some different implications. First, by permitting a firm to choose its debt maturity, optimal leverage is lower when the firm is financed by shorter-term debt. They also provide explanation of why a firm issues shortterm debt given that the longer-term debt generates higher firm value. They find that short-term debt reduces the asset substitution agency problem. While short-term debt does not exploit tax benefit of debt, it can balance against bankruptcy and agency costs in determining the optimal maturity of the capital structure. This is a consequence that short-term debt 11

If the payout rate (ı) is equal to 0, then it can be shown that 2 =2// 2 2 2 1=2 a C z D .rı. C Œ.a / C2r D 2r2 . This is exactly the 2 2 x defined in Equation (60.6) by Leland (1994). However, we should note that the introduction of the payout rate relaxes the unreasonable assumption assumed by Leland (1994) that net cash outflows associated with the choice of leverage such as coupons after-tax benefits must be financed by selling additional equity.

60 Alternative Methods to Determine Optimal Capital Structure: Theory and Application

holders need not protect themselves by demanding higher coupon rates and thus equity holders will also benefit. Second, for bond portfolio management, they find that Macaulay duration overstates true duration of risky debt, especially for junk bonds that may have negative duration. Furthermore, convexity can become concavity.

60.4 Recent Development of Capital Structure Models In Sect. 60.3, we have reviewed in detail two of the most well-known capital structure models – Leland (1994), and Leland and Toft (1996) models. These two models have been served as benchmark models in the optimal capital structure literature. Their stationary debt structures and time-independent setting have been widely adopted in the literature. However, compared with historical average, the optimal leverage implied in the Leland (1994) model is unreasonably high, while the model’s predicted yield spread is too low given the suggested high leverage, especially for the unprotected debt. Furthermore, unreasonably large bankruptcy costs as high as 50% is required to lower the optimal leverage to this level. In addition to the issues above, several implications by the Leland (1994) model are also not in accordance with the empirical observations. In the Leland (1994) model, the firm will issue equity to finance contractual debt obligation until equity value is driven to zero in the absence of reorganization option. Hence, bankruptcy boundary coincides with the liquidation barrier. However, in practice, it is normal for management to file for bankruptcy when equity value is still positive. Some researchers argue that a more realistic bankruptcy procedure may affect the valuation of corporate securities and capital structure decisions. Therefore, subsequent studies attempt to extend Leland’s (1994) model to a more realistic setting and hope to make the model-implied yield spread and leverage ratio to be more consistent with the observed average in practice. The first branch of these extensions is to permit deviations of absolute priority rule (APR) in Chap. 14 bankruptcy proceedings. The importance of violations from APR has been addressed in the literature. Eberhart and Weiss (1998) clearly point out that APR is explicitly or implicitly assumed in many seminal finance models, such as Merton (1974) and Myers (1977). The APR is adhered since shareholders receive nothing until debt holders have been fully paid. The APR is also assumed in most of the early structural models. However, empirical studies indicate that the APR violation is a common phenomenon (see Franks and Torous1989; Eberhart et al. 1990; Weiss 1990). Therefore, it has of great interest to researchers to analyze the cause of APR deviation

941

and its impact on capital structural decision under rational expectation framework. Structural models can demonstrate the violation of APR by incorporating different default boundaries. Leland (1994) has a discussion (in Sect. VI.C) regarding deviation of absolute priority, in which bondholders do not receive all of the remaining asset value and stockholders also receive a fraction of the remaining asset value. Nevertheless, it cannot explain the behavior of debt holders’ concession by simply changing the boundary condition since their claims will be reduced. Therefore, several recent models permit the debt renegotiation by private workouts or by courtsupervised debt renegotiation under Chap. 14 bankruptcy proceedings. For example, Fan and Sundaresan (2000) propose a game-theoretic setting that incorporates equity’s ability to force concessions and varying bargaining powers to the debt holders and equity holders. Later on, Francois and Morellec (2004) extend this bargaining game to explicitly model Chap. 14 protection and court-supervised renegotiation. By introducing the possibility of renegotiating the debt contract, the default can occur at positive equity value. This is in contrast to the Leland’s (1994) model that the default occurs as the equity value reaches zero. As a consequence of that, issuing new equity is costless, and the APR is respected. Another branch of the extension is the dynamic capital structure. Early capital structure models assume that the debt issuance is a static decision. However, in practice, firms adjust outstanding debt levels in response to changes in firm value. Early studies of dynamic capital structure include works by Kane et al. (1984, 1985) and Fischer et al. (1989). Recent works consist of the Goldstein et al. (2001) and Ju and Ou-Yang models (2006), and many others. In general, the dynamic feature of these models predicts a lower leverage ratio than those static capital structure models and is more in line with the observed leverage in practice. There are still other researchers who seek to explore the capital structure problem of investment risks and agency costs, the timing of default under more realistic bankruptcy proceedings, alternative process, and other issues. We briefly categorize these studies as follows (the list below is by no means exclusive): Dynamic Capital Structure:12 Fischer et al. (1989); Goldstein et al. (2001); Ju et al. (2004); Ju and Ou-Yang (2006). Strategic Default and Renegotiation: Anderson and Sundaresan (1996); Mella-Barral and Perraudin (1997); Mella-Barral (1999); Fan and Sundaresan (2000); Francois and Morellec (2004). Risk Management for Firms: Ross (1996); Leland (1998). 12

Taurén (1999) and Collin-Dufresne and Goldstein (2001) approach this issue by proposing alternative models to reflect the firms’ tendency to maintain a stationary leverage ratio. However, their studies focus on the risky corporate debt pricing and the implications of credit spreads.

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Bankruptcy Procedures (Chapter 7 versus Chapter 11): Broadie et al. (2007). Optimal Investment: Mauer and Triantis (1994), Childs et al. (2005); Hackbarth (2006). Stochastic Interest Rate: Kim et al. (1993); Acharya and Carpenter (2002); Huang, Ju and Ou-Yang (2003); Ju and Ou-Yang (2006). Jump: Dao and Jeanblanc (2006). The Fan and Sundaresan (2000) model as well as the Goldstein et al. (2001) model are presented in detail in Sects. 60.4.1 and 60.4.2, respectively. We choose these two papers that, in our opinion, are the most important or representative of that stream of research. The remaining research will be presented in less detail and we will briefly outline their key assumptions and results. Readers may refer to the original papers for details.

60.4.1 The Fan and Sundaresan (2000) Model Fan and Sundaresan (2000) propose a game-theoretic setting that incorporates equity’s ability to force concessions and various bargaining powers to the debt and equity holders. Two cases are discussed in their study where the firm’s asset value falls below a certain boundary: Debt-equity swap and strategic debt service. In the case of debt-equity swap, the debtholders exchange their claims for equity. In strategic debt service, borrowers stop making the contractual coupon and start servicing debt strategically until the firm’s assets returns to its original level. Some key assumptions under their framework include the following: (1) During the default period, the tax benefits are lost. (2) Asset sales for dividend payments are prohibited. (3) The firm can be liquidated only at a cost. The proportional cost is ˛.0  ˛  1/ and the fixed cost is K.K 0/. Note that debt holders have strict absolute priority upon liquidation. (4) The asset value of the firm follows the lognormal diffusion process dV D .  ˇ/V dt C VdBt

(60.21)

where  is the instantaneous expected rate of return on the firm gross of all payout, and ˇ is the firm’s cash payout ratio. Here we present only the results of strategic debt service. If the equity and debt holders can reach an agreement of temporary coupon reduction when the firm is under financial distress, the firm will not lose its potential future tax benefits. At the trigger point for the strategic service VQS , both parties will bargain the total value of the firm v.V /. The driving force behind strategic behavior is the presence of proportional and fixed costs of liquidation. The bargaining power of equity holder clearly depends on the liquidation cost ˛VS C K. Fan and Sundaresan (2000) endogenize both the reorganization boundary and the optimal sharing rule between equity and debt holders upon default.

The total value of the firm is 8 T C ˆ VQS Q r C   r C VS v.V / D ˆ :V C  TC C V ; when V  VQS C   r VQS

(60.22)

where  D 0:5 

rˇ 2

˙

rh

0:5 

rˇ 2

i2

C

2r . 2

For any V  VQS , Fan and Sundaresan set the optimal sharing rule between shareholders and debtholders as Q Q Q / D v.V Q / D .1  /v.V E.V / and D.V /:

(60.23)

The Nash solution to the bargaining game can be characterized as o n Q /0 Q  D arg max v.V n o1 Q

.1  /v.V /  max Œ.1  ˛/V  K; 0   .1  ˛/V  K ; (60.24) D min    v.V / where  denotes bargaining power of shareholders. At equilibrium, Fan and Sundaresan (2000) show that debt holders accept less than the contractual coupon and still permit shareholders to run the company. This results in deviations from APR, in which the shareholders will get a bigger share and the debtholders will get a less proportion of the firm. But both parties will be better off. Therefore, the concession of debtholders is well explained under the Fan and Sundaresan’s framework. They then solve the trigger point for strategic debt service (K D 0 case) as c.1  TC C TC /  1 VQ D r 1   1  ˛

(60.25)

and the strategic service amount paid to debt holders S.V / D .1  ˛/ˇV . It can be shown that the equity value satisfies the following differential equations:13 1 2 2Q Q  V EVV .V; t/ C .r  ˇ/V EQ V .V; t/  r E.V; t/ 2 CˇV  c.1  TC / D 0; when V > VQS I 1 2 2Q Q  V EVV .V; t/ C .r  ˇ/V EQ V .V; t/  r E.V; t/ 2 CˇV  S.V / D 0; when V > VQS I (60.26)

13

See Dixit and Pindyck (1994).

60 Alternative Methods to Determine Optimal Capital Structure: Theory and Application

with the boundary conditions: lim E.V / D V 

V "1

c.1  TC / r

cT C  C   r V #VQS

C  cT C lim EV .V / D  ˛  C   r VQS V #VQS lim E.V / D  ˛ VQS 

943

Note that when V > VQS , the contractual coupon is paid and the tax shield is in effect. By contrast, when V  VQS , the tax shield is lost because the equity holders are strategically servicing the debt according to the cash flow generated by the firm. Both the strategic debt service amount S.V / and the trigger level VQS are determined endogenously. The equity value is

8  

.1  TC /C  .1  C / TC C V .1  TC /C < C  ; V  Q E.V / D r .1   /r .C   /.1   / r VS : v.V /  .1  ˛/V;

The Fan and Sundaresan model shows that debt renegotiation encourages early default and increases credit spreads on corporate debt, given that shareholders can renegotiate in distress to avoid inefficient and costly liquidation. It might be the interest of debt holders to forgive part of the debt service payments if they can avoid the wasteful liquidations, which can be shared by the two claimants. If shareholders have no bargaining power, no strategic debt service takes place and the model converges to the Leland model. Furthermore, by introducing the possibility of renegotiating the debt contract, the default can occur at positive equity value. This is in contrast to the Leland’s (1994) model in that the default occurs when the equity value reaches zero as a consequence of that issuing new equity is costless and the APR is respected.

dı D P dt C dz ı

Z



1

dsıs e rs t

14 Although unlevered firm value may not be traded, trading of equity or other contingent claims allows the use of unlevered firm value as the state variable (Ericsson and Reneby 2004a).

(60.27)

(60.28)

The value to the claim of entire period is the discounting expected cash flows (unlevered firm) Q

Goldstein, Ju, and Leland propose an EBIT-based model in that the state variable is earnings before interests and taxes (EBIT) instead of commonly used unlevered firm value.14 This approach is intuitively appealing: the EBIT-generating machine runs independently of how the EBIT flow is distributed among its claimants. The advantage of taking the claim on future EBIT is that all contingent claimants, including the government claim, to future EBIT flows are treated in a consistent fashion. More precisely, the government claim is no longer treated as a cash inflow in a form of tax benefit, but as a cash outflow on EBIT in a form of tax. In other words, each dollar paid out, whether to interest payments, dividends, or taxes, affects the firm in the same way. Another advantage is that unlevered and levered equity never exist simultane-

when V  VQS

ously. In contrast to the unlevered firm value, the claim to future EBIT flow should be more reasonable to changes in capital structure. Furthermore, compared with previous models, this characterization also has assumptions that are more in accordance with empirical findings.15 The single technology of a pure exchange economy produces a payout flow that is specified by the (EBIT) process

V .t/ D Et

60.4.2 The Goldstein et al. (2001) Model

when V > VQS

D

ıt r 

(60.29)

where  D .P  / is the risk-neutral drift of the payout flow rate: dı D dt C dzQ (60.30) ı Since r and  are constants, V and ı share the same dynamics: dV (60.31) D dt C dzQ V

15

Fischer et al. (1989) assume that the following: (i) The value of an optimally levered firm can only exceed its unlevered value by the amount of transaction costs incurred. It eliminates the arbitrage opportunity. (ii) A firm that follows an optimal financing policy offers a fare riskadjusted return. Thus, unlevered firms offer a below-fair expected rate of return. Goldstein et al. (2001) argue that these key assumptions by Fischer et al. (1989) are not realistic. First, the arbitrage strategy is not feasible because a bidder must pay a large premium in order to gain control of management. They also argue that for publicly traded assets, rational investors will push down security prices so that fair expected return is obtained on any traded assets regardless of the policies employed by the management.

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S.-S. Chen et al.

Accordingly,

with boundary conditions dV C dı D rdt C dzQ : V

lim Vsolv D V and Vsolv D 0 as V D VB

(60.32)

V !1

Thus, Vsolv D V  VB pB .V /

60.4.2.1 Optimal Static Capital Structure Assume that a firm issues a consol bond with a constant coupon payment C as long as the firm remains solvent. As in Leland (1994), any claim must satisfy the ordinary differential equation (60.33)

where P is the payout flow. The solution to the above equation can be written as the sum of a particular solution to the full inhomogeneous equation and the general solution to the corresponding homogeneous equation. The general solution of VFV C 2 2 V FVV  rF D 0 (no intertemporal cash flows): 2 F D A1 V " where x D



2

2 2

y

C



2 4   yD 2

C A2 V r



x

2 2

(1.3) Define Vint as the value of the claim to interest payments. Vint D Since lim Dc D

2 2 V FVV C P  rF D 0 VF V C 2

V !1

Vint D

2

3, s

2 2  C 2r 2 5  2 2

Note that x is positive and y is negative. Since A1 V y goes to infinity while V becomes large, A1 equals zero for all claims of interest.

(60.38)

Dsolv .V / D .1  Ti /Vint

(60.39)

Gsolv .V / D Teff .Vsolv  Vint / C Ti Vint

(60.40)

.1  Teff / D .1  Tc /.1  Td / (1.5) Define Vdef .V / as the present value of the default claim. Vdef .V / D A1 V y C A2 V x with boundary conditions lim Vdef .V / D 0 and Vdef .V D VB / D VB

V !1

lim pB .V / D 0 and lim pB .V / D 1

V !1

Thus,

V !1

pB .V / D

(60.37)

where Ti is the personal tax rate of interest payments, Td is the effective rate for dividends, TC is the tax rate for corporate profits, and the effective rate Teff is

with boundary conditions

Thus,

C Œ1  pB .V / r

Esolv .V / D .1  Teff /.Vsolv  Vint /

(1.1) Define pB .V / as the present value of a claim that pays $1 contingent on firm value reaching VB . pB .V / D A1 V y C A2 V x

, A1 D 0. This claim vanishes at V D

(1.4) Finally, separating the value of continuing operations among equity, debt, and government, we have

#, C 2r 2

C r

C C A1 V y C A2 V x r

VB , one can obtain

(60.34)

2

(60.36)

V VB

Vdef .V / D VB pB .V /

x (60.35)

(1.2) Define Vsolv .V / a claim entitled to the entire payout ı as long as the firm remains solvent; that is, firm value remains above VB . Vsolv D V C A1 V y C A2 V x

(60.41)

Note that Vsolv .V / C Vdef .V / D V , the present value of the claim to EBIT. Therefore, value is neither created nor destroyed by changes in the capital structure. Similar to the claims to funds during solvency, the default claim can be divided among bankruptcy costs, debt, and government, and: BCdef .V / D ˛Vdef .V /

(60.42)

60 Alternative Methods to Determine Optimal Capital Structure: Theory and Application

Ddef .V / D .1  ˛/.1  Teff /Vdef .V /

(60.43)

Gdef .V / D .1  ˛/Teff Vdef .V /

(60.44)

Note that Goldstein et al. (2001) assume that the portion of the EBIT-generating machine not lost to bankruptcy cost can be sold to competitors in the event of bankruptcy. Thus, the government can have a claim in bankruptcy due to the taxes on future EBIT from competitors. (1.6) The restructuring costs are assumed to be proportional to the initial value of the debt issuance: RC.V0 / D qŒDsolv .V0 / C Ddef .V0 /

PU .V / D A1 V y C A2 V x with boundary conditions pU .VB / D 0 and lim pB .V / D 0

C r

V !1



(60.46)

 . where  D Note that while most comparative statics are similar to those in Leland (1994), equity is a decreasing function of effective tax rate. This is because an increase in Teff increases the government claim at the expense of equity. In contrast, previous capital structure models use unlevered firm value as state variable and treat government claim as cash inflow. As a result, the tax benefits increase with tax rate and in turn increase equity value. However, empirical evidence indicates that equity prices are a decreasing function of the tax rate. Under the characterization of Goldstein et al. (2001), even though the tax benefit is an increasing function of TC , equity is a decreasing function of TC . This feature solves the problem of the previous capital structure models and is supported by empirical studies. x xC1

60.4.2.2 Optimal Upward Dynamic Capital Structure Strategy16 Goldstein et al. (2001) assume that there will be a threshold VB , where the firm will optimally choose bankruptcy, and there be a threshold VU , where management will call the outstanding debt issue at par, and sell a larger new issue.17 They 16

(2.1) Define PU .V / as the present value of a claim that pays $1 contingent on firm value reaching VU . This claim receives no dividend and therefore has the form

@V V DVB

VB D  

then use backward induction scheme to show that it is optimal for each period’s coupon payments, restructuring threshold, and bankruptcy threshold to increase by the same factor  D VU =V0 that the initial EBIT value increases by each period. All claims will also scale up by the same factor. Because of this scaling feature, the firm will be identical at every future restructuring date, except that all factors will be scaled up by  .

(60.45)

The restructuring costs are deducted from the proceeds of the debt issuance before distribution to equity holders. ˇ D 0, (1.7) Solving the smooth-pasting condition @E ˇ one can obtain

945

They mention that downward restructuring can be incorporated into their model. However, they argue that transaction costs can discourage debt reduction outside Chap. 14 (see Gilson 1997). In addition, due to the neglected issues such as asset substitution, asymmetric information, equity’s ability to force concessions, and Chap. 14 protection, they are skeptical about the results for the downward restructuring. 17 Goldstein et al. (2001) claim that given management’s strategy, the value of the debt issue would be the same even if it were not callable, as long as all debt issues receives the same recovery rate in the event of bankruptcy.

Thus, y

PU .V / D  y

VBx y V V C B V x † †

(60.47)

y

where †  VB VUx  VU VBx . (2.2) Similarly, define pB .V / as the present value of a claim that pays $1 contingent on firm value reaching VB . y

pB .V / D  y

VUx y V V C U V x † †

(60.48)

y

where †  VB VUx  VU VBx . (2.3) From (2.1) and (2.2), all period 0 claims can be written in terms of pU .V / and pB .V /. 0 Vsolv .V / D V  pU .V /VU  pB .V /VB

(60.49)

(current firm value remains between VU and VB ) 0 .V / D Vint

C0 Œ1  pU .V /  pB .V / r

(60.50)

0 .V / D pB .V /VB Vdef

(60.51)

0 Vres .V / D pU .V /VU

(60.52)

0 0 0 .V / C Vdef .V / C Vres .V / D V . Note that the sum Vsolv

(2.4) Assuming full-loss offset, the values of period 0 claims after the restructuring: 0 0 .V / C .1  ˛/.1  Teff /Vdef .V / d 0 .V / D .1  Ti /Vint (60.53) 0 0 .V /  Vint .V / e 0 .V / D .1  Teff /ŒVsolv

(60.54)

946

S.-S. Chen et al. 0 0 0 g 0 .V / D Teff ŒVsolv .V /  Vint .V / C Ti Vint 0 C.1  ˛/Teff Vdef .V /

(60.55)

0 bc0 .V / D ˛Vdef .V /

(60.56)

(2.5) Define e.V / as the equity claim to intertemporal EBIT flows for all periods. Note that e.V / is not the total equity claim. Since the claims to future debt issues are currently owned by equity and restructuring costs, future debt holders will have to pay fair price for these claims in the future. As a result, there are also cash transfers between the claimants at each restructuring date.

The total equity claim, E(V), just before the initial debt issuance, is the sum of the intertemporal claims of debt and equity, less the claim of restructuring costs: E.V0 / D

e 0 .V0 / C d 0 .V0 /  qD 0 .V0 / 1  PU .V0 /

(60.62)

And after the initial debt issuance during period 0, E.V / D PU .V /E.V0 / C e 0 .V /  PU .V /D 0 .V0 / (60.63) This reduces to E.V0C / D E.V0 /  .1  q/D 0 .V0 /

(60.64)

The scaling feature inherent in the model implies It also reduces to

d.V0 / g.V0 / e.V0 / D 0 D 0 0 e .V0 / d .V0 / g .V0 /

E.VU  / D E.V0 /  D 0 .V0 /

1 V0 D D 0 1  PU .V0 / V0  VU .V0 /PU .V0 / (60.57) The equity claim to intertemporal cash flows is e 0 .V0 / 1  PU .V0 / ˚  D e 0 .V0 / 1 C PU .V0 / C ŒPU .V0 / 2 C : : : 9 8 1 = <X ŒPU .V0 / j D e 0 .V0 / (60.58) ; :

e.V0 / D

j D0

(2.6) Next, they determine the portion of the future debt claims that currently belong to equity and restructuring costs. The management will call the outstanding debt issue at par at restructuring date. Therefore, the initial value of the debt claim equals the sum of the claim to period 0 cash flows plus the present value of the call value D 0 .V0 / D d 0 .V0 / C pU .V0 /D 0 .V0 / Hence, D 0 .V0 / D

d 0 .V0 / 1  pU .V0 /

(60.59)

(60.60)

D 0 .V0 / is the fair value debt holders must pay at the time of issuance. The proceeds are then divided between equity and restructuring cost. By the scaling argument, the total restructuring claim is ˚  RC.V0 / D qD .V0 / 1 C PU .V0 / C ŒPU .V0 / 2 C : : : 0

D

qD 0 .V0 / 1  PU .V0 /

(60.61)

(60.65)

just before the next issuance.  VB =V0 Finally, managers choose C ,   VU =V0 , in order to maximize the wealth of equity holder, subject to limited liability. In the Goldstein, Ju, and Leland model, the optimal initial leverage ratio is substantially lower than that obtained in the static model and is more in line with the historical average. This is a result of that the firm has the option to increase its leverage in the future. Thus, it will wait until it becomes optimal to exercise it. The possibility of increasing leverage in the future decreases the initial optimal leverage. In addition, the bankruptcy triggered level VB also drops significantly since a firm that has an option to adjust its capital structure is more valuable than the one without this option. Moreover, the model also implies reasonable and lower bankruptcy cost for the optimal leverage. The tax benefits to debt also increase significantly over the prediction of static models.

60.4.3 Other Important Extensions In this section, we will briefly present some other extensions with the introduction of investment risk, alternative bankruptcy procedures, as well as alternative stochastic processes, and will outline key results of these models.

60.4.3.1 Investment Risk – Leland (1998) One possible agency problem, asset substitution, has been discussed in Leland (1994) and Leland and Toft (1996). Leland (1998) explicitly models this issue by allowing the firm to choose its risk strategy and examines the interaction between capital structure and risk choice. To capture the

60 Alternative Methods to Determine Optimal Capital Structure: Theory and Application

agency cost, Leland (1998) assumes that risk choices are made ex post; that is, after debt is in place. The risk choices cannot be precontracted in the debt covenants or credibly precommitted. The environment with ex post risk choice can be contrast with the hypothetical ex ante, in which the firm simultaneously chooses its risk strategy and its debt structure to maximize initial firm value. The measure of agency costs is therefore the difference in maximal values between the ex ante and ex post cases. This reflects the loss in value follows from the risk strategies chosen to maximize equity value rather than firm value. In this paper, optimal capital structure reflects both the tax advantages less default costs and the agency costs resulting from asset substitution (Jensen and Meckling 1976). The joint determination of capital structure and investment risk is examined. Leland (1998) finds that agency costs restrict leverage and debt maturity and increase yield spreads.

60.4.3.2 Chapter 11 Protection – Francois and Morellec (2004) Francois and Morellec (2004) extend the Fan and Sundaresan (2000) model to incorporate the possibility of Chap. 14 filings. Under their setup, shareholders hold a Parisian down-and-out option on the firm’s asset; that is, shareholders have a residual claim on the cash flows generated by the firm unless the value of these assets reaches the default threshold and remains below that threshold for the exclusive period. It is generally acknowledged that there are two types of defaulting firms: First, firms that are economically sound but default only due to temporary financial distress, and recover under Chap. 14. Second, firms that are economically unsound, keep on losing value under Chap. 14, and are eventually liquidated under Chap. 10. The modeling philosophy comes from the empirical studies that show that most firms emerge from Chap. 14. Only a few are eventually liquidated under Chap. 10 after filing Chap. 14. Following the Nash bargaining game of Fan and Sundaresan’s approach,18 Francois and Morellec presume the firm renegotiates its debt obligations whenever the asset value falls below a constant threshold VB . However, a major

18 Note that the specification for the bargaining game within Francois and Morellec’s framework is slightly different from that of Fan and Sundaresan (2000). Francois and Morellec focus on Chapter 11 filings – court-supervised debt renegotiation in contrast to the private workouts by Fan and Sundaresan. Therefore, the automatic stay of assets prevents shareholders from liquidating the firm’s assets. Nevertheless, renegotiation plan requires each participant to receive a payoff that exceeds the liquidating value of its claims. As a result, bondholders’ payoff has to exceed .1  ˛/VB and this results in the difference of the specification for the bargaining game.

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difference from Fan and Sundaresan’s model is that Francois and Morellec assume that a proportional costs ' are borne by the company during the renegotiation process. The costs of financial distress incurred by the firm filing Chap. 14 are ignored in the prior research. Even though the costs implied by private workouts are generally low, Chap. 14 filings are associated with large costs of financial distress that may affect shareholder’s default decision. Note that Leland (1994) only allows liquidation while Fan and Sundaresan (2000) only permit private workouts. Francois and Morellec solve the endogenous default barrier by maximizing equity value and providing closed-form solutions for corporate debt and equity values. They find that the possibility to renegotiate the debt contract has ambiguous impact on leverage choices and increases credit spreads on corporate debt. The sharing rule of cash flows during bankruptcy has a large impact on optimal leverage, while credit spread on corporate debt shows little sensitivity to the varying bargaining power.

60.4.3.3 Chapter 11 vs. Chapter 7 – Broadie et al. (2007) Broadie et al. (2007) propose framework in which optimal debt and equity values are determined in the presence of Chaps. 10 and 14 under the U.S. bankruptcy code. They explicitly consider two separate barriers for default and liquidation for the more realistic bankruptcy procedures. They model the key feature of bankruptcy proceedings by automatic stay provision – suspending the dividends while entering bankruptcy (Chap. 14) and the imposition of APR (Chap. 10) upon liquidation. To facilitate the analysis, they follow the EBIT process by Goldstein et al. (2001)19 and assume the firm issues a single consol bond. They model the critical characteristics of the U.S. bankruptcy proceeding – grace period, automatic stay, debt forgiveness, absolute priority, and transfer of control rights from equity to debt holders. Some prior models, for example, such as the Francois and Morellec (2004) model, also make a distinction between default and liquidation. However, Broadie et al. (2007) argue that in their model the firm cannot be liquidated even if the unlevered firm value is too low during the grace period. As a result, limited liability could be violated. Therefore, under the setup of Broadie, Chernov, and Sundaresan, liquidation could happen as the firm value either

19 Note that they assume the same process for EBIT before and after default. Therefore, their framework models financial rather than economical distress since bankruptcy by itself does not cause poor performance. The work by Goldstein et al. (2001) does not need this assumption since there is no difference between default (bankruptcy) and liquidation under their setup.

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reaches the liquidation barrier or stays under the bankruptcy barrier for longer than the grace period. In addition, they explicitly model the automatic stay provision by recording the accumulated unpaid coupons plus interest in arrears, denoted as At . When a firm recovers from Chap. 14, the debt holders will forgive a fraction 1  , 0    1 of arrears and receive an amount At . Moreover, the firm does not pay dividends to shareholders while it is in the default state. The entire EBIT while in default is accumulated in a separated account St . If St is not sufficient to repay the arrears, equity holders must raise the remaining at the cost of diluting equity. If St > At , the leftover is distributed to shareholders.20 In this paper, they focus on the issues of bankruptcy proceedings and the optimal choice of these two boundaries driven by different objectives.21 They show that the first-best outcome, the total firm value maximization ex-ante upon filing Chap. 14, is different from the equity value maximization outcome. They also show that the first-best outcome can be restored in large measure by giving creditors either the control to declare Chap. 14 or the right to liquidate the firm once it is taken to Chap. 14 by the equity holders. This serves as the threat from debtholders to prevent equity holders from filing for Chap. 14 too soon to get debt relief. Finally, they also find that on average the firms are more likely to default and are less likely to liquidate relative to the benchmark model of Leland (1994).

60.4.3.4 Optimal Debt Maturity – Ju and Ou-Yang (2006) In the work by Leland and Toft (1996), debt maturity is shown to be crucial to the leverage ratio and credit spreads. In practice, the optimal capital structure and optimal maturity structure are interdependent decisions. However, in Leland and Toft (1996), debt maturity is exogenously specified. In addition, Goldstein et al. (2001) show that the optimal capital structure is very sensitive to the input level of the interest rate. Therefore, Ju and Ou-Yang (2006) propose a dynamic model in which the optimal structure and an optimal debt maturity are jointly determined in a Vasicek (1977) meanreverting interest rate process.22 20

Note that in order to accommodate the complex feature of bankruptcy proceedings, numerical approach is necessary to solve the model. The detailed binomial lattice methodology is developed by Broadie and Kaya (2007). 21 To determine if a distressed firm would choose the reorganization option under nearly perfect market assumption, they first ignore taxes to emphasize the impact of bankruptcy and return to this issue later in their paper. 22 We should note that the default boundary is exogenously specified in their model for tractability. The design of recapitalization of the firm’s

S.-S. Chen et al.

In their model, the tradeoff between the gains from dynamically adjusting the debt amount and the restructuring costs if adjusting it yield the optimal debt maturity. In other words, the tax rate and the transaction costs are the most two important parameters in determining the optimal maturity. They find that a higher transaction cost yields a longer debt maturity since it is more expensive to rebalance a firm’s capital structure. On the other hand, a higher tax rate yields a shorter optimal debt maturity because it is more valuable to recapitalize the capital structure. They also find that the initial level and long-run mean of the interest rate process are key variables in determining both optimal capital structure and optimal maturity structure. When the interest rate is higher (lower), coupon and principal are higher (lower). The coupon and principal are determined in such a way that the market value of debt is independent of the spot interest rate level. This is in sharp contrast to the fact that the level of spot interest rate is a key parameter in the constant interest rate model. Finally, the volatility of the interest rate process and the correlation between the interest rate process and the firm asset value process play important roles in determining the debt maturity structure.

60.5 Application and Empirical Evidence of Capital Structure Models Capital structure models are intended to quantitatively examine the capital structure theories. In capital structure models, the corporate debt value and capital structure are interlinked variables. Therefore, the valuation of corporate securities and financial decisions are jointly determined. Through the structural approach, researchers can conduct a detailed analysis of the behavior of bond prices and optimal leverage ratios as financial variables (such as taxes, interest rates, payout rates or bankruptcy costs) change. While theoretically elegant, capital structure models must be able to explain observed capital structure and bond prices in practice. As a result, the most commonly used measures for a capital structure model are the predicted leverage ratio and yield spreads. Most of the capital structure models illustrate the performance by numerical examples using the historical average as model input parameters. In general, the early capital structure models predicted optimal leverage rates that were too high. The recent models such as those incorporating dynamic capital structure settings have predicted optimal leverage more in line with the capital structure is different from that of Goldstein et al. (2001). In contrast to the recapitalization barrier assumed in Goldstein et al. (2001), Ju and Ou-Yang (2006) assumes that the firm rebalances its capital structure periodically as long as the firm is solvent.

60 Alternative Methods to Determine Optimal Capital Structure: Theory and Application

observed average under reasonable bankruptcy cost. Nevertheless, capital structure models (as well as corporate bond pricing models using structural approach) still predict credit spreads too low, especially for the short-term debt. There are only a few empirical studies on the models employing structural approach. Among them, a large portion of the empirical tests of structural models focus on the risky debt pricing performance of the bond pricing models, such as Longstaff and Schwartz (1995). The empirical performance in pricing risky debt is generally unsatisfactory (see Wei and Guo 1997; Anderson and Sundaresan 2000; Delianedis and Geske 2001; Huang and Huang 2003, among others.). The empirical investigations for capital structure models are even rare due to their complexity. The most well-known study including capital structure models is by Eom et al. (2004). Eom et al. (2004) carry out an empirical analysis of five structure models including Merton (1974), Geske (1977), Longstaff and Schwartz (1995), Leland and Toft (1996), and Collin-Dufresne and Goldstein (2001). They test these models with bond data of firms with simple capital structure on the last trading day of each December from 1986 to 1997. They calibrate these models using the book value of total liabilities in the balance sheet as the default boundary, and calculate the corresponding asset value as the sum of the market value of equity and the book value of total debt. Then they estimate the asset return volatility using bond-implied volatility as well as six equity return volatilities measured by different time horizons before and after the bond price observation. In contrast to previous studies that have suggested that structural models generally predict yield spreads too low, the result of Eom et al. (2004) shows more complicated phenomena, although all of these models make significant errors in predicting the credit spread. The Merton (1974) and Geske (1977) models underestimate the spreads, while the Longstaff and Schwartz (1995), Leland and Toft (1996), and Collin-Dufresne and Goldstein (2001) models all overestimate the spreads. Huang and Huang (2003) use several structural models to predict yield spread, including the Longstaff and Schwartz (1995) model, the strategic model, the endogenousdefault model, the stationary leverage model as well as two models they propose: one with a time-varying asset risk premium and one with a double exponential jump-diffusion firm value process. They calibrate inputs, including asset volatility, for each model so that target variables, including leverage, equity premium, recovery rate, and the cumulative default probability at a single time horizon, are matched. They show that the models make quite similar predictions on yield spreads. In addition, the observed yield spreads relative to Treasury bonds are considerably greater than the predicted spreads, especially for highly rated debt. Hence, they conclude that additional factors such as liquidity and taxes must be important in explaining market yield spreads.

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By adopting a different estimation method, Ericsson et al. (2006) use the MLE approach proposed by Duan (1994) to perform an empirical test on both CDS spreads and bond spreads, including three structural models – Leland (1994), Leland and Toft (1996), and Fan and Sundaresan (2000). In contrast to previous evidence from corporate bond data, CDS premia are not systematically underestimated. Also, as expected, bond spreads are systematically underestimated, which is consistent with the fact they are driven by significant non-default factors. In addition, they also conduct regression analysis on residuals against default and non-default proxies. Little evidence is found for any default risk components in either CDS or bond residuals, while strong evidence, in particular, an illiquidity premium, is related to the bond residuals. They conclude that structural models are able to capture the credit risk price in the markets but they fail to price corporate bonds adequately due to omitted risks. Also using the MLE approach, Ericsson and Reneby (2004b) estimate yield spreads between 1 and 50 months out of sample by an extended version of the Leland (1994) model, which allows for violation of the absolute priority rule and future debt issues. In addition to the stock price series, they also include the bond price and dividend information in estimating their model. The bond samples consist of 141 U.S. corporate issues and a total of 5,594 dealer quotes. Their empirical results show that, for the 1 month-ahead prediction, a mean error is merely 2 basis points. This is similar to the fitting error of those reduced-form models in Duffee (1999). Therefore, they conclude that the inferior performance of structural models may result from the estimation approaches used in the existing empirical studies. Finally, another potentially important application of capital structure model is to predict the credit quality of a corporate security. For example, Leland (2004) examines the default probabilities predicted by the Longstaff and Schwartz (1995) model with the exogenous default boundary, and the Leland and Toft (1996) model with endogenous default boundary. As Leland stated We focus on default probabilities rather than credit spreads because (i) they are not affected by additional factors such as liquidity, tax differences, and recovery rates; and (ii) prediction of the relative likelihood of default is often stated as the objective of bond ratings.

Leland uses Moody’s corporate bond default data from 1970 to 2000 in his study and follows a similar calibration approach by Huang and Huang (2003). Rather than matching the observed default frequencies, Leland instead chooses common inputs across models to observe how well they match observed default statistics. The empirical results show that when costs and recovery rates are matched, the exogenous and the endogenous default boundary models fit observed default frequencies equally. The models predict longer-term default frequencies quite accurately, while

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shorter-term default frequencies tend to be underestimated. Thus, he suggests that a jump component should be included in asset value dynamics.23 In summary, the empirical performance of capital structure models in pricing risky debt is not satisfactory. Even with the incorporation of jump process, the illiquid corporate bond market still hinders structural models from accurately pricing risky debt. However, predicting the credit quality of a corporate security could be a good application of capital structure models because they are less affected by the micro structure issues. Since recent capital structure models put numerous efforts on the event of bankruptcy, prediction of default probabilities or default events shall be potentially important applications.24

60.6 Conclusion In this paper, we review the most important and representative capital structure models. The capital structure models incorporate contingent claim valuation theory to quantitatively analyze prevailing determinants of capital structure in corporate finance literature. In capital structure models, the valuation of corporate securities and financial decisions are jointly determined. Most of the capital structure models provide closed-form expressions of corporate debt as well as the endogenously determined bankruptcy level, which are explicitly linked to taxes, firm risk, bankruptcy costs, risk-free interest rate, payout rates, and other important variables. The behavior of how debt values (and therefore yield spreads) and optimal leverage ratios change with these variables can thus be investigated in detail. While theoretically elegant, capital structure models do not perform well empirically in risky corporate bond pricing. Researchers have been attempting to resolve the yield spread underestimates by introducing jumps and liquidity premium. On the other hand, since recent capital structure models put numerous efforts on the event of bankruptcy, we suggest that prediction of default probabilities or default events shall be potentially important applications. Furthermore, some researchers argue that the past poor performance of capital structure models may come from the estimation approaches historically used in the empirical studies (Ericsson and Reneby 2005, Ericsson, Reneby, and Wang 2006). And we have seen some innovative estimation methods for solving

23 We should note that Zhou (2001) show that jumps can explain why yield spreads remain strictly positive even as maturity approaches zero. 24 In default prediction, some researchers have conducted empirical investigation on a company basis analysis such as Chen et al. (2006) and Chen et al. (2008), though the models they tested are mainly bond pricing models.

S.-S. Chen et al.

the estimation problem in models employing structural approach (see Bruche 2005) and the reference therein). Thus, we hope to see more rigorous empirical studies for capital structure models in the future.

References Acharya, V. and J. Carpenter. 2002. “Corporate bond valuation and hedging with stochastic interest rates and endogenous bankruptcy.” Review of Financial Studies 15, 1355–1383. Anderson, R. and S. Sundaresan. 1996. “Design and valuation of debt contract.” Review of Financial Studies 9, 37–68. Anderson, R. and S. Sundaresan. 2000. “A comparative study of structural models of corporate bond yields: an exploratory investigation.” Journal of Banking and Finance 24, 255–269. Black, F. and J. C. Cox. 1976. “Valuing corporate securities: some effects of bond indenture provisions.” Journal of Finance 31, 351–367. Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 81, 637–654. Brennan, M. and E. Schwartz. 1978. “Corporate income taxes, valuation, and the problem of optimal capital structure.” Journal of Business 54, 103–114. Briys, E. and F. de Varenne. 1997. “Valuing risky fixed rate debt: an extension.” Journal of Financial and Quantitative Analysis 32, 239–248. Broadie, M. and O. Kaya. 2007. “A binomial lattice method for pricing corporate debt and modeling chapter 11 proceedings.” Journal of Financial and Quantitative Analysis 42, 279–312. Broadie, M., M. Chernov, and S. Sundaresan. 2007. “Optimal debt and equity values in the presence of chapter 10 and chapter 14.” Journal of Finance 62, 1341–1377. Bruche, M. 2005. Estimating structural bond pricing models via simulated maximum likelihood, Working paper, London School of Economics. Chen, R., S. Hu, and G. Pan. 2006. Default prediction of various structural models, Working paper, Rutgers University, National Taiwan University, and National Ping-Tung University of Sciences and Technologies. Chen, R., C. Lee, and H. Lee. 2008. Default prediction of alternative structural credit risk models and implications of default barriers, Working paper, Rutgers University. Childs, P., D. Mauer and S. Ott. 2005. “Interaction of corporate financing and investment Decisions: the effects of agency conflict.” Journal of Financial Economics 76, 667–690. Collin-Dufresne, P. and R. S. Goldstein. 2001. “Do credit spreads reflect stationary leverage ratios?” Journal of Finance 56, 1929–1957. Dao, B. and M. Jeanblanc. 2006. Double exponential jump diffusion process: a structural model of endogenous default barrier with rollover debt structure, Working paper, The Université Paris Dauphine and the Université d’Évry. Delianedis, G. and R. Geske. 2001. The components of corporate credit spreads: default, recovery, tax, jumps, liquidity, and market factors, Working paper, UCLA. Duan, J. C. 1994. “Maximum likelihood estimation using pricing data of the derivative contract.” Mathematical Finance 4, 155–167. Duffee, G. 1999. “Estimating the price of default risks.” Review of Financial Studies 12, 197–226. Durand, D. 1952. “Costs of debt and equity funds for business trends and problems of measurement,” Conference on Research in Business Finance, National Bureau of Economic Research, New York, pp. 215–247.

60 Alternative Methods to Determine Optimal Capital Structure: Theory and Application Eberhart, A., W. Moore, and R. Roenfeldt. 1990. “Security pricing and deviations from the absolute priority rule in bankruptcy proceedings.” Journal of Finance, December, 1457–1469. Eberhart, A. C. and L. A. Weiss. 1998. “The importance of deviations from the absolute priority rule in chapter 11 bankruptcy proceedings.” Financial Management 27, 106–110. Elizalde, A. 2005. Credit risk models II: structural models, Working paper, CEMFI and UPNA. Eom, Y. H., J. Helwege, and J. Huang. 2004. “Structural models of corporate bond pricing: an empirical analysis.” Review of Financial Studies 17, 499–544. Ericsson, J. and J. Reneby. 2004a. “A note on contingent claims pricing with non-traded assets.” Finance Letters 2(3). Ericsson, J. and J. Reneby. 2004b. “An empirical study of structural credit risk models using stock and bond prices.” Journal of Fixed Income 13, 38–49. Ericsson, J. and J. Reneby. 2005. “Estimating structural bond pricing models.” Journal of Business 78, 707–735. Ericsson, J., J. Reneby, and H. Wang. 2006. Can structural models price default risk? Evidence from bond and credit derivative markets, Working paper, McGill University and Stockholm School of Economics. Fan, H. and S. Sundaresan. 2000. “Debt valuation, renegotiations and optimal dividend policy.” Review of Financial Studies 13, 1057–1099. Fischer, E., R. Heinkel, and J. Zechner. 1989. “Dynamic capital structure choice: theory and tests.” Journal of Finance 44, 19–40. Francois, P. and E. Morellec. 2004. “Capital structure and asset prices: some effects of bankruptcy procedures.” Journal of Business 77, 387–411. Franks, J. and W. Torous. 1989. “An empirical investigation of U.S. firms in reorganization.” Journal of Finance 44, 747–767. Geske, R. 1977. “The valuation of corporate liabilities as compound options.” Journal of Financial and Quantitative Analysis 12, 541–552. Hackbarth, D. 2006. A real options model of debt, default, and investment, Working paper, Olin School of Business, Washington University. Harris, M. and A. Raviv. 1991. “The theory of capital structure.” The Journal of Finance 46(1), 297–355. Harrison, J. M. 1985. Brownian motion and stochastic flow systems, Wiley, New York. Huang, J. and M. Huang. 2003. How much the corporate-treasury yield spread is due to credit risk? Working paper, Penn State University and Stanford University. Huang, J., N. Ju, and H. Ou-Yang. 2003. A model of optimal capital structure with stochastic interest rates, Working paper, New York University. Jensen, M. and W. Meckling. 1976. “Theory of the firm: managerial behavior, agency costs and capital structure.” Journal of Financial Economics 3, 305–360. Ju, N. and H. Ou-Yang. 2006. “Capital structure, debt maturity, and stochastic interest rates.” Journal of Business 79, 2469–2502. Ju, N., R. Parrino, A. Poteshman, and M. Weisbach. 2004. “Horses and rabbits? Trade-off theory and optimal capital structure.” Journal of Financial and Quantitative Analysis 40, 259–281.

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Kane, A., A. Marcus, and R. MacDonald. 1984. “How big is the tax advantage to debt.” Journal of Finance 39, 841–852. Kane, A., A. Marcus, and R. MacDonald. 1985. “Debt policy and the rate of return premium to leverage.” Journal of Financial and Quantitative Analysis 20, 479–499. Kim, I., K. Ramaswamy, and S. Sundaresan. 1993. “Does default risk in coupons affect the valuation of corporate bonds? A contingent claims model.” Financial Management 22, 117–131. Lee, C. F., John C. Lee, and Alice C. Lee. 2009, Financial analysis, planning and forecasting, 2nd ed. Hackensack, NJ, World Scientific Publishers. Leland, H. E. 1994. “Corporate debt value, bond covenants, and optimal capital structure.” Journal of Finance 49, 1213–1252. Leland, H. E. 1998. “Agency cost, risk management, and capital structure.” Journal of Finance 53, 1213–1243. Leland, H. E., 2004. “Prediction of default probabilities in structural models of debt.” Journal of Investment Management 2(2), 5–20. Leland, H. E. and K. B. Toft. 1996. “Optimal capital structure, endogenous bankruptcy, and the term structure of credit spreads.” Journal of Finance 51, 987–1019. Longstaff, F. and E. Schwartz. 1995. “A simple approach to valuing risky fixed and floating rate debt and determining swaps spread.” Journal of Finance 50, 789–819. Mauer, D. and A. Triantis. 1994. “Interactions of corporate financing and investment decisions: a dynamic framework.” Journal of Finance 49, 1253–1277. Mella-Barral, P. 1999. “The dynamics of default and debt reorganization.” Review of Financial Studies 12, 535–578. Mella-Barral, P. and W. Perraudin. 1997. “Strategic debt service.” Journal of Finance 52, 531–566. Merton, R. C. 1974. “On the pricing of corporate debt: the risk structure of interest rates.” Journal of Finance 28, 449–470. Modigliani, F. and M. Miller. 1958. “Corporate income taxes and the cost of capital.” American Economic Review 53, 433–443. Myers, S. C. 1977. “Determinants of corporate borrowing.” Journal of Financial Economics 5, 147–175. Myers, S. C. 1984. “The capital structure puzzle.” Journal of Finance 39, 575–592. Rubinstein, M. and E. Reiner. 1991. “Breaking down the barriers.” Risk Magazine 4, 28–35. Taurén, M. 1999. A model of corporate bond prices with dynamic capital structure, Working paper, Indiana University. Vasicek, O. 1977. “An equilibrium characterization of the term structure.” Journal of Financial Economics 5, 177–188. Wei, D. and D. Guo. 1997. “Pricing risky debt: an empirical comparison of the Longstaff and Schwartz and Merton models.” Journal of Fixed Income 7, 8–28. Weiss, L. 1990. “Bankruptcy costs and violation of claims priority.” Journal of Financial Economics 27, 285–314. Zhou, C. 2001. “The term structure of credit spreads with jump risk.” Journal of Banking and Finance 25, 2015–2040.

Chapter 61

Actuarial Mathematics and Its Applications in Quantitative Finance Cho-Jieh Chen

Abstract We introduce actuarial mathematics and its applications in quantitative finance. First, we introduce the traditional actuarial interest functions and use them to price different types of insurance contracts and annuities. Based on the equivalence principle, risk premiums for different payment schemes are calculated. After premium payments and the promised benefits are determined, actuarial reserves are calculated and set aside to cushion the expected losses. Using the similar method, we use actuarial mathematics to price the risky bond, the credit default swap, and the default digital swap, while the interest structure is not flat and the first passage time is replaced by the time of default instead of the future life time. Keywords Actuarial risks r Interest r Life insurance r Annuity r Risk premium rActuarial reserve rRisky bond price r Credit spread r Credit default swap r Default digital swap

61.1 Introduction In this article, we introduce actuarial mathematics and its applications in quantitative finance. Actuarial risks are the risks that insurance companies face. The famous actuary C.L. Trowbridge coined the term “C-1” risk to denote the risk of asset defaults and decreases in market values of investments (see Panjer et al. 1998, Sect. 4.3). We can generally use the term “C-1” risk to denote the market risk, credit risk, and liquidation risk of assets. Trowbridge used the term “C-2” risk to denote the mortality and morbidity risks. We can use the term “C-2” risk to denote the risk of pricing insufficiency to cover underlying liabilities. Trowbridge used the term “C-3” risk to denote the risk of interest rate fluctuation. Subsequently, the term “C-4” risk is used to denote

C.-J. Chen () University of Alberta, Edmonton, Alberta, Canada e-mail: [email protected]

the operational risk, such as the risk of loss resulting from inadequate or failed internal processes, people and systems, or from external events. The accounting, managerial, social, and regulatory risks are C-4 risks.

61.2 Actuarial Discount and Accumulation Functions Let the bank account process be denoted by fB.t/I t 0g, where B.t/ represents the time-t accumulated value of one dollar deposited at time zero in a riskless account with B.0/ D 1. When B.t/ is a linear increasing function of the time t, then B.t/ D 1 C it; t 0: The interest accrued of this pattern is called simple interest. When lnŒB.t/ is proportional to the time t, then B.t/ can be written as B.t/ D .1 C i /t : The interest accrued of this pattern is called compound interest. The effective rate of interest during the n-th period is defined as the ratio of the capital increment to the investment at the beginning of the period as in D

B.n/  B.n  1/ ; for n 1; n 2 N: B.n  1/

The effective rate of discount can be similarly defined as the ratio of the capital increment to the capital at the end of the period as dn D

B.n/  B.n  1/ ; for n 1; n 2 N: B.n/

When the interest rate is a fixed constant, the subscripts of the interest rate and discount rate can be dropped. The discount factor is defined by 1 vD : 1Ci

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_61, 

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C.-J. Chen

The relationship among i , v, and d can be expressed by the following equations. The first,

The relationship between the interest functions can be given by

iv D d

1 C i D e ı D v1 D .1  d /1

m

n i .m/ d .n/ D 1C D 1 : m n

(61.1)

Equation (61.1) can be interpreted as the interest i of one dollar for one term, if payable at the beginning of a term, equals i times the discount factor v. This amount is called the discount rate d . The second equation,

As m approaches infinity, lim

m!1

1C

i .m/ m

m

D lim e i

.m/

m!1

D

ı

1 i D1 D 1  v: d D iv D 1Ci 1Ci

e . It can be shown that (61.2)

lim i .m/ D ı;

m!1

Equation (61.2) can be interpreted as the discount rate d is the difference between one dollar payable at time zero and time-zero value of one dollar payable at time one. When we borrow money from a bank, we are usually given the annual percentage rate (APR) instead of the effective rate. This interest is usually paid more frequently than once per period. This interest rate is called the nominal rate of interest and is denoted by i .m/ if the interest is payable m times per period. The nominal rate i .m/ is divided by m then is compounded m times for a period as

m i .m/ 1Ci D 1C : m When a credit card company charges an APR of 18% convertible monthly, the effective rate of interest been charged 12   1 D 19:56%, which is higher than the is i D 1 C 0:18 12 rate that we are told. The effective rate of discount can be defined similarly as

m d .m/ 1d D 1 : m The rate of the instantaneous interest movement at a specific time t is defined by ıt D

B 0 .t/ d ln B .t/ D ; t 0; B .t/ dt

where ı, is called the force of interest in actuarial science and is called the short rate in quantitative finance. The bank account process can be written as Z

t

B.t/ D exp

ıs ds :

0

If the force of interest is a constant, then the bank account process equals B.t/ D e ıt :

and lim d .m/ D ı:

m!1

An annuity is consisted of a series of payments made at equal time intervals. The present value of an annuity is denoted by a. The present value of an annuity paying $1 at the beginning of each year for n years with interest rate i has a standardized actuarial notation as 2 n1 aR nji D N D 1CvCv C  Cv

1  vn 1  vn D : 1v d

(61.3)

This annuity is called an n-year annuity-due in actuarial science. An annuity-due has payments due at the beginning of each term. The accumulated value of the n-year annuitydue is n n1 C .1 C i /n2 C    C .1 C i /1 sRnji N D .1 C i / C .1 C i /

D

.1 C i /n  1 D .1 C i /n aR nji N : d

It equals the present value of the annuity in Equation (61.3) times the accumulation factor .1 C i /n . The present value of an annuity paying $1 at the end of each year for n years with interest rate i has a standardized actuarial notation as 2 3 n anji N D vCv Cv CCv D

1  vn v.1  vn / D : 1v i (61.4)

This annuity is called an n-year annuity-immediate. The accumulated value of the annuity-immediate is n1 snji C .1 C i /n2 C .1 C i /n3 C    C 1 N D .1 C i /

D

.1 C i /n  1 D .1 C i /n anji N : i

Comparing Equation (61.3) with Equation (61.4), we can find that both present values share the same numerator but

61 Actuarial Mathematics and Its Applications in Quantitative Finance

have different denominators. The n-year annuity-due has d in the denominator while the n-year annuity-immediate has i in the denominator. An annuity sometimes pays more frequently than one time per year. Let a $1 annual nominal payment be payable at the beginning of each m-th of the year with each payment equal 1/m, where m 1 and m is an integer (m D 12, 4, and 2 representing monthly, quarterly, and semi-annually payments, respectively). The number of payment periods in years is n. The present value of this annuity-due is .m/

D aR nji N

1 1 2 1 1 C v m C v m C    C vn m m

955

In general, if an annuity provides payment of g.t/ at time t for a period of n with non-constant force of interest, the present value of the annuity is Zn aD

g.t/ dt D B.t/

0

Zn



g.t/e

Rt

ıs ds

0

dt:

0

For more complicated actuarial interest functions, please consult Kellison (1991).

61.3 Actuarial Mathematics of Insurance

1

D

1  v m nm 1  vn 1 1  vn h i

D D : 1 1 m d .m/ 1  vm m 1  .1 C i / m (61.5)

This m-th payment n-year annuity-due in Equation (61.5) is similar to the n-year annuity-due in Equation (61.3) except the denominator of the present value is replaced by d .m/ . The accumulated value of this annuity-due is .m/

.m/

D .1 C i /n aR nji D sRnji N N

.1 C i /  1 : d .m/ n

If this annuity is payable at the end of each m-th of a year, its actuarial present value is

.m/

anji N



1 1

v m 1  v m nm 1 2 1 1 v m C v m C    C vn D

D 1 m m 1  vm

A life insurance policy provides financial security to offset the financial impact caused by unexpected death. Payments of a life insurance policy are triggered by the termination of the life of the insured. Let a newborn’s age-at-death be denoted by X . The cumulative distribution function (c.d.f.) is FX .x/ D PrfX  xg; x 0; which is the probability that the newborn will die before age x. The probability that the newborn will survive age x is called the survival function of X and is denoted by SX .x/ D PrfX > xg D 1  FX .x/; x 0: If a newborn survives age x, we use (x) to denote a lifeage-x. The future life time of (x) is denoted by T .x/. The probability that (x) will die by age x C t is t qx

1

D

v m .1  vn / 1  vn 1  vn iD h D .m/ : 1 1 i d .m/ =v m m 1  .1 C i / m

D PrfT .x/  tg D Prfx < X  tjX > xg D

SX .x/  SX .t/ SX .x/

D

FX .t/  FX .x/ : 1  FX .x/

The accumulated value of this annuity-immediate is .m/ snji N

D .1 C

.m/ i /n anji N

.1 C i /n  1 D : i .m/

When m approaches infinity, the annuity provides continuous payments. The present value of a continuous annuity is .m/

anj D lim anji D lim N m!1

1v 1v : D .m/ d ı n

m!1

vm  vnCm : ı

t px

D PrfT .x/ > tg D 1  t qx :

n

If an annuity does not provide any payments for the first m years then provides payments for a period of n years after the first m years, this m-year deferred n-year continuous annuity has a present value of mjaN njN D vm aN nj D

The probability that (x) will survive age x C t is

Theprobabilitythat(x)willdiebetweenagexCt andxCt Cuis t ju px

D Prft < T .x/  t C ug D t Cu qx  t qx D t px  t Cu px D t px u qxCt :

(61.6)

Equation (61.6) can be interpreted as the probability that (x) will die between age x Ct and x Ct Cu equal the probability that (x) will survive age xCt times the probability that (xCt) will die before or equal to age x C t C u.

956

C.-J. Chen

The force of mortality is defined by Prfx < X  x C xjX > xg

x!0

x fX .x/ x=SX .x/ fX .x/ d ln SX .x/ D lim D D :

x!0

x SX .x/ dx

.x/ D lim

The benefit of an insurance contract is a function of the future life of an insured. Let the benefit of an insurance contract be denoted by bt when T .x/ D t; t 0, for an insured (x). Let the present value of the benefit be denoted by Z and Z D vt bt if the interest is assumed to be constant. The expected present value of the insurance is Z1

Therefore,

EŒZ D ZxCt



.s/ds D ln x

SX .x C t/ D ln t px ; SX .x/

and t px

De



xCt R

.s/ds

x

:

The c.d.f. of T .x/ is

0

d @1  e d t qx D fT .x/ .t/ D dt D e



xCt R

xCt R

1

x

.s/ds

Zn e

A

2

h i A1 xWnj D EŒZ 2 D E v2T .x/ IfT .x/  ng Zn

Z1 t t px .x C t/dt

0

where the second moment of the present value equals the first moment calculated using the force of interest of 2ı. The variance of the present value of the insurance is

0

D

2 VarŒZ D 2 A1 xWnj  A1 xWnj :

Z1 ST .x/ .t/dt D

0

t px dt: 0

It is also called the complete-expectation-of-life. Let bT .x/c be the integer part of T .x/ [or the largest integer less than or equal to T .x/] so T .x/bT .x/c represents the fractional part of T .x/. The expected value of bT .x/c is called the curtateexpectation-of-life and equals ex D E fbT .x/cg D

1 X

k  k px  qxCk D

kD0

D k.k px / j1 0 

1 X kD0

1 X

For a term insurance contract, the “1” on the top of x represents that the $1 benefit is payable only on the death of (x). If the “1” is on top of nj, it means that the policy provides survival benefit of $1 at the insured’s age x C n if the insured (x/ survives age x C n. This insurance is called an n-year pure endowment insurance and is denoted by Z1

k .k px /

1 AxWnj

1 X kD1

e n fT .x/ .t/dt

D EŒv IfT .x/ > ng D n

n

kD0

.kC1 px / D

e .2ı/t fT .x/ .t/dt D A1 xWnj@2ı0

D

ı

Z1

vt t p x .x C t/dt:

The second moment of the present value of the payment is

Œ.x C t/ D t px .x C t/:

ex D E ŒT .x/ D

fT .x/ .t/dt D 0

The expected value of T .x/ is denoted by e x and equals ı

ıt

0

dt

x

0

Zn D

.s/ds

zt px u.x C t/dt:

An n-year term life insurance pays death benefit of $1 at the moment of death if the death is at or before the age x C n of (x). The present value of the payment is Z D vT .x/ IfT .x/  ng where I fT .x/  ng is an indicator function taking value of 1 if T .x/  n and taking value of 0 otherwise. The expected present value of the payment is

The probability density function (p.d.f.) is 

zt fT .x/ .t/dt D

  A1 xWnj D EŒZ D E vT .x/ IfT .x/  ng

FT .x/ .t/ D PrfT .x/  tg D t qx :

0

Z1

Z1 k px :

D

vn t p x .x C t/dt D vn n px : n

61 Actuarial Mathematics and Its Applications in Quantitative Finance

The variance of the present value of the payment is Z1 v2n t p x .x C t/dt  Œvn n px 2

VarŒZ D n

2 1 1 D v2n n px  v2n n px2 D 2 AxWnj  AxWnj :

957

When the benefit is payable at the end of the m-th of the year of death, the expected present value of a whole life insurance is   dmT.x/e .m/ Ax D EŒZ D E v m

D

An n-year endowment insurance provides both death and survival benefit of $1. The payment of the insurance is bt D 1. The present value of the insurance is Z D vT .x/^n . The expected present value of an n-year endowment insurance is

kC 1 m1 X X Z

j C1 m

vkC

AxWnj D

1 C AxWnj :

The whole-life insurance provides death benefit over the whole life span. The expected present value of the whole life insurance is Ax D

1 AxW1j

Z1 D EŒv

T .x/

e ıt fT .x/ .t/dt

D 0

Z1 D

vt t p x .x C t/dt:

D

1 m1 X X

vkC

j C1 m

If the insurance benefit is payable at the end of the year of death instead of the moment of death, the actuarial symbol is different. The expected present value of the benefit for an n-year term insurance providing death benefit of $1 at the end of the year death is h i A1xWnj D EŒZ D E vdT .x/e IfT .x/  ng

kD0

where the ceiling of T .x/ is defined as the smallest integer that is greater than or equal to T .x/. For a whole life insurance with benefit payable the end of the year of death, the expected actuarial present value is kC1 1 Z i X h dT .x/e D vkC1 fT .x/ .t/dt Ax D E v

D

1 X kD0

vkC1 k px  qxCk :

px  1 qxCkC j : m

m

If we assume a uniform distribution of deaths (U.D.D.) in each year of age, the survival function follows SX .n C s/ D .1  s/SX .n/ C sSX .n C 1/; where 0  s < 1 and n is an integer. Let the fractional part of T .x/ be denoted by S D T .x/  bT .x/c. Under the U.D.D. assumption, S follows a uniform Œ0; 1/ distribution. The life expectancies follow

Under the U.D.D. assumption, the expected actuarial present values of the whole life insurance follow i h   Ax D E vT .x/ D E vdT .x/e.1S / i  h  D E vdT .x/e E .1 C i /1S Z1 D Ax

kC1 n1 Z n1 X X vkC1 fT .x/ .t/dt D vkC1 k px  qxCk ; D

kD0t Dk

j

kC m

1 ı e x D EŒT D E ŒbT c C S D E ŒbT c C E ŒS D ex C : 2

0

kD0 k

fT .x/ .t/dt

kD0 j D0 j t DkC m

kD0 j D0 1 AxWnj

j C1 m

e ıu du D Ax

i eı  1 D Ax ; ı ı

0

where U D 1  S also has a uniform distribution over the unit interval and     dmT.x/e dmT.x/e .m/ dT .x/edT .x/eC m m Ax D E v DE v  h i  dmT.x/e D E vdT .x/e E .1 C i /dT .x/e m Z1 .1 C i /dt e

D Ax

dmte m

dt

0

D Ax

m1 X j D0

.j C1/=m Z

.1 C i /dt e j=m

dmte m

dt

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C.-J. Chen

8 ˆ Z2=m < Z1=m dmte dt e dmte m dt C D Ax .1 C i / .1 C i /dt e m dt ˆ : 0

1=m

m=m Z

.1 C i /dt e

CC

dmte m

dt

.m1/=m

1=m

m=m Z

CC

m

.1 C i /1 m dt

.m1/=m

9 > = > ;

D

i .m/

at j ST .t/j1 0

C

Z1 d a

tj

dt

ST .t/dt

0

Z1

> ;

1 2 1 n D Ax  .1 C i /1 m C .1 C i /1 m m o m C    C .1 C i /1 m   .1 C i /m=m  1  D Ax   m .1 C i /1=m  1

i

ax D

9 > =

8 ˆ Z2=m < Z1=m 1 2 1 m D Ax .1 C i / dt C .1 C i /1 m dt ˆ : 0

Using integration by parts, we can simply it as

D

Z1 vt ST .t/dt D

0

Z1 vt t px dt D

0

0

where t Ex D vt t px . If an annuity provides a maximum payment period of n years, it is called an n-year temporarily annuity. The present value of the payment is Z D aT .x/^nj ; where T .x/^n D min.T .x/; n/. The expected present value of the payment is

axWnj

h i Zn D E aT .x/^nj D at j  t px .x C t/dt C anjn px : 0

Applying integration by parts, we obtain Zn axWnj D

a t j t px jn0

C

Ax :

Zn v  t px dt C anj n px D

Z1 Ax D

e ıt e t dt D

 : Cı

0

0

If an annuity provides payments as long as the insured is still alive with guaranteed payments for n years, it is called an n-year certain and life annuity. The present value of the payment is Z D aT .x/_nj , where T .x/ _ n D max.T .x/; n/. The expected present value of the payment is Z D aT .x/_nj , where i

h

For more complicated insurance products, please consult Bowers et al. (1997).

vt  t px dt:

t

0

If we assume the force of mortality is constant, then the expected present value of a whole life insurance is

t Ex dt;

Zn

axWnj D E aT .x/_nj D

a nj  t px .x C t/dt 0

Z1 C

61.4 Actuarial Mathematics of Annuity A whole life annuity provides benefits while the insured is alive. The present value of the payment is aT .x/j . The expected present value of the annuity is Z1 i Z1 h ax D E aT .x/j D a t j fT .t/dt D at j  t px .x C t/dt: 0

0

at j  t px .x C t/dt n

Z1 D anj  n qx  at j 

1 t px j n

C

vt  t px dt n

Z1 D anj  n qx C anj  n px C

vt  t px dt n

Z1 D anj C

vt  t px dt: n

(61.7)

61 Actuarial Mathematics and Its Applications in Quantitative Finance

959 n i h X D aR nj  aR nC1j n px C 1 C vk k px

An n-year deferred whole life annuity provides benefits if the insured is alive but without any payments for the first n years. The present value of the payment is Z D vn aT .x/nj IfT .x/ > ng. The expected present value is

kD1

D vn n px C v0 0 px C

Z1 nj ax

D

D

n

n1 X

vn asj nCs px .x C n C s/ds

If an annuity provides survival benefits at the end each year with a maximum payment period of n years, it is called an n-year temporarily life annuity-immediate. The present value of the benefits is Z D abT .x/c^nj . The expected present value is i h axWnj D E abT .x/c^nj

0

Z1 vn asj n px  s pxCn .x C n C s/ds 0

Z1 D v n px

asj s pxCn .x C n C s/ds

n

0

D

D n Ex axCn :

(61.8)

The relationship between the n-year deferred whole life annuity in Equation (61.8) and the n-year certain and life annuity in Equation (61.7) is i h h i axWnj D E aT .x/_nj D E anj C aT .x/nj IfT .x/ > ng D anj C nj ax :

i h aR xWnj D E aR dT .x/e^nj D

n1 X

aR kC1j k px  qxCk C aR nj n px :

kD0

By applying summation by parts ( n P

n P

g.x/ f .x/ D f .x/

f .x C 1/ g.x/, where f .x/ D f .x C

xD0

n1 X

akj k px  qxCk C anj n px D

n X

vk k px :

kD1

If an annuity of $1 per year provides benefits of 1=m at each m-th of a year, it is called an m-thly payment n-year temporarily life annuity-due. The present value is Z D .m/ aR mT.x/ ˇˇ . The expected present value is d

e

m

^nˇˇ

 D E aR dmT.x/e m

 ^nj

n1 m1 X X 1 j vkC m kC j px : D m m j D0 kD0

The relationship between an n-year temporarily life annuitydue and an n-year endowment insurance is aR xWnj

h

i

"

1  vdT .x/e^n D E aR dT .x/e^nj D E d   1  AxWnj 1  E vdT .x/e^n D : D d d

#

xD0

1/  f .x//, we obtain aR xWnj D

n1 X kD0

.m/ aR xWnj

If an annuity provides survival benefits at the beginning each year with a maximum payment period of n years, it is called an n-year temporarily life annuity-due. The present value of the benefits is Z D aR dT .x/e^nj . The expected present value is

 g.x /jnC1 0

vk k px :

kD0

Z1

D

vk k px

kD1

vn at nj t px .x C t/dt

D

n X

aR kC1j .k px / C aR nj n px

61.5 Actuarial Premiums and Actuarial Reserves 61.5.1 Actuarial Premiums

kD0 n1 ˇn X ˇ D aR kC1j .k px /ˇ  .kC1 px / aR kC1j C aR nj n px 0

D aR nC1j n px C 1 C

kD0 n1 X kD0

kC1 px v

kC1

C aR nj n px

In previous sections, we obtain expected present values of insurances and annuities. These are expected costs for an insurer. In return, an insurer receives risk premiums to cover these losses. In this section, we discuss how to use the equivalence principle to obtain premiums.

960

C.-J. Chen

Because an insurer receives premiums and pays claims, the present value of the loss of an insurer is L D present value of claims present value of premium incomes: By letting EŒL D 0, the expected claim is equal to the expected gain. This is called the equivalence principle. The premium obtained is called the pure premium or the benefit premium and is denoted by P . In practice, the insurer charges more than P in order to cover expenses, commissions, fees, and losses due to uncertainties. The extra charge is called the security loading. The gross premium, G, usually follows

Consider that an insured purchases an n-year term insurance whose $1 death benefit is payable at death, by paying premiums at the beginning of each year when the insured is still alive with a maximum payment period of h, h  n. The premium is AN N N N / D xWnj : h P .AxWnj aR xWhjN For an n-year term insurance whose death benefit is payable at death purchased by continuous premium payments when the insured is still alive with a maximum payment period of n, the premium is PN .ANxWnjN / D

G D .1 C /P; where  is called the relative security loading and P is called the security loading. Consider that an insured purchases a whole life insurance of $1 payable at the end of the year of death by paying premiums at the beginning of each year when the insured is still alive. The present value of the loss of the insurance company is L D vdT .x/e  P aR dT .x/ej : Based on the equivalence principle, we have

0 D E.vdT .x/e /  P E aR dT .x/ej D Ax  P aR x :

Px D

L D vT .x/^n 

:

P

maR dT .x/e^hj : m

The annual premium is

Ax : aR x

hP

Consider that an insured purchases an n-year term insurance whose death benefit of $1 is payable at the end of the year of death, by paying premiums at the beginning of each year when the insured is still alive with a maximum payment period of h, h  n. The present value of the loss of the insurance company is

aN xWnjN

When the payment period and insurance period are the same, the subscript h in front of P can be neglected. In practice, premiums are very often paid by monthly payments. As men.m/ represents the expected tioned in the previous section, aR xWnj present value of an annuity of $1 per year providing benefits of 1/m at each m-th of a year for n years. Consider an n-year term insurance of $1 payable at death with continuous m-thly payments of Pm when the insured is still alive with a maximum payment period of h, h  n. The present value of the loss of the insurance company is

(61.9)

After rearrangement of Equation (61.9), we denote this specific premium by Px as

ANxWnjN

.m/

.ANxWnjN / D

ANxWnjN .m/ N xWhj

aR

:

At each m-th of the year, the insured pays periodic level premium of

hP

.m/ .A / xWnj

m

.

61.5.2 Actuarial Reserve

L D vdT .x/e^n  P aR dT .x/e^hj : Based on the equivalence principle, we have



0 D E vdT .x/e^n  P E aR dT .x/e^hj D AxWnj  P aR xWhj : This specific premium is denoted by h PxWnj and equals h PxWnj

D

AxWnj aR xWhj

:

The equivalence principle holds at time 0. As time moves on, the equivalence principle does not necessarily hold. Let us consider the loss of the insurer at time t > 0 for selling a whole life insurance with continuous premium payments. If the individual died at time t  , with 0 < t  < t, the loss at time t  is the benefit payment of $1 minus the premium income as t L

D 1  PN .ANx /Nst j :

61 Actuarial Mathematics and Its Applications in Quantitative Finance

The time-t value of the loss is tL

  D .1 C i /t t  PN .ANx /  sNt N j  .1 C i /t t :



The loss or gain is realized at time t . There is a loss at time t  if 1 > PN .ANx /Nst  j and a gain if 1 < PN .ANx /s t  j . Let us consider the loss for an insured .x/ who survives time t. At time t, the insured attains age x C t. The insurer is responsible to pay for the future death benefit and will receive premiums from time t to the death of the insured. The loss evaluated at time t is tL

.Ax / D EΠt Lj T .x/ > t D AxCt  P .Ax /axCt : (61.10)

If a n-year term insurance is purchased by h-year payments, for 0 < t < h < n, the time-t loss is tL

D vT .xCt /^.nt /  h P .AxWnj /aT .xCt /^.ht /j:

The reserve is h t

If the insurance AxWnj is purchased by h-year payments at the beginning of each year, the reserve becomes h t

V .ANxWnjN / D ANxCt Wnt j  h P .AxWnj /aR xCt Wht j :

If the insurance AxWnj is purchased by h-year payments at the beginning of each year, the reserve becomes h t

VxWnj D AxCt Wnt j  h PxWnjN aR xCt Wht j :

61.6 Applications in Quantitative Finance

D vT .xCt /  P .Ax /aT .xCt /j :

To cushion the possible loss in the future, the insurer has to put aside capital of an amount of EŒt Lj T .x/ > t . This expected loss is called the benefit reserve and is tV

961

V .AxWnj / D EΠt Lj T .x/ > t D AxCt Wnt j  h P .AxWnj /axCt Wht j :

Actuarial mathematics can be used not only for insurance valuation, but also for some applications in quantitative finance. Li (1998) use actuarial mathematics to obtain the term structure of credit risk. Consider a risky zero coupon bond paying $1 at maturity T . We assume that there exists a risk neutral probability measure Q. Let the time-t price of this risky bond be denoted by V .t; T /. Let time-t price of the riskless zero-coupon bond be denoted by P .t; T /. Let the time of default be denoted by . Assume that the bondholder will receive a fractional amount ı at maturity if default occurs. This recovery scheme is referred to as the recovery-of-treasury-value scheme. Under the risk neutral measure and the RTV scheme, the time-t price of this risky bond can be written by the Jarrow and Turnbull (1995) pricing formula as 

(61.11) V .t; T / D

In Equations (61.10) and (61.11), reserves are obtained by looking prospectively into the future on the loss minus the income. This method is called a prospective formula. Alternatively, we can look retrospectively into past on the contributions that the insured has made minus the insurance protection that the insured has used. The reserve based on the retrospective formula is V .AxWnj / D h P .AxWnj / t h

axWt j t Ex

1



AxCt Wnt j t Ex

D h P .AxWnj /s xWt j 

t Ex

; (61.12)

where h PN .ANxWnj /s xWt j is the accumulated premium contribu1

A

 B.t/ .ı  If  T g C 1  If > T g/ ; B.T /

Q

where Et represents time-t expected value under the Q-measure based on information available up to time t. If we assume the bank account process is independent of the default process, we have Q

V .t; T / D P .t; T /Et Œ.ı  If  T g C 1  If  T g/ : (61.13) After rearrangement of Equation (61.13), the risky bond price becomes

1

AxCt Wnt j

Q Et

Wnt j is the insurance protions by the survivorship and xCt t Ex tection that has been used by the survivorship. It is not difficult to find out that Equation (61.11) equals Equation (61.12).

V .t; T / Q D Et Œ.1  .1  ı/If  T g/ D 1.1ı/.T t / qt : P .t; T / The probability of default becomes .T t / qt

D

P .t; T /  V .t; T / : .1  ı/P .t; T /

(61.14)

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C.-J. Chen



If we know ı; P .t; T1 /; P .t; T2 /;    P .t; Tn / and V .t; T1 /; V .t; T2 /;    V .t; Tn / with that t  T1  T2      Tn , we can obtain .T1 t / qt ; .T2 t / qt ;    ; .Tn t / qt . Consider that .T2 t / qt

D .T1 t / qt C .T1 t / pt .T2 t T1 / qt CT1 :

(61.15)

Equation (61.15) shows that the probability of default .T2 t / qt for the time period .t; T2 equals the probability of default for the time period .t; T1 plus the probability of survival for the time period .t; T1 times the probability of default for the time period .T1 ; T2 . Recursively using this method, we can obtain the whole term structure of default for the whole time period .t; Tn . For example, assume that a firm is estimated to have a recovery rate of 60% if default were to occur. The riskless zero-coupon bond prices are given as P .2008; 2009/ D 0:9792; P .2008; 2010/ D 0:9550, P .2008; 2011/ D 0:9221; and P .2008; 2012/ D 0:8905. The zero-coupon risky bond prices of a public firm are quoted as V .2008; 2009/ D 0:9654; V .2008; 2010/ D 0:9268; V .2008; 2011/ D 0:8823; and V .2008; 2012/ D 0:8390: Based on Equation (61.14), the probability of default can be obtained as 1 q2008

D 0:03523; 2q2008 D 0:0738;

3 q2008

D 0:1079 and 4 q2008 D 0:1446:

Based on Equation (61.15), we can obtain the 1-year default probability for the time period (2008, 2012] as 1 q2008

D 0:03523; 2q2009 D 0:03999;

3 q2010

D 0:03680 and 4 q2011 D 0:41111:

.m/ R bm.T t1 /c : .t1 t /j a tW m

Let us consider the put-call parity here. Let a European call and a European put on a non-dividend-paying stock have the same strike price K with maturity T . Let the time-t price of the European call and the European put be denoted by ct and pt . The relationship between ct and pt is  ct  pt D

B.t/.ST  K/C B.t/.K  ST /C  B.T / B.T /

 (61.16)

D St  K P .t; T /; where AC D A if A 0 and AC D 0 if A < 0. If the stock provides a dividend yield of q, the put-call parity in Equation (61.16) becomes ct  pt D St e q.T t /  K P .t; T /:

(61.17)

Assume that the stock provides periodic dividends m times per year with each dividend equal d . Assume the current time is t and the next coupon payment will be made at t1 . The putcall parity in Equation (61.16) becomes ct  pt D St  d m

.m/

R .t1 t /j a

.m/

ˇ

bm.T t1 /c ˇ ˇ m

 K P .t; T /; (61.18)

h

B.t / / B.t / D m1 B.t C B.t1B.t C1=m/ C B.t1 C2=m/ 1/ i / C    C B.bm.TB.t . t1 /c=m/ We can extend actuarial equivalence principle to pricing credit default swap. Let the credit default swap issuer provide the loss given default of the reference party to the insured for a period of n. The insured pays periodic payments for a maximum period of h for this protection. If default occurs, the issuer pays the insured an amount of 1-random recovery rate (R) and the insured no longer has to pay the payment. We assume that the random recovery rate R is independent of the bank account process. Therefore, the loss of the issuer is

where

R .t1 t /j a

ˇ

bm.T t1 /c ˇ ˇ m

B.t/ I.  t C n/  P aR  ^hj : B./

By applying equivalence principle, we have Q

Q

1

0 D Et ŒL D .1  Et ŒR /At Wnj  P aR t Whj :

(61.19)

Rearranging Equation (61.19), we can obtain the periodic payment 1 Q .1  Et ŒR /At Wnj : P D aR t Whj

V .t; T / D P .t; T /Œ1  .1  ı/.T t / qt

Q Et

B.t/.ST  K/ B.T /

L D .1  R/

Assume that the bond provides periodic coupons m times per year with each coupon equal c. Assume the current time is t and the next coupon payment will be made at t1 . The pricing formula of Equation (61.14) is then modified to

Cc m

Q

D Et



If the issuer pays the insured at the end of the period if default occurs, then the premium is Q

P D

.1  Et ŒR /A1

t Wnj

aR t Whj

:

(61.20)

For example, we assume the insured pays two annual premiums to obtain a 4-year protection. Based on the default

61 Actuarial Mathematics and Its Applications in Quantitative Finance

963

probabilities of the previous example, the periodic premium can be obtained using Equation (61.20) as

Q

P D

.1  E2008 ŒR /A1

2008W4j

aR 2008W2j

B.2008/ B.2008/ B.2008/ B.2008/ q2008 C p2008 q2009 C 0:4

2 p2008 q2010 C 3 p2008 q2011 B.2009/ B.2010/ B.2011/ B.2012/ D B.2008/ 1C p2008 B.2009/ D

0:4 Œ0:9792  0:03523 C 0:9550  .1  0:03523/  0:03999 1 C 0:9792  .1  0:03523/ C

D

0:4 ŒC0:9221  .1  0:07382/  0:03680 C 0:8905  .1  0:10790/  0:04111 1 C 0:9792  .1  0:03523/

0:4  0:135441 D 0:027859: 1:9447

Using actuarial mathematics, we can obtain clean and nice closed-form solutions for some problems in quantitative finance.

method. In Equations (61.10) and (61.11), reserves are calculated prospectively as the expected current value of future liabilities. In Equation (61.12), the reserve is calculated retrospectively as the expected current value of the unused contributions. Both methods reach the same result. Actuarial mathematics can be used not only for insurance valuation, but also for some applications in quantitative finance. When we replace the time of death by the time of default and assume that interest rate is not flat, we can use actuarial mathematics to obtain the term structure of credit risk from zero-coupon bonds (like Li 1998). The bond price for coupon-paying bond is also obtained in a similar manner in this chapter. In the last part of this chapter, we used actuarial mathematics to price the credit default swap and default digital swap. In general, actuarial mathematics and quantitative finance could be integrated in more areas.

61.7 Conclusion

References

In this chapter, we introduce actuarial mathematics and actuarial interest functions, insurance, annuities, premiums, and reserves. The pure premium is calculated based on the equivalence principle. The real premium that the insurance company charges is called the loaded premium, which is the pure premium plus a loaded amount. After premiums are collected by the insurer, the reserve is set aside to cushion the expected loss. We introduce two methods of finding the actuarial reserve: the prospective method and the retrospective

Bowers, N. L. et al. 1997. Actuarial mathematics, 2nd Edition, Society of Actuaries, Itasca, IL. Jarrow, R. A. and S. M. Turnbull. 1995. “Pricing derivatives on financial securities subject to credit risk.” Journal of Finance 50, 53–85. Kellison, S. G. 1991. The theory of interest, 2nd Edition, IRWIN, Homewood, IL. Li, D. 1998. Constructing a credit curve, Credit Risk: Risk Special Report, November, 40–44. Panjer, H. H. et al. 1998. Financial economics, The Actuarial Foundation, Schaumburg, IL. Schonbucher, P. J. 2003. Credit derivatives pricing models, Wiley, New York.

For a default digit swaps, (see Schonbucher 2003, Sect. 5.8.1) the insured will receive a predetermined fixed amount s if default occurs. The periodic payment becomes P D

sA1

t Wnj

aR t Whj

:

If a default digit swap pays $0.50 for per dollar exposure, then the periodic payment is P D

0:5  0:135441 D 0:034823: 1:9447

Chapter 62

The Prediction of Default with Outliers: Robust Logistic Regression Chung-Hua Shen, Yi-Kai Chen, and Bor-Yi Huang

Abstract This paper suggests a Robust Logit method, which extends the conventional logit model by taking outliers into account, to implement forecast of defaulted firms. We employ five validation tests to assess the in-sample and out-of-sample forecast performances, respectively. With respect to in-sample forecasts, our Robust Logit method is substantially superior to the logit method when employing all validation tools. With respect to the out-of-sample forecasts, the superiority of Robust Logit is less pronounced. Keywords Logit r Robust Logit r Forecast r Validation test

62.1 Introduction In recent years a large number of researchers and practitioners have worked on the prediction of business defaults. This prediction is important because not only can it reduce nonperforming loans but it can help to determine capital allocation. As required by the Basel Committee on Banking Supervision, the prediction of business default is the first step to fulfill the requirement of the internal rating-based (IRB) of Basel II. Large banks are therefore eager to develop their default prediction system to monitor credit risk. One of the issues regarding credit risk assessment is the model or method of default prediction used. Altman and Saunders (1998) have traced development of the literatures in risk measurement for 20 years. Currently, the common model used includes discriminant analysis (Altman 1968), logit and probit models (Ohlson 1980; Westgaard and Wijst 2001), multiC.-H. Shen () Department of Finance, National Taiwan University, Taipei, Taiwan e-mail: [email protected] Y.-K. Chen Department of Finance, National University of Kaohsiung, Kaohsiung, Taiwan e-mail: [email protected] B.-Y. Huang Department of Business Management, Shih Chien University, Taipei, Taiwan e-mail: [email protected]

group hierarchy model (Doumpos et al. 2002; Doumpos and Zopounidis 2002), neural network (Atiya 2001; Piramuthu 1999; Wu and Wang 2000) and option type models such as KMV to name a few. The works of Lennox (1999) for the comparison of the first three models and Dimitras et al. (1996) for the discussion of the first five models have become the standard reference for the financial research. While there are many methods to estimate the probability of default, none of them have taken the outliers into account when there is a discrete dependent variable. Outliers that can seriously distort the estimated results have been well documented in the conventional regression model. For example, Levine and Zervos (1998) employ 47 countries data and confirm that the liquidity trading is positively related to economic growth. Zhu et al. (2002), however, reject this positive relation when they employ the econometric methods to minimize the outlier distortion effects caused by Taiwan and South Korea. Although methods and applications that take outliers into account are well known when the dependent variables are continuous (Rousseeuw 1984; Rousseeuw and Yohai 1984), few have conducted empirical studies when the dependent variable is binary. Atkinson and Riani (2001), Rousseeuw and Christmann (2003), and Flores and Garrido (2001) have developed the theoretical foundations as well as the algorithm to obtain consistent estimator in logit models with outliers, but they do not provide applied studies. If outliers indeed exist when the dependent variable is binary, the conventional logit model might be biased. The aim of this paper is to predict default probability with the consideration of outliers. This is a direct extension of the logit estimation method and is referred to as the Robust Logit model hereafter. We apply the forward search method of Atkinson and Cheng (2000) and Atkinson and Riani (2001) to Taiwan data. To the best of our knowledge, our paper is the first to use the Robust Logit model for actual data. Once estimated coefficients are obtained, we assess the performances of the logit and the Robust Logit methods by using five validation tools; that is, contingency table (cross-classifications table), cumulative accuracy profile (CAP), relative or receive operation characteristics (ROC), Kolmogorov-Smirnov (KS), and Brier score.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_62, 

965

966

The paper proceeds as follows. In addition to the first section, the next section provides literatures of the logit and the Roubst Logit regression model. Section 62.3 introduces the five validation models. Section 62.4 discusses the empirical model and estimated results. Section 62.5 provides the conclusion.

62.2 Literature Review of Outliers in Conventional and in Logit Regression Conventional linear regression analysis taking outliers into account has been utilized since 1960. The methods, such as Least Median of Squares (LMS), Least Trimmed Squares (LTS) (Rousseeuw 1983, 1984), which exclude the effect of outliers on linear regression, are now standard options in many econometric softwares. Until 1990, however, the literature was slow in the consideration of outliers when the logit model is involved. Furthermore, most development tends to center on the theoretical derivations of outliers in logit method in the fields of statistics and actuaries.

C.-H. Shen et al.

62.2.2 Outliers in Logit Regression: Robust Logistic Regression Our Robust Logistic (RL) regression is based on Atkinson and Riani (2001) forward approach, which include five steps. <Step 1> Choice of the initial subset of observations Randomly choose k C 1 observations where k C 1 is equal to one-third of total observations as our starting sample size.1 The corresponding estimated coefficient vector of logit method is denoted as ˇO .kC1/ and the

value predicted .kC1/ O , where of the observed company is yOi D F xi ˇ i D 1:::;N. <Step 2> Obtain the median of the errors Calculate the probability of accurate rate of the prediction of the default companies as p .kC1/;i : ( p .kC1/;i

D yOi ; if yi D 1 D 1  yOi ; if yi D 0:

Corresponding to the accurate rate of the prediction, the probability of the nonaccurate rate of the prediction e .kC1/;i is calculated as e .kC1/;i D 1  p .kC1/;i of all observations. Then take an ascending order of all e .kC1/;i , i.e., e .kC1/;1 < e .kC1/;2 <    < e .kC1/;N ;

62.2.1 Outliers in Conventional Regression Cook and Weisberg (1982) suggest a backward procedure to exclude outliers in the conventional linear model. By using the whole sample as the first step, Cook and Weisberg (1982) detect one outlier and remove it. Then, they go on detecting the second outlier and removing it, followed by the third and so on. Repeating this step, they remove all outliers. While the backward approach appears straightforward, it, however, has suffered from a masking effect. Namely, the statistical properties of the detected outliers, are affected by the outliers remaining in the sample. Barrett and Gray (1997) and Haslett (1999) suggest a multiple outliers’ method to overcome this problem but the speed of detection is slow. Atkinson (1985, 1994) proposes a forward search algorithm (forward approach), which uses only a small subset without outliers as the first step. Then, another subset is added and examine in the next observation. Continuing this step, he claims that the forward approach can remove outliers without the masking effect. When the number of outliers is not known, Atkinson and Riani (2006) indicate a method that makes efficient use of the individual simulations to derive the simultaneous properties of the series of tests occurring in the practical data-analytical case. Riani and Atkinson (2007) provide easily calculated bounds for the statistic during the forward search and illustrate the importance of the bounds in inference about outliers.

and obtain the median of all e .kC1/;i which is e .kC1/;med . <Step 3> Proceed forward search algorithm Add an additional observation in the subset. Employing k C 2 observations to yield coefficients ˇO .kC2/ . Yet, these k C 2 observations are the observations corresponding to the smallest errors of (k C 2) observations in Step 2; that is, observations corresponding to e .kC1/;1 ; e .kC1/;2 ; : : : ; e .kC1/;kC2 . This is equivalent to removing the outliers. Then, repeating Step 2 and we can obtain the median of e .kC2/;i , which is e .kC2/;med . <Step 4> Obtain all estimated coefficients and corresponding error median Add an additional observation again. It means that repeat Step 3 by adding another observation and use k C 3 observations corresponding to the smallest k+3 errors of e .kC2/;i in Step 3 We then similarly obtain ˇO .kC3/ and median e .kC3/;med . Repeat the above steps by adding one additional observation in each estimation and the process is done until all samples are used. We thus obtain estimated coefficients ˇO .kC4/ ; ˇO .kC5/ ; : : : ; ˇO N and the corresponding median e .kC4/;med ; e .kC5/;med ; : : : ; e N;med . <Step 5> Outlier is found

1

Atkinson uses k D C1 as the number of parameters C1 as the starting sample size. We do not adopt his suggestion because the small sample size often is full of zeros without one, invalidating the logit model.

62 The Prediction of Default with Outliers: Robust Logistic Regression

  Calculating e ;med D min e .kC1/;med ; e .kC2/;med ; : : : e N;med and its corresponding ˇO  , which is the estimator of RL method.2 Although this forward search method is intuitively appealing, it encounters three problems in the actual application. First, the random sampling may pick all zeros or ones as dependent variable, which fail the estimation. Next, the selected initial set of samples affects the estimation results. Thus, repeated sampling of the initial set become necessary. Third, companies identified as outliers may be those extremely good and bad companies. They are statistical outliers but not financial outliers.

967

the sum of TP% and TN% is referred to as the hit rate, whereas FP% and FN% are referred to as Type II and Type I errors, respectively. Furthermore, TP% C FN% D 1 and FP% C TN% D 1. The weakness of this table is that only one cutoff is chosen to decide these ratios. Typically, the selection of this cutoff is based on the average rule (the cutoff is then 0.5) or the sample proportion rule (the cutoff is then the number of default/total number firms). More cutoffs may be needed, which motivates the development of the following validation tests.

62.3.2 Cumulative Accuracy Profile (CAP) 62.3 Five Validation Tests Once we obtain the estimated results from two methods, we compare their forecasting ability based on the following five validation tests for the assessment of discriminatory power. The validation methods introduced here are mostly based on the work by Sobehart and Keenan (2004), Sobehart et al. (2000) and Stein (2002).

62.3.1 Contingency Table (Cross-Classification Table) Contingency Table, also referred as the Cross-Classification Table, is the most often used validation tool in comparing the power of prediction. Let TP% and TN% be the ratios of success in predicting default and non-default firms, whereas FP% and FN% be the ratios of failure in predicting default and non-default firms (see Table 62.1). In conventional terms, Table 62.1 Contingency table (cross-classification table) Non-default Default XX XXXPredicted companies companies XXX Actual X .yi D 0/ .yi D 1/ TP%D

FP%N

Non-default

companies (yOi D F xi ˇO  < cutoff)

FN%D

TN%N

Notations are taken from Sobehart et al. (2000). TP true positive means that companies are default and are accurately predicted; FN false negative means companies are not default and not correctly predicted; FP false positive means companies are default and not correctly predicted; TN true negative means that companies are not default and correctly predicted; D is number of default and N is number of non-default companies 2

We could further repeat Step 1 to start different set of observations.

Perfect Turning points =

Fraction of Defaulted Companies

Default companies

(yOi D F xi ˇO   cutoff)

CAP curve is a visual tool which graph can easily be drawn if two representative samples of scores for defaulted and nondefaulted borrowers are available. The shape of the concavity of the CAP is equivalent to the property that the conditional probabilities of default, given the underlying scores, form a decreasing function of the scores (default probability). Alternatively, non-concavity indicates suboptimal use of information in the specification of the score function. Researchers typically calculate the accuracy ratio, which is the area under the rating model divided by the area under the perfect model, to examine the performance of model. This is equivalent to A/B graphed in Fig. 62.1. Figure 62.1 plots CAPs of a perfect rating and random model. A perfect rating model will assign the lower estimated scores to the defaulters. In this case the CAP is increasing linearly and then staying at one. For a random model without any discriminative power, the fraction x of all debtors with the lower scores contain x percent of all defaults. Applied rating systems will be somewhere in between these two extremes. Statistically, the comparison ratio is defined as the

D Perfect Model

N +D

B Rating Model A Random Model

45␱ Fraction of All Companies

Fig. 62.1 CAP curve

968

C.-H. Shen et al.

ratio of A/B, where A is the area between the CAP of the rating model being validated and the CAP of the random model, and B is the area between the CAP of the perfect rating model and the CAP of the random model. The calculation of area under rating model is as follows. First, a descending order of the estimated default rates is ranked. Then, it takes the top s% number of firms that have the higher estimated default rates, making these numbers equal to G D s% .N C D/ where N and D are the number of non-defaulting and default companies in the data set. Within G firms, it then calculates the number of firms that are actually default and are divided by G to yield y%. Repeating the above process, we obtain a sequence of s% and y%. Plotting y% (y-axis) against s% (x-axis) yields CAP. The shape of the rating model depends on the proportion of solvent and insolvent borrowers in the sample.

Hit Rate 1.0 FN％

FP%

0.8 A 0.6 TP%

0.4

B

0.2

TN%

0.2

0.4

0.6

0.8 1.0 False Alarm Rate

Fig. 62.2 ROC curve

obtain the first set of (FP%, TP%/ D .5%; 8%). Continuing this process, we can get many sets of FP% and TP%, which generate ROC.

62.3.3 Receiver Operating Characteristic (ROC) 62.3.4 Kolmogorov-Smirnov (KS) ROC curve uses the same information as CAP to answer the question: What percentage of non-defaulters would a model have to exclude? (Stein 2002). It generalizes Contingency Table analysis by providing information on the performance of a model at any cutoff that might be chosen. It plots the FP% rate against the TP% rate for all credits in a portfolio. In particular, ROCs are constructed by scoring all credits and ordering the non-defaults from the worst to the best on the x axis and then plotting the percentage of defaults excluded at each level on the y axis. The area under the rating model is ACBD

kDn1 X

.XkC1  Xk /.YkC1 C Yk /  2;

The KS-test tries to determine if two datasets differ significantly. The KS-test has the advantage of making no assumption about the distribution of data. It also enables us to view the data graphically. KS plots the cumulative distribution of default and non-default firms, denoted as F 1 and F 2, respective by and then calculates the maximum distance between these two curves as KS D max.F 1  F 2/ The large KS suggest the rejection of the null hypothesis of equality of distributions.

kD0

where A and B are the area under rating model and 45ı , respectively and n is the number of intervals. The ROC curve is demonstrated in Fig 62.2. For example, assuming that the number of default firm D D 50, and non-default firm N D 450. Then, similar to the CAP method, a descending order of the estimated default probability is ranked. Next, giving a fixed type II error, FP%, and finding the corresponding cutoff of c%, we can calculate the corresponding TP%. To illustrate this, if FP% is first chosen to be 5%, then 23 non-default firms are misjudged as default (450 5% D 22:5). At the same time, the cutoff c% is decided. Based on this c%, if we successfully predict four defaulted firms, making TP% D 8%.4=50 D 8%). Thus we

62.3.5 Brier Score Brier score computes the mean squares error between the estimated and actual default rate. The Brier Score is estimated as

2 Pn bi  Ii P i D1 ; BD n where POi is the predicted value and Ii is the actual 0 and 1. From the above definition, it follows that the Brier score is always between zero and one. The closer the Brier score is to zero the better is the forecast of default probabilities.

62 The Prediction of Default with Outliers: Robust Logistic Regression

62.4 Source of Data and Empirical Model 62.4.1 Source of Data To ensure the reliability of financial statements, our samples are actual listed companies on the Taiwan Stock Exchange. Default firms are defined as those stocks that require full delivery; that is, transaction with cash in Taiwan Stock Exchange. These firms include (1) check bouncing of the CEOs, board directors and supervisors of companies; (2) firms that request for financial aid from the government, due to restructuring, bankruptcy, liquidation, ongoing uncertainty, acquisitions, tunneling, trading halts, and credit crunch by banks. In our sample, there are 52 default companies in the period 1999–2004. For each default company, we search for the three additional companies with a similar size of assets in the same industry, resulting in 156 non-default companies. Hence, 208 companies in total are in our sample. The names and codes of all the companies are reported in Table 62.2 as well as the reasons of their defaults. We also reserve 20% of our sample for out-of-sample forecast. That is, there are 42 reserved firms.

62.4.2 Empirical Model Our empirical model is on the basis of Altman’s z-score, which contains five variables. Four of them are financial accounting variables and one of is market variable .X4 /. Because of multicollinearity, we choose only four of them;3 that is our model is Yt D f .X1t 1 C X2t 1 C X3t 1 C X4t 1 C 1:0X5t 1 / where Y is the binary variable with 1 and 0 and 1 denotes defaulted and zero otherwise, X1 D operation capital/total asset (operating capital D liquid asset  liquid liability), X3 D earnings before interest and tax (EBIT)/total asset, X4 D stock value/total liability, X5 D net sales revenue/total asset (net sales revenues D sales  redemption and discount). All signs are expected to be negative.

62.5 Empirical Results The left and right parts of Table 62.3 report the estimated results using the logit and the Robust Logit models, respectively. When the logit model is used, all coefficients show the expected negative sign and all are significant except for coefficient of X5 . By contrast, when the Robust Logit model 3

We omit X2 D retained earnings/total assets.

969

is employed, all coefficients not only show the expected signs but also are significantly different from zero. Alongside this, the pseudo-R2 is 0.3918 for the logit model but is higher up to 0.9359 for the Robust Logit model, suggesting that in-sample fitting is much better in the Robust Logit model than in the logit model. Figure 62.3 plots the curve of CAP, ROC, and KS using in-sample forecasts. The curves generated by the Robust Logit method is more concave to the southeast than the logit method shown in the CAP and ROC. With respect to KS method, the maximum distance between non-default and default is also bigger for the Robust Logit method than for the logit method. Figure 62.4 is similar to Fig. 62.3 but an out-of-sample forecast is used. The CAP and ROC curves generated by the two methods are twisted with each other to some extent and the area under the curves can be hardly evaluated by the human eye. With respect to KS method, the maximum distance of non-defaults and defaults clearly show that the Robust Logit method is superior to the logit method. Table 62.4 reports the five validation tests by using the in-sample forecast. With respect to the Contingency Table, when the logit method is used, the TP% and TN% are about 77% but are higher up to 97.67 and 93.67%, respectively, when the Robust Logit method is undertaken. Thus, the Robust Logit method defeats the logit method when the validation is based on the Contingency Table. The KS is 5.288 and 6.410 for the two methods, respectively, again supporting the superiority of the Robust Logit method. The CAP ratio also reaches the similar conclusion, where they are 0.7040 and 0.8308 for the two methods, respectively. Not surprisingly, ROC ratios also support the same conclusion as the two ratios are 0.8447 and 0.9867, respectively. Finally, the Brier score, whose definition is opposite to the previous validation tests, is smaller if the performance of the method is superior. The scores for two models are respectively 0.1207 and 0.0226. Accordingly, all validation tests suggest that the Robust Logit method is superior to the logit method in insample prediction. Table 62.5 reports the validation tests by using the outof-sample forecast. The superior performance of the Robust Logit method in in-sample forecast becomes less pronounced here. When Contingency Table is employed, the TP% and TN% yielded by the logit model are about 75%, which is similar to their in-sample counterparts reported in Table 62.4. The values, however, change dramatically when the Robust Logit is used. The TP% becomes 100.0% but TN% is only about 48%. This implies that the Robust Logit method is aggressive in the sense that it has a greater tendency to assign companies as default. The use of KS test still support the conclusion reached by the in-sample case, i.e., the logit method performs worse than the Robust Logit method. The differences between the two methods in CAP test become trivial as the logit method is 0.6566 and the Robust Logit is

970

C.-H. Shen et al.

Table 62.2 All sample companies Code of failing Names companies of companies

Default date

Types of default

Matching samples of non-default companies

9913

MHF

1999/1/18

G

9911(SAKURA), 9915(NienMade), 9914(Merida)

1998

2005

U-Lead

1999/1/24

G

2002(CSC), 2006(Tung Ho Steel), 2007(YH)

1998

2539

SAKURAD

1999/3/22

G

2501(CATHAY RED), 2504(GDC), 2509(CHAINQUI)

1998

2322

GVC

1999/4/1

O

2323(CMC), 2324(Compal), 2325(SPIL)

1998

2522

CCC

1999/4/18

C

2520(KINDOM), 2523(DP), 2524(KTC)

1998

1431

SYT

1999/5/21

H

1432(TAROKO), 1434(F.T.C.), 1435(Chung Fu)

1998

1808

KOBIN

1999/5/24

H

1806(CHAMPION), 1807(ROMA), 1809(China Glaze)

1998

9922

UB

1999/10/5

G

9918(SCNG), 9919(KNH), 9921(Giant)

1998

1206

TP

1999/11/2

H

1216(Uni-President), 1217(AGV), 1218(TAISUN)

1998

1209

EHC

2000/3/23

N

1201(Wei-Chuan), 1203(Ve Wong), 1207(CH)

1999

2528

CROWELL

2000/4/28

G

2526(CEC), 2527(Hung Ching), 2530(DELPHA)

1999

1462

TDC

2000/7/11

G

1458(CHLC), 1459(LAN FA), 1460(EVEREST)

1999

2703

Imperial

2000/9/5

H

2702(HHG), 2704(Ambassador), 2705(Leo Foo)

1999

1422

MICDT

2000/9/6

C

1417(CARNIVAL), 1418(TONG-HWA), 1419(SHINKO.SPIN.)

1999

1505

YIW

2000/9/6

G

1503(Shihlin), 1504(TECO), 1506(Right Way)

1999

2334

KFC

2000/9/6

C

2333(PICVUE), 2335(CWI), 2336(Primax)

1999

2518

EF

2000/9/6

G

2514(LONG BON), 2515(BES), 2516(New Asia)

1999

Data year

(continued)

62 The Prediction of Default with Outliers: Robust Logistic Regression

971

Table 62.2 (continued) Code of failing Names companies of companies

Default date

Types of default

Matching samples of non-default companies

2521

HCC

2000/9/8

G

2505(ky), 2509(CHAINQUI), 2511(PHD)

1999

2019

Kuei Hung

2000/9/16

G

2020(MAYER PIPE), 2022(TYCOONS), 2023(YP)

1999

2011

Ornatube

2000/10/13

C

2012(CHUN YU), 2013(CSSC), 2014(CHUNG HUNG)

1999

9906

Corner

2000/10/27

C

9905(GCM), 9907(Ton Yi), 9908(TGTG)

1999

1222

Yuan Yi

2000/11/2

C

1224(HSAFC), 1225(FOPCO), 1227(QUAKER)

1999

2902

Choung Hsim

2000/11/29

H

2903(FEDS), 2910(CHUN YUAN STEEL), 2912(7-ELEVEN)

1999

2517

CKA-LT

2000/11/30

G

2520(KINDOM), 2523(DP), 2524(KTC)

1999

2537

Ezplace

2001/1/12

G

2534(HSC), 2535(DA CIN), 2536(Hung Poo)

2000

1408

CST

2001/4/2

G

1410(NYDF), 1413(H.C.), 1414(TUNG HO)

2000

1407

Hualon

2001/5/22

G

1402(FETL), 1416(KFIC), 1409(SSFC)

2000

2540

JSCD

2001/5/25

G

2533(YUH CHEN UNITED), 2538(KeeTai), 2530(DELPHA)

2000

2304

A.D.I.

2001/7/28

C

2301(LTC), 2302(RECTRON), 2303(UMC)

2000

1438

YU FOONG

2001/8/10

E

1435(Chung Fu), 1436(FUI), 1437(GTM)

2000

1450

SYFI

2001/8/24

G

1451(NIEN HSING), 1452(HONG YI), 1453(PREMIER)

2000

2318

Megamedia

2001/9/28

G

2315(MIC), 2316(WUS), 2317(HON HAI)

2000

2506

PCC

2001/10/16

G

2514(LONG BON), 2515(BES), 2516(New Asia)

2000

Data year

(continued)

972

C.-H. Shen et al.

Table 62.2 (continued) Code of failing Names companies of companies

Default date

Types of default

Matching samples of non-default companies

1613

Tai-I

2001/10/22

E

1614(SANYO), 1615(DAH SAN), 1616(WECC)

2000

2512

Bao-Chen

2002/4/16

C

2504(GDC), 2509(CHAINQUI), 2511(PHD)

2001

1805

KPT

2002/6/2

G

1802(TG), 1806(CHAMPION), 1807(ROMA)

2001

1602

PEW

2002/9/6

G

1601(TEL), 1603(HwaCom), 1604(SAMPO)

2001

1221

CCI

2003/3/6

C

1217(AGV), 1234(HEYSONG), 1231(Lian Hwa Foods)

2002

2342

MVI

2003/4/18

G

2344(WEC), 2345(ACCTON), 2347(Synnex)

2002

3053

DING ING

2003/4/26

E

3045(TWN), 3046(AOpen), 3052(APEX)

2002

2329

OSE

2003/6/30

G

2330(TSMC), 2331(Elitegroup), 2332(D-LINK)

2002

1212

SJI

2003/9/30

G

1204(jingjing), 1216(UniPresident), 1218(TAISUN)

2002

3001

KIM

2004/3/5

C

3010(WAN LEE), 3011(JH), 3018(TUNG KAI)

2003

2525

Pao Chiang

2004/3/20

G

2520(KINDOM), 2523(DP), 2524(KTC)

2003

2494

Turbocomm

2004/4/15

E

2489(AMTRAN), 2488(HANPIN), 2492(WTC)

2003

2398

Procomp

2004/6/15

H

2382(QCI), 2388(VIA), 2409(AUO)

2003

3021

Cradle

2004/7/26

C

3020(USTC), 3022(ICP), 3023(Sinbon)

2003

2491

Infodisc

2004/8/23

G

2308(DELTA), 2311(ASE), 2312(KINPO)

2003

2490

Summit

2004/9/15

C

2349(RITEK), 2350(USI), 2351(SDI)

2003

Data year

(continued)

62 The Prediction of Default with Outliers: Robust Logistic Regression

973

Table 62.2 (continued) Code of failing Names companies of companies

Default date

Types of default

Matching samples of non-default companies

3004

NAFCO

2004/9/23

H

2356(INVENTEC), 2357(ASUSTEK), 2358(MAG)

2003

1534

Tecnew

2004/9/24

C

1531(SIRUBA), 1532(CMP), 1533(ME)

2003

9936

Compex

2004/10/20

E

9933(CTCI), 9934(GUIC), 9935(Ching Feng)

2003

Data year

The number is the code listed in Taiwan Stock Exchange Type of Default: C check bounce; E concern of continuing operation; G financial Aid from government; O substantial loss (low net worth)

Table 62.3 Estimated results: logit vs. Robust Logit

Logit Methods

Coefficients

Robust Logit t -value

Coefficients 

t -value

Constant

0:3600

0:7506

17.0487

2.1627

X1

1:6195

1:1766

10:2357

1:7913

X3

13:1535

4:1651

234:2707

2:2311

X4

0:5519

2:0683

2:1146

2:3319

0:5858



X5

0:4227

Log likelihood

61:4865

9:510

12:3312

Average likelihood

0.6905

0.9200

Pseudo-R-square

0.3918

0.9359

Number of sample

166

114

Number of default companies

43

43

Number of non-default companies

123

71

Medium of residuals

–

6.42811e-04

  

,

,

2:0225

denote significant at 10, 5 and 1% level, respectively

only 0.6812. Similar results occur in ROC test as the former test is 0.6717 but the latter one is 0.6883. Thus, based on CAP and ROC, they are in a tie. Last, to our surprise, the Robust Logit method is defeated by the logit method as its Brier score is higher than the logit method. Thus, when the out-ofsample forecast is implemented, the differences between two methods are hard to distinguish.

62.6 Conclusion We compare the forecast ability between logit and Robust Logit methods, where the latter take the possible outliers into account. Six validation tests are employed when the

in-sample forecasts are compared, i.e., pseudo-R square, Contingency Table, CAP, ROC, KS and Brier score, whereas the latter five validation tests are undertaken for the out-ofsample forecast. With respect to the in-sample forecasts, Robust Logit method is substantially superior to the logit method when using all validation tests here. With respect to the out-of-sample forecasts, Robust Logit method yields less type II but large type I errors than the logit method when Contingency Table is used, suggesting that Robust Logit is more aggressive in assigning firms as default. Robust Logit is marginally better than the logit method when CAP, ROC, and KS, are adopted but worse when the Brier score is used. Thus, the superiority of Robust Logit is less pronounced or even disappears in the out-of-sample forecasts.

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C.-H. Shen et al.

a. CAP curve

b. ROC curve

c. Kolmogorov-Smirnov

Fig. 62.3 In-sample forecast: Logit and Robust Logit

62 The Prediction of Default with Outliers: Robust Logistic Regression

a. CAP curve

b. ROC curve

c. Kolmogorov-Smirnov

Fig. 62.4 Out-of-sample forecast: logit and Robust Logit forecast

975

976

Table 62.4 Validation tests (in-sample forecast)

C.-H. Shen et al.

Methods

Logit method

Robust Logit method

Cross-classification

TP% TN% TP% TN% 76.74% 78.05% 97.67% 94.37%

KS CAP ROC Brier score

5.288 0.7040 0.8447 0.1207

6.410 0.8308 0.9867 0.0226

CAP: cumulative accuracy profile; ROC: receiver operating curve; KS: Kolmogorov-Smirnov

Table 62.5 Estimated results of cross-classification, KS and Brier (out-of-sample forecast)

Methods

Logit method

Robust Logit method

Cross-classification

TP% TN% TP% TN% 77.78% 72.73% 100.00% 48.48%

KS CAP ROC Brier score

1.45 0.6566 0.6717 0.1319

1.558 0.6812 0.6883 0.3756

CAP: cumulative accuracy profile; ROC: receiver operating curve; KS: Kolmogorov-Smirnov

References Altman, E. I. 1968 “Financial ratios, discriminate analysis and the prediction of corporate bankruptcy.” Journal of Finance 23(4), 589– 609. Altman, E. I. and A. Saunders 1998 “Credit risk measurement: developments over the last 20 years.” Journal of Banking & Finance 21(11–12) 1721–1742. Atiya, A. 2001 “Bankruptcy prediction for credit risk using neural networks: a survey and new results.” IEEE Transactions on Neural Networks 12(4), 929–935. Atkinson, A. C. 1985 Plots, transformations, and regression: an introduction to graphical methods of diagnostic regression analysis, Oxford University Press, New York. Atkinson, A. C. 1994 “Fast very robust methods for the detection of multiple outliers.” Journal of the American Statistical Association 89(428), 1329–1339. Atkinson, A. C. and T. -C. Cheng 2000. “On robust linear regression with incomplete data.” Computational Statistics & Data Analysis 33(4), 361–380. Atkinson, A. C. and M. Riani 2001 “Regression diagnostics for binomial data from the forward search.” The Statistician 50(1), 63–78. Atkinson, A. C. and M. Riani 2006 “Distribution theory and simulations for tests of outliers in regression.” Journal of Computational Graphical Statistics 15(2) 460–476. Barrett, B. E. and J. B. Gray 1997 “Leverage, residual, and interaction diagnostics for subsets of cases in least squares regression.” Computational Statistics and Data Analysis 26(1) 39–52. Cook, R. D. and S. Weisberg. 1982. Residuals and influence in regression, Chapman & Hall/CRC, London. Dimitras, A. I., S. H. Zanakis and C. Zopoundis 1996 “A survey of business failures with an emphasis on prediction methods and industrial application.” European Journal of Operation Research 90(3), 487–513. Doumpos, M. and C. Zopounidis 2002. “Multi-criteria classification and sorting methods: a literature review.” European Journal of Operational Research 138(2), 229–246.

Doumpos, M., K. Kosmidou, G. Baourakis and C. Zopounidis 2002 “Credit risk assessment using a multicriteria hierarchical discrimination approach: a comparative analysis.” European Journal of Operational Research 138(2), 392–412. Flores, E and J. Garrido 2001 “Robust logistic regression for insurance risk classification.” Business Economics Universidad Carlos III, Departamento de Economía de la Empresa Working Papers wb016413. Haslett, J. 1999. “A simple derivation of deletion diagnostic results for the general linear model with correlated errors.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 61(3), 603–609. Lennox, C 1999 “Identifying failing companies: a reevaluation of the logit, probit and DA approaches.” Journal of Economics and Business 51(4), 347–364. Levine, R. and S. Zervos 1998 “Stock markets, banks, and growth.” American Economic Review 88(3), 537–558. Ohlson J. T. 1980 “Financial ratios and the probabilistic prediction of bankruptcy.” Journal of Accounting Research 18(1), 109–131. Piramuthu, S. 1999 “Financial credit-risk evaluation with neural and neurofuzzy systems.” European Journal of Operational Research 112(2) 310–321. Riani, M. and A. Atkinson 2007 “Fast calibrations of the forward search for testing multiple outliers in regression.” Advances in Data Analysis and Classification 1(2) 123–141. Rousseeuw P. J. 1983 “Regression techniques with high breakdown point.” The Institute of Mathematical Statistics Bulletin 12, 155. Rousseeuw, P. J. 1984 “Least median of squares regression.” Journal of the American Statistical Association 79(388), 871–880. Rousseeuw, P. J. and A. Christmann 2003 “Robustness against separation and outliers in logistic regression.” Computational Statistics & Data Analysis 43(3) 315–332. Rousseeuw P. J. and V. J. Yohai 1984 “Robust regression by means of S-estimators,” in Robust and nonlinear time series analysis, Vol. 26, W. H. Franke and R. D. Martin (Eds.). Springer, New York, pp. 256–272. Sobehart, J. R. and S. C. Keenan 2004 “Performance evaluation for credit spread and default risk models,” in Credit risk: models and

62 The Prediction of Default with Outliers: Robust Logistic Regression management Second Edition, D. Shimko (Ed.). Risk Books London, pp. 275–305. Sobehart, J. R., S. C. Keenan and R. M. Stein 2000 “Rating methodology: benchmarking quantitative default risk models: a validation methodology.” Moody’s Investors Service, Global Credit Research, New York. Stein, R. M. 2002 “Benchmarking default prediction models: pitfall and remedies in model validation.” Moody’s KMV, Technical Report, No. 030124.

977 Westgaard, S and N. Wijst 2001 “Default probabilities in a corporate bank portfolio: a logistic model approach.” European Journal of Operational Research, 135(2), 338–349. Wu, C. and X. M. Wang 2000 “A neural network approach for analyzing small business lending decisions.” Review of Quantitative Finance and Accounting, 15(3), 259–276. Zhu A. M. Ash and R. Pollin 2002 “Stock market liquidity and economic growth: a critical appraisal of the Levine/Zervos model.” International Review of Applied Economics 18(1), 1–8.

Chapter 63

Term Structure of Default-Free and Defaultable Securities: Theory and Empirical Evidence Hai Lin and Chunchi Wu

Abstract In this chapter, we survey modern term structure models for pricing fixed income securities and their derivatives. We first introduce bond pricing theory within the dynamic term structure model (DTSM) framework. This framework provides a general modeling structure in which most popular term structure models are nested. These include affine, quadratic, regime switching, jump-diffusion, and stochastic volatility models. We then review major studies on default-free bonds, defaultable bonds, interest rate swaps and credit default swaps. We outline the key features of these models and evaluate their empirical performance. Finally, we conclude this chapter by summarizing important findings and suggesting directions for future research. Keywords Risk-neutral and physical measures r Latent factors r Default intensity r Recovery r Liquidity r Counterparty risk r Default correlation r Yield spreads r Structural and reduced-form models

63.1 Introduction This article provides a survey on term structure models designed for pricing fixed income securities and their derivatives.1 The past several decades have witnessed a rapid development in the fixed income markets. A number of new fixed-income instruments have been introduced successfully into the financial market. These include, to mention just a few, strips, debt warrants, put bonds, commercial mortgagebacked securities, payment-in-kind debentures, zero-coupon convertibles, interest rate futures and options, credit default swaps, and swaptions. The size of the fixed income market

H. Lin Department of Finance, School of Economics & Wang Yanan Institute of Studies in Economics, Xiamen University, Xiamen, 361005, China C. Wu () Department of Finance, Robert J. Trulaske, Sr. College of Business, University of Missouri-Columbia, Columbia, MO 65211, USA e-mail: [email protected]

has greatly expanded. The total value of the fixed income assets is about two thirds of the market value of all outstanding securities.2 From the investment perspective, it is important to understand how fixed income securities are priced. The term structure of interest rates plays a key role in pricing fixed income securities. Not surprisingly, a vast literature has been devoted to understanding the stochastic behavior of term structure of interest rate, the pricing mechanism of fixed income markets, and the spread between different fixed income securities. Past research generally focuses on (1) modeling the term structure of interest rates and yield spreads; (2) providing empirical evidence; and (3) applying the theory to the pricing of fixed income instruments and risk management. As such, our review centers on alternative models of term structure of interest rates, their tractability, empirical performance, and applications. We begin with the basic definitions and notations in Sect. 63.1. We provide clear concepts of term structure of interest rates that are easily misunderstood. Section 63.2 introduces bond pricing theory within the dynamic term structure model (DTSM) framework. This framework provides a general modeling structure in which most of the popular term structure models are nested. This discussion thus helps understand the primary ingredients to categorize different DTSMs; that is, the risk-neutral distribution of the state variables and the mapping function between these state variables and instantaneous interest rate. Section 63.3 provides a literature review of the studies on default free bonds. Several widely used continuoustime DTSMs are reviewed here, including affine, quadratic, regime switching, jump-diffusion, and stochastic volatility (SV) models. We conclude this section with a discussion of empirical performance of these DTSMs, where we discuss some open issues, including the expectation puzzle, the linearity of state variables, the advantages of multifactor and nonlinear models, and their implications for pricing and risk management. 1

For a survey on term structure models, see Dai and Singleton (2002b), Dai and Singleton (2003), and Maes (2004). 2 See The 2008 Statistical Abstract, U.S. Census Bureau.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_63, 

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H. Lin and C. Wu

The studies of defaultable bonds are explored in Sect. 63.4. We review both structural and reduced-form models, with particular attention given to the latter. Several important issues in reduced form models are addressed here, including the specification of recovery rates, default intensity, coupon payment, other factors such as liquidity and taxes, and correlated defaults. Since it is convenient to have a closed-form pricing formula, it is important to evaluate the tradeoff between analytical tractability and the model complexity. Major empirical issues are related to uncovering the components of yield spreads and answering the question whether the factors are latent or observable. Section 63.5 reviews the studies on two popular interest rate derivatives: interest rate swap and credit default swap. Here we present the pricing formulas of interest rate swap and credit default swap based on risk-neutral pricing theory. Other risk factors, such as counterparty risk and liquidity risk are then introduced into the pricing formula. Following this, we review important empirical work on the determinants of interest rate swap spread and credit default swap spread. Section 63.6 concludes the paper by providing a summary of the literature and directions for future research. These include the following: (1) the economic significance of DTSM specification on pricing and risk management; (2) the difference of interest rate dynamics in the risk neutral measure and physical measure; (3) the decomposition of yield spreads; and (4) the pricing of credit risk with correlated factors.

which maps time to maturity  into the yield of the zerocoupon bond with that maturity at time t. The price of the zero-coupon bond can be calculated from its yield by D.t; T / D expŒ.T  t/r.t; T /

(63.3)

63.2.3 Instantaneous Interest Rate The instantaneous interest rate at time t, rt is defined as: rt D lim

T !t

 ln D.t; T / T t

(63.4)

63.2.4 Forward Rate The forward rate at time t, ft T1 !T2 , is the interest rate between two future time points T1 and T2 , which is settled at time t. Specifically, ft T1 !T2 D

ln D.t; T1 /  ln D.t; T2 / T2  T1

(63.5)

Remark: if T1 D t, ft t !T2 D r.t; T2 /.

63.2 Definitions and Notations 63.2.5 Instantaneous Forward Rate 63.2.1 Zero-Coupon Bonds A default-free zero-coupon bond (or discount bond) with maturity date T and face value 1 is a claim that has a nonrandom payoff of 1 for sure at time T and no other payoff before maturity. The price of a zero-coupon bond with maturity date T at time 0  t  T is denoted by D .t; T /.

The instantaneous forward rate at time t with an effective date T , f .t; T / is defined as f .t; T / D lim ft T !T2 D lim T2 !T

D

T2 !T

ln D.t; T /  ln D.t; T2 / T2  T

@ ln D.t; T / @T

(63.6)

63.2.2 Term Structure of Interest Rates Consider a zero-coupon bond with a fixed maturity date T . The continuously compounded yield on this bond is r.t; T / D 

1 ln D.t; T / T t

(63.1)

The zero-coupon yield curve or term structure of interest rates at time t is the function  ! r.t; t C / W Œ0; 1 ! u A./ut ;  t

(101.6)

(101.1) d

rt  lim

(101.3)

t

1 y.ut ; /   ln P .ut ; /: 

101.3 The Model

1

We further assume that there exists a risk-neutral measure, or a martingale measure, P , under which the bond price can be written as

 ln P .ut ; / : 

(101.2)

For technical details, see, for example, Jacod and Shiryaev (Jacod).

where A./ is a positive definite matrix so that all spot rates are bounded from below at zero. As the asymmetric part of A has zero contribution to the spot rate, we also assume that A is symmetric with no loss of generality. In principle, interest-rate positivity can be guaranteed either through a quadratic form or through an exponential function. However, the exponential family is not consistent with any diffusion dynamics for the state vector (Björk and Christensen 1999; Filipovi´c 1999, 2000). Furthermore, the history of interest rates across the world (witness Switzerland and, in recent times, Japan) shows that we must allow an interest rate of zero to be reachable. Zero is not reachable if interest rates are specified as exponential functions of the state variable, but can be reached under our quadratic specification by letting the state vector u approach the vector of zeros. The fact that u can be small argues against the inclusion of linear terms, since the linear term would dominate when the state vector is small, thus potentially allowing negative interest rates.

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Proposition 1 (Bond Pricing). Under the assumptions of diffusion state dynamics in Equation (101.1) and the quadratic form in Equation (101.6) for interest rates, the term structure of zero-coupon bonds is given by P .rt ; / D exp .c./rt / ;

(101.7)

where rt is the instantaneous interest rate. The pricing is as if rt follows a square-root process under the risk-neutral measure P , p drt D ›rt dt C  rt d wt ;

(101.8)

with › 2 R;  2 RC being constant parameters and wt being a newly defined scaler Wiener process. The maturity coefficient c./ is determined by the following Riccati equation: 1 c 0 ./ D 1  ›c./   2 c./2 ; 2

(101.9)

with the boundary condition: c.0/ D 0. Although we start with a d -dimensional state vector, the dimension of the term structure collapses to one. The proof of the bond pricing formula follows standard argument. We solve for the coefficients c./ by applying the Feynman–Kac formula and the principle of matching. Proof. Applying the Feynman–Kac formula to the zero price function in Equation (101.3) yields: r .u/ P .u;/ D

@P .u;/ C L P .u; / ; @t

(101.10)

where L denotes the infinitesimal generator under the riskneutral measure P and is given by  L P .u; / D

@P @u

>

1 C tr 2

 .u/ 

@2 P @u@u>



 † .u/ † .u/> :

The quadratic specification for the spot rate in Equation (101.6) implies that the instantaneous interest rate also has a quadratic form: r .u/ D u> A0 .0/ u:

into Equation (101.10), we have u> A0 .0/u D u> A0 ./ u  2u> A ./  .u/

i h t r A ./ † .u/ † .u/>   C2 u> A ./ †.u/†.u/> A ./ u ; (101.12) which should hold for all maturity  and states u. To maintain the quadratic nature of the Equation (101.12), we need the diffusion term † .u/ to be independent of the state vector u. Let V  ††> denote a positive definite symmetric constant matrix. Indeed, via a rotation of indices, we can set V D I with no loss of generality. Equation (101.12) becomes u> A0 .0/u D u> A0 ./ u  2u> A ./  .u/ t r .A ./ V / C 2u> A ./2 V u

(101.13)

Furthermore, to balance the power of the equation, we decompose the drift function  into two parts,  .u/ D 1 .u/  Bu, where B denotes a constant matrix and is assumed to be symmetric with no loss of generality. The first part 1 .u; t/ satisfies the equality:  2u> A ./ 1 .u/  t r .A ./ V / D 0:

(101.14)

That is, the role of 1 .u/ is to cancel out the constant term on the right-hand side of Equation (101.13). However, since the drift term 1 .u/ cannot depend on maturity , for the equality (Equation (101.14)) to hold, we must be able to factor out the maturity dependence A ./ D a ./ D;

(101.15)

where a ./ is a scalar and D is a positive definite symmetric matrix independent of . This maturity separation determines the most important result of this article: the collapse of dimensionality. Given the maturity separation, Equation (101.14) becomes  2u> D1 .u/  t r .DV / D 0:

(101.16)

Equation (101.13) becomes a0 .0/ u> Du D a0 ./ u> Du C a ./ 2u> DBu C a ./2 2u> D 2 V u:

(101.17)

(101.11) For this equation to hold for all states u 2 Rd , we need

Plugging the quadratic specifications for the spot rate in Equation (101.6) and for the short rate in Equation (101.11)

a0 .0/ DDa0 ./ DC2a ./ DB C2a ./2 D 2 V:

(101.18)

101 Taking Positive Interest Rates Seriously

1493

After rearrangement, we have a0 ./ I D a0 .0/ I  2a ./ B  2a ./2 DV:

(101.19)

Since the equation needs to hold for all elements of the matrix, we must have 2B D ›I I

DV D

1 vI: 4

(101.20)

We hence obtain the ordinary differential equation, 1 a0 ./ D a0 .0/  ›a ./  va ./2 : 2

(101.21)

Furthermore, let x D u> Du, the zero price can then be written as  ln P D u> A ./ u Da ./ u> Du Da ./ x:

(101.22)

Next, given the state vector process d u D .1 .u/  Bu/ dt C

p V d z;

by Itô’s lemma, we obtain the process for x under P , dx D 2u> D .d u/ C t r .DV / dt   D 2u> D1 .u/ C t r .DV /  2u> DBu dt p C2u> D V d z p D ›xdt C vxd w: We obtain the last equality by applying Equation (101.16) and by defining a new Wiener process w: p p p 2u> D V d z 2u> D V d z 2u> D V d z D p D p : dw D p vx 4u> DVDu vu> Du The instantaneous interest rate is rt D a0 .0/ xt . A rescaling of index c./ D a./=a0 .0/;

D

p va0 .0/;

(101.23)

gives us  ln P D c ./ rt ;

(101.24)

with 1 c 0 ./ D 1  ›c ./   2 c ./2 ; 2 p drt D ›rt dt C  rt d w:

(101.25) (101.26)

The initial condition c.0/ D 0 is determined by the fact that P .rt ; 0/ D 1.

Under our model, due to the maturity separability, the dimension of the state space collapses to one. Bonds are priced as if there is only one dynamic factor. Instantaneously, this one dynamic factor follows a two-parameter square-root process under the risk-neutral measure P . We leave the dynamics of this factor under the physical measure P unspecified. The specification of the physical dynamics can be separately determined to match the time-series properties of interest rates while satisfying the constraints implied by the Girsanov theorem. The two parameters of the square-root process determine both the instantaneous risk-neutral dynamics of the single dynamic factor, and the shape of the yield curve via the ordinary differential equation in Equation (101.25). Over time, these two parameters can change and will change, to accommodate changes in the shape of the yield curve. Nevertheless, the bonds are priced as if the two parameters are constant. We hence label them as static factors. One way to think of this is that there is only one source of market risk that is priced on the term structure. Bond pricing is done as if other factors were constant. The instantaneous market risk is seen to be a multiplicative factor for all rates. We obtain a three-factor term structure model with one dynamic factor and two static factors; yet, this three-factor structure is not a result of exogenous specification, but of a collapse of dimensionality due to the seemingly innocuous contention that all rates are bounded below from zero via the quadratic form in Equation (101.6). Our three-factor model also contrasts sharply with traditional three-factor models in that the three factors in our model summarize everything that is uncertain about the shape of the term structure. Traditional three-factor models often contain many parameters in addition to the three factors. The estimates of these parameters often exhibit large standard errors. Therefore, such models are subject to parameter risk. Under our specification, there are no other parameters to be estimated and hence no other risks to be concerned with that will affect the shape of the term structure. Treating › and  as constants, we can solve the term structure coefficients c./ analytically:   2 1  e 2 ;  c ./ D 4  .2  ›/ 1  e 2

(101.27)

p with  D 12 ›2 C 2 2 . We can see immediately that c./ > 0 for all  > 0. Furthermore, since the short rate follows a square-root process under the risk-neutral measure, it is bounded below from zero. By absolute continuity, the short rate should be bounded below from zero as well under the statistical measure. Therefore, all spot rates are bounded below from zero.

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Although we start with a quadratic specification for the spot rates, the final bond pricing formula says that spot rates are proportional to one dynamic factor. The square-root riskneutral dynamics of the short rate brings our model very close to the traditional term structure model of Cox et al. (1985). The key difference lies in the absence of a constant term in the drift of the risk-neutral dynamics and the absence of a constant term in the affine structure of the bond yields. A constant term in the affine structure drives the boundary away from zero and hence violates our assumption that all rates are bounded from zero. We solve the coefficients c./ treating › and  as constants. Yet, in our application, we allow the two parameters to vary every day to fit the current yield curve. Thus, there seems to be inconsistency between the two practices. However, the inconsistency is only an illusion since we treat › and  not as time-inhomogeneous parameters, but as static factors. We explicitly recognize the risk associated with the time variation of these factors and hedge the risk away by forming portfolios that are first-order neutral to their variation. Due to the low dimensionality of the factor structure, neutrality can be achieved with a maximum of only four instruments. In contrast, in a traditional three-factor model with more than ten parameters, making a portfolio first-order neutral to all parameters and state variables is impractical due to transaction costs. Our practice is also decisively different from traditional time-inhomogeneous specifications as often applied under the framework of Heath et al. (1992). In these specifications, the model parameters are allowed to vary over time in such a way that we can always fit the current observed term structure perfectly. Thus, these models have little to say about the fair pricing of the yield curve. Furthermore, accommodating the whole yield curve often necessitates accepting an infinite dimensional state space, which create difficulties for hedging practices.

101.4 The Hump-Shaped Forward Rate Curve The term structure of the long forward rates has been persistently downward sloping (Brown and Schaefer 2000). Given the initial upward sloping term structure in most observations, the downward slopes in the very long term imply a hump-shaped term structure for the forward rates. Our model captures very nicely the hump shape of the forward rate curve. We can rotate the system and redefine the three factors explicitly on the hump shape of the forward rate curve. Formally, we let F denote the maximum of the instantaneous forward rate (the peak of the hump), M the maturity at which the forward rate reaches its maximum, and  some measure

E. Pan and L. Wu

of the curvature of the forward rate curve at the maximum. Then, the instantaneous forward rate at maturity  is given by f ./ D F sech2 Œ.  M / :

(101.28)

Appendix 1 provides a derivation for the transformation. The parameter  is related to the curvature of the forward rate curve at the maximum by: ı.M / 

f 00 .M / D 22 : f .M /

(101.29)

The new triplet ŒF; M;  defines the same term structure as the original triplet Œr; ›;  . They are linked by,

› 1 ›2 ; F D r 1 C 2 ; M D  arctanh 2  2 1p 2 D › C 2 2 : r D F sech2 .M / ; 2 › D 2 tanh .M / ;

 2 D 22 sech2 .M / : (101.30)

The new formulation defines the forward rate curve by controlling the exact shape of the curve at the hump. Thus, if we observe a forward rate curve, we can determine the value of the three factors very easily. In our estimation, we model T  1= instead of , because it has a natural interpretation of time scale. In contrast, the original triplet of factors Œr; ›;  define the instantaneous risk-neutral dynamics of the interest rates. They also define the level, the slope, and the curvature of the forward rate curve at the short end ( D 0): f 0 .0/ D r;

f 0 .0/ D ›; f .0/

ı.0/ D

f 00 .0/ D ›2   2 : f .0/ (101.31)

The relation shows clearly the interaction between the instantaneous risk-neutral dynamics of the short rate and the shape of the forward rate curve. The drift parameter › controls the initial slope of the forward rate curve. The initial curve is upward sloping when › is negative. On the other hand, the instantaneous volatility term  contributes to the curvature of the forward rate curve. The larger the variance, the more concave the forward rate curve. Furthermore, the two points of the forward rate curve at t D 0 and t D M are linked by a unit-free quantity  D tanh .M /: f .0/  f .M / D  2; f .M /

ı.M /  ı.0/ D 3 2 : ı.M /

These observations dramatically simplify the calibration of the forward rate curve as the factors can be directly mapped to the level and shape of the forward rate curve.

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101.5 Fitting the US Treasury Yields and US Dollar Swap Rates To investigate the model’s performance, we calibrate the model to two sets of data. One is US Treasury constant maturity par yields and the other is US dollar swap rates of the same maturities. We investigate the goodness of fit of the model on the two sets of data. We also extract the three factors from the two markets for each day and analyze the time series variation of these risk sources.

1  P ./ ; h .Xt ; / D 200 P2 i D1 P .i=2/

101.5.1 Data and Estimation We obtain both the swap rate data and the constant maturity Treasury yields from Lehman Brothers. The maturities include 2, 3, 5, 7, 10, 15, and 30 years. The data are weekly (Wednesday) closing mid quotes from December 14th, 1994 to December 28th, 2000 (316 observations). Table 101.1 reports the summary statistics of the swap rates and Treasury par yields. We observe an upward-sloping mean term structure for both swaps and US Treasuries. The standard deviation for both the levels and the first differences exhibit a hump-shaped term structure with the plateau coming at 3-year to 5-year maturities. Interest rates are highly persistent. The excess skewness and kurtosis estimates are small for both levels and first differences. We are interested not only in the empirical fit of the model on the yield curves of different markets, but also in the time series behaviors of the three factors X  ŒF; M; T at each date. The choice of ŒF; M; T over Œr; ›;  in the estimation is purely due to numerical consideration. If we can forecast the three factors, we will be able to forecast the yield curve. A natural way to capture both the daily fitting of the crosssection of the term structure and the forecasting of the time series of interest rates is to formulate the framework into a state space system and estimate the system using Kalman (1960) filter. For the estimation, we assume that the three factors can be forecasted via a simple VAR(1) system: Xt D A C ˆXt 1 C ©t ;

(101.32)

where © denotes the forecasting residuals. We use this forecasting equation as the state propagation equation, with © as the state propagation error with covariance matrix Q. We then construct the measurement equations based on the valuation of the par yields on the Treasury and swap market, respectively, St ./ D h .Xt ; / C et ;

where h .Xt ; / denotes the model-implied value of the par yield of maturity  as a function of the factors Xt and et denotes the measurement error, which we assume has a covariance matrix of R. In the estimation, we assume that the measurement errors on each series are independent, but bear distinct variance. Thus, R is a diagonal matrix, with each element denoting the goodness of fit on each corresponding series. Since the US Treasury par bond and the US dollar swap contract both have semiannual payment intervals, the modelimplied par yield is given by

(101.33)

(101.34)

where P ./ denotes the model-implied value of the zero coupon bond (discount factor) and is given in Equation (101.7). Since the measurement equation is nonlinear in the state vectors, we apply the extended Kalman Filter, under which the conditional variance update is based on a first-order Taylor expansion. The parameters of the state space system include those that control the forecasting time series dynamics and the covariance matrices of the state propagation errors and measurement errors ‚  ŒA; ˆ; Q; R . We estimate these parameters using a quasilikelihood method assuming that the forecasting errors of the par yields are normally distributed. Appendix 2 provides more details for the estimation methodology. Table 101.2 reports the estimates (and standard errors in parentheses) of the state space estimation on both the US dollar swap market and the US Treasury market.

101.5.2 Model Performance Table 101.3 reports the summary properties of the pricing errors on the swaps and Treasury par yields. We define the error as the difference between the market-observed rates and the model-implied rates, in basis points. The fitting is good despite the simple model structure. Overall, the mean absolute error is within a few basis points. The maximum error is only 28 basis points for the swap rates and 41 basis points for the Treasury par yields. An inspection of the error properties across different maturities indicates that the key difficulty of the model lies in fitting interest rates at short maturities (2 years). The mean error on the 2-year rates is 7:5 basis points for swaps and 4:5 for Treasuries, implying that the observed 2-year rates are on average lower than those implied by the model. Figure 101.1 plots the time series of the pricing errors on the swap rates (left panel) and the Treasury par yields (right panel) at selected maturities: two, five, ten, and 30 years.

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Table 101.1 Summary statistics of US dollar swap rates and US treasury par yields Swap Treasury Mat 2 3 5 7 10 15 30 2 3 5 7 10 15 30

Mean Levels 6.190 6.303 6.454 6.560 6.681 6.817 6.889 Differences 0.007 0.007 0.007 0.007 0.007 0.007 0.007

Std

Skew

Kurt

Auto

Mean

Std

Skew

Kurt

Auto

0.656 0.657 0.642 0.629 0.615 0.591 0.576

0.343 0.266 0.133 0.061 0.022 0.056 0.050

0.293 0.224 0.004 0.147 0.300 0.388 0.247

0.971 0.971 0.971 0.971 0.971 0.969 0.971

5.799 5.857 5.947 5.994 6.059 6.111 6.266

0.631 0.654 0.681 0.672 0.669 0.653 0.624

0.050 0.071 0.143 0.081 0.008 0.052 0.243

0.858 0.798 0.672 0.522 0.238 0.152 0.207

0.969 0.969 0.969 0.970 0.971 0.971 0.974

0.120 0.123 0.123 0.122 0.119 0.117 0.106

0.328 0.349 0.214 0.264 0.188 0.304 0.445

0.977 0.834 0.627 0.593 0.410 0.393 0.545

0.027 0.020 0.013 0.019 0.043 0.024 0.018

0.008 0.008 0.009 0.009 0.009 0.008 0.008

0.119 0.121 0.124 0.120 0.118 0.114 0.102

0.201 0.316 0.178 0.282 0.233 0.348 0.415

0.642 0.746 0.600 0.592 0.353 0.319 0.528

0.001 0.013 0.008 0.013 0.040 0.015 0.033

The table presents summary statistics of US dollar swap rates and US Treasury par yields. Mean, Std, Skew, Kurt, and Auto denote, respectively, the sample estimates of the mean, standard deviation, skewness, kurtosis, and first-order autocorrelation. The data are weekly closing mid quotes from Lehman Brothers, from December 14th, 1994 to December 28th, 2000 (316 observations)

We observe that except at short maturities, the pricing errors are normally within ten basis points. The magnitude of these pricing errors is comparable to those reported in much more complicated models.

101.5.3 The Time Series Behavior of the Interest-Rate Factors Through the state space estimation, we obtain estimates on the three interest-rate factors each day that fit the observed Treasury and swap rate curves, respectively. In this section, we analyze the time series behavior of the three factors estimated from both markets, and compare how the three factors relate to one another and how they differ across the two markets.

101.5.3.1 The Dynamic Level Factor Under our model structure, the level of the yield curve can be represented by the instantaneous short rate r. The left panel of Fig. 101.2 plots the extracted instantaneous interest rate from the swap market (solid line) and the Treasury market (dashed line). The right panel of Fig. 101.2 depicts the difference (swap spread) between the two short rates. The average spread on the two short rates over this sample period is 34.19 basis points. Overall, the two short rates move very closely to each other. However, the swap spread does change over time. Before 1998, the spread is in general within 40 basis

points. The spike in the swap spread in late 1998 and early 1999 corresponds to the hedge fund crisis during that time. The swap spread during year 2000 is also unusually high, corresponding to the reduced supply in the US Treasury as a result of the budget surplus at that time. Thus, although the spread spike in early 1999 can be attributed to a credit event, the spread plateau in 2000 is mainly due to liquidity factors.

101.5.3.2 The Slope and Curvature Factors The slope of the forward rate curve is closely related to the drift parameter › of the short rate risk-neutral dynamics. The slope is positive when › is negative. In contrast, the instantaneous volatility  of the short rate dynamics is closely related to the curvature of the forward rate curve. The higher the volatility, the more concave the forward rate curve. Figure 101.3 plots the time series of › (left panel) and  (right panel) as an illustration of the slope and curvature dynamics of the yield curve. The solid lines depict the factors extracted from swap market and the dashed lines depict the factors from the US Treasury market. The two markets move closely together as their shape (slope and curvature) of the forward rate curves also move together. Furthermore, comparing the time series of the short rate to that of the slope and curvature factors, we see that the slope and curvature factors tend to move in a direction opposite to the level factor. When the short rate is high, the forward rate curve tends to be flat. The two spikes in the slope and curvature time series correspond to the two dips in the short rate.

101 Taking Positive Interest Rates Seriously

Table 101.2 Summary statistics of the three factors from swaps and US strips

Data

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Swap

Treasury

State propagation equation: Xt 2 0:1831 6 AD 4 3:0572 7:9858 2 6 6 6 6 6 6 6 6 4

ˆD

2 6 6 6 6 6 6 6 6 4

S© D

D A C ˆXt1 C ©t ; 3 .0:1233/ 7 .0:6830/ 5 .1:4326/

3 0:9761 0:0052 0:0027 7 .0:0133/ .0:0307/ .0:0081/ 7 7 0:0163 0:9005 0:0110 7 7 .0:0081/ .0:0164/ .0:0046/ 7 7 7 0:0386 0:1895 0:9350 5 .0:0164/ .0:0425/ .0:0115/

2

3

2

1:1173 0 0 .0:0552/ 0:0634 0:7012 0 .0:0577/ .0:0413/ 0:7486 0:3839 1:5043 .0:1723/ .0:1862/ .0:1305/

Measurement equation: St D h.Xt / C et ; 3 2 2 2 0:1106 .0:0115/ 6  7 6 0:0512 .0:0051/ 6 3 7 6 7 6 6 6 5 7 6 0:0114 .0:0012/ 7 6 6 6 7 7 D 6 0:0130 .0:0007/ 7 6 6 7 6 6 6 10 7 6 0:0188 .0:0007/ 7 6 6 4 15 5 4 0:0295 .0:0017/ 30 0:0127 .0:0026/

L.103 /

S© S©> D C ov.©/ 2 3 0:2405 .1:4971/ 7 6 4 4:0164 .2:2390/ 5 4:4539 .3:9432/ 3 0:9660 0:0018 0:0046 6 7 6 .0:1497/ .0:0057/ .0:0051/ 7 6 7 6 0:0358 0:9673 0:0122 7 6 7 6 .0:0303/ .0:0123/ .0:0106/ 7 6 7 6 7 0:9299 5 4 0:0136 0:0286 .0:0490/ .0:0233/ .0:0202/ 1:1418 0 0 6 6 .0:0726/ 6 6 0:1898 2:3490 0 6 6 .0:2869/ .0:2455/ 6 6 2:3074 3:6332 4 1:5145 .0:6367/ .0:6562/ .0:3997/

7 7 7 7 7 7 7 7 5

  Cov.e/ D diag i2 ; 3

3 7 7 7 7 7 7 7 7 5

i D 2; 3; 5; 7; 10; 15; 30 3 0:1468 .0:0180/ 6 0:0928 .0:0112/ 7 6 7 6 7 6 0:0314 .0:0016/ 7 6 7 6 0:0003 .0:0785/ 7 6 7 6 7 6 0:0409 .0:0021/ 7 6 7 4 0:0498 .0:0039/ 5 2

7 7 7 7 7 7 7 7 7 7 5

0:0285 .0:0027/

5.6517

4.7706

The table reports the parameter estimates (standard deviations in parentheses) of the state space system. The state propagation captures the dynamics of the three factors Xt ŒFt ; Mt ; Tt , where Ft is represented in one thousandth, and M and T are in years. The standard deviation of the measurement error (i ) captures the model’s performance in fitting the constant maturity yields or swap rates of the denoted maturities. The standard deviation is measured in annual percentages. The model is calibrated to both the US dollar swap rates (left panel) and the US Treasury constant maturity par yields, both of which are weekly data from December 14th, 1994 to December 28th, 2000 (316 observations)

Table 101.3 Summary statistics of pricing errors on US dollar swap rates and US treasury par yields Swap Treasury Mat 2 3 5 7 10 15 30

Mean

Std

Mae

Max

Auto

Mean

Std

Mae

Max

Auto

7:524 2:681 0:608 0:843 0:022 0:879 0:445

7.641 4.053 1.053 1.323 1.837 2.446 0.763

8.611 3.948 0.796 1.087 1.279 2.052 0.554

28:290 13:102 5:635 7:859 10:430 8:118 6:676

0.893 0.751 0.158 0.257 0.245 0.629 0.160

4:358 1:731 1:327 0:723 0:249 3:423 1:341

13:970 9:243 3:400 1:401 4:315 2:983 2:503

12:669 8:077 2:602 0:857 3:221 3:947 1:758

41:425 31:094 14:568 8:929 13:536 12:308 17:064

0.923 0.871 0.531 0.111 0.674 0.434 0.468

The table presents summary statistics of the pricing errors on US dollar swap rates and US Treasury par yields. We define the pricing error as the difference, in basis points, between the market observed rates and the model implied rates. Mean, Std, Mae, Max, and Auto denote, respectively, the sample estimates of the mean, standard deviation, mean absolute error, max absolute error, and first-order autocorrelation. The market observed rates are weekly closing mid quotes from Lehman Brothers, from December 14th, 1994 to December 28th, 2000 (316 observations). We compute the model-implied rates based on the state space system estimated in Table 101.2

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20

20

Pricing Error, Bps

30

10 0 −10

−30 Jan97

Jan98

Jan99

−40 Jan95

Jan00

40

5 year Swap

30

30

20

20

Pricing Error, Bps

Pricing Error, Bps

−10 −20

40

10 0 −10

−30

40

Jan98

Jan99

40

10 year Swap

30

20

20

Pricing Error, Bps

30

10 0 −10

−30

40

Jan98

Jan99

40

30 year Swap

30

30

20

20

10 0 −10

−30 Jan98

Jan99

Jan00

Jan96

Jan98

Jan99

Jan00

Jan98

Jan99

Jan00

Jan97

30 year Treasury

0

−20

Jan97

Jan00

−10

−30 Jan96

Jan99

10

−20

−40 Jan95

Jan98

10 year Treasury

−40 Jan95

Jan00

Jan97

0

−30 Jan97

Jan96

−10 −20

Jan96

Jan00

10

−20

−40 Jan95

Jan99

5 year Treasury

−40 Jan95

Jan00

Jan98

−10 −20

Jan97

Jan97

0

−30 Jan96

Jan96

10

−20

−40 Jan95

Pricing Error, Bps

0

−30 Jan96

2 year Treasury

10

−20

−40 Jan95

Pricing Error, Bps

40

2 year Swap

30

Pricing Error, Bps

Pricing Error, Bps

40

−40 Jan95

Jan96

Jan97

Fig. 101.1 Swap rate pricing errors. Lines report the time series of the pricing errors on swap rates and Treasury par yields. The pricing error is in basis points, defined as the difference between the market-observed rate and the model-implied rate

101.6 Extensions: Jumps in Interest Rates Our model is derived under three important assumptions: the positivity of interest rates guaranteed via a quadratic form, a finite-dimensional state representation, and diffusion state dynamics. Interest rate positivity is a necessary condition to guarantee no arbitrage, as long as we are allowed to hold cash for free. The zero lower bound assumption for

rates of all maturities matches empirical observation. A finite state representation is also necessary for complete hedging to be feasible in practice in the presence of transaction costs. However, the assumption on pure-diffusion state dynamics is more for convenience and tractability than for reasonability. We do observe that interest rates move discontinuously (jumps) every now and then. In this section, we explore whether incorporating a jump component by itself violates

101 Taking Positive Interest Rates Seriously

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The Level Factor

The Level Spread 90 Difference in Short Rate, Bps

7.5

Short Rate, %

7 6.5 6 5.5 5 4.5

80 70 60 50 40 30 20 10

4 Jan95

Jan96

Jan97

Jan98

Jan99

0 Jan95

Jan00

Fig. 101.2 The short rates and swap spreads. The left panel depicts the instantaneous interest rate (in percentages) implied from the swap market (solid line) and the US Treasury market (dashed line). The right

Jan96

Jan97

Jan98

Jan99

Jan00

panel depicts the spread, in basis points, between the short rate from the US Treasury market and the short rate from the Treasury market

The Curvature Factor

x 10−3

The Slope Factor 4

0.05

3.5 Curvature, 2λ2

Slope, −κ

0.04 0.03 0.02

3 2.5 2 1.5

0.01

1

0 Jan95

0.5 Jan96

Jan97

Jan98

Jan99

Jan00

Jan95

Jan96

Jan97

Jan98

Jan99

Jan00

Fig. 101.3 The slope factor › and the curvature factor  . Lines depict the slope factor (›, left panel) and the curvature factor ( , right panel) extracted from the swap market (solid line) and the US Treasury market (dashed line)

the assumptions on positive interest rates and finite state dynamics and if not, how jumps can be incorporated into the state dynamics. We start with the degenerating case that the jump component has zero weight in the state dynamics. Then, our previous analysis indicates that zero prices can be written as  ln P .rt ; / D c ./ rt ;

(101.35)

Af .x/ D

R0C

.m .dy/ C x .dy// ; where a0 D a C

p drt D ›rt dt C  rt d wt ;

(101.36)

and the coefficient c ./ satisfies a Riccati equation. As we discussed before, this model serves as a special example of a one-factor affine model. Duffie et al. (2000) incorporate Poisson jumps in the affine structure. Filipovi´c (2001) incorporates more general jumps in a one-factor affine structure. Since we are dealing with a one factor structure, we consider the more general jump specification in Filipovi´c (2001). Filipovi´c (2001) proves that under the general affine framework, the positive short rate rt is a CBI-process (Conservative Branching Process with Immigration), uniquely characterized by its generator

R R0C

(101.37)

.1 ^ y/ m .dy/ for some numbers

 ; a 2 RC ; › 2 R and nonnegative Borel measures m .dy/ and  .dy/ on R0C (the positive real line excluding zero) satisfying 2

where the short rate rt follows a square-root dynamics with a zero mean:

  1 2 00  xf .x/ C a0  ›x f 0 .x/ 2 Z   C f .x C y/  f .x/  f 0 .x/ .1 ^ y/

Z

Z .1 ^ y/ m .dy/ C R0C

R0C

  1 ^ y 2  .dy/ < 1:

(101.38) We can obtain our current model by setting the jump part to zero and the constant part of the drift of the square root process to zero (a D 0). The two Borel measures define two jump components. The jump component defined by m .dy/ is a direct addition to the diffusion process. The jump component defined by  .dy/ is specified as proportional to x. Hence, we label the former as a constant jump component and the latter a proportional jump component. In essence, the arrival rate of jumps in the “constant” component does not depend on the short rate level, but the arrival rate of the

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“proportional” component is proportional to the short rate level. Condition (101.38) requires that the jump component defined by m.dy/ exhibit finite variation and the jump component defined by .dy/ exhibit finite quadratic variation. Under the specification in Equation (101.37), the zero prices are given by  ln P .rt ; / D A ./ C B ./ rt

(101.39)

with A ./ and B ./ solve uniquely the generalized Riccati equations B 0 ./ D R .B .// ; B .0/ D 0 Z  A ./ D F .B .s// ds;

(101.40) (101.41)

0

where R and F are defined as 1 R ./  1  ›   2 2 2 Z   1  e y   .1 ^ y/  .dy/ I C R0C

(101.42)

Z

  1  e y m .dy/ :

F ./  a C R0C

To guarantee that all rates are bounded from zero, we need to set A ./ D 0 for all , which we obtain by setting a D 0 and m .dy/ D 0. The condition a D 0 is already known. The second condition m .dy/ D 0 says that we cannot add a constant jump component while maintaining that all rates are bounded from zero. Nevertheless, we can incorporate a proportional jump component. Since B ./ is positive for all , all interest rates are bounded from zero. In the absence of the proportional jump component, R ./ is reduced to our Riccati equation for the diffusion case. The last term in Equation (101.42) captures the contribution of the proportional jump component.

101.7 Conclusion In this paper, we contend that all interest rates should be bounded from below at zero. Such a seemingly innocuous contention, together with the assumption of continuity, results in a dramatic collapse of dimensionality. The conditions lead to a term structure model that has only one dynamic factor and two static factors. Even more surprising, there are no other parameters in the model that affect the shape of the term structure. Therefore, model calibration becomes a

trivial problem and there no longer exists a distinction between out-of-sample and in-sample performance. Furthermore, risks from the three factors can be hedged away easily with only a few instruments. Since there are no more parameters, the model is not subject to any parameter risk. To put the model into practical application, we cast the model in a state space framework and estimate the three states via quasi maximum likelihood together with an extended Kalman filter. We apply this estimation procedure to both the US Treasury market and the US dollar swap market. Despite its extreme simplicity, the model performs well in fitting the daily term structures of both markets. A time series analysis of the extracted factors from the two markets provides us with some interesting insights on the evolution of the interest rate market. A potential application of the model, which can be explored in future research, is to forecast the term structure of interest rates. Recently, Diebold and Li (2003) and Diebold et al. (2004) illustrate how the Nelson–Siegel framework can be applied successfully to forecasting the term structure of Treasury yields. Yet, the inherent inconsistency of the Nelson–Siegel model is well-documented in Björk and Christensen (1999) and Filipovi´c (1999, 2000). Our model provides a parsimonious but consistent alternative to the Nelson–Siegel framework. Another line for future research is to explore the model’s implications for option pricing. Acknowledgment We thank Cheng F. Lee (the editor), Yacine Ait-Sahalia, Peter Carr, and Massoud Heidari for insightful comments. All remaining errors are ours.

References Ahn, D.-H., R. F. Dittmar, and A. R. Gallant. 2002. “Quadratic term structure models: theory and evidence.” Review of Financial Studies 15(1), 243–288. Babbs, S. H. 1997. Rational bounds, Working paper, First National Bank of Chicago. Backus, D., S. Foresi, A. Mozumdar, and L. Wu. 2001. “Predictable changes in yields and forward rates.” Journal of Financial Economics 59(3), 281–311. Balduzzi, P., S. Das, S. Foresi, and R. Sundaram. 1996. “A simple approach to three-factor affine term structure models.” Journal of Fixed Income 6, 43–53. Beaglehole, D. R. and M. Tenney. 1991. “General solution of some interest rate-contingent claim pricing equations.” Journal of Fixed Income 1, 69–83. Beaglehole, D. R. and M. Tenney. 1992. “A nonlinear equilibrium model of term structures of interest rates: corrections and additions.” Journal of Financial Economics 32(3), 345–454. Björk, T. and B. J. Christensen. 1999. “Interest rate dynamics and consistent forward rate curves.” Mathematical Finance 9(4), 323–348. Black, F. 1995. “Interest rates as options.” Journal of Finance 50(5), 1371–1376. Brandt, M. and D. A. Chapman. 2002. Comparing multifactor models of the term structure, Working paper, Duke University.

101 Taking Positive Interest Rates Seriously Brandt, M., and A. Yaron. 2001. Time-consistent no-arbitrage models of the term structure, Working paper, University of Pennsylvania. Brown, R. H. and S. M. Schaefer. 2000. Why long term forward interest rates (almost) always slope downwards, Working paper, Warburg Dillion Read and London Business School, UK. Chen, R.-R. and L. Scott. 1993. “Maximum likelihood estimation of a multifactor equilibrium model of the term structure of interest rates.” Journal of Fixed Income 3, 14–31. Constantinides G. M. 1992. “A theory of the nominal term structure of interest rates.” Review of Financial Studies 5(4), 531–552. Cox, J. C., J. E. Ingersoll, and S. R. Ross. 1985. “A theory of the term structure of interest rates.” Econometrica 53(2), 385–408. Dai, Q. and K. Singleton. 2000. “Specification analysis of affine term structure models.” Journal of Finance 55(5), 1943–1978. Dai, Q. and K. Singleton. 2002. “Expectation puzzles, time-varying risk premia, and affine models of the term structure.” Journal of Financial Economics 63(3), 415–441. Dai, Q. and K. Singleton. 2003. “Term structure dynamics in theory and reality.” Review of Financial Studies 16(3), 631–678. Diebold, F. X. and C. Li. 2003. Forecasting the term structure of government bond yields, Working paper, University of Pennsylvania. Diebold, F. X., L. Ji, and C. Li. 2004. “A three-factor yield curve model: non-affine structure, systematic risk sources, and generalized duration,” in Memorial volume for Albert Ando. Cheltenham, L. Klein (Ed.). Edward Elgar, UK. Duffee, G. R. 2002. “Term premia and interest rate forecasts in affine models.” Journal of Finance 57(1), 405–443. Duffie, D. 1992. Dynamic asset pricing theory, 2nd Edition, Princeton University Press, Princeton, NJ. Duffie, D. and R. Kan. 1996. “A yield-factor model of interest rates.” Mathematical Finance 6(4), 379–406. Duffie, D., J. Pan, and K. Singleton. 2000. “Transform analysis and asset pricing for affine jump diffusions.” Econometrica 68(6), 1343–1376. El Karoui, N., R. Myneni, and R. Viswanathan. 1992. Arbitrage pricing and hedging of interest rate claims with state variables: I theory, Working paper, University of Paris. Filipovi´c, D. 1999. “A note on the Nelson–Siegel family.” Mathematical Finance 9(4), 349–359. Filipovi´c, D. 2000. “Exponential–polynomial families and the term structure of interest rates.” Bernoulli 6(1), 1–27. Filipovi´c, D. 2001. “A general characterization of one factor affine term structure models.” Finance and Stochastics 5(3), 389–412. Flesaker, B. and L. Hughston. 1996. “Positive interest.” RISK 9(1), 46–49. Goldstein, R. and W. Keirstead. 1997. On the term structure of interest rates in the presence of reflecting and absorbing boundaries, Working paper, Ohio-State University. Gorovoi, V. and V. Linetsky. 2004. “Black’s model of interest rates as options, eigenfunction expansions and Japanese interest rates.” Mathematical Finance 14(1), 49–78. Heath, D., R. Jarrow, and A. Morton. 1992. “Bond pricing and the term structure of interest rates: a new technology for contingent claims valuation.” Econometrica 60(1), 77–105. Heidari, M. and L. Wu. 2003. “Are interest rate derivatives spanned by the term structure of interest rates?.” Journal of Fixed Income 13(1), 75–86. Jacod, J. and A. N. Shiryaev. 1987. Limit theorems for stochastic processes, Springer, Berlin. Jamshidian, F. 1996. “Bond, futures and option valuation in the quadratic interest rate model.” Applied Mathematical Finance 3, 93–115. Jin, Y., and P. Glasserman. 2001. “Equilibrium positive interest rates: a unified view.” Review of Financial Studies 14(1), 187–214.

1501 Kalman, R. E. 1960. “A new approach to linear filtering and prediction problems.” Transactions of the ASME – Journal of Basic Engineering 82(Series D), 35–45. Knez, P. J., R. Litterman, and J. Scheinkman. 1994. “Explorations into factors explaining money market returns.” Journal of Finance 49(5), 1861–1882. Leippold, M. and L. Wu. 2002. “Asset pricing under the quadratic class.” Journal of Financial and Quantitative Analysis 37(2), 271–295. Leippold, M. and L. Wu. 2003. “Design and estimation of quadratic term structure models.” European Finance Review 7(1), 47–73. Litterman, R. and J. Scheinkman. 1991. “Common factors affecting bond returns.” Journal of Fixed Income 1(1), 54–61. Longstaff, F. A. 1989. “A nonlinear general equilibrium model of the term structure of interest rates.” Journal of Financial Economics 23, 195–224. Longstaff, F. A., P. Santa-Clara, and E. S. Schwartz. 2001. “Throwing away a million dollars: the cost of suboptimal exercise strategies in the swaptions market.” Journal of Financial Economics 62(1), 39–66. Norgaard, M., N. K. Poulsen, and O. Raven. 2000. “New developments in state estimation for nonlinear systems.” Automatica 36(11), 1627–1638. Pan, E. 1998. “Collpase of detail.” International Journal of Theoretical and Applied Finance 1(2), 247–282. Rogers, L. C. G. 1995. Mathematical finance, vol. IMA Volume 65, Springer, New York. Rogers, L. C. G. 1996. “Gaussian errors.” RISK 9, 42–45. Rogers, L. C. G. 1997. “The potential approach to the term structure of interest rates and foreign exchange rates.” Mathematical Finance 7, 157–176.

Appendix 101A Factor Representation The term structure is determined by the following ordinary differential equation: 1 c 0 ./ D 1  ›c./   2 c./2 ; 2

(101.43)

with c.0/ D 0. One solution of this Riccati equation is given in Equation (101.27). Another way of solving the equation is through the following change of variables: c./ 2 C › ; ./  p ›2 C 2 2

D

1p 2 › C 2 2 ; 2

(101.44)

where ›2 C 2 2 defines the discriminant of the ordinary differential equation. Then the ordinary differential equation (101.43) is transformed into the elementary problem 0

  ./ D  1  ./2

with .0/ D ›=.2/. The solution of Equation (101.45) is ./ D tanh Œ .  M / ;

(101.45)

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St D HXt C et ;

where M is defined by the boundary condition .0/ D tanh .M / D That is,

Translating gives c./ D

› : 2

the Kalman Filter provides the efficient a posteriori update on the conditional mean and variance of the state vector:

› 1 M D  arctanh :  2 ./ back to the bond pricing coefficients c./

2 Œtanh  .t  M / C tanh M 2

22 r sech2  .  M / 2



›2 22 r Dr 1C 2 ; F D 2 2

bt D .I  Kt H / V t ; V

  St  H X t Xt C et ;

is the maximal forward rate and M is the corresponding maturity.

where 

H Xt

Appendix 101B Extended Kalman Filter and Quasilikelihood The state space estimation method is based on a pair of state propagation and measurement equations. In our application, the state vector X propagates according to VAR(1) processes specified in Equation (101.32). The measurement equation is given in Equation (101.33), which is based on the valuation of the par yield. Let X t denote the a priori forecast of the state vector at time t conditional on time t  1 information and V t the corresponding conditional covariance matrix. Let b t denote the a posteriori update on the time t state vecX bt the corretor based on observations (St ) at time t and V sponding a posteriori covariance matrix. Then, based on the VAR(1) specification, the state propagation equation is linear and Gaussian. The a priori update equations are:



 ˇ @h X t ˇˇ D ˇ @Xt ˇ

:

(101.49)

(101.50)

Xt DX t

Thus, although we still use the original pricing relation to update the conditional mean, we update the conditional variance based on this linearization. For this purpose, we need to numerically evaluate the derivative defined in Equation (101.50). We follow Norgaard et al. (2000) in updating the Cholesky factors of the covariance matrices directly. Using the state and measurement updates, we obtain the one-period ahead forecasting error on the par yields,   et D St  S t D St  h X t : Assuming that the forecasting error is normally distributed, the quasi log-likelihood function is given by L .S/ D

T X

lt ;

(101.51)

t D1

b t 1 I X t D A C ˆX bt 1 ˆ> C Q: V t D ˆV

(101.48)

where S t and ANt are the a priori forecasts on the conditional mean and variance of the observed series and R are the covariance matrix of the measurement errors. However, in our application, the measurement equation in Equation (101.33) is nonlinear. We apply the Extended Kalman Filter (EKF), which approximates the nonlinear measurement equation with a linear expansion:

D F sech2  .  M / where

ANt D H V t H > C R  1 Kt D V t H ANt I   b t D X t C Kt St  S t I X

(101.46)

The instantaneous forward rate is given by f ./ D c 0 ./r D

St D HXtI

(101.47)

The filtering problem then consists of establishing the conditional density of the state vector Xt , conditional on the observations up to and including time t. In case of a linear measurement equation,

where ˇ ˇ 1 >  1 1 lt D  log ˇANt ˇ  e ANt et ; 2 2 t where the conditional mean S t and variance ANt are given in the EFK updates in Equation (101.48).

Chapter 102

Positive Interest Rates and Yields: Additional Serious Considerations* Jonathan Ingersoll

Abstract Over the past quarter century, mathematical modeling of the behavior of the interest rate and the resulting yield curve has been a topic of considerable interest. In the continuous-time modeling of stock prices, one only need specify the diffusion term, because the assumption of risk-neutrality for pricing identifies the expected change. But this is not true for yield curve modeling. This paper explores what types of diffusion and drift terms forbid negative yields, but nevertheless allow any yield to be arbitrarily close to zero. We show that several models have these characteristics; however, they may also have other odd properties. In particular, the square root model of Cox–Ingersoll–Ross has such a solution, but only in a singular case. In other cases, bubbles will occur in bond prices leading to unusually behaved solutions. Other models, such as the CIR three-halves power model, are free of such oddities.

bound. Section 102.3 reviews the Cox et al. (1985) and Pan Wu (2006) models and shows that the zero lower bound in the latter is due to the interest rate process, which has an absorbing barrier at zero and no finite-variance steady-state distribution. Section 102.4 derives the bubble-free solution to the Pan Wu model when the risk-neutral and true processes differ in their behavior at zero. Section 102.5 develops a two-state variable extension to the Cox, Ingersoll, Ross model which permits a lower bound of zero for all yields with no price bubbles and with no absorption of the interest rate at zero. Section 102.6 discusses non-affine models of yields and shows it is possible to have interest rate models with a finite-variance steady-state distribution, no absorption at zero, and a lower bound of zero for all yields. Section 102.7 briefly discusses other constant lower bounds for yields.

Keywords Term structure r Interest rates r Price bubbles Positivity

102.2 A Non-Zero Bound for Interest Rates

r

102.1 Introduction In a recent paper Pan and Wu (2006) have posited that (nominal) interest rates of all maturities should have a lower bound of exactly zero – that is, yields arbitrarily close to zero should be possible for bonds of all maturities. They derived the “unique” model in which this assumption together with continuity and the absence of arbitrage are satisfied. This paper questions that desirata and highlights some curiosities that models possessing this property also have. In addition, it shows that, far from there being a unique model with this property, there are in fact countless other models that satisfy these conditions even if only one source of risk is assumed. Section 102.2 of this paper briefly discusses the relation among yields and highlights the question of a zero lower

J. Ingersoll () Yale School of Management, New Haven, CT, USA e-mail: [email protected]

Pan and Wu argue: “Asserting that an interest rate can be negative or cannot be lower than, say, 3%, is equally absurd. For example, no rational traders are willing to offer free floors at any strictly positive level of interest rates.” We contend that these claims are vastly different. The presence of cash alone requires that nominal interest rates of all maturities be nonnegative. On the other hand, the lack of interest rate floors is just common sense. It is true that the Cox–Ingersoll–Ross interest rate model and other similar models absolutely prohibit yields to various maturities below certain non-zero levels so that interest rate floors at some positive interest rate should be cost-free. But why would any trader offer a zero-cost floor at a rate he knew the interest rate could never reach? True, he could not lose on such a contract, but there would be no possibility of gain either if they were being given for free. Conversely, no buyer would be willing to pay any positive

*

The author has benefited from his discussions with his colleagues. This paper is reprinted from Advances in Quantitative Analysis of Finance and Accounting, 7 (2009), pp. 219–252.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_102, 

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price and would completely indifferent about receiving such a floor at a zero cost. There would simply be no market for such contracts. In any case we must not forget that interest rate models are models, that is, simplifications of the world. By their very nature, all models make absolute claims of one type or another. The model that Pan and Wu derived, for example, requires that the 5-year yield to maturity always be greater than (or less than depending on the model’s two parameters) the 3-month yield to maturity. But no trader would quote swap rates based on this guarantee. If a model that is based on reasonable assumptions makes a surprising prediction that should be taken as a good sign. We want our models to tell us things we didn’t know or otherwise lead to new intuitions. The model’s assumptions may turn out to be wrong, but the logical process that leads us to the surprising conclusion is valuable nonetheless. So the appropriate question becomes: Is the conclusion that yields to maturity possess strictly positive bounds a surprising one? I do not think so and believe just the opposite in fact. Historically, yield curves have been downward sloping when the spot rate is high and upward sloping when the spot rate is low. This is generally explained by saying that long rates are related to expected spot rates and that the spot rate has some long-run or steady state distribution so that the expected rate is always closer to the long-run average than is the current spot rate. This notion can be made precise as follows. Continuous-time interest rate models generally assume that bonds can be priced using a risk-neutral (or equivalent martingale) process1 and a form of the expectations hypothesis for discounting. In particular the price of a zero-coupon bond is given by  Z Et exp  Pt .r; £/ D b

If the instantaneous rate is bounded below by zero but free to move above zero, the usual properties of averages would seem to guarantee that long-term yields would be bounded away from zero with the exact bound depending on the properties of the stochastic process generating the evolution of the interest rate and the maturity of the bond in question. Apparently it would not be surprising to find models with yields bounded away from zero; quite the contrary, we should expect just that property. In fact were the average in Equation (102.2) an arithmetic one, this property would be universally true and long rates would be bounded away from zero whenever positive interest rates remained possible in the future. Geometric averages are different, though. A negative geometric average is never greater than the corresponding arithmetic average, and if there is any variation in the random variable it is strictly less.2 So yields may have lower bounds of exactly zero even when the instantaneous rate is guaranteed to be remain positive. Before going on to explore positive interest rates in more detail we review the Cox et al. (1985) and the special case of it, which is the Pan and Wu term (2006) structure models in the next section.

102.3 The Cox–Ingersoll–Ross and Pan–Wu Term Structure Models Though derived in a different fashion, the Pan–Wu (henceforth PW) model is a special case of the Cox–Ingersoll–Ross (henceforth CIR) affine term structure mode in which the risk-neutral dynamics of the instantaneous interest rate are



t C£

rs ds

:

p ^ drt D ›.™  rt /dt C ¢ rt d¨t :

(102.1)

(102.3)

t ^

Here b Et denotes the expectation at time t with respect to the equivalent martingale process, and rs is the instantaneous spot rate prevailing at time s. When bonds are priced like this, the yield to maturity is the negative geometric expectation of the instantaneous rates over the time interval 1 Yt .r; £/   `n Pt .r; £/ £   Z t C£

 1 D  `n b rs ds : (102.2) Et exp  £ t

The symbol D is used as a reminder that the dynamics in Equation (102.3) are the equivalent-martingale “risk-neutral” dynamics, the pricing equation for all zero-coupon bonds is 0 D 12 ¢ 2 rPrr C ›.™  r/Pr  rP  P£

where £  T  t is the time until the maturity date T . The CIR solution is P .r; £/ D A.£/ expŒB.£/r 2

1

When pricing bonds and other fixed-income assets in the presence of interest rate uncertainty, the equivalent martingale process that allows discounting at the interest rate does not result from assuming riskneutrality on the part of investors as it does in the Black–Scholes model. Nevertheless, the term “risk-neutral” is still commonly applied. See Cox et al. (1981) for further discussion of this matter.

(102.4)



(102.5)

In discrete time, the yield in Equation (102.2) is 1 C Y 1=n



O Œ..1 C r1 /    .1 C rn //1 E . By Jensen’s inequality this is less O Œ..1 C r1 /    .1 C rn //1=n . So one plus the yield to maturity is than E

less than the expectation of the geometric average of one plus the future prevailing spot rates. Since a geometric average is never larger than the corresponding arithmetic average, the yield to maturity must be less than the risk-neutral expected spot rate prevailing in the future.

102 Positive Interest Rates and Yields: Additional Serious Considerations

Fig. 102.1 Lower bound for CIR yield curves. In the CIR model, the yield to maturity for any bond is Y .r; £/ D `nŒA.£/ =£ C B.£/r=£. The £ period yield is increasing in the spot rate, r, and has a lower bound of Y .£/ D `nŒA.£/ =£, which is achieved when the spot rate is zero. For the PW model, A.£/ 1 and all yields have a lower bound

where B.£/ 

2.1  e ”£ / 2” C .›  ”/.1  e ”£ / 

2”e .›”/£=2 A.£/  2” C .›  ”/.1  e ”£ / p ”  ›2 C 2¢ 2 :

2›™=¢ 2

The PW derivation is substantially different from that used by CIR. They start with a general vector diffusion, ut , and then to ensure positive interest rates assume that yields of all maturities have a quadratic form y.ut ; £/ D 1£ u0t W.£/ut where W is a symmetric positive definite matrix. They go on to show that the only model satisfying these conditions is equivalent to the one-factor, two-parameter model given in Equation (102.3) and (102.5) with ™ D 0. In particular, for the PW model, A.£/  1. In the CIR and PW models, the yield to maturity for a zero-coupon bond of maturity £ is `n A.£/ B.£/ 1 C r: Y .r; £/   `n P .r; £/ D  £ £ £

(102.6)

Since A.£/  1; B.£/ 0, and the interest rate cannot go below zero for the assumed dynamics, the £-maturity yield is clearly bounded below by Y .£/  `n.A.£//=£, which is strictly positive except when ™ D 0, the PW case.

1505

of zero. The parameter ¢ D 0:078 gives a standard deviation of changes in the interest rate equal to 1.9% point at the mean interest rate level of ™ D 6%. The parameter › measures the rate of return towards the mean level. After t years the interest rate will on average have moved the fraction e›t of the distance back toward ™

The limiting yield to maturity for an infinitely lived zerocoupon bond is 2›™=.› C ”/, which is somewhat less than ™. Since this quantity is independent of the current interest rate, it is also the lower bound for the infinite-maturity zerocoupon rate, which is constant in the CIR model. As shown in Fig. 102.1, the lower bounds for shorter maturity rates increase from zero (at £ D 0) to approach this level asymptotically. The rate of increase in the lower bounds is governed primarily by the parameter › with larger values producing a faster approach and generating higher lower bounds for yields of all maturities.3 The difference between the lower bounds in the two models is due to the behavior of the interest rate at zero. For the CIR model with ™ > 0, zero is a natural reflecting barrier of the interest rate process. If the interest rate hits zero, the uncertainty momentarily vanishes, and the then certain change in the interest rate is an increase – immediately moving it back to a positive value. For the PW model, r D 0 is an absorbing barrier. The uncertainty still vanishes, but now the expected future change is also zero, so the interest rate remains stuck at that level. Once the instantaneous interest rate reaches zero, it and the yields of all maturities are zero

3

The lower bounds for all yields also increase with › because the asymptotic value, Y1 D 2›™=.› C ”/, does. See Dybvig, et al. (1996) for an analysis of the asymptotic long rate.

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forever. It is obvious, therefore, that no yield can have a positive lower bound. Furthermore, this property is true in any model of interest rates in which zero is an accessible absorbing barrier for the interest rate.4 A closely related feature of the PW model is that the values of zero-coupon bonds do not approach zero for long maturities. B.£/ is an increasing function but is bounded above by 2=.› C ”/, so no zero-coupon bond has a price less than expŒ2r=.› C ”/ regardless of its maturity. In other words, no zero-coupon bond has a value less than the value that a 2=.› C ”/ year bond would have were the yield curve flat at r. For the parameters estimated by PW this “maximal” maturity is about 50 years. This limit also means that the prices for annuities and coupon bonds with fixed coupons will become unbounded as the maturity increases. This aspect of the model is certainly problematic in many other contexts as well since transversality will be violated for many valuation problems. Since the interest rate can be trapped at zero, it is not surprising that long bond prices do not vanish. However, this problem does not follow directly from the interest rate being trapped at zero. It can be true in other models as well. For example, in Dothan’s (1978) model of interest rates, long-term bond prices also do not approach zero, although the interest rate has a lognormal distribution for which zero is inaccessible.5 It is also true in the two-factor version of the CIR model discussed in Sect. 102.4 below.

102.4 Bubble-Free Prices An interesting question remains. Can yields have a lower bound of zero, even when the interest rate cannot be trapped at zero and the values of zero-coupon bonds do become vanishingly small for long maturities? At first the answer to this question would seem obviously to be yes. Pricing is based on the risk-neutral process, but whether or not the interest rate is trapped at zero is a property of the true process. Apparently all that would be required is that the true process not have an absorbing state at zero while the risk-neutral process did. For example, the true and risk-neutral processes could be p drt D .r/dt C ¢ rt d¨t p ^ drt D ›rt dt C ¢ rt d¨t : 4

.0/ > 0

With .0/ > 0, zero will be a natural reflecting barrier for the interest rate process just as it is in the CIR model. Whenever the interest rate reaches zero, it will immediately become positive again. As always, the diffusion term in the risk-neutral process is identical to that in the true process, but the drift term is altered. Unfortunately, the true and risk-neutral processes in Equation (102.7) are not equivalent as is required for a proper risk-neutral process. The true process results in a continuous probability density for r defined over all nonnegative values.6 The risk-neutral process also has a continuous distribution over all positive r – in particular a non-central chi squared distribution, but there is an atom of probability at zero as well. As shown in the appendix, the risk-neutral probability that the interest rate will have reached zero and been trapped there at or before time T is  ı  PbrfrT D 0jrt D rg D exp 2›r Œ¢ 2 .e ›.T t /  1/ : (102.8) There is a corresponding positive state price atom (not simply a positive state price density) associated with the state rT D 0. The Arrow–Debreu price for the state (atom) rT D 0 when rt D r is7 Q.r; t/ D exp

˚

 ı  ›  ” cothŒ 12 ”.T  t/ r ¢ 2 :

But under the true process, rT > 0 with probability one, so the true state price (like the true probability) cannot have a positive atom for rT D 0. As previously stated the problem here is a lack of equivalence between the true and risk-neutral measures. They do not possess the same probability-zero sets of states. There are, in fact, two opposite situations to consider. If zero is inaccessible under the true process, then the risk-neutral process clearly has a larger set of possibilities. On the other hand, if zero is accessible under both processes, then the

6

The drift term, ./, must satisfy mild regularity conditions. If  is continuous and the stochastic process is not explosive so that r D 1 is inaccessible, then the density function for r 2 .0; 1/ will exist for all future t with a limiting steady state distribution of c

(102.7)

See, for example, Longstaff (1992), which solves the bond-pricing problem in the CIR square-root framework with r D 0, an absorbing barrier. 5 In Dothan’s 1978 model, interest prates.evolves

. as dr D ¢rd¨. The p 8r ¢ ¢ > 0, where K1 is the asymptotic bond price is 8r K1 modified Bessel function of the second kind of order one. See, in particular, Fig. 102.2 on p. 66.

(102.9)

¢ 2r

 Z exp 2¢ 2



r

.x/x 1 dx

where c is chosen to ensure it integrates to unity. The density function may of course be zero for some values. 7 The function coth.x/ is the hyperbolic cotangent: coth x .e x C e x /=.e x  e x /. The hyperbolic functions are related to the standard circular functions as: sinh x D i sin ix; cosh x D cos ix; tanh x D i tan ix; and coth x D i cot ix:

102 Positive Interest Rates and Yields: Additional Serious Considerations

true process has a large set of possibilities – namely those in which the interest rate reaches the origin and becomes positive again. The latter case, when zero is accessible but not an absorbing barrier under the true process, is irreconcilable with the assumed risk-neutral process. The risk-neutral process completely specifies the partial differential equation and this cannot be altered to modify the probabilities for “interior” states after the origin has been reached and left. In the former case, when zero is inaccessible, the risk-neutral process can be reconciled with the true process making the measures equivalent by assigning zero probability to the offending rT D 0 state. This can be done because the offending state is at a boundary of the distribution so the assignment can be handled by an appropriate boundary condition to the partial differential Equation (102.4) leaving it and therefore the risk-neutral diffusion process itself unchanged. In the remainder of this section we assume that zero is inaccessible under the true process. A sufficient condition for zero to be inaccessible is that .r/ 12 ¢ 2 for all r sufficiently small.8 The boundary condition applied at r D 0 to the bond pricing problem clearly has the form P .0; £/ D p.£/ for some function p.£/. In the PW model, p.£/ D 1 is used, but other assumptions can be made without changing the local no-arbitrage condition inherent in the partial differential equation. Furthermore, since zero is inaccessible, the condition will never actually apply. But what function is logically consistent? Fix a particular boundary condition, p.£/, by conjecture. Now consider a contract that grants 1=p.£/ zero-coupon bonds when the interest rate is zero and then terminates. Such a contract is clearly worthless, if r D 0 is truly inaccessible. However, the risk-neutral pricing procedure will assign a price equal to the expected present value of receiving Œ1=p.£/  p.£/ D 1 when r reaches zero. This is just the value of Q as given in Equation (102.9) above. Therefore, we must have p.£/ D 0. We shall refer to this boundary condition and the resulting solution as bubble-free for reasons to be explained below. As shown in Appendix 102A, the value of a zerocoupon bond under the true and risk-neutral processes in Equation (102.7) and the correct no-arbitrage boundary condition P .0; £/ D 0 when zero is inaccessible is P BF .r; £/ D expŒrB.£/ .1  expŒrŸ.£/ / where Ÿ.£/ 

2” 2 Œ” sinh.”£/ C › cosh.”£/  › 1 : ¢2 (102.10)

8 This condition is sufficient because zero is inaccessible for the CIR p process dr D k.r  r/dt C ¢ rd¨ if 2kr  ¢ 2 , and by assumption, the specified process with drift ./  12 ¢ 2 for small r dominates the CIR process for sufficiently small r. This model is a special case of Heston, et al. (2007).

1507

The function B.£/ and the parameter ” are the same as in the CIR and PW models so this formula is just the PW price multiplied by the factor 1  expŒŸ.£/r . Since Ÿ.£/ > 0 for £ > 0, the value in Equation (102.10) is strictly less than the PW model price at any time before the bond matures. The yield to maturity computed from Equation (102.10) is 1 Y BF .r; £/   `n P BF .r; £/ £ 1 B.£/ r  `n .1  expŒrŸ.£/ / : D £ £ (102.11) Since bond prices are lower than in the PW model, yields are correspondingly higher. However, as shown in Fig. 102.2, the yield curves are very similar for short maturities particularly when the spot rate is high. The PW and bubble-free yield curves match better at high rates because the boundary behavior then has less of an effect. As in the PW model, the yield curve is humped as a function of maturity when › < 0, but for realistic parameters, the peak occurs at longer maturities than in their model.9 Unlike the PW model, bubble-free yields are not monotonic in the short rate, but have a U-shape, and the lower bound is not zero. For the £-period yield, the lower bound is Y BF .£/ D

Ÿ.£/ B.£/ B.£/ C Ÿ.£/ 1 `n  `n > 0; £Ÿ.£/ B.£/ £ B.£/ C Ÿ.£/ (102.12)

which is achieved when the interest rate is rmin .£/ D

B.£/ C Ÿ.£/ 1 `n : Ÿ.£/ B.£/

(102.13)

Figure 102.3 plots the yields for various maturities as a function of the spot rate. For interest rates above 4%, yields of all maturities through 20 years are nearly proportional to the spot rate just as they are (exactly) in the PW model. Below a spot rate of about 4%, yields have a strong U-shape in the spot rate. The minimum yield for each maturity is not zero, and for longer maturities can be quite high. A side-effect of the bubble-free structure is that it restores the property that the values of zero-coupon bonds go to zero as £ goes to infinity. This can be verified by Equation (102.10) but it must be true since yields for all positive maturities are bounded away from zero, and P .r; £/  expŒY .£/£ . Perhaps a more surprising result is that yields of all maturities become unboundedly large as the interest rate itself

9

In Figs. 102.2 through 102.6, the parameters used, › D 0:03; ¢ D 0:04, are the midpoints of the estimates by PW, though they were fitting their bond pricing function not the bubble-free function.

1508

J. Ingersoll

Fig. 102.2 Pan–Wu and bubble-free yield curves. The PW yield is given ı in Equation (102.6) as rB.£/=£ with B.£/ 2.1  e ”£ / Œ2” C .›  ”/.1  e ”£ / . The bubble free yield is £1 ŒB.£/r  `n.1  expŒrŸ.£/ with Ÿ.£/ 2” 2 ¢ 2 Œ” sinh.”£/ C

› cosh.”£/  › 1 as given in Equation (102.11). In each case the riskp ^ neutral evolution of the interest rate is dr D ›rdt C ¢ r  d¨. The parameter choices › D 0:03 and ¢ D 0:04 correspond to the middle of the range estimated by Pan and Wu by fitting yield curves

Fig. 102.3 Bubble-free ble free yield is given expŒrŸ.£/ with Ÿ.£/ The parameter choices to the middle of the ting yield curves. All

proportional to the spot rate when the latter exceeds 4%. The lower bound for the £-period yield is given in Equation (102.12) D B.£/ =Œ£Ÿ.£/ .`nŒB.£/ C Ÿ.£/  `n B.£//  as Y .£/ .`nŸ.£/  `nŒB.£/ C Ÿ.£/ /= £, which is achieved at a spot rate of `n Œ1 C Ÿ.£/=B.£/ = Ÿ.£/

yield as a function of the spot rate. The bubin Equation (102.11): £1 ŒB.£/r  `n.1  2” 2 ¢ 2 Œ” sinh.”£/ C › cosh.”£/  › 1 . › D 0:03 and ¢ D 0:04 correspond range estimated by Pan and Wu by fityields through 20 years are approximately

102 Positive Interest Rates and Yields: Additional Serious Considerations

approaches zero. The intuition for this surprising result is found in the bond’s risk premium. The bond’s risk premium,  .r; £/, can be determined by Ito’s Lemma

˜.r; £/ 

1509

 B.£/ @P=@r D B.£/ C Ÿ.£/fexpŒrŸ.£/  1g1 P

PW BF (102.17)

1 2 ¢ rPrr C .r/Pr  P£ EŒdP D 2 dt: Œr C  .r; £/ dt  P P (102.14)

Comparing Equation (102.14) to the pricing equation, which uses the risk-neutral process, we have for the risk-neutral and true dynamics given in Equation (102.7)  .r; £/ D

@P = @r Œ.r/ C ›r : P

(102.15)

The semi-elasticity, .@P =@r/=P , also determines each bond’s return risk. Again by Ito’s Lemma   dP dP @P = @r p E D ¢ rd¨: P P P

(102.16)

This, of course, is no coincidence. The absence of arbitrage requires that assets whose returns are perfectly correlated have risk premiums proportional to their standard deviations. The relations Equation (102.15) and (102.16) are true for both the PW and BF prices, though the semi-elasticities differ. Under the PW and bubble-free solutions the semielasticities are

Fig. 102.4 Semi-elasticity of bubble-free bond prices. The semi=P D B.£/ C elasticity is given in Equation (102.17): ˜.r; £/ Pr ı Ÿ.£/fexpŒrŸ.£/  1g1 with B.£/ 2.1  e ”£ / Œ2” C .›  ”/ .1e ”£ / and Ÿ.£/ 2” 2 ¢ 2 Œ” sinh.”£/C› cosh.”£/› 1 . The parameter choices › D 0:03 and ¢ D 0:04 correspond to the middle of

Figure 102.4 shows the semi-elasticity function, ˜.r; £/  Pr =P , for the bubble-free prices. (In the PW model, the semi-elasticity is constant as a function of r at the asymptote shown.) As the interest rate increases, the semi-elasticity approaches B.£/, and both the risk and term premium are approximately independent of the interest rate. Where this flattening occurs depends on the bond’s maturity. For bonds with maturities less than 2 years, ˜ is nearly constant for all interest rates above 1.5%. For bonds with maturities in excess of 10 years, ˜ does not flatten out until the interest rate is above 5%. This leads to the unbounded risk premiums and a number of other unusual effects. Under the PW formula, the risk premium at low interest rates is negative since at r D 0 the premium is  .0; £/ D B.£/.0/ < 0. Because the bond price can never exceed one and, for the PW model, the price approaches one as r nears zero, the expected change in the bond price, and therefore its risk premium, must be negative near r D 0. For the bubble-free price, the semi-elasticity and therefore the risk premium becomes unboundedly large driving the bond price to zero as r approaches zero.

the range estimated by Pan and Wu bypfitting yield curves. Any bond’s return standard deviation is j˜.r; £/j ¢ r with a negative value of ˜ indicating that bond’s price decreases with an increase in the interest rate. Any bond’s term premium is ˜.r; £/Œ.r/ C ›r

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J. Ingersoll

Fig. 102.5 Risk premiums of bubble-free bond prices. The risk premium is the product of the semi-elasticity in Equation (102.17): ˜.r; £/ Pr =P D B.£/ C Ÿ.£/fexpŒrŸ.£/  1g1 and the difference between the true and risk-neutral expected changes, .r/ C ›r.

It is plotted for .r/ D 0:01.0:06  r/. The other parameter choices › D 0:03 and ¢ D 0:04 correspond to the middle of the range estimated by Pan and Wu by fitting yield curves

At some interest rate levels, longer maturity bonds are less risky and have smaller term premiums than shorter maturity bonds. For example, when the spot rate is 2%, 10-year bonds are less risky than 2-year bonds. At low enough interest rates levels, the risk changes sign and bond prices are increasing in the interest rate. For example, this reversal occurs around 1.5% for 5-year bonds. The change in sign of the semi-elasticity means that the risk premiums also change in sign. In fact in many cases the term premium changes in sign twice. This is true for example for all zero-coupon bonds if the true process mean is .r/ D k.r  r/ with k > ›. Typical term premiums are illustrated in Fig. 102.5. Both the PW and BF prices are self-consistent. That is, if zero coupon bonds traded at either model’s prices, they would move in response to changes in interest rates just as the model predicts, and the risk premiums indicated in Equations (102.15) and (102.17) would be earned. However, this is always true of prices with bubbles. It is only in comparison to an alternate price that the bubble is obvious. The magnitude of the PW bubble is

arbitrage would exist by selling the bonds and replicating them at a cost equal to this lower value according to the replicating hedge inherent in the partial differential equation. Suppose bonds sold at the higher PW price. An investment matching the lower BF price could be achieved via a portfolio that holds n.r; £/ D PrBF =PrPW bonds and invests the residual, P BF  nPPW , in instantaneous lending. The change in value of this portfolio is d Port D n.r; £/dPPW C ŒP BF  n.r; £/P PW r  dt   p D n.r; £/ Œr C ˜PW .r/ P PW dt C PrPW ¢ rd¨ CŒP BF  n.r; £/P PW r  dt D

PrBF PrPW PW P .r/dt C P BF r  dt PrPW P PW

PrBF PW p P ¢ rd¨ PrPW r   p D r C ˜BF .r; £/.r/ P BF dt C PrBF ¢ rd¨ D dP BF C

(102.19) P P W  P BF D e rŒB.£/CŸ.£/ :

(102.18)

Unlike most bubbles, this bubble disappears at a fixed point in time, when the bond matures and is always between 0 and 1 in magnitude. Therefore, this bubble is finite in duration and bounded in size. If bonds sold at the higher PW price, an

which exactly matches the change in the BF formula, so this portfolio would always be self-financing and always equal in value to P BF . A long position in this portfolio and a short position in the bond (at the PW price) would be an arbitrage. It would have a negative cost and be guaranteed to

102 Positive Interest Rates and Yields: Additional Serious Considerations

1511

Fig. 102.6 Price bubble in Pan–Wu bond prices. The difference between the Pan–Wu and Bubble-Free price is a price bubble. The magnitude of the price bubble is e rŒB.£/CŸ.£/ where B.£/

2.1  e ”£ / =Œ2” C .›  ”/.1  e ”£ / and Ÿ.£/ 2” 2 ¢ 2 Œ” sinh.”£/ C › cosh.”£/  › 1 . Unlike most price bubbles this bubble is bounded in value between 0 and 1 and disappears after a fixed finite

be worth zero when the bond matured. Furthermore, since P PW > P BF , the arbitrage will never have a negative value at any point during the arbitrage. The PW model is valid and gives a lower bound of zero for all yields only when zero is an absorbing barrier under both the true and risk-neutral interest rate processes. The next obvious question is can we have a lower bound of zero for all yields even when the interest rate process does display mean reversion? The answer is yes as we show in the next two sections of this paper.

claim is not true. There are other continuous term structure models in which there is no arbitrage and all interest rates are bounded by zero exactly. In fact there are infinitely many other affine term structure models with a single (excluding parameter) source of risk that have a lower bound of zero for yields. The simplest such model extends the CIR dynamics so that the mean interest rate level, ™, is no longer a constant but is a weighted average of past interest rates. Specifically let xt be an exponentially smoothed average of past spot rates with an average lag of 1=•11 Z 1 xt D • e •s rt s ds: (102.20)

102.5 Multivariate Affine Term-Structure Models with Zero Bounds on Yields Pan and Wu (2006) claim: “Positivity and continuity, combined with no arbitrage, result in only one functional form for the term structure with three sources of risk.”10 But this

0

The dynamics of xt are locally deterministic, and the evolution of the state space is p drt D ›.xt  rt /dt C ¢ rt d¨t dxt D •.rt  xt /dt

10

Pan and Wu refer to their model as a three-factor (i.e., r; ›; ¢) model with a single dynamic factor, r. In fitting their model they allow the parameters to vary over time, hence adding two additional sources of risk. This “stochastic-parameter” method has been widely used in practice since being introduced to term-structure modeling by Black et al. (1990). As Pan and Wu point out, this is inconsistent with their derivation, which assumes the parameters to be constant. Were the parameters actually varying, then bond prices would not be given by Equation (102.5) or (102.10). A true multifactor model giving results similar to PW would be a special case of the Longstaff and Schwartz (1992) multifactor extension to the CIR model with thep constants in the drift terms set to zero. That is, dsi D ›i si dt C ¢i s i d¨i and r D s1 C s2 CPs3 . The zero-coupon yield to maturity in this model is Y .s; £/ D £1 Bi .£/si . This Longstaff–Schwartz model is an imme-

(102.21)

with a single source of risk, d¨.

diate counterexample to PW’s claim that their formula is unique, but as in the PW model each of the state variables can be trapped at zero, and the interest rate becomes trapped at zero once all three state variables are so trapped. R1 11 The average lag in the exponential average is • 0 se•s ds D •1 . An exponentially smoothed average is the continuous-time equivalent of a discrete-time geometrically smoothed average xt D .1  ˜/ P ˜s rts . Geometrically smoothed averages were first suggested in interest rate modeling by Malkiel (1966).

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J. Ingersoll

Unlike in the PW model, the interest rate cannot be trapped at zero under the process in Equation (102.21). Since x is a positively weighted average of past values of r, it clearly must remain positive. Even were r to reach zero,12 x would only decay towards zero at the rate •. But when r reaches zero, its diffusion term is zero so the immediate change in r is dr D ›x  dt, and as x is still positive at this point, dr > 0; r immediately becomes positive again. Assuming the factor risk premium is linear in the state variables,  .r; x; £/ D .§0 r C §1 x/Pr =P , then the bondpricing equation is13 1 0 D ¢ 2 rPrr C Œ.›  §1 /x  .› C §0 /r Pr 2 C •.r  x/Px  rP  P£ :

(102.22)

The price of a zero-coupon bond has the form P .r; x; £/ D expŒb.£/r  c.£/x :

(102.23)

This pricing formula can be easily verified by substituting the partial derivatives of P in Equation (102.23) into the pricing Equation (102.22). The terms proportional to r and x must separately sum to zero so the functions b and c are the solutions to the linked ordinary first-order differential equations

r terms W 0 D 12 ¢ 2 b 2 .£/ C .› C §0 /b.£/  •c.£/  1 C b 0 .£/ with b.0/ D 0 x terms W 0 D .§1  ›/b.£/ C •c.£/ C c 0 .£/ with c.0/ D 0:

(102.24)

While a closed-form solution to Equation (102.24) is not known, b.£/ and c.£/ can be easily computed by numerically

12

It is irrelevant for this discussion whether or not zero is accessible; if zero is not accessible for r then clearly neither r nor x can become negative. By comparison to the CIR process, however, we can determine that 0 is accessible for r (though not x). Specifically compare the CIR process with ™ < 2¢ 2 =› to the process in Equation (102.21). The diffusion terms are identical and the excepted change under the bivariate process is smaller than for the CIR process whenever both r and x are less than ™. Since 0 is accessible for the CIR process, it must be accessible for the dominated bivariate process. The fact that r and x can be larger than ™ does not alter this conclusion as the accessibility of 0 depends only on the behavior of r and x near 0. 13 Only r is locally stochastic, so the risk premium is proportional to Pr =P and independent of Px =P . The risk-neutral and true processes are equivalent if and only if §1 < › so that r remains positive under the risk-neutral process as well.

integrating the two equations.14 Note that this calculation need only be done once for a given set of parameters; bond prices at each interest rate level and maturity need not be separately computed as in a finite difference or binomial model. For this model, the yield to maturity on a £-period zerocoupon bond is b.£/ c.£/ 1 rC x: Y .r; x; £/   `n P .r; £/ D £ £ £ (102.25) Since both r and x can be arbitrarily close to zero, yields to maturity have lower bounds of exactly zero for all maturities as in the PW model even though the interest rate exhibits mean reversion as in the CIR model. It should be noted, however, that the lower bound of zero cannot be approached immediately. In the PW model, yields are proportional to the interest rate, and since r can be arbitrarily close to zero at any time, so can the yields. In this two factor model r can also be arbitrarily close to zero at any time, but xt cannot be smaller than x0 e•t ; therefore, the £-period yield cannot be less than £1 c.£/x0 e•t at time t. Figures 102.7 and 102.8 display the yield curve for the two-factor model. Figure 102.7 compares it to the CIR yield curve for the same parameters and when xt D ™. The twofactor yield curve is less steeply sloped than the CIR yield curve because x and r both tend to move towards each other rather than r simply moving toward ™. This lessens future expected movements in the interest rate. Figure 102.8 illustrates the effect on the yield curve of different values of •. The larger is •, the less steeply sloped is the yield curve as x then moves more strongly towards r. Both figures also show that the long zero-coupon yields go to zero as the maturity lengthens regardless of the parameter values (provided • is not zero which is the CIR model). This is true even though the interest rate can never be trapped at zero as in the PW model verifying that the latter is not a prerequisite to the former. One problem with this model, apart from the lack of a simple closed-form expression for the solution, is that the longterm behavior of the interest rate is unrealistic. The model displays short-term mean reversion with r staying near x, but over long periods of time, the interest rate is likely to become

14

In particular, b.£ C £/  b.£/    12 ¢ 2 b 2 .£/ C .› C §0 /b.£/  •c.£/  1 £

b.0/ D 0

c.£ C £/  c.£/  Œ.§1  ›/b.£/ C •c.£/ £

c.0/ D 0:

102 Positive Interest Rates and Yields: Additional Serious Considerations

1513

Fig. 102.7 Cox–Ingersoll–Ross and two-factor affine yield curves. This figure shows the yields to maturity for the CIR and two-factor model in Equation (102.25). The parameters are › D 0:7; • D 0:5; ¢ D

0:078. The two factor yield curve is less steeply sloped than the CIR yield curve and is always downward sloping to zero for large maturities

Fig. 102.8 Two-factor yield curves illustrating dependence on •. This figure shows the yields to maturity for the two-factor model in Equation (102.25). The parameters are › D 0:7 and ¢ D 0:078. The

yield curve is plotted for various values of •, the reciprocal of the average lag in the central tendency mean

very large as there is no steady state distribution for the interest rate. In particular, the expected interest rate at time t,

does not reach a limit independent of the current state, and the variance of rt

  › .r0  x0 / 1  e .›C•/t ›C• •r0 C ›x0 ; (102.26) ! t !1 •C›

EŒrt jr0 ; x0 D r0 

VarŒrt2 

¢ 2 •2 .•r0 C ›x0 /t C o.t/: 2›.› C •/3 (102.27)

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J. Ingersoll

becomes unbounded. So, in the limit, the distribution of rt is completely diffuse over all positive values. Of course, a similar criticism can also be leveled against the PW model. Those dynamics also have no steady state. The interest rate variance becomes unbounded, and the spot rate is either trapped at zero or has an infinite expectation.15 Also as in the PW model, the prices of very long zerocoupon bonds are bounded away from zero as £ becomes large. For given values of r and x, the smallest that a bond price can be is 8£ P .r; x; £/ > P  expŒb1 r  c1 x p .§0 C §1 /2 C 2¢ 2  §0  §1 where b1  ¢2 ›  §1 b1 : c1  (102.28) • This can be verified with Equation (102.24). The derivatives c 0 and b 0 are positive (zero) whenever c is less than (equal to) .›  §1 /b=• and •c is greater than (equal to) 1  .› C §0 /b  1 2 2 0 2 ¢ b , respectively. Since b.0/ D c.0/ D c .0/ D 0 and 0 b .0/ > 0, both b and c increase monotonically staying in the range bounded by the b 0 D 0 and c 0 D 0 curves approaching b1 and c1 as £ ! 1. Similar properties are true for all affine models with multiple state variables. Consider the general multiple statevariable extension to the CIR model with a risk-neutral evolution of p ^ dr D .r; O x/dt C ¢ rd¨

X p ^ D KO C ›O 0 r C ›O i xi dt C ¢ rd¨

X •ji xi dtj D 1; : : : I: dxj D j .r; x/dt D j C •j 0 r C (102.29)

The same analysis leading to Equation (102.24) shows the yield curve will be affine Y .r; x; £/ D a.£/ C b.£/r C

X

ci .£/xi :

102.6 Non-Affine Term Structures with Yields Bounded at Zero Outside of the affine class, many models of the term structure will have zero lower bounds for all rates and still be wellbehaved with mean reversion and a finite-variance steadystate distribution with no atom of probability at r D 0. One such model that admits to a closed form solution for bond prices is the three-halves power model in which the interest rate’s evolution is 3=2

drt D ›rt .™  rt /dt C ¢rt d¨t ;

(102.31)

with ›; ™; ¢ > 0.17 This structure was introduced in the original CIR (1985) paper to model the rate of inflation. Like the CIR process, this diffusion displays mean reversion with a central tendency of ™, and the local variance vanishes when rt D 0 so negative rates are impossible.18 The interest rate has a finite-variance steady state distribution with a density function, mean, and variance of

(102.30)

so all yields will have a lower bound of exactly zero only if a.£/  0 and all state variables remain nonnegative. The former requires that the constant terms, KO and all j , are zero. The latter requires additionally that all ›O i .i ¤ 0/ and all •ij .j ¤ i / are nonnegative and that all state variables are

15

nonnegative initially.16 The interest rate will never be trapped at zero so long as one of the parameters, ›O i .i ¤ 0/, is positive. However, as in the two-variable case, there will not be a finite-variance steady-state distribution, and the prices of all zero-coupon bonds will be bounded away from zero even as the maturity grows without bound. Our search for an affine model in which the interest rate is well behaved in the long term and yields of all maturities are bounded below exactly by zero has not been successful. But such models are possible outside of the affine structure. One such model has already appeared in the literature. It is the three-halves power model used in the two-factor CIR model. This model is discussed in the next section.

If ›  0 in the PW model, then the interest rate is eventually trapped at zero with probability one. If › < 0, then the expected interest rate ›t ! 1; VarŒrt jr0 D and variance  ›t become ı infinite, EŒrt jr0 D r0 e 2 2›t e › ! 1, and there is an atom of probability for r0 ¢ e r1 D 0 equal to exp.2›r0 =¢ 2 /.

16 In addition, the risk-neutral process must also be equivalent to the true process so all state variables must remain nonnegative under the latter as well. 17 The central tendency parameter, ™, and the adjustment parameter, ›, can be zero just as in the PW model. For the three-halves process, the origin remains inaccessible even in these cases, and yields are still bounded below by zero. There is, however, no finite-variance steadystate distribution. 18 Since the drift term is zero at an interest rate of zero, rt D 0 is technically an absorbing state. However, zero is inaccessible for all parameter values so r is never trapped there. To verify this define z D 1=r. pThen using Itô’s Lemma, the evolution of z is dz D .›C¢ 2 ›™z/dt¢ zd¨. Since z D 1 is inaccessible for the square root process with linear drift, zero is inaccessible for r D 1=z. Note also that 2.› C ¢ 2 / > ¢ 2 so zero is inaccessible for z guaranteeing that 1 is inaccessible for r in Equation (102.31).

102 Positive Interest Rates and Yields: Additional Serious Considerations

f .r; 1/ D r 1  EŒr1 D

Œ.“  1/™ 1C“ “ .“1/™=r r e .“ C 1/

“1 ™ “

Var Œr1 D

The three-halves power model may fit the data better than the CIR model and other proposed models. Chan et al. (1992) tested the general specification for interest rate evolution EΩt D 0

2”

EŒ©2t D ¢ 2 rt : (102.33)

Using GMM they found a best unrestricted fit of ” D 1:4999 with a standard error of 0.252. This, of course, is not a direct test of Equation (102.31), which has a quadratic rather than linear form for the expected interest rate, but the estimated

P .r; £/ D

(102.32)

value of ” is almost exactly what this model calls for. The CIR-square-root, Merton (1975, 1990), and Vasicek (1977) models are all well outside the usual confidence intervals, and the Brennan and Schwartz (1982) and Dothan (1978) models are just at the 5% significance level.19 We assume that the risk premium is of the form  .r; £/  r D .§1 r C §2 r 2 /Pr =P so that the risk-neutral process also has the form in Equation (102.31). The risk-neutral drift is O then ›O r.™r/ with ›O  ›C§2 and ™O  .›™§1 /=.›C§2 /.20 Zero-coupon bond prices for this model can be determined from Cox et al. (1985) as

.  •/ Œc.£/=r • M.•; ; c.£/=r/ ./

1 ›™  §1  .›™§1 /£ e 1 ; “O  1 C 2.› C §2 /= ¢ 2 2 ¢ O •  12 .“O 2 C 8 =¢ 2 /1=2  12 “;   1 C .“O 2 C 8 =¢ 2 /1=2 ;

c.£/  2

where

./ is the gamma function, and M./ is the confluent hypergeometric function.21

Y1 





lim  1£ `n P .r; £/ £!1

( 

• `n £



 e .›™§1 /£  1 C O.£1 / • £ o.£/

See Abramowitz and Stegum (1964) for the properties of the gamma and confluent hypergeometric functions.

(102.34)

The asymptotic long rate is

! £!1 ! £!1

19

21

¢ 2 .“  1/2 ™2 ¢2 2 D r 2› “2 2› 1

“  1 C 2›=¢ 2:

where

rt D ’ C “rt 1 C ©t

1515

O .›™  §1 /• D ›O ™• 0

for ›O > 0 for ›O  0: (102.35)

The volatility parameter is ” D 0 for Merton (1990) and Vasicek (1977), ” D 12 for CIR, and ” D 1 for Brennan and Schwartz (1982), Dothan (1978) and Merton (1975). Each of these models with the exception of Merton’s (1975) does have a linear form for the expected change in r. 20 We require that §2  .› C 12 ¢ 2 / so that the true and risk-neutral processes are equivalent. If this condition is not satisfied then the riskneutral process is explosive, and the interest rate can become infinite in finite time. As shown in footnote 18, the risk-neutral process for z O O Œdz D .O› C¢ 2  ›O ™z/dt. So if §2 violates the condition given, 1=r has E 2 2.O› C ¢ / < ¢ 2 , and 0 is accessible for z implying that 1 is accessible in finite time for r under the risk-neutral (though not true) process.

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J. Ingersoll

When §2 < › (i.e., ›O > 0), Y1 is a positive constant a bit less O 22 The asympthan the risk-neutral central tendency level ™. totic bond price, P .r; 1/, is also zero when §2  ›.23 For short maturities

P Substituting P .r; £/ D expŒ yi .£/r i and its derivatives into the bond pricing equation gives 0D

Y .r; £/ D r C 12 rŒ›™  §1  .› C §2 /r £ C O.£2 / D r C 12 ›O r.™O  r/ £ C O.£2 /

(102.36)



so the yield curve is upward sloping at £ D 0 whenever the interest rate is below the risk-neutral central tendency point, O < ™O since ›O • < 1; therefore, O The long rate is Y1 D ›O ™• ™. the yield curve is humped shape whenever the spot interest O rate is between the values ›O •™O and ™. Even though the asymptotic yield to maturity is a positive constant (when ›O > 0), the yield to maturity on any finite maturity bond is bounded below exactly by zero. As shown in the Appendix, the £-period yield to maturity is an analytic function of r and can be expressed as a power series with no leading constant term, 2r 2.› C §2 /r 2 1  C : Y .r; £/   `n P .r; £/ D 2 £ ¢ £c.£/ ¢ 4 £c 2 .£/ (102.37) Therefore, for any ı finite maturity, the yield to that maturity is less than 2r Œ¢ 2 £c.£/ when r is sufficiently small and approaches its lower bound of zero as r does.24 These properties can also be verified for many models even when a closed-form solution to the bond pricing problem cannot be found. Suppose that the risk-neutral expected change and variance in the interest rate process and the yield to maturity are analytic functions of the interest rate at r D 0; that is, they can be expressed as the infinite power series ¢ 2 .r/ D

1 X

.r/ O D

si r i

i D0

Y .r; £/ D

1 X

mi r i

i D0 1 X

£1 yi .£/r i :

(102.38)

i D0

A yield to maturity will have the desired lower bound of exactly zero if and only if the lead term in its expansion is zero; that is, y0 .£/  0. Holding ¢ constant, ›O • increases from 0 to 1 when “O ranges from 1, its lowest value, to 1. 23 Whenı §2 > ›, the asymptotic bond price is P .r; 1/ D .  •/ ./Œ2O›=¢ 2 • M.•; ; 2O›=¢ 2 / > 0. 24 The lower bound for any yield is zero since for every finite £, there exists an interest rate r£ such that Y .r; £/ < © for all r < r£ . The bound is not a uniform one for all £, however, in that r£ depends on £. The bound cannot be uniform since the asymptotic long rate is a positive constant.

i2 1 X i hX si r .i C 1/yi C1 .£/r i 2

X  .i C 1/.i C 2/yi C2 .£/r i

D

X

mi r i

hX

i hX i .i C 1/yi C1 .£/r i r C yi0 .£/r i

 1  2 s0 y1 .£/  2y2 .£/  m0 y1 .£/ C y00 .£/ 2 C r  Œ   C r 2  Œ   C   

(102.39)

The terms in each power of r must be identically zero, so y0 .£/ will be constant (and therefore 0 since yi .0/ D 0) if and only if s0 D m0 D 0. If s0 D 0, then zero is a natural barrier for the interest rate process and negative rates will be precluded if .0/ 0.25 If m0 also is zero, then rt D 0 is an absorbing barrier; however, the barrier might be inaccessible as in the three-halves power model. Therefore, models like CIR (1985) or Vasicek (1977) with a linear expected change in the interest rate, .r/ D ›.™ r/, will not have yields with a lower bound of zero. Conversely, a model like that in Merton (1975) with ¢.r/ D vr and .r/ D ar  br2 will have all yields bounded below by exactly zero.

102.7 Non-Zero Bounds for Yields All of the results presented here apply to other constant bounds for yields that derive from restricted interest rate processes. Suppose the interest rate is restricted so that it can never go below r. Under what conditions will all yields have this same lower bound of r ? This question can be easily answered by defining the modified interest rate, ¡  r  r. Since ¡ and r are linearly related, the dynamics for ¡ will be identical to those for r translated down by r, and the modified rate, ¡, can never be negative. All yields will have an identical lower bound of r whenever yields in the modified economy are all bounded below by zero. To verify this claim write the interest rate dynamics as dr D .r/dt C ¢.r/d¨. Now express the bond prices in the original economy as P .r; £/ D e r£ ‚.¡; £/. The bond pricing equation can then be reexpressed as

22

1 2 ¢ .r/Prr C .r/Pr  rP  P£ 2 1 D ¢ 2 .¡ C r/‚¡¡ C .¡ C r/‚¡  ¡‚  ‚£ 2

0D

25

(102.40)

Zero can also be an inaccessible natural barrier for the process if .0/ D 1 even if ¢.0/ ¤ 0.

102 Positive Interest Rates and Yields: Additional Serious Considerations

This is just the regular bond pricing equation for an interest rate ¡ with expected change and standard deviation of ¡ .¡/  .¡ C r/ and ¢¡ .¡ C r/. So all yields in the original economy will be larger than those in the modified economy by exactly r. In particular, all yields will have a lower bound of r if, and only if, they have a lower bound of 0 in the modified economy. For example, in Sundaresan (1984),26 the dynamics of the real interest rate are   dr D .’  1/.r C ¢ 2 / .r  • C 12 ’¢ 2 /dt C ¢d¨ (102.41) with r > ¢ . Rewriting this in terms of ¡  r C¢ , we have 2

2

  1 d¡ D .’  1/¡ .¡  • C .’  2/¢ 2 /dt C ¢d¨ : 2 (102.42) which is identical to Merton’s (1975) model. All yields have a lower bound of 0 in Merton’s model; therefore, all yields in Sundaresan’s model are bounded below by exactly ¢ 2 .

102.8 Conclusion This paper has established the properties of interest-rate models in which all yields have a lower bound equal to the lower bound of the interest rate itself. In particular, when the interest rate must remain positive, it is possible for all yields to have a lower bound of zero as well. Yields are not simple arithmetic expectations (or even risk-neutral expectations) of future short rates; therefore, they can be as low as the lowest possible interest rate even when the interest rate process displays mean reversion and has a finite-variance steady-state distribution. This paper also illustrates the problems that can arise when then true and risk-neutral stochastic processes for the interest rate have different boundary behaviors. Price bubbles can be introduced unless events that are impossible under the true distribution are assigned zero probability under the riskneutral process as well. In some cases, such as the Pan Wu model, alternate bubble-free prices can be derived.

26 Equation (102.41) fixes a typo in the second unnumbered equation on p. 84 of Sundaresan (1984).

1517

References Abramowitz, M. and I. Stegum. 1964. Handbook of mathematical functions, Dover Publications, New York. Black, F., E. Derman, and W. Toy. 1990. “A one-factor model of interest rates and its application to treasury bond options.” Financial Analysts Journal 46, 33–39. Brennan, M. J. and E. S Schwartz. 1982. “An equilibrium model of bond pricing and a test of market efficiency.” Journal of Financial and Quantitative Analysis 41, 301–329. Chan, K. C., G. Andrew Karolyi, F. Longstaff, and A. Sanders 1992. “An empirical comparison of alternative models of the short-term interest rate.” Journal of Finance 47, 1209–1227. Cox, J. C., J. E. Ingersoll, and S. A. Ross. 1981. “A re-examination of traditional hypotheses about the term structure of interest rates.” Journal of Finance 36, 769–799. Cox, J. C., J. E. Ingersoll, and S. A. Ross. 1985. “A theory of the term structure of interest rates.” Econometrica 53, 385–408. Dothan, U. 1978. “On the term structure of interest rates.” Journal of Financial Economics 6, 59–69. Dybvig, P. H., J. E. Ingersoll, and S. A. Ross. 1996. “Long forward and zero-coupon rates can never fall.” Journal of Business 69, 1–25. Heston, S. L., M. Lowenstein, and G. A. Willard. 2007. “Options and bubbles.” Review of Financial Studies 20, 359–390. Longstaff, F. A. 1992. “Multiple equilibria and term structure models.” Journal of Financial Economics 32, 333–344. Longstaff, F. A. and E. S. Schwartz. 1992. “Interest rate volatility and the term-structure of interest rates: a two-factor general equilibrium model.” Journal of Finance 47, 1259–1282. Malkiel, B. 1966. The term structure of interest rates: expectations and behavior patterns, Princeton University Press, Princeton. Merton, R. C. 1975. “An asymptotic theory of growth under uncertainty.” Review of Economic Studies 42, 375–393. Merton, R. C. 1990. “A dynamic general equilibrium model of the asset market and its application to the pricing of the capital structure of the firm,” chapter 11 in Continuous-time finance Basil Blackwell Cambridge, MA. Pan, E. and L. Wu. 2006. “Taking positive interest rates seriously.” chapter 14, Advances in Quantitative Analysis of Finance and Accounting 4, Reprinted as chapter 98 herein. Sundaresan, M. 1984. “Consumption and equilibrium interest rates in stochastic production economies.” Journal of Finance 39, 77–92. Vasicek, O. 1977. “An equilibrium characterization of the term structure.” Journal of Financial Economics 5, 177–188.

Appendix 102A 102A.1 Derivation of the Probability and State price for r T = 0 for the PW Model Let H.r; t/ be probability that rT D 0 conditional on rt D r for the square root stochastic process with no mean reversion: dr D ›rdt C ¢r 1=2 d¨. Let Q.r; £/ be the state price for the state rT D 0; that is, Q.r; £/ is the value at time t when the interest rate is r of receiving $1 if the interest rate is zero at time T D t C £. This section of the appendix verifies Equations (102.8) and (102.9) in the text that

1518

J. Ingersoll

ı where h.£/  2› ¢ 2 .e ›£  1/ ı where q.£/  Œ›  ” coth. 12 ”£/ ¢ 2 :

H.r; tI T / D expŒh.£/r Q.r; £/ D expŒq.£/r

(102A.1)

The probability H satisfies the Kolmogorov backward equation and boundary conditions

0 D 12 ¢ 2 rHrr  ›rHr  H£

subject to H.0; tI T / D 1 and H.r; T I T / D 0:

The condition H.0; t/ D 1 must be satisfied because zero is an absorbing state so once r reaches zero before time T , it will be there at time T with probability one.

Hr D  expŒrh.£/ h.£/

(102A.2)

For the solution given, H.0; t/ D 1 and H.r; T / D 0 are readily confirmed. The solution itself can be verified by differentiating and substituting into Equation (102A.2). The partial derivatives we need are

Hrr D expŒrh.£/ h2 .£/

H£ D  expŒrh.£/ rh0 .£/

with h0 .£/ D 2›2 ¢ 2 .e ›£  1/2 e ›£ D  12 ¢ 2 e ›£ h2 .£/:

(102A.3)

So we have   1 2 ›£ › 1 2 1 2 2 ¢ rHrr  ›rHr  H£ D expŒrh.£/ rh .£/ ¢ C  ¢ e 2 2 h.£/ 2   1 2 1 2 ›£ 2 1 2 ›£ D expŒrh.£/ rh .£/ ¢ C 2 ¢ .e  1/  2 ¢ e D0 2

as required establishing the first part of Equation (102A.1). The state price for the state rT D 0 has the same value as a contract that pays $1 the first time that the interest rate hits zero because the interest rate is then trapped at zero, and

0 D 12 ¢ 2 rQrr  ›rQr  rQ  Q£

(102A.4)

the state rT D 0 will be realized for sure, and with a zero interest rate there is no further discounting. The value of this therefore asset satisfies the usual pricing partial differential equation

subject to Q.0; £/ D 1

and Q.r; 0/ D 0:

(102A.5)

The boundary conditions are confirmed since coth.0/ D 1. Again differentiating and substituting into Equation (102A.5) gives27

27

The derivative of the hyperbolic cotangent is the negative hyperbolic cosecant function d coth x=dx D csch2 x D 4.e x  e x /2 :

The third equality in Equation (102A.6) uses the identity coth2 x  csch2 x 1. The fourth equality follows from the definition of ”.

102 Positive Interest Rates and Yields: Additional Serious Considerations

1519

1 2 ¢ rQrr  ›rQr  rQ  Q£ 2  rQ  D 2 12 Œ›  ” coth. 12 ”£/ 2  ›Œ›  ” coth. 12 ”£/  ¢ 2  12 ” 2 csch2 . 12 ”£/ ¢

 1 2 rQ  rQ 2 1 2 2 1 2 1 D 2  ›  ¢ C 2 ” Œcoth . 2 ”£/  csch . 2 ”£/ D 2  12 ›2  ¢ 2 C 12 ” 2 D 0 ¢ 2 ¢

as required establishing Equation (102A.1).

the

second

part

(102A.6)

of setting the bond price to zero when the interest rate is zero. This boundary condition is required to make the risk-neutral and true processes equivalent.

102A.2 Bond Price When r t = 0 Is Accessible for Only the Risk-Neutral Process As discussed in the body of the paper, the pricing equation is the standard one with only an altered boundary condition

0 D 12 ¢ 2 rPrr ›rPr rPP£

P .r; 0/D1

P .0; £/ D 0: (102A.7)

The price of the bond is

P .r; £/ D expŒrB.£/ .1  expŒrŸ.£/ / where

Ÿ.£/ 

2” 2 ¢ 2 Œ” sinh.”£/ C › cosh.”£/  ›

B.£/ 

2 2.1  e ”£ / D : 1 2” C .›  ”/.1  e ”£ / ” coth. 2 ”£/ C ›

B.£/ is the same function as found in the CIR solution. The maturity condition are satisfied since B.0/ D 0; Ÿ.0/ D 1, and the boundary condition at r D 0 is clearly

(102A.8)

satisfied. That the solution satisfies the pricing partial differential equation can be verified by substituting the derivatives

Pr D B.£/P .r; £/ C Ÿ.£/ exp .rŒB.£/ C Ÿ.£/ / Prr D B 2 .£/P .r; £/  Ÿ.£/ŒŸ.£/ C B.£/ exp .rŒB.£/ C Ÿ.£/ / P£ D rB 0 .£/P .r; £/ C rŸ0 .£/ exp .rŒB.£/ C Ÿ.£/ /

(102A.9)

into the partial differential equation and collecting terms

0 D rP .r; £/

1

2¢

2

   B 2 C ›B  1 C B 0  r exp .rŒB.£/ C Ÿ.£/ / 12 ¢ 2 Ÿ2 C .› C 12 ¢ 2 B/Ÿ C Ÿ0 :

(102A.10)

1520

J. Ingersoll

The first term is zero because B.£/ is the same function as in the CIR model. So we need only verify that the final term in brackets is also zero. The derivative of Ÿ is

Ÿ0 .£/ D  D

2” 2 Œ” sinh.”£/ C › cosh.”£/  › 2 Œ” 2 cosh.”£/ C ”› sinh.”£/ ¢2 ¢2 2 Ÿ .£/Œ” 2 cosh.”£/ C ”› sinh.”£/ : 2” 2

(102A.11)

So the final term in brackets in Equation (102A.10) is  ı  1 2 2 1 1 ¢ Ÿ C .› C ¢ 2 B/Ÿ C Ÿ0 D ¢ 2 Ÿ2 1 C .2›= ¢ 2 C B/ Ÿ  cosh.”£/  .›= ”/ sin.”£/ 2 2 2 

 1 2 1 2 2 2 ›C ¢ B Œ” sinh.”£/ C › cosh.”£/  ›  cosh.”£/  .›= ”/ sin.”£/ D ¢ Ÿ 1C” 2 2   1 1 D ¢ 2 Ÿ2 ¢ 2 ” 2 BŒ” sinh.”£/ C › cosh.”£/  › C .›2 = ” 2  1/Œcosh.”£/  1 2 2   1 4 2 2 1 BŒ” sinh.”£/ C › cosh.”£/  ›  cosh.”£/ C 1 : (102A.12) D ¢ Ÿ ” 2 2

Substituting for B.£/ and using the “half-angle” identity,28   1 2 2 1 2 1 4 2 2 ” sinh.”£/ C › cosh.”£/  › 0 ¢ Ÿ C .› C ¢ B/Ÿ C Ÿ D ¢ Ÿ ”  cosh.”£/ C 1 2 2 2 ”Œcosh.”£/ C 1 = sinh.”£/ C › " # 1 4 2 2 ” sinh2 .”£/  ” cosh2 .”£/ C ” D ¢ Ÿ ” D 0: 2 ”Œcosh.”£/ C 1 C › sinh.”£/

102A.3 Properties of the Affine Exponentially Smoothed Average Model The multivariate affine model with a exponentially smoothed average has a Kolmogorov backward equation of 0 D 12 ¢ 2 rFrr C›.x r/Fr C•.r x/Fx F£ :

The “half-angle” identity is coth 12 x D .cosh x C 1/=sinh x. The final equality in Equation (102A.13) follows from the identity cosh2 x  sinh2 x D 1.

28

equation subject to various boundary conditions. In partic  ular, the expected values EŒrt C£ and E rt2C£ are solutions with boundary conditions F .r; x; 0/ D r and r 2 , respectively. The expected value of the interest rate is EŒrt C£ D

(102A.14)

The joint probability distribution for r and x, and other probably functions are the solutions to this partial differential

(102A.13)

rt Œ• C ›e .›C•/£ C ›xt Œ1  e .›C•/£ ›C• (102A.15)

The expected value of the square of the interest rate has a quite messy formula, but for times far in the future, its asymptotic behavior is EŒrt2C£ D

¢ 2 •2 .•rt C ›xt /£ C o.£/: 2›.› C •/3

(102A.16)

102 Positive Interest Rates and Yields: Additional Serious Considerations

Since the mean value converges to EŒr1 D .•rt C ›xt /= .• C ›/, the variance of rt C£ also diverges at the rate £.

where C is the constant of integration required to ensure the density integrates to one. For the three-halves power process the steady state density is  Z C f .r1 / D 2 3 exp 2 ¢ r

102A.4 Properties of the Three-Halves Power Interest Rate Process

f .x1 / D

C ¢ 2 .x/

x

 .z/= ¢ 2 .z/d z

r

 ›z.™  z/= ¢ z d z 2 3

2

2

D .2›™ =¢ 2/2C2›=¢ r 32›=¢ exp.2›™ =¢ 2 r/:

The steady-state distribution for a diffusion on .0; 1/ with inaccessible boundaries and evolution dx D .x/dt C ¢.x/d¨ is  Z exp 2

1521

(102A.18) For the risk-neutral dynamics 3

dr D rŒ›™  §1  .› C §2 /r dt C ¢r 2 d¨ (102A.17)

3

D ›O r.™O  r/dt C ¢r 2 d¨;

(102A.19)

the bond price is given in Cox et al. (1985) as

P .r; £/ D

.  •/ Œc.£/=r • M.•; ; c.£/=r/ ./

1 2O›™O  .›™§1 /£ e 1 ; “O  1 C 2O› = ¢ 2 2 ¢ O •  12 .“O 2 C 8 =¢ 2 /1=2  12 “;   1 C .“O 2 C 8 =¢ 2 /1=2 ;

where

c.£/ 

To help analyze the three-halves power model, we first present some preliminary results. Define the function

(102A.20)

m.zI a; b/  za M.a; b; 1=z/ where M.a; b; x/ is the confluent hypergeometric function.29 Then

m0 .z/ D aza1 M.a; b; 1 =z/ C za2 .a =b/M.a C 1; b C 1; 1 =z/ D aza1 M.a C 1; b; 1 =z/:

(102A.21)

By repeated application of Equation (102A.21), the nth derivative is m.n/ .z/ D .1/n a.a C 1/    .a C n  1/zan M.a C n; b; 1 =z/ D .1/n with

.a C n/ an z M.a C n; b; 1=z/ .a/ m.n/.0/ D .1/n

.a C n/.b/ : .a/.b  a  n/

(102A.22)

Since derivatives of all orders exist, m is analytic and can be expressed as the power series m.z/ D

1 X nD0

1 X ı zn m.n/ .0/ nŠ D .z/n nD0

29 .a C n/.b/ The derivative of the confluent hypergeometric function is : @M.a; b; x/=@x D .a=b/M.a C 1; b C 1; x/. The asymptotic behavior nŠ.a/.b  a  n/ as x ! 1 is M.a; b; x/ D .b/=.b  a/x a Œ1 C O.1=x/ . See (102A.23) Abramowitz and Stegum (1964).

1522

J. Ingersoll

The bond price in Equation (102A.20) therefore can be written as the power series30

.  •/ X .• C n/ .  •/ m .c.£/=rI •; / D P .r; £/ D Œr =c.£/ n .•/ .•/ nD0 nŠ.  •  n/ 1

D 1  Œr =c.£/ •.  •  1/ C 12 Œr =c.£/ 2 •.  •  1/.1 C •/.  •  2/     D 1

2r ¢ 2 c.£/

C

2.1 C ›O /r 2 C ¢ 4 c 2 .£/

(102A.24)

And since `n.1 C x/ D x  12 x 2 C    , the yield to maturity can be expressed as the power series 1 2r 2.O› C 1/r 2 2r 2 Y .r; £/   `n P .r; £/ D 2  4 2 C 4 2 C £ ¢ £c.£/ ¢ £c .£/ ¢ £c .£/ D

2O›r 2 2r  C O.Œr =c.£/ 3 /: ¢ 2 £c.£/ ¢ 4 £c 2 .£/

(102A.25)

At short maturities Y .r; £/ D r C 12 ›O r.™O  r/£ C O.£2 /:

30

Note that •.  •  1/ D ¢ 4

h 

›O C 12 ¢ 2

2

C 2¢ 2

(102A.26)

i1=2

C ›O C 12 ¢ 2

 h 

›O C 12 ¢ 2

2

C 2¢ 2

i1=2

  ›O  12 ¢ 2 D 2¢ 2 and   2•  2 D 2O›=¢ 2

Chapter 103

Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds* Cheng-Few Lee, Dilip K. Patro, and Bo Liu

Abstract This paper proposes a generalized functional form CAPM model for international closed-end country funds performance evaluation. It examines the effect of heterogeneous investment horizons on the portfolio choices in the global market. Empirical evidences suggest that there exist some empirical anomalies that are inconsistent with the traditional CAPM. These inconsistencies arise because the specification of the CAPM ignores the discrepancy between observed and true investment horizons. A comparison between the functional forms for share returns and NAV returns of closed-end country funds suggests that foreign investors may have more heterogeneous investment horizons compared to the U.S. counterparts. Market segmentation and government regulation does have some effect on the market efficiency. No matter which generalized functional model we use, the empirical evidence indicates that, on average, the risk-adjusted performance of international closed-end fund is negative even before the expenses. Keywords Closed-end country fund r Functional transformation r Performance evaluation

103.1 Introduction and Motivation A nonlinear functional form is more appropriate when returns are measured over an interval different from the “true” homogeneous investment horizon of investors that is assumed by the CAPM (Jensen 1969). Levy (1972) and D.K. Patro () OCC, 250 E Street SW, Washington, DC 20219, USA e-mail: [email protected] C.-F. Lee Department of Finance and Economics, Rutgers Business School, 94 Rockafeller Road, Piscataway, NJ 08854, USA e-mail: [email protected] B. Liu Citigroup Global Market Inc., 390 Greenwich Street, New York, NY 10013, USA e-mail: [email protected]

Levhari and Levy (1977) demonstrate that if the assumption of the holding period is different from the “true” investment horizon, there will be a systematic bias of the performance measurement index as well as the beta estimate. Lee (1976) has proposed a non-linear model to investigate the impact of investment horizon on the estimate of beta coefficients. Lee et al. (1990) have theoretically derived the relationship between heterogeneous investment horizons and capital asset pricing model. Empirically, Fabozzi et al. (1980) have used a generalized functional form approach developed by Zarembka (1968) to investigate the open-ended mutual fund return generation process. Chaudhury and Lee (1997) use the functional form model for international stock markets to investigate whether the international stock markets are integrated or segmented. The U.S. traded international closed-end funds (CEFs) provide a useful testing ground for the market efficiency in the framework of functional transformation (see Patro 2008). Mutual funds provide investors with heterogeneous investment horizons. As mentioned in Johnson (2004), investors’ liquidity needs are primarily revealed by their investment horizons. His empirical result shows that investors’ liquidity needs can vary significantly across individual investors, which leads to an obvious heterogeneous investment horizons phenomenon according to his duration model. The closed-end funds are unique in that they provide contemporaneous and observable market-based rates of returns for both funds and underlying asset portfolios. Moreover, for most funds the value of the underlying portfolio is known with considerable accuracy since the component assets are listed on the stock market. However, close-end funds typically trade at a substantial discount to the underlying value of their holdings (the net asset value (NAV) of the fund). The discount value is not constant, and varies considerably over time. Unlike domestic closed-end funds, international * A different version of the chapter was published in volume three of Advances in Financial Planning and Forecasting 2008, pp 235–278. The views expressed in this article are solely that of the authors and does not represent necessarily views of the OCC or Citigroup. We would like to thank Ren-Raw Chen, Ben Sopranzetti, Louis Scott and participants in ASSA 2005 Annual Meeting in Philadelphia for helpful comments.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_103, 

1523

1524

closed-end funds shares and underlying assets are traded in different markets. Therefore, share returns and NAV returns may display totally different distribution characteristics according to the corresponding markets. Investors from different markets will also have different consumption patterns and investment horizons, especially those investors from emerging markets. Prior research of Chang et al. (1995) finds that except for the Mexico Fund, the shares of a sample of 15 closed-end country funds did not outperform the Morgan Stanley Capital International (MSCI) world market index for the 1989–1990 periods. Patro (2001) also provides evidence of inferior performance of risk-adjusted share returns and NAV returns of 45 U.S.-based international closed-end funds over the 1991–1997 period. The main purpose of this chapter is to propose a generalized functional form model for closed-end mutual fund performance evaluation. First, we want to investigate whether the negative performance of international closed-end funds documented by previous studies is due to an incorrect specification of the return generating process. Second, we want to test whether international closed-end funds investors have heterogeneous investment horizons. Third, we want to provide a comprehensive analysis of the relationship between the functional form of the returns and fund characteristics, with special attention given to the difference between emerging funds and developed funds, single country funds and regional funds. We also consider the short sale effect on the return generating process of international CEFs. Short sale plays an important role in the determinant of the market efficiency. On the one hand, short sale facilitates efficiency of the price discovery; on the other hand, short sale may also facilitate severe price declines in individual security. A common conjecture by regulators is that short-sales restriction can reduce the severity of price declines. Hong and Stein (2003) developed a model linking short sale constraints to market crashes, where they find investors with negative information cannot reveal their information until the market begins to drop if they are constrained from short selling and their activities will further aggravate market declines and lead to a crash. Most research suggests that short sale constraints have an adverse effect on the market efficiency. Empirical evidence from both the U.S. and non-U.S. markets also supports the theoretical view that constraining short sale hinders price discovery – particularly when the news is bad. Aitken et al. (1998) provide evidence that short sale trades reflect significant bad news about companies. Jones and Lamont (2002) show that stocks with expensive short sale cost have higher valuations and thus lower subsequent returns, which are consistent with their hypothesis that stocks that are difficult to short sale are overpriced. Bris et al. (2004) find that there exists significant cross-sectional variation in equity returns in the markets where short selling is allowed or practiced con-

C.-F. Lee et al.

trolling for a host of other factors, while no such evidence is displayed in the non-short-selling markets. Although we doubt closed-end country fund managers will undertake short sale in their portfolio investment, short sales does affect investors’ behaviors in the local markets, which will further affect the prices of the stocks in the local market. Taking a look at the data-generating process of international CEFs, especially the movement of NAV returns, will provide a better understanding of the effect of government regulations on the financial markets. In this chapter, we use the generalized functional form approach to investigate performance of 90 closed-end country funds. Both linear and loglinear assumption of the return generating process are rejected for nearly 25% of closedend country funds. The wide distribution of estimated transformation parameters indicates that investors from foreign countries (especially emerging countries) have more heterogeneous investment horizons compared to their U.S. counterparts. The government regulation (short sale) effect on the functional form only affects the NAV returns. Further, we find that most of the international closed-end country funds are not integrated with the world market, even though the global risk factor is significant in several models. Consistent with previous research, after the consideration of the functional form transformation of share returns, international closed-end country funds generate negative risk-adjusted returns, no matter which benchmark we use. Moreover, in the framework of generalized CAPM, consistent with previous research, all factors in Carhart (1997) four-factor model remain as important pricing factors. However, when we include the global return into the model, the momentum factor no longer plays an important role. This chapter is organized as follows. In Sect. 103.2, we investigate the appropriateness of the functional form used by previous research. In Sect. 103.3, a generalized international capital asset pricing model is used to test whether the closed-end fund is integrated with the world market or not. Section 103.4 describes the data and testing methodology used in the empirical analysis. Finally, the empirical results are summarized in Sect. 103.5 and the conclusion is given in Sect. 103.6.

103.2 Literature Review 103.2.1 CES Functional Form of the CAPM The traditional CAPM assumes that all investors have the same single-period investment horizon. However, this assumption is unlikely to be true. In reality, individual investors have multiple investment horizons depending on their consumption patterns. The explicit consideration of

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds

multiperiod investment horizons generates several important implications on the empirical estimation of the systematic risk and the risk-return relationship. One of the multiperiod investment analyses is the nonlinear functional form of CAPM. Tobin (1965) pioneered the study of multiperiod investment CAPM. He analyzed the effect of the heterogeneous investment horizon on portfolio choices and developed a relationship between the risk and return measures of the single-period investment horizon and those of the multiperiod investment horizon. Following Tobin’s work, Jensen (1969) was the first to investigate the effect of investment horizon on the estimation of the systematic risk. Based on the instantaneous systematic risk concept, he concluded that the logarithmic-linear form of the CAPM could be used to eliminate systematic risk. However, Jensen did not include the investment horizon parameter in his model. Lee (1976) extended Jensen’s (1969) work and derived the CES functional form of the CAPM that introduces the functional form investment parameter into regression directly and then estimates the systematic risk beta in a homogeneous investment horizon framework. Lee (1976) showed that based on a homogeneous mean-variance preference structure and an equilibrium market, a risk-return relationship can be defined as: H H / D .1  ˇpH /.1 C RftH / C ˇpH E.1 C Rmt / E.1 C Rpt (103.1)

where H is the investment horizon assumed by the CAPM; H 1 C Rpt is the holding period return on the pth portfolio; 1 C H Rmt is the holding period return of the market portfolio; 1 C RftH is the holding period return of the risk-free asset; ˇpH is the systematic risk. Equation (103.1) implies that the risk return tradeoff is linear only when the investment horizon is the same as assumed by the CAPM. If the observed horizon is defined by N, which is not the same as H, then (1) can be rewritten as:

  n   o1=   N N  D 1  ˇpH 1 C RftN CˇpH E 1 C Rmt E 1CRpt (103.2) where  D H=N , In order to get the full nonlinear estimation in the above equation, using logarithm and Euler expansion, Equation (103.2) can be rewritten as:

1525

This implies that the CAPM should include a quadratic excess market return if H is not trivial. Following the standard regression method, both ˇpH and pH can be estimated. The adjusted coefficient of determination will reflect whether the new parameter of pH can improve the explanatory power of the CAPM. If the quadratic market return is significantly different from zero and is arbitrarily omitted, then the estimated systematic risk may be subjected to the specification bias.

103.2.2 Generalized Functional Form of the CAPM Using the Box and Cox (1964) transformation technique, Lee (1976) and Fabozzi et al. (1980) developed a generalized model to describe the mutual fund return-generating process: .1 C Rpt /./  .1 C Rft /./ D ˛p C ˇp ..1 C Rmt /./ .1 C Rft /./ C "pt (103.4) where .1CRpt /./ D .1 C Rmt /./ D

..1CRpt / 1/ ; .1CRft /./ 

D

..1CRft / 1/ , 

  ..1 C Rmt /  1/ ; "pt  N 0; "2 

 is the functional transformation parameter to be estimated across the pth mutual fund’s time series rates of return. Equation (103.4) includes both the linear and log-linear functional forms as special cases: When  D 1; Rpt  Rft D ˛p C ˇp .Rmt  Rft / C "pt When  D 0; ln.1 C Rpt /  ln.1 C Rft / D ˛p C ˇp .ln.1C Rmt /  ln.1 C Rft / C "pt Box and Cox (1964) used the maximum likelihood method to determine the functional form parameters. ln Max./ D N lnŒO " ./ C .  1/

N X

ln.1 C Rpt /

t D1



N Œln.2 / C 1 2

(103.5)

where N is the number of observation and O " ./ is the estimated regression residual standard error of Equation (103.4). After Equation (103.4) is estimated over a range of values for



2 , Equation (103.5) is used to determine the optimum value N  ln 1 C RfNt C pH ln E.1 C Rmt /  ln 1 C RfNt for  that maximizes the logarithmic likelihood over the pa(103.3) rameter space. McDonald (1983) found a generalized model to be ap

1 H H H propriate in a significant number of cases of his sample of where  D H ˇ 1  ˇ



   N N  ln 1 C RfNt D ˇpH ln E 1 C Rmt ln E 1 C Rpt

p

2

p

p

1526

C.-F. Lee et al.

1,164 securities, although the bias of the CAPM beta did not appear to be material. Generalized functional form has also been found in the international stock market. Chaudhury and Lee (1997) found that the linear (loglinear) empirical return model could be rejected for more than half of the international sample of 425 stocks from ten countries.

103.2.3 Translog Functional Form of the CAPM Jensen (1969), Lee (1976), Levhari and Levy (1977), and McDonald (1983) investigated the empirical implications of multiperiod investment, but none of them has provided a generalized asset pricing model for the equilibrium risk-return relationship under heterogeneous investment horizons. In order to control the systematic skewness from the square-term of a market excess return in the CES functional form of the CAPM, Lee et al. (1990) proposed the translog functional form of the CAPM in a heterogeneous investment horizon framework:

 j fj    N mj D 1  ˇj 1 C RftN Cˇj E 1 C Rmt E 1 C RjtN (103.6) H

H

H

where j D Nj ; j D Nfj ; mj D Nmj and H and N are investors’ weighted time horizon and observed time horizon, respectively. Note each portfolio has different set of . When i D fi D mi , it reduces to homogeneous generalized functional form CAPM.

The translog function form provides a generalized functional form that is local second-order approximation to any nonlinear relationship. For many production and investment frontiers employed in econometric studies, the translog function often provides accurate global approximations. Moreover, the translog model permits greater substitution among variables. Thus, it provides a flexible functional form for risk estimation. Finally, this model can be estimated and tested by relatively straightforward regression methods with a heterogeneous investment horizon. All of the three alternative functional CAPM models can reduce the misspecification bias in the estimates of the systematic risk and improve the explanatory power of the CAPM.

103.3 Model Estimation 103.3.1 Generalized Functional Form Model for Closed-End Fund Using the technique of Box and Cox (1964), Lee (1976) and Fabozzi et al. (1980) developed a generalized model to describe the mutual fund return-generating process. Following their approach, we define the generalized Box-Cox functional form of the international closed-end fund return as follows:

.1 C Rjk .t//./  .1 C Rf .t//./ D ˛jk C ˇjk Œ.1 C Rmk .t//./  .1 C Rf .t//./ C "jk .t/ where .1 C Rjk .t//./ D .1CRf .t // 1 , 

.1 C Rmk .t//./ D

.1CRjk .t // 1 ; 

.1 C Rmk .t//  1 ;  

.1 C Rf .t//./ D

(103.7)

Equation (103.7) can be rewritten as: .1 C Rjk .t//./ D ˛jk C .1  ˇjk /.1 C Rf .t//./

"jk .t/  N.0; j2 /

 is the functional form parameter to be estimated across the mutual funds; Rjk .t/ is the monthly rate of return (shares and NAV) for the closed-end fund j, which invest in country k in period t; Rmk .t/ is the monthly market rate of return of country k in period t; Rf .t/ is risk free rate of interest in period t.

Cˇjk .1 C Rmk .t//./ C "jk .t/ (103.8) Equation (103.8) is a constrained or restricted regression. Equation (103.7) includes both linear and log-linear functional forms as special cases: When  D 1, it reduces to the linear case: Rjk .t/  Rf .t/ D ˛jk C ˇjk ŒRmk .t/  Rf .t/ C "jk .t/ when  approach zero, it reduces to the log-linear case:

In.1 C Rjk .t//  In.1 C Rf .t// D ˛jk C ˇjk ŒIn.1 C Rmk .t//  In.1 C Rf .t// C "jk .t/

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds

Following Box and Cox (1964), we use the maximum likelihood method to determine the functional form parameter: Max l./ D 

f˛;ˇ;;g



N X N  2  In.1 C Rpt / In O i ./ C .  1/ 2 t D1 N X N 1 2 In.2 /  e 2 2i2 i t D1

where N is the number of observation and O "2 ./ is the estimated regression residual standard error and ei D .1 C Rjk .t//./ Œ˛jk C.1ˇjk /.1CRf .t//./ Cˇjk .1CRmk .t//./ . Given the unrestricted estimates of parameters .˛; ˇ; ; /, a model that is linear . D 1/ or log-linear . D 0/ is a simple parametric restriction and can be tested with a likelihood ratio statistic. The test statistics is 21 D 2 fIn L . D 1 or  D 0/  In L . D MLE/g : This statistics has a chi-squared distribution with one degree of freedom and can be referred to the standard table (5% critical value D 3:84). We can use the difference of these two likelihood values to test whether  of each fund is significantly different from zero or one.

103.3.2 Functional Form of the International Closed-End Country Fund Model In the context of international asset pricing, if the world capital market is integrated, ex ante risk premium on a security equals ex ante risk premium of global market portfolio times the security’s systematic risk with respect to the global portfolio (Solnik 1974a). However, Solnik (1974b, c) and Lessard (1974) also report that there are strong national factors that present in the price generating process of individual securities. Hence, Solnik (1974a) suggested using a twofactor model for individual securities. A direct extension to Solnik’s model would lead us to the generalized functional form of the international capital asset pricing model for closed-end funds: .1 C Rj k .t//.œ/ D ˛j C ˇj k .1 C Rmk .t//./ Cˇjg .1 C Rg .t//.œ/ C "j .t/: (103.9) where .1 C Rjk .t//./ D .t // 1

.1CRmk 

.1CRmk .t // 1 ; .1 

C Rmk .t//./ D

,

.1 C Rg .t//./ D

.1 C Rg .t//  1 ; "j .t/  N.0; "2 / 

1527

.1 C Rjk .t//.œ/ and .1 C Rmk .t//.œ/ are generalized transformation of the monthly CEF return (both share returns and NAV returns) and the monthly market return of country k. .1 C Rg .t//.œ/ is the transformed monthly return on the global market portfolio. Both Rmk .t/ and Rg .t/ are commonly proxied by the return on the country stock index and the global stock index. In an integrated world market, the pure national risk ˇjk would not be priced since it can be diversified away through international investments. On the other hand, in an extreme segmented market, the international risk ˇjg would not be priced (Errunza and Losq 1985; Jorion and Schwartz 1986). In a mildly segmented market (Errunza and Losq 1989), the pure national risk of securities in which foreign investment is not allowed would be priced as in the case of complete segmentation, but their global systematic risk would also be priced as in the case of complete integration. In addition to the two-factor generalized global CAPM model, we also consider the generalized single factor model with only global index as the pricing factor. Pioneering work by Carhart (1997) has shown that the four-factor model, which includes a momentum factor, is superior to both the CAPM and the Fama-French three-factor model in explaining the cross-sectional variation in fund returns. We further extend our generalized CAPM model into the framework of Carhart four-factor model. In this chapter, we focus on the empirical specification of return generating process in the context of international asset pricing. We would explore whether it is worthwhile to include a global index in the generalized two-factor model. Specifically, we will compare the generalized functional form of CEF between developed markets and emerging markets, between single country funds and regional funds, and between short sale allowed country funds and short sale not allowed country funds.

103.4 Data and Methodology 103.4.1 Data The total sample of data includes 90 U.S.-traded international closed-end country funds with complete monthly data from July 1965 to December 2003.1 These returns are adjusted for capital distribution, stock splits, dividends, and right offerings. All of the data are in the U.S. dollars. Table 103.1 lists the fund name, fund code, IPO date, and the total number of observations. The market indices of different countries and regions are also included in Table 103.1. 1

Most funds report their NAVs weekly on Fridays. We use the data for the last Friday of the month for the monthly observations.

1528

C.-F. Lee et al.

Table 103.1 Information of international closed-end country funds traded in the U.S Developed/ Single country No. Code Fund name Index emerging fund

Short sale allowed

IPO date

Observations (month)

1

JPN

Japan Fund Inc

Japan

Dev

S

Y

5-15-75

265

2

ASA

ASA Limited

South Africa

Em

S

Y

9-1-58

462

3

MXF

Mexico Fund Inc

Mexico

Em

S

Y

6-3-81

270

4

KF

Korea Fund Inc

Korea

Em

S

N

8-22-84

232

5

ZSEV

Z-Seven Fund Inc

World

World

R

Y

12-29-83

195

6

IAF

First Australia Fund Inc

Australia

Dev

S

Y

12-12-85

216

7

ITA

Italy Fund Inc

Italy

Dev

S

Y

2-26-86

203

8

FRN

France Fund

France

Dev

S

Y

5-30-86

42

9

GER

Germany Fund

Germany

Dev

S

Y

7-18-86

209

10

VLU

Worldwide Value Fund

World

World

R

Y

8-19-86

130

11

TWN

Taiwan Fund Inc

Taiwan

Em

S

N

12-16-86

204

12

EMF

Templeton Emerging Mkts Fund

EM

Em

R

B

2-26-87

202

13

APB

Asia Pacific Fund Inc

AP

Em

R

N

4-24-87

200

14

MF

Malaysia Fund Inc

Malaysia

Em

S

G

5-8-87

199

15

SAF

Scudder New Asia Fund Inc

Asia

B

R

B

6-18-87

198

16

CLM

Clemente Global Growth Fund

World

World

R

Y

6-23-87

198

17

UKM

United Kingdom Fund

United Kingdom

Dev

S

Y

8-6-87

139

18

SWZ

Swiss Helvetia Fund

Europe

Dev

S

Y

8-19-87

196

19

TTF

Thai Fund Inc

Thailand

Em

S

G

2-17-88

190

20

BZF

Brazil Fund Inc

Brazil

Em

S

N

3-31-88

189

21

IBF

First Iberian Fund

Spain C Portugal

Dev

R

B

4-13-88

130

22

SNF

Spain Fund Inc

Spain

Dev

S

N

6-21-88

186

23

IGF

India Growth Fund Inc

India

Em

S

N

8-12-88

176

24

ROC

ROC Taiwan Fund

Taiwan

Em

S

N

5-12-89

175

25

OST

Austria Fund Inc

Austria

Dev

S

Y

9-21-89

150

26

CH

Chile Fund Inc

Chile

Em

S

N

9-26-89

171

27

PGF

Portugal Fund Inc

Portugal

Dev

S

Y

11-1-89

169

28

FPF

First Philippine Fund Inc

Philippines

Em

S

N

11-8-89

162

29

TKF

Turkish Investment Fund Inc

Turkey

Em

S

N

12-5-89

168

30

GF

New Germany Fund

Germany

Dev

S

Y

1-24-90

167

31

NEF

Scudder New Europe Fund

Europe

Dev

R

Y

2-9-90

113

32

GSP

Growth Fund of Spain Inc

Spain

Dev

S

N

2-14-90

105

33

CEE

Central European and Russia Fund

EE

Em

R

N

2-27-90

166

34

IF

Indonesia Fund Inc

Indonesia

Em

S

N

3-1-90

141

35

JOF

Japan OTC Equity Fund

Japan

Dev

S

Y

3-14-90

165

36

GTF

GT Greater Europe Fund

Europe

Dev

R

Y

3-22-90

113

37

FRG

Emerging Germany Fund Inc

Germany

Dev

S

Y

3-29-90

108

38

IRL

Irish Investment Fund

Ireland

Dev

S

Y

3-30-90

165

39

ANE

Alliance New Europe Fund Inc

Europe

Dev

R

Y

3-27-90

10

40

JGF

Jakarta Growth Fund

Indonesia

Em

S

N

4-10-90

119



World-AC world index; EM-Emerging Market; AP-Asian Pacific; EE-EM Eastern Europe; EAFEEM-AC EAFE C EM; Latin-EM Latin American.

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds

1529

Table 103.1 (continued) No.

Code

Fund name

Index

Developed/ emerging

Single country fund

Short sale allowed

IPO date

Observations (month)

41

EF

Europe Fund Inc

Europe

Dev

R

Y

4-26-90

164

42

PEF

Pacific-European Growth Fund

Eafeem

Dev

R

Y

4-20-90

27

43

FRF

France Growth Fund Inc

France

Dev

S

Y

5-10-90

163

44

TC

Thai Capital Fund

Thailand

Em

S

G

5-22-90

128

45

EWF

European Warrant Fund Inc

Europe

Dev

R

Y

7-17-90

161

46

SGF

Singapore Fund

Singapore

Dev

S

N

7-24-90

161

47

LAM

Latin America Investment Fund

Latin

Em

R

N

7-25-90

161

48

MXE

Mexico Equity and Income Fund

Mexico

Em

S

Y

8-14-90

160

49

MEF

Emerging Mexico Fund

Mexico

Em

S

Y

10-2-90

101

50

AF

Argentina Fund Inc

Argentina

Em

S

N

10-11-91

121

51

LAQ

Latin America Equity Fund

Latin

Em

R

N

10-22-91

108

52

MSF

Morgan Stanley Emerging Mkts

Em

Em

R

N

10-25-91

146

53

KIF

Korean Investment Fund Inc

Korea

Em

S

N

2-13-92

116

54

BZL

Brazilian Equity Fund Inc

Brazil

Em

S

N

4-3-92

140

55

LDF

Latin American Discovery Fund

Latin

Em

R

B

6-16-92

138

56

CHN

China Fund Inc

China

Em

S

N

7-10-92

137

57

GCH

Greater China Fund Inc

China

Em

S

N

7-15-92

137

58

JFC

Jardine Fleming China Region

China

Em

S

N

7-16-92

137 136

59

JEQ

Japan Equity Fund Inc

Japan

Dev

S

Y

7-17-92

60

ISL

First Israel Fund Inc

Israel

Em

S

N

10-22-92

134

61

TCH

Templeton China World Fund Inc

China

Em

S

N

9-9-93

118

62

GSG

Global Small Cap Fund Inc

World

World

R

Y

10-6-93

74

63

GRR

Asia Tigers Fund Inc

Asia

Em

R

N

11-18-93

121

64

KEF

Korea Equity Fund

Korea

Em

S

N

11-24-93

121

65

PKF

Pakistan Investment Fund Inc

Pakistan

Em

S

N

12-16-93

89

66

SHF

Schroder Asian Growth Fund Inc

Asia

B

R

B

12-22-93

50

67

TGF

Emerging Tigers Fund Inc

Em

Em

R

N

2-25-94

26

68

SBW

Salomon Brothers Worldwide

World

World

R

Y

12-22-93

120

69

GTD

GT Global Developing Markets

Em

Em

R

N

1-11-94

44

70

AFF

Morgan Stanley Africa

Em

Em

R

N

2-3-94

97

71

IFN

India Fund Inc

India

Em

S

N

2-14-94

118

72

IIF

Morgan Stanley India Inv Fund

India

Em

S

N

2-17-94

118

73

SOA

Southern Africa Fund Inc

South Africa

Em

S

Y

2-25-94

118

74

JFI

Jardine Fleming India Fund Inc

India

Em

S

N

3-3-94

117

75

NSA

New South Africa Fund Inc

South Africa

Em

S

Y

3-4-94

62 (continued)

1530

C.-F. Lee et al.

Table 103.1 (continued) No.

Code

Fund name

Index

Developed/ emerging

Single country fund

Short sale allowed

IPO date

Observations (month)

76

FAE

Fidelity Advisor Emerging

Em

Em

R

N

3-18-94

62

77

EMO

TCW/DW Emerging Markets Opp Tr

Em

Em

R

N

3-23-94

44

78

TEA

Templeton Emerging Markets

Em

Em

R

N

4-29-94

100

79

CUBA

Herzfeld Caribb

Latin

B

R

Y

4-22-94

115

80

TYW

Taiwan Equity Fund Inc

Taiwan

Em

S

N

7-18-94

69

81

APF

Morgan Stanley Asia Pacific

AP

B

R

B

4-16-96

113

82

TVF

Templeton Vietnam Opportunity

Asia

Em

S

N

9-15-94

95

83

TDF

Templeton Dragon Fund Inc

Asia

B

R

N

9-21-94

111

84

CRF

Czech Republic Fund Inc

Czech Republic

Em

S

Y

9-23-94

96

85

EME

Foreign and Colonial Emerging

Em

Em

R

N

10-28-94

67

86

LLF

Lehman Bros Latin America Fund

Latin

Em

R

N

10-28-94

50

87

FAK

Fidelity Advisor Korea Fund

Korea

Em

S

N

10-25-94

68

88

TRF

Templeton Russia Fund Inc

Russia

Em

R

N

6-15-95

99

89

RNE

Morgan Stanley Russia and New Europe

EE

Em

R

N

9-24-96

87

90

DGE

Dessauer Global Equity Fund

World

World

R

Y

5-30-97

22

This table reports sample of the closed-end country funds. Index is the MSCI country or regional index corresponding to underlying assets of each fund. Index used for regional fund is the regional index that mostly covers the investing countries of the fund; global fund uses the MSCI global index; Dev/Em display whether the fund invests in developed countries or emerging countries, World-world fund, B-fund invests in both developed and emerging countries; Single country fund indicates whether the fund is single country fund or regional fund; Short sale gives the information whether or not the short sale is allowed in the investing country of each fund, Y-Short sale allowed, N-Short sale not allowed, B-funds invest in countries where short sale allowed and countries where short sale not allowed, G-funds invest in country where short sale policy changed during our sample period

Our sample includes 55 single country closed-end funds and 35 regional closed-end funds, with their underlying assets traded in 31 different countries. We divide those funds into two groups: developed market funds and emerging market funds, according to the country of their underlying asset. Among these funds 19 funds invest in developed markets, 53 funds invest in emerging markets, while the other 18 funds invest in both. We also classify each fund according to the regulation policy of short sale in the invested countries.2 Among these countries, the first group includes most developed markets such as the UK, Australia, Austria, Belgium, Canada, Czech Republic, Denmark, France, Germany, Ireland, Italy, Japan, Mexico, Netherlands, Portugal, South Africa, and Switzerland where short sale is allowed and practiced; the second group includes the countries where short sale is not allowed or not practiced, they are Colombia, Greece, Indonesia, Pakistan, Singapore, South Korea,

2

Information of short sale policy comes from Bris et al. (2004).

Taiwan, Argentina, Brazil, Chile, Finland, India, Israel, New Zealand, Philippines, China, and Spain; and the last group includes three countries (regions) that short sale regulation and practice changed during our sample period, they are Hong Kong, Malaysia, and Thailand. All together there are 35 funds that invest in countries or regions where short sale is allowed or practiced, and other 46 funds that invest in countries or regions where short sale is not allowed or practiced. The market indices for different countries, regions and the world market used in this study are from MSCI of Morgan Stanley. The U.S. market index used is the monthly return of the CRSP value weighted composite index. The problem of assuming a riskless rate in an international context is well known, and it is especially troublesome for international capital market where a domestic market determined short-term rate similar to the U.S. T-bills is generally not available. Since we have no domestic market short rate information, instead we use 30-day U.S. T-bill rate as a proxy. All information of the Fama-French three factors and the momentum factor is downloaded from Kenneth French’s website. Table 103.2 reports the descriptive statistics for the share returns and the NAV returns of international closed-end

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds

Table 103.2 Summary statistics of the U.S. international closed-end country fund Share return No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

1531

NAV return

Ticker

Min

Max

Mean

STD

Min

Max

Mean

STD

AF AFF ANE APB APF ASA BZF BZL CEE CH CHN CLM CRF CUBA DGE EF EME EMF EMO EWF FAE FAK FPF FRF FRG FRN GCH GER GF GRR GSG GSP GTD GTF IAF IBF IF IFN IGF IIF IRL ISL ITA JEQ JFC JFI JGF JOF JPN KEF

0:342 0:250 0:244 0:422 0:183 0:276 0:388 0:394 0:376 0:305 0:294 0:437 0:321 0:299 0:247 0:277 0:229 0:360 0:174 0:452 0:238 0:300 0:349 0:203 0:253 0:330 0:341 0:266 0:272 0:196 0:323 0:222 0:159 0:479 0:383 0:248 0:375 0:199 0:247 0:242 0:264 0:177 0:274 0:282 0:325 0:264 0:320 0:246 0:276 0:295

0.351 0.153 0.237 0.409 0.194 0.514 0.351 0.376 0.188 0.335 0.587 0.196 0.195 0.515 0.145 0.219 0.453 0.286 0.210 0.340 0.252 0.397 0.444 0.239 0.161 0.197 0.517 0.444 0.206 0.241 0.409 0.239 0.195 0.214 0.429 0.538 0.566 0.323 0.395 0.322 0.195 0.272 0.542 0.327 0.375 0.258 0.629 0.485 0.595 0.404

0:002 0:004 0:019 0:013 0:001 0:014 0:016 0:008 0:008 0:015 0:017 0:008 0:003 0:005 0:006 0:008 0:010 0:014 0:000 0:006 0:002 0:005 0:002 0:007 0:007 0:013 0:013 0:010 0:002 0:002 0:013 0:013 0:004 0:004 0:009 0:014 0:006 0:010 0:008 0:012 0:008 0:007 0:009 0:004 0:010 0:003 0:005 0:007 0:023 0:001

0.101 0.071 0.128 0.110 0.074 0.111 0.119 0.116 0.080 0.100 0.121 0.075 0.079 0.100 0.081 0.070 0.085 0.100 0.074 0.118 0.089 0.129 0.108 0.071 0.069 0.091 0.117 0.092 0.080 0.094 0.085 0.079 0.074 0.093 0.083 0.102 0.137 0.107 0.108 0.103 0.073 0.077 0.101 0.096 0.115 0.104 0.130 0.114 0.093 0.111

0:278 0:202 0:137 0:421 0:132 0:499 0:520 0:309 0:322 0:245 0:229 0:241 0:228 0:204 0:179 0:199 0:138 0:340 0:169 0:271 0:203 0:323 0:280 0:178 0:180 0:231 0:265 0:234 0:215 0:191 0:184 0:168 0:116 0:356 0:449 0:296 0:421 0:217 0:450 0:480 0:163 0:182 0:357 0:136 0:961 0:188 0:429 0:205 0:681 0:317

0.244 0.123 0.053 0.333 0.173 0.409 0.484 0.351 0.216 0.183 0.312 0.134 0.879 0.123 0.179 0.134 0.176 0.198 0.096 0.225 0.249 0.720 0.555 0.155 0.174 0.157 0.383 0.189 0.161 0.169 0.268 0.147 0.107 0.159 0.148 0.193 0.514 0.260 0.421 0.891 0.151 0.134 0.258 0.241 22.075 0.313 0.254 0.446 0.225 0.481

0:001 0:002 0:014 0:007 0:002 0:006 0:015 0:004 0:004 0:007 0:007 0:001 0:005 0:000 0:016 0:001 0:002 0:005 0:001 0:005 0:003 0:005 0:004 0:000 0:003 0:007 0:006 0:001 0:002 0:003 0:008 0:009 0:000 0:002 0:004 0:005 0:000 0:011 0:000 0:013 0:006 0:003 0:001 0:001 0:165 0:005 0:013 0:005 0:006 0:001

0.081 0.055 0.082 0.087 0.055 0.107 0.137 0.120 0.074 0.074 0.080 0.054 0.101 0.052 0.089 0.052 0.048 0.075 0.055 0.091 0.070 0.148 0.090 0.055 0.054 0.074 0.102 0.070 0.063 0.070 0.064 0.057 0.051 0.070 0.064 0.070 0.122 0.098 0.100 0.135 0.050 0.066 0.070 0.069 1.918 0.087 0.100 0.090 0.079 0.112 (continued)

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Table 103.2 (continued) Share return

NAV return

No.

Ticker

Min

Max

Mean

STD

Min

Max

Mean

STD

51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 Average

KF KIF LAM LAQ LDF LLF MEF MF MSF MXE MXF NEF NSA OST PEF PGF PKF RNE ROC SAF SBW SGF SHF SNF SOA SWZ TC TCH TDF TEA TGF TKF TRF TTF TVF TWN TYW UKM VLU ZSEV

0:307 0:304 0:385 0:391 0:471 0:444 0:449 0:537 0:318 0:363 0:457 0:326 0:463 0:332 0:183 0:230 0:314 0:543 0:278 0:389 0:500 0:255 0:190 0:239 0:385 0:281 0:469 0:274 0:260 0:288 0:096 0:402 0:615 0:371 0:266 0:326 0:301 0:211 0:235 0:267 0:315

0.569 0.514 0.277 0.255 0.291 0.171 0.381 0.833 0.243 0.369 0.545 0.156 0.329 0.645 0.161 0.343 0.400 0.330 0.408 0.257 0.508 0.513 0.198 0.875 0.271 0.234 0.550 0.339 0.311 0.228 0.149 0.704 0.543 0.714 0.315 0.732 0.278 0.211 0.168 0.325 0.363

0:017 0:002 0:014 0:008 0:013 0:012 0:013 0:009 0:011 0:013 0:017 0:011 0:007 0:004 0:003 0:007 0:009 0:017 0:001 0:010 0:016 0:005 0:009 0:013 0:011 0:008 0:002 0:007 0:012 0:006 0:005 0:013 0:026 0:010 0:000 0:014 0:011 0:011 0:007 0:002 0:007

0.123 0.125 0.103 0.101 0.102 0.098 0.128 0.136 0.091 0.102 0.129 0.067 0.096 0.109 0.067 0.082 0.117 0.127 0.104 0.095 0.083 0.097 0.078 0.111 0.085 0.065 0.149 0.097 0.092 0.084 0.065 0.149 0.175 0.145 0.097 0.137 0.098 0.063 0.057 0.067 0.099

0:327 0:306 0:323 0:321 0:451 0:323 0:460 0:372 0:267 0:464 0:534 0:162 0:420 0:905 0:116 0:211 0:590 0:454 0:262 0:274 0:458 0:208 0:177 0:228 0:304 0:397 0:653 0:513 0:286 0:245 0:094 0:558 0:512 0:280 0:224 0:388 0:315 0:196 0:260 0:640 0:320

0.645 0.571 0.171 0.217 0.239 0.122 0.188 0.341 0.211 0.206 0.516 0.145 0.298 0.322 0.055 2.687 1.127 0.259 0.213 0.252 0.157 0.290 0.107 0.194 0.291 0.154 1.801 0.703 0.257 0.147 0.126 0.490 0.343 0.333 0.321 0.540 0.262 0.163 0.113 0.155 0.571

0:013 0:001 0:005 0:006 0:008 0:013 0:005 0:002 0:006 0:008 0:010 0:006 0:000 0:012 0:004 0:013 0:005 0:012 0:002 0:005 0:004 0:002 0:003 0:002 0:006 0:003 0:016 0:006 0:004 0:000 0:009 0:020 0:020 0:007 0:002 0:007 0:010 0:004 0:004 0:002 0:005

0.107 0.126 0.080 0.083 0.101 0.074 0.113 0.089 0.079 0.089 0.124 0.047 0.094 0.126 0.044 0.220 0.167 0.104 0.085 0.070 0.074 0.069 0.049 0.068 0.083 0.057 0.208 0.113 0.080 0.058 0.047 0.175 0.133 0.106 0.078 0.111 0.096 0.050 0.048 0.073 0.107

Min

Max

Mean

STD

0:140 0:007

0.113 0.0135

Global index monthly return One month T-bill

0:0074 0:0046

0.0427 0.0023

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds

Global Carhart GCAPM Model: .1 C Rpt /./ D ˛p C j ˇpm .1 C Rmt /./ C ˇpg /.1 C Rgt /./ C ˇp;mkrf .1 C ./ MKRFt / C ˇp;smb .1 C SMBt /./ C ˇp;hml .1 C HMLt /./ C ˇp;umd .1 C UMDt /./ C "pt

funds. The price return and the NAV return are calculated as follows: p

Rj k .t/ D

Pj k .t/  Pj k .t  1/ nov Rj k .t/ Pj k .t  1/

NAVj k .t/  NAVj k .t  1/ D NAVj k .t  1/ where Pjk .t/ and NAV jk .t/ are share price and NAV for fund j investing in country k during period t. The average monthly share return is 0.73%, which is higher than 0.53% of the average monthly NAV return. However, the average standard deviation of the share return is only 9.93% compared to 10.7% of the average standard deviation of the NAV return. This partly reflects the higher risk of the underlying foreign countries’ assets of CEFs, especially of those emerging markets.

103.4.2 Methodology To determine the appropriate functional form of the CEF, for the share return, we use both local country index and the U.S. CRSP value weighted composite index return as the proxy of the market return; for the NAV return, we only use the monthly return of the local country index as the proxy of the market return. The use of respective local market indices as benchmarks may be motivated by noting that the funds are constrained to invest most of their assets in equity securities of one country or region, and if the fund managers are market timers, they are more likely to time the respective local market. Therefore, it is of interest how the funds compare to the passive market portfolios. Since all of these funds are traded in the U.S. market, and their price movements are also affected by the U.S. market, therefore, using the CRSP value weighted composite index as the benchmark for the share return is reasonable Table 103.4. The generalized functional models used for the share return and the NAV return of international closed-end funds are as follows: Share return

GCAPM Model: .1 C Rpt /./ D ˛p C ˇp .1 C Rft /./ C k ./ / C "pt .1  ˇp /.1 C Rmt GCAPM-CRSP Model: .1 C Rpt /./ D ˛p C ˇp .1 C  crsp ./ Rft /./ C .1  ˇp / 1 C Rmt C "pt Global Index Model: .1 C Rpt /./ D ˛p C ˇpg .1 C Rgt /./ C "pt Global GCAPM Model: .1 C Rpt /./ D ˛p C   k ./ C ˇpg /.1 C Rgt /./ C "pt ˇpm 1 C Rmt Global GCAPM-CRSP Model: .1 C Rpt /./ D ˛p C  crsp ./ ˇpm 1 C Rmt C ˇpg /.1 C Rgt /./ C "pt

1533

NAV return

GCAPM Model: .1CNAV pt /./ D ˛p Cˇp .1CRft /./ C   k ./ .1  ˇp / 1 C Rmt C "pt Global Index Model: .1 C NAV pt /./ D ˛p C ˇpg .1 C Rgt /./ C "pt Global GCAPM Model: .1 C NAV pt /./ D ˛p C   k ./ C ˇpg .1 C Rgt /./ C "pt ˇpm 1 C Rmt crsp

k where Rmt is the monthly return of country k’s index, Rmt is the monthly return of the CRSP value weighted composite index, Rgt is the monthly return of the MSCI global market index.3 All together, we use six different generalized models for share returns and three different generalized models for NAV returns. The objective of this chapter is to determine whether the global index needs to be included in the empirical return model in addition to the national or domestic market index. MLE estimation of generalized functional model provides the alternative pairs of coefficient estimates and t-ratios. We wish to see whether the coefficient of the global index is significant, and more importantly, the choice of functional form matters with respect to the significance of the global index. Additionally, as the underlying assets and closed-end funds are traded at different markets, investors from different markets may have different investment horizons. Even for the same closed-end fund the functional form model of the share returns and the NAV returns may be different. Therefore, we want to investigate the difference of the functional form model between share returns and NAV returns of closed-end funds, as well as the difference between emerging market funds and developed market funds, between single country funds and regional funds. To test whether funds’ characteristics will affect the functional form specification, we use two approaches for the comparison. First, we run the grouped regression that pools across all CEFs based on different grouping criterions. We divide all of CEFs into six groups: single country funds versus regional funds, emerging market funds versus developed market funds, short sale allowed funds versus short sale not allowed funds. Second, we pool all data into one group across time and funds. In addition, we add different fund characteristics as dummy variables in the estimation. Pooling data across funds and time allows us to test the determinants of

3

Morgan Stanley started the emerging market index in 1988; therefore our global index (including developed and emerging markets) starts from January 1988.

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C.-F. Lee et al.

the cross-sectional variation in the functional transformation. Several specifications of the pooled model are estimated. In order to control the heteroskedasticity problem, we use White t-statistics that are based on the standard error robust to heteroskedasticity Table 103.5. For all of our empirical tests with pooling data for international CEFs share returns and NAV returns we use the different specification of the following model respectively: j

.1 C Rpt /./ D ˛p C ˇf .1 C Rf /./ C ˇm .1 C Rmt /./ C ˇg .1 C Rgt /./ C ˇmkrf .1 C MKRF t /./ C ˇsmb .1 C SMBt /

./

C ˇhml .1 C HMLt /

./

C ˇumd .1 C UMDt /./ C ˇe EM P C ˇr REGp C ˇs SSp C "pt

j

.1 C NAV pt /./ D ˛p C ˇf .1 C Rf /./ C ˇm .1 C Rmt /./ C ˇg .1 C Rgt /./ C ˇe EM p C ˇr REGp C ˇs SSp C "pt ./

where ././ D . /  1 and Em, Reg, SS are dummy variables of funds’ characteristics for emerging markets funds or developed market funds, regional funds or single country funds, short sale allowed country funds or not allowed country funds. In general, we summarize the test hypothesis of our empirical study as follows: Test 1: The coefficients of the global index return as well as the national market index return are significant. ! To test the market segmentation. Test 2: The functional form of CEF does matter (functional transformation parameter lambda is significant) in the context of global asset pricing model. ! To test the heterogeneous investment horizons. Test 3: There exists significant difference of the functional form between share returns and NAV returns of international closed-end funds. ! To test the difference of investors’ behavior from the U.S. market and foreign markets. Test 4: There exists significant difference of the functional form between closed-end funds investing in emerging markets and developed markets. ! To test the market maturity’s effect on the functional form. Test 5: There exists significant difference of the functional form between single country closed-end funds and regional closed-end funds. ! To test whether there exists cross-market effect. Test 6: There exists significant difference of the functional form between international CEFs investing in short

sale allowed countries and those investing in short sale not allowed countries. ! To test the government regulation’s effect on the market efficiency.

103.5 Empirical Results 103.5.1 Generalized Functional Form for International Closed-End Fund Unlike McDonald (1983) and Chaudhury and Lee (1997), we find the transformation parameter  to be positive on average. In Table 103.3, we find the average  of all share returns in GCAPM is 0.218, which is smaller than the average  of NAV returns, and this similar pattern remains no matter which model we use. The absolute magnitude of  seems a little higher for NAV returns compared to the share returns. Another noticeable feature is the dispersion of  across the funds in our sample. The standard deviation of  changes from 1.248 to 2.717, and the minimum and maximum value of  range from 11 to 13. These dispersion measures are rather large relative to the mean and median  values. This indicates the fund-specific nature of the transformation parameter and raises questions about the conventional use of the same transformation parameter (usually 1 for linear and 0 for loglinear specification) for all securities. Moreover, from Fig. 103.1 we find the distribution of  of the NAV returns is more disperse compared to that of the share returns, and there exists some skewness in most of the distributions of . The large dispersion of  for NAV return indicates that foreign investors have more heterogeneous investment horizons compared to the U.S. investors. In Table 103.4 we report the rejection rate of the hypothesis of linear and loglinear return. Irrespective of which generalized model we use, the loglinear return hypothesis is rejected for at least 30% of our sample funds, and the linear return hypothesis is rejected for more than 45% of the funds. These numbers increase to 58.4 and 55% of the NAV returns for global GCAPM model. The GCAPM model of the share return has the lowest rejection rate, where neither linear nor loglinear hypothesis can be rejected for 60% of funds. When we substitute the market return with the CRSP value weighted index return, more funds are rejected for the linear or loglinear hypothesis (rejection rate for both hypothesis increases to 20%). This reflects that the heterogeneous investment horizons for U.S. investors are more significant when we treat international closed-end fund in a local environment rather than in the global environment. However, when we include global market return in the model specification, the rejection rates also increase. This implies that in order to explore the heterogeneous investment horizon in

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds

Table 103.3 Transformation parameter summary

 GCAPM (share return)

GCAPM_CRSP (share return)

Global index (share return)

Global GCAPM_CRSP (share return)

Global Carhart GCAPM (share return)

GCAPM (NAV return)

Global index (NAV return)

Global GCAPM (NAV return)

t ./

jj

t .jj/

Mean

0:218

0:369

0:994

1:577

Median

0:282

0:495

0:801

1:335

STD

1:248

1:928

0:778

1:158

Min

3:798

6:310

0:001

0:000

Max

3:162

4:380

3:798

6:310

0:026

0:029

1:030

1:668

0:164

0:290

0:831

1:355 1:213

Mean Median STD

1:349

2:070

0:864

Min

5:320

5:610

0:006

0:020

Max

2:936

4:090

5:320

5:610

Mean

0:107

0:182

1:098

1:724

Median

0:366

0:370

0:942

1:600

STD

1:405

2:086

0:875

1:174

Min

5:780

5:780

0:013

0:020

2:814

4:080

5:780

5:780

0:047

0:007

0:992

1:572

Median

0:130

0:110

0:817

1:330

STD

1:269

1:977

0:774

1:168

Min

3:642

6:440

0:000

0:000

Max

2:395

4:100

3:642

6:440

Mean

0:147

0:199

1:096

1:703

Median

0:237

0:400

0:870

1:540 1:186

Max Global GCAPM (share return)

1535

Mean

STD

1:460

2:085

0:956

Min

5:561

5:700

0:000

0:000

Max

2:894

4:200

5:561

5:700

Mean

0:021

0:047

0:961

1:547

Median

0:127

0:190

0:756

1:360

STD

1:237

1:946

0:771

1:187

Min

3:675

6:660

0:001

0:000

Max

2:402

4:020

3:675

6:660

Mean

1:269

1:758

2:183

2:881

Median

1:213

1:700

1:739

2:160

STD

2:769

3:331

2:116

2:415

Min

10:978

10:220

0:005

0:040

Max

12:630

11:190

12:630

11:190

Mean

1:185

1:786

2:303

2:617

Median

1:599

1:880

1:770

2:030

STD

3:833

2:735

3:278

1:944

Min

29:013

8:790

0:012

0:110

Max

10:668

10:590

29:013

10:590

Mean

1:567

1:760

2:204

2:792

Median

1:441

1:730

1:571

2:120

STD

2:717

3:194

2:201

2:303

Min

5:758

10:240

0:000

0:000

Max

12:852

11:380

12:852

11:380

This table reports the summary of estimated transformation parameters of each model

Model  ¤ 0;  ¤ 0;  D 0;  D 0; Sum  ¤ 0;  ¤ 0;  D 0;  D 0; Sum

¤1 D1 ¤1 D1

¤1 D1 ¤1 D1

GCAPM_CRSP (%) 18(20%) 12(13.3%) 30(33.3%) 30(33.3%) 90 – – – – –

Global Global index (%) 20(22.5%) 11(12.4%) 26(29.2%) 32(36.0%) 89 26(29.2%) 24(27%) 14(15.7%) 25(28.1%) 89

Global GCAMP (%) 15(16.9%) 13(14.6%) 26(29.2%) 35(39.3%) 89 35(39.3%) 17(19.1%) 14(15.7%) 23(25.8%) 89

Global GCAMP_CRSP (%) 21(23.6%) 11(12.4%) 25(28.1%) 32(36%) 89 – – – – –

This table reports the rejection rate of linear and loglinear hypothesis of share return and NAV return for each model

NAV returns

Share returns

GCAPM (%) 13(14.4%) 16(17.8%) 29(32.2%) 32(35.6%) 90 36(40%) 12(13.3%) 14(15.6%) 28(31.1%) 90

Table 103.4 Comparison of functional factor Global Carhart GCAPM (%) 19(21.3%) 10(11.2%) 23(25.8%) 37(41.6%) 89 – – – – –

Total (%) 106(19.8%) 73(13.6%) 159(29.7%) 198(36.9%) 536 97(36.2%) 53(19.8%) 42(15.7%) 76(28.4%) 268

Total (%) (share return NAV return) 203(25.2%) 126(15.7%) 201(25%) 274(34.1%) 804

1536 C.-F. Lee et al.

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds

Table 103.5 Correlation coefficient of dummy variables Em Reg SS EM Reg SS

1 0.06675 -0.69308

0.06675 1 0.02349

-0.69308 0.02349 1

This table reports the correlation coefficient of three dummy variables of each fund

the context of international asset pricing, the global market return should be included. In the next section we also test whether the global risk is priced in the framework of functional form CAPM. From Table 103.5 we also find an obvious pattern that, on average, the rejection rate of the NAV returns is higher than the rejection rate of the share returns for the corresponding model. This is reasonable since compared to the U.S. market (where share funds traded), the foreign countries’ investor (where the NAV of funds is determined) have more heterogeneous investment horizons. The linear or loglinear specification of security return may be more inappropriate for international financial markets, especially for those emerging markets. We will explore the difference between developed markets and emerging markets further in the following section. When we compare the intercept of the generalized models, the average excess return (alpha) after controlling for the market risk premium is only 0:11% for the share returns and 0:33% for the NAV returns, while the average excess return of the share returns for the conventional linear model is 0.05% and 0:13% for the loglinear model, and the corresponding number of the NAV return is 0:14% and 0:44%, respectively. This interesting result indicates that the functional form of returns does affect the performance evaluation. Consistent with Patro (2001), the NAV return seems to perform slightly worse compared to the shares when performance is measured using the local market portfolios.

103.5.2 Generalized Global Model for International Closed-End Fund Using a local market index to evaluate performance is appropriate only in fully segmented international capital markets. Therefore, using a benchmark that is likely to be meanvariance efficient in a global pricing context is important. Although the evidence in favor of the domestic CAPM is ambiguous, several authors such as Cumby and Glen (1990), Harvey (1991), Ferson and Harvey (1994) and Chang et al. (1995) provide evidences in favor of the mean-variance efficiency of the MSCI world market index. When we include the global index in the pricing model, we find the rejection

1537

rate of the conventional linear or the loglinear model is higher compared to the model with only local market index. Consistent with previous research, the global market risk has been priced in most of the international closed-end funds. We find that the coefficient of global return is significant for all of developed country funds and most of emerging country funds, no matter if we use share returns or NAV returns. This may partly provide evidence that all developed market and most emerging markets are integrated markets, at least mildly segmented markets.4 When we compare the one-factor generalized global index model and the two-factor generalized global model, we find an interesting feature. After we introduce the local market return into the global index model, risk premium ˇg of the global factor reduces and the risk premium of the one-factor generalized global index model ˇg is nearly equal to the summation of the risk premium of the local market return and the global return in the two-factor generalized global model.

103.5.3 Comparison of Functional Form Model Between Developed Market Funds and Emerging Market Funds, Between Regional Funds and Single Country Funds, Between Short Sale Allowed Country Funds and Short Sale Not Allowed Country Funds As our expectation, there exist significant different patterns of transformation parameters between emerging market funds and developed market funds no matter which model we use. From Panel A of Table 103.6, we find developed markets generally have negative transformation parameters while emerging markets have positive transformation parameters. The only exception is the GCAPM model for the share return, where  is positive and insignificantly different from 0. This implies that the conventional loglinear return can be used in the pricing model of share returns for the developed markets. However, this is not true for the emerging markets, which have a significant positive  and the factor loading of the market return is also smaller than that of the developed markets. Results of the NAV return from Table 103.7 tell us another story. The transformation parameters of the developed market funds change to positive while the emerging market funds do not have consistent . The significant different patterns of transformation parameters also exist between

4

Although we admit that all of international CEFs in our sample are U.S. traded securities, the evidence from NAV return still provides some support for the market integration test of foreign markets.

1538

C.-F. Lee et al.

GCAPM(CRSP)_Share

GCAPM_Share

GlobalIndex_Share

22

20 18

18

16

12

16

14

14

12

10 8 6

Frequency

20 14

Frequency

Frequency

16

12 10 8

2 0

4

4

2

2 0

0 −4

−3

−2

−1

0

1

2

3

−6 −5 −4 −3 −2 −1

4

Lambda

0

1

2

3

−4

4

Frequency

10 8 6

16

18

14

16

12

14

10 8 6

6 4

2

2

0

0 2

3

3

8

2

1

2

10

4

0

1

12

4

−1

0

GlobalCarhart_Share 20

Frequency

12

−2

−1

GlobalGCAMP(CRSP)_Share 18

14

−3

−2

Lambda

16

−4

−3

Lambda

GlobalGCAPM_Share

Frequency

8 6

6

4

10

0 −6 −5 −4 −3 −2 −1

Lambda

0

1

2

3

−4

4

−3

−2

Lambda

−1

0

1

2

3

Lambda

GCAPM(ALL)_Share

GCAPM_NAV

18 16

16

14

14

12 Frequency

Frequency

12 10 8 6

10 8 6

4

4

2

2

0 −4

−3

−2

−1 0 Lambda

1

2

0 −12 −10 −8 −6 −4 −2 0 2 4 Lambda

3

6

8 10 12 14

GlobalGAPM_NAV

GlobalIndex_NAV 20 16

16

14

14

12 Frequency

Frequency

18

12 10 8 6

8 6 4

4

2

2 0 −12 −10 −8 −6 −4 −2 0 2 4 Lambda

10

0 6

8 10 12

−8 −6 −4 −2

0

2 4 6 Lambda

8

10 12 14

Fig. 103.1 Histogram of estimated functional form parameter. This figure shows the distribution of estimated lambda of each model for international closed-end fund share return and NAV return. The curve in each graph is the standard normal distribution with mean 0 and variance 1

0.645(26.0) 0.681(26.8) 0.651(26.3)

4:70.4:2/ 2:43.2:05/ 4:37.3:91/

5:49.4:85/ 2:99.2:56/ 4:99.4:52/

Model 5

Model 6

0.197(3.87) 0.138(2.62) 0.187(3.68)

0.573(16.4) 0.502(13.9) 0.562(16.2)

0.778(36.5) 0.808(36.9) 0.782(36.8)

5:12.3:84/ 1:46.1:04/ 4:16.3:13/

Model 4

0.560(10.1) 0.538(9.43) 0.556(10.0)

1.244(35.4) 1.181(31.9) 1.224(34.8)

0.645(18.41) 0.583(15.28) 0.637(18.32)

4:20.3:87/ 1:70.1:53/ 3:88.3:62/

Model 3

0.666(27.02) 0.701(27.78) 0.670(27.26)

1.312(47.62) 1.273(44.34) 1.300(47.31)

0.0424

5:09.4:22/ 1:38.1:11/ 4:21.3:55/

0.971(55.38) 0.970(53.67) 0.971(55.41)

0.0615

Em D 0

Model 2

0.029(1.67) 0.030(1.66) 0.029(1.66)

0.00161

0.00766

2:17.1:98/ 0:146.0:13/ 2:20.2:04/

Std. dev.

0.00877

Model 1

0.00397

Mean

Table 103.6 Cross-sectional regressions for share return of international CEFs by group Panel A: comparison between emerging market funds and developed market funds Intercept .103 / Rf Rm Rg Rmkrf

0.245(7.4) 0.233(6.85) 0.243(7.34)

0.178(5.4) 0.169(5.0) 0.176(5.4)

0.234(6.0) 0.221(5.5) 0.230(5.9)

Rsmb

0.235(5.83) 0.199(4.76) 0.229(5.67)

0.229(5.59) 0.194(4.57) 0.224(5.46)

0.382(7.93) 0.339(6.75) 0.370(7.65)

Rhml

0.014(0.61) 0.010(0.39) 0.013(0.57)

0.0160(0.69) 0.013(0.52) 0.016(0.67)

0.0395(1.44) 0.0378(1.29) 0.0390(1.40)

Rumd

0:194.2:45/ 1 0

0:169.2:12/ 1 0

0:361.4:11/ 1 0

0:142.1:79/ 1 0

0:308(81.3) 1 0

0.020(0.26) 1 0



(continued)

0.0895

0.00848

Share return

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds 1539

0.625(24.1) 0.598(22.6) 0.633(24.6)

0.627(27.04) 0.601(25.57) 0.636(27.57)

0.649(72.2) 0.660(73.0) 0.646(72.4) 0.628(69.3) 0.640(70.2) 0.625(69.5)

6:54.5:44/ 1:76.1:47/ 6:90.5:93/

2:69.3:05/ 1:10.1:26/ 3:10.3:50/

5:37.4:30/ 0:486.0:38/ 5:54.4:57/

3:07.3:36/ 1:79.1:89/ 3:36.3:70/

3:93.4:33/ 2:56.2:72/ 4:22.4:68/

Model 2

Model 3

Model 4

Model 5

Model 6

0.643(71.58) 0.652(72.21) 0.639(71.73)

0.756(90.15) 0.756(89.38) 0.753(90.24)

0.563(12.0) 0.553(11.7) 0.566(12.1)

0.167(3.37) 0.136(2.86) 0.163(3.49)

1.310(40.6) 1.263(37.9) 1.312(40.9)

1.393(50.76) 1.359(48.43) 1.397(51.23)

0.244(29.16) 0.244(28.80) 0.247(29.6)

0.580(0.64) 1.70(1.85) 0.041(0.04)



0.00756 0.0419

Model 1



0.00782 0.1088

0.00368

Em D 1

Rmkrf

0.00134

Rg

Std. dev.

Rm

Mean

Rf

Table 103.6 (continued) Intercept .103 /

0.251(9.82) 0.241(9.34) 0.254(9.92)

0.200(7.84) 0.191(7.45) 0.202(7.93)

0.433(12.8) 0.412(12.0) 0.434(12.8)

Rsmb

0.211(6.60) 0.207(6.35) 0.212(6.65)

0.227(7.03) 0.224(6.79) 0.228(7.09)

0.321(7.44) 0.285(6.46) 0.323(7.49)

Rhml

0.013(0.73) 0.006(0.33) 0.014(0.80)

0.013(0.71) 0.006(0.35) 0.014(0.78)

0:097.4:14/ 0:117.4:81/ 0:096.4:12/

Rumd



0.163(3.11) 1 0

0.175(3.31) 1 0

0.0342(0.61) 1 0

0.187(3.56) 1 0

0.0696(1.24) 1 0

0.325(6.19) 1 0



(continued)

0.1115

0.00852

Share return

1540 C.-F. Lee et al.

0.0711 0.899(59.51) 0.948(40.70) 0.906(61.17)

0.00137

0.101(6.72) 0.052(2.25) 0.094(6.32)

1.73(1.58) 15:9.9:57/ 0.379(0.35)

2:06.1:65/ 0:992.0:81/ 4:58.3:73/

Model 1

Model 2

0.00738

Std. dev.

0.00376

4:51.5:43/ 2:81.3:23/ 4:41.5:33/

Model 6

Mean

0.615(68.1) 0.637(70.0) 0.616(69.2)

3:53.4:19/ 1:95.2:21/ 3:46.4:13/

Model 5

1.3172(45.49) 1.3039(45.16) 1.3587(47.12)

0.0418

0.00799

Reg D 1

0.076(1.87) 0.039(0.94) 0.074(1.82)

0.596(25.1) 0.557(22.7) 0.594(25.2)

0.648(72.6) 0.669(74.4) 0.649(73.7)

6:19.5:51/ 0:309.0:26/ 4:73.4:29/

Model 4

0.632(15.7) 0.625(15.2) 0.631(15.7)

1.212(41.8) 1.161(37.8) 1.196(41.0)

0.632(29.89) 0.596(27.54) 0.630(30.03)

3:40.4:21/ 1:50.1:78/ 3:34 .4:15/

Model 3

0.627(69.80) 0.647(71.42) 0.628(70.73)

1.321(54.62) 1.286(51.06) 1.309(54.14)

7:47.7:0/ 1:63.1:51/ 6:08.5:86/

0.755(90.69) 0.761(90.15) 0.753(90.92)



0.00745 0.0422

Model 2

0.245(29.37) 0.239(28.26) 0.247(29.76)



0.00807 0.0999

0:525.0:63/ 0.965(1.14) 0:747.0:90/

Std. dev.

Rmkrf

Model 1

0.0038

0.00149

Mean

Reg D 0

Table 103.6 (continued) Panel B: comparison between regional funds and single country funds Intercept .103 / Rf Rm Rg

0.256(10.8) 0.244(10.0) 0.255(10.7)

0.193(8.1) 0.182(7.5) 0.192(8.1)

0.368(11.8) 0.346(10.8) 0.362(11.6)

Rsmb

0.189(6.44) 0.183(6.02) 0.189(6.42)

0.207(7.0) 0.201(6.5) 0.207(6.9)

0.293(7.5) 0.255(6.3) 0.283(7.2)

Rhml

0.0182(1.12) 0.012(0.72) 0.0179(1.11)

0.022(1.33) 0.017(0.97) 0.022(1.32)

0:058.2:73/ 0:078.3:42/ 0:063.2:92/

Rumd



0.6950(9.18) 1 0

0.657(7.92) 1 0

0:057.1:15/ 1 0

0:0399.0:80/ 1 0

0:336.5:71/ 1 0

0:034.0:68/ 1 0

0:309.5:32/ 1 0

0:131.2:68/ 1 0



(continued)

0.0925

0.00943

0.1061

0.00801

Share return

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds 1541

0.656(26.26) 0.634(24.91) 0.661(26.56)

0.732(80.05) 0.732(79.22) 0.730(80.15)

7:13.5:6/ 2:34.1:84/ 7:13.5:76/

2:63.2:77/ 1:00.1:06/ 2:83.2:99/

Model 2

Model 3

0.612(62.77) 0.621(63.24) 0.610(63.01)

1.3857(47.73) 1.3580(45.54) 1.3857(47.93)

0.10869

0.00133

0.268(29.25) 0.268(28.97) 0.270(29.6)

0.04198

Rmkrf SS D 0

0.812(0.82) 2.00(1.99) 0.436(0.44)

0.00750

Rg

Model 1

0.00663

Rm

Std. dev.

0.00367

Rf

Mean

Intercept .103 /

Panel C: comparison between short sale allowed country funds and short sale not allowed country funds

0.494(9.09) 0.478(8.74) 0.514(9.46)

0.672(33.5) 0.675(33.2) 0.673(34.1)

2:42 .2:17/ 1:34 .1:20/ 3:45 .3:16/

Model 6

0.173(2.95) 0.166(2.81) 0.176(3.00)

0.618(17.9) 0.597(17.2) 0.641((18.7)

0.698(38.6) 0.701(38.3) 0.698(39.2)

2:16.1:95/ 1:12.1:00/ 3:18.2:92/

Rmkrf

Model 5

Rg 0.509(14.76) 0.495(14.30) 0.528(15.39) 1.318(38.1) 1.296(37.5) 1.364(40.0)

Rm 0.695(34.59) 0.696(34.26) 0.698(35.37)

2:66.2:01/ 1:39.1:06/ 4:75.3:69/

Rf

Model 4

Table 103.6 (continued) Intercept .103 / Model 3 0:713.0:66/ 0.337(0.31) 2:02.1:89/

Rsmb

0.229(7.23) 0.221(7.0) 0.236(7.40)

0.209(6.75) 0.203(6.55) 0.215(6.92)

0.338(9.32) 0.329(9.1) 0.354(9.68)

Rsmb

Rhml

0.282(7.12) 0.268(6.74) 0.297(7.50)

0.283(7.14) 0.269(6.76) 0.298(7.52)

0.375(8.03) 0.358(7.66) 0.406(8.69)

Rhml

Rumd

0:002.0:07/ 0:006.0:26/ 0.003(0.14)

0:001.0:06/ 0:005.0:23/ 0.003(0.15)

0:026.0:99/ 0:031.1:16/ 0:017.0:64/

Rumd

0.105(1.81) 1 0

0.0010(0.02) 1 0

0.241(4.17) 1 0



0.484(6.0) 1 0

0.488(6.03) 1 0

0.6162(8.06) 1 0

 0.550(6.75) 1 0

(continued)

0.10912

0.00784

Share return

Share return

1542 C.-F. Lee et al.



0.458(15.69) 0.429(14.61) 0.473(16.31)

2:50.2:55/ 1:00.1:08/ 3:13.3:29/

Model 3

0.717(38.27) 0.726(38.16) 0.714(38.3)

1.1748(45.53) 1.1494(44.24) 1.1907(46.62)

2:17.1:94/ 0:125.0:11/ 3:20.2:91/

0.904(63.24) 0.866(67.27) 0.908(63.92)



0.00774 0.04222

Model 2

0.0957(6.69) 0.134(10.43) 0.092(6.51)

0.00948 0.06577

1:27.1:33/ 1.00(1.18) 2:00.2:14/

Std. dev.

Model 1

0.00390

0.00156

Mean

SS D 1

0.125(2.47) 0.104(2.01) 0.128(2.52)

0.598(60.9) 0.608(61.6) 0.596(61.2)

3:74.3:83/ 2:39.2:35/ 3:89.4:0/

Model 6

0.617(12.0) 0.613(11.7) 0.619(12.0)

0.639(23.0) 0.617(21.6) 0.642(23.3)

0.623(64.0) 0.633(64.8) 0.621(64.4)

Rmkrf

2:80.2:83/ 1:55.1:51/ 2:94.3:0/

Rg

Model 5

Rm 1.274(37.2) 1.233(34.6) 1.273(37.3)

Rf

5:64.4:25/ 0:78.0:57/ 5:50.4:24/

Model 4

Table 103.6 (continued) Intercept .103 /

0.275(10.0) 0.270(9.74) 0.276(10.1)

0.219(7.99) 0.216(7.79) 0.220(8.01)

0.424(11.8) 0.408(11.2) 0.423(11.8)

Rsmb

0.208(6.05) 0.208(5.90) 0.208(6.06)

0.226(6.5) 0.226(6.4) 0.226(6.5)

0.298(6.5) 0.266(5.63) 0.287(6.47)

Rhml

0.005(0.29) 0:002.0:08/ 0.006(0.32)

0.005(0.27) 0:0013.0:07/ 0.006(0.30)

0:102.4:1/ 0:121.4:66/ 0:102.4:12/

Rumd



0.319(4.38) 1 0

0.3289(4.36) 1 0

0.393(5.49) 1 0

0.089(1.54) 1 0

0.099(1.72) 1 0

0:031.0:5/ 1 0



(continued)

0.08843

0.00883

Share return

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds 1543

0.220(7.63) 0.208(7.17) 0.225(7.77)

0.186(6.54) 0.175(6.13) 0.190(6.68)

0.285(8.42) 0.271(7.95) 0.291(8.58)

Rsmb

0.234(6.60) 0.215(6.0) 0.242(6.82)

0.236(6.6) 0.217(6.01) 0.243(6.84)

0.364(8.56) 0.339(7.89) 0.373(8.8)

Rhml

0.034((1.71) 0.032(1.54) 0.035(1.78)

0.035(1.74) 0.033(1.58) 0.036(1.8)

0.0497(2.06) 0.0474(1.92) 0.051(2.11)

Rumd

0.255(3.52) 1 0

0.262(3.6) 1 0

0.2549(3.29) 1 0



Share return



C ˇsmb .1 C SMBt /./ C ˇhml .1 C HMLt /./ C ˇumd .1 C UMDt /./ C "pt

.1 C Rpt /./ D ˛p C ˇf .1 C Rf /./ C ˇm .1 C Rmt /./ C ˇg .1 C Rgt /./ C ˇmkrf .1 C MKRF t /./

j

Significant at 5% level. Significant at 10% level. This table reports the grouped cross-sectional regression of share returns of international closed-end country funds. Rf is the 1-month T-bill of the U.S. market; Rm j is the MSCI country or regional index monthly return corresponding to each international CEF. Rg is the MSCI world index monthly return; EM is the dummy variable, which is one for emerging country fund and zero for developed country fund; Reg is the dummy variable, which is one for regional fund and zero for single country fund; SS is the dummy variable, which is one for fund investing in countries where short sale is allowed, and equal to zero for fund investing in countries where short sale is not allowed. MKRF, SMB, HML, and UMD are monthly returns of Carhart four factors. The data spans from 1965 to 2003. White heteroscedasticity consistent t-statistics is in the parentheses. The model estimated is:



0.294(6.34) 0.262(5.60) 0.306(6.63)

0.695(37.1) 0.706(37.1) 0.692(37.2)

4:08.4:11/ 2:72.2:69/ 4:55.4:66/

Model 6

0.295(6.11) 0.286(5.88) 0.297(6.16)

0.508(16.7) 0.471(15.3) 0.523(17.4)

0.743(43.5) 0.753(43.4) 0.740(43.5)

Rmkrf

3:73.3:75/ 2:41.2:38/ 4:20.4:29/

Rg

Model 5

Rm 1.171(37.2) 1.135(35.5) 1.186(38.3)

Rf

3:05.2:55/ 1:00.0:83/ 3:76.3:21/

Model 4

Table 103.6 (continued) Intercept .103 /

1544 C.-F. Lee et al.

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds

single country funds and regional funds. All single country funds’ share returns produce significant negative  while regional funds’ share returns have significant positive . Although the NAV return does not have this obvious positivenegative pattern, the value of  is really smaller for the single country funds compared to the regional funds. A comparison between short sale allowed country funds and short sale not allowed country funds also gives us a similar pattern with regional funds. Short sale allowed country funds have consistent positive  for both share returns and NAV returns. Results from the pooled regression, as reported in Tables 103.6 and 103.7, are also supportive of a significant global risk premium. No matter which model we use for both share returns and NAV returns, the factor the loadings of the global risk is significant positive for both developed market funds and emerging market funds, regional funds or single country funds, short sale allowed funds or short sale not allowed funds. Moreover, in the framework of the generalized CAPM, consistent with previous research, all factors in Carhart (1997) four-factor model remain significant, including the size factor after controlling for the momentum factor. And the factor loadings of each model are consistent with our expectation. However, when we include the global return or the local market return into the model, the momentum factor no longer plays an important role. As reported in Tables 103.8 and 103.9, results of crosssectional pooling regression give us another view to compare the differences among different model specifications for the functional form of international CEFs. First, all of the transformation parameters are significantly positive, with larger  for share returns than  for NAV returns. Consistent with the grouped regression, the global risk has been priced in each of generalized model, even controlling for the local market returns. Although the transformation parameters are significant for each model, the factor loadings between the generalized model and the linear or the loglinear model are similar, especially for the NAV returns. The results from generalized model are more close to the models with loglinear returns, which implies that the conventional use of loglinear return is still reasonable. Controlling for the risk factors in the generalized model, the dummy variables do not produce significant result for share returns. However, the dummy variables of Em and SS are significant for NAV returns. This implies that the difference between the emerging market and the developed market, and the difference between short sale allowed country funds and short sale not allowed country funds are more significant for NAV returns. This is consistent with the characteristics of international closed-end funds, underlying assets are traded in foreign markets while share returns are determined by the U.S. investors. Hence NAV returns should reflect the difference among different funds more significantly.

1545

103.5.4 Performance Evaluation As motivated by Jensen (1969), and Lee (1976) among others, one of the important applications of the generalized CAPM is to give a more unbiased and efficient performance evaluation of mutual funds. In Table 103.10 we report the risk-adjusted returns of each model for share returns and NAV returns of individual fund. Consistent with previous research such as Patro (2001), the average performance of international CEFs is negative, even before the exclusion of fund expenses irrespective of the model used. The grouped regressions and cross-sectional pooling regressions give us similar results.

103.6 Conclusion In this chapter, we investigate the functional form specifications for 90 U.S.-based international closed-end country funds. Our results suggest that the frequently used linear and loglinear specifications of security return may be inappropriate for many funds. The use of a generalized functional form should be the rule rather than an exception. The evidence against the linear functional form in our sample indicates that the investment horizon is likely different from 1 month. For most funds, the true investment horizon is not instantaneous either. Similar to McDonald (1983) and Bubnys and Lee (1989), we encounter many negative estimates for the horizon parameter in our sample. Lee and Wei (1988) have shown that the estimated  will be negative instead of positive if the security return and the market return had a bivariate lognormal distribution. Similar for most funds, some of the NAV returns seem to perform slightly worse compared to the share returns. Our empirical results imply that the functional transformation does matter for international CEFs. The global risk factor is priced in the framework of generalized CAPM. The momentum factor is no longer an important factor in asset pricing of share returns of international CEFs after controlling for the risk factors of the global market return and the local market return. No matter which benchmark we use, international CEFs have negative risk adjusted performance. Moreover, there exists significant different patterns of functional forms for emerging market funds and developed market funds, single country funds and regional funds, and funds for which short sales are allowed. A comparison between the functional form share returns and NAV returns of closed-end country funds suggests that foreign investors, especially those investors from emerging markets, may have more heterogeneous investment horizons vis-à-vis their U.S. counterparts.

Mean Std. dev. Model 1

Reg D 0

Model 3

Model 2

Mean Std. dev. Model 1

Em D 0

4:36.4:32/ 0:123.0:04/ 4:58.4:5/

6:04.6:20/ 6:04.5:34/ 7:88.6:33/ 5:69.4:50/ 4:58.3:43/ 7:38.5:29/ 5:70.5:05/ 4:93.4:09/ 6:88.5:40/

Intercept .103 /

0.00378 0.0015 0.239(23.55) 0.288(9.5) 0.238(23.26)

0.00396 0.00162 0.188(10.81) 0.190(10.33) 0.188(9.34)

Rf

0.00826 0.0996 0.761(74.81) 0.712(23.5) 0.762(74.56)

0.743(28.96) 0.749(27.49) 0.723(24.39)

0.00878 0.0612 0.812(46.54) 0.810(44.02) 0.812(40.25)

Rm

0.00763 0.0421

0.856(29.15) 0.848(27.28) 0.872(26.97) 0.121(3.31) 0.109(2.80) 0.152(3.52)

0.00776 0.0423

Rg

0.073(10.18) 1 0

0.610(31.14) 1 0 0.561(25.05) 1 0 0.574(28.85) 1 0

œ

Table 103.7 Cross-sectional regressions for NAV return of international CEFs by group

0.00685 0.2732

0.00189 0.0833

NAV return

2:32.2:82/ 2:98.3:64/ 3:84.4:54/

Reg D 1

2:70.2:61/ 1.60(0.47) 2:76.2:64/ 81:8.27:18/ 1.39(0.67) 6:61.4:80/ 2:58.2:56/ 1:27.0:85/ 2:61.2:25/

Em D 1

Intercept .103 /

0.00373 0.00138 0.283(24.12) 0.266(23.13) 0.252(21.76)

0.00366 0.00136 0.247(25.71) 0.298(9.4) 0.247(25.69)

Rf

0.00747 0.0710 0.717(60.98) 0.734(63.76) 0.748(64.7)

0.748(68.45) 0.744(48.81) 0.748(68.48)

0.00806 0.1084 0.753(78.46) 0.702(22.13) 0.753(78.51)

Rm

0.00814 0.0416

0.807(16.80) 0.831(17.13) 0.944(29.26) 0.0616(2.91) 0.0204(0.53) 0.0618(2.35)

0.00781 0.0418

Rg

1.737(16.97) 1 0

0.00349 0.0725

0.00836 0.2905

NAV return

(continued)

0.0125(1.64) 1 0 0:391.35:5/ 1 0 0.0137(1.69) 1 0

œ

1546 C.-F. Lee et al.

2:27.1:96/ 2.37(0.57) 2:30.1:98/ 6:74.4:50/ 0:342.0:14/ 6:87.4:59/ 2:167.1:95/ 3.70(0.88) 1:80.1:57/

5:93.4:70/ 0:336.0:22/ 6:55.5:15/ 3:50.3:38/ 1.40(0.45) 3:80.3:57/

0.00366 0.00134 0.268(25.08) 0.323(8.52) 0.268(25.07)

Rf

0.723(64.78) 0.680(16.0) 0.717(59.6)

0.00656 0.1083 0.732(68.5) 0.677(17.83) 0.732(68.54)

0.742(64.17) 0.708(20.38) 0.743(63.93)

Rm

0.918(26.22) 0.826(14.37) 0.919(26.27) 0.070(2.41) 0:023.0:21/ 0.080(2.61)

0.00762 0.0419

0.859(29.26) 0.801(22.98) 0.865(29.21) 0.078(2.87) 0.0056(0.07) 0.081(2.93)

Rg

0.012(1.52) 1 0 0.021(2.07) 1 0 0.0129(1.59) 1 0

0.099(10.09) 1 0 0.068(9.47) 1 0

œ

0.00799 0.3139

NAV return

6:23.6:6/ 5:52.5:59/ 7:40.6:99/ 4:96.4:35/ 3:76.3:17/ 6:77.5:49/ 5:490.5:58/ 4:70.4:51/ 6:60.6:07/

SS D 1

2:11.2:13/ 4:35.4:44/ 6:38.6:28/ 2:40.2:9/ 3:20.3:84/ 4:10.4:78/

Intercept .103 /

0.00389 0.00157 0.1979(13.87) 0.204(13.58) 0.189(11.94)

Rf

0.741(38.57) 0.738(36.28) 0.746(34.98)

0.00959 0.0656 0.802(56.25) 0.796(53.13) 0.811(51.19)

0.615(39.32) 0.641(41.2) 0.662(42.2)

Rm

0.870(32.68) 0.860(30.99) 0.896(31.12) 0.1318(4.34) 0.125(3.97) 0.138(4.14)

0.00796 0.0421

0.940(41.0) 0.963(41.56) 0.992(41.32) 0.250(9.53) 0.230(8.67) 0.220(8.03)

Rg

0.594(30.74) 1 0 0.571(25.14) 1 0 0.561(28.69) 1 0

2.200(19.35) 1 0 1.864(17.54) 1 0

œ

0.00291 0.0848

NAV return



C ˇg .1 C Rgt /./ C "pt where . /./ D

. /./  1 

.1 C NAV pt /./ D ˛p C ˇf .1 C Rf /./ C ˇm .1 C Rmt /./

j

Significant at 5% level. Significant at 10% level. j This table reports the grouped cross-sectional regression of NAV returns of international closed-end country funds. Rf is the 1-month T-bill of the U.S. market; Rm is the MSCI country or regional index monthly return corresponding to each international CEF. Rg is the MSCI world index monthly return; EM is the dummy variable, which is one for emerging country fund and zero for developed country fund; Reg is the dummy variable, which is one for regional fund and zero for single country fund; SS is the dummy variable, which is one for fund investing in countries where short sale is allowed, and equal to zero for fund investing in countries where short sale is not allowed. The data spans from 1965 to 2003. White heteroscedasticity consistent t-statistics is in the parentheses. The model estimated is:



Model 3

Model 2

Mean Std. dev. Model 1

SS D 0

Model 3

Model 2

Intercept .103 /

Table 103.7 (continued)

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds 1547

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 Model 10 Model 11 Model 12

3.73e–3 1.39e–3

Rf

1.51(2.24) 0:89.1:35/ 1:53.1:26/ 1:36.1:69/ 0:662.0:72/ 1:083.0:50/ 0:649.0:73/ 2:39.0:84/ 1:67.2:40/ 1:62.0:74/ 2:31.3:35/ 2:56.1:18/ 0.486(14.6) 0.552(14.9)

0.674(84.7) 0.661(78.8) 0.641(78.2) 0.629(73.3)

0.494(15.1) 0.557(15.3)

0.660(83.8) 0.652(78.4) 0.626(77.2) 0.619(72.8)

0.574(32.17) 0.606(32.25) 0.574(32.16) 0.568(30.52) 0.605(31.28)

0.605(34.4) 0.638(34.47) 0.605(34.39) 0.593(32.27) 0.631(33.08)

7.64e–3 0.042

Rg

0.788(108.8) 0.642(79.26) 0.644(78.64) 0.642(79.25) 0.633(74.93) 0.634(74.83)

7.59e–3 0.0917

Rm

0.212(28.85) 0.788(107.5) 0.654(80.00) 0.656(79.29) 0.654(80.01) 0.641(75.28) 0.642(75.10)

0.189(0.28) 0.212(29.32) 2:65.4:09/ 3:50.2:96/ 3:0.3:77/ 2:10.2:34/ 2:60.1:21/ 4:75.5:47/ 5:70.2:05/ 3:18.4:74/ 2:88.1:34/ 3:93.5:89/ 3:86.1:83/

Panel B  D 1

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 Model 10 Model 11 Model 12

Panel A  D 

Mean Std. dev.

Intercept .103 /

1.207(51.5) 1.213(46.3) 0.577(29.2) 0.595(27.5) 0.184(5.5) 0.143(3.87)

1.254(55.5) 1.256(49.5) 0.613(31.7) 0.623(29.5) 0.215(6.58) 0.170(4.69)

Rmkrf

0.340(13.9) 0.355(13.0) 0.190(9.91) 0.190(9.02) 0.239(12.4) 0.244(11.6)

0.359(14.9) 0.372(13.8) 0.200(10.5) 0.196(9.39) 0.250(13.1) 0.252(12.0)

Rsmb

Table 103.8 Cross-sectional pooled regressions for share return of international CEFs

0.289(9.24) 0.289(8.30) 0.225(9.20) 0.221(8.22) 0.213(8.78) 0.206(7.77)

0.324(10.6) 0.320(9.38) 0.237(9.91) 0.227(8.66) 0.225(9.51) 0.213(8.23)

Rhml

8.03e–5(0.04)

0.0006(0.42)

0.668 0.471

Em 0.444 0.497

SS

9:34e-4.0:70/ 3:85e–4.0:23/ 8:64e–4.0:43/

9.79e–4(0.73)

0.342 0.474

Reg

1:87e–4.0:09/ 0:062.3:55/ 0:057.2:92/ 1.11e–3(0.41) 0.0097(0.71) 0.017(1.12) 4:01e–4.0:19/ 6.45e–3(0.47) 0.014(0.94) 2:47e–4.0:12/

3.25e–4(0.22)

4.72e–4(0.28)

2.45e–4(0.14)

3.16e–5.0:01/

3.61e–4(0.21)

1.38e–3(1.01)

(continued)

0.280(6.82) 0.132(3.18) 0.108(2.50) 0.132(3.19) 0.182(4.06) 0.148(3.19) 0:011.0:24/ 0.032(0.64) 0.110(2.66) 0.135(2.89) 0.097(2.34) 0.123(2.64)



1 1 1 1 6:93e–4.0:51/ 1 5:96e–4.0:29/ 1 1 1.72e–3(0.65) 1 1 7:37e–4.0:36/ 1 1 5:6e–4.0:28/ 1

0:048.2:83/ 0:046.2:43/ 3:68e–4.0:14/ 3.97e–4(0.18) 2.37e–3(0.92) 0.015(1.16) 0.021(1.44) 8.70e–6(0.00) 5:58e–4.0:33/ 1:05e–3.0:53/ 0.013(0.98) 0.019(1.30) 4.29e–5(0.02) 2:89e–4.0:18/ 8:31e–4.0:42/

Rumd

1548 C.-F. Lee et al.

0:317.0:48/ 2:93.4:56/ 3:75.3:18/ 3:25.4:11/ 2:46.2:73/ 2:84.1:34/ 4:70.5:53/ 5:80.2:09/ 3:38.5:07/ 3:06.1:43/ 4:11.6:2/ 4:04.1:91/

0.214(29.71)

Rf

0.612(35.03) 0.644(35.01) 0.612(35.03) 0.602(32.93) 0.638(33.64)

0.496(15.1) 0.558(15.3)

0.658(84.1) 0.650(78.6) 0.624(77.5) 0.617(73.0)

Rg

0.786(109.1) 0.639(79.49) 0.642(78.93) 0.639(79.48) 0.630(75.01) 0.631(74.98)

Rm

1.253(55.8) 1.258(49.9) 0.619(32.3) 0.630(30.0) 0.219(6.7) 0.174(4.81)

Rmkrf

0.359(14.9) 0.372(13.8) 0.201(10.6) 0.197(9.45) 0.252(13.2) 0.253(12.1)

Rsmb

0.324(10.6) 0.321(9.43) 0.239(10.0) 0.229(8.73) 0.227(9.6) 0.215(8.29)

Rhml

0:048.2:84/ 0:045.2:41/ 0.016(1.21) 0.021(1.48) 0.013(1.03) 0.019(1.34)

Rumd

7.03e–5(0.03)

5.85e–5(0.03)

4:21e–4.0:16/

1.11e–4(0.05)

6.25e–4(0.44)

Em

4:06e–4.0:25/

6:95e–4.0:42/

4.05e–4(0.19)

5:29e–4.0:32/

9.11e–4(0.68)

Reg

8:76e–4.0:44/

1:11e–3.0:56/

2.39e–3(0.92)

9:83e–4.0:73/ 9:19e–4.0:46/

SS

0 0 0 0 0 0 0 0 0 0 0 0



C ˇhml .1 C HMLt /./ C ˇumd .1 C UMDt /./ C ˇe Emp C ˇr Regp C ˇs SSp C "pt where . /./ D

. /./  1 l 

.1 C Rpt /./ D ˛p C ˇf .1 C Rf /./ C ˇm .1 C Rmt /./ C ˇg .1 C Rgt /./ C ˇmkrf .1 C MKRF t /./ C ˇsmb .1 C SMBt /./

j

Significant at 5% level. This table reports the cross-sectional pooled regression of share returns of international closed-end country funds. Rf is the 1-month T-bill of the U.S. market; Rm j is the MSCI country or regional index monthly return corresponding to each international CEF. Rg is the MSCI world index monthly return; EM is the dummy variable, which is one for emerging country fund and zero for developed country fund; Reg is the dummy variable, which is one for regional fund and zero for single country fund; SS is the dummy variable, which is one for fund investing in countries where short sale is allowed, and equal to zero for fund investing in countries where short sale is not allowed. MKRF, SMB, HML, and UMD are monthly returns of Carhart four factors. The data spans from 1965 to 2003. White heteroscedasticity consistent t-statistics is in the parentheses. The model estimated is:



Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 Model 10 Model 11 Model 12

Panel C  D 0

Intercept .103 /

Table 103.8 (continued)

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds 1549

4:16.5:71/ 3:63.4:85/ 6:49.4:56/ 3:68.4:01/ 1:88.1:75/ 4:01.1:52/

1:07.0:52/ 0:086.0:04/ 4:12.1:00/ 1.04(0.40) 3.02(0.95) 0.682(0.09)

4:34.5:91/ 3:81.5:06/ 6:68.4:67/ 3:91.4:24/ 2:08.1:91/ 4:22.1:60/

0.241(30.01)

0.283(12.71)

0.242(30.33)

3.71e–3 1.40e–3

0.759(94.65) 0.731(77.80) 0.737(74.88) 0.731(77.79) 0.721(71.35) 0.725(69.24)

0.717(32.14) 0.699(26.63) 0.704(25.01) 0.699(26.62) 0.688(23.43) 0.692(22.34)

0.758(94.89) 0.730(78.02) 0.736(75.11) 0.730(78.02) 0.720(71.59) 0.724(69.48)

7.81e–3 0.0915



0.109(5.31) 0.102(4.57) 0.109(5.31) 0.114(5.18) 0.110(4.64)

0.064(1.12) 0.051(0.80) 0.0646(1.13) 0.059(0.92) 0.0488(0.70)

0.107(5.27) 0.100(4.52) 0.107(5.27) 0.112(5.12) 0.108(4.59)

7.80e–3 0.0419

0.00243(0.96)

0.00416(2.39)

0.00404(0.54)

0.00641(1.29)

0.00245(0.98)

0.00417(2.41)

0.671 0.470

Em

5:22e  4.0:25/

3.01e–4(0.19)

0:00477.0:78/

0:00329.0:75/

6:92e  4.0:34/

1.42e–4(0.09)

0.341 0.474

Reg

0:00423.2:61/ 0:00284.1:14/

0:00673.1:42/ 0:00452.0:61/

0:00424.2:64/ 0:00283.1:15/

0.441 0.496

SS

0.076(12.06) 0.072(11.31) 0.071(10.75) 0.072(11.30) 0.072(10.96) 0.070(10.51)



5.88e–3 0.229

NAV return

C ˇe Emp C ˇr Regp C ˇs SSp C "pt where . /./ D

. /./  1 

.1 C NAV pt /./ D ˛p C ˇf .1 C Rf /./ C ˇm .1 C Rmt /./ C ˇg .1 C Rgt /./

j

Significant at 5% level.  Significant at 10% level. This table reports the cross-sectional pooled regression of NAV returns of international closed-end country funds. Rf is the 1-month T-bill of the U.S. market; Rmj is the MSCI country or regional index monthly return corresponding to each international CEF. Rg is the MSCI world index monthly return; EM is the dummy variable, which is one for emerging country fund and zero for developed country fund; Reg is the dummy variable, which is one for regional fund and zero for single country fund; SS is the dummy variable, which is one for fund investing in countries where short sale is allowed, and equal to zero for fund investing in countries where short sale is not allowed. The data spans from 1965 to 2003. White heteroscedasticity consistent t-statistics is in the parentheses. The model estimated is:



Model 1 Model 2 Model 3 Model 4 Model 5 Model 6

Panel C  D 0

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6

Panel B  D 1

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6

Mean Std. dev. Panel A  D 

Table 103.9 Cross-sectional pooled regressions for NAV return of international CEFs Intercept .103 / Rf Rm Rg

1550 C.-F. Lee et al.

0:85 0:78 1.18 3:69 1.62

0:75 0:51 1.11 3:80 1.31

Mean Median STD Min Max

Global index (share return)

0:90 0:91 1.14 4:00 1.66

0:33 0:38 0.52 2:59 1.24

Mean Median STD Min Max

GCAPM (NAV return)

0:85 0:74 1.11 3:65 2.03

0:80 0:51 1.11 3:87 1.41

Mean Median STD Min Max

GCAPM_CRSP (share return)

t .˛/ 0:13 0:31 0.85 1:77 2.17

˛.%/

Mean Median STD Min Max

GCAPM (share return)

0:11 0:16 0.61 2:80 1.54

 D 

Table 103.10 Performance evaluation of international CEFs ˇm

0.77 0.79 0.19 0.20 1.47

1.16 1.12 0.31 0.39 2.15

0.87 0.89 0.25 0.21 1.76

15.05 15.19 7.14 1.34 33.55

6.84 7.06 2.39 1.42 13.90

12.20 12.33 4.72 1.02 27.40

t .ˇm /

1.32 1.33 0.36 0.37 2.33

ˇg

8.10 7.87 3.06 1.48 16.92

t .ˇg /

0:32 0:11 0.89 3:55 1.62

0:14 0:39 1.84 2:29 16.24

0:38 0:17 0.95 4:12 1.70

0.05 0.05 0.59 2:00 1.88

˛.%/

D1

0:30 0:21 0.99 3:53 1.63

0:91 0:95 1.24 4:00 1.69

0:34 0:26 1.02 3:69 3.28

0.13 0.12 0.86 1:77 2.53

t .˛/

0.76 0.80 0.27 1:05 1.51

1.13 1.12 0.30 0.34 2.12

0.88 0.90 0.24 0.20 1.64

ˇm

14.72 14.40 7.38 0:72 31.82

6.58 6.46 2.35 1.28 13.82

12.08 12.33 4.63 1.05 25.45

t .ˇm /

1.30 1.34 0.35 0.33 2.31

ˇg

7.78 7.49 2.99 1.50 16.56

t .ˇg /

0:73 0:45 0.96 4:32 1.53

0:44 0:45 0.62 3:38 0.77

0:82 0:55 0.98 4:98 1.37

0:13 0:12 0.60 2:44 1.65

˛.%/

D0 t .˛/

0:84 0:73 1.01 4:00 1.61

1:04 1:08 1.16 4:14 1.70

0:90 0:79 0.98 4:16 2.81

0:16 0:17 0.84 1:82 2.33

ˇm

0.79 0.80 0.19 0.27 1.49

1.19 1.20 0.32 0.37 2.29

0.89 0.90 0.25 0.23 1.71

14.93 13.95 7.57 1.22 34.99

7.04 7.02 2.49 1.46 14.52

12.33 12.37 4.71 1.19 26.92

t .ˇm /

8.18 8.04 3.06 1.64 16.95

t .ˇg /

(continued)

1.34 1.35 0.37 0.36 2.47

ˇg

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds 1551

Mean Median STD Min Max

Mean Median STD Min Max

Mean Median STD Min Max

Global GCAPM (share return)

Global GCAPM_CRSP (share return)

Global GCAPM (NAV return)

0:83 0:75 1.15 3:79 1.66 0:83 0:71 1.22 7:61 1.60

0:30 0:30 0.52 2:40 1.50 0.72 0.74 0.25 0.04 1.80

0:07 0:01 0.59 2:44 1.18 11.81 10.94 6.84 0.20 30.55

0:14 0:04 1.45 3:42 3.10

8.30 8.31 4.53 1:62 23.27

0.11 0.08 0.26 0:94 0.98

1.40 1.40 0.67 0:57 4.25

0.56 0.56 0.41 0:70 1.86

0.73 0.73 1.92 10:50 6.45

4.08 3.99 2.13 1:19 10.67

3.45 3.57 1.77 1:98 8.52

0:06 0:33 2.40 2:52 21.63

0:31 0:11 0.91 3:48 1.12

0:19 0:12 0.70 2:67 1.64

0:16 0:27 2.40 2:91 21.18

0:76 0:45 1.10 4:02 1.30

0.68 0.66 0.37 1:21 2.27

6.36 6.37 2.76 0:30 13.74

0:59 0:66 0.97 3:30 1.62

t .ˇg /

0:42 0:35 0.79 3:88 1.32

0.91 0.92 0.37 0:13 2.94

˛.%/

ˇg

0:59 0:61 0.96 2:74 1.49

0:41 0:32 0.84 3:91 1.90

t .ˇm /

t .˛/

˛.%/

ˇm

D1

 D 

0:90 0:80 1.51 10:55 1.64

0:27 0:14 0.97 3:16 1.33

0:23 0:26 0.93 2:57 1.99

0:67 0:60 0.93 3:50 1.26

t .˛/

0.72 0.75 0.28 0:28 1.84

0:09 0:05 0.59 2:26 1.20

0.70 0.72 0.34 1:14 2.04

ˇm

11.57 10.82 6.97 0:18 27.56

0:21 0:10 1.43 3:13 2.67

8.95 8.40 4.91 1:62 26.16

t .ˇm /

0.05 0.07 0.69 5:79 1.00

1.39 1.38 0.66 0:51 3.97

0.52 0.50 0.39 0:55 1.75

0.84 0.93 0.80 6:04 1.61

ˇg

0.63 0.61 1.80 9:08 4.15

3.99 3.79 2.11 1:08 10.87

2.86 3.12 2.21 7:78 8.52

6.27 6.24 2.93 1:47 13.96

t .ˇg /

This table gives the summary statistics of estimated alphas and betas for functional form, linear and loglinear specification of different models

Mean Median STD Min Max

Global index (NAV return)

Table 103.10 (continued)

0:43 0:41 0.64 3:70 0.86

0:74 0:45 0.97 4:37 1.00

0:40 0:34 0.74 3:00 1.55

0:75 0:52 0.75 3:73 0.29

˛.%/

D0

1:04 0:96 1.44 10:37 1.65

0:83 0:66 0.98 3:69 1.18

0:58 0:58 0.95 2:81 1.78

1:08 0:92 0.83 3:73 0.30

t .˛/

0.74 0.77 0.27 0.08 1.86

0:06 0:01 0.60 2:35 1.27

0.68 0.69 0.36 1:22 2.14

ˇm

11.69 10.91 7.06 0.30 31.47

0:13 0:01 1.45 3:33 2.74

8.85 8.72 4.98 1:68 26.48

t .ˇm /

0.11 0.07 0.30 0:94 1.10

1.41 1.39 0.66 0:27 4.10

0.57 0.52 0.40 0:62 1.89

0.93 0.94 0.33 0:08 1.63

ˇg

0.56 0.59 1.78 8:43 4.68

4.05 3.96 2.10 0:52 10.67

3.20 3.26 1.99 2:36 8.49

6.37 6.34 2.86 0:35 13.95

t .ˇg /

1552 C.-F. Lee et al.

103 Functional Forms for Performance Evaluation: Evidence from Closed-End Country Funds

References Aitken, M., F. Alex, M. S. McCorry and P. L. Swan. 1998. “Short sales are almost instantaneously bad news: evidence from the Australian stock exchange.” Journal of Finance 53, 2205–2223. Box, G. E. P. and D. R. Cox. 1964. “An analysis of transformation.” Journal of the Royal Statistical Society Series B 26(2), 211–252. Bris, A., W. N. Goetzmann and N. Zhu. 2004. “Efficiency and the bear: short sales and markets around the world.” Working paper, Yale School of Management. Bubnys, E. L. and C. F. Lee. 1989. “Linear and generalized functional form market models for electric utility firms.” Journal of Economics and Business 41, 213–223. Carhart, M. 1997. “On persistence in mutual fund performance.” Journal of Finance 52, 57–82. Chang, E., C. Eun and R. Kolodny. 1995. “International diversification through closed-end country funds.” Journal of Banking and Finance 19, 1237–1263. Chaudhury, M. M. and C. F. Lee. 1997. “Functional form of stock return model: some international evidence.” The Quarterly Review of Economics and Finance 37(1), 151–183. Cumby, R. and J. Glen. 1990. “Evaluating the performance of international mutual funds.” Journal of Finance 45(2), 497–521. Errunza, V. and E. Losq. 1985. “International asset pricing under mild segmentation: theory and test.” Journal of Finance 40, 105–124. Errunza, V. and E. Losq. 1989. “Capital flow controls, international asset pricing, and investors’ welfare: a multi-country framework.” Journal of Finance 44, 1025–1037. Fabozzi, F., J. Francis and C. F. Lee. 1980. “Generalized functional form for mutual fund returns.” Journal of Financial and Quantitative Analysis 15, 1107–1119. Ferson, W. and C. Harvey. 1994. “Sources of risk and expected returns in global equity markets.” Journal of Banking and Finance 18, 775–803. Harvey, C. 1991. “The world price of covariance risk.” The Journal of Finance 46(1), 111–157. Hong, H. and J. Stein. 2003. “Differences of opinion, short-sales constraints and market crashes.” Review of Financial Studies 16(2), 487–525. Jensen, M. C. 1969. “Risk, the pricing of capital assets, and the evaluation of investment portfolio.” Journal of Business 2, 167–247.

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Johnson, T. W. 2004. “Predictable investment horizons and wealth transfers among mutual fund shareholders.” Journal of Finance 59(5), 1979–2012. Jones, C. and O. Lamont. 2002. “Short sale constraints and stock returns.” Journal of Financial Economics 66(2), 207–239. Jorion, P. and E. Schwartz. 1986. “Integration vs. segmentation in the Canadian stock market.” Journal of Finance 41, 603–613. Lee, C. F. 1976. “Investment horizon and the functional form of the capital asset pricing model.” Review of Economics and Statistics 58, 356–363. Lee, C. F. and K. C. J. Wei. 1988. “Impacts of rates of return distributions on the functional form of CAPM.” Working paper, Rutgers University. Lee, C. F., C. Wu and K. C. J. Wei. 1990. “The heterogeneous investment horizon and the capital asset pricing model: theory and implications.” Journal of Financial and Quantitative Analysis 25, 361–376. Lessard, D. 1974. “World, national, and industry factors in equity returns.” The Journal of Finance 29(2), 379–391. Levhari, D. and H. Levy. 1977. “The capital asset pricing model and the investment horizon.” Review of Economics and Statistics 59, 92– 104. Levy, H. 1972. “Portfolio performance and the investment horizon.” Management Science 36, 645–653. McDonald, B. 1983. “Functional forms and the capital asset pricing model.” Journal of Financial and Quantitative Analysis 18, 319–329. Patro, D. K. 2001. “Measuring performance of international closed-end funds.” Journal of Banking and Finance 25, 1741–1767. Patro, D. K. 2008. “Stock market liberalization and emerging market country fund premiums.” Journal of Business 78(1), 135–168. Solnik, B. 1974a. “An equilibrium model of international capital market.” Journal of Economic Theory 36, 500–524. Solnik, B. 1974b. “An international market model of security price behavior.” Journal of Financial and Quantitative Analysis 10, 537– 554. Solnik, B. 1974c. “The international pricing of risk: an empirical investigation of the world capital market structure.” Journal of Finance 29, 365–377. Tobin, J. 1965. “The theory of portfolio selection,” in The theory of interest rates, F. Hain and F. Breechling (Eds.). MacMillan, London, 3–51. Zarembka, P. 1968. “Functional form in the demand for money.” Journal of the American Statistical Association 63, 502–511.

Chapter 104

A Semimartingale BSDE Related to the Minimal Entropy Martingale Measure* Michael Mania, Marina Santacroce, and Revaz Tevzadze

Abstract An incomplete financial market model is considered, where the dynamics of the assets price is described by an Rd -valued continuous semimartingale. We express the density of the minimal entropy martingale measure in terms of the value process of the related optimization problem and show that this value process is determined as the unique solution of a semimartingale backward equation. We consider some extreme cases when this equation admits an explicit solution. Keywords Semimartingale backward equation r Contingent claim pricing r Minimal entropy martingale measure r Incomplete markets

104.1 Introduction Using the dynamic programming technique, we study the structure of the minimal entropy martingale measure, which is related to the problem of derivative pricing in incomplete markets. We assume that the dynamics of the price process of the assets traded on the market is described by an Rd valued continuous semimartingale X D .Xt ; t 2 Œ0; T / defined on a filtered probability space .; F ; F D .Ft ; t 2 Œ0; T /; P / satisfying the usual conditions of right-continuity and completeness, where F D FT and T < 1 is the fixed time horizon. Suppose that the market contains also a riskless M. Mania () A. Razmadze Mathematical Institute, M. Aleksidze St. 1, Tbilisi, Georgia and Georgian–American University, 3, Alleyway II, Chavchavadze Ave. 17, A, Tbilisi, Georgia e-mail: [email protected] M. Santacroce Politecnico di Torino, Department of Mathematics, C.so Duca degli Abruzzi 24, 10129 Torino, Italy R. Tevzadze Institute of Cybernetics, S. Euli St. 5, Tbilisi, Georgia and Georgian–American University, 3, Alleyway II, Chavchavadze Ave. 17, A, Tbilisi, Georgia

bond with discounted price equal to 1 at all times. Denote by Me the set of equivalent martingale measures of X , i.e., a set of measures Q, equivalent to the basic measure P , such that X is a local martingale under Q. In complete markets every contingent claim is replicable and the price of the claim is determined using the uniquely defined martingale measure. It is also well known that if the market is incomplete the set of martingale measures is not in general a singleton and there is an interval of arbitrage-free prices, associated to different pricing (martingale) measures [see El Karoui and Quenez (1995)]. The choice of the pricing measure is no longer preference-free and depends on the utility functions of investors or on the criterion relative to which the hedging error is measured. The most popular choices of pricing measures are the minimal martingale measure, introduced in Föllmer and Schweizer (1991), and the variance-optimal martingale measure (Delbaen and Schachermayer 1996; Schweizer 1996). The latter is determined by minimizing the L2 -norm of densities of the martingale measures with respect to the basic measure P among all signed martingale measures and represents the best choice when the quality of the hedging strategies is measured by the quadratic criterion [see Gourieroux et al. (1998); Pham et al. (1998)]. Another possibility is the minimal entropy martingale measure which minimizes the relative entropy of a martingale measure with respect to the measure P and is known to be closely related to the exponential hedging problem (Delbaen et al. 2002; Rheinländer 1999; Rouge and El Karoui 2000). For the economic interpretation of the minimal, variance optimal and minimal entropy martingale measures see Hoffman et al. (1992), Delbaen et al. (1997), and Frittelli (2000), respectively. It is known [see Frittelli (2000); Miyahara (1996)] that for a locally bounded process X the minimal entropy martingale measure always exists, is unique and if there is a martingale measure with finite relative entropy then the minimal entropy martingale measure is equivalent to P .

* This paper is reprinted from Finance and Stochastics, 7 (2003), No. 3, pp. 385–402.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_104, 

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Our aim is to give the construction of the minimal entropy We assume the following conditions to be satisfied: martingale measure when the dynamics of the discounted as(a) All .F; P /-local martingales are continuous. sets price process is governed by a continuous semimartin(b) There is an equivalent martingale measure Q such that gale. We obtain a description of the minimal entropy marQ Q EZT lnZT < 1, i.e. tingale measure in terms of the value function of a suitable problem of an optimal equivalent change of measure and MeEnt ¤ ;: (104.1) show that this value process uniquely solves the corresponding semimartingale backward stochastic differential equation Note that conditions (a) and (b) imply that X is a continu(BSDE). We show that in two specific extreme cases [already ous semimartingale satisfying the structure condition. This studied in Biagini et al. (2000), Laurent and Pham (1999), means that X admits the decomposition Pham et al. (1998) in relation to the variance-optimal martingale measures] this semimartingale BSDE admits an explicit (104.2) Xt D X0 C ƒt C Mt ; solution which gives explicit construction of the minimal entropy martingale measure. In particular, we give a necessary where M is a continuous local martingale and there exists and sufficient condition when the minimal entropy martin- a predictable Rd -valued process  such that dƒ D d hM i R 0 gale measure coincides with the minimal martingale measure with KT D 0T s d hM is s < 1, where 0 denotes the transas well as with the martingale measure appearing in the sec- position. The process K is called the mean–variance tradeoff ond above mentioned extreme case. process of X [see Schweizer (1994) for the interpretation of BSDEs have been introduced in Bismuth (1973) for the the process K]. linear case as the equations for the adjoint process in the Since X is continuous, any element Q of Me is given by stochastic maximum principle. In Chitashvili (1983) and the density ZtQ which is expressed as an exponential martinPardoux and Peng (1990) the well-posedness results for gale of the form BSDEs with more general generators were obtained [see (104.3) Et .  M C N / also El Karoui et al. (1997) for references and related rewhere N is a local martingale strongly orthogonal to M and sults]. The dynamic programming approach in relation to the notation   M stands for the stochastic integral. mean–variance and exponential hedging was first used in If the local martingale ZO D E.  M / is a true martinLaurent and Pham (1999) and Rouge and El Karoui (2000) gale, dPO =dP D ZO T defines an equivalent probability mearespectively, for diffusion models. The dynamic programsure called the minimal martingale measure for X . ming method was also applied in Mania and Tevzadre (2000) We denote by NEnt .X / the class of local martingales N and Mania et al. (2002) to determine the variance optimal and strongly orthogonal to M such that the process .Et .  p-optimal martingale measures in a semimartingale setting. M C N /; t 2 Œ0; T / is a strictly positive P -martingale with The semimartingale backward equation, as a stochastic verEET .  M C N /lnET .  M C N / < 1. Then sion of the Bellman equation in an optimal control problem,  was first derived in Chitashvili (1983) [see also Chitashvili dQ e MEnt D Q  P W jF and Mania (1987, 1996)]. dP T For all unexplained notations concerning the martingale o theory used below we refer the reader to Dellacherie and D ET .  M C N /; N 2 NEnt .X / : (104.4) Meyer (1980), Jacod (1979), Lipster and Shiryayev (1986). We recall the definition of BMO-martingales and the reverse Hölder condition. The square integrable continuous martingale M belongs to the class BMO iff there is a constant C > 0 such that

104.2 Some Basic Definitions, Conditions, and Auxiliary Facts

E.hM iT  hM i jF /  C 2

(104.5)

Q

For any equivalent martingale measure Q we denote by Z the density process of Q relative to the measure P and let M Q be a P -local martingale such that Z Q D E.M Q / D .Et .M Q /; 0  t  T /; where E.M / is the Doleans-Dade exponential of M . Let n o Q Q MeEnt D Q 2 Me W EZT lnZT < 1 :

for every stopping time . The smallest constant with this property is called the BMO norm of M and is denoted by jjM jjBMO . Let Z be a strictly positive uniformly integrable martingale. Definition 1. The process Z satisfies REnt .P / inequality if there is a constant C1 such that

104 A Semimartingale BSDE Related to the Minimal Entropy Martingale Measure

E

ZT ZT ln jF Z Z

 C1 ;

(104.6)

1557

Let us introduce the following notations Et T .M Q / D

for every stopping time . For the proof of the following assertion the reader is referred to Rheinländer (1999) [see Doleans-Dade and Meyer (1979) or Kazamaki (1994) for the case x p ; p > 1]. Proposition 1. Let E.M / be an exponential martingale associated to the continuous local martingale M . Then if E.M / is a uniformly integrable martingale and satisfies the REnt .P / inequality then M belongs to the class BMO. Let us recall also the concept of relative entropy [see Csiszar (1975) about the basic properties of relative entropy]. The relative entropy, or Kullback–Leibler distance, I.Q; R/ of the probability measure Q with respect to the measure R is defined as

ET .M Q / ; hM Q it T D hM Q iT  hM Q it Et .M Q /

and let Vt D essinf E.Et T .M Q /lnEt T .M Q /jFt / Q2MeE nt

D

  E Q lnEt T .  M C N /jFt

essinf

(104.9)

N 2NE nt .X /

be the value process corresponding to the problem (104.8). Let us introduce also the process Vt D

1 essinf E Q .hM Q it T jFt /: 2 Q2MeE nt

(104.10)

(104.7)

Remark 2. We shall see later, that Vt D V t if there exists an equivalent martingale measure satisfying the REnt inequality.

if Q  R and I.Q; R/ D 1 otherwise. Note that I.Q; R/ 0 and that I.Q; R/ D 0 iff Q D R. The minimal entropy martingale measure Q is a solution of the optimization problem

The optimality principle, which is proved in a standard manner [see, e.g., El Karoui and Quenez (1995), Elliott (1982), Laurent and Pham (1999)], takes in this case the following form:

I.Q; R/ D E R

inf

Q2Mabs

dQ dQ ln ; dR dR

I.Q; P / D I.Q; P /;

where Mabs is the set of measures Q absolutely continuous with respect to P such that X is a local martingale under Q. Proposition 2. If X is locally bounded and there exists Q 2 Mabs such that I.Q; P / < 1 then the minimal entropy martingale measure exists and is unique. Moreover if I.Q; P / < 1 for some Q 2 Me then the minimal entropy martingale measure is equivalent to P . Remark 1. This assertion is proved in Frittelli (2000) assuming that X is bounded, defining the class Me as a set of equivalent measures Q such that X is a martingale (and not a local martingale) under Q. The proof is the same if X is locally bounded and Me is defined as in the Introduction. Since any continuous process is locally bounded, under assumptions A) and B) the minimal entropy martingale measure always exists and is equivalent to the basic measure P . Therefore, hereafter we shall consider only equivalent martingale measures and focus our attention on the construction and properties of optimal martingale measures. Thus, we consider the optimization problem inf

Q2MeE nt

EET .M Q /lnET .M Q /:

(104.8)

Proposition 3. (A) There exists an RCLL semimartingale still denoted by Vt such that for each t 2 Œ0; T Vt D essinf E Q .lnEt T .M Q /jFt /: Q2MeE nt

Vt is the largest RCLL process equal to 0 at time T such that Vt C lnEt .M Q / is a Q-submartingale for every Q 2 MeEnt : (B) The following properties are equivalent: (i) Q is optimal, i.e. , V0 D infQ2MeE nt E Q lnET .M Q / D   E Q lnET .M Q /;  (ii) Q is optimal for all conditional criteria, i.e., for each t 2 Œ0; T 



Vt D E Q .lnEt T .M Q /jFt / a:s: 

(iii) Vt C lnEt .M Q / is a Q -martingale. The following statement, proved in Delbaen et al. (2002), is a consequence of Proposition 3(B). Corollary 1. If there exists an equivalent martingale measure QQ whose density satisfies the REnt .P / inequality, then the density of the minimal entropy martingale measure also satisfies the REnt .P / inequality.

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Proof. It follows immediately, since for any stopping time 





E.E T .M Q /lnE T .M Q /jF / D essinf E.E T .M Q /lnE T .M Q /jF / Q2MeE nt

Q

Q

 E.E T .M Q /lnE T .M Q /=F /  C:u t

104.3 Backward Semimartingale Equation for the Value Process

Y of Equations (104.11) and (104.12) belongs to the class BMO and jjLjjBMO  .2C C 1/2 jjM Q jjBMO ;

We say that the process B strongly dominates the process A and we shall write A  B, if the difference B  A 2 AC loc , i.e., is a locally integrable increasing process. Let .AQ ; Q 2 Q/ be a family of processes of bounded variations, zero at time zero. Denote by essinf.AQ / the largest process of finite Q2Q

variation, zero at time zero, which is strongly dominated by the process AQ for every Q 2 Q, i.e., this is an “ess inf” of the family .AQ ; Q 2 Q/ relative to the partial order . Let us consider the following semimartingale backward equation Yt D Y0  essinf

1 2

Q2MeE nt



hM it C hM ; Lit C Lt ; t < T; Q

Q

(104.11)

where C is an upper bound of the process Y . Proof. Using the Itô formula for YT2  Y2 and the boundary condition YT D 0 we have that Z

(104.12)

Yt D Y0 C Bt C Lt ; B 2 Aloc ; L 2 M2loc ;

(104.13)

such that YT D 0 and Bt D  essinf

Q2MeE nt

2

 hM it C hM ; Lit : Q

Ys d.Bs C Ls /  0

Q

(104.14)

(104.17)



for any stopping time . Since Y satisfies Equation (104.11) 1 Bt C hM Q it C hM Q ; Lit 2 AC loc 2

(104.18)

and, therefore, Equation (104.17) implies that Z

We say that the process Y is a solution of Equations (104.11) and (104.12) if Y is a special semimartingale with respect to the measure P with canonical decomposition

1

T

hLiT  hLi C 2

with the boundary condition YT D 0:

(104.16)

Z

T

hLiT  hLi C 2

Ys dLs  Z



2

T

Ys d hM Q is 

T

Ys d hM Q ; Lis  0:

(104.19)



Without loss of generality we may assume that L is a square integrable martingale, otherwise one can use localization arguments. Therefore, if we take conditional expectations in Equation (104.19), having inequality jYt j  C in mind, we obtain E.hLiT  hLi jF /  CE.hM Q iT  hM Q i jF /

Z T (104.20) jd hM Q ; Lis jjF  0  2CE 

Let Z

t

Lt D 0

0 s dMs

C LQ t ;

Q M i D 0; hL;

(104.15)

Now using the conditional Kunita–Watanabe inequality from Equation (104.20) we have 1=2

E.hLhT hLi jF /2C jjM Q jjBMO E 1=2 .hLiT hLi jF /

be the Galtchouk–Kunita–Watanabe decomposition (G–K– W) of L with respect to the martingale M .

C jjM Q jjBMO  0:

Lemma 1. If there exists Q 2 MeEnt such that M Q 2 BMO, then the martingale part L of any bounded solution

Solving this quadratic inequality with respect to x D E 1=2 .hLiT  hLi jF /, we obtain the estimate

(104.21)

104 A Semimartingale BSDE Related to the Minimal Entropy Martingale Measure

E.hLiT  hLi jF /  .2C C 1/2 jjM Q jjBMO :

and as Vt C lnEt .M Q / is a Q-submartingale for every Q 2 MeEnt , we have that

Since the right-hand side does not depend on , the estimate (Equation (104.16)) also holds and L belongs to the space BMO. t u The value process of the problem (104.8) defined by Equation (104.9) is a special semimartingale with respect to the measure P with the canonical decomposition 2 ; A 2 Aloc : Vt D V0 C mt C At ; m 2 Mloc

(104.22)

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1 At C hM Q it C hm; M Q it 2 AC loc 2

(104.26)

for every Q 2 MeEnt . On the other hand, according to Proposition 2 the optimal martingale measure Q exists and is equivalent to P . There fore, by the optimality principle the process Vt C lnEt .M Q / will be a Q -martingale and using again Girsanov’s theorem we obtain that

Let Z

t

mt D 0

's0 dMs C m Q t ; hm; Q Mi D 0

(104.23)

Theorem 1. Let conditions (A) and (B) be satisfied. Then: (a) the value process V is a solution of the semimartingale backward equations (104.11) and (104.12). Moreover, a martingale measure Q is the minimal entropy martingale measure if and only if it is given by the density dQ D  ET .M Q /dP , where Q

(104.27)

Relations (104.26) and (104.27) imply that

be the G–K–W decomposition of m with respect to M . Now we formulate the main statement of the paper.

Mt

1 At C hM Q it C hm; M Qit D 0: 2

Z

t

D 0

0s dMs  m Q t:

 At D  essinf

Q2MeE nt

(104.28)

hence, the value process V satisfies Equation (104.11) and it is evident that VT D 0. Equality (Equation (104.27)) implies that the processes At and, hence, Vt are continuous. Let us show now that the optimal martingale measure Q is given by Equation (104.24). From Equation (104.28) we have

(104.24)

1 At D  h  M it C h  M; mit 2

1 hN it C hN; mit  essinf 2 N 2NE nt .X /

(b) If, in addition, the minimal martingale measure exists and satisfies the reverse Hölder REnt -inequality, then the value process V is the unique bounded solution of Equations (104.11) and (104.12). Proof. (a) By Condition (B) there exists QQ 2 MeEnt and according to Proposition 3 the process Zt D Vt ClnEt .M QQ / is a Q Q-submartingale, hence a P -semimartingale by Girsanov’s theorem. Since ET .M QQ / is strictly positive and continuous the process lnEt .M QQ / is a semimartingale and, consequently, the value process V will be also a semimartingale under P . Condition (A) implies that any adapted RCLL process is predictable [see Revuz and Yor (1999)], hence any semimartingale is special. So, V is a P -special semimartingale admitting decomposition (Equation (104.22)). The processes m  hm; M Q i and M Q  hM Q i are Q-local martingales by Girsanov’s theorem. Therefore, since

 1 hM Q it C hM Q ; mit ; 2

1 1 D  h  M it C h  M; mit C hmi Q t 2 2 

1 essinf .hN C mi Q t/ 2 N 2NE nt .X /

1 1 Q t; D  h  M it C h  M; mit C hmi 2 2 (104.29) since Q t / D 0: essinf .hN C mi

(104.30)

N 2NE nt .X /

To prove relation (104.30) let us define a sequence of stopping times

Zt D Vt C lnEt .M Q / 1 Q D V0 C mt C At C Mt  hM Q it 2 Q

(104.25)

D V0 C .mt  hm; M Q it / C .Mt  hM Q it / 1 C At C hM Q it C hm; M Q it 2

n D inf ft W Et .NQ /

1 or n

Et .  M  m/ Q ng ^ T;

where NQ is a local martingale from the class NEnt .X /, which exists by condition B). It is not difficult to see that the local

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martingale N n D m Q n C NQ  NQ n belongs to the class NEnt .X / and n " T . Therefore Q t /  hN n C mi Q t essinf .hN C mi N 2NE nt .X /

D hm Q m Q n C NQ  NQ n it  2.hmi Q t  hmi Q t ^n C hNQ it  hNQ it ^n / for each n 1 and Equation (104.30) holds, since the righthand side of the latter inequality tends to zero as n ! 1. Here, as previously, m Q is the orthogonal martingale part of m in the G–K–W decomposition (Equation (104.23)) and Q n ^t ; t 2 Œ0; T / is a stopped martingale. m Q n D .m  By the optimality principle Vt C lnEt .M Q / is a Q martingale. Since V solves Equation (104.11), this implies that   1 1   hM Q it C hM Q ; mit D hM Q it ChM Q ; mit : essinf 2 2 Q2MeE nt (104.31)

C

1 hM Q it C hL; M Q it 2  ! 1 hM Q it C hL; M Q it :  essinf 2 Q2MeE nt

Therefore, the Girsanov theorem implies that Yt C lnEt .M Q / is a Q-local submartingale for every Q 2 MeEnt . Thus, the process Yt Et .M Q / C Et .M Q /lnEt .M Q / is a local P -submartingale. Since .Et .M Q /; t 2 Œ0; T / is a martingale with EET .M Q /lnET .M Q / < 1, the process Et .M Q /lnEt .M Q / will be from the class D, as a submartingale bounded from below (by the constant 1=e). On the other hand, the process Yt Et .M Q / is also from the class D, since Y is bounded and Et .M Q / is a martingale [see, e.g., Dellacherie and Meyer (1980)]. Thus, Yt Et .M Q / C Et .M Q /lnEt .M Q / is a submartingale from the class D, hence from the boundary condition we have that Yt Et .MQ /CEt .MQ /lnEt .M Q /  E.ET .M Q /lnET .M Q /jFt /

Q

Since M is represented in the form   M C N  for  some N 2 NEnt .X /, it follows from Equations (104.29) Q and, hence, the and (104.31) that the processes N  and m  Q are indistinguishable. So, the processes M Q and M  m minimal entropy martingale measure is unique and admits representation (Equation (104.24)). (b) It is easy to see that the value process satisfies the twosided inequality for all t 2 Œ0; T 0  Vt  C a:s::

(104.32)

The positivity of V follows from the Jensen inequality. On the other hand, if there exists a martingale measure QQ satisfying the reverse Hölder REnt inequality, we have that V is bounded above, since

Q2MeE nt

Q

 E.Et T .M Q /lnEt T .M Q /jFt /  C: Thus, V is a bounded solution of Equations (104.11) and (104.12). Uniqueness. Let Y be a bounded solution of Equations (104.11) and (104.12). Let us show that the processes Y and V are indistinguishable. Since Y solves Equation (104.11) we have 1 Q Yt C lnEt .M Q / D Y0 C Lt C Bt C Mt  hM Q it 2 Q

D Y0 C .Lt  hL; M Q it / C .Mt  hM Q it /

EŒEt T .M Q /lnEt T .M Q /jFt D Vt : Yt  essinf e Q2ME nt

(104.34) Let us show the inverse inequality. Similarly to Equation (104.29) we have 1 1 Q Bt D  h  M it C h  M; Lit C hLi t 2 2

(104.35)

and the infimum is attained for the martingale Nt D LQ t ;

(104.36)

where LQ is the orthogonal martingale part of L in the G–K–W decomposition (Equation (104.15)). 0 Q Since the minimal martingale Let M Q D   M  L. measure satisfies the REnt .P / condition, Proposition 1 implies that   M 2 BMO. On the other hand for any s  t

Vt D essinf E.Et T .M Q /lnEt T .M Q /jFt / Q

for all Q 2 MeEnt and

(104.33)

Q s  hLit  hLis ; Q t  hLi hLi 0

hence Lemma 1 implies that M Q 2 BMO. Therefore, from 0 Kazamaki (1994) it follows that the process .Et .M Q /; t 2 0 Œ0; T / is a martingale, hence dQ0 D ET .M Q /dP defines an absolutely continuous martingale measure. 0 It is easy to see that Yt C lnEt .M Q / is a local martingale under Q0 . Indeed, Equations (104.35) and (104.15) imply that

104 A Semimartingale BSDE Related to the Minimal Entropy Martingale Measure

1 1 Q 0 Yt ClnEt .M Q / D Y0 CLt  hM it ChM; Lit C hLi t 2 2

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be the decomposition of the value process. Then the triple Q is a solution of the martingale equation .V0 ; '; m/ Z

1 1 Q  .  M /t  LQ t  h  M it  hLi t D Y0 C ..  /  X /t 2 2 (104.37)

cC

which is a Q0 -local martingale, by Girsanov’s theorem. 0 0 0 Therefore, Zt D Yt Et .M Q / C Et .M Q /lnEt .M Q / is a P -local martingale. Let us show that Q0 2 MeEnt and that the process Z is a martingale. It is easy to see that

Q such that c 2 RC and Conversely, if a triple .c; ; L/, M; LQ 2 BMO, solves Equation (104.41), then the process Y defined by

Zt C Et .M

Q0

0

0

0

Yt Et .M Q / C Et .M Q /lnEt .M Q / E.YT ET .M Q / 0

0

C ET .M Q /lnET .M Q /jFt /: Therefore, from Equations (104.12) and (104.34) we obtain that E.Et T .M

Q0

/lnEt T .M

Q0

/jFt /  Yt  Vt  C: 0

(104.38) 0

The latter inequality implies that EET .M Q /lnET .M Q / < 1 and that Q0 is optimal, hence by Proposition 2, Q0 is equivalent to P and Q0 2 MeEnt . Using the same arguments as before, we have that Z is a local martingale of class D, therefore it is a martingale [see, e.g., Dellacherie and Meyer (1980)]. Now, the martingale property and the boundary condition imply that 0

0

Yt D E.Et T .M Q /lnEt T .M Q /jFt /:

(104.39)

Since Q0 2 MeEnt , the equality Yt D Vt a.s. for all t 2 Œ0; T results from Equations (104.34) and (104.39) , hence V is the unique bounded solution of Equations (104.11) and (104.12). u t Now we formulate Theorem 1(b) in the following equivalent martingale form as a Proposition 4. Let the conditions of Theorem 1(b) be satisfied and let Z

t

Vt D V0 C At C 0

1 1 Q hM iT hM; M iT  hLi T 2 2 0 (104.41) Q 2 BMO. and c 2 RC ,'  M; m

's0 dMs C m Q t ; hM; mi Q D0

(104.40)

0 Q s dMs C LT

Yt D E

1 / : e

Thus, Z is a local martingale majorizing a uniformly integrable martingale, hence it is a supermartingale and we have that 0

T

D

1 h  M it T  h  M; 2

 M it T

1 Q  hLit T jFt ; 2 (104.42)

is a bounded solution of Equations (104.11) and (104.12), and coincides with the value process. Proof. Relation (104.29) implies that Equations (104.11) and (104.12) is equivalent to the backward semimartingale equation Z t 1 1 Q Yt D Y0  hM it ChM; M it C hLi tC 2 2 0 YT D 0:

0 Q s dMs CLt ;

(104.43) (104.44)

Since V solves Equation (104.43), using the boundary condition (104.12) we obtain from Equation (104.43) that the triple .V0 ; '; m/ Q satisfies Equation (104.41). Besides, it follows from Lemma 1 that '  M; m Q 2 BMO. Q Conversely, let the triple .c; ; L/ solves Equation (104.41) and Y be the process defined by Equation (104.42). Using the martingale properties of the Q we see that the martingale BMO-martingales  M and L, part of Y [defined by Equation (104.42)] coincides with Rt V0 C 0 s0 dMs C LQ t , hence Y satisfies Equations (104.11) and (104.12). As  M; LQ 2 BMO, the conditional Kunita– Watanabe inequality and Equation (104.42) imply that Y is bounded, therefore Y coincides with the value process by Theorem 1(b). t u It is well known (Frittelli 2000; Rheinländer 1999) that martingale measure if and only if Q is the minimal entropy RT Q cC 0 h0s dXs (i) ET .M / D e for some constant c and an X -integrableR h, RT  T (ii) E Q 0 h0s dXs D 0 and E Q 0 h0s dXs 0 for any Q 2 MeEnt . The sufficiency part of this assertion is hard to verify, since condition (ii) involves the optimal martingale measure. The following corollary of Theorem 1 shows that the integrand h of the minimal entropy martingale measure can be

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expressed in terms of the value process V and since V solves Equations (104.11) and (104.12) condition (ii) is automatically satisfied. Corollary 2. A martingale measure Q is the minimal entropy martingale measure if and only if the corresponding density admits representation 

ET .M Q / D e V0 C

RT 0

.'s s /0 dXs

;

(104.45)

where ' is the integrand in the G–K–W decomposition of the martingale part m of the value process. Proof. It follows from Theorem 1 and relation (104.29) that V satisfies equation 1 1 Q t C.' M /t C m Q t: Vt D V0  hM it ChM; ' M it C hmi 2 2 Taking the exponentials of the both sides of the latter equation, using the definitions of the process X and of the Doleans-Dade exponential we obtain that Q V0 C e Vt D Et1 .  M  m/e

Rt

0 .'s s /

0 dX

s

(104.46)

Z E

T

Q

.'s  s /0 dXs D E Q lnET .  M  m/ Q  V0

0     ElnET .M Q / ET .M Q /  ET .M Q / 0 (104.49) t for any Q 2 MeEnt , hence (ii) is satisfied. u

Corollary 3. If there exists a martingale measure QQ whose density satisfies the reverse Hölder inequality REnt .P /, then Vt D V t

(104.50)

Proof. We denote by REnt .X / the set of martingale measures Q whose densities Z Q satisfy the REnt .P / inequality. By Corollary 1 the minimal entropy martingale measure Q is in REnt .X /. Therefore Vt D essinf E Q .lnEt T .M Q /jFt / Q2MeE nt

D essinf E Q .lnEt T .M Q /jFt / Q2RE nt .X /



1 Q D essinf E Q Mt T  hM Q it T C hM Q it T jFt 2 Q2RE nt .X / D

1 essinf E Q .hM Q it T jFt / 2 Q2RE nt .X /

and from the boundary condition (104.12) we have ET .  M  m/ Q De

V0 C

RT 0

.'s s /0 dXs

:

(104.47)

Now, since by Theorem 1 Q is the minimal entropy martingale measure if and only if it satisfies Equation (104.24), the representation (Equation (104.45)) follows from Equations (104.24) and (104.47). Note that for the process '   condition (ii) is satisfied. Indeed, Equation (104.46) implies that Z

t

Q D V0 C Vt ClnEt .M  m/

.'s s /0 dXs

(104.48)

0

Rt and by the optimality principle . 0 .'s  s /0 dXs ; t 2 Œ0; T / R  T is a Q -martingale, hence E Q 0 .'s  s /0 dXs D 0. e For any Q 2 MEnt and x 2 Œ0; 1 let us define Qx D  xQ C.1 x/Q . Then ZTx D xET .M Q /C.1 x/ET .M Q / is the corresponding density and according to Lemma 2.1 of Frittelli (2000) the function f .x/ D EZTx lnZTx is differentiable at x and d     EZTx lnZTx jxD0 D ElnET .M Q / ET .M Q /  ET .M Q / : dx 0. Therefore, from Moreover, Q is optimal iff Equation (104.48) and the latter inequality we obtain 

d f jxD0 dx

since Q 2 REnt .X / implies that M Q 2 BMO (Proposition 1) and according to Proposition 7 of Doleans-Dade and Meyer (1979) from M Q 2 BMO.P / we have that the process M Q  hM Q i is a BMO martingale with respect to the Q measure Q and, hence, E Q .Mt T  hM Q it T jFt / D 0: We recall that Mt T D MT  Mt and hM it T D hM iT  t hM it . u This expression of the value process enables us to determine easily the minimal entropy martingale measure in some particular cases. Proposition 5. Assume that the minimal martingale measure Qmi n belongs to the class MeEnt and   X is a martingale with respect to any Q 2 MeEnt . Then the following assertions are equivalent (1) the minimal entropy martingale measure Q coincides with the minimal martingale measure Qmi n , (2) the mean variance tradeoff admits representation Z

T

h  M iT D c C 0

0 s dXs

(104.51)

for some process such that R T constant c and X -integrable RT E mi n 0 s0 dXs D 0 and E Q 0 s0 dXs 0 for any Q 2 MeEnt .

104 A Semimartingale BSDE Related to the Minimal Entropy Martingale Measure

Proof. (1)) (2). Let Q D Qmi n. Then by Corollary 2 ET .  M / D e V0 C

RT 0

.'s s /0 dXs

;

(104.52)

where ' is defined by Equation (104.40). It follows from Equation (104.52) that 

Z

exp  

T

0

 1  h  M iT 2  Z T 0 0 ; 's dXs  s dMs  h  M iT

0s dMs Z

T

D exp V0 C 0

0

which implies 1 h  M iT D V0 C 2

Z

's0 dXs ;

(104.53)

and, hence, Equation (104.51) is satisfied with D 2' and c D 2V0 . RT Since E Q 0 0s dXs D 0 for any Q 2 MeEnt , it RT follows from Equation (104.49) that E Q 0 's0 dXs RT E Q 0 0s dXs D 0 for any Q 2 MeEnt . Besides Equation (104.46) implies that Z

Remark 3. Condition (104.51) is satisfied in the case of “almost complete” diffusion models [see, e.g., Pham et al. (1998)], where the market price of risk is measurable with respect to the filtration generated by the asset price process. Corollary 5. The mean variance tradeoff h  M iT is deterministic if and only if the minimal entropy martingale measure coincides with the minimal martingale measure and ' D 0 hM i -a.e., where ' is defined by Equation (104.23) and hM i is the Dolean measure of hM i:

Proposition 6. Assume that the minimal martingale measure exists and satisfies the reverse Hölder REnt -inequality. Then the density of the minimal entropy martingale measure is of the form  Q

ZT

D

Z

0

.'s  s / dXs 0

(104.54)

0

Rt and by the optimality principle . 0 .'s  s /0 dXs ; t 2 RT Œ0; T / is a Qmi n -martingale, hence E mi n 0 's0 dXs D R T E Q 0 0s dXs D 0. (2)) (1). If Equation (104.51) is satisfied, then  1 0s dXs C h  M iT 2 0

0   Z T 1 c C D exp   dX s s s 2 2 0 T

RT and it is evident that E Q 0 . 12 s  s /0 dXs 0 for any R T Q 2 MeEnt and E mi n 0 . 12 s  s /0 dXs D 0, hence Q D mi n by the sufficiency condition of Theorem 2.3 in FritQ telli (2000), corresponding to the previously given condition (ii). u t Corollary 4. Assume that the mean variance tradeoff h  M iT is bounded. Then Q D Qmi n if and only if Equation (104.51) is satisfied for Rsome constant c and X t integrable process such that . 0 s0 dXs ; t 2 Œ0; T / is mi n Q -martingale. The proof follows from Proposition 5, since the boundedness of hM iT implies that X is a martingale with respect to any Q 2 MeEnt . Besides, if equality (Equation (104.51))



T

exp  0  Z E exp 

t

Vt C lnEt .  M / D V0 C

 Z ET .  M / D exp 

Rt is satisfied and if . 0 s0 dXs ; t 2 Œ0; T / is a martingale with respect to some Q 2 MeEnt then this process is bounded and will be a martingale with respect to any Q 2 MeEnt .

The proof immediately follows from Proposition 5.

T 0

1563

0s dXs 

(104.55)

DcCm OT

(104.56)

T

0s dXs

if and only if 

1 exp  h  M iT 2



for some constant c and a martingale m O strongly orthogonal to M . Proof. Let ZT .Q / is of the form, Equation (104.55). By Equation (104.24) we have that   Z T 0s dXs exp  0 ET .  M  m/ Q D  Z T  E exp  0s dXs 0

which implies that     1 1 Q T Q T  hmi exp  h  M iT D c exp m 2 2 Z T D cET .m/ Q DcCc Es .m/d Q m Q s; 0

where the martingale m, Q orthogonal to M , is defined by Equation (104.23) and belongs to the class BMO according to Lemma R t1. Therefore, Equation (104.56) is satisfied with m O t D c 0 Es .m/d Q m Q s , which is a martingale according to Kazamaki (1994).

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Conversely, let Equation (104.56) be satisfied. Then using the Itô formula for ln.c C m O t / from Equation (104.56) we have Z

T

lnc C 0

1 1 dm O s D  h  M iT cCm Os 2 Z :  1 1 C dm Os 2 0 cCm Os T

lnc;

Z

 0

(2) By Corollary 2, Q is the minimal entropy martingale measure if and only if the corresponding density admits representation

1 dm Os c Cm Os



is a solution of the martingale equation (104.41). The martingale cC1 mO  m O belongs to the class BMO, since by Equation (104.56) c C m O t  1 and Proposition 1 with the Jensen inequality imply that 1

cCm O t D E.e  2 hM iT =Ft / E.e

 12 hM it T

=Ft / e

 Q where M Q D   M  L:



dQ D ET .M Q /dP;

which implies that the triple D 0; LQ D 

LQ is a real-valued local martingale strongly orthogonal to M . We give two equivalent characterizations of the minimal entropy martingale measure in terms of the solution of Equation (104.57): (1) By Theorem 1, Q is the minimal entropy martingale measure if and only if

ET .M Q / D e Y0 C

RT

e

1 2C

s s /

.

RT 0

s0 dXs

:

Since solution of Equation (104.41) is unique in the class RC BMO BMO, we obtain that ' D 0. Therefore, it follows from Corollary 2 that ZT .Q / is of the form, Equation (104.55). t u

In an incomplete market model where the dynamics of the discounted assets price process is described by an Rd -valued continuous semimartingale, we characterize the minimal entropy martingale measure in terms of the value function of a suitable problem of an optimal equivalent change of measure. This value process has been expressed (under certain conditions) as the unique bounded solution of the BSDE 1 1 Q Yt D Y0  h  M it C h  M;  M it C hLi tC 2 2 Z t 0 Q C YT D 0; (104.57) s dMs C Lt ; 0

which is equivalent to Equations (104.11) and (104.12) according to Equation (104.29). We say that the solution of this equation is a special semimartingale Y , keeping in mind that  M C LQ is the martingale part of Y . One can say Q also that a solution of Equation (104.57) is a triple .Y; ; L/, d where is R -valued predictable M -integrable process and

s

:

(104.58)

over all 2 ˘; (104.59)

where ˘ is the class of all Rd -valued predictable M -integrable processes such that  X is a martingale with respect to all Q 2 MeEnt . The optimal strategy of problem (104.59) can be expressed in terms of the BSDE (Equation (104.57)) as  D

104.4 Conclusions

0 dX

We point out that, by duality, one can also find the optimal portfolio related to the exponential utility maximization problem to maximize EŒe ˛

 12 E.hM it T =Ft /

0

1 .  ˛

/:

This can be shown using Equation (104.58) in a standard way, since i h h RT 0 RT 0 RT  E e ˛ 0 s dXs D E Q e ˛ 0 s dXs Y0  0 .

s s /

0 dX

i s

 e Y0 by Jensen’s inequality and this value is attained for  D 1 ˛ .  /. We also consider two specific extreme cases for which the BSDE admits an explicit solution and in such cases we explicitly construct the minimal entropy martingale measure. In particular, we give necessary and sufficient conditions for the minimal entropy martingale measure to coincide with the minimal martingale measure as well as to coincide with the martingale measure (expressed by Equation (104.55)) appearing in the second above mentioned extreme case. Acknowledgment The authors are grateful to an anonymous referee for valuable remarks and suggestions. This research is supported by INTAS Grant 97-30204.

104 A Semimartingale BSDE Related to the Minimal Entropy Martingale Measure

References Biagini, F., P. Guasoni and M. Pratelli. 2000. “Mean variance hedging for stochastic volatility models.” Mathematical Finance 10, 109–129. Bismut, J. M. 1973. “Conjugate convex functions in optimal stochastic control.” Journal of Mathematical Analysis and Applications 44, 384–404. Chitashvili, R. 1983. “Martingale ideology in the theory of controlled stochastic processes,” in Lecture Notes in Mathematics, Vol. 1021, Springer, Berlin, pp. 73–92. Chitashvili, R. and M. Mania. 1987. “Optimal locally absolutely continuous change of measure: finite set of decisions.” Stochastics Stochastics Reports 21, 131–185 (part 1), 187–229 (part 2). Chitashvili, R. and M. Mania. 1996. “Generalized Ito’s formula and derivation of Bellman’s equation,” in Stochastic processes and related topics, Stochastics Monograph Vol. 10, H. J. Engelbert et al. (Eds.). Gordon & Breach, London, pp. 1–21. Csiszar, I. 1975. “I divergence geometry of probability distributions and minimization problems.” Annals of Probability 3, 146–158. Delbaen, F. and W. Schachermayer. 1996. “The variance-optimal martingale measure for continuous processes.” Bernoulli 2, 81–105. Delbaen, F., P. Monat, W. Schachermayer, M. Schweizer, and Ch. Stricker. 1997. “Weighted norm inequalities and hedging in incomplete markets.” Finance and Stochastics 1, 181–227. Delbaen, F., P. Grandits, Th. Rheinländer, D. Samperi, M. Schweizer, and Ch. Stricker. 2002. “Exponential hedging and entropic penalties.” Mathematical Finance 12, 99–123. Dellacherie, C. and P. A. Meyer. 1980. Probabilités et potentiel. Chapitres V a VIII. Théorie des martingales, Hermann, Paris. Doleans-Dade, K. and P. A. Meyer. 1979. “Inégalités de normes avec poinds,” in Séminaire de Probabilités XIII, Lecture Notes in Mathematics, Vol. 721, Springer, Berlin, pp. 204–215. El Karoui, N. and M. C. Quenez. 1995. “Dynamic programming and pricing of contingent claims in an incomplete market.” SIAM Journal on Control and Optimization 33, 29–66. El Karoui, N., S. Peng, and M. C. Quenez. 1997. “Backward stochastic differential equations in finance.” Mathematical Finance 75, 1–71. Elliott, R. J. 1982. Stochastic calculus and applications, Springer, New York. Föllmer, H. and M. Schweizer. 1991. “Hedging of contingent claims under incomplete information,” in Applied stochastic analysis, Stochastics monographs Vol. 5, M. H. A. Davis and R. J. Elliott (Eds.). Gordon & Breach, London, pp. 389–414.

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Frittelli, M. 2000. “The minimal entropy martingale measure and the evaluation problem in incomplete markets.” Mathematical Finance 10, 39–52. Gourieroux, C., J. P. Laurent, and H. Pham. 1998. “Mean–variance hedging and numeraire.” Mathematical Finance 8, 179–200. Hoffman, N., E. Platen, and M. Schweizer. 1992. “Option pricing under incompleteness and stochastic volatility.” Mathematical Finance 2, 153–187. Jacod, J. 1979. Calcule Stochastique et problèmes des martingales, Lecture Notes in Mathematics, Vol. 714, Springer, Berlin. Kazamaki, N. 1994. Continuous exponential martingales and BMO, Lecture Notes in Mathematics, Vol. 1579, Springer, Berlin. Laurent, J. P. and H. Pham. 1999. “Dynamic programming and mean– variance hedging.” Finance and Stochastics 3, 83–110. Liptser, R. Sh. and A. N. Shiryayev. 1986. Martingale theory, Nauka, Moscow. Mania, M. and R. Tevzadze. 2000. “A semimartingale Bellman equation and the variance-optimal martingale measure.” Georgian Mathematical Journal 7, 765–792. Mania, M., M. Santacroce, and R. Tevzadze. 2002. “A semimartingale backward equation related to the p-optimal martingale measure and the lower price of a contingent claim,” in Stochastic processes and related topics, Stochastics Monograph Vol. 12, Taylor & Francis, London, pp. 189–212. Miyahara, Y. 1996. “Canonical martingale measures of incomplete assets markets,” in Probability theory and mathematical statistics: proceedings of the seventh Japan–Russia symposium, World Scientific, Tokyo, pp. 343–352. Pardoux, E. and S. G. Peng. 1990. “Adapted solution of a backward stochastic differential equation.” Systems & Control Letters 14, 55–61. Pham, H., Th. Rheinländer, and M. Schweizer. 1998. “Mean–variance hedging for continuous processes: new proofs and examples.” Finance and Stochastics 2, 173–198. Revuz, D. and M. Yor. 1999. Continuous Martingales and Brownian motion, Springer, Berlin. Rheinländer, Th. 1999. Optimal martingale measures and their applications in mathematical finance, Phd Thesis, Berlin. Rouge, R. and N. El Karoui. 2000. “Pricing via utility maximization and entropy.” Mathematical Finance 10, 259–276. Schweizer, M. 1994. “Approximating random variables by stochastic integrals.” Annals of Probability 22, 1536–1575. Schweizer, M. 1996. “Approximation pricing and the variance optimal martingale measure.” Annals of Probability 24, 206–236.

Chapter 105

The Density Process of the Minimal Entropy Martingale Measure in a Stochastic Volatility Model with Jumps (Reprint)* Fred Espen Benth and Thilo Meyer-Brandis

Abstract We derive the density process of the minimal entropy martingale measure in the stochastic volatility model proposed by Barndorff-Nielsen and Shephard (Journal of the Royal Statistical Society, Series B 63:167–241, 2001). The density is represented by the logarithm of the value function for an investor with exponential utility and no claim issued, and a Feynman–Kac representation of this function is provided. The dynamics of the processes determining the price and volatility are explicitly given under the minimal entropy martingale measure, and we derive a Black and Scholes equation with integral term for the price dynamics of derivatives. It turns out that the price is the solution of a coupled system of two integro-partial differential equations. Keywords Stochastic volatility r Lévy processes r Subordinators r Minimal entropy martingale measure r Density process r Incomplete market r Indifference pricing of derivatives r Integro-partial differential equations

105.1 Introduction In this paper we derive the density process of the minimal entropy martingale measure in a Black and Scholes market with a stochastic volatility model given by Barndorff-Nielsen and Shephard (2001). We apply our results to find the minimal entropy price of derivatives in this market, and present a system of integro-partial differential equations (integro-PDEs) that determines the price. The knowledge of the density process also enables us to describe the price dynamics of the market under the minimal entropy measure. Barndorff-Nielsen and Shephard (2001) propose a geometric Brownian motion where the squared volatility is modeled by a non-Gaussian Ornstein–Uhlenbeck process as the price dynamics for a financial asset. In their model, the volatility level will revert toward zero, with random upward

F.E. Benth () and T. Meyer-Brandis University of Oslo, Oslo, Norway e-mail: [email protected]; [email protected]

shifts given by the jumps of a subordinator process (being an increasing Lévy process). Due to the stochastic volatility, this asset price model leads to an incomplete market, and the arbitrage theory does not provide a unique price for derivatives written on the asset due to the existence of a continuum of equivalent martingale measures. Thus, the risk preferences of the market participants need to be included in the price formation of the derivatives. Utility indifference pricing (see Hodges and Neuberger Hodges and Neuberger (1989)) gives an alternative to the arbitrage theory to derive the fair premium of derivatives in incomplete markets. One considers an investor trying to maximize his exponential utility by either entering into the market by his own account, or issuing a derivative and investing his incremental wealth after collecting the premium. The indifference price of the claim is then defined as the premium for which the investor becomes indifferent between the two investment alternatives. It is well-known (see e.g., Fritelli (2000), Rouge and El Karoui (2000), and Delbaen et al. (2002)) that the zero risk aversion limit of the indifference price corresponds to the minimal entropy martingale measure price. The zero risk aversion limit is of particular interest, since this is the only price for which the buyer and seller agree on the indifference price. We state the density of the minimal entropy martingale measure by appealing to general results by Grandits and Rheinländer (2002) and verification results by Rheinländer (2005). The density process is introduced via a function H which is related to the solution of the portfolio optimization problem of the investor having an exponential utility function and not issuing any claim. In fact, it arises from the logarithmic transform of the value function in a similar fashion as demonstrated in Musiela and Zariphopoulou (2003). This function is represented as an expectation of the exponential of a ratio between the drift and squared volatility, and shown to be the Feynman–Kac solution of an integro-PDE.

*

This paper is reprinted from Finance and Stochastics, 9 (2005), No. 4, pp. 563–575.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_105, 

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It provides us with the scaling of the jumps when considering the minimal entropy dynamics of the stochastic volatility model. We apply our results to find the minimal entropy price of a class of claims with payoff given by a function of the underlying at maturity of the contract. The price is written as an expected value of the payoff, where we have complete knowledge of the dynamics of the asset and volatility processes. Furthermore, we state the integro-PDE for the pricing equation, which will become a Black and Scholes partial differential equation with an additional integral term arising from the stochastic volatility. This integral term will also include the function H , and thus to solve it we need to consider a coupled system of two integro-PDEs. Related papers studying the minimal entropy martingale measure for stochastic volatility markets are Hobson (2004), Becherer (2003, 2004), and Benth and Karlsen (2005). There is a huge interest in applying entropy measures in pricing of derivatives in incomplete markets. We list here some related works. Fujiwara and Miyahara (2003) investigate the minimal entropy martingale measure for geometric Lévy processes, while Kim and Lee (2007) study it for the specific CGMY model. A general approach to minimal entropy martingale measures in incomplete markets based on a semimartingale backward stochastic differential equation is found in Mania et al. (2003). Monoyios (2007) studies the minimal entropy measure and the Esscher transform in an incomplete market generated by two Brownian motions. The relations between the minimal martingale measure and the minimal entropy martingale measure in incomplete markets are studied by Arai (2001). Finally, we mention Branger (2004) where cross-entropy is analyzed in connection with derivatives pricing. Entropy-based methods have been applied in numerous papers for inferring implied probability densities from market price data. We only mention a few papers here, including Avellaneda (1998), Buchen and Kelly (1996), Guo (2001), Matsuba and Takahashi (2003), Mogedey and Ebeling (2000), and Palatella (2005). Other interesting financial applications of entropy based ideas can be found in Tang et al. (2006) where decision making problems and financial management are studied. Further, McCauley (2003) provides a comparison of ideas in thermodynamics and finance where entropy plays a role. The paper is organized as follows: In Sect. 105.2 we define our financial market, and in Sect. 105.3 we study the density of the minimal entropy martingale measure. Section 105.4 is devoted to the density process and the analysis of the function H , while in Sect. 105.5 we apply our results to the minimal entropy pricing of claims.

F.E. Benth and T. Meyer-Brandis

105.2 The Market Given a probability space .; F ; P / and a time horizon T , consider a financial market consisting of a bond and a risky asset with prices at time 0  t  T denoted by Rt and St , respectively. Assume without loss of generality that the bond yields a risk-free rate of return equal to zero, i.e., dRt D 0;

(105.1)

together with the convention that R0 D 1. In this paper we will consider the stochastic volatility model introduced by Barndorff-Nielsen and Shephard (2001), but let us mention that our results can be achieved for more general stochastic volatility models (under appropriate integrability conditions) as long as the volatility driving process Lt is independent from Bt (see Equations (105.2) and (105.3) below). In the Barndorff-Nielsen and Shephard model the price of the risky asset is evolving according to the following dynamics dSt D ˛.Yt /St dt C .Yt /St dBt ; d Yt D Yt dt C dLt ;

S0 D s > 0 (105.2)

Y0 D y > 0;

(105.3)

where Bt is a Brownian motion and Lt a pure jump subordinator (that is, an increasing pure jump Lévy process with no drift) with Poisson random measure denoted by N.dt; d z/. In this paper we assume Bt and Lt to be independent. The Lévy R1 measure .d z/ of Lt satisfies 0 min.1; z/ .d z/ < 1. Further, we denote by fFt gt 0 the completion of the filtration .Bs ; Ls I s  t/ generated by the Brownian motion and the subordinator such that .; F ; Ft ; P / becomes a complete filtered probability space. In this paper we will assume the following specification of the parameter functions ˛ and : ˛.y/ D  C ˇy;

.y/ D

p

y;

(105.4)

with  and ˇ being constants. The process Yt models the squared volatility, and will be an Ornstein–Uhlenbeck process reverting toward zero, and having positive jumps given by the subordinator. An explicit representation of the squared volatility is Z

t

Yt D y exp .t/ C

exp ..t  u// dLu :

(105.5)

0

The scaling of time by  in the subordinator is to decouple the modeling of the marginal distribution of the (log)returns of S and their autocorrelation structure. We note that in Barndorff-Nielsen and Shephard (2001) it is proposed to use a superposition of processes Yt with different speeds of

105 The Density Process of the Minimal Entropy Martingale Measure in a Stochastic Volatility Model

mean-reversion. However, in this paper we will stick to only one process Yt , but remark that there are no essential difficulties in generalizing to the case of a superposition of Y ’s. The modeling idea is to specify a stationary distribution of Y that implies (at least approximately) a desirable distribution for the returns of S . Given this stationary distribution, one needs to derive a subordinator L. In Barndorff-Nielsen and Shephard (2001), several examples of such distributions and their associated subordinators are given in the context of financial applications. We denote by ./ the cumulant function of Lt , which is defined as the logarithm of the characteristic function ./ D ln E Œexp .iL1 / ;

 2 R:

1569

länder (2002) the density of QME for general stochastic volatility models driven by an independent noise process is shown to be of the form Z ZT D c  exp  Z D c  exp 

T 0 T 0

˛.Yt / 1 dS S t  2 .Yt / t

Z T 2 ˛.Yt / ˛ .Yt / dBt  dt : 2 .Yt / 0  .Yt / (105.12)

Here, c is the normalizing constant given by  Z c 1 D E exp 

(105.6)

T

0

From the Lévy–Kintchine Formula we have Z

1

./ D

˚ i z  e  1 .d z/:

(105.7)

0

We suppose that the Lévy measure satisfies an exponential integrability condition, that is, there exists a constant k > 0 such that Z 1 ekz .d z/ < 1: (105.8)

E Œexp .L1 / D exp . .//

(105.9)

with

./ D

.i/:

(105.10)

Note that Lt is also a subordinator, and the cumulant function of this is  ./. The process Lt has the decomposition Z tZ

Z tZ

1

Lt D 0

0

 Z E exp

T 0

˛ 2 .Ys / ds  2 .Ys /

 < 1:

(105.13)

Then ZT as defined in (105.12) is the density of the minimal entropy martingale measure QME . Proof. Referring to the results in Rheinländer (2005), it is enough to verify the following three statements: (i) (ii) (iii)

The expectation EŒZT is equal to one. The measure induced by ZT , denoted by QME , has finite entropy. We have Z

z.N.d z; dt/.d z/dt/; 0

:

Proposition 1 Suppose we have

T

1

z .d z/dt C



However, their boundedness assumptions on ˛ and  are not covering our assumptions in model (Equations (105.2) and (105.3)). Thus, we give a short proof that ZT in Equation (105.12) is indeed the density we are looking for by appealing to the sufficient conditions developed by Rheinländer (2005).

1

Later we will be more precise about the size of k (see Proposition 2 and the examples following), and relate it to parameters in the specification of the Lévy measure. Under condition (105.8), the moment generating function is defined for all jj  k, and

˛ 2 .Yt / dt 2 2 .Yt /

0



˛.Yt / 1 S  2 .Yt / t

2 d ŒS t 2 Lexp .P /;

(105.14)

0

(105.11) where the second integral on the right-hand side is a martingale. The reader is referred to Applebaum (2004), Bertoin (1996), Protter (2003), and Sato (1999) for more information about Lévy processes and subordinators.

where ŒS t is the quadratic variation process of St and Lexp .P / is the Orlicz space generated by the Young function exp./. (i) Define Z

105.3 The Minimal Entropy Martingale Measure In this section we derive the density of the minimal entropy martingale measure of the model (Equations (105.2) and (105.3)), which we denote by QME . In Grandits and Rhein-

ZT0

D exp  0

T

˛.Yt / dBt  .Yt /

Z

T 0

! 1 ˛ 2 .Yt / dt : 2  2 .Yt / (105.15)

Then by assumption (105.13) and the Novikov condition, we know that Zt0 is a true martingale. We denote its corresponding probability measure by Q0 and note that Yt has the same dynamics under P and Q0 . Hence, we get

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F.E. Benth and T. Meyer-Brandis

 EŒZT D cE

ZT0

Z exp  0

 Z D cEQ0 exp 

T 0

 ˛ 2 .Yt / dt 2 2 .Yt /

 ˛ 2 .Yt / dt D 1: (105.16) 2 2 .Yt /

T

(ii) Using the same arguments as in i), we see that  Z E ŒZT j ln ZT j D EQ0 exp   Z D EQ0 exp 

T 0 T 0

ˇ Z T

ˇ Z T 2 ˇ ˇ ˛ 2 .Yt / ˛.Yt / ˛ .Yt / ˇ dt ˇ dBt C dt ˇˇ 2 .Y / 2 2 .Yt / .Y /  t t 0 0 ˇ

ˇZ T 2 ˇ ˛ .Yt / ˛.Yt / e ˇˇ dt ˇˇ d B t ˇ < 1; 2 2 .Yt / .Y t/ 0 Proposition 2 If

et is the Brownian motion under Q0 . where B (iii) Since we have Z exp 0

T



˛.Yt / 1 S  2 .Yt / t

!

2

 ˇ2 .1  exp.T // z  1 .d z/ < 1; exp  0 (105.18) then ZT defined in (105.12) is the density of the minimal entropy martingale measure QME . Z

Z

T

D exp

d ŒS t

0

˛ 2 .Yt / dt ;  2 .Yt /

assumption (105.13) implies condition (105.14).

1





Proof. Since Yt y exp.T /, we have

Proposition 2 gives a sufficient condition for assumption (105.13) stated in terms of the Lévy measure of L1 . Moreover, it determines an exact constant k in the exponential integrability condition (105.8):

2 ˛ 2 .Yt / D C 2ˇ C ˇ 2 Yt  C C ˇ 2 Yt 2  .Yt / Yt for a positive constant C . But this gives

  Z T

 ˛ 2 .Yt / 0 2 dt  C E exp ˇ Y dt t 2 0  .Yt / 0

  2 Z T ˇ y.1  exp.T // C D C 0 E exp .1  exp..T  t/// dLt  0 Z TZ 1

D C 00 exp  .exp.f .t/z/  1/ .d z/ dt ;

 Z E exp

(105.17)

T

0

(105.19)

0

where C 0 , C 00 are positive constants, and f .t/ ˇ 2 .1  exp..T  t// =.

D t u

Let us consider some examples of the process Lt that are relevant in finance, and state sufficient conditions for the density of the minimal entropy martingale measure. If we choose the stationary distribution of Yt to be an inverse Gaussian law with parameters ı and  , that is Yt  IG.ı;  /, the Lévy measure of L becomes

1 ı 3=2 .1 C  z/ exp   z d z: .d z/ D p z 2 2 2

Hence, the exponential integrability condition in Prop. 2 is satisfied whenever ˇ 2 .1  exp.T //
1=2 (inverse Gaussian law) or ˛ > 1=2 (the variance gamma law).

105.4 The Density Process Section 105.3 determines the minimal entropy martingale measure QME for the model (Equations (105.2) and (105.3)). It is of interest (for example in pricing of derivatives) to know the dynamics of the processes St and Yt under QME . In this section we identify the density process of QME as a certain stochastic exponential, which then by means of the Girsanov theorem gives us the dynamics of St and Yt under QME . A key ingredient in the description of the density process of the minimal entropy martingale measure is the function H.t; y/ defined as follows:

  Z ˇ 1 T ˛ 2 .Yu / ˇ d u Yt D y ; H.t; y/ D E exp  2 t  2 .Yu /

where RC D .0; 1/. We remark that our motivation for considering the function H comes from portfolio optimization with an exponential utility function. It turns out that the difference between the value function of the utility maximization problem and the utility function itself can be represented as H . This can be seen by, e.g., considering the Hamilton–

Lemma 3 For all .t; y/ 2 Œ0; T RC it holds that   exp a.t/y 1 C b.t/y C c.t/  H.t; y/  1; (105.22) where a.t/ D 

 .exp..T  t//  1/ ; 2

ˇ2 b.t/ D  .1  exp..T  t/// ; 2 Z T c.t/ D ˇ.T  t/ C 

.b.u// d u: t

Proof. The upper bound of 1 is clear (which is reached for t D T ). Denote

.t; y/ 2 Œ0; T RC ;

(105.20)

Jacobi–Bellman of the stochastic control problem. We refer to Musiela and Zariphopoulou Musiela and Zariphopoulou (2003) for more on this in a different market context than ours. Using the time-homogeneity of the Lévy process, we can rewrite H.t; y/ as

  Z ˇ 1 T t ˛ 2 .Yu / ˇY 0 D y ; d u H.t; y/ D E exp  2 0  2 .Yu / This function will describe the change of the jump measure of the Lévy process under the minimal entropy martingale measure. However, before considering this in more detail, we study some simple but useful properties of H.t; y/: The function satisfies the following bounds:

1571

.t; y/ 2 Œ0; T RC :

(105.21)



2 2 C 2ˇ C ˇ y : y (105.23)

1 1 ˛ 2 .y/ D g.y/ WD  2 2  .y/ 2

Using the explicit representation of Yu in (105.5), its lower bound Yu y exp..u  t// and the fact that Z

T



Yu d u D YT  Yt  .LT  Lt /;

t

it is straightforward to derive   H.t; y/ exp a.t/y 1 C b.t/y  ˇ.T  t/

  Z ˇ2 T .1  exp..T  u/// dLu :

E exp  2 t (105.24) The lower bound follows.

1572

F.E. Benth and T. Meyer-Brandis

Later we shall make explicit use of the differentiability of H.t; y/ with respect to t and y, proved in the following proposition: Proposition 4 The function H.t; y/ is continuously differentiable in t and y, i.e., H 2 C 1;1 .Œ0; T RC /. Proof. The random variable Z



T t

Xt;T WD exp

g.Yu / d u 0

with g.y/ as in (105.23) and Y0 D y, is obviously differentiable with respect to y. The derivative is given by 1 2

Z

T t



0

2 2 eu d u  Xt;T  ˇ Yu2

which can be bounded by C C C 0 =y 2 for two positive constants C; C 0 after appealing to the inequality Yu y exp.u/. Hence, the dominated convergence theorem implies that H.t; y/ is differentiable with respect to y. Moreover, by similar arguments we find that @H=@y is continuous, which proves the first part of the Proposition. Concerning the differentiation with respect to t, we rewrite H.t; y/ as Z H.t; y/ D E T t

D





T s

Z

0 T s

E exp

0



@H LY H D y C @y

g.Yu /d u g.Ys / ds C 1;

0

(105.25) where we have appealed to the Fubini–Tonelli theorem together with the exponential integrability conditions on L to interchange integration and expectation. Because Yu is a Lévy diffusion and the compensating measure of a Lévy process is diffuse with respect to time, the integrand in (105.25) is continuous in s. Hence H.t; y/ is continuously differentiable in t.

(105.27)

Z

1

fH.t; y C z/  H.t; y/g .d z/: 0

(105.28)

The function H in (105.20) solving Equations (105.26) and (105.27) plays a crucial role in the derivation of the density of the minimal entropy martingale measure. In general, Equation (105.20) is rather difficult to calculate explicitly. However, if we consider the special case ˛.y/ D ˇy, i.e.,  D 0 in Equation (105.2), a direct calculation using the moment generating function of L1 gives the following explicit solution of the integro-PDE (Equations (105.26) and (105.27)): Corollary 5 Suppose ˛.y/ D ˇy. Then the solution of Equations (105.26) and (105.27) is given as H.t; y/ D exp .b.t/y C c.t// ;

(105.29)

where b and c are defined as ˇ2 .1  exp..T  t/// ; 2 Z T c.t/ D 

.b.u// d u;

b.t/ D 

 

y 2 RC :

Here

g.Yu /d u g.Ys /ds C 1

exp 0

Z

Z

T t

H.T; y/ D 1;

t

We recall that is the log moment generating function of L1 defined in Equation (105.10). Setting  D 0 in (105.2) corresponds to an expected log-return of .ˇ  12 /y of the risky asset St . If we, for instance, specify the stationary distribution of Y to be inverse Gaussian, then the log-returns will be approximately normal inverse Gaussian distributed (see Barndorff-Nielsen and Shephard (2001)), and choosing this to be symmetric corresponds to ˇ D 12 , that is, with  D 0 we have zero expected log-return. Now we introduce the notation

We get from the theory of Markov processes that H.t; y/ is the Feynman–Kac representation of the solution of the following integro-PDE:

ı.y; z; t/ WD

H.t; y C z/ H.t; y/

(105.30)

and define the following stochastic exponentials

˛ 2 .y/ @H  2 H C LY H D 0; @t 2 .y/

.t; y/ 2 Œ0; T / RC ; (105.26)

with terminal data Z Zt0 WD exp  Z tZ Zt00

D exp

t 0

˛.Ys / dBs  .Ys / Z tZ

1

1

ln ı.Ys ; z; s/N.d z; ds/ C 0

0

Z

t 0

1 ˛ 2 .Ys / ds 2  2 .Ys /

! (105.31) !

.1  ı.Ys ; z; s// .d z/ds : 0

0

(105.32)

105 The Density Process of the Minimal Entropy Martingale Measure in a Stochastic Volatility Model

We identify the density process in question as follows.

where the right hand side of the above equation is the density in Equation (105.12). Since we have

Theorem 6 Suppose condition (105.13) is fulfilled. Then

dSt D ˛.Yt /dt C .Yt /dBt ; St

Zt WD Zt0 Zt00 is the density process of the minimal entropy martingale measure QME .

T

0

we get Z ln.ZT0 / D 

Proof. We want to show that Z 0 00 ZT ZT D c  exp 

1573

˛.Yt / dBt  .Yt /

Z

T 0

Z ln.ZT0 / C ln.ZT00 / D 



2

˛ .Yt / dt ;  2 .Yt /

T 0

˛.Yt / 1 1 St dSt C 2  .Yt / 2

Z 0

T

˛ 2 .Yt / dt:  2 .Yt / (105.33) 2

t/ Now, substituting in Equation (105.33) for 12 ˛ 2 .Y the ex.Yt / pression we get from the integro-PDE Equation (105.26), we end up with

Z T @y H.t; Yt / @t H.t; Yt / ˛.Yt / 1 St dSt C  Yt dt H.t; Yt / H.t; Yt / 0 .Yt / 0 Z TZ 1 C .ln H.t; Yt C z/  ln H.t; Yt // N.d z; dt/: T

0

(105.34)

0

Note that we have used the short-hand notation @t H for @H=@t and @y H for @H=@y. Since H 2 C 1;1 from Proposition 4, we can apply Itô’s formula on h.t; Yt / D ln H.t; Yt / to derive Z

T



h.T; YT / D h.0; Y0 / C Z

T

C 0

Finally, substitution of Equation (105.34) yields ZT0 ZT00

Equation

T 0

1

.ln H.t; Yt C z/  ln H.t; Yt // N.d z; dt/:

(105.35)

0

(105.35)

in

˛.Yt / 1 St dSt 2 0  .Yt /

Z T 2 ˛.Yt / ˛ .Yt / dBt  dt ; 2 .Yt / 0  .Yt /

Z D exp  ln H.0; y/  Z D c  exp 

0

Z

@y H.t; Yt / @t H.t; Yt / dt  Yt H.t; Yt / H.t; Yt /

T

(105.36) such that ZT0 ZT00 is indeed the density of QME . Finally, the orthogonality of Zt0 and Zt00 together with the fact EŒZT0 ZT00 D 1 (point i) in the proof of Proposition 1) yields that Zt D Zt0 Zt00 is a martingale.

105.5 The Entropy Price of Derivatives and Integro-Partial Differential Equations As an application of our results, we consider the price of derivatives written on the asset S under the minimal entropy martingale measure. We derive the corresponding integroPDE for claims having a payoff given by the asset price ST at maturity of the contract, a typical example being a European call option on S . Consider a contingent claim with payoff f .ST / at maturity time T , where we suppose that f is of linear growth and f .ST / 2 L1 .QME /. Then the entropy price of the claim at time t given St D s and Yt D y, denoted by ƒ.t; y; s/, is

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F.E. Benth and T. Meyer-Brandis

defined as the conditional expectation under the minimal entropy martingale measure. We thus get ˇ   ƒ.t; y; s/ D EQME f .ST / ˇ Yt D y; St D s : (105.37) Knowing the density process of QME we are now able to determine the dynamics of St and Yt under QME . Define the et by two processes e S t and Y   et e et ; St dB de St D  Y

(105.38)

et D Y et dt C d e dY Lt ;

(105.39)

et is Brownian motion and e Lt is a pure jump Markov where B process with the predictable compensating measure e  .!; d z; dt/ D

et .!/ C z/ H.t; Y .d z/dt: et .!// H.t; Y

(105.40)

Observe that the state-dependent jump measure e .d z/ becomes deterministic when  D 0: Indeed, from Corollary 5 we find that e  .!; d z; dt/ D eb.t /z .d z/ dt; where b.t/ is given in Corollary 5. Hence, for  D 0, e L is an independent increment process (see e.g., Sato (1999)). et and e Lt and the Girsanov By using the independence of B theorem for Brownian motion and random measures (see Jacod and Shiryaev (1987)), respectively, we see that ˇ   ƒ.t; y; s/ D EQME f .ST / ˇ Yt D y; St D s   ˇ  et D y; e DE f e ST ˇ Y St D s : This representation allows us to set up an integro-PDE for the entropy price. Like in Bensoussan and Lions (1982), Chapter 3, Theorem. 8.1, using the bounds of H.t; y/, we get that ƒ.t; y; s/ is the Feynman–Kac representation of the solution of the following integro-PDE

1 @ƒ @2 ƒ @ƒ C  2 .y/s 2 2  y @t 2 @s @y Z 1 H.t; y C z/ .d z/ D 0; .t; y; s/ 2 Œ0; T / R2C ; C .ƒ.t; y C z; s/  ƒ.t; y; s// H.t; y/ 0

with terminal condition ƒ.T; y; s/ D f .s/; .y; s/ 2 R2C :

(105.42)

Note that in order to solve this integro-PDE, we need to consider the Equations (105.26) and (105.27) for H as well. Thus, the minimal entropy price of a claim in the BarndorffNielsen and Shephard model is given as the solution of a coupled system of two integro-PDEs.

105.6 Conclusions In incomplete markets, there exist many risk neutral pricing measures, and tools to single out one for the purpose of derivatives pricing is called for. A natural choice seems to be given via utility indifference pricing, where the limiting pricing measure for zero risk aversion plays a crucial role, namely the minimal entropy martingale measure. This is the measure minimizing the relative entropy, and can be viewed financially as the measure yielding a price which both the buyer and the seller “agree” on.

(105.41)

We have studied the minimal entropy martingale measure for the stochastic volatility model of Barndorff-Nielsen and Shephard (2001). The novel feature of their model is that the squared volatility process follows an Ornstein–Uhlenbeck process driven by a subordinator, ensuring a positive volatility and a very flexible modeling of observed financial time series. We have found the density process of the Radon– Nikodym derivative of the minimal entropy martingale measure in terms of an expectation functional solving a firstorder integro-partial differential equation. The proof is based on a general characterization result by Rheinländer (2005). We also present the characteristics of the subordinator under the minimal entropy martingale measure, which turns out to have state-dependent jumps and thereby loosing the independent increment feature under the objective probability. The price under the minimal entropy martingale measure of a derivative with payoff depending on the underlying asset at some exercise time will solve a Black–Scholes type integropartial differential equation. We finally remark that Rheinländer and Steiger (2006) recently generalized our results to the case of an asset price dynamics with BNS-volatility and leverage.

105 The Density Process of the Minimal Entropy Martingale Measure in a Stochastic Volatility Model

The characterization of the minimal entropy martingale measure goes via the solution of a very involved partial differential equation. Acknowledgments We are grateful to Kenneth Hvistendahl Karlsen and Thorsten Rheinländer for interesting and fruitful discussions. This chapter is a reprint of the paper Benth and Meyer-Brandis (2005) (with a slightly extended reference list and amended conclusions). Two anonymous referees and an associate editor are thanked for their careful reading of an earlier version of that paper, leading to a significant improvement of the presentation.

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Chapter 106

Arbitrage Detection from Stock Data: An Empirical Study Cheng-Der Fuh and Szu-Yu Pai

Abstract In this paper, we discuss the problems of arbitrage detection, which is known as change point detection in statistics. There are some classical methods for change point detection, such as the cumulative sum (CUSUM) procedure. However, when utilizing CUSUM, we must be sure about the model of the data before detecting. We introduce a new method to detect the change points by using Hilbert–Huang transformation (HHT) to devise a new algorithm. This new method (called the HHT test in this paper) has the advantage in that no model assumptions are required. Moreover, in some cases, the HHT test performs better than the CUSUM test, and has better simulation results. In the end, an empirical study of the volatility change based on the S&P 500 is also given for illustration. Keywords Arbitrage detection r Hilbert–Huang transformation r Volatility r CUSUM

106.1 Introduction 106.1.1 Background In the stock market, investors always pursue the goal of finding change points promptly. These changes may come from the alteration of company policies or from an economic recession. We hope to detect the changes as soon as they appear. There is already a term to describe these problems: change point detection or arbitrage detection. Many results, such as the CUSUM test, exist in the change point detection of previous studies. The first result can be found in Page (1954) who constructed classical CUSUM tests. This test provides a widely accepted procedure to detect change points. C.-D. Fuh () National Central University and Academia Sinica, Taipei, Taiwan e-mail: [email protected] S.-Y. Pai National Taiwan University, Taipei, Taiwan e-mail: [email protected]

However, there are still some problems to be found in the CUSUM test in Pollak and Siegmund (1985) and Siegmund (1985). For example, it is necessary to ascertain the distribution of the data. Without the distribution of the data, the CUSUM test cannot be implemented. Therefore, we introduce a new method of arbitrage detection: the Hilbert– Huang transformation (HHT) test. The HHT was discussed in Huang et al. (2003), Huang and Shen (2005) and Wu and Huang (2004) and has already been used extensively in other fields. Such as, studies in geophysics, structural safety, and operating research that make use of it. However, the HHT is rarely applied in finance. In this respect we try to design a test using HHT on some finance problems due to the fact that many financial data are nonlinear, nonstationary time series with unknown models, which is exactly what the HHT can analyze. This paper is organized as follows. In this section, we introduce a new method (called the HHT test in this paper) to detect the change points. In Sect. 106.2, we describe some cases where the HHT test performs better than the CUSUM test and has better simulation results. In Sect. 106.3, we find some weakness of the HHT test in the change of the mean. In Sect. 106.4, an empirical study of the volatility change based on the S&P 500 is also given for illustration.

106.1.2 Previous Studies in Arbitrage Detection Consider an infinite sequence of observation: x1 ; x2 ; x3 ; : : : ; xi ; : : :. These variables represent the stock price at time i . At some unknown time v, either the company altered its policy or it was the beginning of a period of economic recession, such as the dot-com bubble in 2000. These types of events have a common characteristic, in that they transfer the whole structure of the market. In other words, the distribution of the time series changes parameters, like the mean or the variance. Therefore, what we seek is a stopping rule T , which detects the change promptly. When mentioning change point detection, the CUSUM test cannot be left unnoticed. We will use the above

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assumption to explain the idea of CUSUM. First, x1 ; x2 ; x3 ; : : : ; xv1 are independent and identically distributed with the probability density function f0 , whereas xv ; xvC1 ; : : : are independent and identically distributed with the probability density function f1 , for some v > 1. Let Pi denote the probability as the change from f0 to f1 occurs at the i th observation; Ei .T / denotes the expectation of a stopping rule T when the change occurs at time i . If i D 0, there is no change. A stopping rule  can be described as: Minimize sup Ev .  v C 1j v/

(106.1)

v1

subject to E0  B

(106.2)

for some given (large) constant B. A special method to solve the above problem is the following. Assume x1 ; : : : ; xn have been observed. Consider for 1  v  n Hv W x1 ; : : : ; xv1  f0 I

xv ; xvC1 ; : : : ; xn  f1

against Then the log likelihood ratio statistic can be written down: max .sn  sk / D sn  min sk ; 0kn

where Sn D

n X j D1

 log

 f1 .xj / : f0 .xj /

(106.3)

(106.4)

Then we can get a stopping rule:

  D inf n W sn  min sk b :



0kn

106.1.3.1 Hilbert–Huang Transformation A new instrument on arbitrage detection, HHT, has already been extensively used in engineering. We choose to use the HHT instead of other spectrum analysis methods, such as Fourier transformation and Wavelet transformation, for the following reasons: First, the HHT can be utilized on nonlinear, nonstationary

H0 W x1 ; : : : ; xn  f0 :

0kn

HHT. In this way, we focus on the high frequency part while detecting volatility change because high frequency means short period. Hence the long period part, which is useless in detecting volatility change, can be omitted. By using the above methods we have several simulation studies. The first simulation study analyzes the time series from normal distribution, which represents the basic model. Following, we analyze the time series from Brownian motion models and geometric Brownian motion models, which are close to real financial data. The last simulation study analyzes the time series from Markov switch models and unlike the previous models, in this one, we do not know when the distribution of the data will change. Therefore, the HHT test can detect the change points, although the change timing is unknown.

(106.5)

106.1.3 The Use of Hilbert–Huang Transformation on Arbitrage Detection The HHT was used to analyze data in two ways. The first one is using the whole time–frequency–amplitude plot. This is the classical way to utilize the HHT. Its advantage is that we can collect the information from all of the data; however, the disadvantage of the classical method is it is less sensitive than method two. The second way is using a part of the time–frequency– amplitude plot. This method is a new application for the

time series. Although Fourier transformation has a wide application on spectrum analysis, it can be only used on stationary time data; Wavelet transformation can be applied on nonstationary data, but it is ineffective on nonlinear data. However, financial data are usually nonlinear and nonstationary time series. Therefore, the HHT becomes the first choice. Second, although Fourier transformation is useful on spectrum analysis, it can only transform a function from time domain into frequency domain. In other words, we do not know the frequency at the specific timing (Table 106.1). Third, compared with Wavelet transformation, the HHT is more precise. In Huang et al. (1998), the time–frequency– amplitude plots from HHT show more details than the Wavelet transformation.

106.1.3.2 The Process of Hilbert–Huang Transformation Empirical Mode Decomposition (EMD) 1. Let X.t/ denote a time series, identify local maxima and local minima of X.t/, and then connect all the local maxima by a cubic spline line named the upper envelope.

106 Arbitrage Detection from Stock Data: An Empirical Study

Table 106.1 The comparison between Fourier transformation and Hilbert–Huang transformation

Fourier transformation

Hilbert–Huang transformation

Symbol

F ./

H./

Transformation of domain

From time domain to frequency domain

From time domain to time and frequency domain

Function

F .X.t // D A.!/

H.X.t // D A.t; !/

Repeat the above process for local minima to produce the lower envelope. max.t/ denotes the upper envelope and min.t/ denotes the lower envelope. Let their mean be M1 .t/ D

1579

Œmax.t/ C min.t/ ; 2

(106.6)

2. The analytic signal is defined as Zj .t/ D Cj .t/ C iYj .t/ D Aj .t/e ij .t / ; where Aj .t/ D

and the difference between the data and M1 .t/ is X.t/  M1 .t/ D H1 .t/:

(106.7)

2. Repeat step 1 to obtain Mj .t/, Hj .t/: H1 .t/  M2 .t/ D H2 .t/;

(106.8)

Hk1 .t/  Mk .t/ D Hk .t/;

(106.9)

and

q

ŒCj2 .t/ C Yj2 .t/ ;

Yj .t/ : j .t/ D arctan Cj .t/

Hk .t/ D C1 .t/;

(106.10)

T X ŒHk1 .t/  Hk .t/ 2 t D1

2 Hk1 .t/

(106.11)

is smaller than a predetermined value (about 0.2–0.3 in Huang et al. 1998). 3. Let (106.12) X.t/  C1 .t/ D R1 .t/; and do the above procedure again. We can obtain C2 .t/; C3 .t/; : : : and so on. 4. Let Cj .t/ be the intrinsic mode functions (IMFs) (Fig. 106.1).

dj .t/ : dt

(106.17)

3. The original time series can be expressed as following: X.t/ D
bg; (106.21) where M is the mean of the three maximum amplitudes in the Hilbert spectra (to avoid the extreme value), subject to (106.22) E0 Π0 > B: Compared with CUSUM, we obtain Table 106.2. It is apparent that CUSUM yields better results when the change is large. However, the HHT test performs better with slight change, and the average stopping time in the HHT test grows slower compared with the CUSUM test where the change becomes minor.

106.2.2 Volatility Change in Geometric Brownian Motion Models

where W . t/ is standard Brownian motion. Let x1 D 100, and then we can use the above model to generate x1 ; x2 ; x3 ; : : :, and so on. After generating all xi , we let yi D xi  100 for all i to reduce the influence of the start point. In this model, we could focus on high frequency parts of time–frequency–amplitude plots because high frequency means short period. The larger  implies the larger amplitude of the short period part. Therefore, we focus on the frequency from 0.375 to 0.5, in other words, in the period from 2 to 2.7 days. In Fig. 106.3, we cannot see the difference directly from the Hilbert amplitude spectrum, but in Fig. 106.3(15) we can see the mean of maximum amplitude touch 0.5 after t > 100. We use the same idea to make a criterion, and we have Table 106.3. The result is similar to Sect. 106.2.1. The HHT performs better than CUSUM, when  changes slightly.

106.2.3 Volatility Change in Markov Switch Models

After detecting change points in normal distribution, we apply the HHT test to geometric Brownian motion (GBM) models, which is always used to describe stock prices. In option pricing, modeling stock prices by GBM models has a well-known problem: the implied volatility is not a constant. In practice, stock prices also have the volatility clustering property. Here we do not seek to reduce the inaccuracy. On the contrary, we aim to detect volatility clustering. First of all, assume x1 ; x2 ; x3 ; : : : ; x100  GBM.r D 0:08;  D 1 /;

The simulation study here analyzes the time series from Markov switch models, and unlike the previous models, in this one, we do not know when the distribution of the data will change. Therefore, the HHT test can detect the change points although the change timing is unknown. The Markov switch model can be described as

and

(106.23) x101 ; x102 ; x103 ; : : : ; xn  GBM.r D 0:08;  D 2 /: (106.24) In other words, xi D xi 1 e .r0:5

2 / t CW . t /

;

(106.25)

Yt D ˛St Yt 1 C "t ;

(106.26)

  "t  N u D 0; S2t ;

(106.27)

St 2 f1; 2g

(106.28)

where represent two states. In state 1, let ˛1 D 0:1;

(106.29)

1 D 0:1:

(106.30)

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Fig. 106.3 (1)–(14) Hilbert spectra. (15) The means of maxima in Hilbert spectra. (16) The original data

Table 106.3 The comparison between CUSUM test and HHT test in volatility change of Geometric Brownian Motion Models

Change of volatility

Stopping time of the CUSUM test

Stopping time of the HHT test

sigma from 0.1 to 0.3 sigma from 0.1 to 0.2 sigma from 0.1 to 0.15 sigma from 0.1 to 0.125 sigma from 0.1 to 0.11 sigma from 0.1 to 0.105 sigma from 0.1 to 0.1

3.59 10.51 39 214.32 642.7 766.38 853.46

30.67 158.75 340.59 591.21 663.24 702.8 840.55

106.2.4 A Brief Summary

In state 2, let ˛2 D 0:1;

(106.31)

2 D :

(106.32)

Let the transition matrix be " # 0:99 0:01 pD : 0:001 0:999

(106.33)

The result of Markov switch models is shown in Table 106.4 and Fig. 106.4. From the above table we can still find the good characteristics of the HHT test. Although we cannot quickly detect the large change, the stopping times is begin to slow.

In volatility change, we can find a detection rule from the ideas of the HHT. The HHT is a method to decompose a time series and produce Hilbert spectra, which are time– frequency–amplitude plots. The plots can be used to determine whether or not the volatility changes. In detecting volatility change, we focus on the amplitude change. The stopping rule is that the maxima of the amplitude are larger than a constant b subject to small type 1 error. In other words, the expectation of stopping time must be larger than a constant B in null hypothesis. Here we adopt B D 800, which is usually utilized in practice. Then the criterion b can be produced subject to B D 800.

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Fig. 106.4 (1)–(14) Hilbert spectra. (14) The means of maxima in Hilbert spectra. (15) The state of the data. (16) The original data

Table 106.4 The volatility changes of Markov switch models Stopping time of the HHT test Change of volatility sigma from 0.1 to 0.3 sigma from 0.1 to 0.2 sigma from 0.1 to 0.15 sigma from 0.1 to 0.125 sigma from 0.1 to 0.11 sigma from 0.1 to 0.105 sigma from 0.1 to 0.1

38.8 180.4 431.0 549.7 666.3 770.1 889.5

In both the HHT and the CUSUM tests, we can find that as the change becomes minor, the stopping times of the HHT test and the CUSUM test increase. However, the increasing speed is different for the two tests. The stopping time of CUSUM test increases faster than that of the HHT test. Therefore, the HHT test performs better than the CUSUM test when the change is slight.

106.3 Arbitrage Detection: Mean Change 106.3.1 Mean Change in Normal Distribution Models

Let x1 ; x2 ; x3 ; : : : ; x100  N.u D u1 ;  2 D 1/; (106.34) x101 ; x102 ; x103 ; : : : ; xn  N.u D u2 ;  2 D 1/: (106.35) The stopping time of the HHT test is increasing faster than that of the CUSUM test, showing in Fig. 106.5 and Table 106.5. This means that the HHT test is weak in detecting the change of the mean. Reviewing the procedure of the HHT may explain the reason. It treats the low frequent IMFs as unimportant components. However, in the change of the mean, low frequent IMFs represent the trend of the data. Therefore, we are not satisfied with the results of the HHT test in the change of the mean.

106.3.2 Mean Change in Brownian Motion Models After the unsatisfying results in Sect. 106.3.1, we try another model, the Brownian motion model, which cumulates the value of u. The model we use is x1 ; x2 ; x3 ; : : : ; x100  BM.u D u1 ;  2 D 1/; (106.36)

The change of the mean is an important topic in finance. It represents the trend of the stock prices. Let us start with the essential model: the normal distribution.

x101 ; x102 ; x103 ; : : : ; xn  BM.u D u2 ;  D 1/: 2

(106.37)

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Fig. 106.5 (1)–(14) Hilbert spectra. (15) The means of maxima in Hilbert spectra. (16) The original data

Table 106.5 The comparison between CUSUM test and HHT test in mean change of Brownian Motion Models Stopping time of the Stopping time of HHT test CUSUM test Change of mean mean from 0 to 2 mean from 0 to 1 mean from 0 to 0.5 mean from 0 to 0.25

14.7 38.7 136.8 589.7

3.81 12.54 45.69 174.25

In Fig. 106.6(15), we find that when the value of the time series decreases, the amplitude of mean of maximum will increase; when the time series start to increase (the change point), the amplitude of the maximum mean will stop changing. We cannot find a good criterion for the phenomenon, but we can detect it visually. In Fig. 106.7, we can observe that after about t D 110 the value stops increasing. Conservatively, we choose t D 118 to be the stopping time. In Fig. 106.8, we find the same phenomenon, and we detect the change on t D 127. In Fig. 106.9, we can detect the change at t D 137, and we find another special phenomenon, which is when u2 becomes smaller, the “shake” after the change point becomes larger.

In Fig. 106.10, we can detect the change at t D 176, and the range of the special phenomenon, the “shake” increases from 70–80 to 90–120. In Fig. 106.11, we cannot see the change any more. In the case of u1  u2 > 0, the means of maximum continue increasing. We should analyze the data xi xi 1 instead in this case. In Table 106.6 we can see the HHT test performs well in the case when u1  u2  0. However, it is useless in the case when u1  u2 > 0.

106.3.3 A Brief Summary In this section, we discuss the problem of the change of the mean. Unfortunately the HHT test does not perform well in this case. In normal distribution models, the HHT test loses its positive characteristics mentioned in Sect. 106.2, which is that the stopping time increases slowly when the change is slight. Therefore, the HHT test cannot perform better than the CUSUM test under such condition.

106 Arbitrage Detection from Stock Data: An Empirical Study Fig. 106.6 (1)–(14) Hilbert spectra. (15) The means of the maxima in Hilbert spectra. (16) The original data

Fig. 106.7 Change from BM(1,1) to BM(2,1)

Fig. 106.8 Change from BM(1,1) to BM(1,1)

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Fig. 106.9 Change from BM(1,1) to BM(0.5,1)

Fig. 106.10 Change from BM(1,1) to BM(0,1)

Fig. 106.11 Change from BM(1,1) to BM(0.5,1).

In Brownian motion models, it is hard to find worthwhile criteria to detect the change, but we still can find some phenomena in plots of the maxima means. When the time series starts to change, the value of the maximum means will stop at the same level; when u2 is small, the “shake” of the amplitude of mean of maximum will increase. However, this detection rule is useless while u1  u2 > 0. The reason for the weakness of the HHT test in detecting the change of the mean may be the procedure of the HHT, which focuses on high frequency parts but neglects trend changes.

106.4 Empirical Studies This section discusses the empirical studies of the HHT test. The CUSUM test cannot be applied here because we do not have any model assumptions for the S&P 500 Index. However, when utilizing the HHT test, we do not need any assumptions, and the empirical studies of the HHT test lead us to a positive conclusion.

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Table 106.6 Stopping time of the HHT test in mean change of Brownian Motion Models Change of mean Stopping time of the HHT test

106.4.1 Volatility Change Data (Subprime Mortgage Crisis in 2007)

u from 1 to 2 u from 1 to 1 u from 1 to 0.5 u from 1 to 0 u from 1 to 0.5

106.4.1.1 Using Stock Prices Directly Let us use our new method on empirical studies. In 2007, the global market faced a serious crisis, and suffered an unprecedented credit risk. At the same time, stock prices underwent acute vibration. This event is a good example in which to test our new method. First, using the S&P 500 Index from January 3, 2006 to April 9, 2008, we obtain 570 daily data. Second, we use the data of VIX Index, which is the implied volatility from S&P 500 to check our result. When we apply our new method to detect change points, is it efficient? If a change does exit, how quickly can we detect it?

In Fig. 106.12(16), S&P 500 Index has an acute vibration after t D 400. The same phenomenon can be found in Fig. 106.12(15), the VIX Index. After t D 400, VIX Index are all approximately larger than 20%.

vix(%)

VIX

Time

Figure 106.12(15)

Amplitude

Mean of Maxima

Time

Figure 106.12(14)

18 27 37 76 Cannot detect

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Fig. 106.12 (1)–(13) Hilbert spectra. (14) The means of the maxima in Hilbert spectra. (15) The VIX Index. (16) The S&P 500 Index

VIX

VIX(%)

Table 106.7 tells us that the change points can be detected efficiently exclusive of Event 1. However, the changes of volatility in Event 2 and Event 3 still continue for 20 days and 53 days, respectively. Only a minority of indexes have their own implied volatility index. The S&P 500 Index is one of them and its implied volatility index is called VIX. Some markets do not have big enough option trading volumes. Therefore the new method can be utilized on these markets. We can regard trading options as trading the volatility. Accordingly, in the market with low option trading volume, when we detect the increase of the volatility, the prices of options are probably undervalued. Contrarily, when the volatility decreases, the prices of options are possibly overvalued.

Time

Figure 106.13(15) Mean of Maxima

In this section, the log returns of stock prices are analyzed. The use of the log returns of stock prices allows us to focus on the volatility change without the influence of the trend. Therefore, analyzing the log returns of data yields better results than analyzing the original data. Figure 106.13(16) is the log return of the S&P 500 Index from January 3, 2006 to April 9, 2008.

Amplitude

106.4.1.2 Using Log Return of Stock Prices

Time

Figure 106.13(14)

106 Arbitrage Detection from Stock Data: An Empirical Study

Volatility change event

Start date

End date

Date of the HHT test detection

Stopping time of the HHT test

Event 1

393

430

436

43

Event 2

465

495

475

10

Event 3

501

570

517

16

From Fig. 106.13(14) and Table 106.8 the results of the detection show that the method in this section can detect the changes faster then in Sect. 106.4.1.1. Therefore, we can make investment decisions more promptly by applying the results in this section. All the three results of events in this section are better than in Sect. 106.4.1.1. In Event 1, stopping time D 43 when using the data of S&P 500 Index directly, however, we have stopping time D 9 here. In Event 2, stopping time D 10 in Sect. 106.4.1.1, which contrasts with stopping time D 4 here. Only in Event 3 the stopping time D 16 in Sect. 106.4.1.1 which is almost the same as stopping time D 15 in this section. It means that by using our detection results we can react more quickly to the change of the volatility.

106.4.2 Volatility Change Data (Dot-Com Bubble in 2000) Another economic recession in 2000 is considered here. In 1998–2000, the stock prices of dot-com companies rose quickly. However, most dot-com companies had not even made a profit in that period. Hence, after irrational investors spent all the wealth they had buying shares, the stock prices of dot-com companies began to fall, and were accompanied by the rise in the volatility. We adopt the data of S&P 500 Index from July 3, 2000 to December 31, 2001, including 374 daily prices. As a result of the better results when analyzing log return of data in Sect. 106.4.1, we prefer log return of data here (Fig. 106.14).

VIX%

VIX

Time

Figure 106.14(15)

Mean of Maxima

Amplitude

Table 106.7 The stopping times of the HHT test in subprime mortgage crisis (using raw data)

1589

Time

Figure 106.14(14) Here we try to find the change points by sight. The conclusion is Table 106.9. In Event 1, because the change is about 16–30%, the stopping time is short. In Event 2 and 3, the change is about 20– 30%, so the stopping time is longer than in Event 1, but still quite small when compared to the length of duration of the change. In other words, the HHT test can detect the change points promptly so we can make investment decisions before the changes finish in the dot-com bubble case.

106.4.3 A Brief Summary In empirical study, we obtain two conclusions. First, the new method, the HHT test, is useful on empirical data. In both the subprime mortgage crisis and the dot-com bubble, we can detect the change points correctly and quickly. In the subprime mortgage crisis, we can find three events in which the volatility changes from less than 20% to more than 20%. We can detect all the changes promptly in the subprime mortgage crisis. In the dot-com bubble, there exist four events and all of them can be detected quickly, too. Therefore, the results we get from data of both the subprime mortgage crisis and the dot-com bubble are outstanding. Second, due to a shorter stopping time, using the log return of data is better than using the original data. The reason is that in log return of data, we can focus on the volatility change without the influence of the trend.

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Fig. 106.13 (1)–(13) Hilbert spectra. (14) The means of the maxima in Hilbert spectra. (15) The VIX Index. (16) The log return of the S&P 500 Index

Table 106.8 The stopping times of the HHT test in subprime mortgage crisis (using log return of stock prices)

Volatility change event

Start date

End date

Date of the HHT test detection

Event 1

393

430

402

Event 2

465

495

469

4

Event 3

501

570

516

15

106.5 Conclusions and Further Researches

Stopping time of the HHT test 9

Second, the HHT test performs better than the CUSUM

test in some cases.

106.5.1 Conclusions In this paper we reach four conclusions: First of all, we introduce a new method to arbitrage de-

tection. This method can be utilized without any model assumptions. HHT can produce Hilbert spectra, which represent frequency–time distribution of the amplitude. By utilizing high frequency parts of Hilbert spectra, we can devise the HHT test. Moreover, this test can be applied without model assumptions. In the classic method of change point detection, using the CUSUM test, we need to know the distribution of the data before and after change to compute the log likelihood ratio statistic. The above information is needless in the HHT test.

In Sect. 106.2, although the CUSUM test performs well in large change problems, the HHT test is good handling those cases with slight changes. Therefore, if dealing wit slight changes, we can consider the HHT test first. Third, in practice, we still obtain good results.

In Sect. 106.4, we review the data of the dot-com bubble in 2000 and the subprime mortgage crisis in 2007, respectively. Both of the two financial crises are well known and extensively influential. The HHT test can detect these two crises promptly and efficiently, and we can even make a profit from the successful detection. Fourth, the advantage and disadvantage of the HHT test.

In Sect. 106.3, we learn the characteristics of the HHT test. When detecting the change of the mean, it yields a mediocre result. Although the HHT test is sensitive to volatility change, it is insensitive to the change of the mean.

106 Arbitrage Detection from Stock Data: An Empirical Study

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Fig. 106.14 (1)–(13) Hilbert spectra. (14) The means of the maxima in Hilbert spectra. (15) The VIX Index. (16) The log return of the S&P 500 Index

Table 106.9 The stopping times of the HHT test in the dot-com bubble

Volatility change event

Start date

End date

Date of the HHT test detection

Stopping time of the HHT test

Event 1

63

158

76

13

Event 2

159

226

194

35

Event 3

291

373

309

18

In this paper, we become familiar with some features of the new method, the HHT test. It is a good method in dealing with the volatility change problems, but it has some difficulties when facing the changes of the mean. These imperfect parts of the HHT test still need to be solved.

congenitally in mean change case. However, if there are some means that can overcome the obstacles of this weakness, then the HHT test could become a comprehensive method for arbitrage detection.

References 106.5.2 Further Research The evidence presented above indicates that there are some persuasive reasons for preferring the HHT test in some cases. Therefore, we indicate some open problems here. First, some strict proof of the HHT test surpassing the CUSUM needs to be provided. Because the HHT does not have theoretical bases, the proof may be the hardest part of further research. Second, why does the HHT test yield better results of detection of volatility change? Why is it insensitive when dealing with the change of the mean? Third, can we utilize the HHT test in other fields? We have a good conclusion in finance, so the next step should be doing some research on other kinds of data. Fourth, how can we settle the weakness of the HHT test in mean change? The HHT test may have some limits

Huang, N. E. and S. P. Shen. 2005. Hilbert–Huang transform and its applications, World Scientific, London. Huang, N. E., Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Lin. 1998. “The empirical mode decomposition and the Hilbert spectrum for nonlinear and nonstationary time series analysis.” Proceedings of the Royal Society of London A 454, 903–995. Huang, N. E., M. C. Wu, S. R. Long, S. P. Shen, W. Qu, P. Gloersen, and K. L. Fan. 2003. “A confidence limit for the empirical mode decomposition and Hilbert spectral analysis.” Proceedings of the Royal Society of London A 459(2037), 2317–2345. Page, E. S. 1954. “Continuous inspection schemes.” Biometrika 41, 100–115. Pollak, M. and Siegmund, D. 1985. “A diffusion process and its applications to detecting a change in the drift of Brownian motion.” Biometrika 72(2), 267–280. Siegmund, D. 1985. Sequential analysis, Springer-Verlag, New York, NY. Wu, Z. and Huang, N. E. 2004. “A study of characteristics of white noise using the empirical mode decomposition method.” Proceedings of the Royal Society of London 460, 1597–1611.

Chapter 107

Detecting Corporate Failure Yanzhi Wang, Lin Lin, Hsien-Chang Kuo, and Jenifer Piesse

Abstract This article introduces definitions of the terms bankruptcy, corporate failure, insolvency, as well as the methods of bankruptcy, and popular economic failure prediction models. We will show that a firm filing for corporate insolvency does not necessarily fail to pay off its financial obligations as they mature. Moreover, we will assume an appropriate risk monitoring system centered by well-developed failure prediction models, which is crucial to various parties in the investment world as a means to look after the financial future of their clients or themselves. Keywords Corporate failure r Bankruptcy r Distress r Receivership r Liquidation r Failure prediction r Discriminant analysis r Conditional probability analysis r Hazard model r Misclassification cost model

107.1 Introduction The financial stability of firms is always of concern to many agents in a society, including investors, bankers, governY. Wang Department of Finance, Yuan Ze University, 135 Yuan-Tung Rd., Jung-Li, Taiwan 32003, R.O.C. e-mail: [email protected] L. Lin () Department of Banking and Finance, National Chi-Nan University, 1 University Rd., Puli, Nantou Hsien, Taiwan 545, R.O.C. e-mail: [email protected] H.-C. Kuo Department of Banking and Finance, National Chi-Nan University and Takming University of Science and Technology, 1 University Rd., Puli, Nantou Hsien, Taiwan 545, R.O.C. e-mail: [email protected] J. Piesse Management Centre, School of Social Science and Public Policy, King’s College, University of London, 150 Stamford St., London SE1 9NN, UK University of Stellenbosch, Stellenbosch, Western Cape, RSA e-mail: [email protected]

mental/regulatory bodies, and auditors. The credit rating of listed firms is an important indicator not only to the stock market for investors to adjust the stock portfolio they hold, but also to the capital market for lenders to calculate the costs of loan default and to consider the borrowing terms for their clients. It is also the duty of the governmental and regulatory organizations to monitor the general financial status of firms in order to make proper economic and industrial policies. Moreover, in the interest of the public, the auditors of firms need to maintain a watching brief over the going concern of their clients for the foreseeable future and fairly present their professional view in the audit report attached to each of their client’s financial statements. Thus, it is understandable that the financial stability of firms attracts so much attention from so many different parties. A single firm’s bankruptcy will influence a chain of members of society, especially its debtors and the employees. But, if a group of firms in an economy simultaneously face financial failure, it will not only leave scars on this particular economy, but also its neighbors. The latest evidence is demonstrated by the financial storm clouds gathered over Thailand in July 1997, which caused immediate damage to most Asia–Pacific countries. For these reasons, the development of bankruptcy theory and bankruptcy prediction models, which can protect the market from unnecessary losses, is essential. This can also help governmental organizations make appropriate policies in time to maintain industrial cohesion and minimize the damage caused by widespread corporate bankruptcy to the economy as a whole. As a popular definition of “distress,” the financial terms “near bankruptcy or corporate failure,” “bankruptcy” (see Brealey et al. 2001: 621; Ross et al. 2002: 858), or “financial failure,” can be referred to in a variety of circumstances, including: 1. the market value of assets of the firm is less than its total liabilities; 2. the firm is unable to pay debts when they come due; 3. the firm continues trading under court protection.

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_107, 

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Of these, the second condition; that is, the insolvency of the company, has been of a main concern in the majority of early bankruptcy studies in literature, because insolvency cannot only be explicitly identified but has also served as a legal and normative definition of the term “bankruptcy” in many developed countries. However, the first definition is more complicated and subjective in the light that different accounting treatments of asset valuation usually give a different market value of a company’s assets. Meanwhile, the court protection legislation varies among countries.

107.2 The Possible Causes of Bankruptcy Insolvency problems can result from decisions taken within the company or a change in the economic environment, essentially outside the firm. Some general causes of insolvency are noted in Table 107.1. Apart from those listed above, a new company is usually thought to be riskier than those with longer history. Blum (1974: 7) confirmed that “other things being equal, younger firms are more likely to fail than older firms.” Hudson (1987), studying a sample between 1978 and 1981, also pointed out that companies liquidated through a procedure of creditor’s voluntary liquidation or compulsory liquidation during that period of time were mainly 2–4 years old and three-quarters of them less than 10 years old. Moreover, Walker (1992: 9) also found that “many new companies fail within the first three years of their existence.” This evidence suggests that the distribution of the failure likelihood against company age is skewed to the right. However, a clear cut point in age structure has so far not been identified to distinguish “new” from “young” firms in a business context, nor is there any convincing evidence with respect to the propensity to fail by firms of different ages. In consequence, the age characteristics of liquidated companies can only be treated as an “observation” or “suggestion,” rather than a “theory.”

Table 107.1 Some possible causes of insolvency (Rees 1990: 394) 1: Low and declining real profitability 2: Inappropriate diversification – into unfamiliar industries or not away from declining ones 3: Import penetration into the firm’s home markets 4: Deteriorating financial structures 5: Difficulties controlling new or geographically dispersed operations 6: Overtrading in relation to the capital base 7: Inadequate financial control over contracts 8: Inadequate control over working capital 9: Failure to eliminate actual or potential loss-making activities 10: Adverse changes in contractual arrangements

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Although the most common causes of bankruptcy have been noted above, they are not sufficient to explain or predict corporate failure. In other words, a company with any of these characteristics is not doomed to bankruptcy in a predictable period of time. This is because some factors such as government intervention may play an important part in the rescue of distressed firms. Therefore, as Bulow and Shoven (1978) noted, the conditions under which a firm goes through liquidation are rather complicated. On the whole, as Foster (1986: 535) put it, “there need not be a one-toone correspondence between the non-distressed/distressed categories and the non-bankrupt/bankrupt categories.” It is noticeable that this ambiguity is even more severe in the notfor-profit sector of the economy.

107.3 The Methods of Bankruptcy As the corporate failure is not only an issue for those people involved as company owners and creditors but also influences the economy as a whole, many countries legislate for formal bankruptcy procedures for the protection of public interests from avoidable bankruptcy, such as Chaps. 10 and 14 in the US, and the Insolvency Act in the UK. The objectives of legislation are to “[firstly] protect the rights of creditors. . . [secondly] provide time for the distressed business to improve its situation. . . [and finally] provide for the orderly liquidation of assets” (Pastena and Ruland 1986: 289). In the UK where a strong rescue culture prevails, the Insolvency Act contains six separate procedures that can be applied to different circumstances to prevent either creditors, shareholders, or the firm as a whole from unnecessary loss, thereby reducing the degree of individual as well as social loss. They will be briefly described in the following section.

107.3.1 Company Voluntary Arrangements A voluntary arrangement is usually submitted by the directors of the firm to an insolvency practitioner, “who is authorized by a recognized professional body or by the Secretary of State” (Rees 1990: 394) when urgent liquidity problems have been identified. The company in distress then goes through the financial position in detail with the practitioner and discusses the practicability of a proposal for corporate restructuring. If the practitioner endorses the proposal, it will be put to the company’s creditors in the creditors’ meeting, requiring an approval rate of 75% of those attending. If this restructuring report is accepted, those notified will thus be bound by this agreement and the practitioner becomes the supervisor of the agreement. It is worth emphasizing that a

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voluntary arrangement need not pay all the creditors in full but a proportion of their lending (30% in a typical voluntary agreement in the UK) on a regular basis for the following several months. The advantage of this procedure is that it is normally much cheaper than formal liquidation proceedings and the creditors usually receive a better return.

holders and other preferential creditors by selling the assets of the businesses at the best prices. The whole business may be sold as a going concern if it is worthy. As in an administration order, the receiver must advise creditors of any progress by way of a creditors’ meeting, which will be convened in a short period of time after the initial appointment.

107.3.2 Administration Order

107.3.4 Creditors’ Voluntary Liquidation

It is usually the directors of the insolvent firm who petition the court for an administration order. The court will then assign an administrator who will be in charge of the daily affairs of the firm. However, before an administrator is appointed, the company must convince the court that the making of an order is crucial to company survival or a better realization of company assets than would be the case if the firm was declared bankrupt. Once it is rationalized, the claims of all creditors are effectively frozen. The administrator will then submit recovery proposals to the creditors’ meeting for approval within 3 months of the appointment being made. If this proposal is accepted, the administrator will then take the necessary steps to put it into practice. An administration order can be seen as the UK’s version of the American Chap. 11 in terms of the provision of a temporary legal shelter for companies in trouble to escape future failure without damaging their capacity to continuously trade (Counsell 1989). This does sometimes lead to insolvency avoidance (Homan 1989).

In a creditor’s voluntary liquidation, the directors of the company will take the initiative to send an insolvency practitioner an instruction that will lead to the convening of creditors’ and shareholders’ meetings. In a shareholders’ meeting, a liquidator will be appointed and this appointment will be asked for ratification in a creditors’ meeting later. Creditors have the right of final say as to who acts as liquidator. A liquidator will start to find potential purchasers and realize the assets of the insolvent firm to clear its debts. Unlike receivers who have wide ranging powers in the management of the businesses, a liquidator’s ability to continue trading is restricted to such that a beneficial winding up will be promised. As Rees (1990) stated, it is the most common method used to terminate a company.

107.3.3 Administrative Receivership An administration receiver has very similar power and functions of an administrator but is appointed by the debenture holder (the bank) secured by a floating or fixed charge after the directors of the insolvent company see no prospect for improving its ability to repay debts. In some cases, before the appointment of an administration receiver, a group of investigating accountants will be empowered to examine the real state of the company. The investigation normally includes the estimation of the valuable assets and liabilities of the company. If this investigation team finds that the company has no other choices but to be liquidated, an administration receiver who works in partnership with the investigation team will thus be entitled to take over the management of the company. The principal aim is to raise money to pay debenture

107.3.5 Members’ Voluntary Liquidation The procedure for a member’s voluntary liquidation is rather similar to that of a creditors’ voluntary liquidation. The only difference is that in a members’ voluntary liquidation the directors of the firm must swear a declaration of solvency to clear debts with fair interest within 12 months and creditors should not be involved in the appointment of a liquidator. In other words, a company’s announcement of a members’ voluntary liquidation by no means signals its insolvency, but only means a shutdown along with the diminishing necessity of its existence.

107.3.6 Compulsory Liquidation A compulsory liquidation is ordered by the court to wind up a company directly. This order is usually initiated by the directors of the insolvent firm or its major creditors. Other possible petitioners include the Customs and Excise, the Inland

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Revenue, and local government (Hudson 1987: 213). The whole procedure is usually started with a statutory demand made by creditors who wish to initiate a compulsory liquidation. If the firm fails to satisfy their request in certain working days, this failure is sufficient grounds to petition the court to wind up the firm. Once the order is granted, Official Receiver will take control of the company instantly or a liquidator will be appointed by Official Receiver instead. The company then must cease trading and liquidation procedure starts. However, an interesting phenomenon is that many valuable assets may be removed or sold prior to the control of the liquidator or even during the delivery of the petition to the court. Therefore, it probably leaves nothing valuable for the liquidator to deal with. In this sense, the company initiating a compulsory liquidation has been practically terminated far before a court order is granted.

107.4 Prediction Model for Corporate Failure Because corporate failure is not simply a closure of a company but has impacts on the society and economy where it occurs. Therefore, it makes good business and academic sense to model corporate failure for prediction purposes. If bankruptcy can be predicted properly, it may be avoided and the firm restructured. By doing so, not only the company itself but also employees, creditors, and shareholders can all benefit. Academics have long believed that corporate failure can be predicted through the design of legal procedure (Dahiya and Klapper 2007) or financial ratio analysis. It is because that using financial variables in the form of ratios can control for the systematic effect of size of, and industry effects on, the variables under examination (Lev and Sunder 1979: 187– 188), to facilitate cross-sectional comparisons in attempt to objectively discover the “symptoms” of corporate failure. In addition to the cross-sectional comparisons, Theodossiou (1993) builds up the time series analysis to investigate the discovery of the corporate failure. In consequence, financial ratio analysis for decades is not only preferred when the interpretation of financial accounts is required, but it has also been extensively used as inputs to the explicit formulation of corporate bankruptcy prediction models. Other than the financial ratio analysis, the application of the Merton (1974) Model (e.g., the KMV Model) that is widely applied by practitioners also predicts corporate failure. Differing from the financial ratio analysis, the Merton (1974) Model examines the possibility of bankruptcy with the stock market valuation using the concept of “distance to default,” and the possibility of bankruptcy is negatively related to the distance to default, which measures how far the firm valuation is from the critical value of default in the firm value distribution.

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107.4.1 Financial Ratio Analysis and Discriminant Analysis The use of ratio analysis for the purpose of predicting corporate failure at least can be dated back to Fitzpatrick (1932), but this method did not attract enough attention until Beaver (1966) proposed his famous univariate studies. Beaver (1966) systematically categorized 30 popular ratios into six groups, and then found that some ratios under investigation, such as the cash flow/total debt ratio, demonstrated excellent predictive power in the corporate failure model. His results also showed the deterioration of the distressed firms prior to their financial failure, including a sharp drop in their net income, cash flow, and working capital, coupled with an increase in total debt. Although useful in predicting bankruptcy, univariate analysis was later criticized for its failure to incorporate more causes of corporate failure measured by other ratios into a single model. This criticism prompted interest in the multivariate distress prediction model that simultaneously includes different financial ratios with better combined predictive power for corporate failure. In the 1980s, with the increasing attention placed on multiratio analysis, multivariate discriminant analysis (MDA) began to dominate the bankruptcy prediction literature. MDA determines the discriminant coefficient of each of the characteristics chosen in the model on the basis that these will discriminate between failed and nonfailed ones in an efficient manner. Then, a single score for the firms in study will be generated. A cutoff point will be determined to minimize the dispersion of scores associated with firms in each category and the chance of overlap. An intuitive advantage of using MDA techniques is that the entire profile of the characteristics under investigation and their interaction are considered. Another advantage of using MDA lies in its convenience in application and interpretation (Altman 1983: 102–103). One of the most popular MDA must be the Z-score model developed by Altman (1968). On the basis of their popularity and relevancy to corporate failure, 22 carefully selected financial ratios were further classified into five bankruptcyrelated categories. In Altman’s sample 33 bankrupt and 33 nonbankrupt manufacturing companies during the period 1946–1965 were included. The best and final model in terms of failure prediction contained five variables that are still frequently used in banking and business sectors to date. This linear function is Zscore D 1:2Z1 C 1:4Z2 C 3:3Z3 C 0:6Z4 C 0:999Z5 ; (107.1) where Z-score is the overall index; Z1 the working capital/total assets; Z2 the retained earnings/total assets; Z3 the earnings before interest and taxes/total assets; Z4 the market value equity/book value of total debt; and Z5 the sales/total assets.

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Altman (1968) also tested the cutoff point to balance the Type I and Type II Error, and found that in general it was highly possible for a company with its Z-score smaller than 1.8 to go bankrupt in the next few years of the examining period, and one with its Z-score higher than 2.99 was comparatively much safer. This Z-score model is so well known that it is still one favorite indicator of credit risk to fund suppliers in the new millennium. Although these statistical discrimination techniques are popular in predicting bankruptcy, they suffer from some methodological problems. Some methodological problems identified stem directly from the use of financial ratios (also see Agarwal and Taffler 2008). For example, proportionality and zero-intercept assumptions are the main two factors that are crucial to the credibility of the ratio analysis. The proportionality and zero-intercept assumption determines the form of any ratio. The basic form of a ratio is usually assumed to be y=x D c, where y and x are two accounting variables, which are assumed to be different but linearly related, and c can be interpreted as the value of this specific ratio. This raises three questions. First, is there an error term in the relationship between the two accounting variables? Second, is it possible that an intercept term exists in this relationship? And finally, what if the numerator and denominator are not linearly related? With regard to the first question, Lev and Sunder (1979) proved that if there is an additive error term in the relationship between y and x suggested by the underlying theory, that is, y D ˇx C e or y=x D ˇ C e=x, the comparability of such ratios will be limited. It is because “the extent of deviation from perfect size control depends on the properties of the error term and its relation to the size variable, x” (Lev and Sunder 1979: 191). The logic behind their argument can be summarized as follows. Considering the error term is homoscedastic, e=x is then smaller for the large firms than for the small firms because x as a size variable for large firms will on average be greater than that of small firms. In other words, the ratio y=x for large firms will be closer to the slope term ˇ than that for small firms. Thus, because the variance of the ratio y=x for smaller firms is greater than that of larger firms, it proves that the ratio y=x of two groups (i.e., large and small firms) are statistically drawn from two different distributions. This certainly weakens the comparability of the ratios with such an underlying relationship. It seems that to place an additive error term in the relationship between denominator and numerator in a ratio is not adequate in terms of size control. However, if heteroscedasticity in y is the case, it may result in the homoscedasticity of y=x. But it is also possible that this heteroscedastic problem of y=x is unchanged. In fact, Lev and Sunder (1979) also pointed out that this problem may be ameliorated only when the error term is

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multiplicative in the relationship; that is, y D ˇxe or y=x D ˇe. It is because the deviation of y=x now has no mathematical relationship with size variable x. As a result, this format of the ratio is more appropriate for comparison purpose. The above discussion can also be applied to the case where an intercept term exists in the relationship between two ratio variables, represented by y D ˛ C ˇx or y=x D ˇ C ˛=x. It is obvious that the variance of ratio y=x for smaller firms will be relatively larger than that of larger firms, under the influence of the term ˛=x. Again, suffice to say that the underlying formation of ratio is not acceptable to be used in the comparison of company performance. If two variables are needed to control the market size of the variable y, such as y D ˛CˇxCız or y D ˛CˇxCıx 2 , and if the underlying relationship is nonlinear, considerable confusion will be caused in the interpretation of ratios, not to mention the results of the ratio analysis. All those problems cast doubts on whether the normally used form of ratios is appropriate in all circumstances. Theoretically, it seems that the use of ratios is less problematic if and only if highly restrictive assumptions can be satisfied. Empirically, Whittington (1980) claimed that the violation of proportionality assumption of the ratio form is the problem researchers will most frequently encounter in the use of financial data in practice, especially in a time series study of an individual firm. McDonald and Morris (1984: 96) found that the proportionality assumption is better satisfied when a group of firms in a simple homogeneous industry was analyzed, otherwise, some amendment of the form of ratios will be necessary. However, they do not suggest the replacement of the basic form of ratio with a more sophisticated one. On the contrary, they commented that, on average, the basic form of ratio empirically performed quite satisfactorily in its application to ratio analysis. Keasey and Watson (1991: 90) also suggested that possible violations of the proportionality assumptions can be ignored. Since then, due to lacking further theoretical developments or the improvement of ratio forms, the basic form of ratio is still prevailingly used in the bankruptcy studies. In addition to the flaws in the designs of financial ratios, there are some methodological problems associated with the use of MDA. Of them, non-normality, inequality of dispersion matrices across all groups, and nonrandom sampling are the three problems that haunt the model users. The violation of normality assumption of MDA has given rise to considerable discussion since the 1970s (Kshirsagar 1971; Deakin 1976; Eisenbeis 1977; Amemiya 1981; Frecka and Hopwood 1983; Zavgren 1985; Karels and Prakash 1987; Balcaena and Ooghe 2006). Violation of the normality is the ultimate cause of biased tests of significance and estimated error rates. Studies on univariate normality of financial ratios found that ratio distributions tend to be

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skewed (Deakin 1976; Frecka and Hopwood 1983; Karels and Prakash 1987). If the ratios included in the model are not perfectly univariate normal, their joint distribution will, a priori, not be multivariate normal (Karels and Prakash 1987). Therefore, a good variable set for bankruptcy modeling should be able to minimize multivariate non-normality problems. A traditional but prevalent stepwise procedure apparently cannot satisfy this requirement. However, despite quite a few complementary studies on data transformation and outlier removal for ratio normality (Eisenbeis 1977; Ezzamel et al. 1987; Frecka and Hopwood 1983), their suggestion is rarely used in later research on the generation program of MDA models (Shailer 1989: 57). Because all these techniques are imperfect, McLeay (1986) advocated that selecting a better model is more straightforward than the removal of outliers or data transformations. In comparing the problems of non-normality to inequality of dispersion matrices across all groups, the latter does not seem crucial to the estimation procedure of MDA. In theory, the violation of the equal dispersion assumption will affect the appropriate form of the discriminating function. After testing the relationship between the inequality of dispersions and the efficiency of the various forms of classification models, a quadratic classification rule seems to outperform a linear one in terms of the overall probability of misclassification when the variance–covariance matrices of the mutually exclusive populations are not identical (Eisenbeis and Avery 1972; Marks and Dunn 1974; Eisenbeis 1977). More importantly, the larger the difference between dispersions across groups, the more the quadratic form of discriminating function is recommended. One of the strict MDA assumptions is random sampling. However, the sampling method that MDA users prefer for bankruptcy prediction studies is choice-based, or state-based, sampling, which prepares an equal or approximately equal draw of observations from each population group. Because corporate failure is not a frequent occurrence in an economy (Altman et al. 1977; Wood and Piesse 1988), such sampling techniques will cause a relatively lower probability of misclassifying distressed firms as nondistressed firms (Type I Error) but a higher rate of misclassifying nondistressed firms as distressed firm (Type II Error) (Lin and Piesse 2004; Kuo et al. 2002; Palepu 1986; Zmijewski 1984). Therefore, the high predictive power of MDA models claimed by many authors appears to be suspect. As Zavgren (1985: 20) commented, MDA models are “difficult to assess because they play fast and loose with the assumptions of discriminant analysis.” When doubt is cast on the validity of the results of MDA models, scholars begin to look at more defensible approaches, such as conditional probability analysis (CPA). The change in these mainstreams is also reviewed by Balcaena and Ooghe (2006).

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107.4.2 Conditional Probability Analysis Since the late 1970s, the use of discriminant analysis was gradually replaced by the use of CPA, which is different from MDA in that CPA produces the “probability of occurrence of a result, rather than producing a dichotomous analysis of fail/survive as is the norm with basic discriminant techniques” (Rees 1990: 418). CPA primarily refers to logit and probit techniques and has been discussed or used rather widely in academics (Keasey and Watson 1987; Martin 1977; Mensah 1983; Ohlson 1980; Peel and Peel 1987; Storey et al. 1987; Zavgren 1985, 1988; Sun 2007). Its popularity highlights the various responses of the model users to the risk of failure. But the main value of using CPA is that CPA application does not depend on demanding assumptions as does MDA (Kennedy 1991, 1992). However, logit CPA is not always better than MDA under all conditions. If the multivariate normality assumption is met, the maximum likelihood estimator (MLE) of MDA is more efficient asymptotically than MLE of logit models. In any other circumstance, the MLE of MDA may not remain consistent, but MLE of logit models will (Amemiya 1981; Judge et al. 1985; Lo 1986). However, as a rejection of normality in bankruptcy literature is very common, the logit model is appealing. Methodologically speaking, suffice to say that it seldom goes wrong if logit analysis is used in distress classification. One most commonly cited and used CPA was developed by Ohlson 1980). Different from Altman’s (1968) sample with an equal number of bankrupts and nonbankrupts, his sample included 105 bankrupt and 2058 nonbankrupt industrial companies during 1970–1976. Using logistic analysis, his failure prediction models with accuracy rate of above 92% contained nine financial ratios and suggested that the company size, capital structure, return on assets, and current liquidity were the four most powerful bankruptcy indicators. His model looked like Y D 1:3  0:4Y1 C 6:0Y2  1:4Y3 C 0:1Y4  2:4Y5 1:8Y6 C 0:3Y7  1:7Y8  0:5Y9 ;

(107.2)

where Y is the overall index; Y1 the log(total assets/GNP price-level index); Y2 the total liabilities/total assets; Y3 the working capital/total assets; Y4 the current liabilities/current assets; Y5 the one if total liabilities exceed total assets, zero otherwise; Y6 the net income/total assets; Y7 the funds provided by operations/total liabilities; Y8 the one if net income was negative for the last 2 years, zero otherwise; and Y9 the measure of change in net income. It is interesting to find that Ohlson (1980) chose 0.5 as cutoff point, implicitly assuming a symmetric loss function across the two types of classification error. He later ex-

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plained that the best cutoff point should be calculated using data beyond his examining period (i.e., 1976), but unnecessary in his paper because the econometric characteristics of CPA and his large sample size would neutralize the problems (Ohlson 1980: 126). He also pointed out the difficulty of comparing his results with others due to the differences in the design of lead time, selection of predictors and observation periods, and finally the sensitivity of these results to choice of estimation procedures. As far as the predictive accuracy rates of MDA and CPA are concerned, Ohlson 1980) found that the overall results of logit models were not an obvious improvement on those of MDA. Hamer (1983) tested the predictive power of MDA and logit CPA, and then concluded that the both performed comparably in the prediction of business failure for a given variable set. However, with the knowledge that the predictive accuracy rates were overstated in previous MDA papers mainly due to the employment of choice-based sampling technique, the above comparison may be biased and the inferences from them could favor CPA. Apart from this, there do exist some factors that vary among previous papers and may erode such comparisons, including differences in the selection of predictors, the firm matching criteria, the lead time, the estimation and test time periods, and the research methodologies. Unless these factors are deliberately controlled, any report about the comparisons between CPA and MDA in terms of the predictive ability will not be robust. In conclusion, not only can CPA provide what any other technique can do in terms of user and ease of interpretation, but more importantly, it has no strict assumptions in the way MDA does, from which biased results primarily stem (Keasey and Watson 1991: 91). Even more recently, the dynamics of firm-specific and macroeconomic covariates can be captured by the new CPA modeling proposed by Duffie et al. (2007). With these appealing advantages, CPA is believed to be superior and preferred to MDA for bankruptcy classification.

107.4.3 Three CPA Models: LP, PM, and LM

where y is a dichotomous dummy variable that takes the value of 1 if the event occurs and 0 if it does not, and Pr./ represents the probability of this event. F ./ is a function of a regressor vector x coupled with a vector ˇ of parameters to govern the behavior of x on the probability. The problem arises as to what distribution model best fits the above equation. Derived from three different distributions, LP, PM, and LM are then chosen to fit into the right-hand side of the equation. LP is a linear regression model, which is easy to use but has two main problems as far as its application to generate the probability of an outcome is concerned. The first problem is the heteroscedastic nature of the error term. Recall the form of an ordinary LP, Y D X 0 ˇ C ", where Y is the probability of an outcome and X is a column of independent variables, ˇ is the parameter vector, and " is the error term. When an event occurs, Y D 1, " D 1X 0 ˇ, but when it does not occur, Y D 0, " D .X 0 ˇ/. It is noticeable that this error term is not normally distributed, so feasible general least squares estimation procedure should be used to correct this heteroscedasticity problem (Greene 1997: 87). The second problem, which may be the real difficulty with the LP model, is that LP cannot constrain Y to lie between 0 and 1 as a probability should. Amemiya (1981: 1486) then suggested the condition that Y D 1 if Y > 1 and Y D 0 if Y < 0. But this may produce unrealistic and nonsensical results. Therefore, it is not surprising that LP is used less frequently now. Hence, it will not be employed in this study either. In the discussion of qualitative response models, academics seem to be more interested in the comparisons between logit and probit models. Although logit models are derived from logistic density and probit models generated from Normal density, these two distributions are almost identical except that the logistic distribution has thicker tails and a higher peak in the middle (Cramer 1991: 15). In other words, the probability at each tail and in the middle of logistic distribution curve will be larger than that of the Normal distribution. However, one of the advantages of using the logit model is its computational simplicity. A look at the formula of these two models listed below will help one to appreciate this merit: Z

There are three commonly cited CPA models: the linear probability model (LP), the probit model (PM), and the logit model (LM). For CPA estimates the probability of the occurrence of a result in study, the general form of a CPA equation can be easily set as Pr.y D 1/ D F .x; ˇ/; Pr.y D 0/ D 1  F .x; ˇ/;

Probit Model W

1

1 2 p e t =2 dt 2

D ˆ.ˇ 0 x/; Logit Model W

Prob.Y D 1/ D D

(107.3)

ˇ0 x

Prob.Y D 1/ D

(107.4)

exp.ˇ 0 x/ 1 C exp.ˇ 0 x/ 1 ; 1 C exp.ˇ 0 x/ (107.5)

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where the function ˆ./ is the standard normal distribution. This mathematical convenience of logit models is one of the reasons for its popularity in practice (Greene 1997: 874). As far as the classification accuracy of CPA models is concerned, some comparisons of the results produced from these two models have been made and, as discussed above, generally suggest that they are actually indistinguishable in cases where the data are not heavily concentrated in the tails or the middle (Amemiya 1981; Cramer 1991; Greene 1997). This finding is consistent with the difference in the shape of the two distributions from which PM and LM are derived. It is alsopproven that the coefficients of the logit model are about = 3  1:8 times as large as those of the probit model, implying that the slopes of each variable in both models are very similar. In other words, “the logit and probit models results are nearly identical” (Greene 1997: 878). Choice of sampling methods is also a validity factor of CPA. The prevailing sampling method in the bankruptcy literature is to draw a sample with an approximately equal number of bankrupts and nonbankrupts, usually referring to the state-based sampling technique, as an alternative of random sampling. Although the econometric estimation procedure usually assumes random sampling, the use of state-based sampling has an intuitive appeal. As far as bankruptcy classification models are concerned, corporate failure is an event with rather low probability. Hence, a random sampling method may result in the inclusion of a very small percentage of bankrupts but a very high percentage of nonbankrupts. Such a sample will not be an effective one for any econometric model to produce efficient estimators (Palepu 1986: 6). In contrast, state-based sampling is an “efficient sample design” (Cosslett 1981: 56) that can effectively reduce the required sample size without influencing its provision of efficient estimators if an appropriate model and modification procedure are employed. In short, the information content of a state-based sample for model estimation is relatively better than that of a random sampling. A state-based sample for CPA results in the understatement of the Type I Error but the overstatement of the Type II Error (Palepu 1986; Lin and Piesse 2004). Manski and McFadden (1981) suggested several alternatives that can negate the drawbacks of using statebased sampling. They include the weighted exogenous sampling maximum likelihood estimator (WESMLE) and modified version by Cosslett (1981), the nonclassical maximum likelihood estimator (NMLE), and the conditional maximum likelihood estimator (CMLE). They compare and report these estimation procedures, which can be summarized into the following four points: 1. All these estimators are computationally tractable, consistent, and asymptotically normal. 2. The weighted estimator and conditional estimator avoid the introduction of nuisance parameters.

Y. Wang et al.

3. The nonclassical maximum likelihood estimators are strictly more efficient than the others in large samples. 4. In the presence of computational constraints, WESMLE and CMLE are the best; otherwise, NMLE is the most desirable. Accordingly, by using any one of the above modification methods, the advantages of using state-based sampling technique can be maintained, whereas the disadvantages can be mostly removed. What can also be inferred from this comparison is that the selection of modification method depends on two factors: the sample size and the computational complexity. Of them, CMLE is most frequently referred to in the bankruptcy literature for three reasons. First, this method has been more extensively demonstrated in Cosslett (1981) and Maddala (1983) studies with its application to the logit model. Second, this method has been used in the development of the acquisition prediction model by Palepu’s (1986), merger/insolvency choice model by BarNiv and Hathorn (1997), and bankruptcy classification models by Lin and Piesse (2004). Finally, because CMLE only changes the constant term of the model produced by normal MLE procedures, but has no effects on other parameters, this procedure is relatively simple and straightforward for CPA users. In a word, the merits of using CMLE with CPA include computational simplicity and practical workability. Without biases caused by the choice of sampling methods, modified CPA can almost correct all possible methodological flaws that MDA can have.

107.4.4 Time Series Analysis: CUSUM Model One application regarding the time series analysis on corporate distress is provided by Theodossiou (1993). He explores the idea that the difference between healthy and distressed firms by using the firm’s change in characteristics in time series could help to discover the corporate distress. In the procedure from normal status to the distress, the financial ratios deteriorate over time, which provides the information for early warning. In Theodossiou’s (1993) framework, we may set up Xi;1 ; : : : ; Xi;T be a sequence of important financial variables for firm i . The unconditional mean of Xi;t in the group for healthy companies is stationary over time; i.e., E.Xi;t jh/ D uh . For the group of distressed firms, the mean of Xi;t diverges from uh and gradually moves toward to mean uf which is denoted as the mean of the distressed firms; that is, uh ! uf;s !    ! uf;1 ! uf D uf;0 , in which uf;m  E.Xi;t jf; m/ for m D 0; 1; 2; : : :; s being the mean of Xi;t , m reporting periods before failure. Next, the deviations of a firm’s indicator vectors from their means are expressed as a vector-autoregressive-movingaverage (VARMA) process of order p and q:

107 Detecting Corporate Failure

Xi;t  uh D

p X

1601 q X

ˆk .Xi;t k  uh / C "i;t 

‚s "i;t s ;

sD1

kD1

(107.6) X p

Xi;t  uf;m D

ˆk .Xi;t k  uf;mCk / C "i;t

kD1



q X

m D 0; 1; 2; : : : ; (107.7)

‚s "i;t s ;

sD1

E."i;t / D 0; D D 0 for i ¤ j and/or t ¤ s, i; j D 1; 2; : : :,N and N D Nf C Nh . Based on the time series processes of financial ratios, Theodossiou (1993) shows a cumulative sums (CUSUM) model that conveys the information regarding the deterioration on the distress firm’s financial status: ˙; E."i;t "0j;s /

Ci;t D min.Ci;t 1 C Zi;t  K; 0/ < L;

for K; L > 0; (107.8)

Ci;t and Zi;t are cumulative(dynamic) and an quarterly(static) time series performance score for the i th company at time t. K and L are sensitivity parameters required as positive values. Accordingly, score Zi;t is a function of the attribute vector Xi;t accounting for serial correlation in the data, and is estimated by Zi;t D ˇ0 C ˇi Xi;t 

p X

ˆk Xi;t k C

q X

# ‚s "i;t s ;

sD1

kD1

(107.9) # 0

" p X 1 .uh  uf /  ˇ0   ˆk .uh  uf;k / 2D kD1 # " p X 1 ˙ ˆk .uh C uf;k / ; .uh C uf /  kD1

"

ˇ1  .1=D/ .uh  uf / 

p X

#0 ˆk .uh  uf /

˙ 1 ;

kD1

and " D D .uh  uf /  2

" ˙

1

Pf D prob.Ci;t > Ljdistress firm ands D 1/; Ph D prob.Ci;t  Ljhealthy firm/

E."i;t "0i;t /

"

thus the Ci;t scores are equal to zero. In contrast, any distress firm that has Zi;t lower than K makes the cumulative score of Ci;t negative. In particular, two parameters K and L determine the discovery ability of CUSUM model and relate to the Type I and Type II Errors. Generally, the larger the value of K is, the less likely Type I Error and the more likely Type II Error might occur. The opposite is true with the parameter L:

p X

#0 ˆk .uh  uf /

kD1

.uh  uf / 

p X

# ˆk .uh  uf / :

kD1

Based on the CUSUM model, we are able to measure the overall performance on the firm’s financial distress possibility by the cumulative score Ci;t . If one company is typically a healthy firm, then Zi;t scores are positive and greater than K,

(107.10)

are, respectively, the percentages of distress and healthy country in the population not classified accurately by the prediction of CUSUM model, which also termed as Type I and Type II Errors. In means of the determination of the optimal values of K and L, we solve the dynamic optimization problem: Min D EC D wf Pf .K; L/ C .1  wf /Ph .K; L/; K;L

(107.11) where wf and wh D 1  wf are investor’s specific weights attached to the probabilities Pf and Ph . EC is the expected error rate.

107.4.5 Merton Model In the Merton (1974) Model, the probability of a firm going into default is measured by the stock market valuation on firm value. Conceptually, the firm value (on the market valuation basis) less than the firm’s obligation falls into distress. Yet, the market valuation on a firm (equal to sum of debt and equity) is not provided directly; only the stock market valuation on equity is available. Merton (1974) attempts to estimate the mean and standard deviation of the firm value indirectly through the option formula. One simple way refers to the Black and Scholes (1973) option formula, the mean and volatility of the firm value could be estimated from the option model. The measure of “distance to default” serves as the proxy for the possibility of corporate financial distress where distance to default is defined as the distance from the actual firm value to the critical value of the distress firm, given the confident level of Type I Error in determining the firm’s default. In Fig. 107.1, the distance to default is expressed as the DD. More precisely, the market valuation and standard deviation of the firm equity could be obtained from the Merton (1974) Model as VE D VA N.d1 /  De  r N.d2 /; E D

N.d1 /VA A ; VE

(107.12)

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Fig. 107.1 Firm value distribution and distance to default. Note. This figure is from Fig. 8.1 in Crosbie and Bohn (2002): (1) the current asset value; (2) the distribution of the asset value at time H ; (3) the volatility of the future assets value at time H ; (4) the level of the default point, the book value of the liabilities; (5) the expected rate of growth in the asset value over the horizon; and (6) the length of the horizon, H

where d1 D

ln.VA =D/ C .r C .1=2/A2 / p ; A 

p d2 D d1  A ;

VE is the market value of the firm’s equity, E is the standard deviation of the equity, VA is the market valuation on firm value, A is the standard deviation of the firm value, r is risk-free rate,  is time interval between the current date and the option expiration date, D is the total liability, N is the cumulative density function for Normal. From Equation (107.9) and (107.10), we solve unknown parameters VA and A . Based on the estimates for VA and A , the distance to default is DD D

ln.VA =D/ C .  A2 =2/ p ; A 

the Merton-based KMV Model with a naïve alternative. Let market value of each firm’s debt equal to the face value of debt; that is, naïve D D F: Then, Bharath and Shumway (2008) approximate the volatility of each firm’s debt by following naïve equation: naïve D D 0:05 C 0:25E : The 5% is a naïve estimation on term structure volatility, and 25% multiplying the equity volatility, which represents the tendency of default risk. Bharath and Shumway (2008) use this simple approximation and estimate the total volatility of the firm:

(107.13)

in which  is the growth rate of assets and DD is the distance to default. Generally, the smaller DD, the higher likelihood this firm will meet the default in near future. One vital question in the estimation of the distance to default is how to compute the standard deviation of the return on equity price. One way is to estimate the standard deviation by daily returns with some autocorrelation adjustments. In the real world, KMV Corporation applies Merton (1974) to build up the financial distress forecast model that is usually termed as KMV Model. In the estimation of equity return standard deviation, KMV Corporation adopts estimations on VA and A other than the option approach. By and large, the different method of measuring standard deviation of equity return predicts different financial distress model. To model the equity return distribution (particularly the standard deviation), Bharath and Shumway (2008) compare

naïve A D

VE naïve D E C naïve D VE C naïve D VE C naïve D D

VE F E C .0:05 C 0:25E /: VE C F VE C F (107.14)

Next, they set up the expected return process by the firm’s stock return in previous year: naïve  D ri t 1 : Finally, the naïve distance to default is naïve DD D

lnŒ.VE C F /=F C .ri t 1  0:5naïve A2 / p : naïve A  (107.15)

107 Detecting Corporate Failure

1603

Under this naïve DD, Bharath and Shumway (2008) argue that this alternative is slightly better than the traditional Merton-based KMV Model.

107.5 The Selection of Optimal Cutoff Point The final issue with respect to the accuracy rate of a bankruptcy classification model is the selection of an optimal cutoff point, especially for the scoring model. As Palepu (1986) noted, traditionally the cutoff point determined in most early papers was an arbitrary cutoff probability, usually 0.5. This choice may be intuitive, but lacks theoretical backing. Joy and Tollefson (1975), Altman and Eisenbeis (1978), and Altman et al. (1977) developed an equation to calculate the optimal cutoff point in their ZETA model. Two elements in the calculation of the optimal cutoff point can be identified as (1) the costs of Type I and Type II Errors and (2) the prior probability of failure and survival. These two essentials had been ignored in most previous studies. However, Kuo et al. (2002) adopted the points by use of Fuzzy theory to enhance credit decision model. Although their efforts were such a big breakthrough, there were still several unsolved problems. The first problem is the subjectivity in deciding the costs of Type I and Type II Error. Altman et al. (1977: 46) claimed that bank loan decisions will be approximately 35 times more costly for Type I Errors than for Type II Errors. This figure certainly cannot be applied to other decision models used by different parties such as investors and in different periods of time. In this case, such an investigation on costs of Type I and Type II Errors may be a one-off case. With constrains in reality, the subjectivity of selecting convenient cutoff figures in academic studies seems inevitable. The second problem is the subjectivity of selecting a prior bankruptcy probability. Wood and Piesse (1988) criticized Altman et al. (1977) for choosing a 2% higher failure rate than the average annual failure rate of 0.5%, suggesting spurious results from Altman et al. and necessitating a correction that was taken up in the later research. The final problem is that the optimal cutoff score produced may not be “optimal” in the light of the violation of assumptions with respect to multinormality and equal dispersion matrices (Altman et al. (1977: 43, footnote 17), which is apparently a common methodological problem in this data analysis. The optimal cutoff equation in Maddala (1983: 80) seems to be less problematic. It firstly develops the overall misclassification cost model as Z Z f1 .x/dx C C2 P2 f2 .x/dx; (107.16) C D C1 P1 G2

G1

where C is the total cost of misclassification; C1 the cost of misclassifying a failed firm into nonfailed one (Type I Error); C2 the cost of misclassifying a nonfailed firm into failed one (Type II Error); P1 the proportion of the failed firms to the total population firms; P2 the proportion of the nonfailed firms to the total population firms; G1 the failed firm group; G2 the nonfailed firm group; x a vector of characteristics x D .x1 ; x2 ; : : :; xk /; f1 .x/ the joint distribution of the characteristics x in failed firm group; f2 .x/ the joint distribution of x in nonfailed firm group; and P1 C P2 D 1. However, Z

Z

*

f1 .x/dx C G2

f1 .x/dx D 1:

(107.17)

G1

From Equations (107.3) and (107.4), we then have Z ) C D C1 P1 1 

Z f1 .x/dx C C2 P2 G1

Z D C1 P1 C

f2 .x/dx G1

ŒC2 P2 f2 .x/  C1 P1 f1 .x/ dx: G1

(107.18) To minimize the total cost of misclassification, min C , we have to let C2 P2 f2 .x/  C1 P1 f1 .x/  0 or

f1 .x/ C2 P2 : f2 .x/ C1 P1

(107.19)

(107.20)

Assume the expected costs of Type I Error and Type II Error are the same, C2 P2 D C1 P1 , the condition to minimize the total misclassification cost will be f1 .x/ 1: f2 .x/

(107.21)

This result is consistent with the one proposed by Palepu (1986) under the assumption of equal costs of Type I and II Errors. Therefore, the optimal cutoff point should be the probability value where the two conditional marginal densities, f1 .x/ and f2 .x/, are equal. In this equation, what can be found is that there is no need to use the prior failure rate to calculate the optimal cutoff point, but instead, the ex post failure rate (i.e., sample failure rate). Palepu (1986) more clearly illustrates this convenience by the application of Bayes’ formula. Instead of using the costs of Type I and Type II Errors, the expected costs of these errors are still required in this formula. Unfortunately, the subjectivity of deciding the relationship of these two types of expected costs still cannot be removed. There is no theory suggesting they shall be the

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same; that is, C2 P2 D C1 P1 . However, compared to the traditionally arbitrary 50% cutoff point, this assumption is more neutral and acceptable in an economic sense. This application of this procedure for the determination of bankruptcy cutoff probability can be found in Palepu (1986) and Lin and Piesse (2004).

107.6 Recent Development While MDA and CPA are classified as static analyses, dynamic modeling is becoming the mainstream in bankruptcy literature as other academic schemes. Shumway (2001) criticized static bankruptcy models for their observation of only the status of each bankrupt company 1 year prior to their failures and their ignorance of firms’ changes from year to year. In contrast, Shumway (2001) proposed a simple hazard dynamic model, which contained a survivor function and a hazard function to determine a firm’s failure possibility at each point in time. Given the infrequency of corporate failure in reality, hazard model users will avoid the small sample problem because this dynamic model requires all available time series information of firms. As this hazard model takes the duration dependence, time-varying covariates, and data sufficiency problems into consideration, it is in methodology superior to the models of MDA and CPA families. More empirical evidence is needed to support its excellency in prediction power. Studies following similar concepts can also be found in Whalen (1991) and Helwege (1996).

107.7 Conclusion In all the frequently used methods of bankruptcy, it is clear that the reasons a firm may file for corporate insolvency does not necessarily include the inability to pay off its financial obligations when they mature. For example, a solvent company can also be wound up through members’ voluntary liquidation procedure to maximize the shareholders’ wealth when the realized value of its assets exceeds its present value in use. Bulow and Shoven (1978) modeled the potential conflicts among the various claimants to the assets and income flows of the company (e.g., bondholders, bank lenders, and equity holder) and found that a liquidation decision should be made when “the coalition of claimants with negotiating power can gain from immediate liquidation” (Bulow and Shoven 1978: 454). Their model also considered the existence of some asymmetrical claimants of the firm. These results confirmed the complicated nature of the bankruptcy decision and justify the adoption of members’ voluntary liquidation procedure to determine a company’s value (see Brealey et al. 2001: 622; Ross et al. 2002: 857).

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As to the development of failure prediction models, new models are methodologically superior, but the increase of their prediction power does not seem to trade off the increase of their modeling complexity, which casts doubts to their real values in practice. In addition, the costs of bankruptcy vary with different bankruptcy codes in different countries (see Brealey et al. 2001: 439–443; Ross et al. 2002: 426). It implies that bankruptcy prediction models with universally applicable factors and cutoff probability does not exist. Acknowledgments We would like to express our heartfelt thanks to Professor C. F. Lee who provided valuable suggestions in structuring this article. We also owe thanks to many friends at University of London (UK) and National Chi Nan University (Taiwan) for valuable comments. We also want to thank our research assistant Chiu-Mei Huang for preparing the manuscript and cheerfully proofreading several drafts of the manuscript. Last, but not least, special thanks go to the Executive Editorial Board of the Handbook of Quantitative Finance and Risk Management at Kluwer Academic Publishers, who encouraged us to write the article, expertly managed the development process, and superbly turned the final manuscript into a finished product.

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Chapter 108

Genetic Programming for Option Pricing N.K. Chidambaran

Abstract This chapter describes the Genetic Programming methodology and illustrates its application for the pricing of options. I describe the various critical elements of a Genetic Program – population size, the complexity of individual formulas in a population, and the fitness and selection criterion. As an example, I implement the Genetic Programming methodology for developing an option pricing model. Using Monte Carlo simulations, I generate a data set of stock prices that follow a Geometric Brownian motion and use the Black–Scholes model to price options off the simulated prices. The Black–Scholes model is a known solution and serves as the benchmark for measuring the accuracy of the Genetic Program. The Genetic Program developed for pricing options well captures the relationship between option prices, the terms of the option contract, and properties of the underlying stock price. Keywords Genetic programming r Option pricing r Non– parametric r Monte-Carlo simulation

108.1 Introduction Genetic Programming is a powerful computational tool that has seen widespread applications – from the determination of gene sequencing to searching for technical trading rules in financial futures markets. This chapter describes Genetic Programming and illustrates its application for developing an option pricing model. Genetic Programming is a non-parametric data-driven methodology. Genetic Programming requires minimal assumptions to implement and easily adapts to changing and uncertain economic environments.

N.K. Chidambaran () Graduate School of Business, Fordham University, 1790 Broadway, New York, NY 10023, USA e-mail: [email protected]

Theoretical option pricing models based on risk-neutral pricing theory, such as the seminal Black–Scholes model, rely on strict assumptions that do not hold in the real world. The Black and Scholes (1973) model has been shown, for example, to exhibit systematic biases from observed option prices (Rubinstein 1985; Macbeth and Merville 1979; Macbeth and Merville 1980) and researchers have attempted to explain the systematic biases as an artifact of its assumptions. The most often challenged assumption is the normality of stock returns. Merton (1976) and Ball and Torous (1985) propose a Poisson jump-diffusion returns processes. French et al. (1987) and Ballie and DeGennaro (1990) advocate GARCH (Bollerslev 1986) processes. While closed-form solutions for the option price cannot be obtained for all these models, pricing formulas can be obtained numerically. The difficulty in finding an analytical closed-form parametric solution has led to non-parametric approaches. Rubinstein (1997) suggests that we examine option data for the implied binomial tree to be used for pricing options. Chidambaran and Figlewski (1995) use a quasi-analytic approximation based on Monte Carlo simulation. Hutchinson et al. (1994) build a numerical pricing model using neural networks. Chidambaran et al. (1999) propose Genetic Programming to develop an adaptive evolutionary model of option pricing that is also data driven and non-parametric. They show that this method offers some advantages over learning networks. In particular, it can operate on small data sets, circumventing the large data requirement of the neural network approach noted by Hutchinson et al. (1994). The philosophy underlying Genetic Programming is to replicate the process by which genetic traits evolve in offspring in the biological world, through a random combination of the genes of the parents. A random selection of equations of the option contract terms and basic statistical properties of the underlying stock price will have among them some elements that will ultimately make up the true option pricing formula. By selectively breeding the equations, presumably these elements will be passed onto future generations of equations that can price options more accurately. The essence of the method is the selection of equation components; that is, genetic traits, that parents pass on to the next

C.-F. Lee et al. (eds.), Handbook of Quantitative Finance and Risk Management, c Springer Science+Business Media, LLC 2010 DOI 10.1007/978-0-387-77117-5_108, 

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generation. Since it is impossible to determine which element is the best ex ante, the focus is on choosing parents that seem to be the fittest. The genes to be propagated to the next generation are thus selected on the basis of the pricing errors of the equations. An important advantage of the Genetic Programming approach over other numerical techniques is its ability to incorporate known approximate solution as a starting point for the approximation. That is, the algorithm can seed the initial population of equations with a particular equation or individual. This has two effects on the efficiency of the program. One, it starts with an individual member in the population that gives a good fit to the data. Two, the elements of this equation will add to the “gene pool” to be used in evolving future generations. When developing the option pricing model using options prices, for example, we can start with the Black–Scholes model in the initial gene pool. The genetic programming should quickly converge to the included Black–Scholes model when data is consistent with the assumptions of the Black–Scholes model. When the data is from an environment that violates the Black–Scholes model, the included Black–Scholes model serves as a useful starting point. The program can converge to the true pricing model more efficiently as it begins the search from a locally optimum solution. There are many factors that determine the efficiency of Genetic Programming. Important specifications include the size of the population, the method of selecting equations with their embedded “genetic traits” to serve as parents, the number of mutations that are allowed, the size of the data set used for training the program. I examine these parameters and explore the efficiency of the Genetic Program. The chapter proceeds as follows. In Sect. 108.2, I discuss the elements of a genetic programming and highlight its advantages over other non-parametric methods. In Sect. 108.3, I assess the ability of Genetic Programming in learning the Black–Scholes model, given data that are simulated according to the assumptions of the Black–Scholes world. In Sect. 108.4, I discuss extensions of the Genetic Program to non-Black–Scholes world and show how Genetic Programming can adapt the Black–Scholes model to its specifications. In Sect. 108.5, I conclude.

N.K. Chidambaran

108.2.1 Basic Approach Genetic Programming is an offshoot of Genetic Algorithms. Genetic Algorithms have been used to successfully develop technical trading rules by Allen and Karjalainen (1999) for the S&P 500 Index and by Neely et al. (1997) for foreign exchange markets. Genetic Programming has also been used in heterogeneous multi-agent economies by Marimon et al. (1990), in multi-agent financial markets by Lettau (1997), and in multi-agent games by Ho (1996). I use a variant of Genetic Programming called Genetic Regression, where the desired program is a function that relates a set of inputs such as share price, option exercise price, and so forth, to one output, the option price. The set of data on which the program operates to determine the relationship between input parameters and the options price is called the training set. The set of data on which the resulting formula is tested is called the test set. The procedure of the basic approach is as follows. Individuals: Given a problem to be solved and a training

set of matched inputs and outputs, an initial set of possible formulas is randomly generated. These formulas are functions of some or all of the independent variables and randomly generated constants. The programmer specifies the allowable operations on the variables and constants, for example, mathematical operations and the cumulative probability function for a standard normal distribution. Each formula is an individual and the elements of the randomly generated formulas are the genes of the individual. A formula is represented in a binary tree. The nodes of a tree represent either a value or an operation. If the node represents an operation, then the operation is executed from the left branch to the right branch of the tree as shown in Fig. 108.1. All formulas, even complex nonlinear formulas, can be specified in terms of these binary trees. Figure 108.2 shows the binary tree representation for the Black–Scholes Formula.

108.2 Genetic Program Elements Genetic Programming is a technique that applies the Darwinian theory of evolution to develop efficient computer programs. In this section I describe the mechanics of the approach and the various ways to improve its efficiency.

Fig. 108.1 A binary tree representation of the function ln(S) X C T

108 Genetic Programming for Option Pricing

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Fig. 108.2 A binary tree representation of the Black–Scholes Model

Population: A set of individuals is called the population.

In the initial step all the individuals in the population are randomly generate. The larger the population, the larger is the diversity in the gene pool as a greater number of possible combinations are observed. The size of the population is usually kept at a constant throughout the programming step and population size is a control variable for optimizing the modeling process. Fitness: Every individual in the population is evaluated to test whether it can accurately price options in the training data set. A smaller mispricing for the training data set indicates a better fit. The program assigns a fitness measure that can be used to rank the performance of each individual formula in fitting the input data. The fitness measure and ranking will be used to select the individual formulas that best fit the data. Selection: Based on a fitness measure, a subset of the population is selected to act as the parents for the next generation of the population of formulas. The logic behind the selection criteria is that the individual formula that best explains the data consists of elements (the genes) that will ultimately constitute the elements of an option pricing formula. Procreation: A pair of the parents generates a pair of offspring. Components of the parent formulas are crossed to generate offspring formulas. A random point is selected in each parent tree. The sub-trees below that random point are switched between the two parent formulas. This operation creates a new pair of individuals, the offspring. It is possible that no crossover is performed and the parents themselves are placed in the new population (a clone).

The process of selection and crossover is repeated until the new generation is completely populated. Generations: Each successive population is referred to as a generation. Through the process of selection and procreation each new generation has a set of individual formulas that are an overall better fit of the options data (the input data). Analogous to numerical methods, we have to set the maximum number of iteration steps that the Genetic Program is allowed to run, which is implemented by specifying a maximum limit on the number of Generations. I usually use a large number, for example, 1,000 generations, but I find that the Genetic Program converges quickly in about 100 generations and improvements are fewer and far between in subsequent generations. Evolutionary pressure in the form of fitness-related selection combined with the crossover operation eventually produces populations of highly fit individuals. The program keeps track of the best-fit individual found in each generation step throughout this process. The researcher can control the accuracy of the Genetic Program by specifying the maximum permissible error in fitting the data. Once the Best Formula in a generation meets the error criteria specified, the Genetic Program has converged to a solution and the Best Formula in the final generation is the desired option pricing formula. Since Genetic Programming is a computational tool, it is common to run the program multiple times (i.e., multiple trials) and use the mean (median) of the several runs as the final result. The solutions obtained via a Genetic Program is a formula, but it is up to the researcher to gather insight from the elements of the final formula obtained from a Genetic

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Program. On the one hand, a Genetic Program can be viewed as a Black-Box that gives the value of an option given the inputs. On the other hand, the Genetic Program results in a program and it may be useful to examine the elements of the final equation obtained.

108.2.2 Fitness and Selection Criteria The method of selecting parents for the next generation can affect the efficiency of genetic programs. I examine different selection methods: Best, Fitness, Fitness-overselection, Random, Tournament with four individuals and Tournament with seven individuals. These methods represent various attempts to preserve a degree of randomness in the evolutionary process. In the Best method, individuals are ranked in terms of their fitness, ascending in the order of magnitude of their errors. The individuals with the smallest errors are thus picked to serve as parents of the next generation. In the Fitness method, individuals are selected randomly with a probability that is proportional to their fitness. In the Fitnessoverselection method, individuals are classified into two groups. Group 1 has best-fit individuals and Group 2 has the remainder. Individuals are selected randomly with an 80% probability from Group 1 and a 20% probability from Group 2. In the Random method, the fitness of the individuals is completely ignored and parents are chosen at random from the existing population. Finally, in the Tournament method, n individuals are selected at random from the population and the best-fit individual is chosen to be a parent. I examine Tournament method with n D 4 and n D 7.

108.2.3 Advantages of Genetic Programming An important advantage of Genetic Programming is its capability of incorporating a known analytical approximation to the solution into the program. I experiment with including the Black–Scholes model as an initial parameter; that is, part of the initial gene pool, for the algorithm. Since the method begins with a known approximation, it increases the probability of finding the true pricing formula and reduces computing time. Genetic Programming requires smaller training sets than Neural Networks, which is a popular alternative adaptive learning algorithm (see Hutchinson et al. 1994, and Koza 1992). Since most options, especially those that are deep-in-the-money and deep-out-of-the-money, are thinly traded, Genetic Programming is an ideal tool for option pricing.

N.K. Chidambaran

The methodology can also be made robust to changing environmental conditions and can operate on data sets generated over a range of possible conditions. I make the population robust by stochastically changing the training sets in the middle of the evolution. Only individuals with the desirable characteristics that are well adapted to changing environments will survive. The problem of over-fitting, in particular, is easily resolved by this approach. Further, new formulas can evolve out of previously optimal solutions when the data set contains structural changes rather than requiring retraining from scratch like in learning networks. Since genetic programs are self-learning and self-improving, they are an ideal tool for practitioners.

108.2.4 Convergence Characteristics of Genetic Algorithms and Programs The implementation of the Genetic Programming is effectively a search over the space of functions that can be constructed from a user-defined set of base variables and operations. This space of functions is generally infinite. However, the Genetic Programming algorithms are aided by the fact that the search space can be limited and that the search is a parallel search. The complexity of the problem can be controlled by the maximum depth size of the binary trees used to represent formulas. In my work on option pricing, I use a maximum depth size of 17 for the binary trees, which mirrors the parameter commonly used to limit the size of tree sizes Koza (1992). Practically, I chose the maximum depth size possible without running into excessive computer run times. Note that the Black–Scholes formula is represented by a tree of depth size 12. A depth size of 17, therefore, is large enough to accommodate complicated option pricing formulas and works in practice. The search space is, however, still very large and it is computationally inefficient to examine every possible tree. The implicit parallelism of Genetic Programming, however, ensures that the search is efficient. The central idea behind the parallelism of Genetic Programming is that each of the formula elements defines hyperplanes; that is, sub-regions of the search space. In the population of candidate formulas, all the elements are present, and the fitness of each formula is a function of how many of the elements of the true pricing formula is present in the individual being evaluated. All formulas that contain a particular element will have similar errors and an evaluation of the formulas in the population is a parallel search for the hyperplanes containing the elements that make up the true option-pricing model. For example, the Black–Scholes formula is: C D SN.d1/  Xe r N.d 2/

(108.1)

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where, p p d1 D Œln.S=X / C .r C  2 =2/ =  and d 2 D d1    (108.2) N (d1) and N (d2) are the cumulative standard normal values for d1 and d2, S is the current stock price, X is the exercise price, r is the risk free rate, t is the option time to maturity and s is the volatility of the underlying stock. We can treat the formula to be the point at which the hyperplanes containing the term S N (d1) and -X e-rt N (d2) intersect. Searching over a randomly generated set of formulas is, therefore, a parallel search over a set of hyperplanes. The true option pricing formula will consist of many different elements that form a set of hyperplanes and this is called its schemata. The individual sub-regions formed by the hyperplanes are the schema. If an individual equation contains elements that represent a superior region of the search space, it will generally be reflected as better fitness for the equation. This will increase the individual’s chance to reproduce and pass on its schema to the next generation. When used to solve problems that involves a search for the sequence of elements that make up a gene, or any problem that involves a search for a sequence of numbers, Holland (1975) and Koza (1992) and show that the schemata of the Genetic Algorithm search process is extremely efficient and the algorithm converges. In this paper, I implicitly test whether such an approach will also work when searching for a closed-form option-pricing model.

rived for each simulated option, using the Black–Scholes equation. I thus have a sample of simulated options data. I adopted many of the simplifications suggested by Hutchinson et al. (1994) in generating the data sample; for example, I hold the annual volatility  and riskless rate r constant throughout. Figure 108.3 shows a stock price path generated by Geometric Brownian motion and Figure 108.4 shows the distribution of associated option prices. Table 108.1 describes the specifications of the Genetic Programming model. I use the four basic mathematical operations, the log function, the exponential function, the square root function, and the cumulative Normal distribution. The basic division operation is protected against division by zero and the log and square root functions are protected against negative arguments. The current stock price, option exercise price, option intrinsic value, and option time-to-maturity are input parameters. The functional representation of a formula is assumed to be 17-step deep. I implement ten trial runs; that is, I generate ten different Genetic Programming option pricing formula. Table 108.2 shows the genetic programming parameters used in each run. For each training set the price path of a stock with starting

108.3 Black–Scholes Example In this section, I present an example of how Genetic Programming can learn the Black–Scholes model, paralleling the study by Hutchinson et al. (1994) and Chidambaran et al. (1999). Data to train the Genetic Programming is generated through Monte-Carlo simulation. For each data set, price paths of the underlying stock with initial value S0 D 50 are simulated for 504 days (24 months 21 days/month). Stock returns are assumed to follow a diffusion process dS(t)/S(t)D mdt C sdW(t) with annual continuously compounded expected return  D 0:10, standard deviation  D 0:20, and risk-free rate r D 0:05. Stock price at time t is calculated as: t P

S.t/ D e

i D1

Zi

I

t D 1; ::; 504:

Fig. 108.3 Sample geometric brownian stock price process using Monte Carlo simulation

(108.3)

I then generate a sample of call options for each stock price realization. CBOE rules (Hull 1993) were used to create call options with varying strikes and maturity for each day of the simulated price path. Option prices are de-

Fig. 108.4 Distribution of option prices derived from the stock price path in Fig. 108.1

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N.K. Chidambaran

Table 108.1 Training variables and arithmetic operations

Name

Source

Definition

S

Option contract

Stock price

X

Option contract

Exercise price

S/X

Part of Black–Scholes

Option moneyness

S

Option contract

Time to maturity (years)

max(S-X)

Boundary condition

Option intrinsic value Max (S-X,0)

C

Standard arithmetic



Standard arithmetic

Subtraction



Standard arithmetic

Multiplication

%

Standard arithmetic

Protected division:

Addition

x%y D 1 if y D 0 D x/y otherwise Exp

Black–Scholes component

Exponent: exp(x) D ex

Plog

Black–Scholes component

Protected Natural log:

Psqrt

Black–Scholes component

Protected Square root:

plog.x/ D ln.jxj/ psqrt(x) D sqrt(jxj) Ncdf

Table 108.2 Genetic programming parameters Sum of absolute dollar errors and percentage errors Fitness criterion Population size

100–50,000

Sample size

5% – dynamically sampled

Number of generations

100–1,000

Mutations

10–50%

value S0 D 50 was simulated through twenty-four 21-day months as described earlier. Options were created according to CBOE rules and valued using the Black–Scholes formula. Each training set consisted of the daily values of these options. Formula populations were exposed to dynamically sampled subsets, about 5% of the entire sample. The data set is stochastically changed in the middle of the training run to prevent over-fitting. I find that evaluating the population formulas on such stochastic subsets of the data set resulted in reduced training times and better out-of-sample performance. Only robust formulas can survive the constantly changing environment and pass on their “traits” to the next generation. In my work I have experimented with population size from 100 to 50,000 individual formulas and varied the level of mutations between 10 and 50%. I have found that the larger the initial population, or in biological terms the larger the diversity of the gene pool, the greater is the accuracy and the efficiency of the genetic program. The criterion for selecting the surviving formulas is a linear combination of the absolute pricing errors and the

Black–Scholes component

Normal Cumulative Distribution Function

percentage pricing errors. I found (Chidambaran 2003) that the formulas consistently made relatively small absolute errors when pricing out-of-the-money options and relatively large absolute errors when pricing in-the-money options. The pattern in the magnitudes of the percentage error is just the opposite. Linear combination of these two error measurements leads to a more efficient selection rule. For example, if the true price of an option is $2.00 and one of the Genetic Programming formulas gives a price of $2.20, then percentage error is small (10%) but the dollar error is $0.20, which is economically significant. On the other hand, if the true price is $0.10 and the formula gives a price of $0.07, dollar error is small ($0.03) but the price is off by 30%. Our error measure is then 30 (10% C $0.20) in the first case and 33 (30% C $0.03) in the second case. In the classic Genetic Programming fashion, I define the fitness of a formula to be: 1 1C

# fitnesscases P

(108.4) "i

i D1

where ©i is the training error for the ith case. This training error is defined as the sum of percentage and dollar errors if the Black–Scholes value was greater than $0.01 and just the dollar error if the Black–Scholes value was less than $0.01. It should be noted that the restrictions on Genetic Programming are far fewer than those required for Neural Networks. Only the variables needed for pricing options have to be specified. I need not make assumptions on the

108 Genetic Programming for Option Pricing

smoothness or complexity of the formulas beyond the maximum allowable depth (tree size) for representing a formula. I measure the performance of Genetic Programming on an out-of-sample two-dimensional options grid of option maturities and strike prices. Each cell in the table is the average pricing errors across ten different Genetic Programming formulas. My results indicate a pattern of performance of the Genetic Programming model consistent with Chidambaran et al. (1999). First, the dollar pricing errors are small for short-maturity options as opposed to long-maturity options. However, the percentage pricing errors are just the opposite. Obviously this is because the magnitude of option prices varies substantially across option maturities. Second, the errors vary across the option strike. Once again this is because option prices are very small for out-of-the money options and much higher for in-the-money options. The fitness criterion I use balances the two effects by minimizing a combination of the absolute and percentage pricing errors. I found that this allows me to better control the pricing errors for out-of-the money and in-the-money options without adversely affecting the errors for at-the-money options. While all the parent selection methods yield similar qualitative results, they vary widely in efficiency. Results indicate that the Fitness-overselection method and the Tournament method (n D 7) providing the best results and that Genetic Programming gives a good numerical approximation to the Black–Scholes model.

108.4 Extensions An important advantage of the Genetic Programming approach over other numerical techniques is its ability to incorporate known approximate solution as a starting point for the approximation. That is, we can seed the initial population of equations with a particular equation or individual. This has two effects on the efficiency of the program. One, I start with an individual member in the population that gives a good fit to the data. Two, the elements of this equation will be added to the “gene pool” to be used in evolving future generations. This approach can reach the true pricing model more efficiently as it begins the search from a locally optimum solution. In the work above using the data simulated using a Geometric Brownian motion and Black–Scholes option prices, the Black–Scholes equation will exactly fit the data. I verify this by seeding the program with the Black–Scholes model and find that the program immediately identifies the Black–Scholes equation as the Best Formula. Chidambaran et al. (1999) extend the application of Genetic Programming and illustrate how the Genetic

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Programming model can adapt and outperform the Black– Scholes model in a jump-diffusion world described by Merton (1976). Since the closed form solution for the option prices in a jump-diffusion world is available, they can measure the pricing errors from the Genetic Programming model and the Black–Scholes model in such a world. They show that Genetic Programming performs very well, beating the Black–Scholes and Neural Networks in a jump-diffusion world as well. The efficiency and accuracy of the routine improves even more when the initial population is seeded with the Black–Scholes model and the program operates to improve on the known elements that seem to work; that is, the elements of the Black–Scholes model.

108.5 Conclusion In this chapter I have described a procedure to use Genetic Programming to develop an option pricing model using only data. The results, from a controlled simulation, suggest that Genetic Programming works well in practice. Genetic Programming has many advantages over other numerical techniques. First, it is a non-parametric data driven approach and requires minimal assumptions about the stock price process. Many researchers attribute the systematic biases in Black– Scholes prices to the assumption that stock prices follow a diffusion process and have developed alternatives to Black– Scholes by considering other stock price processes. However, no single model explains all of the Black–Scholes biases and closed form solutions are elusive (Rubinstein 1985). Genetic Programming uses options data and extracts the implied pricing equation without making specific assumptions about the stock price process. Second, the Genetic Programming method requires less data than other numerical techniques such as Neural Networks (Hutchinson et al. 1994). I show this by simulation studies (Chidambaran et al. 1999) that use smaller subsets of the data and by using both genetic programs and neural networks to price relatively thinly traded equity options. Indeed, in some cases the programs are run on as few as 50 data points. The time required to train and develop the genetic programming formulas is also relatively short. Third, Genetic Programming can incorporate known analytical approximations in the solution method. For example, we can use the Black–Scholes model as a parameter in the genetic program to build the option pricing model when developing extensions of the Black–Scholes model. The final solution can then be considered to be an adaptation of the Black–Scholes model to conditions that violate the underlying assumptions. The flexibility in adding terms to the parameter set used to develop the functional approximation can also

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be used to examine whether factors beyond those used in this study, for example, trading volume, skewness and kurtosis of returns, and inflation, are relevant to option pricing. Finally, since the Genetic Programming method is self-learning and self-improving, it is an ideal too for practitioners. The selflearning and self-improving feature also makes the method robust to changing economic environments.

References Allen, F. and R. Karjalainen. 1999. “Using genetic algorithms to find technical trading rules.” Journal of Financial Economics 51(2), 245–271. Ball, C. A. and W. N. Torous. 1985. “On jumps in common stock prices and their impact on call option pricing.” Journal of Finance 40(March), 155–173. Ballie R. and R. DeGennaro. 1990. “Stock returns and volatility.” Journal of Financial and Quantitative Analysis 25(June), 203–214. Black, F. and M. Scholes. 1973. “The pricing of options and corporate liabilities.” Journal of Political Economy 81, 637–654. Bollerslev, T. 1986. “Generalized autoregressive conditional heteroskedasticity.” Journal of Econometrics 31(April), 307–327. Chidambaran, N. K. 2003. Genetic programming with Monte Carlo simulation for option pricing, Unpublished working paper, Rutgers University. Chidambaran, N. K. and S. Figlewski. 1995. “Streamlining Monte Carlo simulation with the quasi-analytic method: analysis of a pathdependent option strategy.” Journal of Derivatives 3(2), 29–51. Chidambaran, N. K., C. H. Jevons Lee, and J. Trigueros. 1999. “An adaptive evolutionary approach to option pricing via genetic programming,” in Computational Finance – Proceedings of the Sixth International Conference, Y. S. Abu-Mostafa, B. LeBaron, A. W. Lo, and A. S. Weigend (Eds.). MIT Press, Cambridge, MA.

N.K. Chidambaran French, K. R., G. W. Schwert, and R. F. Stambaugh. 1987. “Expected stock returns and volatility.” Journal of Financial Economics 19(September), 3–29. Ho, T. H. 1996. “Finite automata play repeated prisoner’s dilemma with information processing costs.” Journal of Economic Dynamics and Control 20(January–March), 173–207. Holland, J. H. 1975. Adaptation in natural and artificial systems, The University of Michigan Press, Ann Arbor. Hull, J. 1993. Options, futures, and other derivative securities, 2nd Edition, Prentice-Hall, Englewood Cliffs, NJ. Hutchinson, J., A. Lo, and T. Poggio. 1994. “A nonparametric approach to the pricing and hedging of derivative securities via learning networks.” Journal of Finance 49(June), 851–889. Koza, J. R. 1992. Genetic programming, MIT Press, Cambridge, MA. Lettau, M. 1997. “Explaining the facts with adaptive agents.” Journal of Economic Dynamics and Control 21, 1117–1147. Macbeth, J. D. and L. J. Merville. 1979. “An empirical estimation of the Black-Scholes call option pricing model.” Journal of Finance 34(December), 1173–1186. Macbeth, J. D. and L. J. Merville. 1980. “Tests of the Black-Scholes and Cox call option valuation models.” Journal of Finance 35(May), 285–301. Marimon, R., E. McGrattan, and T. J. Sargent. 1990. “Money as a medium of exchange in an economy with artificially intelligent agents.” Journal of Economic Dynamics and Control 14, 329–373. Merton, R. C. 1976. “Option pricing when underlying stock returns are discontinuous.” Journal of Financial Economics 3(January–March), 125–144. Neely, C., P. Weller, and R. Dittmar. 1997. “Is technical analysis in the foreign exchange market profitable? A genetic programming approach.” Journal of Financial and Quantitative Analysis 32(4), 405–426. Rubinstein, M. 1985. “Nonparametric tests of alternative option pricing models.” Journal of Finance 40(June), 455–480. Rubinstein, M. 1997. “Implied binomial trees.” Journal of Finance 49, 771–818.

Chapter 109

A Constant Elasticity of Variance (CEV) Family of Stock Price Distributions in Option Pricing, Review, and Integration Ren-Raw Chen and Cheng-Few Lee)

Abstract One of the important issues in option pricing is to find a stock return distribution that allows the stock rate of return and its volatility to depend on each other. Cox’s (Notes on option pricing I: constant elasticity of diffusions, unpublished draft, Stanford University, 1975) Constant Elasticity of Variance (CEV) diffusion generates a family of distributions for such a purpose. The main goal of this paper is to review and show the procedures of how such process and its resulting option pricing formula are derived. First, we show how the density function of the CEV diffusion is identified and we demonstrate the option formula by using the Cox and Ross (Journal of Financial Economics 145–166, 1976) methodology. Then, we transform the solution into a non– central chi-square distribution. Finally, a number of approximation formulas are provided. Keywords Non-central chi-square r CEV diffusion r Lognormal process r Gamma function r Option pricing model

109.1 Introduction Since the simultaneity of the world’s first options exchange, Chicago Board of Options Exchange (CBOE), and the most powerful option pricing model to date, the Black–Scholes (BS) model, in 1973, interest and research on the option area have grown explosively. People have found that options, so-called derivative (or redundant) securities, can serve as a perfect hedging tool. The BS model also provides a synthetic way of hedging if certain options are not traded. Furthermore, a number of corporate finance areas find option theories useful in either pricing (e.g., convertible/callable bonds), or carrying out important characteristics of the problem (e.g., investment policies).

R.-R. Chen and C.-F. Lee () Rutgers University, New Brunswick, NJ, USA e-mail: [email protected]

The major drawback of the BS model is its assumption of the constant volatility of the stock return distribution. Due to the dynamic nature of our economy, the volatility is changing over time. This important property is not characterized in the stock return assumptions of the BS model. To overcome this problem, people have come up with a number of suggestions. Hull and White (1987), Johnson and Shanno (1987), Scott (1987), and Wiggins (1987) set the volatility to vary over time. This direct modeling of the randomness of the volatility gives flexibility to the stock return distribution but pays a high price of costly numerical calculations. The non-existence of a closed form solution prevents their models from being popular. Geske’s compound option model (1979) also allows the volatility to change over time. However, his model introduces randomness only when an option can be exercised early. If the option cannot expire prematurely, his model will not allow the volatility to change.1 Cox (1975) proposes a constant elasticity of variance (CEV) diffusion that is complex enough to allow changing volatility and simple enough to generate a closed form solution for options. The CEV diffusion not only has the flexibility to allow changing volatility but also preserves the property of non-negative values of the state variable as does in the log-normal diffusion used in the BS model. Beckers (1980), MacBeth and Merville (1980), Christie (1982), Emanuel and MacBeth (1982), Ang and Peterson (1984), Schaefer and Schwartz (1984), Choi and Longstaff (1985), Tucker et al. (1988), Jacklin and Gibbons (1989), and 

This paper was previously published in the Journal of Financial Studies, 1 (1993), No. 1, pp. 29–51. ) Assistant Professor and Distinguished Professor and Chairperson, respectively. This paper adopts from our previous paper titled “Distribution Family of Stock Prices for Option Pricing Models” presented at the 1991 Annual Conference for Accounting and Quantitative Finance in Buffalo, NY. We would like to thank Joe Ogden and Todd Patzel for their helpful comments. 1 Bachelier (1900), a French mathematician, was believed to be the first person who used the Brownian Motion to model stock prices dynamically. There are other static analyses on the stock price distributions (e.g., Mandelbrot [1963] and Fama [1965]). For a detailed discussion on the distributions of the stock returns, please refer to Fama (1976, Chapter 1).

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R.-R. Chen and C.-F. Lee

Lauterbach and Schultz (1990) have successfully used CEV models to perform empirical studies. Due to its flexibility, the CEV diffusion can be extended to allow mean-reversion, a necessary characteristic in interest rates. As a consequence, a modified CEV, the square root process, has led to a most used term structure model by Cox et al. (1985). Their term structure model is better than the one from the log-normal diffusion used by Dothan (1978), which does not allow mean reversion, and the one from the Ornstein-Uhlenbeck diffusion used by Vasiced (1977), which permits negative interest rates. The main purpose of the paper is to review and show the procedure of deriving Cox’s CEV models. First, the CEV diffusion is defined and all necessary derivations are shown. Then it is shown that the transition density of the CEV diffusion can be transformed into a non-central chi-square distribution. This paper is organized as follows: The next section shows the CEV diffusion and its transition density. Section 109.3 derives the resulting option pricing formula. The implementation of the non-central chi-square probabilities is introduced in Sect. 109.4. Finally, in sections, the paper is summarized.

109.2 The CEV Diffusion and Its Transition Density In this section, we show how the Constant Elasticity of Variance (CEV) model proposed by Cox (1975) can be used to determine the stock price distribution. A process, S.t/, is said to follow a CEV diffusion if it satisfies the following stochastic differential equation: ˇ

ds D Sdt C S 2 dW

(109.1)

Where W .t/ is a standard Wiener process,  and ¢ are constants representing instantaneous mean and standard deviation, and “ (also constant) is known as the CEV parameter that determines the stock price distribution, which includes normal and log-normal as special cases. It should be noted that the instantaneous volatility on the ˇ stock return, S 2 , is a function of the stock price. The direction and size of the impact of the stock price on the volatility clearly depend on the magnitude of “. If “ is less than two, the higher the stock price, the lower the volatility. On the other hand, if “ is equal to two, the stock price has no influence on the volatility. In other words, the volatility will be a constant ¢ over time regardless of the stock price. Recently, Jacklin and Gibbons (1989) are trying to conduct studies on this CEV diffusion directly with the stock prices. It has been well recognized that option markets are more efficient than stock markets. Furthermore, with the possibility of perfect hedging between stocks and options, given

stock prices, option prices should be determined efficiently. Thus, focusing on option markets should provide less noisy results than on stock markets. In the next section, we show the resulting option formulas. It is clear that when “ D 2, it is nothing but a log-normal diffusion used by Black and Scholes. The transition density can be shown in the following theorem. Theorem 1. The log-normal process, dS D Sdt C ¢SdW, has the following transition density function: f .ln S.T /jln S.t//



2 .T  t/;  2 .T  t/ D N ln S.t/ C   2 where N.m; V / is a normal density function with mean m and variance V Proof. Divide the diffusion equation by S , we get: dS D dt C dW S Now, both the drift and the diffusion coefficient are constants. Therefore, we know that dS/S should be normally distributed. Define X be lnS , by Itô’s Lemma, we can get: 1 dX D XS dS C XSS .dS/2 2 1 1 2 2 1 D .Sdt C SdW/   S dt S 2 S2

2 dt C dW D  2 Since both   ¢ 2 =2 and ¢ are constants and dW is normally distributed with mean 0 and variance dt,



2 .T  t/;  2 .T  t/ f .X.T /jX.t// D N X.t/ C   2 Substituting back lnS completes the proof. If “ ¤ 2, then the transition is more complex. Cox has given the transition density for “ < 2 as the following theorem:2 Theorem 2. If “ in Equation (109.1) is strictly less than two, then the process has the following transition density: f .S.T /jS.t// 1

1

D .2  ˇ/k 2ˇ .xz12ˇ / 42ˇ e xz I

1 2ˇ

p Œ2 xz

(109.2)

2 Note that “ < 2 is the usual case found empirically. Empirical evidence has suggested that the relationship between the volatility and the stock price is negative.

109 A Constant Elasticity of Variance (CEV) Family of Stock Price Distributions

where  is a set of all possible states .¨/ at time t C t. Applying Taylor’s series expansion to the first term of the density (from t to t C t) around t, we can get:

where kD

1617

2  2 .2  ˇ/.e .2ˇ/.T t /  1/

ft !T .S / D

x D kS.t/2ˇ e .2ˇ/.T t /

 Z  @ft !T .S /

t C o. t/ ft !T .S / C @t  ft C t !T .S /d!

z D kS.T /2ˇ

Z

D Œ1 C 0 C o. t/  and Iq ./ is the modified Bessel function of order q.

Z D

3

Proof. By Chapman–Kolmogorov equation, we have: ft !T .S /  f .S.T /jS.t// Z D ft !t C t .S /  ft C t !T .S /d! 

ft C t !T .S /d! 

ft C t !T .S /d! 

This is a single dimensional expansion because S.t/ is not stochastic. Next, we perform a two dimensional expansion on the integram: once on time .t/ and twice on the underlying variable .S /.

 Z  @ft !T .S / @2 ft !T .S / . S /2 @ft !T .S /

t C

S C C o. t/ d! ft !T .S / C @t @S @S 2 2   Z  ˇ @ft !T .S / @ft !T .S / @2 ft !T .S /  2 S ˇ

t C .S t C S 2 W / C

t C o. t/ d! ft !T .S / C D  @t @S @S 2 2   Z  ˇ @ft !T .S / @2 ft !T .S /  2 S ˇ @ft !T .S / @ft !T .S /

t C .S t/

t C o. t/ C S 2 Wd! D ft !T .S / C  2 @t @S @S 2 @S 

ft !T .S / D

D ft !T .S / C

@ft !T .S / @2 ft !T .S /  2 S ˇ @ft !T .S /

t C .S t/

t C o. t/  @t @S @S 2 2

Divide t both sides and let it go to zero, we have the partial differential equation that f has to satisfy: 0D

@2 ft !T .S / . 2 S ˇ / @ft !T .S / @ft !T .S / S C C  @t @S @S 2 2 (109.3)

Solving this PDE is not an easy task; but one can verify that Equation (109.2) satisfies this PDE. A sketch of the verification is provided in Appendix 109A. Note that the proof of Theorem 1 cannot apply here because no matter what transformation we make, we will never

have both the drift and the diffusion coefficient constant.4 Thus, we know that the transition density cannot be normal. The proof of Theorem 2 is a standard procedure to identify the transition density, if it exists.5 Equation (109.3) is called the forward equation of the density function f .6 It should be noted that Equation (109.3) does not have a unique solution unless relevant boundary conditions are specified. These conditions normally depend on how the stochastic process evolves. A full development of this process can be seen in Feller (1951). In Theorem 2 where “ is less than 2, this process will hit zero and once it hits zero, it will stay there forever. Hence Theorem 2 only describes the part of the distribution where S is alive. There is a nontrivial probability 4

3

For example, see Breiman (1986, p. 199) for the Chapman– Kolmogorov equation.

For the transformation to qualify Gaussian, drift term has to be linear in X (transformed S) and the diffusion coefficient needs to be constant. 5 See Breiman (1986, p. 205–207) for this standard procedure. There are other methods but they all involve solving a partial differential equation. 6 See Karlin and Tayler (1981, p. 214) for details.

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R.-R. Chen and C.-F. Lee

that S will be dead at some time between t and T (i.e., S hits zero and stays there) as follows: f .S.s/ D 0jS.t// D G

1 ;x 2ˇ

(109.4)

v

u m1

e u du and .m/

./ is a Gamma function. It is well understood that in the log-normal diffusion .“ D 2/, S will never hit zero. Hence, it is not difficult for us to conjecture that zero will keep being S ’s absorbing barrier until “ D 2. When “ > 2, the process will not hit zero and its transition density describes a complete behavior of S . Schroder (1989) shows that Theorem 2 is essentially a form of non-central chi-square density if we make an appropriate change of variable as indicated in the following theorem. Theorem 3. The density function in Theorem 2 is a noncentral chi-square density if the changes of variables, z D kS.T /2“ and y D 2z, are made. Proof. The Jacobian of the transformation, z D kS.t/2“ , is jJ j D

1 S.T /ˇ1 .2  ˇ/k

Plugging the Jacobian into Equation (109.2), we obtain: ˇ1

1

k 2ˇ S.T /ˇ1 .xz12ˇ / 42ˇ e xz I

1 2ˇ

p Œ2 xz

1 42ˇ

Rewrite the middle term .xz12ˇ / and replace z with kS.T /2“ . Thus, the transformed density becomes: .x=z/

1 42ˇ

e

xz

I

1 2ˇ

p Œ2 xz

Now, do the second change of variable, y D 2z (with the Jacobian, jJ j D 1=2). Then the density further becomes:

1  p  2x 42ˇ 2xy e 2 I 1 2 xz 2ˇ y 1

 p  1 y 42ˇ 2xy e 2 I 1 2 xz or 2ˇ 2 2x 1 2



Note that this expression is close to a non-central chi-square density. It differs from the non-central chi-square density in that the sign of the power of y=2x does not match with that of the order of the Bessel function. Fortunately, if the order of the modified Bessel function is an integer, we have a property of the Bessel function: Iq ./ D Iq ./;

1  p  2xy 1 y 42ˇ e 2 I 1 2 xz 2ˇ 2 2x



Rm for some S 2 .t; T / where G.m; v/ D

which will make the above equation be rewritten as:

This is a non-central chi-square density function with 2  2=.“  2/ degrees of freedom and 2x non-centrality.7 We denote such a density function X2 .2  2=.“  2/; 2x/. We also denote its distribution function. ZyN

1  p  2xy 1 y 42ˇ e 2 I 1 2 xz dy 2ˇ 2 2x

0

x 2 yI N 2

2 ; 2x 2ˇ



If q is not an integer, the density will not be precisely a non-central chi-square density but the distribution function can still be expressed as a non-central chi-square distribution. Comparing the integram above and the original densityfunction, one can find that the original density acts like8 a non-central chi-square density with 2x being the random variable, y contributing to the degree of non-centrality, and 2  2=.“  2/ being the degrees of freedom. Thus, the same integral (without q becoming q) will be represented by another non-central chi-square distribution function: ZyN 0

1  p  2xy 1 y 42ˇ e 2 I 1 2 xz dy 2ˇ 2 2x

D 1  x 2 2xI 2 

2 ; yN 2ˇ



It is easy to see that for 1=.2“/ to be a non-negative integer, “ must be one. If “ D 1, then the Equation (109.1) will reduce to the square root process, a process used in Cox and Ross (1976) and Cox et al. (1985). This is a convenient choice in expressing the option pricing formulas in terms of two noncentral chi-square distributions. Although the empirical evidence has shown that the relationship between the stock price and its return volatility is negative, in order to allow flexibility in testing this hypothesis, the transition density9 for “ > 2 is given by Emanuel and MacBeth (1982) through a similar manner. With the following theorem derived by Emanuel and MacBeth, the CEV process [Equation (109.1)] is fully characterized. 7 See Johnson and Kotz (1970) or Equation (109.11) later on in the paper for the non-central chi-square density formula. 8 We use the term, “acts like,” because the density is never a non-central chi-square one. It can be represented as a non-central chi-square only in the form of distribution functions. 9 Not in the form of non-central chi-square.

109 A Constant Elasticity of Variance (CEV) Family of Stock Price Distributions

Theorem 4. If “ in Equation (109.1) is strictly greater than two, then the process has the following transition density: 1

1

f .S.T /jS.t// D .ˇ2/k 2ˇ .xz12ˇ / 42ˇ e xz I

1 ˇ2

 p  2 xz

where k, x, z, and Iq ./ are as defined. Furthermore, this is a non-central chi-square density for all values of “. x2 2 C

1  p  2xy 1 y 42ˇ 2 ; 2x D e 2 I 1 2 xy 2ˇ ˇ2 2 2x

Proof. The procedure is precisely identical to that of Theorem 3 except that 1=.“  2/ needs not be an integer.10 It can be seen that Theorem 4 provides a more direct link between the transition density function of the process [Equation (109.1)] and the non-central chi-square density. Thus, the distribution function is directly a non-central chisquare distribution function. Note that the PDE that this transition entity must satisfy is the same one11 but with a different boundary condition. Note that when “ < 2, S D 0 is an absorbing barrier while here it is inaccessible.

109.3 The CEV Option Pricing Models

1619

condition max fK  S; 0g leads to the put option formula. Note that this PDE is similar to the PDE that the density function has to satisfy. The differences are (1) this PDE has an extra term on the left hand side,  C ; and (2) this PDE is “risk-neutralized” so that  is replaced by the risk-free rate,  . The first difference is caused by the fact that Equation (109.6) is not a forward equation but a backward equation for the Kac functional [see Karlin and Tayler (1981, p. 224–4)]. The second difference results from an arbitrage argument used by Black and Scholes (1973) and Cox et al. (1979).12 Therefore, one can understand that the CoxRoss method is no easier than the traditional Black–Scholes method since both of them have to solve essentially the same PDE. However, the Cox-Ross method is convenient if one changes the payoff slightly. Since the density function is already calculated, it is just a matter of finding the expected value, while in the BS method, one has to solve another similar but different PDE. Given the density function, we can find the European call option formula by replacing C.T / in Equation (109.5) with max fS.T /  K; 0g. The put formula can be obtained from either a similar replacement (max fK  S.T /; 0g) or imposing the put-call parity. Cox shows the following call option pricing model under the density described by Theorem 2. Theorem 5.

The European option formula can be derived by taking a riskneutralized expectation of the option’s end payoff at the maturity date:

C.t/ D S.t/

1 e x x n G n C 1 C X

1 2ˇ 2ˇ ; kK

.n C 1/

nD0

1 X e x x 2ˇ G.n C 1; kK 2ˇ /

K 1  n C 1 C nD0 2ˇ nC1

C.t/ D EO t Œe .T t / C.T /

where EO t Πis the conditional expectation (conditional on S.t/) taken under the risk-neutralized process of the stock price. The risk-neutralized process is the same as the original process except that the drift parameter, , is replaced by an instantaneous riskless rate,  (see Cox and Ross [1976]). In their original work, Black and Scholes used an arbitrage-free argument to price C.t/. If the same argument applies there, the PDE that the option has to satisfy is: C D

@C @2 C 2 ˇ @C C S C  S @t @S @S 2

e

(109.5)

.T t /

(109.7) Proof. C.t/ D EO t Œe .T t / maxfS.T /  K; 0g Z1 De

.T t /

S.T /f .S.T /jS.t//dS.T / K

(109.6)

Solving this PDE with the boundary condition max fS  K; 0g leads to the call option formula and with the

10 Note that the Jacobian for the first change of variable in the proof of Theorem 3 is -dz /dS. This is because z is now an inverse function of S. Details can be seen in Emanuel and MacBeth (1982). 11 It can be easily seen that it is the same forward equation that it has to satisfy.

Z1 e

.T t /

K

f .S.T /jS.t//dS.T / K

12 A formal discussion of this so-called “risk-neutralized” expectation starts from Harrison and Kreps (1979) where they introduced the equivalent martingale measure. They show that no arbitrage is equivalent to the existence of an equivalent martingale measure, which can be reached by using the Girsonov’s theorem. In the case of Black and Scholes, the stock return (risky) will be transformed into the riskless rate.

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R.-R. Chen and C.-F. Lee

Using Equation (109.1) and writing out the Bessel function explicitly, we can write the integram of the second integral as follows. .2  ˇ/k

1 X nD0

1 2ˇ

.x

12ˇ

/

1 42ˇ

e

xz

zn x n .n C 1/.n C 1 C

.zx/

Integrating the quantity leads to: Z1 X 1

1 42ˇ

kK 2ˇ

nD0

1 X

Making change of variable z D kS.T / and rearranging terms, we obtain the following expression:

nD0

nD0

1 2ˇ

dz

1 R1 e z zn dz e x x nC 2ˇ kK 2ˇ .n C 1/

1  n C 1 C 2ˇ

  1 1 X e x x nC 2ˇ G n C 1; kK 2ˇ

D 1  n C 1 C 2ˇ nD0

1

e xz zn x nC 2ˇ .n C 1/ n C 1 C

1 2ˇ

The final expression can be obtained by interchanging the integral sign and summation sign:

1 2ˇ /

2“

1 X

1

e xz zn x nC 2ˇ .n C 1/ n C 1 C

The first integral can be done in a similar manner. The integram is:

1 1 X X e .T t / S.T /e xz zn x nC 2ˇ e .T t / S.T /e xz znC 2ˇ z 2ˇ x n x 2ˇ

D

1 1 .n C 1/ n C 1 C 2ˇ nD0 .n C 1/ n C 1 C 2ˇ nD0 1

1

1

1

1

1

1

1 e .T t / S.T /e xz znC 2ˇ k 2ˇ S.t / x n k 2ˇ e .T t / X S.T /

D 1 .n C 1/ n C 1 C 2ˇ nD0

D

1 X

Similarly, integrating this integram and interchanging integration and summation will lead to: 1 e x x n G n C 1 C X nD0

1 2ˇ 2ˇ ; kK

1

e xz znC 2ˇ x n S.t/

1 nD0 .n C 1/ n C 1 C 2ˇ

.n C 1/

This completes the proof. Since the Cox-Ross method can be imposed and the transition density can be expressed as a non-central chi-square density. Schroder shows that Equation (109.8) can be simplified into a formula in terms of non-central chi-square distribution functions. Theorem 6. Equation (109.7), with proper transformations, can be expressed in terms of non-central chi-square

distributions and the option pricing formula takes the same form as the Black–Scholes formula.  2 C.t/ D S.T / 1   2zI 2 C e .T t / K2 2xI

2 ; 2x 2ˇ



2 ; 2z 2ˇ (109.8)

Where x and z as defined before and 2 .¨I ¤; œ/ is a cumulative non-central chi-square distribution function with ¨, ¤, and œ being the upper limit of the integral, degrees of freedom, and noncentrality, respectively.

109 A Constant Elasticity of Variance (CEV) Family of Stock Price Distributions

1621

Proof. As in Theorem 5, we have the call option formula expressed in terms of two integrals:

C.t/ D EO t Œe .T t / maxfS.T /  K; 0g Z1 D

Z1 e

.T t /

S.T /f .S.T /jS.t//dS.T /  e

.T t /

K

K

f .S.T /jS.t//dS.T / K

The integram of the first integral can be simplified as follows.

 1 1  e .T t / S.T /.2  ˇ/k 2ˇ xz12ˇ 42ˇ e xz I 1

D e .T t / S.T /.2  ˇ/k 2ˇ 1

D e .T t / S.T /.2  ˇ/k 2ˇ

z x

1 z 42ˇ

Recall that z D kS.T /2“ and x D kS.t/e .2“/.T t / in the risk-neutralized world. Thus, the equation can be simplified as follows: S.t/.2  ˇ/kS.T /1ˇ

1 z 42ˇ

x

e xz I

1 2ˇ

 p  2 xz

1  p  z 42ˇ xz @z D S.t/ e I 1 2 xz 2ˇ @S.T / x

Plugging into the first integral, we obtain:

S.t/

Z1 1  p  z 42ˇ xz e I 1 2 xz dz 2ˇ x

y

Letting y D 2z, we arrive at the non-central chi-square distribution we desire:  S.t/ 1  2 2y  I 2 C

2 ; 2x 2ˇ

x 2 z2ˇ



x

1 2ˇ

1

42ˇ

1

 p  2 xz

e xz I

1 2ˇ

 p  2 xz

ˇ

x 2ˇ z 2ˇ e xz I

1 2ˇ

 p  2 xz

if 1=.2  “/ is an integer; or e .T t / K2 2xI 2 

2 ; 2y  2ˇ



if 1=.2  “/ is not an integer. This completes the proof. Similar to Theorems 5 and 6, in the case where, “ > 2, we can show solutions in a similar manner. Thus, we simply state the results in the following theorem without proof. Theorem 7. When “ > 2, we may adopt the density in Theorem 4 and react the call option formula as follows.   1 1 X e x x nC 2ˇ G n C 1; kK 2ˇ

C.t/ D S.t/ 1  n C 1 C 2ˇ nD0

1 e x x n G n C 1 C 1 ; kK 2ˇ X 2ˇ e .T t / K .n C 1/ nD0 (109.9) in its original form and

where y  D kK 2“ . From Theorem 3, the second integral is clearly a non-central chi-square distribution after proper transformations. Thus, the second term is: .T t / 2 e K 1   2y  I 2  

2 ; 2x 2ˇ



 C.t/ D S.T / 1  2 2xI e

.T t /

 2 ; 2z ˇ2

K 2zI 2 C 2

2 ; 2x ˇ2

(109.10)

if expressed in term of non-central chi-square forms.

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R.-R. Chen and C.-F. Lee

When p D 2, the density described in Theorem 1 will yield the well-known Black–Scholes formula as follows: p C.t/ D S.T /N.d /  e .T t / KN.d   T  t / (109.11) ln.

S.t /

2

/C. C  /.T t /

K p 2 where d D .  T t It should be noted that although the option formulas can be expressed much more simply in the form of non-central chi-square distributions, which are easier to interpret, it does not make the computations any easier. Unlike other wellknown distributions, a non-central chi-square probability involves an infinite sum whose convergence, in some cases, is very slow. In the following section, we discuss how to compute such probabilities.

with 1 degree of freedom and one non-central 2 with 1 degree of freedom and  noncentrality. The sum of   12 (l) is nothing but 2 .  1/; and the last term is 2 .l  / but this is nothing but a squared N.; 1/. It will be clear later on that this separation is helpful since calculating a non-central chi-square’s tail probabilities can be very difficult. Johnson and Kotz (1970) have shown the density of a noncentral chi-square variable as: f .y/ D

Equations (109.8) and (109.11) involve non-central Chi-square probabilities. Unlike other known central distributions, each non-central chi-square probability involves an infinite sum of Poisson-weighted central chi-square probabilities. Thus, to obtain an accurate estimate, one has to make sure that he has summed up enough terms. It is found that with a large value of degrees of freedom or non-centrality, the number of terms to be summed up is huge. In this section, we present the potential problems involved in computing noncentral chi-square probabilities and how to overcome them.

109.4.1 A Formula for Non-Central Chi-Square Distribution, An Infinite Sum of Poisson-Weighted Central Chi-Squares Let Ui0 s be independent unit normal variables and ıi0 s be constants. Then the distribution of  X

.Uj C ıj /2

j D1

is non-central 2 with  degrees of freedom and parameter P  D j D1 ıj2 . It is clear then that if all  D 0, which means all ı will have to be zero, the distribution will be a central 2 distribution with  degrees of freedom and we denote it 2 .; /. From the definition, we know that a 2 .; / can be regarded as the sum of a series of (  1 terms) central 2 ’s

(109.12)

All the tail probability can be expressed as a weighted sum of central chi-squares with Poisson probabilities being the weights: 2 .xI ; /

109.4 Computing the Non-Central Chi-Square Probabilities

p y 1 y v2 4 I v2 y e  2 : 4 2 

D

3   j 2 Zx y 1 v 2 4 v   y 2 Cj 1 e  2 dy5 jŠ 2 2 Cj  2v C j

 1 X e 2

j D0

0

(109.13) The first two cumulants of this distribution are  C  and 2. C 2/. It is clear that the first part is a Poisson probability with mean =2 and the second part is a central 2 .C2j /. Therefore the non-central 2 probability depends on two probabilities. From Johnson and Kotz, we also know that if we increase either the degrees of freedom or the noncentrality, the distribution will shift to the right. This is because when either parameter gets large, both Poisson and central 2 distribution will shift to the right. As a consequence, we have to add a big number of terms to get a non-central 2 probability. Summing up a big number of terms is not a problem with modern computer technology. The problem lies in each probability to be summed up is extremely small. When the degrees of freedom and noncentrality get real big (this is the case in some options in our sample) even double precision is not able to detect such probability. Therefore, although a non-central distribution looks like all the other well-known distributions, it is much harder to compute its tail probabilities, because we not only have to compute the integral, but to compute the infinite sum. Unfortunately, for most stock options, degrees of freedom and noncentrality tend to be very large. In such a case, we have to go for the approximation methods described below.

109.4.2 Various Approximation Formulas Johnson and Kotz suggest a normal approximation by Sankaran (1963) be a good approximation when  and

109 A Constant Elasticity of Variance (CEV) Family of Stock Price Distributions

 are large. This approximation is also recommended by Schroder (1989). Sankaran suggests that

y C

h (109.14)

 C 2 1 C h.h  1/ . C /2



. C /2 h.h  1/.2  h/.1  3h/ 2. C /4



and variance



 C 2  C 2 h2 1  .1  h/.1  3h/ . C /2 . C /2 where 1  23 . C /. C 3/. C 2/2 . There are two other approximations in Abramowitz and Stegun (1970): P .Y < y/ D N.y  / where y  D

q

2y 1Cb



q

2a 1Cb

Stock options tend to have large  and possibly . Recall that  in Cox’s model is either 2z or 2x, each of which is an increasing function of k. which, in turn, is a sensitive function of “. As “ gets closer to 2, k will grow tremendously. 2 Even if “ D 0, k will carry a value of 2 2 .e2.T t / 1/ , which is already quite large.14 This leads us to use the approximations rather than direct calculation base on Equation (109.13).

is approximately normal with expected value

(109.15)

 1,

a D  C  and b D

 ; C

and P .Y < y/ D N.y  / (109.16) q q 2y 2a D 1Cb  1Cb  1 and a and b are defined as

where y  above. Since when either  or  approaches infinity, the standardized variable will have unit normal distribution, we have: P .Y < y/ D N.y  /

(109.17)

p . where y  D y.C/ 2C4 Theoretically, when  and  are not so large, it is not difficult to sum up terms from Equation (109.13). FORTRAN IMSL mathematical library has a routine to compute such probabilities following precisely Equation (109.13). When  and  are large, a non-central 2 variable will tend to be normally distributed and therefore normal approximation will be accurate.13

109.5 Conclusion The CEV process can result in a whole class of stock price distributions by changing the “ parameter. It is known that this class of processes does not permit negative values for the state variable; it allows the volatility to change over time; and it also allows the volatility to be either positively or negatively related to the stock price. Furthermore, it can be extended to allow mean reversion without changing its basic properties.15 This paper fully reviews and integrates the CEV diffusion and its resulting option models. All necessary theorems are proved. It is hoped that the reader can use this paper as a starting point to understand stochastic processes, their transition densities, stock price and interest rate dynamics, and further be able to price contingent claims. This line of study has a much room for future research. Empirical studies on the process itself have so far been very limited. Due to the nonlinear nature of the process and nonstationary data, a direct test of the distribution has not yet been done. Empirical studies on the resulting option formula are still very limited. In its theoretical aspects, the CEV class of processes are useful in modeling other financial variables, which by definition cannot be negative, such as volatility, interest rates, and other macro/micro economic variables. Given the growing popularity of the contingent claim pricing methodology, a number of innovative securities traded over the counter (e.g., swaps) and corporate finance areas (e.g., capital budgeting) are perfect candidates for the CEV process.

Appendix 109A In this appendix, we show that the forward equation [Equation (109.3)], 0 D ft C Sf s C ¢s2 S “ fss ; 14

13

1623

However, we find very little support that either method is reliable. The IMSL routine will occasionally give probabilities greater than 1 and the approximations do not provide close answers among themselves. One needs to be very careful about what approximation to use.

For interest rate options, both  and  are quite small. Hence, Equation (109.13) is plausible to use. 15 The mean reversion feature has been explicitly incorporated in the square root process (“  1) in studying the term structure of interest rates. See Cox et al. (1985) for details.

1624

R.-R. Chen and C.-F. Lee

will be satisfied by the density function [Equation (109.2)]. f .S.T /jS.t//  1 1  D .2  ˇ/k 2ˇ xz12ˇ 42ˇ e xz I

 p  2 xz :

1 2ˇ

We first derive the following necessary partial derivatives: fs

fss

ft

1 2ˇ

1 2ˇ

1 2ˇ

D fx xs  D f

1 2ˇ



Cf

1 1 2ˇ

 xs

D fxx xs xs C fx xss







 1 1 1 1 f 1 f 1 Cf  2 .xs / 2ˇ 2ˇ 2ˇ 2ˇ 



 1 1 C f Cf  1 xss 2ˇ 2ˇ

 D f



D fx xt C fk kt C fz zt 









1 1 Cf  1 xt C 2ˇ 2ˇ



  1 1 Cf C 1 zt C f 2ˇ 2ˇ

D f

where xs D .2  ˇ/kS.t/1ˇ e .2ˇ/.T t / xss D .1  ˇ/.2  ˇ/kS.t/ˇ e .2ˇ/.T t / xt D  .2  ˇ/kS.t/1ˇ e .2ˇ/.T t / C kt S.t/2ˇ e .2ˇ/.T t / kt D 

22  2 .e .2ˇ/.T t /  1/2

zt D S.T /2ˇ kt D 

e .2ˇ/.T t /

22 S.T /2ˇ  2 .e .2ˇ/.T t /  1/2

The detailed derivation of the above result can be obtained from the authors on request. The simplification depends on the recursive properties of the Bessel function: @Iq Œ! q D IqC1 Œ! C Iq Œ! @! ! 2q Iq1 Œ! D IqC1 Œ! C Iq Œ! !

ft

1 2ˇ

k

kt

We now plug the partial derivative results into the forward equation:  2S ˇ fss D 0 ft C Sf s C 2 or fx xt C fk kt C fz zt C Sf x xs C

  2S ˇ  2 fx xss C fx xss D 0 2

Note that Sf s will cancel the first term of xt (multiplied by 2 ˇ fx ) and the first two terms in  2S fss (multiplied by .xs /2 ) will cancel the second term of xt (multiplied by fx ). For the remaining terms, we first have to express all density func1 1 or .2ˇ/  1 order of the Bessel tions in terms of either .2ˇ/ function. This involves the use of the second recursive formula of the Bessel function:

1 f C1 2ˇ



x 1 1 1 D f 1  f z 2ˇ .2  ˇ/z 2ˇ

109 A Constant Elasticity of Variance (CEV) Family of Stock Price Distributions

1 2 2ˇ



z 1 ˇ1 1 D f  f 1 x 2ˇ .2  ˇ/x 2ˇ

f

After the all expressions been made in a consistent manner, we can cancel terms directly and are left with the following three terms: f

1 2ˇ



e .2ˇ/.T t / Œ.2  ˇ/  1  .1  ˇ/ D 0 .e .2ˇ/.T t /  1/

and this completes the verification.

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Author Index

A Aadland, R., 1042 Abdel-khalik, A.R., 1307 Abraham, J.M., 1206 Abramowitz, M., 1057, 1059, 1515, 1521, 1623 Acharya, V.V., 667, 942, 988, 1000, 1001 Ackert, L., 137, 140 Acs, Z.J., 829–831, 839, 840 Adams, J., 873 Adenstedt, R.K., 1410 Admati, A.R., 1173 Afonso, A., 641, 659 Agarwal, V., 1597 Aggarwal, R., 843, 1349, 1351 Agrawal, D., 884, 885, 991, 995 Agrawal, V., 807 Aguilar, O., 1109 Aharony, J., 1406 Ahn, C.M., 984, 1121, 1128 Ahn, D.-H., 587, 717, 718, 983, 985–987, 1121, 1128, 1490 Aigner, D.J., 1093 Aitken, M., 1524 Ait-Sahalia, Y., 532, 985–987 Akgiray, V., 1333 Akhter, S.H., 1308 Alam, P., 1308 Al-Eryani, M.F., 1308 Alex, F., 1524 Alexander, G.J., 73, 265 Alexe, G., 641, 642 Alexe, S., 641 Alizadeh, S., 1109, 1118, 1273, 1274, 1276, 1279 Allayannis, G., 679 Allen, F., 1223, 1608 Allen, L., 138, 139, 671, 759, 819–822, 824, 825 Altiok, T., 1136 Altman, E.I., 193, 649, 665, 807, 965, 1061, 1270, 1327, 1328, 1596–1598, 1603 Ambarish, R., 1400, 1401 Amemiya, T., 1203, 1597–1600 Amihud, Y., 293, 294, 340, 342, 686, 1093 Amin, K.I., 533, 547, 587, 1191 Amram, M., 375, 392, 418 Ananthanarayanan, A.L., 458 Anas, A., 1206 Andersen, L., 703, 726, 727

Andersen, T.G., 526, 548, 715, 725, 737, 904, 985, 987, 1174, 1278, 1334, 1335, 1338, 1341, 1344 Anderson, D.W., 988 Anderson, J., 1036 Anderson, R., 668, 941, 949 Andricopoulos, A.D., 506 Ang, A., 185, 984, 985, 1122 Ang, J.S., 38, 137–163, 1615 Angbazo, L., 686 Angel, J., 342 Angrist, J.D., 1365, 1366 Ansel, J.P., 465 Aoki, M., 830 Applebaum, D., 1569 Aragó, V., 1174 Arai, T., 1568 Archer, S.H., 79, 83 Arditti, F.D., 38–40, 1224 Arnold, L., 451 Arnott, R.D., 185, 193, 305–307 Arora, N., 668, 670 Arzac, E., 1223 Asai, K., 1197 Ash, M., 965 Ashizawa, R., 1374 Asmussen, S., 47, 1445, 1446, 1448, 1451, 1454, 1457, 1461 Asquith, P., 1383, 1406 Atiya, A., 965 Atkinson, A., 966 Atkinson, A.C., 880, 965, 966 Aumann, R., 1382 Ausin, M.C., 1376, 1378 Avellaneda, M., 1568 Avery, R.B., 1598 Avram, F., 47, 1445, 1446, 1448, 1451, 1454, 1457, 1461 Avramov, D., 192 B Baba, Y., 1283–1285 Babbs, S.H., 1483, 1490 Bachelier, L., 1055, 1615 Back, K., 347, 1017 Backus, D., 986, 1490 Bae, K., 221 Bagehot, W., 340 Bai, J., 1347, 1348 Bailey, A.D., 1307

1685

1686 Baillie, R.T., 1175, 1394, 1395, 1420 Bajari, P., 1206 Baker, G.P., 139, 143 Bakshi, C., 524 Bakshi, G.S., 533, 537, 547–571, 575, 987 Balakrishnan, N., 854 Balakrishnan, S., 831 Balcaena, S., 1597, 1598 Balduzzi, P., 986, 987, 1490 Bali, T.G., 985 Ball, C.A., 725, 737, 985, 1273, 1607 Ballie, R., 1607 Bank, P., 1007, 1008 Bansal, R., 717, 984, 1121, 1126, 1128 Bantwal, V.J., 756, 757 Banz, R.W., 109, 378, 1091, 1093 Bao, J., 670 Baourakis, G., 965 Barber, B.M., 1235, 1236, 1244 Barberis, N., 204, 289, 293, 301 Barclay, E.I., 1025 Barclay, M., 342 Barclay, M.J., 1028, 1153 Barle, S., 515, 520 Barles, G., 1007 Barndorff-Nielsen, O.E., 49, 515, 527, 528, 854, 873, 874, 876, 877, 1273, 1278, 1420, 1567–1570, 1572, 1574 Barnea, A., 27, 36, 1025, 1036 BarNiv, R., 1600 Barone-Adesi, G., 523, 552, 762, 1098, 1191 Barr, R.S., 643 Barraquand, J., 587 Barrett, B.E., 966 Barrett, C.B., 854, 855 Barry, D., 1347 Bartels, L.M., 1359, 1362 Bartter, B.J., 395, 409, 418, 505 Baruch, S., 341 Bas, J., 984 Baskin, J., 1036 Bassett, G., 836, 1411 Basu, S., 109 Bates, D., 526, 528, 532, 533, 547, 549, 550, 552, 555–558, 575, 1109 Bates, J.M., 42, 193, 199, 1476–1479 Baufays, P., 1347 Baum, D., 1007, 1008 Baumler, J.V., 1308 Baumol, W.J., 82, 1319 Bauwens, L., 1278, 1424 Bawa, V.S., 87, 192, 286 Baxter, M., 451 Bayer, P., 1206 Beaglehole, D.R., 754, 984, 1490 Beatty, R., 186 Beaver, W.H., 102, 117, 1227, 1596 Becher, D.A., 1427 Becherer, D., 1568 Beck, T., 226, 1007, 1367 Becker, B.E., 140, 144 Beckers, S., 1273, 1279, 1397, 1615 Beecher, A., 1598 Begley, J., 1328 Beja, A., 339, 606 Bekaert, G., 223, 226, 233, 984, 986, 1122, 1236, 1237, 1394 Beladi, H., xlv Belkaoui, A., 1098

Author Index Ben-Ameur, H., 584 Benchekroun, H., 1381 Beneish, M., 193 Ben-Horin, M., 68 Benkard, C.L., 1206 Benninga, S., 182, 619, 620, 623, 626 Bensoussan, A., 1574 Benston, G., 810 Benth, F.E., 1567–1575 Benveniste, L.M., 844 Benzoni, L., 715 Bera, A.K., 1333 Beranek, W., 1224 Bergantino, S., 802 Berger, A.N., 767, 770, 775, 1366, 1438 Berger, P.G., 863, 864, 869 Berk, J., 1223, 1237 Berkman, H., 676, 679 Berkowitz, J., 515, 525, 528 Bernard, R.H., 1319 Berndt, A., 996 Bernhard, R., 1224 Bernstein, P.L., 305–307, 342 Berry, M.J.A., 1324 Bertoin, J., 1447, 1569 Bertrand, P., 321, 324 Bertsekas, D., 91 Bessembinder, H., 342 Bessler, W., 342 Best, M.J., 205 Bey, R.P., 1098 Bhagat, S., 831 Bhamra, H.S., 670 Bharath, S., 669, 1602, 1603 Bhaskar, K.A., 1319 Bhatia, A.V., 649 Bhatia, M., 697, 699 Bhattacharya, M., 458 Bhattacharya, R. N., 401 Bhattacharya, S., 1400 Bhattacharya, U., 1236 Bhojraj, S., 186 Bhu, R., 802 Biagini, F., 1556 Biais, B., 339, 342, 1153 Bibby, B.M., 874 Bielecki, T.R., 671, 990, 1128, 1129 Bierman, H., Jr., 138 Bikbov, R., 715 Bingham, N.H., 874 Bishnoi, U., 802 Bismuth, J.M., 1556 Bitler, M.P., 1366 Bjerksund, P., 523, 524 Bjork, T., 448, 826, 1491, 1500 Black, F., 25–27, 79, 107, 188, 191, 199, 204, 267, 273, 274, 321, 324, 327, 355, 378, 385, 447, 454, 464, 465, 532, 576, 606, 611, 612, 665–667, 670, 728, 729, 763, 768, 809, 810, 910, 933, 934, 937, 985, 988, 1091, 1110, 1191, 1396, 1472, 1483, 1490, 1511, 1567, 1568, 1601, 1616, 1619 Blæsid, P., 877 Blair, B.J., 1333–1344 Blair, R.D., 682 Blalock, H.M., 1303 Blanco, R., 1000 Bliss, R.R., 292, 522, 986

Author Index Blomquist, G., 1206 Blum, M., 1594 Blume, M.E., 102, 108, 117, 610, 884, 886 Boardman, C.M., 1319 Bodhurta, J., 375 Bodie, Z., 172 Bodnar, G.M., 679 Bodurtha, J.N., 465 Boe, W.J., 1307 Boehmer, E., 1153 Bognanno, M.L., 138, 140, 143 Bogue, M., 605 Bohl, M.T., 1174 Bohn, H., 1236, 1237, 1247 Bohn, J.R., 668, 670 Bohn, J., 1602 Bollerslev, T., 526, 853, 899, 904, 1109, 1175, 1180, 1181, 1273, 1278, 1279, 1290, 1333–1336, 1341, 1344, 1347–1349, 1409, 1418–1421, 1607 Bollwerslev, T., 1394, 1395 Bonates, T.O., 641 Bondarenko, O., 534 Bongaerts, D., 670 Bontemps, C., 1206 Bookstaber, R.M., 375, 392, 418 Booth, L., 1223, 1224 Booth, N.B., 1347 Borison, A., 1042 Boros, E., 641–643 Borucki, L.S., 1465, 1467 Bose, I., 1324 Boser, B., 1269 Bossaerts, P., 290, 294 Botteron, P., 1041 Bouaziz, L., 585 Bouchet, M.H., 664 Boudoukh, J., 292, 587, 986, 987 Boudreaux, K.J., 1224 Bourke, P., 648 Bouye, E., 698 Bowden, R.J., 1206 Bower, D.H., 1465, 1467 Bower, R.S., 1465, 1467 Bowers, N.L., 958 Bowman, R.G., 68 Box, G.E.P., 1205, 1347, 1525–1527 Boyarchenko, S.I., 1446 Boyce, W.M., 1027 Boyd, J.H., 885 Boyle, P., 402, 405, 584 Bradbury, M.E., 676, 679 Bradley, M., 831, 836 Brailsford, T.J., 1333 Brancheau, J.C., 1316 Brander, J., 1381, 1399, 1403 Brandimarte, P., 182 Brandt, A., 1129 Brandt, M.W., 471, 718, 1109, 1118, 1273, 1274, 1276–1279, 1490 Branger, N., 1568 Bratley, P., 1137, 1139 Brav, A., 289 Brealey, R., 464, 1123 Brealey, R.A., 1593, 1604 Breeden, D.T., 283, 518, 737 Breen, W., 1068, 1075 Breidt, F.J., 1409, 1410

1687 Breiman, L., 1617 Brennan, M.J., 204, 289–318, 534, 667, 769, 778, 915, 916, 921, 937, 987, 1041, 1191, 1236, 1237, 1247, 1515 Brennan, S., 1000 Brenner, R., 737 Breton, M., 584 Brewer, E., 678, 686 Brewer, T.L., 648, 659 Brick, I.E., 1025–1027, 1223–1232 Bris, A., 1524, 1530 Britten-Jones, M., 458 Briys, E., 585, 666, 934, 988 Broadie, M., 505, 506, 667, 942, 947, 948, 1168, 1170 Brock, W.A., 447, 885 Brockett, P.L., 1065 Brockwell, P.J., 1410 Broll, U., 1179 Brooks, C., 853, 854 Brooks, R., 854 Brorsen, B., 853 Brotherton-Ratcliffe, R., 726, 727 Broto, C., 1421 Brown, D., 995 Brown, J.R., 1366 Brown, K.C., 999 Brown, R.H., 987, 1494 Brown, S.J., 46, 125, 127, 227, 249, 278, 283–287, 987, 1098 Bruce, P., 1062 Bruche, M., 515–528 Bruner, R.F., 273, 1223 Brunner, B., 522 Bubnys, E.L., 1465, 1545 Buchen, P.W., 1568 Bühler, W., 999 Bulirsch, R., 1033 Bulow, J., 1403, 1594, 1604 Burke, S.P., 853, 854 Burmeister, E., 1092 Buss, M.D.J., 1316 C Cadle, J., 319–332 Caginalp, G., 138 Cai, J., 1421, 1447 Cakici, N., 515, 520 Calantone, R., 1061, 1062 Campbell, J.Y., 289, 290, 292, 299, 896, 910, 986, 1068, 1080, 1087, 1283, 1474 Camerer, C., 137 Campa, J.M., 533, 1366 Campbell, J., 471, 668, 1174, 1176, 1393, 1394 Campbell, T.S., 677 Campello, M., 1366 Canina, L., 533, 1333, 1335 Cantor, R., 648 Cao, C., 524, 533, 537, 547–574, 670, 987, 1397 Cao, H., 1236, 1237, 1247 Capaldi, R.E., 290 Capobianco, E., 1420 Caporale, M.G., 1284 Cappiello, L., 1278 Carelton, W.T., 1025 Carey, M., 1280 Carey, M.S., 639, 640 Carhart, M.M., 1237, 1244, 1524, 1527, 1545

1688 Cario, M.C., 1136 Carlsson, C., 1196 Carpenter, J.N., 667, 942, 988 Carr, P., 526, 547, 556, 565, 575, 667 Carter, D., 695 Carverhill, A.P., 585 Casassus, J., 717, 916, 917, 1041 Casella, G., 1109, 1372 Cason, T., 138 Castellacci, G.M., 584 Cathcart, L., 988 Cebenoyan, A.S., 677, 679, 686 Cecchetti, S.J., 1122 Celec, S.E., 1319 Cesa-Bianchi, N., 1257 Cesari, R., 328–329 Cetin, U., 667 Chacko, G., 670, 714, 1128 Chakravarty, S., 1397 Chalamandaris, G., 447–470 Chalasani, P., 49, 587, 588, 590, 592, 593, 596–602 Chan, C.K., 1072, 1324 Chan, K., 221, 548, 557, 1374, 1396, 1397 Chan, L.K.C., 185, 193, 273, 832 Chan, R., 1179 Chan, S.H., 829, 831, 834 Chan, S.P., 1190 Chan, T., 1446 Chan, Y.L., 1394 Chance, D.M., 458, 465 Chang, C.-C., 226, 587, 762, 763, 767–778, 1028 Chang, E., 547, 556, 565, 1524, 1537 Chang, J.S., 233, 762, 763 Chang, J.-R., 697–711 Chang, K.H., 533 Chang, N., 1325 Chang, S., 1427, 1428 Chao, C., 1381 Chao, J.C., 1365, 1366 Chapman, D.A., 718, 986, 1490 Charupat, N., 137, 140 Chaudhury, M.M., 1523, 1526, 1534 Chauvin, K.W., 835 Chava, S., 996, 1397 Cheh, J., 1324 Cheh, J.J., 1325 Chen, A.-C., 697–711 Chen, C.R., 1301–1306 Chen, C.-S., 829–840 Chen, C.-J., 953–963 Chen, F., 186, 189 Chen, H., 641, 647, 670 Chen, H.-Y., 69–91, 125–135, 491–503 Chen, J., 1347 Chen, J.D., 802 Chen, L., 670, 1366 Chen, N.-F., 667, 1072, 1446, 1478 Chen, R., 725, 737, 950, 982, 1483–1486 Chen, R.-R., 439–446, 471, 473, 477, 531–545, 670, 990, 1055–1060, 1483–1488, 1490, 1615–1625 Chen, S., 1407 Chen, S.N., 98 Chen, S.-S., 873, 877, 933–950, 1160 Chen, W.-H., 641, 647 Chen, W.-P., 165–184, 807–818 Chen, Y.-T., 47, 1293, 1297–1300, 1445–1464

Author Index Chen, Y.-K., 965–976 Chen, Y.-D., 807–818 Chen, Z., 524, 533, 537, 547–575, 987, 1397, 1424, 1425 Cheng, J.K., 1319 Cheng, M., 341 Cheng, T.-C., 965 Cheng, X., 990 Cheng-Few, L., 1151–1162 Cheridito, P., 987, 1128 Chernov, M., 667, 715, 725, 947, 1109 Chesney, M., 1041 Chetty, V., 192 Cheung, C.S., 132 Cheung, Y., 1273 Chevalier, J., 1235, 1237, 1244, 1247 Chew, B.K., 1179 Chiang, R.C., 1025, 1036 Chiang, T.C., 853–862 Chiappori, P.-A., 1156, 1161 Chib, S., 986, 1109–1113 Chidambaran, N.K., 1607–1614 Childs, P., 942 Chiou, W., 226 Chiou, W.-J.P., 221–233 Chiras, D.P., 458, 1333 Chitashvili, R., 1556 Cho, D., 1281 Cho, M.-H., 1366 Chod, J., 1407 Choi, K.-W., 654 Choi, S., 843 Choi, T.S., 1319 Choo, E.U., 653, 1063 Chordia, T., 342 Chou, H., 1273–1280 Chou, R., 1273, 1274, 1277–1279 Chou, R. Y., 1273–1280, 1333 Chou, W., 1381 Chow, Y.F., 1179 Christensen, B.J., 1093, 1333–1335, 1343, 1491, 1500 Christensen, K., 1274, 1279 Christensen, M., 1211 Christie, W., 342 Christmann, A., 965 Christoffersen, P., 515, 524, 525, 528, 1276, 1278 Christopherson, J.A., 1068, 1070, 1085 Chu, C.C., 103–105, 1476, 1479 Chu, C.-S.J., 1347, 1349 Chuang, H.W., 575–581 Chui-Yu, C., 1190 Chung, H., 165–184, 807–815 Chung, S.L., 42, 505–513 Chung, Y.P., 1396 Church, B., 137 Churchill, G.A. Jr., 1474 Citron, J.T., 648, 659 Claessens, S., 863, 864 Clark, E., 660 Clark, P.K., 1175 Clarke, J., 1152 Clemen, R., 193, 194 Clewlow, L.J., 585 Cliff, M.T., 1366 Cochran, W.G., 859 Cochrane, J.H., 283–285, 290–292, 294, 533, 981, 986, 1067, 1069, 1070, 1082, 1086, 1394

Author Index Cohen, K.J., 339–341 Cohen, R., 610 Cohn, R.A., 1466 Collin-Dufresne, P., 666, 668, 670, 714–717, 721, 722, 884, 885, 910, 916, 917, 941, 949, 985, 988, 995, 996 Collins, A., 1206 Conley, T.G., 986 Conrad, J., 1071, 1174, 1366 Conroy, P., 843 Constantinides, G.M., 984, 1017, 1229, 1490 Cook, R.D., 966 Cooper, I.A., 996, 1225 Cooperman, E.S., 677, 679, 686 Copeland, M., 677 Copeland, T.E., 339, 340, 677, 1173, 1176 Cornell, B., 886 Corrado, C., 1279 Corrado, J.C., 587 Cortazar, G., 1041 Cosset, J., 221, 226 Cossin, D., 670 Cosslett, S.R., 1600 Costabile, M., 587 Costanigro, M., 1206 Costinot, A., 994 Cotton, P., 1118 Coulson, N., 194 Counsell, G., 1595 Court, A.T., 1201 Courtadon, G.R., 465 Cox, D., 1205 Cox, D.R., 453, 1525–1527 Cox, J., 15, 397, 471–473, 531, 547, 549, 550, 575, 576, 612, 617, 666, 667, 670, 754, 764, 1128, 1274, 1374, 1483, 1615, 1616, 1618, 1619, 1623 Cox, J.C., 399, 409, 415, 418, 425, 427, 453, 458, 464, 466, 483, 505, 579, 769, 826, 933, 934, 937, 982, 988, 1071, 1190, 1191, 1494, 1503, 1504, 1515, 1521 Cox, S.H., 757–759, 761 Crabbe, L., 886 Crack, T., 195 Cragg, J.G., 1364, 1367, 1368 Crama, Y., 641 Cramer, J.S., 1599, 1600 Crato, N., 1409, 1410 Crawford, A.J., 686 Cremers, K., 42, 187, 192, 199 Cremers, M., 668, 669 Cremonini, D., 328–329 Crosbie, P., 1602 Crouhy, M., 585 Csiszar, I., 1557 Cui, J., 235–245 Culp, C.L., 676 Cumby, R., 1537 Cumming, J.D., xlvi Curran, M., 584, 587 Curry, T., 643 Cutler, D.M., 138 Cvitanic, J., 1007, 1008 Czyzyk, J., 646 D D’Antonio, L.J., 21, 873, 878 D’Arcy, S.P., 1466 D’haeseleer, P., 1325

1689 Dahiya, S., 1596 Dahya, J., 866 Dai, Q., 671, 713, 717, 718, 912, 979, 981, 983, 984, 986, 987, 1121, 1126–1129, 1490 Daigler, R.T., 1173 Daines, R., 1366 Damodaran, A., 186, 192, 198, 1152 Daniel, K., 199, 273, 274, 831 Dao, B., 942, 1446 Daouk, H., 912 Das, S., 208, 665, 714, 1121, 1128, 1490 Dassios, A., 763 David, A., 670 Davidson, G.K., 1162 Davidson, R., 1363, 1365, 1478 Davis, J.L., 273 Davis, M., 994 Davis, P., 334, 343–345 Davis, R.A., 1410 Davydenko, S.A., 668 Dawid, A.P., 1260, 1264 Day, T., 548, 555 De Jong, A., 873 De Jong, F., 226, 670 De Lima, P., 1409, 1410 De Long, B., 1395 De Roon, F., 873 De Santis, G., 221, 1235 De Schepper, A., 586 de Servigny, A., 641, 644 de Varenne, F., 666, 934, 988 Deakin, E.B., 1597, 1598 DeAngelo, H., 1230 Deaton, A., 42, 197, 199 Deaves, R., 137, 140 DeGennaro, R., 1607 Del Guercio, D., 865 Delbaen, F., 465, 1012, 1018, 1555, 1557, 1567 Delianedis, G., 668, 949 Dellacherie, C., 1556, 1560, 1561 DeLong, G.L., 1427, 1439 DeLong, J.B., 138 DeMarzo, P., 677, 684, 1223 Demirgüç-Kunt, A., 226 Demmel, J., 1033 Demsetz, H., 340, 1041 Demsetz, R.S., 683, 767, 770, 775 Denis, D.J., 1366 Dennis, P., 533 Dennis, S., 1366 Dentcheva, D., 247–257 Denuit, M., 586, 593, 594 Deo, R.S., 1414 Derman, E., 458, 464, 985, 1511 DeRoon, F.A., 173 Derrien, F., 844, 847 Desai, M.A., 1366 Deshmukh, S.D., 678, 686 Detemple, J., 505, 506 Deuskar, P., 734 Devetsikiotis, M., 1136 Dewachter, H., 985, 987 Dhaene, J., 586, 593, 594 Diaconis, P., 1137 Diamandis, P., 1352 Diamond, D.W., 678, 686

1690 Diamond, D.B.Jr., 1206 Diavatopoulos, D., 137–163 Diba, B.T., 139 Diebold, F.X., 526, 885, 1175, 1273, 1274, 1278, 1279, 1421, 1500 Diebold, F., 193, 194, 1395 Diener, F., 505 Diener, M., 505 Diewert, W.E., 1206 Dillon, W., 1061, 1062 Dimitras, A.I., 965 Dimson, E., 273, 1333 Ding, Z., 1409, 1411 Dittmar, R., 1614 Dittmar, R.F., 746, 1001, 1133, 1500 Dixit, A., 1381 Djankov, S., 226 Do, B.H., 321, 328 Doherty, N.A., 1466 Doleans-Dade, K., 1556, 1557, 1562 Domowitz, I., 340 Donald, S.G., 1364, 1367, 1368 Dong, G.N., 1209–1220 Dongarra, J., 1033 Doob, J.L., 1262 Dothan, U., 1506, 1515 Douglas, A.V.S., 1028 Douglas, G.W., 107 Doukas, J., 829, 831, 834 Doumpos, M., 965, 1061 Dow, S., 830 Dowd, K., 1411 Downing, C., 1397 Dreher, A., 1368 Dreiman, M., 1206 Driessen, J., 221, 226, 993, 995 Driffill, J., 1122 Duan, C.W., 915–931 Duan, J.C., 533, 670, 754, 819, 823, 825, 949 Duarte, J., 715, 717, 729, 743, 744, 746, 987, 1128 Dubitsky, R., 802 Dubois, D., 1184 Duck, P.W., 513 Duckstein, L., 1330 Duff, I., 1033 Duffee, G.R., 714, 815, 949, 987, 993, 995, 1490 Duffie, D., 342, 447, 464, 665, 667, 671, 677, 684, 714, 716–718, 723, 726, 734, 748, 981–985, 988, 990, 993, 995–997, 999, 1000, 1012, 1016, 1121, 1128, 1270, 1490, 1491, 1499, 1599 Dufresne, F., 1445, 1453, 1454 Dufwenberg, M., 138 Dugan, M.T., 1328 Duiffie, G., 1128 Duin, R., 40 Dumas, B., 515, 522, 532, 533, 542, 547, 555, 556, 1237, 1244 Dunbar, C., 1162 Dunn, O.J., 1598 Duong, H.N., 1279 Dupire, B., 458, 518 Durand, D., 144 Durand, R., 934 Durbin, J., 1363 Durham, G.B., 986 Durrleman, V., 711 Dybvig, P.H., 208, 987, 1505 Dyl, E.A., 80

Author Index E Eades, K.M., 273, 1223 Easley, D., 340, 1397 Ebeling, W., 1568 Ebens, H., 1278, 1334, 1335, 1338 Eberhart, A., 941 Eberlein, E., 533, 874 Eckardt, W., 375 Eckbo, B.E., 1400, 1406 Economides, N., 341 Ederington, L.H., 995 Ederington, L., 883 Edwards, A.K., 844 Edwards, S., 1283 Eftekhari, B., 274 Egriboyun, F., 49, 587, 588, 590, 592, 593, 599–602 Ehrenberg, R.G., 138, 140, 143 Eisenbeis, Robert A., 1597, 1598 Eisenbeis, Robert O., 1603 El Karoui, N., 1490, 1555–1557 Eldridge, S.W., 1324, 1326, 1328, 1329 Elerian, O., 986 Eliasson, A., 641, 648 El-Jahel, L., 988 Elliott, G., 1280 Elliott, R.J., 1128, 1129, 1557 Ellison, G., 1235, 1237, 1244, 1247 Elton, E.J., 46, 82, 125, 127, 128, 130, 132, 133, 208, 227, 235, 236, 249, 285, 884, 885, 991, 995, 1465, 1466 Emanuel, D., 471, 477, 532, 1615, 1618, 1619 Embrechts, P., 698, 994 Emery, W.G., 1025 Engle, R., 342, 738, 1274, 1277, 1278 Engle, R.F., 532, 885, 901, 903, 909, 1109, 1122, 1176, 1180, 1181, 1273, 1284, 1285, 1291, 1347, 1348, 1409, 1417–1419, 1421, 1422, 1424 Eom, Y., 986, 987 Eom, Y.H., 667, 668, 670, 949, 995, 999 Epple, D., 1206 Epps, M., 1173–1175 Epps, T., 1173–1175 Eraker, B., 725 Ericsson, J., 668, 670, 943, 949, 950, 995 Errunza, V., 226, 1235, 1527 Ervine, J., 375 Estep, T., 330 Estrella, A., 985 Eum, S.J., 1206 Eun, C., 1524, 1537 Evans, M.D., 1122 Evans, J.L., 79 Ewing, B.T., 1352 Ezzamel, M., 1598 Ezzell, J.R., 686 Ezzell, R., 1223–1227, 1229, 1231 F Fabozzi, F., 912 Fabozzi, F.J., 98, 1523, 1525, 1526 Faccio, M., 1427, 1428 Fadil, M.W., 700 Faff, R.W., 321, 328, 1333 Fairchild, J.R., 757–759 Fairley, William B., 1466 Fama, E., 193, 448, 453, 853, 864, 1088, 1394, 1615 Fama, E.G., 273

Author Index Fan, K.L., 1577 Fan, R., 714 Farber, A., 1223 Farragher, E.J., 1319 Faulkender, M., 1366 Fawson, C., 854, 855 Feenstra, R.C., 1206 Fehle, F., 999 Feller, W., 42, 297, 471–473, 478, 479, 1617 Feltham, G., 185 Fernandes, M., 1278 Fernandez, P., 1225 Ferreira, F., 1206 Ferri, G., 645, 648, 654, 664 Ferruz, L., 267–280 Ferson, W.E., 680, 1067–1089, 1072, 1082, 1085, 1537 Figlewski, S., 505, 533, 558, 1333, 1335, 1607 Filipovi, D., 983, 985, 987 Filipovic, D., 987, 1128, 1499 Findlay, M.C., 247 Finger, C.C., 697, 699 Fink, K., 1221 Finnerty, J., 375, 392, 418 Finnerty, J.E., 69–91, 93–105, 111–122, 377–392 Firoozi, F., 1381–1389 Firth, M., 644 Fischer, E., 941, 943 Fishburn, P.C., 247, 249, 873 Fisher, F.M., 1303 Fisher, L., 1067 Fisher, R., 1062 Fisher, S., 915, 916, 920 Fitzpatrick, P.J., 1596 Flanagan, C., 802 Flannery, B., 1271 Flannery, M.J., 676–680, 686, 694, 819, 823, 825 Flannery, M., 884 Fleming, M.J., 884, 885, 912 Fleming, J., 233, 515, 522, 1070, 1333–1335, 1343, 1529 Flesaker, B., 1490 Fletcher, J., 221 Flood, R.P., 138 Flood, R., 1393 Flores, E., 965 Fluck, Z., 1028 Fogler, H.R., 1303 Foley, C.F., 1366 Follain, J.R., 1202, 1206 Follmer, H., 1555 Fomby, T.B., 1095 Fong, W.M., 1179, 1396 Fons, J.S., 669 Fontes, D.B.M.M., 1041 Forbes, S.W., 1465, 1466 Foresi, S., 986, 987, 1490 Forrest, S., 1325 Forsyth, P., 1218 Forsythe, R., 140, 145 Foster, D.P., 1265, 1333 Foster, D., 1265 Foster, F.D., 1068, 1069, 1076, 1079 Foster, G., 1594 Foster, K.R., 515, 520 Foucault, T., 341 Foufas, G., 1219 Fouque, J.-P., 1109–1119

1691 Fouse, W., 122 Fox, B.L., 1137, 1139 Fox, I., 831 Francioni, R., 333–347 Francis, J.C., 73, 83, 98, 259–266 Francois, P., 667, 941, 947 Frank, E., 1324 Frankfurter, G., 122 Franks, J., 941 Franses, P.H., 1333 Frecka, T.J., 1224, 1597, 1598 Freed, N., 1062, 1063, 1325 Frees, E.W., 994 French, K., 193, 1088, 1247, 1394 French, K.R., 221, 273, 289, 290, 298, 915, 916, 920, 921, 930, 931, 988, 1068, 1071, 1073, 1075, 1091, 1092, 1094, 1095, 1100, 1102, 1105, 1107, 1467, 1468, 1474, 1607 Frey, R., 995 Fridman, M., 1109 Friedberg, S.H., 1451 Friedman, B.M., 151 Friedman, J., 1069 Friend, I., 108, 610, 863 Fritelli, M., 1567 Froot, K.A., 137, 140, 677, 678, 680, 686, 763, 986, 1236, 1237 Fu, H.C., 587 Fu, M., 585 Fudenberg, D., 1403 Fuh, Cheng-Der, 1577–1591 Fujiwara, T., 1568 Fuller, K., 1427, 1428 Fuller, R., 1196 Fuller, W.A., 1095, 1365 Fung, H.-G., 644 Fung, W., 196 Furbush, D., 676 G G’omez Sanchez-Mariano, E.G., 1377 G’omez-Villegas, E.G., 1377 Gacs, P., 1265 Galai, D., 340, 465, 549, 565 Gale, D., 139 Galindo, J., 641 Gallant, A.R., 717, 718, 983, 986, 1121, 1128, 1421, 1490 Gallant, R., 1109, 1174, 1279 Gallant, S., 1061 Galloway, T.M., 677, 679, 686 Gammerman, A., 1257 Gamst, G., 1063 Ganapathy, S., 1303 Gang, T., 802 Gao, B., 505, 717, 718, 986, 987 Gao, Z., 185 Garber, P., 137 Garcia, R., 984, 1122, 1123 Gardeazabai, J., 1395 Garleanu, N., 1599 Garmaise, M.J., 1366 Garman, M.B., 321, 339, 1273, 1275, 1279 Garrido, J., 965 Garven, J.R., 457 Gastineau, G., 375 Gastwirth, J.L., 250 Geanakoplos, J., 1403 Geczy, C., 679

1692 Gehrlein, W.V., 654 Geman, D., 1371 Geman, H., 585, 586, 761, 762, 1023 Geman, S., 1371 George, C., 447–465 George, E.I., 1372 George, T., 565 Gerard, B., 221, 1235 Gerber, H.U., 47, 1445, 1447, 1453, 1454, 1457, 1462 Geske, R., 41, 439–442, 532, 613, 666–668, 670, 934, 949, 988, 1191, 1615 Ghosh, S.K., 1305 Ghysels, E., 273, 725, 1109, 1420 Gibbons, M., 286, 1615, 1616 Gibbons, M.R., 1069, 1091, 1092 Giesecke, K., 667 Gikhman, I., 451 Gillan, S., 864, 865 Gillet, R., 1223 Gilli, M., 754 Gil-Pelaez, J., 485, 489 Gilroy, B.M., 1041 Gilson, S., 198 Gitman, L., 1223 Givots, S., 375 Glasserman, P., 727, 1490 Glassman, D., 1068, 1070, 1085 Glen, J., 342, 1537 Gloersen, P., 1577 Glosten, L., 340, 341, 471, 854, 1007, 1009, 1223, 1333, 1336, 1340, 1348 Glover, F., 1062, 1063, 1325 Gockenbach, M., 1219 Goettler, R., 341 Goetzmann, W.N., 46, 127, 227, 249, 278, 285, 289, 1524, 1530 Goetzmann, W., 46, 227, 1206 Goh, J., 1151 Goicoechea, A., 1308 Golden, L.L., 1065 Goldman, E., 1373, 1377 Goldman, M.B., 339 Goldstein, R.S., 666, 668, 670, 715, 716, 721, 722, 885, 910, 941–943, 945–949, 985, 986, 988, 1490 Gollinger, T.L., 700 Gombola, M.J., 1151 Gomes, P., 641 Gómez, A.T., 1190, 1197, 1198 Gómez-Bezares, F., 267–280 Gompers, P., 1366 Goovaerts, M.J., 586, 593, 594 Gordon, M.J., 606, 1466 Gordon, R., 1283 Gorovoi, V., 1490 Gorton, G., 138, 139, 197, 676, 677, 683, 686 Goswami, G., 1025, 1036 Gould, J.P., 47, 1293, 1296 Gourieroux, C., 1156, 1555 Goyal, A., 290, 294, 295, 301, 1089 Graham, J., 675–677, 680, 682, 1028, 1224 Grammatikos, T., 1275 Grandits, P., 1555, 1557, 1567, 1569 Granger, C., 42, 193, 199, 342, 607, 1409, 1476–1479 Granger, C.W.J., 44, 45, 1067, 1068, 1072, 1075, 1076, 1279, 1392, 1409–1411, 1417, 1465–1480 Grant, D., 587 Grauer, R.R., 203–218

Author Index Gray, J.B., 966 Gray, S.F., 984, 1122, 1123, 1421 Green, K., 886 Green, R.C., 208, 226, 1030, 1237 Greenbaum, S.I., 678, 686 Greene, W.H., 680, 1303, 1305, 1472, 1599, 1600 Gregory, A.W., 986 Gressis, N., 92 Grice, J.S., 1328 Griffin, J.M., 139, 844 Griffiths, W., 1095, 1472, 1481 Griliches, Z., 1364 Grinblatt, M., 208–210, 273, 274, 996, 997, 1000, 1236 Grinold, R., 187 Groslambert, B., 664 Grossman, H., 139 Grossman, S., 340, 341, 865, 1007, 1009, 1174, 1176 Groznik, Peter., 1236 Gruber, M.J., 46, 82, 125, 127, 128, 130, 133, 135, 208, 227, 235, 236, 249, 285, 884, 885, 991, 995, 1235, 1237, 1244, 1465, 1466 Grunbichler, A., 913 Grundy, K., 273 Guarino, A., 1063 Guasoni, P., 1556 Guedes, J., 1036 Gugler, K., 836 Gulen, Huseyin., 1397 Guo, D., 668, 949 Guo, J., 1168, 1169, 1171 Guo, Jia-Hau, 1165–1171 Guo, W., 1568 Guo, X., 667, 1056 Gupta, A., 714, 996, 1427–1443 Gupta, A.K., 1347 Gupta, M.C., 1025–1037, 1297 Gupta, S., 1381 Guy, J., 117 Guyon, I., 1269 Gyetvan, F., 1330 H Haas, G.C., 515, 517, 523, 524 Haas, M., 1201 Hackbarth, D., 667, 942 Hadar, J., 247, 249 Hafner, R., 522 Hagerman, R., 810 Hahn, J., 1365, 1366 Hahn, W.F., 1206 Hai, W., 1395 Hakanoglu, E., 321 Hakansson, N.H., 203–205, 289, 339, 609 Haldeman, R.G., 1598, 1603 Hall, G.R., 1466 Hall, P., 1414 Hallock, K., 836 Halverson, R., 1205 Hamada, R.S., 35, 683, 684 Hamao, Y.R., 1283 Hambly, B., 517 Hamer, M.M., 1599 Hamidi, B., 324, 329 Hamilton, J.D., 1122–1125, 1397, 1421 Hammeed, A., 1174 Hammer, P.L., 640, 641, 643, 644, 648–650, 652, 654, 655, 657, 658 Hamori, S., 1283

Author Index Han, B., 716, 724, 725, 727, 728 Han, C.-H., 1109–1120 Han, J., 1324 Han, S., 670 Handa, P., 341, 346 Hanke, J., 1276 Hanley, K.W., 843, 844 Hanoch, G., 39, 249 Hansen, B.E., 808, 811, 814, 854 Hansen, L.P., 284–286, 555, 1362 Hao, Q., 844, 845, 847, 849 Hao, X.R., 1308 Haque, N.U., 641, 659 Harbus, F., 1025, 1028, 1036 Härdle, W., 543 Hardouvelis, G.A., 985 Hardy, G.H., 247 Harjes, R., 737 Harlow, W.V., 879, 999 Harrington, S.E., 763, 1466 Harris, C., 1381, 1382 Harris, J.H., 139, 844 Harris, J., 342 Harris, L., 344, 1109, 1173, 1175 Harris, M., 863, 864, 885, 936 Harris, R.S., 30 Harrison, J., 458, 1619 Harshbarger, S., 627 Hart, O., 865 Hart, S., 1382 Hartigan, J.A., 1347 Hartzell, J.C., 864, 865 Harvey, A., 1109, 1277, 1409, 1410 Harvey, C., 342, 556, 1028, 1236, 1237, 1537 Harvey, C.R., 680, 1069, 1070, 1082, 1224, 1334, 1366 Hasbrouck, J., 339, 342, 1395, 1396 Haslett, J., 966 Hastie, T., 1069 Hathorn, J., 1600 Haug, E.G., 465 Haugen, R., 122 Haugen, R.A., 27, 1466 Haushalter, G.D., 677, 679 Hausman, J., 692, 1361, 1363 Hawkins, J., 865 Hayashi, F., 197, 1361 Hayre, L.S., 802 Hayt, G.S., 679 Hayya, J., 92 Hazarika, S., 333–347 He, G., 191 He, H., 999 He, J., 669 Healy, P.M., 1406 Heath, D., 716, 981, 1008, 1490, 1494 Heath, M., 1033 Heaton, J.B., 289 Heckman, J., 1358 Heggestad, A.A., 682 Heidari, M., 714, 1490 Heine, R., 1025, 1028, 1036 Heinemann, F., 144 Heinkel, R., 916 Helman, P., 1325 Helmstadter, E., 841 Helwege, J., 669, 886, 995, 1604

1693 Hendershott, T., 342, 1028, 1153 Henderson, J., 71 Hendricks, D., 1237 Henke, H., 1174 Henry, D., 1028 Hentschel, L., 910 Heravi, S., 853, 854 Hermalin, B.E., 866, 1357 Herold, U., 323, 324, 329 Hertzel, M.G., 1151, 1152 Heston, S., 47, 458, 481, 485, 487, 489, 505, 506, 510, 531, 547, 549–551, 575, 577, 725, 884, 1165 Heynen, R., 533 Hiemstra, C., 1174 Higgins, M.L., 1333 Higgins, R.C., 24, 25, 100 Hilberink, B., 47, 667, 1445, 1446, 1451–1453, 1458 Hill, J.R., 1140 Hill, R., 1095, 1472, 1481 Hill, R.C., 1303, 1305, 1598 Hill, R.D., 1466 Hillebrand, E., 1421 Hilliard, J.E., 587 Hillier, F.S., 1319, 1320, 1323 Hillion, P., 290, 294, 1153 Hines, J.R. Jr., 1366 Hirschey, M., 836 Hirshleifer, D., 199 Hirtle, B., 686 Hjort, N.L., 1412 Hlavka, M., 208 Ho, H.C., 1412, 1413 Ho, K.W., 1160 Ho, K.Y., 165–184 Ho, L.C., 319–332 Ho, T., 324, 329, 335, 1483 Ho, T.S.Y., 988, 1041 Ho, Y.K., 832, 840 Hobson, D., 1568 Hocking, R.R., 1203 Hodges, S.D., 1567 Hodrick, R., 1393, 1394 Hodrick, R.J., 138, 986 Hoeting, J., 42, 192, 199 Hoff, P.D., 1376 Hoffman, C., 523 Hoffman, N., 1555 Hofmeyr, S.A., 1325 Hogan, K., 226, 233, 1235 Hogg, R.V., 854 Holden, C.W., 266 Holderness, C.G., 1153 Holland, J.H., 1611 Hollifield, B., 226 Holt, C.A., 144 Homan, M., 1595 Hong, C.H. Ted, 779–805 Hong, H.G., 138 Hong, H., 1524 Hong, Y., 987 Hopwood, W.S., 1597, 1598 Horvath, L., 1349 Hoshi, T., 834 Hosking, J.R.M., 854, 1410 Hotchkiss, E., 198 Hourany, D., 144

1694 Houston, J.F., 686, 836, 1025, 1427, 1434, 1439, 1441 Houweling, P., 670, 990, 999, 1000 Hovakimian, A., 1028 Howard, C.T., 21, 873, 878 Howe, J.S., 1153, 1154 Howe, K.M., 1319 Howe, M., 916 Howton, S., 679 Hricko, T., 670 Hsieh, D., 196, 1421 Hsieh, D.T., 829–840 Hsieh, J., 1366 Hsieh, P.F., 767–778 Hsing, T., 1414 Hsu, C., 1279 Hsu, C.-J., 641, 647 Hsu, D.A., 1345, 1347, 1349 Hsu, D., 1345–1354 Hsu, J., 185, 193 Hsu, M., 593 Hsu, M.F., 587–602 Hsu, T.L., 165–184 Hsu, Y.L., 471–480, 575–581 Hu, J.N., 885 Hu, W., 669 Hu, Y.-T., 654 Hu, S., 950 Huang, B.-Y., 965–976 Huang, C., 447, 464 Huang, Dar-Yeh Huang, J., 550, 949, 995 Huang, J.-Z., 665–671 Huang, M., 666–670, 949, 988, 995–997 Huang, Ming Huang, N.E., 1577, 1578 Huang, R.D., 808 Huang, R., 342 Huang, Z., 641, 647 Hubbard, R.G., 686 Huberman, G., 172, 173, 203, 883, 1071 Hudson, J., 1594, 1596 Hueng, C.J., 854 Hughes, J., 685 Hughes, J.S., 916 Hughston, L., 1490 Hull, J., 49, 458, 464, 531, 547, 549, 551, 584, 586, 587, 593, 599–602, 670, 698, 727, 982, 996, 999, 1000, 1145, 1191, 1210, 1269, 1483, 1611, 1615 Hulton, C., 1205 Hung, K., 235–245 Hung, Ken Hung, M., 1168, 1169, 1171 Hung, M.-W., 1165–1171 Hung, W.-L., 854 Hurvich, C.M., 293, 294, 1414 Huselid, M.A., 140, 144 Husick, F., 106 Huson, M.R., 1151, 1152 Hutchinson, J., 1607, 1610, 1611, 1613 Hwang, D.Y., 819–826 I Ibaraki, T., 641, 642 Ibbotson, R.G., 68 Ibbotson, R.C., 1100 Ik, K.H., 1407

Author Index Ilmanen, A., 305, 307 Imamura, Y., 1283 Imbens, G.W., 1365, 1366 Inclan, C., 1349, 1351 Ingersoll, J.E., Jr., 483, 549, 754, 764, 1128, 1483, 1503, 1504, 1515, 1521, 1522, 1618, 1623 Ingram, R.W., 1328 Insel, A.J., 1451 Iryna, I., 1284 Isaac, R.M., 138 Isberg, S.C., 829–831, 839, 840 Israel, R., 864 Itô, K., 578 Ivanov, S., 802 Ivkovic, Z., 1366 J Jäckel, P., 1168, 1171 Jacklin, C., 1615, 1616 Jackson, D., 144 Jackson, M., 182 Jackson, R., 92 Jackson, W.E., 686 Jackwerth, J.C., 520, 532, 533, 738 Jacob, N., 122 Jacobs, K., 524, 528 Jacod, J., 1556 Jacques, M., 584–586 Jacquier, E., 1109, 1110, 1421 Jaffe, J.F., 1154 Jagannathan, R., 226, 273, 283–285, 471, 714, 721, 854, 885, 1067, 1068, 1070, 1072, 1075, 1082, 1086, 1333, 1336, 1340, 1348, 1467, 1468 Jagerman, D.L., 1136, 1139, 1141 Jahera, J.S., 1438 Jahnke, W., 122 Jaimungal, S., 758, 759 Jain, K., 641 Jain, P.C., 1151 James, C., 686, 836, 1427, 1439 James, C.M., 676–680, 686, 694 Jamshidian, F., 1490 Jang, B.S., 1284 Jang, H.J., 1153 Jang, J.-W., 763 Janosi, T., 996 Jarrell, G., 836 Jarrow, R.A., 361, 362, 505, 507, 547, 665, 667, 713, 716, 734, 744, 746, 755, 961, 981, 996, 1007–1017, 1490, 1494 Jeanblanc, M., 942, 1446 Jegadeesh, N., 274, 1092 Jelenkovic, P., 1140 Jensen, M., 36, 611, 835, 836, 936, 947 Jensen, P.H., 1206 Jeon, B.N., 1283, 1284 Jha, S., 49, 587, 588, 590, 592, 593, 596–602 Ji, L., 1500 Jimenez, E., 1202 Jin, Y., 1490 Jing, B.-Y., 1414 Joachims, T., 1269 Jobson, J.D., 173, 204 Joe, H., 994 Johannes, M., 984, 987 John, K., 1153, 1154, 1399–1407 John, T.A., 829, 1036, 1400, 1407

Author Index Johnson, H.E., 1191 Johnson, K.M., 1366 Johnson, L.L., 20, 21, 873 Johnson, N., 1618, 1622 Johnson, S.A., 832, 836 Johnson, S.R., 1095 Johnson, T.W., 1523 Johnson, T.C., 1000, 1001 Jones, C., 342, 1273, 1274, 1277, 1279, 1524 Jones, C.M., 884, 885, 1173, 1174, 1176 Jones, C.S., 986 Jones, E.P., 668, 995 Jones, J., 1174 Jones, R., 321 Jordan, B., 1223 Jorge, C.H., 1284 Jorion, P., 204, 289, 533, 703, 1000, 1334, 1527 Jouini, E., 1007 Joy, M.O., 1603 Joyeux, R., 1410 Ju, N., 600, 941, 942, 948 Judge, G.G., 1095, 1303, 1305, 1472, 1481, 1598 Jung, K., 864, 869, 1061, 1312 Jurczenko, E., 332 K Kücher, U., 874 Kaas, R., 586, 593, 594 Kacprzyk, J., 1190 Kaden, O., 341 Kahl, C., 1168, 1171 Kahle, K.M., 1152 Kahn, J., 195, 196 Kahn, R.N., 278–280 Kakade, S., 1265 Kaldor, N., 197, 915 Kalev, P.S., 1279 Kallal, H., 1007 Kalman, R.E., 1495 Kalotay, A.J., 1027 Kamakura, W., 1061 Kamber, M., 1324 Kamien, M.I., 831 Kaminsky, G., 648, 664 Kan, R., 172, 174, 717, 983, 985, 1121, 1128, 1490 Kanatas, G., 678, 686 Kandel, E., 341, 885 Kandel, S., 172, 173, 204, 209, 289, 1070, 1071 Kander, Z., 1347 Kane, A., 172, 941 Kane, E.J., 679, 686 Kang, Q., 471 Kani, I., 458, 464 Kantor, J., 1307 Kaplan, S.N., 185, 198, 769, 1224, 1430 Kaplan, P., 1393 Kaplin, A., 714, 721 Karasinski, P., 985, 1483 Karatzas, I., 447, 448, 460, 517, 1017 Karceski, J., 233, 1446 Karels, G.V., 1597, 1598 Karjalainen, R., 1608 Karlin, S., 483, 1617 Karlsen, K.H., 1568 Karolyi, A.G., 1284, 1379 Karolyi, G.A., 548, 557, 1122, 1515

1695 Karpoff, J.M., 1152 Kashyap, A., 685, 834 Kat, H.M., 1333 Kau, J.B., 1205 Kaufman, H., 342 Kaufmann, D., 649 Kaul, G., 342, 1071, 1173, 1174, 1176 Kavajecz, K.A., 1366 Kavanagh, B.T., 676 Kavee, R.C., 324–327 Kaya, Ö., 1168, 1170 Kazamaki, N., 1557, 1560, 1563 Kealhofer, S., 669 Keane, M.P., 1161 Keasey, K., 1597–1599 Kedia, S., 1366, 1407 Keeley, M.C., 695 Keenan, S.C., 656, 658, 967 Keeton, W.R., 684 Keim, D.B., 109, 289, 884, 886, 1068, 1075, 1091 Keirstead, W., 1490 Keller, U., 533, 874 Kellezi, E., 754 Kellison, S.G., 955 Kelly, F.P., 1141 Kelly, M., 1568 Keloharju, M., 1236 Kelton, D.W., 1137 Kemna, A.G.Z., 533, 584, 585 Kendall, M.G., 1072 Kennedy, D.P., 986 Kennedy, P., 1598 Kensinger, J., 829, 831, 834 Kent, J., 854 Keown, A.J., 1153 Kerfriden, C., 819, 822 Kettler, P., 102, 117 Keynes, J.M., 18, 915, 919 Khindanova, I., 853 Khorana, A., 1237, 1247 Kiang, M., 1061, 1062 Kiesel, R., 654, 874 Kijima, M., 993 Killough, L.N., 1313 Kim, C.J., 1123 Kim, D., 273, 1091–1108, 1467, 1468, 1481 Kim, E.H., 831, 836 Kim, I., 940, 942, 988 Kim, I.-J., 552 Kim, S., 1110–1113 Kim, S.H., 1319 Kim, T.H., 587 Kim, T.S., 299 Kim, Y.C., 864, 869 Kim, Y.S., 1568 Kimmel, R., 987, 1128 King, B., 122 King, D., 1025 King, R.R., 148 Kirby, C., 233, 1070 Kish, R.J., 913 Kishore, V., 1270 Klaoudatos, G., 1136 Klapper, L., 1596 Klarman, S., 185, 198 Klass, M., 1273–1275, 1279

1696 Klassen, T.R., 587 Klebaner, F.C., 503 Kleibergen, F., 1365 Kleidon, A.W., 289 Klein, R., 192 Klemperer, P., 1403, 1594, 1604 Klimberg, R., 1061–1066 Kluger, B., 137, 140 Klugman, S.A., 854 Knez, P.J., 1490 Knight, J., 1417 Knittel, C.R., 1206 Koekebakker, S., 1042 Koenker, R., 836, 1411 Kogan, A., 639–664 Kohlhagen, S.W., 321 Kokoszka, P., 1349 Kolbe, A.L., 1466 Kolmogorov, A.N., 1264, 1267 Kolodny, R., 533, 1524, 1537 Kong, W., 668, 669 Konno, H., 248 Koopman, S., 342 Kopprasch, R., 321 Korajczyk, R., 1072 Korkie, B., 173, 204 Korwar, A.N., 1151 Kosmidou, K., 965, 1066 Kosturov, N., 883–913 Kothare, M., 807 Kothari, S.P., 273, 1068, 1092 Kotz, S., 421, 425, 1260, 1618, 1622 Kou, G., 1326 Kou, S., 666, 667, 1455 Kou, S.G., 47, 727 Koza, J.R., 1610, 1611 Kozicki, S., 985 Kracaw, W.A., 677 Krafet, D., 1283–1285 Kraus, A., 611, 1466 Kreps, D., 139, 550, 826, 981, 1126, 1130, 1619 Krishnamurthy, S., 1152 Kritzman, M., 330 Kroncke, C.O., 1466 Kroner, K., 737, 1181, 1279, 1333, 1349 Kroner, K.F., 1176, 1283–1285 Krueger, A.B., 1365, 1366 Kruse, S., 1168 Kshirsagar, A.M., 1597 Kudbya, S., 1061–1066 Kuehn, L.-A., 670 Kuersteiner, G., 1365, 1366 Kulatilaka, N., 375, 392, 418 Kumar, M.S., 641, 659 Kumar, A., 1244 Kumon, M., 1264 Kunczik, M., 640 Kunitomo, N., 1273, 1275 Kunreuther, H.C., 756, 757 Kuo, H.-C., 1593–1604 Kurbat, M., 669 Kwak, W., 1307–1329 Kwakernaak, H., 1184 Kwan, C.C.Y., 132 Kwan, S.H., 886, 1397 Kyle, A.S., 340, 343, 1007, 1009, 1173

Author Index L La Porta, R., 226 Labys, P., 1273, 1274, 1278 Labys, W.C., 236 Lackner, P., 465 Laderman, E., 802 Laeven, L., 221, 226 Lagnado, R., 204, 289 Lahiri, S.N., 1414 Lai, H.-N., 767–777 Lai, T.L., 1417–1425 Lai, Y., 1118 Lakonishok, J., 185, 193, 233, 1153, 1174, 1446 Lam, K., 1063, 1421 Lam, K.F., 1065 Lam, P., 1122 Lambadaris, I., 1136 Lamberton, D., 447, 460 Lamont, O., 1524 Lamoureux, C., 1180 Lancaster, K.J., 1202 Landen, C., 1128, 1129 Lando, D., 665, 667, 669, 671, 988–990 Landry, B., 47, 1445, 1447, 1462 Landsburg, S., 195, 196, 278–280 Lang, L., 831, 834, 863, 864, 1026, 1028, 1036, 1400, 1401, 1406 Lang, L.H.P., 669, 863, 999, 1153, 1154 Lang, W., 685 Lapeyre, B., 447, 460 Larcker, D.F., 1359, 1366 Laroque, G., 42, 197, 199 Larrain, G., 648 Larson, M., 1219 Last, G., 1129 Lastrapes, W., 1180 Latane, H.A., 1333, 1397 Lathom, G.P., 1324 Launie, J.J., 1466 Laurent, J.P., 1556, 1557 Laurent, S., 1278, 1424 Laury, S.K., 144 Lauterbach, B., 1616 Law, A., 1137 Lawrence, K., 1061–1066 Lawrence, S., 1061–1066 Le, A., 984, 1126 Leal, R., 1349 Leamer, E.E., 1203 Lean, H.H., 1179 Lease, R., 610 Lease, R.C., 1161 LeBaron, B., 1109 LeBaron, B.D., 885 LeCompte, R.L.B., 1430 Lecraw, D.J., 1307 L’Ecuyer, P., 584 Lee, A.C., 53–68, 233, 393–397, 421–438, 619, 623, 1025–1037, 1297, 1465–1481 Lee, B.-S., 1394 Lee, C., 186, 950, 1089 Lee, C.H. Jevons, 1607, 1611, 1613 Lee, C.F., 10, 35, 47, 53–91, 93–109, 111–135, 172, 186, 221–233, 355–375, 377–392, 399–418, 421–446, 471–478, 481–503, 575–581, 619, 623, 670, 873–882, 933–950, 1025, 1151–1162, 1183–1198, 1205, 1224, 1293–1329, 1445–1458, 1465, 1466, 1476, 1479, 1523–1552, 1615–1625

Author Index Lee, D., 1152 Lee, G., 1325 Lee, H., 950, 1307, 1308, 1320, 1322–1324 Lee, H.-H., 233, 933–950, 1028 Lee, H.W., 1151 Lee, I., 1152, 1153 Lee, J., 53–68, 355–375, 393–397, 429–438, 617–636, 883, 903 Lee, J.C., 399–408, 421–428, 481–490, 619, 623, 1191 Lee, J.H., 1568 Lee, J.-P., 753–765 Lee, J.-Y, 873–880 Lee, K.W., 863–872 Lee, P.M., 1366 Lee, R., 1165 Lee, S.S., 877, 1483 Lee, S.C., 819, 821, 822, 825 Lee, S.B., 1041–1054 Lee, T.C., 1095, 1303, 1305, 1472, 1481, 1598 Lee, W.B., 677, 679, 686 Lee, Y.R., 1308 Lefebvre, C., 1307 Lehmann, E., 249 Lei, V., 137 Leippold, M., 714, 718, 748, 1490 Leisen, D.P.J., 505, 506 Lejeune, M.A., 639–664 Leland, H.E., 9, 203, 321, 328, 461, 531, 533, 667–670, 835, 934, 937–959, 988, 995, 1299, 1446, 1451, 1452, 1458 Lemmon, M., 1151, 1152 Lennox, C., 965 Lerner, J., 1366 Lesmond, D.A., 47, 186, 670, 1293, 1297–1300, 1366 Lettau, M., 1067–1070, 1089, 1608 Leung, A.Y.T., 1568 Lev, B., 686, 1224, 1596, 1597 Levary, R.R., 1319 Levendorskii, S., 1446 Levhari, D., 1523, 1526 Levin, L.A., 1265 Levin, N., 1062 Levine, R., 965, 1366 Lévy, E., 587, 596, 597, 600, 601 Levy, H., 39, 79, 249, 1224, 1235, 1236, 1523, 1526 Levy, P., 485 Levy, R., 122 Lewellen, J.W., 190, 273, 289, 610 Lewis, A., 1171 Lewis, B., 1324 Lewis, C., 548, 555 Lewis, C.M., 533, 1025, 1028, 1333, 1335, 1338 Lewis, K., 1122 Li, C., 1500 Li, C.K., 1179 Li, D., 698, 701, 961, 963, 1381, 1382 Li, H., 713–716, 718, 719, 746, 987, 996 Li, J., 853–861 Li, K., 221, 226, 1358 Li, L., 885, 901, 909, 1371-1380 Li, M., 986 Li, S., 1381, 1382 Li, T., 990 Li, W.K., 1421 Li, X., 996, 997, 999, 1000 Liao, A., 1284 Liaw, K.T., 35 Liberatore, M.J., 1319

1697 Lie, E., 187, 198 Lie, H., 187, 198 Lien, D., 873, 879, 882, 1381–1389 Liew, V.K.-S., 1283–1289 Lin, C.S.-M., 409–418 Lin, C.M., 677 Lin, H., 979–1001 Lin, J C., 819, 821, 822, 825, 1153, 1154 Lin, L., 1307, 1593–1604 Lin, T., 926 Lin, T.I., 471–478, 575 Lin, W.T., 915–931 Lin, Y., 1325 Linck, J., 1151, 1152 Lindenberg, E., 286 Lindqvist, T., 138 Lindskog, F., 994 Linetsky, V., 585, 1490 Linoff, G., 1324 Lins, K.V., 1366 Lintner, J., 89, 93, 125, 126, 129, 208, 270, 607, 1091 Lions, J.-L., 1574 Lipson, M.L., 1173, 1174, 1176 Liptser, R.Sh., 1556 Litterman, R., 42, 187, 191, 199, 204, 1484, 1490 Littlewood, J.E., 247 Litzenberger, R., 197, 518, 611, 1400, 1406 Liu, A.L., 999 Liu, A.Y., 1174 Liu, B., 950, 1523–1552 Liu, C.H., 1152 Liu, D., 198, 199 Liu, F.-I., 1409–1414 Liu, H., 1422, 1423 Liu, J., 185, 996, 997, 1056 Liu, L.-G., 645, 648, 654 Liu, N., 1273–1280 Liu, S., 990–992, 995–997 Liu, S.M., 853 Liu, W., 274 Liu, X., 1325, 1371–1380 Liu, Y.H., 1309, 1324 Livingston, M., 913 Livny, M., 1136 Ljungqvist, A., 1366, 1367 Lo, A., 185, 195, 290, 1092, 1607, 1610, 1611, 1613 Lo, A.W., 292, 532, 738, 886, 1068, 1071, 1284, 1474, 1598 Lo, K., 593 Lo, V., 994 Lo, W.A., 1174, 1177 Loayza, N., 1366 Lobato, I.N., 1409 Locke, E.A., 1324 Loeb, M.P., 1153 Logan, B., 1055–1060 Logue, D.E., 1465, 1467 Long, J., 605 Long, M., 831, 836 Long, R., 1062 Long, R.W., 1224 Long, S.R., 1578, 1579 Longin, F., 1283 Longstaff, F., 518, 522, 533, 548, 556, 557, 565, 666, 668, 716, 743, 940, 949, 987, 1122, 1483, 1515, 1615 Longstaff, F.A., 548, 557, 584, 670, 775, 885, 886, 910, 982–988, 990, 991, 993, 995, 999–1001, 1122, 1490, 1506, 1511, 1515

1698 Lookman, A.A., 996 Lopes, H., 1109 Lopes, H.F., 1376, 1378 Lopez, J., 193, 884 Lopez-de-Silanes, F., 226 Lorenz, M.O., 250 Loria, S., 327–328 Lorie, J.H., 1321 Losq, E., 226, 1235, 1236, 1527 Lott, J., 1427, 1429 Loubergé, H., 754 Loughran, T., 843, 1152 Loviscek, A.L., 236 Lowenstein, M., 1507 Lowenstein, R., 203, 204 Lowry, M., 1366 Lubow, J.W., 375, 392, 419 Lucas, A., 342 Lucas, R.E. Jr., 1071 Ludvigson, S., 1068–1070, 1089 Lugosi, G., 1257 Lukas, E., 1041 Lumsdaine, R.L., 885, 1348 Lund, J., 548, 725, 737, 985, 987 Luo, M., 1325 Lusk, E.J., 1307 Lutkepohl, H., 1095, 1303, 1305, 1392, 1472, 1481, 1598 Luttmer, E.G.J., 986 Lyanders, E., 1407 Lyden, S., 668, 995 Lyrio, M., 985 M Ma, C., 1179 Ma, J., 1007, 1008 Ma, T.S., 226 MacBeth, J., 471, 1073, 1615, 1618, 1619 MacDonald, R., 941 Mackinlay, A.C., 195, 273, 292, 886, 1068, 1071, 1092, 1284, 1431, 1474 MacKinnon, J.G., 1365, 1478 Maclennan, D., 1206 MacMillan, L.W., 763 Madan, D., 547, 556, 565, 575, 585, 996 Madan, D.B., 399, 401, 988, 1570 Madariaga, J.A., 276 Maddala, G., 1119 Maddala, G.S., 1067, 1600, 1603 Madhavan, A., 339, 341, 342 Madigan, D., 42, 192, 199 Maenhout, P., 671 Maes, K., 979, 985, 987 Maginn, J.L., 68, 92, 106, 122, 133 Mago, A., 802 Mahanti, S., 670 Mahapatra, R.K., 1324 Mai, J.S.-Y., 125–133, 393–397 Maich, S., 1283 Maier, S., 339–341 Maillet, B., 324, 329 Majluf, N.S., 1383, 1151, 1152 Maksimovic, V., 226, 1028, 1399, 1403, 1405–1406 Malatesta, P.H. 1151, 1152 Malevergne, Y., 260 Malik, F., 1352 Malitz, L., 831, 836

Author Index Malkiel, B.G., 273, 278 Malliaris, A., 447 Malliaris, A.G., 447–470 Mallik, G., 670 Maloney, K.J., 995 Malpezzi, S., 1206 Mamon, R.S., 1128, 1129 Manaster, S., 458, 1333, 1397 Mandelbrot, B., 1273, 1345, 1410, 1615 Mandell, R.E., 996, 997 Mangasarian, O., 1062 Mania, M., 1555–1564 Mankiw, N.G., 1394 Mann, C., 46, 227, 285 Manzur, M., 1179 Mao, J.C.F., 71 Maquieira, C., 1429 Marathe, V., 102, 118 Marcello, J.L.M., 1352 Marcis, R., 1041 Marcus, A., 172, 941 Marimon, R., 1608 Marin, J.M., 1377 Marjit, S., 1381 Mark, N., 641, 659, 1395 Mark, N.C., 1122 Markowitz, H., 20, 111, 125, 221, 226, 236, 259, 261 Marks, S., 1598 Marmol, F., 1075 Mar-Molinero, C., 1598 Marsh, I., 1000 Marsh, P., 1224, 1333 Marsh, T.A., 289 Marshall, A., 221 Marshall, D.A., 986, 1122 Marston, F., 1366 Marston, R.C., 679 Martens, M., 1274, 1278, 1279 Martin, A.D., Jr., 92 Martin, D., 1598 Martin, J., 829, 831, 834 Martin, J.S., 884, 885, 995 Marx, L.M., 342, 1028 Masih, A.M., 1283 Masih, R., 1283 Masoliver, J., 1568 Mason, J.R., 802 Mason, S.P., 668, 995 Massabo, I., 587 Masulis, R., 1400 Masulis, R.W., 1151, 1161, 1230 Masulis, W., 1283 Mathews, J., 1221 Mathews, K.H., 1206 Mathieson, D., 641, 659 Matsuba, I., 1568 Mauer, D., 942 Maurer, R., 329 Mayers, D.F., 1211 Mayhew, S., 533, 1397 Mayo, J., 641 Mayoraz, E., 641–643 McAleer, M., 1274, 1278 McAndrews, J.J., 1028 McBeth, J., 558 McCallum, B.T., 1093

Author Index McCauley, J.L., 1568 McCluskey, J.J., 1206 McConnell, J., 1028, 1427, 1428 McConnell, J.J., 865, 866 McCormick, T., 342, 1028 McCorry, M.S., 1524 McCulloch, R.E., 1422 McDonald, B., 1525, 1526, 1534, 1545 McDonald, J.B., 854, 855 McDonald, R., 458 McElroy, M.B., 1092 McGrattan, E., 1608 McGuire, J., 830 McKibben, W., 108, 118 McLeavey, D.W., 68, 92, 106, 122, 133 McLeay, S., 1598 McMillan, R., 1206 McNamee, P., 1319 McNeil, A., 698, 994, 995 McQueen, G., 883, 884, 1352 Meckling, W., 36, 835, 936, 947 Meckling, W.H., 22 Medeiros, M., 1274, 1278 Meese, R.A., 1394 Megginson, W., 1429 Mehran, H., 863 Mei, J., 1465 Melamed, B., 1135–1149 Melick, W., 515 Melino, A., 549, 555 Mella-Barral, P., 667, 941, 988 Mello, A.S., 667, 996 Mendelson, H., 342, 1093 Mendelson, M., 340 Mendenhall, W., 68, 92, 106, 122, 133 Menkveld, A., 342 Mensah, Y.M., 1319 Mensah, Y., 1598 Mercer, J., 1267 Merton, R., 360, 526, 547, 605, 611–613, 1044, 1457 Merton, R.C., 27, 204, 268, 269, 289, 387, 447, 450, 454, 458, 464, 665–670, 807–809, 819–825, 933, 940, 941, 949, 988, 996, 1071, 1087, 1269, 1515–1517, 1596, 1601, 1602, 1607, 1613 Merton, R.A., 1055 Merville, L., 558, 1615 Merville, L.J., 458, 532, 1307, 1607 Mesnier, M.P., 646 Mester, L., 685 Meulbroek, L.K., 676 Meyer, P.A., 1556, 1557, 1560–1562 Meyer-Brandis, T., 1567–1575 Meyers, L., 1063 Mian, S.L., 676, 679 Miao, J., 667, 942 Mikkelsen, H.O., 1420 Mikkelson, W.R., 1400, 1406 Mildenstein, E., 340 Miles, J., 1223–1227, 1229, 1231 Miles, J.A., 686 Milevsky, M.A., 587, 600, 601 Milgrom, P., 340, 1007, 1009, 1399, 1400 Miller, M., 6, 8, 378, 934, 1007, 1009, 1028, 1223–1225, 1229–1231, 1399–1401 Miller, E., 343 Miller, H.D., 453 Miller, N.H., 1366

1699 Miller, R.B., 1345, 1347, 1349 Milne, F., 399, 401 Milonas, N.T., 915, 916, 920, 921, 930 Miltersen, M., 716, 727, 734 Mincer, J., 103 Ming, J., 1328 Minton, B.A., 999 Minton, B.M., 679 Miranti, P.J. Jr., 1319 Mishkin, F.S., 985 Mishra, B., 1152, 1153, 1400, 1401 Misra, L., 1427–1443 Mitchell, K., 1025, 1036 Mitchell, M.L., 1161 Mithal, S., 670, 991, 995, 999–1001 Mittelhammer, R.C., 1206 Mittnik, S., 515, 517, 523, 524, 853 Mitton, T., 1366 Miyahara, Y., 1555, 1568 Miyakoshi, T., 1174 Mizrach, B., 515–528 Modigliani, F., 6, 8, 9, 26, 27, 33, 677, 934, 1028, 1223–1225, 1230, 1231, 1466 Modigliani, L., 275 Moeller, S., 1427, 1428, 1434, 1439, 1441 Mogedey, L., 1568 Mohnhan, T.F., 1319 Molina, C.A., 1366 Molina, G., 1109–1120 Monat, P., 1555 Monfort, A., 1156 Monoyios, M., 1568 Montero, M., 1568 Montesi, C.J., 873 Moon, C.-G., 685 Moon, M., 768, 769, 772, 775 Moore, E., 138 Moore, P., 185, 193 Moore, T., 1352 Moore, W., 941 Mora, N., 648 Morck, R., 686 Mordecki, E., 47, 1445, 1446, 1454 Moré, J.J., 646 Moreira, M.J., 1365 Morellec, E., 667, 941, 947 Morgan, J.B., 700 Morgan, D.P., 643 Morgan, I.G., 122 Morgenstern, O., 71, 247, 607 Morris, C.S., 684 Morris, M.H., 1597 Morrison, D.G., 1064 Morrison, D.F., 1474 Morse, D. 1174 Mortensen, A., 669 Morton, A., 716, 981, 1008, 1490, 1494 Morton, K.W., 1211 Moser, J.T., 678, 686 Moskowitz, T.J., 1366 Mosler, K., 247 Mossin, J., 93, 203 Mosteller, F., 1411 Mota, B., 1278 Moy, J.W, 1065 Moy, R.L., 375, 392, 419

1700 Mozumdar, A., 986, 1490 Muchnik, I., 641–643 Mukherjee, A., 1381 Muliere, P., 250 Müller, A., 249 Mullins, D., 1383, 1406 Mullins, H.M., 819–821, 824, 825 Munk, C., 1211 Muralidhar, K., 1317, 1319 Muromachi, Y., 993 Musiela, M., 460, 465, 1571 Mussavian, M., 273 Muth, J.F., 1161 Myers, J., 1089 Myers, S., 464, 605, 1383 Myers, S.C., 28, 30, 935, 941, 1042, 1047, 1151, 1152, 1223, 1224, 1226, 1465, 1467 Myneni, R., 1490 N Nagel, S., 139, 273 Nail, L., 1429 Nakamura, L.I., 1028 Nakatsuma, T., 1375 Nance, D.R., 676, 677, 679 Nanda, V., 1237 Nandi, S., 547, 555, 556, 563, 884 Nandy, D., 1366 Nantell, T.J., 879 Napp, C., 1007 Narayanan, M.P., 1025 Narayanan, P., 193, 649, 1598, 1603 Nardari, F., 1109 Nash, R.C., 1162 Nashikkar, A.J., 670 Neave, E.H., 587 Neckelburg, G., 648, 659 Neely, C., 1608 Neftci, S., 447 Neis, E., 670, 990, 991, 993, 995, 999, 1000 Nelken, I., 670, 1000 Nelson, M., Nelson, B.L., 1136 Nelson, C., 1123 Nelson, D., 899, 1109, 1274 Nelson, D.B., 853, 1181, 1333, 1347, 1348, 1409, 1419 Nerlove, M., 885 Nesbitt, S.L., 865 Netter, J., 1427, 1428 Netter, J.M., 1153 Neuberger, A., 458, 1567 Neumann, K., 874 Newbold, P., 103–105, 193, 1068, 1072, 1075, 1076, 1466, 1476–1479 Newby, R., 144 Newey, W., 693, 1068, 1070, 1072, 1075, 1086 Newton, D.P., 506 Ng, A., 221, 226, 233 Ng, D., 1244 Ng, V., 547 Ng, V.K., 533, 1283 Ng, W.-Y., 654 Nguyen, T.T., 1407 Niden, C., 1174 Nieh, C.-C., 1121–1133 Niehans, J., 678 Niehaus, G., 763

Author Index Nielsen, J., 766 Nielsen, L.T, 666, 988 Nieto, L., 1174 Nijman, T.E., 173, 226 Nikeghbali, A., 698 Nison, S., 1273 Nissim, D., 198, 199 Noe, T., 1025, 1036 Nögel, U., 1168 Norden, L., 1000 Norgaard, M., 1502 Nouretdinov, I., 1269 Noussair, C., 137, 138 Novomestky, F., 221 Nyborg, K.G., 1225 Nye, D.J., 1466 O O’Brien, J., 831 O’Brien, T.J., 321, 327 Obreja, I., 996 Obstfeld, M., 221 Odders-White, E.R., 1366 Odean, T., 1235, 1244 Ofek, E., 767, 770, 775, 831, 834, 863, 864, 869, 999, 1026, 1028, 1036 Officer, M., 1427–1429, 1439, 1443 Ogden, J.P., 668, 995 Ogryczak, W., 249, 251, 254 O’Hara, M., 339, 340, 1397 Ohlson, J., 185, 193, 965, 1326–1329, 1598, 1599 Oksendal, B., 447, 448 Oliner, S., 1283 Omberg, E., 299, 316, 505 Ooghe, H., 1597, 1598 Opler, T., 1026, 1028, 1036 Ornstein, L.S., 578 Ortega, J.M., 1033 Ortiz-Molina, H., 1366 Ostdiek, B., 233, 1070, 1333, 1335 Ott, S., 942 Ouederni, B.N., 1323 Ou-Yang, H., 941, 942, 948 Oviedo, R., 670, 949 Ozenbas, D., 342 P Packer, F., 648 Padberg, M.E., 46, 82, 125, 127, 128, 130, 133, 135, 208, 227, 249, 285, 884, 885, 991, 995, 1465 Padmanabhan, P., 226, 1235 Pagano, M., 273, 334, 342–345 Pagano, M.S., 675–694 Page, E.S., 1577 Page, J., 1161 Pai, D., 1061–1066 Pai, Szu-Yu, 1577–1591 Pain, D., 329–332 Palatella, L., 1568 Palepu, K.G., 1406, 1598, 1600, 1603, 1604 Palfrey, T., 140, 145 Palia, D., 686, 1366 Palmon, O., 531–545, 1055–1060 Palmquist, R., 1205 Pan, E., 725, 1489–1502 Pan, G., 950

Author Index Pan, G.G., 670, 950, 990, 1301, 1305 Pan, J., 670, 716, 723, 725, 726, 734, 1397, 1499 Panayi, S., 767, 768 Panigirtzoglou, N., 522 Panjer, H.H., 248, 854, 855, 953 Papageorgiou, E., 1118 Papanicolaou, G., 1110, 1118 Pardoux, E., 1556 Park, S., 1153 Parkinson, M., 1273–1275, 1280 Parlour, C., 339, 341 Paroush, J., 342, 343 Parrino, R., 864, 865, 941, 943, 946, 1151, 1152 Parsons, J.E., 667 Partch, M.M., 1400, 1406 Parulekar, R., 802 Pastena, V., 1593, 1594 Pastor, L., 42, 187, 190–192, 199, 342 Patel, J., 1237 Patel, N., 1062 Patel, S.A., 884, 886 Patro, D.K., 1235–1255, 1523–1552 Patron, H., 1381 Patterson, J.H., 1319 Patton, A.J., 1417 Paul, S., 1152 Pauly, P., 193 Payne, J.E., 1352 Peake, J., 340 Pearson, M., 885 Pearson, N.D., 982, 986 Pedersen, C.S., 274 Pedersen, L., 1599 Pederson, H.W., 757–759 Pedrosa, M., 884 Peel, D.A., 1598 Peel, M.J., 1598 Peker, A., 1174 Peng, L., 1206 Peng, S., 1490, 1556 Peng, Y., 1324, 1325, 1329 Penman, S., 185, 186 Pennacchi, G.G., 686, 819, 823–825 Perello, J., 1568 Perfect, S., 679 Perold, A., 185 Perraudin, W., 667, 941, 988 Perron, P., 984, 1122, 1123, 1347, 1421 Persand, G., 853, 854 Pertman, M., 841 Petersen, M.A., 1366 Peterson, C.L., 802 Peterson, D., 1615 Peterson, S., 985 Petkova, R., 1069, 1070 Petty, W.J., 1307 Pfleiderer, P., 1173 Pham, H., 1555–1557, 1563 Pham, T.M., 327, 328 Philiippatos, G., 92 Philips, T.K., 290 Phillips, P.C.B., 1072, 1075 Phillips, R.D., 676 Phoon, K.F., 1179 Piazzesi, M., 717, 984, 985, 1121 Piesse, J., 1593–1604

1701 Pike, R., 1319 Pinches, G.E., 1098 Pinkerton, J.M., 1153 Pinkowitz, L., 1026 Pinto, J.E., 68, 92, 106, 122, 133 Piotroski, J., 193 Piotroski, J.D., 1153 Piramuthu, S., 965 Pistorius, M.R., 47, 1445, 1446, 1448, 1451, 1454, 1457, 1461 Pitt, L., 802 Pitts, M., 1173, 1175, 1176 Platen, E., 1555 Pliska, S.R., 458, 1015 Plott, C., 140, 143 Plumridge, R.R., 1162 Podolskij, M., 1274, 1278, 1279 Poggio, T., 1607, 1610, 1611, 1613 Pogue, G.A., 22, 68 Pogue, J.A., 92 Poitevin, M., 1381 Pollak, M., 1577 Pollakowski, H.O., 1207 Pollard, D., 1412 Pollard, H., 1451, 1460 Pollin, R., 965 Polson, N.G., 1109, 1110, 1421 Polya, G., 247 Pontiff, J., 1068, 1073 Poon, S.H., 1279, 1333–1344 Poon, W.P.H., 644 Porter, D., 342 Porter, D.P., 137 Posner, S.E., 587, 600, 601 Poterba, J.M. 221, 290, 293, 294, 1247 Poteshman, A., 941 Poulsen, N.K., 1502 Prabhala, N.R., 1333–1335, 1343, 1358 Prade, H., 1184 Prakash, A.J., 1597, 1598 Pratelli, M., 1556 Prause, K., 533, 874 Predescu, M., 670, 1000 Prékopa, A. 250 Press, W. 1271 Price, B., 879 Price, K., 879 Price, L. 274 Prigent, J.L., 321, 324 Pringle, J.J., 30, 676, 679 Pritchard, A.C., 843 Protopapadakis, A., 884 Protter, P., 717, 1007–1024, 1569 Ptittas, N., 1284 Pudet, T., 587 Puri, M.L., 1184 Purnanandam, K., 843 Purschaker, N., 329 Pye, G., 1030 Pyle, D.H, 9, 819–821, 824, 825, 835, 1400 Q Qian, X., 1374 Qiao, Z., 1173–1179, 1283–1289 Qu, W., 1577 Quandt, R., 71 Quandt, R.E., 1319

1702 Quenez, M.C., 1555, 1557 Quenouille, M., 651 Quiggin, J., 247, 248 Quirin, G.D., 1466 Quirk, J.P., 247, 249 Quiros, J.L.M., 1353 Quiros, M.M.M., 1353 R Rabinowitz, N., 197 Rachev, S.T., 853 Radner, R., 1055, 1056 Raftery, A., 42, 199 Ragunathan, V., 1174 Rajan, R.G., 867, 1366 Rajan, U., 341 Ralescu, D.A., 1184 Ramanathan, R., 193, 1303 Ramaswamy, K., 25, 26, 1091, 1466, 1472, 1476 Ramos, R., 802 Ramsden, R., 802 Rand, J., 329–332 Rangel, J.G., 1422 Rao, C., 401 Rao, R.R., 401 Rao, R.K.S., 815 Rasson, J.P., 1347 Ratings, F., 644, 649 Ratti, V., 204 Raven, O., 1501 Ravid, S.A., 1025–1027 Raviv, A., 836, 863, 864, 885 Raviv, J., 1381, 1382 Ray, B.K., 1409 Raymar, S., 1028 Read, J.A., 1466 Rebello, M., 1025, 1036 Rebonato, R., 460, 465 Reck, M., 333–347 Rees, B., 1594, 1595, 1598 Rees, L., 1151 Register, C.A., 677, 679, 686 Reimer, M., 505, 506 Reiner, E., 939 Reinganum, M.R., 109, 1091, 1093 Reinhart, C.M., 648 Reinhart, W.J., 1319 Reisen, H., 648 Remolona, E.M., 884, 885, 912 Ren, Q., 1136 Renault, E., 1156, 1420 Renault, O., 641, 642, 670, 995 Rendleman, R.J., 395, 1397 Reneby, J., 668, 943, 949, 950 Reneby, Joel Rennie, A., 451 Resnick, B.J., 133 Revuz, D., 1447, 1559 Reynolds, C.E., 766 Rheinländer, T., 1555, 1567, 1569, 1574 Rheinländer, Th., 1557, 1561 Rhoades, S.A., 767, 770 Riani, M., 965, 966 Ribeiro, R.A., 1190 Riboulet, G., 698 Richard, S.F., 987

Author Index Richardson, D.H., 1093 Richardson, M., 292 Ricke, M., 138 Riffe, S., 186 Rijken, H.A., 649 Riley, J., 1401, 1402 Ring, S., 1257–1271 Risa, S., 802 Ritchey, R., 515 Ritchken, P., 505, 884 Ritter, J.R., 843, 1152 Rivoli, P., 648, 659 Robbins, H., 1263 Robert, C.P., 1109, 1372 Roberts, J., 1399, 1400 Robicheck, A.A., 68 Robichek, A., 605, 1226 Robin, S., 138 Robins, R., 194 Rocha, G., 1278 Rochet, J.C., 819, 822, 825 Rock, K., 1399–1403 Rockafellar, R.T., 252 Roden, D.M., 677, 679, 687 Röell, A., 342 Roenfeldt, R., 941 Rogers, D.A., 675–678, 680 Rogers, G.T., 290 Rogers, L., 585, 1273, 1275, 1279 Rogers, L.C.G., 47, 587, 591, 593, 596, 597, 667, 1445, 1446, 1451, 1452, 1458, 1490 Rogoff, K., 1394 Roley, V.V., 883, 884 Roll, R., 41, 108, 109, 138, 208, 269, 273, 439, 442, 464, 532, 605, 884, 1475 Rolski, T., 1448 Roman, E., 321 Romano, R., 1207 Rombouts, J.V.K., 1278, 1424 Romer, D., 1394 Roncalli, T., 698, 994 Ronen, J., 843–851 Ronen, T., 1397 Ronn, E.I., 819, 821–825 Roper, A.H., 1366 Rosen, R., 676, 677, 683, 686 Rosen, S., 1202, 1206 Rosenberg, B., 102, 108, 117, 118, 612 Rosenberg, J.V., 532, 738 Rosenfeld, E., 668, 986, 995 Ross, S., 9, 12, 118, 203, 273, 427, 458, 464, 466, 471, 472, 483, 489, 531, 547, 549, 565, 612, 769, 826, 941, 1190, 1191, 1223, 1400, 1466, 1468, 1475, 1593, 1604, 1619, 1620 Rossi, P.E., 1109, 1110, 1174, 1421 Rotemberg, J., 1403 Rothenberg, T.J., 1364 Rother, P., 641 Rothschild, M., 258 Rouge, R., 1555, 1556, 1567 Roulstone, D., 1153 Rousseeuw, P.J., 965, 966 Rouwenhorst, K., 197 Royston, P., 1177 Rozeff, M.S., 1153, 1154 Ruback, R., 185, 198, 769, 1224, 1231, 1430 Rubin, H., 1365

Author Index Rubinstein, M., 35, 104, 283, 320, 324, 415–417, 427, 453, 466, 515, 518, 520, 522, 532, 533, 547, 548, 550, 556, 558, 567, 587–589, 605–616, 939, 1190, 1274, 1607, 1613 Rudd, A., 278–280, 361, 362, 457, 505, 507, 755 Rudebusch, G., 1134 Ruffieux, B., 138 Ruiz, E., 1109, 1277, 1409, 1410, 1421 Ruland, W., 1593, 1594 Runkle, D., 471, 854, 1161, 1333, 1336, 1340, 1348 Russell, W., 247, 249, 668 Russo, E., 587 Rusticus, T.O., 1359, 1366 Ruszczynski, A., 247–257 Rutkowski, M., 460, 465, 671, 990, 1128, 1129 Ryan, S.G., 1227 Rydberg, T., 874, 1570 Ryngaert, M., 777, 1427, 1434, 1439, 1441 S Saá-Requejo, J., 533, 666, 988 Saita, L., 1599 Salanie, B., 1156, 1161 Salop, S., 1406 Samperi, D., 1557, 1567 Samuelson, P., 204, 236, 607, 611, 613, 915, 916, 921, 929–931, 1055 Sánchez, J. de A., 1190, 1197, 1198 Sandas, P., 342 Sanders, A.B., 984, 1122, 1374, 1515 Sandmann, K., 755 Sandroni, A., 1265 Sanghani, J., 144 Sankaran, M., 478, 1622, 1623 Santa-Clara, P., 716, 721, 743, 986 Santacroce, M., 1555–1564 Santhanam, R., 1317, 1319 Santibanez, J., 276 Saposnik, R., 247 Saraf, M., 802 Sarath, B., 843–851 Sargan, J.D., 1362 Sargent, T.J., 1608 Sarig, O., 669, 995 Sarkar, S., 226 Sarkar, A., 647 Sarkissian, S., 1067–1089 Sarnat, M., 38, 39, 79, 1235, 1236 Satchell, S., 1273, 1275, 1279, 1417 Satchell, S.E., 274 Sato, K., 1447, 1569, 1574 Saunders, A., 671, 675, 676, 678, 679, 686, 759, 819–822, 824, 825, 965, 1275 Savage, L.J., 1321 Savarino, J.E., 204 Savin, N.E., 1409 Scarsini, M., 247, 250 Schachermayer, W., 465, 1012, 1018, 1555 Schadt, R., 1067, 1069, 1070, 1072, 1082 Schaefer, S., 668, 987 Schaefer, S.M., 1491, 1494, 1615 Schall, L., 1068, 1073 SchÄonbucher, P.J., 993, 995 Scharfstein, D., 1403 Scheaffer, R.L., 68, 92, 106, 122, 133 Scheffman, D., 1406 Scheinkman, J., 1484, 1490 Scheinkman, J.A., 138

1703 Schink, G.R., 1465 Schlarbaum, G., 610 Schleef, H., 340 Schleifer, A., 138 Schmidli, H., 1448 Schmidt, V., 1095 Schmukler, S.L., 648, 664 Schniederjans, M., 1317, 1319 Scholes, M., 26, 107, 267, 327, 378, 454, 464, 516, 532, 576, 606, 611, 665, 768, 809, 933, 988, 1055, 1098, 1110, 1230, 1466, 1472, 1601, 1619 Schölkopf, B., 1267 Schonbucher, P., 995 Schonbucher, P.J., 963, 993 Schrage, L., 1137, 1139 Schrage, L.E., 1310, 1319, 1322 Schrand, C., 676, 677, 679, 686 Schroder, M., 471, 473, 474, 477, 575, 1618, 1620, 1623 Schubert, D., 995 Schultz, P., 342, 1616 Schwartz, A.L., 587 Schwartz, E.S., 584, 666–668, 670, 716, 768, 769, 772, 775, 777, 885, 886, 910, 916, 934, 937, 940, 949, 982, 985, 987, 988, 995, 1041, 1515, 1527, 1615 Schwartz, N.L., 778, 831 Schwartz, R., 333–352 Schwarz, T., 137–163 Schwebach, R.G., 761 Schweizer, M., 465, 1555, 1556 Schwert, G.W., 883, 1075, 1089, 1352 Scorcu, A.E., 1206 Scott, E., 1615 Scott, L.O., 47, 481, 485, 489, 533, 547–551, 575, 725, 737, 982, 1191, 1483–1488, 1490, 1523, 1615 Scott, W., 694, 879 Seagle, J.122 Sears, S., 375 Seiford, L., 1308 Seligman, J., 190, 191 Senbet, L.W., 27, 36, 1025, 1036 Seneta, E., 1570 Sengupta, B., 1136 Seppi, D., 339, 341, 342 Servaes, H., 865, 1028, 1237, 1247 Seyhun, H.N., 1152, 1153 Shafer, G., 43, 1257–1271 Shailer, G., 1598 Shakarchi, R., 1450, 1458 Shaked, I., 819, 823–825 Shalen, C.T., 1173 Shanken, J., 190, 289, 1067, 1068, 1070, 1085, 1086 Shanmugam, B., 648 Shapiro, A.C., 676, 677 Shapiro, M.D., 1394 Sharda, R., 1325 Sharfman, K., 186 Sharpe, I.G., 111, 112 Sharpe, W.F., 93, 96, 106, 122, 123, 126, 208, 271, 275, 283, 327, 1091, 1366 Shastri, K., 465 Shaw, K.N., 1324 Shaw, W., 1168, 1171 Shea, J., 1364, 1367 Sheard, P., 830 Sheehan, D.P., 1153 Shefrin, H., 399, 401

1704 Shen, C.-H., 965–976, 1124 Shen, F.C., 203–218 Shen, S.P., 1577 Shen, Z., 1578, 1579 Shephard, N., 49, 515, 527, 1273, 1420, 1421, 1567–1569, 1574 Shepp, L., 1055–1060 Sheppard, K., 1278 Sheppard, S., 1202 Sheu, Y.-C., 873–882, 1445–1464 Shi, J., 1293–1300 Shi, Y., 1307–1329 Shi, Z., 585, 587, 591, 593, 596, 597 Shibut, L., 643 Shie, F.S. 819–827 Shih, F., 1400 Shih, H.H., 1579 Shih, P.-T., 42, 505–513 Shih, W.-K., 355–375, 377–392, 491–503 Shiller, R., 289, 1393 Shiller, R.J., 137, 290, 298, 299, 986 Shimizu, H., 1375 Shimko, D., 515, 522, 523, 532, 667 Shiryaev, A.N., 1056, 1491, 1574 Shiu, E.S.W., 1445, 1457 Shleifer, A., 140, 865 Shmueli, G., 1062 Shoven, J., 1595, 1604 Shrestha, K., 873 Shreve, S.E., 447, 448, 460, 517, 981, 982 Shu, J.H., 1274, 1275 Shu, S., 1366 Shum, K.-H., 654 Shumway, T., 669, 1602–1604 Sias, R.W., 864, 865 Sichel, D., 1283 Sick, G.A., 1230 Siclari, J., 584 Sidenius, J., 703 Sieg, H., 1206 Siegel, J., 138, 185 Siegmund, D., 1577 Siems, T.F., 643 Sikorski, D., 1179 Silberberg, E., 236 Silverman, B.W., 543 Sim, A.B., 327, 328 Simin, T., 1067–1089 Simioni, M., 1206 Sims, C.A., 1391, 1392 Singer, R.F., 988 Singleton, K.J., 665, 671, 713, 717, 718, 912, 979, 981–984, 986, 987, 990, 993, 995–997, 999, 1121, 1127–1129, 1270, 1490 Sinha, G., 802 Sinkey, J.F.Jr., 695 Sinquefield, R.A., 1481 Sircar, R., 1118 Sirri, E.R., 1235, 1237, 1238, 1241, 1244, 1247 Sklar, A., 698, 701 Skorokhod, A.V., 451 Skouras, S., 528 Sloan, R.G., 273, 1108 Smidt, S., 339, 1174, 1176 Smirlock, M., 1173, 1176 Smith, A.F.M., 1347 Smith, B., 1206 Smith, C., 676, 677, 1227

Author Index Smith, C.W., 463, 464, 677, 680, 682, 836, 1025 Smith, D.J., 999 Smith, K.V., 122 Smith, M., 865 Smith, P.A., 1366 Smith, R.L., 843, 1151, 1152 Smith, S.D., 676 Smith, T., 1079 Smith, V., 139, 141 Smith, V.L., 137 Smithson, C.W., 676, 677, 679 Smola, A.J., 1267 Smorodinsky, R., 1263 Snedecor, G.W., 859 So, M.K.P., 1421 Sobehart, J.R., 656, 658, 967 Sodal, S., 1042 Sofianos, G., 342 Sola, M., 1122 Sølna, K., 1110, 1118 Solnik, B. Solnik, B., 204, 234, 996, 1235, 1236, 1283, 1527 Sondermann, D., 716, 727, 734 Soner, H.M., 1007 Sopranzetti, B.J., 1201–1207, 1523 Sorensen, D., 1033 Sorensen, M., 1366 Sornette, D., 260, 716, 721, 986 Sortino, F., 274, 328 Souders.T.L., 1313 Spagnolo, N., 1284 Spatt, C., 339, 342 Spence, L.E., 1451 Spence, M., 1381 Spencer, B., 1403 Spindt, P.A., 844 Spong, K.R., 676, 683, 694 Sprenkle, C., 611 Srinivas, P.S., 1397 Sriram, R.S., 647 Stafford, E., 1161 Staiger, D., 1364 Stambaugh, R., 191, 204, 210, 226, 342, 1070 Stambaugh, R.F., 289, 293, 294, 301, 982, 1068, 1072 Stambaugh, R.S., 1072 Stanton, R., 986 Stapleton, R.C., 985 Starks, L.T., 864, 865, 1173, 1176 Staunton, M., 182 Stegemoller, M., 1427, 1428 Stegun, I.A., 1057, 1059, 1623 Steiger, G., 1574 Steil, B., 340, 342 Stein, C., 614 Stein, E.M., 547, 551 Stein, J., 547, 551 Stein, J.C., 678, 680, 684–686 Stein, R.M., 656, 658, 967, 968 Steiner, T., 835 Stensland, G., 523 Stephan, J., 1397 Steuer, R., 1308 Stevens, G.V.G., 266 Stevenson, B.G., 700 Stickel, S., 1174 Stigler, G., 339

Author Index Stiglitz, J.E., 9, 36, 137, 247, 249, 340 Stinchcombe, M., 1347 Stochs, M.H., 1025 Stock, D., 883–912 Stock, J.H., 1175, 1364, 1365, 1367, 1368 Stockman, A., 195, 196 Stoer, J., 1033 Stokey, N., 340 Stolin, D., 1427, 1428 Stoll, H.R., 340, 431, 808, 810, 1041 Stomper, A., 1028 Stone, B., 122 Storey, D.J., 1598 Stout, D.E., 1319 Stoyan, D., 249 Strahan, P.E., 683 Straumann, D., 698 Strebulaev, I.A., 668 Streller, A., 874 Stricker, C., 465 Strock, E., 676, 678, 679, 686 Stulz, R.M., 226, 405, 547, 675, 676, 864, 865, 1236, 1381, 1382 Stutzer, M., 532 Su, T., 587 Subrahmanyam, A., 199, 342 Subrahmanyam, M., 667, 734 Subrahmanyam, M.G., 670, 996 Suchanek, G., 141 Suen, W., 236 Suk, D., 1400 Sul, D., 1395 Sullivan, R.J., 195, 676 Sullivan, W.G., 1323 Summa, J.F., 375, 392, 419 Summers, L.H., 290, 293, 294 Summers, L., 1398 Sun, L., 1598 Sun, T.S., 982, 999 Sundaram, A., 1400, 1407 Sundaram, R., 1490 Sundaram, R.K., 458, 531, 533, 537 Sundaresan, M., 1517 Sundaresan, S., 667, 668, 670, 941–943, 947, 949, 988, 999, 1517 Sunder, S., 98, 137, 138, 140, 143, 1399, 1596, 1597 Sundaram, A., 1399–1407 Sung, T.K., 1325 Suret, J., 221, 226 Surry, Y., 1206 Susmel, R., 1284, 1421 Sverdlove, R., 670 Swaminathan, B., 1089 Swan, P.L., 1524 Swanson, N.R., 1365, 1366 Swary, I., 1406 Switzer, L., 829 Sy, A.N.R., 645, 664 Szafarz, A., 1223 Szego, G., 260 Szewczyk, S.H., 831, 834 T Taffler, R., 1597 Taggart, R.A., 1226, 1230 Takahashi, H., 1568 Takemura, A., 1264 Taksler, G., 668

1705 Taleb, N., 458 Talwar, P.P., 1098 Tam, K., 1061, 1062 Tamayo, P., 641 Tanaka, H., 1197 Tandon, K., 465 Tang, C.M., 168 Tang, X., 1329 Tansey, R., 1062 Tarjan, R.E., 653 Tarpley, R., 193 Tauchen, G., 718, 1173, 1175, 1176 Taurén, M., 666, 941 Tayler, H., 1617, 1619 Taylor, S., 1409, 1410, 1420 Taylor, S.J., 533, 1273, 1333–1344 Teiletche, J., 994 Tejima, N., 667 Telser, L.G., 915 Temin, P., 137, 139 Tenenbaum, M., 1451, 1461 Tenney, M., 984, 1490 ter Horst, E., 1109 Tesar, L., 1236, 1237, 1247 Teugels, J., 1448 Teukolsky, S.A., 1271 Tevzadze, R., 1555–1564 Thanassoulis, E., 1319 Theil, H., 44, 1466, 1476 Theobald, M., 319–332, 1352 Theodossiou, P., 854, 1596, 1600, 1601 Thomadakis, S.B., 915, 916, 920, 921, 930 Thomas, C., 515 Thomas, J., 199 Thompson, D.J., 68 Thompson, G.W.P., 587 Thompson, H.E., 984, 1121, 1127, 1179 Thompson, R., 186 Thompson, S., 715 Thompson, S.B., 290 Tian, Y., 505, 506, 510 Tiao, G.C., 1347, 1350 Tibshirani, R., 1069 Timmermann, A., 195 Tinic, S., 339, 810 Tinsley, P.A., 985 Tirole, J., 139, 1403 Titman, S., 209, 274, 676, 677, 807, 831, 867, 988 Tiwari, A., 341 Tjahjapranata, M., 829, 832, 840 Tkac, P., 1174 Tobin, J., 267, 1525 Toft, K.B., 667, 668, 670, 934, 937, 939–941, 946, 948, 949, 988, 995, 1299 Tollefson, J.O., 1603 Tompkins, R., 517 Tookes, H., 1397 Topaloglu, S., 844 Topkis, D., 1399, 1400 Topper, J., 1221 Torous, W., 725, 737, 741, 988, 1273 Torous, W.N., 303, 985, 1607 Touze, N., 1007 Toy, W., 1002, 1511 Tracie, W., 1152 Trapp, M., 999

1706 Travlos, N., 1427 Travlos, N.G., 676, 678, 679, 686, 866 Treacy, W.F., 639, 640 Trennepohl, G., 375, 392, 419 Trevor, R., 884 Treynor, J., 63, 185, 191 Triantis, A., 942, 1036, 1042 Trigeogis, L., 767, 768, 1041 Trigueros, J., 1013 Trognon., A., 1156 Trolle, A.B., 716 Truong, C., 1279 Trzcinka, C.A., 670 Tsai, C.C.-L., 1447 Tsay, R.S., 1409, 1422 Tse, Y.K., 873, 879, 882 Tsetsekos, G., 829, 831 Tsiolis, A.K., 1136 Tsurumi, H., 1371–1380 Tucker, S., 1615 Tuckman, B., 503 Tufano, P., 665, 676, 679, 686, 1235, 1237, 1238, 1241, 1244, 1247 Tung, C.C., 1579 Turnbull, S., 549, 555, 584, 585, 587, 596, 600, 601 Turner, C.M., 669, 995 Turney, S.T., 1319 Tuttle, D., 37, 68, 92, 106, 122, 133 Tychon, P., 667 Tzeng, G.H., 1190, 1191 U Udell, G., 1438 Uejima, S., 1197 Uhlenbeck, G.E., 578 Unal, H., 665, 676, 677, 679, 686, 984, 988 Underwood, S., 1397 Uno, J., 999 Urias, M., 226 Urich, T., 122 Uryasev, S., 252 V Valdez, E.A., 994 Valieva, E. Van Boening, M., 148 van der Vorst, H., 1033 Van Deventer, D., 667 Van Dijk, D., 1274, 1278, 1279, 1333 Van Dijk, M.A., 1407 Van Horne, J.C., 25, 26, 273, 886 van Ness, J.W., 1410 Vanderbei, R.J., 248, 256 Vapnik, V., 1269, 1271 Vargas, M., 267–280 Varian, H., 1173 Varikooty, A., 49, 587, 588, 590, 592, 593, 596–602 Vasicek, O.A., 103, 667, 948, 982, 984, 988, 1483, 1515, 1516 Vassalou, M., 807, 809, 988 Vecer, J., 585 Veld, C., 873 Venezia, I., 583–586 Venkat, S., 1152 Venkataraman, S., 1025 Verbrugge, J.A., 1438 Verma, A.K., 819, 821–825 Verrecchia, R., 1174

Author Index Vetterling, W., 1271 Vetzal, K., 1218 Viceira, L., 289, 299, 1394 Vidmar, N., 144 Vijh, A.M., 1397 Villani, G., 1042 Ville, J.A., 1260, 1263 Vincete-Lorente, J.D., 831 Vishny, R., 140, 226, 686, 865, 1428 Vishwasrao, S., 1381 Vissing-Jorgensen, A., 1366 Viswanathan, R., 1490 Vohra, R., 1265 Volinsky, C., 42, 192, 199 von Furstenberg, G.M., 1283 von Maltzan, J., 648 von Neumann, J., 71, 247, 249, 254 Vora, G., 587 Vorkink, K., 1352 Vorst, A.C.F., 584, 585 Vorst, T., 533, 670, 990, 999, 1000 Voth, H.J., 137, 139 Vovk, V., 43, 1257–1266, 1269, 1271 Vyncke, D., 586, 593, 594 W Wachter, J., 298, 986 Wadhwa, P., 807 Wahal, S., 1366 Wahl, J.E., 1179 Wakeman, L., 584, 585, 587, 600, 601, 755 Wald, J., 531–541 Waldman, R., 1395 Waldmann, R.T., 138 Walker, I.E., 1594 Walking, R., 831, 834 Walkling, R.A., 1366 Wall, L.D., 679 Wallace, H.A., 1201 Wang, C., 999 Wang, C.-J., 1357–1368 Wang, D., 1274 Wang, H., 47, 666, 949, 950, 1455 Wang, J., 670, 990–992, 995, 1174, 1176, 1177, 1422 Wang, K., 587–602, 819, 821, 822, 825, 873–880, 1599 Wang, K.-L., 854, 855 Wang, Kehluh, 873–882 Wang, L., 1598, 1603 Wang, P.J., 1174, 1352 Wang, R.S., 409 Wang, S.S., 248 Wang, S.Y., 1183–1198 Wang, T., 585, 758, 759 Wang, Wei, 802 Wang, X.M., 965 Wang, Y., 1593–1604 Wang, Z., 1067, 1070, 1072, 1082, 1086 Wang, Z. Jay, 1237 Warachka, M., 1015 Warga, A., 669, 995 Warner, J.B., 807, 1224 Waters, W.R., 1466 Watson, D.J. H., 1308 Watson, M., 1392 Watson, R.D., 759 Watson, R., 1597–1599

Author Index Watts, R., 836 Watts, S., 1328 Weaver, D.G., 1223–1232 Weaver, D., 342 Weber, M., 1000 Webster, E., 1206 Wedel, M., 1061 Weeks, D., 587 Wei, D., 949 Wei, Jason, 670, 1366 Wei, K.C.J., 44, 1191, 1466, 1469, 1545 Wei, S., 1179 Wei, Z., 1366 Weinbaum, D., 668, 669 Weingartner, H.M., 1319 Weinstein, M., 885 Weisbach, M., 866, 941, 1357 Weisbenner, S., 1366 Weisberg, S., 966 Weiss, L., 941 Weiss, L.A., 807, 941 Welch, D., 1028 Welch, I., 290, 294, 295, 301, 831, 1089 Weller, P., 1608 Wells, E., 627 Werker, B.J., 226 Wermers, R., 273, 1237 Werner, I., 1236, 1237 Wessels, R., 831, 867 West, K., 137, 138, 1343 West, M., 1109 West, P.M., 1065 West, R., 810 Westerfield, R., 1223, 1593, 1604 Westgaard, S., 965 Weston, J.P., 679 Wetherbe, J.C., 1316 Whalen, G., 1604 Whaley, R.E., 41, 430, 431, 439, 441, 515, 522, 523, 552, 556, 1191, 1334, 1397 Whitcomb, D., 335, 339–341 White, A., 49, 464, 531, 547, 549, 551, 586, 587, 593, 599–602, 698, 999, 1191, 1483, 1615 White, E., 137, 148 White, H., 195, 587, 1086, 1347, 1439 White, M., 1062 Whitelaw, R.F., 587, 986, 987 Whitmore, G.A., 247 Whittington, G., 1597 Whyte, A., 1305 Wichern, D., 1276 Wichern, D.W., 1345, 1347, 1349 Widdicks, M., 506 Wiggins, J., 547, 551, 1027, 1273, 1275 Wiggins, J.B., 1191, 1615 Wiginton, J.C., 1062 Wijst, N., 965 Wiley, M.K., 1173 Wilhelm, W.J., 843 Wilhelm, W.J. Jr., 1366 Wilks, S.S., 485 Willard, G.A., 1507 Williams, A., 139–141, 148 Williams, J., 606, 1284, 1400, 1401, 1466, 1471 Williams, T., 340 Williamson, O., 831

1707 Williamson, R., 226, 1026 Wilmott, G.E., 1447 Wilmott, J., 1211 Wilmott, P., 451 Wilson, B., 1349 Wilson, R.L., 1325 Wilson, T., 1484 Winkler, R., 194 Wirl, F., 1042 Wise, M., 1325 Wishart, J., 1411 Witten, I.H., 1324 Wolf, A., 342, 343 Wolf, R., 677 Womack, K., 1161 Wong, W.K., 1173–1179, 1283–1289 Wood, D., 1598, 1603 Wood, R., 342 Wooldridge, J., 899, 904 Wooldridge, J.M., 1241, 1336, 1358–1361, 1363 Woolridge, J.R., 829 Working, H., 915 Worley, L., 1026 Wort, D.H., 93–109, 111–122, 172 Worthing, P., 1061, 1062 Wright, J.H., 1365, 1409 Wruck, H.K., 1151, 1153, 1161 Wu, C., 965, 979–1001, 1224, 1526, 1577 Wu, De-Min, 1093 Wu, G., 843 Wu, H.-C., 1190, 1191 Wu, J.-W., 854 Wu, L., 526, 667, 714, 718, 748, 1489–1500, 1504, 1511 Wu, M.C., 1579 Wu, S., 641, 647, 1121–1133, 1603 Wu, T., 1005 Wu, Wei, 1258 Wu, W.-H., 819–827 Wu, Yangru., 1061, 1391–1397 Wu, YiLin., 1151–1162 Wu, Z., 1577 Wycoff, F., 1207 Wynarczyk, P., 1598 X Xia, Y., 289–314, 316 Xie, F., 1366 Xing, H., 1417–1425 Xing, Y., 807, 809, 988 Xiong, W., 138 Xu, W., 1324–1326, 1329 Xu, X., 533, 1334 Xu, Y., 273 Xu, Y.J., 854 Y Yaari, M.E., 247 Yager, R.R., 1190 Yamazaki, H., 248 Yan, A., 1366 Yan, H., 670, 1000, 1001 Yang, C.W., 235–245 Yang, D., 1273, 1275, 1279 Yang, H., 1447 Yang, L., 802 Yang, S.S., 1412, 1413

1708 Yang, Tze-Chen, 703–705 Yang, W., 984 Yap, C.M., 829, 832, 840 Yaron, A., 1490 Yawitz, J., 995, 1036 Ye, G.L., 587 Yee, K., 42, 185–190, 193, 194, 198, 199 Yee, K.K., 185–200 Yen, N.-C., 1578, 1579 Yeo, G.H.H., 863–871 Yermack, D.L., 863, 864, 869 Yildirim, Y., 667, 996 Yilmaz, K., 1395 Yin, G.G., 1129 Yogo, M., 290, 1364, 1365, 1367, 1368 Yohai, V.J., 965 Yong, V.R., 248 Yoo, Y., 198, 199 Yoon, Y., 1275 Yor, M., 585, 1447, 1559 Yoshihara, K., 1411 Young, R., 802 Yu, E., 1381 Yu, F., 670, 993, 994 Yu, H.-C., 829–840 Yu, M.T., 753–764, 819, 823, 825 Yu, P.L., 1308, 1325 Yule, G.U., 45, 1068, 1076 Yunker, P.J., 1307, 1308 Z Zacks, S., 1347 Zadeh, L.A., 1183, 1184, 1188, 1190, 1193 Zahavi, J., 1062 Zajdenweber, D., 754 Zaman, M.A., 1153, 1154 Zanakis, S.H., 965 Zanola, R., 1206 Zantout, Z., 829, 831 Zarembka, P., 1553 Zariphopoulou, T., 1007, 1567, 1571 Zarnowitz, V., 103, 1477 Zavgren, C.V., 1597, 1598

Author Index Zechner, J., 667, 941, 943 Zeckhauser, R.J., 1237 Zeleny, M., 1308 Zellner, A., 192, 1095, 1156 Zeng, Y., 1121, 1133 Zervos, S., 965 Zhang, A., 1349 Zhang, B.Y., 670 Zhang, D., 843 Zhang, G., 1000 Zhang, J.E., 587 Zhang, L., 1069, 1070 Zhang, P., 585 Zhang, S., 1366 Zhang, X., 226, 669 Zhang, Xiongfei Zhang, Y., 1000 Zhao, F., 713–746 Zhao, Y., 323 Zheng, L., 1237, 1244 Zheng, Q., 1578, 1579 Zhong, Z., 670 Zhou, C., 666, 950, 988 Zhou, G., 172, 174, 505, 506, 510 Zhou, H., 668, 670, 717, 984, 1121, 1126, 1128 Zhou, J., 458, 464 Zhou, X., 1391–1397 Zhu, A., 965 Zhu, F., 668, 670 Zhu, H., 668, 670, 1000 Zhu, N., 1524, 1530 Zhu, Y., 324–327 Ziemba, W.T., 323 Zimmermann, H.-J., 1190 Zin, S.E., 986 Zingales, L., 867 Zitzewitz, E., 886 Zivot, E., 1422 Zmijewski, M.E., 1598 Zoido-Lobaton, P., 649 Zopoundis, C., 965 Zwiebel, J., 865

Subject Index

A Accounting beta, 100–105, 117 Accounting information system selection, 48, 1307, 1316–1319, 1329 Acquisition discounts, 1427–1443 Actuarial reserve, 48, 959–961 Actuarial risks, 953 Added constraints, 236 Additive model, 154, 1055, 1056, 1059, 1060, 1201, 1411 Adjustable rate mortgage (ARM), 43, 716, 744–746, 784, 788, 790–793, 796, 797, 805, 854, 1135–1149 Adjusted present value (APV), 29, 30, 32–34, 36, 1223–1225, 1231 After-tax weighted average cost of capital (ATWACOC), 30–33, 36, 1223–1226, 1228–1232 Agency costs, 28, 677–679, 683, 686, 689, 692, 694, 807, 830, 831, 863–865, 867–871, 933, 936, 940, 941, 947, 1036 Alt-A, 782 American option, 14, 41, 356, 360–363, 378, 387, 405, 431–433, 438–446, 506, 510, 512, 517, 523, 533, 551, 552, 589, 780, 1209 Annuity, 32, 460, 954, 955, 958–960 APER. See APparent error rates (APER) APparent error rates (APER), 42, 1064, 1065 Approximately complete markets, 1008, 1017 Approximation of stochastic integrals, 1021–1024 APT. See Arbitrage pricing theory (APT) APV. See Adjusted present value (APV) Arbitrage, 12, 13, 17–19, 27, 47, 118, 140, 144, 149, 185, 217, 286, 339, 343, 360, 367, 395, 433, 436, 438, 447, 455, 459, 461, 464, 465, 483, 489, 490, 522, 523, 576, 585, 619, 620, 726, 748, 826, 886, 920, 943, 981, 1017, 1126, 1131, 1466, 1490, 1503, 1509–1511, 1555, 1567, 1577–1591, 1619 Arbitrage detection, 1577–1591 Arbitrage pricing theory (APT), 3, 12–14, 22, 44, 109, 185, 191, 273, 283, 286, 483, 489, 534, 1007–1024, 1465–1471, 1474–1481 ARCH models, 1174, 1181, 1333–1338, 1343, 1348, 1349, 1418, 1419, 1421 ARM 2/28, 779, 782–785, 788, 793–795 Asian options, 49, 583–602, 755 Asset allocation, 46, 69–91, 166, 171, 184, 185, 192, 203, 206, 210, 217, 221, 222, 224, 226, 248, 290, 304, 305, 323, 327, 328, 330, 1088, 1236, 1273, 1278, 1280, 1393–1394, 1424 Asset pricing, 42, 44, 45, 93, 137–140, 143, 151, 185, 187, 190, 191, 195, 233, 270, 272–274, 280, 283–286, 290, 314, 330, 332, 356–359, 365, 366, 369, 375, 447, 448, 464, 491, 493, 496, 506, 508, 509, 511, 531, 533, 534, 536, 584, 589–595, 600, 677, 853, 861, 1008, 1012, 1014, 1016, 1017, 1067–1089, 1091–1108, 1110, 1121, 1127, 1165, 1229, 1236, 1244, 1274, 1278, 1279,

1349, 1352, 1354, 1371, 1374, 1395, 1466, 1469, 1470, 1476, 1480, 1481, 1526, 1527, 1534, 1537, 1563, 1567, 1573 Asset return predictability, 1394 Autoregressive modular (ARM) forecasting methodology, 43, 1135–1149 modeling methodology, 43, 1135–1149 processes, 43, 1135–1149 B Bank acquisitions, 1427–1443 Bank merger, 767, 771, 775, 777, 1433, 1434, 1439 Bankruptcy, 9, 40, 48, 49, 55, 100, 189, 198, 204, 205, 272, 515, 659, 667, 682, 769, 775–777, 807, 808, 819, 821, 822, 825, 830, 934–942, 944–950, 969, 996, 998, 1000, 1027, 1028, 1045, 1055–1061, 1299, 1307, 1324–1329, 1354, 1381, 1389, 1446, 1451, 1458, 1593–1600, 1603, 1604 Banks, 16, 101, 264, 330, 332, 515, 639–641, 643–649, 659, 660, 675–680, 682–694, 697–699, 703, 705, 710, 713, 729, 767–777, 797, 819–826, 829–840, 883, 884, 953, 954, 961, 962, 965, 969, 984, 997, 1000, 1007, 1008, 1025–1029, 1033–1036, 1061, 1062, 1071, 1121, 1237, 1305, 1329, 1350, 1354, 1367, 1375, 1417, 1465, 1595, 1603, 1604 Bayesian analysis, 190, 192, 199 Bayesian estimation, 192, 1110, 1422, 1424 Behavioral finance, 165, 455, 1152, 1347, 1399 Beta, 11–13, 20, 34–36, 45, 54, 62–64, 93–105, 117–118, 120, 123, 129, 130, 132, 185, 189, 208, 209, 212, 271–277, 279, 280, 285, 286, 338, 366, 533, 534, 680, 689, 692, 854–856, 861, 909, 1067–1070, 1072, 1073, 1080, 1082–1089, 1091–1108, 1231, 1244, 1346, 1466–1469, 1471–1474, 1478, 1481, 1523, 1525, 1526, 1552 Binomial distribution, 41, 385, 393–397, 426, 589, 1161 Binomial model, 49, 395, 396, 441, 442, 505–513, 543, 585–593, 599, 601, 602, 625, 626, 1190, 1191, 1512 Binomial option pricing model, 15, 41, 393–397, 406, 409–418, 425, 426, 505–507, 617–636, 1190, 1194–1196 Bipower variation, 42, 526–528 Bivariate normal distribution, 41, 404, 424, 429–431, 614, 700, 882, 1285, 1373, 1420, 1421 Black-Scholes formula, 329, 385–388, 391, 402, 409, 415, 417, 418, 426, 428, 455, 457, 461–462, 468, 470, 484, 487, 492, 502, 505, 507, 516–518, 534, 548, 577, 579, 584, 606, 768, 809, 825, 1016, 1190, 1191, 1219, 1608, 1610, 1612, 1620, 1622 Black-Scholes model, 41, 47, 386–388, 392, 399–402, 406, 415, 418, 457, 458, 463, 464, 471, 481, 515, 518, 531–535, 537, 538, 540, 541, 543, 547, 548, 558, 576, 584, 715, 718, 809, 916, 1057, 1191, 1504, 1607–1611, 1613, 1615

1709

1710 Black-Scholes option pricing model, 15, 26, 355, 385–387, 397, 409–418, 447–465, 470, 481, 491, 526, 532, 611, 613, 617–627, 1613 Black-Scholes partial differential equation (PDE), 47, 502, 503, 1209, 1211, 1213, 1215, 1218, 1574 Business cycle, 56, 206, 686, 801–802, 915–931, 985, 1121, 1122, 1133 Business model, 1041–1051, 1349 C Calendar spread, 371–372 Call option, 14, 356–358, 360–363, 365, 367, 372, 374, 378–388, 390, 393–397, 405, 409, 425–431, 433, 435–438, 440, 454–458, 461–464, 477, 482, 483, 487, 491–496, 498–502, 506, 508, 510, 518, 526, 533, 536, 538, 548, 550, 552–554, 556, 559–564, 566, 569, 576, 577, 579, 584, 585, 588, 591, 594, 596, 598, 600, 611, 617–624, 626, 717, 749, 754, 761, 762, 780, 796, 809, 810, 821, 824, 886, 915, 916, 919, 920, 922, 933, 934, 1013, 1014, 1016, 1049, 1066, 1170, 1192, 1193, 1219, 1220, 1335, 1446, 1486, 1487, 1573, 1611, 1619, 1621 Capital allocation line (CAL), 171, 172, 175, 178, 180 Capital asset pricing model (CAPM), 3, 10–12, 22, 25, 44, 45, 63, 93–105, 125, 185, 187, 189, 191, 192, 204, 208, 209, 235, 267–280, 283–287, 335–340, 342, 343, 349, 355, 366, 383, 487, 532, 534, 677, 683, 920, 1067–1069, 1089, 1091–1093, 1098–1100, 1102, 1107, 1108, 1258, 1347, 1349, 1466–1481, 1523–1527, 1537, 1545 Capital budgeting, 23, 28–34, 36, 48, 102, 104, 378, 610, 613, 1028, 1041–1051, 1184, 1190, 1197, 1223–1232, 1307, 1319–1324, 1329, 1466, 1623 Capital labor ratio, 35, 101, 102, 1296 Capital market line (CML), 11, 12, 94–97, 105–107, 270, 272, 337–338 Capital standard, 685, 823 Capital structure, 8, 9, 23–25, 27, 28, 34–36, 46, 47, 99, 377, 387–390, 667, 668, 789, 809, 832, 863–871, 933–950, 995, 1025, 1028, 1036, 1043 Capital structure arbitrage, 1041 Capital structure model, 933, 934, 937, 941–950 CAPM. See Capital asset pricing model (CAPM) CARR. See Conditional autoregressive range (CARR) Change-point, 44, 1346–1348, 1350–1352, 1354, 1417, 1421–1424, 1577, 1578, 1581, 1584, 1587–1590 Change-point problem, 1347–1348 Characteristic function, 43, 47, 485, 488, 489, 528, 550, 551, 555, 573–581, 1486, 1569 Closed-end country fund, 44, 1523–1552 Closed fore option pricing model, 47, 485–489, 1611 Co-integration, 43, 1393 Collar, 374 Combinatorial optimization, 659 Combined LTV (CLTV), 45, 783, 784, 795 Complex logarithm, 1165, 1167–1170 Conditional autoregressive range (CARR), 1274, 1276–1278, 1280 Conditional correlation, 1152, 1155–1162, 1274, 1288 Conditional heteroskedasticity, 718, 880, 909, 1122, 1123, 1273, 1417–1422, 1425 Conditional probability analysis (CPA), 103, 1307, 1312–1316, 1329, 1598–1600, 1604 Conditional value at risk, 43, 48, 251–252 Confluent hypergeometric function, 1052, 1057, 1515, 1521 Consistency, 15, 144, 199, 210, 342, 351, 605, 641, 656, 915, 919, 921, 1096, 1098, 1211, 1362 Consolidation values, 767–777

Subject Index Constant elasticity of variance (CEV), 41, 42, 471–478, 575–577, 1191, 1615–1625 diffusion, 471–473, 1615–1619, 1623 model, 474, 476–478, 547, 744, 1616 Constant proportion portfolio insurance (CPPI), 321–324, 327–332 Consumption-based CAPM, 283, 605 Contingent claim pricing, 460, 463, 1623 Continuous time, 45, 204, 290, 292, 385, 411–415, 447–449, 465, 506, 510, 588, 596, 611, 768, 771, 826, 979, 985, 1015, 1055–1060, 1110, 1122, 1128–1133, 1466, 1504, 1511 Convenience yield, 915–931, 997, 1007, 1010 Copula, 42, 43, 697–711, 994–995, 1371, 1376–1379 Corporate bonds, 5, 55, 206, 211, 213, 214, 216, 668–670, 883–912, 934, 949, 950, 988, 991, 995, 996, 999–1001, 1068, 1070, 1229, 1269, 1271, 1397 Corporate failure, 1593–1604 Corporate finance, 3, 23, 27, 28, 30, 46, 104, 144, 675, 677, 678, 684, 694, 950, 995, 1041, 1042, 1051, 1160, 1357, 1358, 1366–1368, 1601, 1615, 1623 Correlation, 8, 11, 20, 21, 42, 45, 48, 59, 61, 70, 77, 78, 81, 91, 96, 103, 106, 117, 132, 133, 139, 144, 166–169, 175, 188, 189, 194, 197, 221, 222, 224–226, 233, 235, 237, 259–264, 271, 277, 278, 290–292, 294, 297–301, 306, 332, 334, 342, 404, 425, 429, 431, 441, 462, 481–483, 486, 490, 550–552, 557, 561, 578, 579, 607, 609, 646, 652, 655–658, 660, 666, 679, 697, 698, 700–703, 705, 710, 714, 716, 718, 721, 723, 725, 728, 734, 737, 748, 756, 757, 760, 765, 770, 772, 774, 775, 789, 792, 797, 798, 812, 813, 856, 868, 870, 871, 877, 885, 886, 896, 917, 921, 926, 930, 948, 987, 990, 993–996, 1001, 1065, 1072, 1093, 1094, 1096, 1098, 1101, 1102, 1109, 1111, 1152, 1154–1162, 1166, 1235–1238, 1241, 1244, 1247, 1258, 1274, 1278, 1285, 1288, 1289, 1305, 1336, 1338, 1340, 1353, 1358, 1359, 1363, 1364, 1376, 1379, 1389, 1417, 1421, 1424, 1472, 1477, 1478, 1483, 1485, 1537, 1601 Cost of equity capital, 25–26, 33, 36, 44, 1465–1481 Cost of equity of the unlevered firm, 1223–1227, 1229, 1231 Counterparty risk, 347, 697, 980, 994, 999 Covered call, 365, 372–374, 933 Cox–Ingersoll–Ross model, 47, 565, 1488, 1503–1506 Cox–Ross–Rubinstein (CRR) lattice binomial approach, 402 Credit crunch, 332, 779, 788, 789, 969 Credit default swaps (CDS), 48, 665, 668–670, 702, 801, 933, 949, 962, 963, 979, 980, 995–1001 Credit derivatives, 330, 331, 670, 997, 998, 1000 Credit risk, 43–45, 47, 639–641, 643, 644, 648, 649, 659, 660, 665–671, 679, 697, 698, 700, 701, 703, 710, 753–757, 759–760, 765, 808, 811, 815, 818, 836, 884, 933, 934, 949, 953, 961, 963, 965, 980, 988, 996, 1001, 1062, 1271, 1446, 1448, 1457, 1458, 1587, 1597 Credit risk rating, 45, 639–660 Credit spread, 668–670, 699, 705, 941, 943, 947–949, 996, 998, 1000, 1001, 1118 Credit VaR, 697–711 Critical line, 87, 91 Crop year, 917–919, 923 Cross-section of stock returns, 1093 CRRA intertemporal CAPM, 605 Cumulative binomial function, 397 Cumulative sum (CUSUM), 1347, 1577, 1578, 1581–1584, 1586, 1590, 1591, 1600–1601 Curtailment, 864 CUSUM. See Cumulative sum (CUSUM) D Data-mining, 44, 45, 48, 194, 200, 274, 641, 1067–1089, 1307–1329 Data snooping, 42, 187, 192, 194–200, 1067, 1092, 1333 DCC. See Dynamic conditional correlation (DCC)

Subject Index Debt financing, 24, 29, 31, 32, 34, 36, 39, 40, 56, 831, 863–865, 870, 938, 979, 1025, 1026, 1028, 1029, 1036, 1224–1226, 1321, 1381, 1382, 1389 Debt-to-income ratio (DTI), 780, 784–786, 800, 803, 804 Decision tree, 397, 617–627, 1031–1033, 1324, 1325 Default, 4, 27, 28, 30, 44, 45, 48, 55, 94, 179–181, 360, 389, 390, 607, 609–611, 613, 623, 639, 641, 644, 648, 664–670, 697–711, 756, 779–802, 807–810, 812–815, 853, 884, 885, 933, 934, 937, 939, 941–944, 947–950, 953, 961–963, 965–976, 979–1001, 1044, 1046, 1070, 1118, 1131–1133, 1224, 1226, 1231, 1232, 1299, 1324, 1353, 1446, 1451, 1593, 1596, 1601, 1602 Default correlation, 697–711, 980, 993–996 Default digital swap, 48, 963 Default intensity, 980, 988–990, 993, 994, 996, 997, 1001 Default risk, 5, 23, 25–28, 43, 44, 55, 640, 649, 753–757, 764, 787, 794, 807–815, 885, 933, 949, 991, 995–997, 999–1001, 1026, 1028, 1032, 1035, 1036, 1121, 1128, 1353, 1602 Default-triggering level, 47, 1293, 1299, 1300, 1451–1453, 1456 Defensive forecasting, 43, 44, 1257–1271 Delinquency, 45, 780, 783, 784, 789, 790, 797–801, 804, 1353 Delta, 47, 321, 327, 329, 455, 457, 491–494, 496–500, 502–503, 517, 522, 548, 549, 552–555, 561–567, 714, 719, 721, 855 Demand/supply shock, 915, 916, 918, 919, 921, 931 Density process, 49, 1556, 1567–1575 Deposit insurance, 683–686, 819–826, 1429 Discriminant analysis (DA), 965, 1061, 1062, 1065, 1066, 1325, 1328, 1596–1598 Displaced log normal, 41, 439–442 Distress, 676–680, 682, 688, 689, 692, 694, 779, 787, 789, 807, 808, 812, 813, 815, 830, 933, 937, 939, 947, 948, 996, 1026, 1036, 1397, 1429, 1593, 1594, 1596, 1598, 1600–1602 Dividend discount model (DDM), 303–306, 310, 314 Duality, 42, 236, 254–257, 286, 866, 1564 Dyl model, 80 Dynamic, 6, 44–47, 49, 138, 193, 194, 205, 221, 223, 233, 289, 290, 297, 299, 300, 302, 303, 314, 321, 323, 324, 327–329, 331, 332, 334, 341, 342, 350, 358, 455, 459, 460, 462, 463, 506, 533, 548–550, 554, 560–565, 567, 585, 605, 665, 666, 670, 713, 715, 717, 719, 723, 725–727, 737, 746, 748, 756, 759, 760, 768, 770, 779–802, 823, 844, 880, 909, 910, 934, 941, 943, 945–946, 948, 950, 979–988, 990, 991, 993, 997, 999–1001, 1009, 1025, 1032, 1036, 1069, 1071, 1089, 1109, 1121, 1122, 1126–1128, 1178, 1220, 1231, 1233, 1236, 1238, 1273, 1274, 1279, 1280, 1284, 1285, 1288–1291, 1312, 1316, 1352, 1382, 1385, 1391, 1392, 1394, 1396, 1417, 1420, 1422, 1424, 1446, 1484, 1489–1500, 1504, 1505, 1509, 1511, 1514, 1516, 1517, 1555, 1556, 1564, 1567–1569, 1571, 1574, 1601, 1604, 1612, 1615, 1623 Dynamic conditional correlation (DCC), 1274, 1278, 1280 E Early exercise, 361, 405, 431–433, 436–440, 552, 1333, 1334, 1488, 1615 Earnings multiples, 7, 199 Edgeworth binomial model, 49, 588–591, 593, 599, 601, 602 Efficient frontier, 46, 48, 60, 61, 78–83, 87–89, 91, 93–95, 111, 115, 120, 121, 123, 125, 126, 128, 165, 167–171, 175–182, 221, 222, 226–228, 233, 236–238, 245, 259, 260, 264, 265, 268–271, 273, 280, 1424 Efficient market hypothesis (EMH), 104, 137, 334, 459, 1258 Efficient portfolio, 46, 48, 60–62, 68, 69, 77–91, 94, 96, 111, 128, 166–172, 175, 180, 192, 245, 248, 259, 260, 265, 266, 272, 284–286 EGB2 distribution, 854–855, 859–861 Electronic markets, 345–347 Empirical analysis, 209, 277–280, 342, 535, 668, 682, 718, 723–725, 738, 844, 859, 917, 949, 1237, 1403, 1417, 1427, 1431, 1524

1711 Empirical evidence, 44, 47, 93, 118, 195, 204, 226, 301, 453, 471, 665, 668–670, 727, 769, 844, 845, 856–859, 863, 945, 948–950, 979–1001, 1015, 1121, 1151, 1241, 1247, 1399, 1401, 1406–1407, 1411, 1421, 1524, 1604, 1616, 1618 Endogeneity, 46, 1357–1368 Equity liquidity, 44, 807–815 Equity risk premium puzzle, 605 Errors-in-variables (EIV) bias, 1092–1099, 1102, 1107 Estimation risk, 10, 46, 203–218, 1092, 1107, 1466, 1472, 1474, 1481 European option, 356, 360–361, 378, 383, 385, 387, 405, 431, 437, 506, 507, 517, 534, 550–552, 555, 556, 585, 589, 769, 1110, 1483, 1487, 1619 Evolution of risk, 1346 Ex-dividend, 8, 28, 41, 296, 430, 432, 433, 436, 437, 439–442, 533–535 Executive stock option, 377 Exercise price, 14, 15, 286, 320, 321, 356–372, 374, 375, 378,–387, 390–393, 397, 399, 402, 403, 405, 415, 425–427, 431, 436–438, 454, 456, 457, 464, 467, 501–502, 557, 562, 583, 593, 618, 619, 621, 758, 761, 762, 920, 1042, 1195, 1196, 1211, 1335, 1608 Exotic options, 453, 465, 583, 584 Expected discounted penalty, 47, 1293, 1298, 1445–1458 Expected utility, 38, 71–74, 91, 208, 209, 217, 247–249, 251, 254–257, 286, 300, 301, 311, 317, 336, 337, 605, 607, 1384 Experimental asset markets, 138 F Failure prediction, 1596, 1598, 1604 Fat tails, 43, 698, 710, 718, 728, 792, 802, 823, 853–856, 861, 873, 896, 1109, 1111, 1285, 1421 Financial applications, 575–581, 961–963, 1183–1198, 1273–1280, 1353, 1568, 1569 Financial crisis, 832, 1122, 1349, 1354 Financial distress costs, 676–680, 682, 689, 692, 694, 807, 808, 812–813, 815, 933, 947, 1026, 1036 Financial markets, 16, 42, 44, 165, 166, 221, 222, 224, 226, 233, 285, 331–332, 342, 345, 450, 605, 606, 613, 639, 648, 738, 779, 782, 998, 1000, 1032, 1173, 1198, 1257, 1279, 1284, 1290, 1347, 1349, 1352–1354, 1383, 1409, 1524, 1537, 1608 Financial signaling, 1399–1407 Finite difference, 49, 1210–1217, 1219, 1220 Finite element, 1218–1219 Finite volume, 49, 1217–1218, 1220 Firm bankruptcy predictions, 1324–1329 Firm size, 835 Fixed operating cost, 1052 Forecast(ing), 5, 8, 11, 18, 23, 42–45, 65, 93–105, 117, 118, 120, 185, 187, 192–194, 198, 199, 217, 271, 290, 294, 295, 301, 303–314, 397, 458, 459, 524–526, 528, 684, 772, 780, 792, 797, 798, 800–802, 810, 831, 853, 855, 861, 907, 909, 910, 912, 967–969, 973–976, 987, 1041, 1135–1149, 1161, 1162, 1192, 1193, 1257–1271, 1274, 1276–1280, 1324, 1333–1344, 1348, 1349, 1378, 1379, 1391–1395, 1401, 1414, 1417–1425, 1432, 1466, 1467, 1476–1481, 1495, 1500, 1502, 1523, 1602 Foreclosure, 787, 797, 803, 1203, 1204, 1353 Foreign exchange volatility model, 686, 1110, 1111, 1113–1118, 1409 Forward measure, 715, 723, 725–728, 732, 734, 737, 738 FTSE100, 42, 175, 182, 533, 1376–1379 Functional transformation, 1523–1525, 1534, 1545 Fundamental analysis, 42, 198, 199, 343 Fundamental asset values, 137 Fundamental theorems of asset pricing, 1008, 1017 Futures option, 359, 362–363, 375, 557, 760, 763, 1488 Fuzziness, 1183, 1184, 1190, 1191, 1197, 1198 Fuzzy set, 41, 48, 1183–1198, 1307–1329

1712 G Game-theoretic probability, 43, 1257, 1258, 1260–1265 Game theory, 1042 Gamma, 473, 496–500, 502–503, 533, 721, 855, 873, 910, 1091, 1373, 1378, 1448, 1570, 1571 Gamma function, 855, 1052, 1515, 1618 GARCH. See Generalized autoregressive conditional heteroscedasticity (GARCH) GARCH effect, 43, 912, 1173–1179, 1285, 1352 Gaussian copulas, 42, 702, 705, 1376, 1378, 1379 Generalized autoregressive conditional heteroscedasticity (GARCH), 43, 44, 329, 518, 533, 715, 823, 853–858, 861, 884, 886, 896, 899, 901, 903–911, 1109, 1122, 1173–1179, 1273, 1274, 1278, 1280, 1284, 1285, 1287, 1288, 1334, 1347–1353, 1374–1376, 1409–1411, 1417–1422, 1424, 1425, 1607 Generalized binomial option pricing model, 43, 395–397, 613 Generalized hyperbolic (GH) distribution, 42, 873, 874, 876, 877, 880 Genetic programming, 1607–1614 Gibbs sampler, 42, 1371, 1373–1375, 1379, 1421 Global recession, 779 Granger-causality test, 1392, 1397 Greek letters, 47, 93, 115, 491–503 H Hansen Jagannathan bounds, 285–286 Hazard model, 1604 Heath–Jarrow–Morton (HJM), 713, 714, 716, 721–723, 727, 746, 981 Hedge ratio, 15, 20–22, 42, 377, 384, 387, 394, 455, 547–549, 552–555, 558, 561, 571, 676, 678, 679, 719, 720, 873–880, 1195 Hedging, 3, 15–22, 47, 233, 301, 320, 328, 329, 331, 332, 365, 366, 377, 383, 384, 493, 535, 543, 547–549, 552–558, 561–568, 570, 571, 583, 585, 668, 675–680, 682–684, 686, 687, 694, 713, 715, 716, 718–724, 729, 744, 746, 753, 756, 765, 873, 877, 878, 884, 909, 1007, 1016, 1118, 1290, 1349, 1351, 1394, 1488, 1494, 1498, 1555, 1556, 1615, 1616 Hedging performance, 47, 329, 547–571, 714, 719–721 Hedonic models, 45, 1201–1206 Heston model, 487–489, 1165–1171 High-frequency returns, 670, 1334, 1344 High/low range, 1278, 1280 High yield (HY), 668, 669, 883–886, 893, 896, 897, 899, 901, 903, 904, 907, 909, 910, 912, 1397 Hilbert–Huang transformation (HHT), 1577–1584, 1586, 1587, 1589–1591 Housing price appreciation (HPA), 45, 780–783, 789–792, 795, 800–803, 805 Housing price index (HPI), 45, 788–792, 801, 803, 804 Human resource allocation, 48, 1307, 1312–1316, 1329 I Illiquid markets, 19, 221 Implied probability densities, 518, 1568 Implied volatility, 42, 328, 329, 457, 458, 517, 518, 522–524, 532–534, 536–540, 544, 545, 548, 549, 552, 555–558, 571, 668, 669, 714, 715, 721, 723, 728–730, 733, 734, 736, 737, 743, 746, 1049, 1110, 1279, 1333–1344, 1487, 1581, 1588 Impulse response, 1392–1393, 1396 Incomplete market, 458, 465, 715, 725, 1008, 1017, 1555, 1564, 1567, 1568, 1574 Indifference curve, 46, 71–77, 79, 81, 169, 171, 268, 270, 280 Indifference pricing of derivatives, 1567 Informational efficiency, 109, 334, 680, 1339, 1343, 1396–1397 Information content of trades, 1395–1396 Information flow, 1000, 1174, 1175, 1177, 1178, 1285, 1397 Information technology, 347, 763, 1048, 1061, 1283–1289, 1326 Instability principle, 1346, 1348

Subject Index Instrumental variable estimation, 1357, 1360, 1361 Insurance, 27, 48, 49, 62, 319–332, 464, 584, 586, 643, 648, 683, 684, 686, 753, 756, 760, 761, 765, 771, 784, 819–826, 830, 953, 955–961, 963, 1086, 1087, 1156, 1184, 1410, 1414, 1445–1447, 1457, 1465–1479, 1481 Integration, 3, 16, 36, 222, 224, 226, 233, 347, 440, 449, 468, 469, 471–478, 491–503, 505, 577, 580, 615, 738, 767, 958, 1011, 1019, 1165, 1168–1171, 1218, 1283, 1324, 1371, 1372, 1379, 1419, 1420, 1449, 1450, 1521, 1527, 1537, 1572, 1615–1625 Integro-differential equation, 1293, 1295, 1298, 1445–1449, 1457 Integro-partial differential equations, 49, 1567, 1568, 1572–1574 Interest, 4, 5, 9, 14–17, 25–33, 36, 45–48, 55, 65, 68, 95, 99, 101, 102, 118, 144, 148, 186, 188, 206, 252, 270, 275, 289, 333, 336, 339, 341, 343, 357, 358, 360–362, 378, 382, 386–390, 418, 426, 432, 433, 436, 437, 485, 515, 516, 518, 522, 561, 605, 640, 641, 644, 660, 665, 678, 684, 700, 717, 753, 768–770, 772, 774–776, 779, 780, 783, 784, 792, 794, 805, 811, 834, 838, 843, 844, 850, 863–865, 878, 910, 916, 920, 930, 933, 934, 936, 938, 941, 943, 944, 953–956, 963, 969, 996–998, 1025, 1027, 1034, 1035, 1062–1064, 1069, 1110, 1113, 1130, 1153, 1192, 1195, 1224–1226, 1229, 1230, 1237, 1273, 1284, 1305, 1326, 1327, 1333, 1346, 1353, 1359, 1361, 1363, 1366, 1376, 1396, 1418, 1435, 1438, 1439, 1533, 1567, 1568, 1571, 1593–1596, 1615 Interest rates, 5, 9, 14–18, 45, 47, 55–57, 108, 178, 185, 205, 217, 226, 270, 273, 275, 276, 290, 292, 299, 303, 310, 314, 316, 321, 329–331, 361, 362, 382, 384–386, 389, 390, 392, 394, 400, 402, 409, 416, 456, 457, 464, 472, 490, 497, 500, 501, 503, 517, 521, 533, 547–571, 578, 588, 601, 607, 609, 611, 619–621, 624, 626, 639, 648, 665–667, 676–692, 703, 705, 713–746, 753–756, 758–760, 762–764, 769, 770, 772, 774, 780, 783, 788, 794, 795, 819, 821–823, 825, 878, 885, 886, 899, 916, 917, 921, 923, 926, 934, 935, 938, 939, 942, 948, 950, 953, 954, 963, 979–981, 983–991, 993, 995–1001, 1013, 1026, 1027, 1030–1032, 1036, 1043, 1067, 1071, 1073, 1075, 1118, 1121–1133, 1136, 1166, 1183, 1190–1197, 1209, 1211, 1224, 1226, 1228–1230, 1296, 1305, 1375, 1379, 1393, 1395, 1417, 1421, 1468, 1483, 1484, 1487–1500, 1503–1517, 1616, 1623 Interest rate volatility, 713, 715, 718, 737, 987, 1122 International capital flows, 1235–1237, 1247 International equity fund, 48, 235, 245, 1238 International mutual funds, 1235–1252 International portfolio management, 10, 46, 221–233 Interstate acquisitions, 1431, 1442 Intertemporal equilibrium, 10, 283–287 Investment constraints, 46, 221, 222, 226, 227, 229–231, 233 Investment model, 48, 203–206, 235–245, 1293, 1295, 1300 Iso-return line, 84, 85, 88 Iso-variance ellipse, 84–88 IT bubble, 1284, 1288, 1289 Iterated cumulative sums of squares algorithm, 1347, 1350 Itô’s Lemma, 451–452, 454, 460, 462, 473, 481, 490, 516, 517, 578, 579, 921, 1493, 1509, 1514, 1616

J Jensen measure (JM), 62–64, 209 Joint normality assumptions, 873, 880 Jump diffusion, 47, 526, 531, 533, 541, 666, 667, 669, 670, 716, 734, 949, 979, 984, 988, 1130–1133, 1191, 1293, 1445–1458, 1607, 1613 Jump risk, 42, 515, 527, 528, 666, 725, 726, 732, 1000, 1129, 1130

K Keiretsu, 829–840

Subject Index L Lagrange multipliers, 81, 169, 173, 236, 237, 254, 255, 257, 269, 1347 Laplace transform, 47, 585, 1136, 1139, 1293–1295, 1298, 1300, 1457, 1458 Latent factors, 985, 1273 Lattice model, 403–406 Law of long time, 1346, 1347 Le Châtelier principle, 48, 235–245 Leverage, 8, 9, 14–16, 35, 46, 56, 99–102, 117, 162, 163, 185, 198, 215, 216, 275, 323, 331, 360, 363, 366, 643, 666, 667, 670, 677, 682–684, 689–692, 725, 734, 801, 823, 825, 829–840, 854, 863–871, 873, 885, 910, 934, 937, 938, 940, 941, 946–950, 996, 1042, 1043, 1048–1051, 1111, 1224, 1225, 1229–1232, 1270, 1277, 1320–1322, 1326, 1352, 1381–1383, 1385–1389, 1396, 1400, 1403–1407, 1417, 1419, 1421, 1433, 1438, 1439, 1574 Levy processes, 526, 1009 Life insurance, 48, 955–958, 960, 1410, 1476 Linear programming approach, 48, 117, 123, 1323, 1324 Linguistic variables, 1190 Lintner’s method of short sales, 129 Liquidation, 20, 39, 40, 186, 188–190, 199, 369, 387, 942, 943, 947, 948, 953, 969, 1008, 1010–1014, 1019, 1594–1596, 1604 Liquidity, 5, 14, 15, 21, 44, 47, 56, 117, 118, 138, 185, 190, 196, 221, 224, 226, 274, 334, 335, 340–343, 345–347, 358, 533, 556, 645, 670, 676, 678, 682, 684, 685, 688, 692, 746, 807–815, 818, 820, 854, 873, 949, 950, 965, 980, 986, 991–993, 995–1001, 1007–1017, 1041, 1279, 1353, 1396, 1397, 1409, 1428, 1429, 1439, 1441, 1490, 1496, 1523, 1594, 1598 Listing effect, 1427–1443 Loan performance, 780, 782, 784–791, 793–797, 800 Loan-to-value (LTV), 783, 788, 792, 795, 800 Local expectations hypothesis, 605 Logarithmic utility, 210, 215, 607, 609 Logical analysis of data (LAD), 45, 640–647, 649–660 Logistic regression, 45, 646, 965–976, 1061–1063, 1065, 1066, 1325 Logit, 45, 965–967, 969, 973–975, 1062–1063, 1307, 1328, 1598–1600 Log normal, 41, 42, 49, 81, 82, 421–424, 426–428, 439–442, 453, 457, 467, 470, 477, 484, 515, 517, 518, 522–524, 534–535, 537–541, 544, 545, 584, 587, 593, 595, 610–613, 728, 754, 762, 942, 985, 1043, 1112, 1490, 1615, 1616, 1618 Log normal distribution, 41, 82, 324, 402–404, 421–428, 440–442, 453, 457, 458, 467, 468, 470, 471, 532, 541, 576, 584, 585, 594–601, 734, 754–756, 761, 854, 855, 1030, 1506, 1545 Log-normal process, 41, 439–440, 1616 Long-memory stochastic volatility model, 48, 1409–1414 Long straddle, 367–369 Long vertical spread, 369–370, 372 Loss severity, 753, 780, 802 M Main bank, 829–840, 1329 Managerial entrenchment, 863, 865 Market beta, 45, 93, 104, 1073, 1091, 1092, 1097, 1107, 1108, 1474 Market efficiency, 109, 143, 195, 334, 999, 1258, 1523, 1524, 1534 Market entry, 1041, 1381 Market makers, 44, 56, 339, 340, 342–344, 346, 351, 360, 807, 808, 815, 1041, 1111, 1398 Market microstructure, 185, 333–347, 351–352, 807, 1007, 1161 Market model, 35, 45, 93, 97–100, 107, 111, 117–119, 128, 271, 350, 680, 727, 734, 737, 738, 1021, 1092, 1094, 1095, 1100–1102, 1104–1108, 1431–1433, 1467–1469, 1471, 1475, 1476, 1564 Market quality, 1395–1396 Market risk premium, 13, 53, 94, 107, 1467, 1468, 1472, 1473, 1537 Markov Chain Monte Carlo (MCMC) algorithms, 42, 49, 1109–1119, 1371–1380

1713 Markowitz model, 69, 79, 81, 82, 90, 91, 112, 130, 132, 165, 166, 236, 239–245, 248, 271 Markowitz quadratic programming model, 235 Martingale property, 42, 880, 1561 Mathematical programming (MP), 42, 1061–1066, 1311, 1319, 1324 Maturity, 4, 5, 14, 15, 17, 18, 20, 27, 56, 323, 327, 331, 358, 360–362, 364, 365, 368, 369, 371, 375, 377, 378, 387, 388, 391, 393–396, 410, 415, 425, 426, 433, 441, 454, 456, 457, 460–462, 493, 494, 506, 508, 517, 523, 532–535, 537–541, 544, 545, 548, 549, 555–557, 559, 562, 565, 571, 585, 594, 596, 599, 600, 610, 666, 667, 669, 678, 679, 685, 698, 699, 703, 705, 714, 716, 717, 721, 723, 727, 728, 732, 734, 739–741, 744, 746, 754–758, 761, 763, 764, 780, 803, 809, 820, 884, 885, 896, 898, 899, 901, 904, 907, 909, 912, 916, 919, 920, 933, 934, 939, 940, 947, 948, 961, 962, 980, 981, 987–991, 995, 998, 1013, 1025–1037, 1128, 1133, 1166–1168, 1170, 1194, 1220, 1229, 1270, 1351, 1451, 1453, 1456–1458, 1467, 1490–1495, 1504–1507, 1509–1516, 1519, 1534, 1568, 1573, 1611, 1613 MCMC algorithms. See Markov Chain Monte Carlo (MCMC) algorithms Mean-variance spanning tests, 10, 46, 165–184 Membership function, 1184–1190, 1193 Mergers, 27, 104, 105, 185, 345, 464, 767, 768, 771, 775, 777, 1153, 1238, 1427–1430, 1433, 1434, 1439, 1443, 1600 Method of payment, 1428, 1430, 1433, 1434, 1439, 1441–1443 Microsoft excel, 41, 175–178, 182, 433–435, 617–636 Minimal entropy martingale measure, 49, 1555–1564, 1567–1575 Minimum variance portfolio, 59, 78, 79, 82, 84, 168–170, 176, 177, 227 Misclassification cost model, 1593, 1603 Mixture of distributions hypothesis (MDH), 1173–1175 Model identification, 1301, 1303 Moment generating function, 404, 423, 481, 485, 575, 1569, 1572 Monotonic convergence, 506–507 Monte-Carlo simulation, 49, 249, 324, 328, 453, 584, 587, 596, 597, 601, 768, 771, 775, 986, 1077, 1098, 1107, 1110, 1118, 1170, 1274, 1488, 1607, 1611 Monty Hall problem, 1371, 1372, 1379 Mortgage, 16, 44, 332, 743, 744, 746, 759, 765, 772, 779, 780, 782, 783, 787, 788, 791–794, 796, 797, 800–803, 805, 1122, 1353, 1354, 1587–1589 Mortgage-backed securities (MBS), 715, 716, 729, 734, 746, 979, 1353 Mortgage market, 716, 743, 744, 746, 779, 783 Multifactor model, 1001, 1109, 1483, 1485, 1511 Multi-index model, 46, 122, 123 Multinomial process, 399 Multiperiod, 203–206, 216, 217, 283, 464, 605, 606, 1025, 1026, 1028, 1033, 1036, 1037, 1042, 1043, 1047, 1119, 1525, 1526 Multiple indexes, 46, 111, 118–121, 124 Multiplicative model, 1055, 1056, 1059 Multivariate GARCH (MGARCH), 1178, 1283, 1284, 1290, 1424, 1425 Multivariate lognormal distribution, 423–425 Multivariate normal distribution, 421, 424, 702, 877 Multivariate volatility, 1274, 1276, 1278, 1290, 1424–1425 N Noisy markets, 185 Non-central chi-square, 41, 1506, 1616–1622 Noncentral chi-square distribution, 473, 476–478 Nonclosed-form option pricing model, 481–485 Non-parametric, 42, 278, 329, 1061–1066, 1433, 1607, 1608, 1613 Nonparametric density, 532, 537, 539, 541 Nonparametric estimation, 42, 535, 734–746 Normal distribution, 38, 41, 66, 81, 82, 275, 276, 278, 321, 329, 385, 386, 397, 404, 416, 421–430, 448, 456–458, 461–463, 470, 484,

1714 519, 522, 537, 587, 614, 615, 679, 697, 700, 702, 705, 706, 710, 725, 759, 778, 809, 820, 827, 853, 855–857, 859, 861, 873, 876, 877, 880, 882, 939, 1016, 1066, 1099, 1123, 1152, 1178, 1276, 1285, 1336, 1346, 1412, 1538, 1578, 1581, 1583, 1584, 1600, 1608, 1611 Numerical analysis, 292 Numerical solution, 29, 49, 405, 485, 1209–1220 O OBPI. See Option-based portfolio insurance Office of Federal Housing Enterprise Oversight (OFHEO), 789–791 On-line prediction, 1257–1258, 1269, 1271 Optimal capital structure, 8, 9, 27, 28, 34, 36, 38–40, 99, 667, 933–950, 1028, 1382, 1451, 1456 Optimal hedge ratio, 873–880 Optimal risky portfolio, 170–172, 175, 178, 180–182 Option, 14–15, 355–360, 377–392, 395–397, 461–462, 589–591, 767–778, 1190–1191 Option-based portfolio insurance (OBPI), 320–321, 327–329 Option pricing, 27, 30, 41–43, 357, 362, 402–406, 415, 416, 421, 439–446, 454–458, 470, 547–574, 584, 611, 612, 625–627, 822, 823, 884, 916, 920, 1007, 1156, 1165, 1192, 1581, 1607–1625 Option pricing model (OPM), 15, 26, 41, 43, 46, 47, 355, 358, 383, 385–387, 393, 395–397, 399–419, 421–440, 447–472, 481–491, 505, 506, 532, 548–555, 564, 571, 575, 576, 613, 617, 621–627, 683, 809, 825, 919, 987, 1190, 1192, 1193, 1581, 1607, 1611, 1613, 1615, 1619–1622 Options, 14, 15, 41, 42, 46, 47, 49, 144, 175, 177, 186, 229, 236, 332, 355–375, 377–397, 403–406, 425, 431–433, 435–439, 442, 453–458, 461, 464, 465, 482, 491–502, 506–508, 510, 512, 515, 517, 518, 520, 523, 524, 526, 531, 533, 534, 537, 538, 548–553, 556–571, 575, 577–580, 583–602, 617–618, 622–624, 667, 682, 713, 727, 729, 732, 738, 744, 746, 749, 753, 755, 760, 762–765, 767, 779, 780, 787, 791, 796, 800, 801, 809, 820, 821, 824, 831, 867, 884, 916, 922, 930, 934, 966, 979, 998, 1015, 1041, 1042, 1047–1049, 1153, 1170, 1175, 1180, 1191, 1196, 1209, 1220, 1229, 1232, 1333–1335, 1339, 1343, 1348, 1349, 1396, 1397, 1405, 1445, 1446, 1483, 1485–1488, 1490, 1588, 1607–1613, 1615, 1616, 1622, 1623 Options pricing, 46, 447, 605–616, 1191, 1349 Ordinary differential equation (ODE), 47, 297, 304, 316, 317, 449, 577, 580, 748, 937, 1167, 1293–1300, 1445–1464, 1493, 1501 P Panel data, 43, 44, 680, 686, 808, 810, 812–814, 1000, 1241, 1368 Penalized internal rate of return for beta index (PIRR for Beta), 276–277 Penalized internal rate of return index (PIRR), 276, 277, 279, 280 Performance evaluation, 38, 44, 208, 1008, 1070, 1072, 1089, 1307, 1523–1552 Performance measure, 46, 53, 62–64, 91, 125–135, 139, 208, 209, 217, 274–278, 280, 329, 567, 1049, 1241, 1245, 1246, 1341 Persistent lagged instruments, 1069, 1082 Phase-type distribution, 1446, 1448, 1449 Poisson distribution, 425–426, 428 Portfolio, 3, 6, 10–13, 16, 20–22, 35, 38, 42–46, 48, 53, 69–91, 93–99, 102, 105–109, 111–135, 138–140, 143, 144, 162, 187, 191, 192, 196, 197, 203–218, 221–233, 235–237, 245, 319–321, 323, 324, 327–329, 333–338, 344, 350, 355, 360, 361, 363, 372, 374, 383–387, 391, 394, 438, 439, 454–456, 458, 464–466, 478, 482, 483, 486, 490, 491, 493, 496–499, 502, 516, 531, 534, 535, 553–555, 562, 563, 566, 571, 576, 578, 585, 606–610, 612, 613, 679, 680, 683–685, 687, 697–703, 705–707, 710, 720, 727, 749, 761, 789, 853, 855, 873, 879, 884, 941, 968, 999, 1008, 1010–1012, 1019, 1030, 1067, 1070, 1083, 1084, 1087, 1088, 1091, 1092, 1094, 1098–1102, 1106, 1107, 1118, 1133, 1235,

Subject Index 1236, 1329, 1348, 1349, 1352, 1394, 1468, 1470, 1475, 1510, 1523–1527, 1593 Portfolio analysis, 48, 57–61, 69, 71, 88, 93, 114–117, 122, 131, 144, 170, 259–266 Portfolio choice, 38, 44, 75, 209, 299, 1525 Portfolio optimization, 42, 46, 91, 165–184, 191, 247–257, 289, 1417, 1424–1425, 1567, 1571 Portfolio planning, 289–318 Portfolio theory, 3, 10, 21, 46, 53–68, 106, 111, 217, 249, 266–280, 283–287, 289, 334, 342, 1008, 1346, 1417 Positivity, 47, 987, 1458, 1489–1491, 1498, 1511, 1560 Power utility, 10, 46, 203–218, 285 Predicting bond yields, 1257–1271 Predicting stock returns, 1069, 1394 Prepayment, 45, 716, 729, 734, 743, 744, 746, 779, 780, 791–797, 800–803, 805 Price bubbles, 137, 140, 148, 150, 156, 158, 162, 1511, 1517 Price discovery, 147, 334, 342, 343, 346, 347, 1000, 1001, 1396, 1397 Price volatility, 5, 14, 187, 289, 342, 350, 368, 382, 482, 508, 772, 1173, 1174, 1194, 1352, 1397, 1421 Pricing performance, 548, 559, 734, 949 Pricing uncertain income streams, 605–616 Prime, 42, 795, 936, 1061–1063, 1065, 1066, 1201–1207, 1223–1232 Principal component analysis, 703, 705, 710, 866, 867, 869 Private placement, 1151–1157, 1161, 1162 Private targets, 1430, 1431, 1433, 1441 Probability of bankruptcy, 205, 775–777, 935 Probability of default (PD), 27, 639, 665, 668, 669, 699, 810, 813, 961, 962 Probit models (PM), 556, 965, 1154–1156, 1161, 1307, 1599–1600 Product market games, 1399–1407 Property/casualty (PC), 43, 44, 1170, 1465–1481 Protective put, 372–374 Publicly traded option, 356, 377, 378 Public offering, 1151, 1152, 1154, 1157, 1162, 1366, 1406 Public targets, 1430, 1433 Put-call parity, 355, 360–363, 367, 375, 433, 437, 438, 584, 620, 625, 721, 962, 1190, 1191, 1487, 1619 Put option, 14, 47, 320, 321, 327, 355, 356, 358, 360–363, 366–368, 372, 374, 375, 379, 380, 385, 391, 393, 405, 427, 433, 436–438, 453, 491, 492, 494, 496, 498, 500–502, 508, 517, 534, 548, 556, 584, 585, 587, 617–625, 627, 631, 636, 749, 757–760, 779, 780, 783, 796, 800, 819–827, 1167, 1335, 1396, 1445, 1446, 1453, 1487, 1619 Q Quadratic forms, 85, 688, 692, 693, 718, 748, 1121, 1360, 1490–1493, 1505, 1598 Quantile, 48, 249–251, 254, 256, 294, 324, 329, 738–743, 745–746, 1241, 1247, 1280, 1409, 1411–1413 Quantile regression, 329, 829–840, 1241, 1247 R Random walk, 291, 294, 334, 342, 387, 448, 453, 481, 543, 588, 589, 605, 607, 609–613, 1068, 1137, 1139, 1174, 1179, 1276, 1375, 1394, 1396 Rank dependent utility, 248, 249 Rate of convergence, 505–507, 1263 Rate of return, 4–7, 9–11, 14, 35, 48, 53, 55–58, 65–68, 81, 83, 86–88, 90, 96, 98, 99, 104–107, 123, 125, 167, 168, 170, 171, 179, 181, 182, 206, 208–210, 236, 259–261, 276, 291, 296, 298, 299, 301, 304–306, 314, 336, 360, 384–386, 388, 389, 391, 394, 399, 402, 415–417, 425–427, 431, 455, 456, 464, 467, 490, 606, 607, 609–613, 821, 824, 920, 937, 939, 942, 943, 1043, 1055, 1059, 1110, 1175, 1319, 1321, 1394, 1466, 1468–1472, 1475–1477, 1481, 1505, 1526, 1568

Subject Index R&D investment, 829–840 Real estate, 45, 66, 77, 330, 355, 779, 1201–1207, 1353, 1438 Real estate own (REO), 787, 803 Real option, 30, 767–777, 1049, 1051, 1184, 1190, 1196, 1197, 1232, 14041–1043 Receivership, 1595 Recession, 99, 224, 643, 686, 701, 732, 743, 779, 1048, 1557, 1559 Recovery, 9, 639, 666, 686, 699, 711, 789, 802, 934, 945, 949, 961, 962, 979, 980, 989–991, 997, 1001, 1270, 1595 Recursive programming, 627 Refunding, 27, 1025–1037 Regimes shift, 45, 904, 984, 1121–1133, 1352 Regime-switching, 44, 45, 717, 904, 907, 909, 912, 979, 984, 987, 990, 1001, 1122–1133, 1352, 1417, 1421–1424 Regression, 20, 25, 42–45, 97, 103, 104, 111, 112, 114, 117–119, 129, 130, 156, 158, 159, 161, 172, 175, 189, 191–194, 199, 208, 217, 272, 273, 279, 284, 289–292, 294–297, 301, 303, 304, 310, 314, 329, 430, 540, 541, 544, 545, 564–568, 571, 610, 646, 654, 656, 658, 660, 670, 676, 679, 680, 687, 689, 691–693, 714, 720, 721, 738, 740–745, 791, 807, 808, 810–815, 817, 829–840, 866, 868–871, 879, 884, 897–899, 901, 917, 921, 923, 926–931, 949, 965–976, 986, 999, 1061–1063, 1065, 1067–1089, 1091–1094, 1097, 1099, 1125, 1135, 1141, 1152, 1154, 1155, 1159, 1161, 1190, 1197–1198, 1201–1207, 1238, 1241, 1244, 1247, 1277, 1284, 1325, 1333, 1338, 1340, 1342, 1343, 1347, 1348, 1358–1360, 1362–1364, 1366, 1368, 1373, 1376, 1379, 1391, 1394, 1417, 1422, 1424, 1425, 1427, 1428, 1430–1433, 1435, 1439–1443, 1467–1481, 1525–1527, 1539–1550, 1608 Residential mortgage-backed securities, (RMBS), 801 Return, 21–22, 57, 66–71, 106–109, 123–124, 166, 206–208, 261, 263–264, 284–285, 296–298, 303–304, 310, 363, 853–862, 886–899, 1041–1054, 1088, 1247, 1333–1344, 1393–1394, 1417–1418, 1588–1589 Return maximization, 115, 120 Return predictability, 289–314, 1393–1394 Reverse-engineering, 45, 190, 199, 641, 647–650, 659 Rho, 277–278, 434, 500–503, 1376 Risk averse, 15, 20, 38, 71, 73–77, 79, 95, 102, 144, 150, 166, 172, 204, 210, 216, 247–257, 260, 285, 319, 336, 337, 349, 466, 586, 605, 606, 610, 613, 683, 1352 Risk-based portfolio insurance (RBPI), 323–324, 329, 331 Risk management, 41–49, 675–694 Risk minimization, 20, 21, 265 Risk-neutral and physical measures, 1001 Risk premium, 4, 5, 11, 13, 18, 19, 27, 29, 30, 48, 53, 62, 66–68, 94, 96, 98, 105, 107, 147, 150, 159, 166, 172, 185, 273, 275, 284, 285, 292, 336–338, 349–350, 483, 533, 534, 551, 668, 669, 679, 725, 726, 728, 732, 759, 764, 819–827, 885, 886, 915, 920, 933, 934, 959, 986, 996, 999, 1094, 1098, 1110, 1126, 1127, 1395, 1466–1476, 1481, 1509, 1510, 1512, 1515, 1527, 1537, 1545 Risky bond price, 962 Robust Logit, 45, 965, 969, 973–975 S Second order complementarities/substitutabilities, 846, 1400, 1403–1405 Securitization, 779, 801, 802, 1353 Security market line (SML), 11, 12, 95–97, 106, 108, 208, 272, 338, 1467 Semi-analytic, 585 Semimartingale backward equation, 49, 1556 Severity, 548, 753, 761, 780, 802, 1203, 1524 Sharpe measure, 46, 62–64, 880 Sharpe performance measure approach, 125–128 Sharpe ratio, 173, 174, 222–224, 227, 229, 232, 274–279, 283, 285, 286, 324, 328, 329

1715 Short sales, 79–83, 125–130, 132, 135, 138, 162, 168, 169, 175–177, 191, 204, 205, 226–230, 270, 286, 310, 316, 464, 844, 1007, 1009, 1524, 1527, 1530, 1533, 1534, 1537–1545, 1547, 1549 Short selling, 46, 79–83, 89–91, 130, 132, 140, 149, 150, 168–171, 176, 178, 179, 182, 221, 226, 231, 270, 323, 336, 360, 385, 490, 843, 844, 851, 1396, 1524 Short straddle, 368–369, 371 Short vertical spread, 370–371 Sigma, 626 Simple binomial option pricing model, 393–395 Simultaneous, 17, 20, 36, 43, 44, 84, 127, 192, 265, 266, 341, 367, 368, 371, 378, 669, 718, 763, 830, 846, 854, 901, 939, 966, 987, 1025, 1033, 1035–1037, 1042, 1091, 1148, 1171, 1308, 1325, 1353, 1357, 1358, 1384, 1396, 1400, 1478, 1479, 1596 Simultaneous equation, 36, 43, 44, 127, 1301–1306, 1366 Single-index model, 46, 111–115, 118, 120–122, 129, 130, 132, 133 Single-price law of markets, 606, 607, 609 2SLS. See Two-stage least squares (2SLS) SP500, 1378 Speculative bubbles, 137–163 Spillover effect, 1174, 1178, 1179, 1284, 1285, 1287–1289, 1352 Standard deviation, 10, 11, 21, 48, 53, 58–62, 65, 66, 70, 71, 77, 81, 89, 94, 96, 97, 100, 102, 103, 106–108, 125, 126, 129, 144, 166–169, 171, 172, 175, 178–182, 188, 190, 197, 208–211, 215–217, 224, 235, 260–262, 267, 268, 274–276, 280, 286, 295, 297, 298, 301, 328, 338, 388, 397, 430, 449, 450, 454, 457, 462, 482, 522, 525–527, 536, 537, 543, 553, 557, 558, 568, 584, 585, 606, 612, 613, 646, 653, 676, 679, 687, 689, 698–700, 710, 754, 757, 759–761, 764, 769, 772, 776, 802, 809, 810, 812–814, 824, 855, 857, 877, 878, 897, 1030, 1065, 1113, 1123, 1136, 1184, 1236, 1241, 1244, 1247, 1258, 1270, 1320, 1335, 1350, 1354, 1375, 1379, 1397, 1413, 1438, 1471–1474, 1477, 1478, 1505, 1509, 1517, 1533, 1534, 1602, 1611 Standard method of short sales, 129 State-prices, 1203, 1506, 1517–1519 State-space estimation, 1495, 1496, 1502 Stochastic calculus, 46, 447, 449, 451, 452, 465 Stochastic discount factor (SDF), 284–287, 986, 1126, 1127, 1130 Stochastic dominance, 38–40, 42, 247–257 Stochastic order, 247 Stochastic volatility, 47, 464, 481–490, 515, 526, 531, 533, 547–575, 577–580, 713–721, 723, 725–734, 744, 746, 979, 985, 1001, 1276, 1279 Stochastic volatility model, 47, 48, 526–528, 531, 533, 547, 548, 573, 718, 728, 730, 731, 985, 1001, 1009–1120, 1165–1171, 1407–1414, 1567–1575 Stock index volatility, 1333–1344 Stock market, 43, 67, 137, 195, 221–224, 262, 272, 274–275, 277, 290, 303, 328, 331, 333, 383, 526, 550, 732, 808–810, 854, 1000, 1067, 1080, 1122, 1174, 1175, 1179, 1190, 1235, 1283–1291, 1302, 1346, 1348–1350, 1352, 1354, 1393, 1394, 1396, 1397, 1523, 1526, 1577, 1596, 1601, 1616 Stock option pricing, 438 Stock return, 43–45, 109, 223, 289, 290, 292–295, 297–299, 301, 303, 314, 453, 478, 481, 483, 485, 531, 533, 536, 549–552, 554, 555, 676–680, 682–687, 809, 810, 812, 831, 853–862, 886, 1000, 1050, 1051, 1067–1073, 1075, 1076, 1086–1089, 1091–1094, 1098, 1099, 1102, 1106–1108, 1153, 1174, 1175, 1177–1179, 1209, 1284–1286, 1335–1336, 1345, 1346, 1349, 1350, 1352, 1393–1394, 1397, 1409, 1421, 1430, 1602, 1607, 1615, 1616, 1619 Strategic substitutes and complements, 1403–1406 Structural and reduced-form models, 980, 988 Structural credit risk models, 665–668, 671, 933 Structural instability, 44, 1345–1354

1716 Structural model, 47, 297, 665, 667–670, 755, 764, 934, 941, 949, 950, 988, 995, 996, 1364 Subordinators, 1567–1569, 1574 Subprime, 45, 332, 765, 779–805, 1122, 1587–1590 Sudden shifts of variance, 1349 Support vector machines, 1257, 1271 Systematic risk, 11, 13, 23, 35, 57, 63, 66, 93, 95–100, 103–105, 107, 108, 112, 123, 133, 270–275, 280, 335, 338, 339, 534, 678, 683, 684, 884, 885, 1127, 1128, 1346, 1429, 1467, 1468, 1472, 1525–1527 T Term structure, 18, 27, 45, 331, 360, 461, 565, 566, 609, 610, 667–670, 705, 714–716, 718, 721, 729, 737–739, 741–744, 755, 764, 788, 922, 923, 962, 963, 979–1001, 1110, 1118, 1121–1133, 1190, 1197, 1337, 1393, 1489–1495, 1500, 1501, 1602, 1623 Term structure models(ing), 45, 47, 549, 565, 566, 667, 668, 713–723, 729, 734, 743, 746, 754, 979, 987, 991, 995, 996, 1001, 1121, 1128–1130, 1133, 1483–1485, 1487, 1490, 1493, 1494, 1511, 1616 Term structure of interest rates (TSIR), 18, 45, 609, 979, 985, 987, 1121–1133, 1190, 1197, 1198, 1489, 1623 Theory of storage, 197, 915–917, 920, 921, 923, 926, 930 Theta, 494–496, 502–503 Three-stage least squares (3SLS), 44, 1301, 1304–1305, 1365 Threshold regression, 44, 808, 811, 814–815 Time-additive utility, 298, 607, 608 Time scales in volatility, 1109–1111, 1113, 1118 Time to maturity, 14, 15, 358, 456, 493, 494, 506, 533, 587, 624, 714, 725, 727, 904, 980, 1029, 1032, 1167, 1170, 1220, 1270, 1277, 1611 Time-varying returns, 1067–1070, 1072, 1080, 1082–1089 Tournament, 138–141, 143, 148–150, 154, 156, 158–160, 1610, 1613 Trading costs, 330, 334, 335, 339, 809 Transfer pricing, 48, 185, 1307–1312, 1329 Treynor measure, 46, 63, 64, 128–132 Turnover, 43, 329, 792–793, 805, 863–866, 917, 919, 1048, 1174–1179, 1315 Two-pass methodology, 1091, 1241 Two-period model, 140, 143, 145, 396, 535 Two-stage least squares (2SLS), 44, 1301–1306, 1357–1366 Two-state option pricing model, 409–415 U Unbiased term structure, 610, 613 Uncollaterized call rate, 42, 1375 Unspanned stochastic volatilities, 713–746, 748–751 Unsystematic risk, 11, 12, 57, 66, 93, 99, 107, 108, 112, 123, 1429 Utility functions, 38, 40, 42, 43, 46, 69, 71–77, 79, 82, 171, 178, 180, 204, 205, 209, 210, 216, 247–249, 251, 254–257, 260, 285, 286, 300, 316, 317, 335, 336, 608, 846, 873, 1555, 1571 Utility theory, 46, 71–77, 180, 247, 250, 254, 255, 257

Subject Index V Validation test, 45, 967–969, 973, 976, 1065 Valuation, 3–8, 10, 14–22, 25, 27, 29, 30, 32, 41, 49, 104, 109, 139, 144, 146–148, 186–190, 193, 194, 198–200, 289, 290, 296, 299, 305, 314, 327, 333, 343, 355, 357, 358, 360, 362, 375, 377–392, 394, 397, 403–406, 429–431, 458, 459, 464, 466, 487, 526, 535, 537, 550, 551, 556, 584–602, 605–616, 667, 668, 699, 723, 753, 756–760, 768, 769, 775, 776, 802, 846, 854, 884–886, 933, 934, 939, 941, 948, 950, 961, 963, 995–999, 1008, 1015, 1016, 1042–1045, 1047–1052, 1055, 1059, 1060, 1224, 1226–1228, 1230, 1334, 1335, 1383, 1400, 1405, 1427, 1428, 1430, 1439, 1442, 1465–1467, 1483, 1502, 1506, 1594, 1596, 1601, 1602 Value-at-risk (VaR), 43, 48, 251–252, 323, 324, 329, 515, 524, 525, 528, 697–711, 853, 861, 987, 1348, 1409–1414 Value investing, 185 VaR. See Value-at-risk Variance-covariance matrix, 61, 262, 657, 1291, 1304, 1347 Variance decomposition, 1393 VBA. See Visual basic for applications Vega, 141, 144, 148–500, 503 Visual basic for applications (VBA), 617, 627–636 Volatility, 481–490, 518, 522, 547–574, 713–746, 748–751, 883–912, 1109–1120, 1165–1171, 1273–1280, 1333–1344, 1349–1352, 1409–1414, 1417–1425, 1567–1574, 1581–1582, 1587–1589 Volatility clustering, 718, 823, 853, 856, 1180, 1181, 1273, 1352, 1409, 1410, 1417, 1581 Volatility smile, 458, 464, 531–533, 541, 548, 567, 727–729, 732, 734 Volume effect, 43, 1174, 1177 W Warrants, 14, 34, 236, 274, 377, 390–392, 583, 940, 979 Weak instruments, 1357, 1359, 1364–1368 Weight, 10, 11, 13, 30–33, 36, 46, 48, 57, 59, 61, 65, 70, 72, 75, 77–81, 83–91, 97, 103, 112–115, 117, 120, 121, 125–130, 132, 133, 135, 166–172, 175–182, 186–194, 196, 199–217, 221–224, 226, 227, 229–232, 237, 256, 261, 263–266, 268, 269, 285, 323, 324, 329, 431, 524, 528, 543, 589, 608, 643, 646, 679, 686, 697, 702, 703, 705, 706, 770, 774, 789, 790, 836, 939, 1044, 1064, 1066, 1083, 1091, 1095, 1096, 1099–1105, 1107, 1112, 1120, 1125, 1140, 1141, 1153, 1187, 1189, 1190, 1215, 1223–1225, 1227, 1229, 1230, 1236–1238, 1241, 1244, 1276, 1280, 1308, 1319, 1320, 1322, 1323, 1335, 1347, 1351, 1360–1362, 1401, 1418, 1421, 1423, 1424, 1470–1472, 1474, 1475, 1477, 1480, 1481, 1499, 1511, 1512, 1526, 1530, 1533, 1534, 1600, 1601, 1622 Whole loan, 803, 8004 Y Yaari’s dual utility, 43, 247, 251, 255, 256 Yield spreads, 55, 565, 566, 666, 668–670, 934, 941, 947–950, 980, 991, 995–996, 1001, 1067, 1070, 1075, 1393