Asset Allocation and International Investments Edited by
Greg N. Gregoriou
ASSET ALLOCATION AND INTERNATIONAL INVESTM...

Author:
Greg N. Gregoriou

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Asset Allocation and International Investments Edited by

Greg N. Gregoriou

ASSET ALLOCATION AND INTERNATIONAL INVESTMENTS

Also edited by Greg N. Gregoriou ADVANCES IN RISK MANAGEMENT DIVERSIFICATION AND PORTFOLIO MANAGEMENT OF MUTUAL FUNDS PERFORMANCE OF MUTUAL FUNDS

Asset Allocation and International Investments

Edited by GREG N. GREGORIOU

Selection and editorial matter © Greg N. Gregoriou 2007 Individual chapters © contributors 2007 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1T 4LP. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The authors have asserted their rights to be identified as the authors of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2007 by PALGRAVE MACMILLAN Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N.Y. 10010 Companies and representatives throughout the world PALGRAVE MACMILLAN is the global academic imprint of the Palgrave Macmillan division of St. Martin’s Press, LLC and of Palgrave Macmillan Ltd. Macmillan® is a registered trademark in the United States, United Kingdom and other countries. Palgrave is a registered trademark in the European Union and other countries. ISBN-13: 978–0–230–01917–1 ISBN-10: 0–230–01917–X This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. A catalogue record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Asset allocation and international investments / edited by Gerg N. Gregoriou. p.cm. — (Finance and capital markets) Includes bibliographical references and index. ISBN 0–230–01917–X 1. Asset allocation. 2. Investments, Foreign. 3. Globalization—Economic aspects. 4. Portfolio management. I. Gregoriou, Greg N., 1956– II. Series. HG4529.5.A83 2006 332.67’3—dc22 2006045369 10 9 8 7 6 5 4 3 2 1 16 15 14 13 12 11 10 09 08 07 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham and Eastbourne

To my mother Evangelia and in loving memory of my father Nicholas

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Contents

Acknowledgments

xi

Notes on the Contributors

xii

Introduction

xvii

1 Time-Varying Downside Risk: An Application to the Art Market

1

Rachel Campbell and Roman Kräussl 1.1 Introduction 1.2 Art as an investment 1.3 Previous empirical studies 1.4 Empirical analysis 1.5 Data 1.6 Methodology 1.7 Results 1.8 Discussion 1.9 Conclusion

2 International Stock Portfolios and Optimal Currency Hedging with Regime Switching

1 3 4 5 6 9 10 11 13

16

Markus Leippold and Felix Morger 2.1 2.2 2.3 2.4 2.5

Introduction The model Estimation results Discussion Conclusion

16 18 21 26 39 vii

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CONTENTS

3 The Determinants of Domestic and Foreign Biases: An Empirical Study Fathi Abid and Slah Bahloul 3.1 Introduction 3.2 Theoretical framework of domestic and foreign biases 3.3 Data and preliminary statistics 3.4 The determinants of domestic and foreign biases 3.5 The empirical analysis 3.6 Additional tests 3.7 Conclusion

4 The Critical Line Algorithm for UPM–LPM Parametric General Asset Allocation Problem with Allocation Boundaries and Linear Constraints

42 42 44 46 56 67 71 74

80

Denisa Cumova, David Moreno and David Nawrocki 4.1 Introduction 4.2 The upside potential–downside risk portfolio model 4.3 An empirical example 4.4 Conclusion

5 Currency Crises, Contagion and Portfolio Selection

80 82 92 94

96

Arindam Bandopadhyaya and Sushmita Nagarajan 5.1 5.2 5.3 5.4 5.5

Introduction Stock market average rates of return and average volatility Stock market correlations Portfolio performance Conclusion

6 Bond and Stock Market Linkages: The Case of Mexico and Brazil

96 97 99 100 101

103

Arindam Bandopadhyaya 6.1 6.2 6.3 6.4

Introduction The estimation equations and data Results Conclusion

7 The Australian Stock Market: An Empirical Investigation

103 105 109 112

114

Adeline Chan and J. Wickramanayake 7.1 7.2 7.3

Introduction Existing evidence Hypothesis

114 115 118

CONTENTS

7.4 7.5 7.6

The data Data analysis and results Conclusion

ix

119 127 132

8 The Price of Efficiency – So, What Do You Think About Emerging Markets? 137 Zsolt Berényi 8.1 8.2 8.3 8.4 8.5

Introduction Higher moment performance analysis – the theory The efficiency gain/loss methodology Testing results Conclusion

9 Liquidity and Market Efficiency Before and After the Introduction of Electronic Trading at the Sydney Futures Exchange

137 138 140 143 149

151

Mark Burgess and J. Wickramanayake 9.1 Introduction 9.2 Review of the literature 9.3 Options data volume as a proxy for liquidity 9.4 Sample design 9.5 Analysis of results 9.6 Conclusion

10 How Does Systematic Risk Impact Stocks? A Study of the French Financial Market

151 152 154 159 165 178

183

Hayette Gatfaoui 10.1 10.2 10.3 10.4 10.5 10.6 10.7

Introduction Theoretical framework Empirical study The impact of systematic risk Further investigation Market benchmark comparison Conclusion

11 Matrix Elliptical Contoured Distributions versus a Stable Model: Application to Daily Stock Returns of Eight Stock Markets

183 185 187 190 195 201 209

214

Taras Bodnar and Wolfgang Schmid 11.1 Introduction 11.2 Small sample tests 11.3 Analysis of the power functions 11.4 Empirical study 11.5 Conclusion

214 216 221 222 224

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12 The Modified Sharpe Ratio Applied to Canadian Hedge Funds

228

Greg N. Gregoriou 12.1 12.2 12.3 12.4 12.5 Index

Introduction Literature review Data and methodology Empirical results Conclusion

228 229 230 231 233 235

Acknowledgments

I would like to thank Stephen Rutt, Publishing Director, and Alexandra Dawe, Assistant Editor, at Palgrave Macmillan for their suggestions, efficiency and helpful comments throughout the production process, as well as Keith Povey (with Elaine Towns) for copy-editing and editorial supervision of the highest order. In addition, I would like to thank the numerous anonymous referees in the US and Europe during the review and selection process of the articles proposed for this volume.

xi

Notes on the Contributors

The Editor Greg N. Gregoriou is Associate Professor of Finance and coordinator of faculty research in the School of Business and Economics at the State University of New York (Plattsburgh). He obtained his PhD (Finance) from the University of Quebec at Montreal and is the hedge fund editor for the peer-reviewed journal Derivatives Use, Trading and Regulation, published by Palgrave Macmillan, based in the UK. He has authored over fifty articles on hedge funds, and managed futures in various US and UK peer-reviewed publications, including Journal of Portfolio Management, Journal of Futures Markets, European Journal of Finance, Journal of Asset Management, European Journal of Operational Research and Annals of Operations Research. He has published four books with John Wiley and Sons Inc. and four with Elsevier.

The Contributors Fathi Abid is a Professor of Finance. He is Director of the research team MODESFI specializing in financial modeling and financial strategy. He lectures frequently on financial market theory and has taught investment and portfolio management at Tunisian and European universities. He has written and co-authored numerous articles in national and international scientific journals, books and proceedings. xii

NOTES ON THE CONTRIBUTORS

xiii

Arindam Bandopadhyaya is the Chairman and Associate Professor of Finance in the Accounting and Finance Department at UMass Boston, USA. He is also the Director of the College of Management’s Financial Services Forum. A recipient of the Dean’s Award for Distinguished Research, Dr Bandopadhyaya has published in journals such as the Journal of International Money and Finance, Journal of Empirical Finance, Journal of Banking and Finance and Review of Economics and Statistics. He has presented his work at national and international conferences such as those of the Financial Management Association, European Finance Association and European Economic Association. He has presented research reports of the Financial Services Form at the Boston Stock Exchange and the Federal Reserve Bank of Boston. Dr Bandopadhyaya teaches corporate finance, international finance and managerial economics. He has received teaching awards from the College of Management, including the Professor of the Year Award and the Betty Diener Award for Teaching Excellence. Slah Bahloul is an Assistant Professor of Finance at Higher School of Business Administration in Sfax, Tunisia. He is a research assistant in the MODESFI team and has taught international finance and financial decision-making. Zsolt Berényi holds an MSc in Economics from the University of Economic Sciences in Budapest, and a PhD in Finance from the University of Munich. His main interests lie in the risk and performance evaluation of alternative investments: hedge funds, CTAs and credit funds. After working for many years for the Deutsche Bank, HypoVereinsbank and KPMG at various locations throughout Europe, Zsolt now leads an independent consultancy in Budapest, Hungary. Taras Bodnar studied Mathematics at the Lviv National University, Ukraine from 1996 to 2001. He received a PhD in Economics in 2004 from the European University Viadrina, Frankfurt (Oder), Germany. Currently, he is a research assistant at the Department of Statistics, European University Viadrina. His fields of interest are quantitative methods in finance, nonstationary time series, elliptical distributions and econometric applications. Mark Burgess currently works in the financial services industry in Australia. He has a Bachelor of Business (Honors) degree from Monash University, Australia. Rachel Campbell completed her PhD on Risk Management in International Financial Markets at Erasmus University, Rotterdam, The Netherlands in 2001. She currently works at the University of Maastricht as an Assistant Professor of Finance. Her work has been published in a number of leading

xiv

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journals, including the Journal of International Money and Finance, Journal of Banking and Finance, Financial Analysts Journal, Journal of Portfolio Management, Journal of Risk, and Derivatives Weekly. She teaches with Euromoney Financial Training on Art Investment, and works as an Independent Economic Adviser for the Fine Art Fund in London, and for Fine Art Wealth Management, UK. Adeline Chan currently works in the financial services industry in Singapore. She has a Bachelor of Business (Honors) degree from Monash University, Australia. Denisa Cumova works in the fund management group at the Berenberg Bank in Hamburg, Germany. She received her PhD in Finance from the University of Technology, Chemnitz, Germany. Hayette Gatfaoui gained a PhD in “Default Risk Valuation of Financial Assets” University Paris 1 in 2000. He taught for five years at the University Paris 1 (Pantheon-Sorbonne) France, and is now an Associate Professor at Rouen Graduate School of Management, France. He is a specialist in applied mathematics (holding a Master’s degree in stochastic modeling for finance and economics). He is currently advising financial firms about risk measurement and risk management topics for asset management, and for credit risk management purposes. Dr Gatfaoui is also a referee for the International Journal of Theoretical and Applied Finance (IJTAF). His current research areas concern risk typology in financial markets, quantitative finance and risk analysis. Roman Kräussl obtained a first-class honors Master’s degree in Economics with a specialization in Financial Econometrics from the University of Bielefeld, Germany, in 1998. He completed his PhD on the Role of Credit Rating Agencies in International Financial Markets at Johann Wolfgang Goethe University, Frankfurtam Main, Germany, in 2002. As the Head of Quantitative Research at Cognitrend GmbH, he was closely involved with the financial industry. Currently he is Assistant Professor of Finance at Vrije Universiteit Amsterdam, The Netherlands and research fellow with the Centre for Financial Studies, Frankfurtam Main. Markus Leippold is Assistant Professor of Finance at the Swiss Banking Institute of the University of Zurich, Switzerland. Prior to moving back to academia he worked for Sungard, Trading and Risk Management Systems, and the Zurich Cantonal Bank. His main research interests are term structure modeling, asset pricing and risk management. He obtained his PhD in financial economics from the University of St. Gallen, Switzerland, in 1999. During his PhD studies, he was a research fellow at the Stern School of Business in New York. He has published in several journals, such as the

NOTES ON THE CONTRIBUTORS

xv

Journal of Financial and Quantitative Analysis, Journal of Economic Dynamics and Control, Journal of Banking and Finance, Review of Derivative Research, Journal of Risk, and Review of Finance. In 2003, he and his colleagues received an award from the German Finance Association for their paper on the equilibrium impacts of value-at-risk regulation, and an achievement award from RISK for their paper on operational risk. In 2004, their research paper on credit contagion won the STOXX Gold Award at the annual conference of the European Financial Management Association. David Moreno holds a PhD degree in Economics from the Universidad Carlos III, Madrid, Spain, and a BSc degree in Mathematics from the Universidad Complutense, Madrid. He is currently Assistant Professor of Financial Economics and Accounting at Universidad Pompeu Fabra, Barcelona and Co-Director of the Master’s Program in Finance. He has previously held teaching and research positions at the Financial Option Research Centre (Warwick Business School, UK), Universidad Carlos III de Madrid, and at the IESE Business School, Barcelona, Spain. His research interests focus on finance in continuous time, with special emphasis on derivatives markets, financial engineering applications, pricing of derivatives, empirical analysis of different pricing models, portfolio management and term structure models. His research has been published in a number of academic journals including Review of Derivatives Research and Journal of Futures Markets, as well as in professional volumes. He has presented his work at a number of international conferences and has given invited talks at many academic and nonacademic institutions. He is associate editor of Revista de Economía Financiera and a member of GARP (the Global Association of Risk Professionals). Felix Morger is a fourth-year PhD student at the Swiss Banking Institute of the University of Zurich, Switzerland. The main part of his thesis is concerned with the theoretical and empirical aspects of Bayesian learning models with Markov switching and their application to asset allocation. Prior to his PhD studies, he worked as a consultant in pension funds. Sushmita Nagarajan is a Senior Associate in the Structured Finance Group at Moody’s Investor Service, New York. Her areas of expertise are rating and monitoring various types of structured derivative products using Moody’s rating methodologies. She also provides quantitative analysis and research surrounding complex derivative products such as asset-backed commercial paper structures. Prior to joining Moody’s she was an intern at State Street Research and Management as a Fixed Income Research Analyst with emphasis on Collateralized Debt Obligations. She graduated summa-cumlaude with a MSc degree in Finance from Boston College, and has an MBA in Finance from Jawaharlal Nehru Technological University, India.

xvi

NOTES ON THE CONTRIBUTORS

David Nawrocki is the Katherine M. and Richard J. Salisbury Jr. Professor of Finance at Villanova University, Villanova, Pa., USA. He is a registered investment adviser and is the director of the Institute for Research in Advanced Financial Technology (IRAFT) at Villanova. Nawrocki’s research includes work on financial market theory, downside-risk measures, systems theory, portfolio theory, and business cycles. He received his PhD in Finance from the Pennsylvania State University, USA. Wolfgang Schmid is a Full Professor at the European University in Frankfurt (Oder), Germany. He received a PhD in Mathematics in 1984 at the University of Ulm, Germany. His fields of major statistical activities are quantitative methods in finance, statistical process control and econometric applications. J. Wickramanayake obtained his PhD in 1994 from La Trobe University, Australia. He completed his Master’s degree at Williams College, Williamstown, Ma., USA in 1982, and did postgraduate studies in the Netherlands in 1978. He has been a member of the Financial Services Institute of Australasia for over ten years. Dr Wickramanayake has more than twenty years’ experience as a financial analyst at a central bank. Currently, he teaches finance at both undergraduate and postgraduate levels at Monash University, Australia. Dr Wickramanayake’s research interests involve banking, financial markets, mergers/acquisitions, bankruptcy and business failures, fund management, superannuation and pension finance.

Introduction

Chapter 1 deals with the economic downturn during 2000 which left many investors with burnt fingers and weary of investing in equities. There has been a continued search for alternative asset classes to fulfill the need for preserving returns while not taking on too high a risk. One such innovative alternative is investing in art as an alternative to stocks, bonds and real estate. This chapter analyses in a detailed empirical study how the risk during the art market bubble increased dramatically before the collapse of the market in the early 1990s. Understanding how deviations from normality in the form of extreme market returns link to the creation of a bubble in asset prices is crucial to our understanding of risk-and-return relationships. Chapter 2 presents a model for strategic asset allocation and currency hedging for an international investor, where the returns on stock indices follow a Gaussian regime-switching model. The authors study a Bayesian investor, who has only partial information on the current regime switching model being active, but updates the investor’s beliefs over time. The results indicate that engaging in optimal currency hedging significantly improves the risk and return characteristics of the Bayesian investor. Chapter 3 describes an empirical study of the determinant factors of domestic and foreign home biases. Using the equity holdings of thirty countries, the authors find that a severe equity home bias exists for both developed and emerging markets. Stock market development, information costs and familiarity factors are found to contribute the most to explaining foreign bias, whereas investor’s behavior has a significant effect on domestic bias. Chapter 4 discusses how human beings have always engaged in different behavior above and below a target rate of return. As a result, reverse S-shaped utility functions have been utilized to describe this human investment behavior, ever since Friedman and Savage (1948) and Markowitz xvii

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INTRODUCTION

(1952). Fishburn (1977) made this approach operational with the lower partial moment, LPM(a, t), model, which detailed risk-seeking and risk-averse behavior below a minimum target return. However, the Fishburn utility measures have attracted criticism, since they assume a linear utility (risk neutral) above the target return. Recently, the upper partial moment/lower partial moment (UPM/LPM) has been put forward as a solution to this problem. This chapter develops a UPM/LPM critical line algorithm that allows this model to be operational. Chapter 5 examines the characteristics of domestic and international portfolios from the perspective of a US investor in Asian emerging markets during a period where the economies have suffered a currency crisis. Among various portfolios constructed, a purely international portfolio posts superior performance compared to a purely domestic one or a combination of domestic and international portfolios in the post-crisis period. Chapter 6 investigates the Brady bond markets of the two largest LatinAmerican economies – Mexico and Brazil. Results indicate that, for the very near future, the yield in each market is determined primarily by past yields in the respective markets. However, over a longer-term horizon, the interrelationships between the bond markets and the stock markets of the two countries become increasingly important. Chapter 7 provides an evaluation and comparison between the explanatory power of the macroeconomic model of Chen et al. (1986) and the three-factor model of Fama and French (1993) in explaining the variation in returns in the Australian equity market for the decade of the 1990s. The empirical results show that firm attributes (Fama and French, 1993) alone are insufficient to explain returns and macroeconomic variables (Chen et al., 1986) can be combined in a better multifactor model to explain the variation in returns. Chapter 8 evaluates inter-market investment efficiency, which may be a complicated task, especially across investment forms with widely differing return characteristics. This chapter offers some new ideas on how to evaluate such investments, using the example of emerging markets. The authors show that replicating the expected return distribution using options, the efficiency of any investment portfolio – for example, not just “emerging market” or “equity” – can be assessed and compared. Chapter 9 examines whether the Sydney Futures Exchange (SFE) in Australia has benefited from the introduction of electronic trading on November 15, 1999. Empirical results in this study show that during the early stage, up to the beginning of August 2000 that the money SPI options were more liquid at times of high volatility after the automation of the SFE. However, the SPI futures were less liquid at times of medium to low market volatility after this event. The authors also found a cointegrating relationship between the Australian Stock Exchange (ASX) and the derivative market (SFE) before

INTRODUCTION

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and after the introduction of electronic trading supporting the semi-strong market efficiency hypothesis. Chapter 10 discusses how many researchers have focused on the common latent component underlying the evolution of stock returns. The authors propose to infer such an unobserved common component while employing the well known Black and Scholes (1973) option pricing formula. Their study is based on the assumption that any small stock market index is a distorted representative of such a latent component. Once this systematic risk factor is exhibited, the authors attempt to assess its impact on a basket of French stock returns. Chapter 11 explores the assumptions of independency and normality which are not appropriate in many situations of practical interest, especially for the data sets from emerging markets. The authors propose to make use of matrix elliptical distribution instead of the normal distribution. Empirically, they show that the assumptions of the elliptical symmetry cannot be rejected for daily returns. Chapter 12 applies the modified Sharpe ratio to a small sample of Canadian hedge funds. Many investors today use the traditional Sharpe ratio to measure risk-adjusted performance, but the proposed modified VaR Sharpe ratio is a superior and more precise method that can deal with the skewed/non-normal returns that hedge fund possess. The results show that the modified Sharpe ratio is more precise when examining non-normal returns.

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CHAPTER 1

Time-Varying Downside Risk: An Application to the Art Market1 Rachel Campbell and Roman Kräussl

1.1 INTRODUCTION The economic downturn during 2000 left many investors with burnt fingers and weary of investing in equities. Since then, there has been a search for alternative asset classes to fulfill the need to preserve returns, while not involving too high a risk. Arising from the media’s continued concern about a potential bubble in the housing market, many investors are showing an increasing interest in alternative asset classes that are not so highly correlated with equities, and provide hedging potential as part of a diversified portfolio of investments. One such innovative alternative asset class to stocks, bonds and real estate is art, which is seen increasingly as not merely items with aesthetic value, but also as attractive investments with a potential capital gain. The planned launch of a Fund of Art Funds by ABN Amro in 2005, aiming to channel money into some existing (and some yet to be launched) independent art funds, serves to highlight this point. It is a well-known fact that investment in art is influenced strongly by income and other fundamental economic factors. The effect on the economy from a collapse in the art market depends on the contagious impact of the art market on the rest of the financial system, predominately through the banking system. Thus, what is the impact of a negative shock in the art market on the overall economy? We argue that the extent to which real effects are likely to occur from a bubble in the art market is likely to 1

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be significantly less because of the type of investment that is made in the art market. There are two main reasons for this. First, as art is a luxury good, investors tend to invest money into the art market that would not necessarily be invested elsewhere in other asset classes beyond holding it as surplus cash. Second, the initial wealth levels of investors typically investing in art markets is higher, and therefore less at the mercy of the banking system, as the banks are unlikely to let such investors become insolvent. We argue that the likelihood of falling prices is only liable to affect the general economy to the extent that the losses made might reduce liquidity in financial markets. Even though booms in other markets, such as in real estate, may lead to a collapse in the initial market followed by a collapse in the banking sector, this is much less likely to be the case in the art market. Although the real effects from a collapse in the art market may be significantly less than in other financial markets, the development of bubbles in the art market is likely to be significantly greater. The rate at which prices in the art market are driven by taste and fashion, predominately via the media, is much greater than in other financial markets, where “value” is a greater function of market fundamentals. The development of a large bubble in the general price of all works of art was well documented in the early 1990s for most classes of art. Indeed, it would appear that there was a severe deviation away from the fundamental valuation of art pieces during this period. This provides an extremely interesting and unique data series with which to analyze the risk to the investor around the period of the bubble’s development. We focus on time-varying downside risk in relation to theory from behavioral finance. This, given our knowledge of the literature, is an area of research that has not been undertaken before. In this chapter we analyze the art market using a measure for time variance in the downside risk, which reflects “bubbliness” in the market. This estimate measures the changing probability of large movements occurring in the return distribution of the historical time series of art price data. Taking such an approach and using techniques developed in extreme value theory (EVT), we are able to provide some new insight into the creation and measurement of risk during times of the development of bubbles in financial markets. We focus on a particularly interesting case: the art market. This market is highly media- and taste-driven, is illiquid and lacks transparency, and thus offers an ideal application in which to observe downside risk with prices that may deviate significantly from fundamental values. This chapter is organized as follows. Section 1.2 briefly surveys the economic literature concerning art as an investment; we explore the financial aspects of art investing by emphasizing similarities and differences among financial assets. Section 1.3 discusses the data and the methodology, and presents the empirical results. Section 1.4 presents some behavioral

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explanations for our results. Section 1.5 concludes and presents an outlook for future research.

1.2 ART AS AN INVESTMENT 1.2.1 The art market in general Financial assets tend to be very liquid, allowing for diversification benefits, and thus reduce risk. Additionally, they are relatively transparent. Most financial assets can be selected on the basis of a fairly small set of objective criteria. Fundamentals do exist and can be analyzed with standard finance tools. Such financial markets are characterized by a large number of individual buyers and sellers, transaction costs are low, and trades in perfectly (or nearly) identical assets are repeated millions of times daily in various exchanges. It goes without saying that, the first impression of the art markets is that they differ significantly from other types of financial markets. Most art markets would appear to be characterized by product heterogeneity, illiquidity, behavioral anomalies, market segmentation, information asymmetries, and almost monopolistic price setting. Moreover, there is no doubt that a substantial amount of the return from art investment is derived not from classical financial returns but rather from intrinsic aesthetic qualities through art as a consumption good. Art works are not liquid assets, and transaction costs are high. Short selling is not possible, and supply is rather inelastic in the short term. There are unavoidable delays between an owner’s decision to sell and the actual sale, since it takes about three to six months to “market a work” – that is, to have it accepted by the auction house, take photographs and print and distribute the catalogue, publish advertisements for the coming auction and so on. Investing in art typically requires substantial knowledge of art and the art market in general, and often a significant amount of capital to acquire a work of a well-known artist. Moreover, the art market is highly segmented and dominated by a few large auction houses. These auction houses, such as Sotheby’s and Christie’s, are used by a restricted number of buyers, mostly wealthy collectors, public museums or private foundations. Informational asymmetries are essential features of these markets. Furthermore, art sells only occasionally. Art objects are created by individuals. Accordingly, there is only a single, unique piece of original work available. This is an extreme case of a heterogeneous commodity. Therefore, financial risk in the art market is related to specific material risk factors associated with the unique physical nature of art works such as theft, fire, water damage, or the possible reattribution to another (less famous) artist. Moreover, the value of an art object is

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determined by a complex and subjective set of beliefs about past, present and future prices. Art has little intrinsic value; its appeal is ultimately dependent on individual tastes and fashion, which can change over time. The future sales price of a piece of art depends on both the number of people who wish to buy the piece when it is put up for sale and the (available) wealth of the individuals or institutions who desire it at that time. The most distinctive difference between financial markets and the art market is that the individual investor’s expected return from investing in art consists not only of a rise in price. It also involves the psychic return from art works through their aesthetic qualities. Most empirical studies have been unable so far to quantify these psychic returns associated with art as a consumption good. Recognizing art as a consumption good helps in part to explain behavioral anomalies less well-known in modern financial markets.

1.3 PREVIOUS EMPIRICAL STUDIES In recent years, an extensive literature has arisen based on calculating the returns on art investments. Starting with Baumol (1986), these include, among others, empirical studies by Goetzmann (1993), Chanel (1995), Mei and Moses (2002) and Campbell (2005). Baumol (1986) and Goetzmann (1993) tend to concur that art is dominated as an investment product by stocks, bonds and real estate. Goetzmann (1993) finds a positive relationship between art investments and the stock market over shorter time periods. He argues that the high and significant positive correlation clearly makes art investment a poor instrument for the purposes of portfolio diversification. Goetzmann (1993) also finds evidence of a significant relationship between aggregate financial wealth and the demand for art. He concludes that this empirical finding is sufficient evidence that the demand for art increases with the wealth of art collectors since, in the twentieth century, art prices tended to follow stock market trends. Chanel (1995) follows this argumentation and concludes that financial markets react quickly to shocks in the economy. Profits generated on financial markets may be invested in art, so that developments in stock markets may be considered as leading indicators for returns in the art markets. Mei and Moses (2002) take a somewhat different view. They argue that a diversified portfolio of works of art play a more important role in portfolio diversification. They base their conclusions on their empirical finding that their art price index has lower volatility and a much lower correlation with other asset classes than was discovered in earlier research. Campbell (2005) focuses on the extent of downside risk, which is less for the art market during periods in which the stock market performs badly. This is highly likely to be driven by issues relating to theories from behavioral finance, caused not only by the low liquidity on the art market, but also to investors being

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anxious not to sell off art works representing a symbol of their reputation and status during falling financial markets. Thus, maintaining art investment remains strong during periods of economic downturn. This helps to drive the hedging connotation of art in the portfolio. The cyclicality of the art and equity markets has been documented in a recent working paper by Bauer et al. (2005), who show that art investments perform well at times when other asset classes are performing badly. Despite this strength during downturns, bubbles are also evident in the art market. The famous bubble in the 1990s occurred because of the excessive demand for works of art by the Japanese. What happens to risk-and-return characteristics during these periods? Is risk time-varying during the expansion of the bubble? Could we have seen the extent to which a bubble was developing in the market? Before answering these questions, a number of preliminary queries need to be addressed on the definition and possible estimation of a bubble.

1.4 EMPIRICAL ANALYSIS 1.4.1 How to define a bubble in the art market? What is a bubble in financial markets? How do we define bubbles ex ante or even ex post? A financial market bubble may be defined loosely as a sharp increase in the price of an asset in a continuous process, with the initial rise generating investors’ expectations of further future increases and thereby attracting new buyers. These buyers are generally speculators, interested in profits from trading in the asset rather than the asset’s earning capacity. Such a definition implies that a high and increasing price is not justified and is fed by momentum investors who buy with the sole purpose of selling quickly to other investors at a higher price. In recent years, economists have tried to give additional substance to the definition of a financial market bubble by linking asset price movements to fundamentals. Fundamentals refer to those economic factors that together determine the price of any asset, such as cash flows and discount rates. For example, Stiglitz (1990, p. 13) defines a bubble in financial markets in the following way: “If the reason that the price is high today is only that the selling price will be high tomorrow – when fundamental factors do not seem to justify such a price – then a bubble exists”. In this context, Siegel (2003) argues that one cannot identify any asset price bubble immediately, because one has to wait a sufficient length of time to determine whether the previous asset prices can be justified by the asset’s subsequent cash flows. Unfortunately, it is not that easy to find an operational definition of a bubble in the art market. If a bubble is defined only as excess changes in prices that are not captured by underlying economic fundamentals, then

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the extent to which bubbles occur in the art market is only by definition. Although some mass media treat the rapid rate of increase in some pieces of art as de facto evidence of a bubble in the art market, our understanding dictates that such rises alone are necessary but not sufficient evidence. Additional evidence is needed that relates current art prices to their fundamental determinants. It is vital to keep in mind that the only constraint limiting the price of a particular piece of art is the wealth of the agents willing to pay it. Economists have identified a number of transmission mechanisms where fluctuations in housing prices can have an effect on the overall economy. One potential effect of a severe home price decline could result from a consumption wealth effect. Although the magnitude of this effect remains controversial in some quarters, a number of empirical studies find significant wealth effects from real estate assets. For example, Case et al. (2001) show that if the magnitude of the wealth effect from housing is around 5 percent, then a severe decline would lead to reduction in consumption of roundabout US$150bn, which is about 2 percent of total personal consumption expenditures. Many analysts argue that the recent increase in home prices is symptomatic of a real-estate bubble that will burst eventually, just as the stock market bubble did in 2000. This would imply the erasing of a significant amount of household wealth. They add that such a decline of disposable income would have sharp adverse macroeconomic effects, as already indebted consumers reduce spending even further to improve their weakened financial situation. Despite the (technical and conceptual) difficulties of defining bubbles in art markets, we believe that the likelihood of falling prices will affect the general economy only to the extent that the losses made may reduce liquidity in financial markets.

1.5 DATA In order to look more specifically at both bull and bear markets, we use almost thirty years’ of monthly data, from January 1976 to December 2004, from the Art Market Research (AMR) database. AMR uses over 800 auction houses to collect sales data for hundreds of individual artists worldwide. This is the most comprehensive data set available for looking at performance during market extremes, since it is available as a monthly index. Indices are constructed for individual artists using the average prices of his or her paintings obtained in the market. A national index is constructed, comprised of a number of chosen artists for each country. Art indices for the USA and the UK art markets are used, covering a large number of artists over several movements and periods in the art scene, as well as a general index which covers the markets that dominate the global market for art.2 Log returns are

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RACHEL CAMPBELL AND ROMAN KRÄUSSL

computed for the art indices over the period January 1976–December 2004 (see Table 1.1). For an art fund, the only dividend received is the pleasure of the painting for the portfolio manager, unless the art fund allows its investors to rent or borrow some of its paintings for display in their own homes, or the works of art are rented out to museums or art collectors on loan, thus providing an additional income stream. The indices do not cover sales by private dealers, or works of art that are bought in – that is, pieces put on the block but not selling; however, these represent a highly significant part of the global art market3 . Figure 1.1 displays the development of the average price indices

Table 1.1 Summary statistics: monthly log return data, January 1976– December 2004 ART 100 Annual average return

US 100

UK 100

5.27%

8.26%

5.12%

17.11%

15.86%

11.10%

Average

0.026

0.041

0.026

Standard deviation

0.121

0.112

0.078

−0.837

−0.817

−0.097

1.694

1.029

−1.083

Annual average standard deviation

Skewness Kurtosis

12,000 10,000

General US British

8000 6000 4000

0

1976 1977/03 1978/06 1979/09 1980/12 1982/03 1983/06 1984/09 1985/12 1987/03 1988/06 1989/09 1990/12 1992/03 1993/06 1994/09 1995/12 1997/03 1998/06 1999/09 2000/12 2002/03 2003/06 2004/09

2000

Figure 1.1 Art indices: International art performance, January 1976–December 2004 Note: The average price indices from AMR for the General Art Market (ART 100): the top US artists (US 100) and the top British artists (UK 100).

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for the general art market, the top 100 US artists and the top 100 UK artists since the later 1970s. From Figure 1.1, it would appear that there was an enormous bubble in the art market during the early 1990s. This has been documented in the literature and is usually considered to have arisen from the effect of the expanding Japanese economy, with wealth flowing directly from this boom into the luxury art market. From the graph, it would appear that prices returned to their fundamental values in the two years following the bubble in 1990. Figure 1.1 also indicates a Japanese phenomenon after the collapse of the Japanese economy, with money flowing directly out of the art market during this period. Interestingly, the causality has been documented between these two markets, as well as between other equity markets and the art market, by Campbell (2005). The restrictive supply of art, recently cited as the reason for increasing prices, and hence returns being made in art investment, is also a driving factor behind the occurrence of bubbles in the art market. More recent developments using art as collateral for credit loans only serve to lengthen the extent and duration of the resulting price rises and the size of the financial bubble. This optimism is exacerbated by the tendency of investors and banks towards myopic disaster behavior. The highly leveraged positions of banks, holding collateral consisting of such opaque assets as real estate and artworks result in downside risk from the real estate and art markets being shifted to the banking sector. This effect is exaggerated by the feeling of wealth created by increases in property prices and art prices feeding on each other. This can have severe implications for the banking sector and the macroeconomy, so this is one reason to evaluate carefully how much of these assets act as collateral on the balance sheet. The extent to which a bubble is able to develop depends on the upward pressure on movements in prices, notably through such mechanisms as greater media coverage, so that a large response to large price changes occurs. The development of a bubble through the occurrence of large price movements should be reflected by greater conditional volatility in financial markets. Movements which occur with a greater probability than conditional volatility would suggest, can be accounted for by the use of a tail index. This is not a new methodology, but its application to the measurement of speculative asset-pricing bubbles, is, to our knowledge, new. Before we observe the relationship empirically between estimates for the probability of larger than conditionally normal movements in prices occurring during the period of development of a speculative bubble, we shall first outline the methodology for estimating a tail index. It is the conditional estimate of the tail index that we estimate using rolling observations, to see how the probability of larger than “normal” movements in prices change over the development and bursting of the bubble in the art market.

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9

1.6 METHODOLOGY In order to analyze the extent to which the market moves away from fundamental values through larger than “normal” probabilities occurring in large movements of the return distribution, we shall apply below the Hill’s (1975) tail index estimator, which was further extended by Huisman et al. (1997). We use EVT to provide us with estimates of tail indices. EVT looks specifically at the distribution of the returns in the tails, and the tail fatness of the distribution is reflected by the tail index. This concept was first introduced by Hill (1975), and measures the speed with which the distribution’s tail approaches zero. The fatter the tail, the slower the speed and the lower the tail index given. An important feature about the tail index is that it equals the number of existing moments for the distribution. A tail index estimate equal to 2 therefore reveals that both the first and second moments exist, in that case the mean and the variance; however, higher moments will be infinite. By definition, the tail index for normal distribution equals infinity, since in that case, all moments exist. Since the number of degrees of freedom reflects the number of existing moments, the tail index can thus be used as a parameter for the number of degrees of freedom to parameterize the student-t distribution. To obtain tail index estimates, we use a modified version of the Hill estimator, developed by Huisman et al. (1997). Their estimator has been modified to account for the bias in the Hill estimator, with the additional advantage of producing almost unbiased estimates in relatively small samples. Specifying k as the number of tail observations, and ordering their absolute values as an increasing function of size, we obtain the tail estimator proposed by Hill. This is denoted by γ, which is the inverse of α: 1 ln(xn−j+1 ) − ln(xn−k ) k k

γ(k) =

(1.1)

j=1

Following the methodology of Huisman et al., we can use a modified version of the Hill estimator to correct for the bias in small samples. The bias in the Hill estimator stems from the fact that it is a function of the sample size. A bias-corrected tail index is therefore obtained by observing the bias of the Hill estimator as the number of tail observations increases up to κ, whereby κ is equal to half of the sample size γ(k) = β0 + β1 k + ε(k),

k = 1, . . . , κ

(1.2)

The optimal estimate for the tail index is the intercept β0 , while the α estimate is the inverse of this estimate. This is the estimate of the tail index that we use to estimate rolling estimations of the degree with which larger

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Table 1.2 Alpha estimates for the All Art Index, January 1976–December 2004 Log returns Alpha

Gamma

SE

Kappa

All Art Index

Both Left Right

1 3.26994 3.18242 3.46838

0.305816 0.314226 0.288319

0.053088 0.081208 0.069766

1 3.26994 3.18242 3.46838

US Art Index

Both Left Right

2 3.87852 3.99025 3.13632

0.25783 0.250611 0.318845

0.044758 0.062053 0.079892

2 3.87852 3.99025 3.13632

UK Art Index

Both Left Right

3 9.60735 9.35147 10.4014

0.104087 0.106935 0.096141

0.018069 0.026795 0.023805

3 9.60735 9.35147 10.4014

Note: This table provides the alpha estimates using the Huisman et al. (1997) estimator for the All Art Index from Art Market Research, using monthly data.

than “conditionally normal” returns occur in the historical distribution of returns over time.

1.7 RESULTS Table 1.2 provides the alpha estimates using the Huisman et al.’s estimator over the period January 1976 to December 2004. We first look at the alpha estimates for the whole sample. We see that there is indeed deviation from the assumption of “normality”, since the alpha estimates are between 2 and 3. This would imply that the tail index is able to capture some of the additional movement occurring in returns beyond that of the assumption of normality, captured by volatility alone. There is a move away from fundamental distribution over time. In Figure 1.2, the inverse alpha estimates (gammas) using the previous eight years’ sample of monthly data are plotted next to the actual monthly returns. We see that, the more the returns fluctuate, the higher the inverse alpha estimate and the greater the movement away from fundamental values. Indeed, the correlation between volatility and alpha is −0.42, which is highly significant at the 95 percent confidence level. It has been shown that this measure increases during periods of instability.4 Therefore, we would expect that the use of the gamma estimate is a good indicator for a movement away from fundamental values, and the development of an asset pricing bubble.

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RACHEL CAMPBELL AND ROMAN KRÄUSSL

General art index gamma Art returns General art index

/0 2

/0 2

04 20

/0 2

03 20

/0 2

02 20

/0 2

01 20

/0 2

00 20

/0 2

99 19

/0 2

98 19

97 19

96 19

95 19

94 19

93 19

91

92 19

19

90 19

89 19

87

88 19

19

02

86

19

19 85 /

84 19

⫺0.1

/0 2

1

/0 2

0

/0 2

2 /0 2

0.1

/0 2

3

/0 2

0.2

/0 2

4

/0 2

0.3

/0 2

5

/0 2

0.4

/0 2

6

Indices value

7

0.5

/0 2

Gamma estimates

0.6

0

Figure 1.2 Art returns and time-varying downside risk, January 1976–December 2004 Notes: Art Index. Returns and Gamma Estimates Monthly Data 96 Rolling Observation for Gamma Estimates for left tail of the Art Index using 96 observations to estimate the downside risk. Based on the data for the art market, we use the eight years’ monthly data available from 1976 to 1984 – a total of 96 observations – to calculate the conditional gamma estimate for the distribution. Obviously, the other moments of the distribution, the mean and the standard deviation are able to change conditionally over time, so that the gamma estimate is able to capture the extent of larger than conditionally normal movements occurring in the return distribution. The results are shown in Figure 1.2. There is an extremely high correlation between the bubble occurring in 1990 and the high values obtained from the tail index estimator over the period of the bubble. The gamma estimates converge to their average values after the bubble bursts in 1991, and maintain a value around the average value over the rest of the sample until the current period.

1.8 DISCUSSION It would appear that the phenomenon of the bubble developing in the art market may be captured through the use of the tail index estimator, which captures the probability of larger than conditionally normal movements in large returns occurring over the return distribution over time. The analysis so far has only been applied to the art market, but there is no reason why

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the methodology may not be used for other financial markets in which it is thought that bubbles have occurred, or are thought to be present. Indeed, the use of a relatively small sample of observations to analyze the tail index estimator provides a robust estimator, which can be applied conditionally over the historical time series of returns. A further issue is that it is not possible to test strictly for efficiency in the art market. We have discussed so far possible reasons for inefficiencies of the art market – for example, information asymmetries. But there are good reasons why particular behavioral anomalies are even larger and more widespread in the art market compared to the financial markets. Many private collectors are not profit-oriented and are particularly prone to the behavioral anomalies that arise from leaving endowments, opportunity costs and sunk cost effects. Circumstantial evidence suggests that private collectors are strongly subject to the endowment effect, which implies that they value an art object owned to a greater extent than one not owned. The result is that people often demand much more to give up an object than they would be willing to pay to acquire it (see Thaler, 1980). This is what Samuelson and Zeckhauser (1988) call a status quo bias; that is, the preference for the current state that biases someone against both buying or selling an object. These anomalies are manifestations of an asymmetry of value that Tversky and Kahneman (1991) call “loss aversion”. Loss aversion means that the disutility of selling an object is greater than the utility associated with buying it. Loss aversion also explains why there is no market for renting art objects. Frey and Eichenberger (1995) argue that the consumption benefits of viewing art should be revealed in the rental fees for art objects. The consumer would pay a fee for enjoying art while being unaffected by price changes in the art market. The reason why such market-revealing pure psychic benefits from art do not exist must be sought in property rights and a corresponding ownership effect. While the decision to buy art might be based on financial calculations, the desire to possess a beautiful and internationally famous work that will impress friends and clients unquestionably adds to the attraction. The owner of a work of art has a monopoly over that specific object, while other assets may be held by many individuals. The major difference between investing in art and in common financial assets is that art is tangible and is associated with a given lifestyle. This implies that an art object yields additional benefits if it is owned and not just rented, because the art object’s aura is also appropriated (Benjamin, 1963). Apart from the endowment effect and its corresponding ownership effect, there is also the opportunity cost effect. This implies that many collectors isolate themselves from considering the returns of alternative uses for their investments. A third behavioral anomaly that plays a large part in the art market is the sunk cost effect. This describes the tendency to be excessively attached to activities (things) for which one has expended resources resulting

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13

from past efforts at building up a (specific) art collection. Additionally, the self-deception theory suggests that the tendency to adjust attitudes to match past actions is a mechanism designed to persuade the individual that he or she is a skillful decision-maker. Are art investors reluctant to realize their losses? Or are investors extremely reluctant to realize their losses in art? Mental accounting is a kind of narrow framing that involves keeping track of gains and losses related to decisions in separate mental accounts. Thaler (1985) argues that individuals reexamine each account only intermittently when it is action-relevant. Mental accounting may explain the disposition effect (Shefrin and Statman, 1985) – that is, the excessive propensity to hold on to assets that have declined in value and to sell the winners. Such a mechanism may even be side-tracked when the individual avoids recognizing losses. Self-deception theory reinforces this argument, since a loss is an indicator of poor decision-making, and a self-deceiver maintains self-esteem by avoiding the recognition of this. Regret avoidance may also reflect a self-deception mechanism designed to protect self-esteem about poor decision ability. Kahneman et al. (1991) show that regret is stronger for individual decisions that involve action rather than passivity. This effect is also known as the “omission bias”. A bequest aspect is also highly relevant. Gifts from parents to their children, or inheritances of family members in the form of art objects are valued more highly by the owner than they would be purely for their monetary value. Frey and Eichenberger (1995) argue that, by selling the object, the owners are transferring with it part of their own “nature”.

1.9 CONCLUSION Using a unique set of data with which to observe and quantify the extent of a bubble in the art market, we have been able to gain a greater insight into the nature of bubbles with respect to the larger than “normal” movements that appear to occur during the build-up and breakdown of financial bubbles. More detailed analysis with regard to return distribution will no doubt enable a richer analysis of the make-up of the asset bubbles, and will be extremely interesting avenues for further research. By defining the degree of “bubbliness” in a market as the degree to which large movements are more likely to occur, the gamma of the distribution of historical returns can be estimated conditionally over time. We see that there is an extremely high correlation between the size of the gamma estimates and prices during the period of the bubble. The larger the gamma estimate, the greater the probability of more extreme movements in the return distribution. This should indeed be constant over time. However, we see that the correlation of the gamma estimates increases during the period of the bubble, and is thereafter fairly constant. We therefore premise that the bubble

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can be defined ex post from a larger probability occurring in the tails of the distribution, observed conditionally over time – from rolling observations used to estimate the degree of “bubbliness” in the market. Although the results presented here are preliminary in nature, they provide an extremely innovative and interesting avenue for further research into the notion of bubbles in financial markets. The use of the art market, which represents a market in which deviations from fundamental values are much more likely, provides a particularly interesting market with which to observe such measures. There are many further areas that may need to be addressed before any definite conclusions can be drawn. For example, the use of this measure on alternative asset classes in which bubbles have been observed. The “dot.com” mania and real estate markets in particular. Although the results are in a preliminary form, they should help to generate further discussion and insight into the determination and measurement of bubbles in financial markets.

NOTES 1. All errors are the responsibility of the authors. Many thanks to participants at the conference on “Art: An Alternative Asset Class” at Sotheby’s, London, for their valuable comments. 2. Figures on the exact numbers of artists per index are available from the authors on request, or from AMR. 3. In a similar manner, the S&P 500 only represents a segment of the whole market for US equities. 4. See Pownall and Koedijk (1999) for an analysis of the Asian financial crises of 1997–8, with the use of the same methodology.

REFERENCES Bauer, R., Campbell, R. A. and Dil, N. (2005) “Art Diversification over the Business Cycle: The Case for the UK”, Working Paper, Maastricht University. Baumol, W. (1986) “Unnatural Value: or Art Investment as a Floating Crap Game”, American Economic Review, 76(2): 10–14. Benjamin, W. (1963) Das Kunstwerk im Zeitalter seiner technischen Reproduzierbarkeit, 4th edn (Frankfurt am Main: Edition Suhrkamp). Campbell, R. A. (2005) “Art as an Alternative Asset Class”, Working Paper, Maastricht University. Case, K. E., Quigley, J. M. and Shiller, R. J. (2001) “Comparing Wealth Effects: The Stock Market versus the Housing Market”, Advances in Macroeconomics, 5(1): 1–32. Chanel, O. (1995) “Is Art Market Behavior Predictable?”, European Economic Review, 39(3–4): 519–27. Frey, B. S. and Eichenberger, R. (1995) “On the Rate of Return in the Art Market: Survey and Evaluation”, European Economic Review, 39(3–4): 528–37. Goetzmann, W. N. (1993) “Accounting for Taste: Art and the Financial Markets over Three Centuries”, American Economic Review, 83(5): 1370–6.

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Hill, B. (1975) “A Simple General Approach to Inference about the Tail of a Distribution”, Annals of Mathematical Statistics, 3(5): 1163–74. Huisman, R., Koedijk, K. G., Kool, C. and Palm, F. (1997) “Fat Tails in Small Samples”, Working Paper, Erasmus University, Rotterdam. Kahneman, D., Knetsch, J. and Thaler, R. (1991) “The Endowment Effect, Loss Aversion, and Status Quo Bias”, Journal of Economic Perspectives, 5(1): 193–206. Mei, J. and Moses, M. (2002) “Art as an Investment and the Underperformance of Masterpieces”, American Economic Review, 92(5): 1656–68. Pownall, R. A. and Koedijk, K. G. (1999) “Capturing Downside Risk in Financial Markets: the Case of the Asian Crisis”, Journal of International Money and Finance, 18(6): 853–70. Samuelson, W. and Zeckhauser, R. (1988) “Status Quo Bias in Decision Making”, Journal of Risk and Uncertainty, 1(1): 7–59. Shefrin, H. and Statman, M. (1985) “The Disposition to Sell Winners Too Early and Ride Losers Too Long: Theory and Evidence”, Journal of Finance, 40(3): 777–90. Siegel, J. J. (2003) “What Is an Asset Price Bubble? An Operational Definition”, European Financial Management, 9(1): 11–24. Stiglitz, J. E. (1990) “Symposium on Bubbles”, Journal of Economic Perspectives, 4(2): 13–18. Thaler, R. H. (1980) “Toward a Positive Theory of Consumer Choice”, Journal of Economic Behavior and Organization, 1(1): 39–60. Thaler, R. H. (1985) “Mental Accounting and Consumer Choice”, Marketing Science, 4(3): 199–214. Tversky, A. and Kahneman, D. (1991) “Loss Aversion in Riskless Choice: A ReferenceDependent Model”, Quarterly Journal of Economics, 106(4): 1039–61.

CHAPTER 2

International Stock Portfolios and Optimal Currency Hedging with Regime Switching Markus Leippold and Felix Morger

2.1 INTRODUCTION Despite the vast literature on optimal currency hedging, there still is considerable disagreement about how international investors should hedge their currency risk. One argument is that investors should fully hedge, since exchange-rate changes in excess of the forward discount rate average out. Therefore, hedging decreases the risk of foreign investment, but does not reduce its expected returns. In the words of Perold and Schulman (1988), currency hedging is a free lunch. However, there is a large branch of literature that does not agree with this viewpoint. As an early example, Froot (1993) argues that the free-lunch argument does not hold in the long run. If exchange rates and asset prices display mean reversion, the optimal hedging policy becomes time-varying. In particular, real exchange rates revert to their means according to the theory of purchasing power parity, and investors should maintain an unhedged foreign currency position. Therefore, for an investor with a long investment horizon, it becomes optimal not to hedge at all. Froot argues that real-exchange rates may deviate from their theoretical fair value over shorter horizons, and currency hedging in this context may become beneficial. As a compromise between these two extreme viewpoints, Black (1989) argues that, using Siegel’s paradox, there is a constant universal hedge 16

MARKUS LEIPPOLD AND FELIX MORGER

17

ratio between zero and one. However, Black has to impose some strong assumptions and because of the time-period sensitivity and significant variability and volatility of input parameters in the optimal hedge ratio, there is a significant dispersion in what constitutes the optimal constant hedge ratio. In contrast, the evidence of Glen and Jorion (1993), who analyze the performance of mean-variance efficient stock and bond portfolios from the G5 countries when hedging the associated currency risk with currency forwards, shows that there is a substantial improvement when using conditional time-varying hedging strategies. It is beyond the scope of this chapter to provide a full account of the existing literature on currency hedging, but we refer, for example, to the recent contribution by Dales and Meese (2001) for an overview. We note that most of the literature builds on simplifying assumptions on the dynamics of the underlying returns. Indeed, there is now ample empirical evidence against the normal distribution for return dynamics and a lot of statistical justification for so-called regime-switching models. For example, Turner et al. (1989), Garcia and Perron (1996), Gray (1996), Perez-Quiros and Timmermann (2000), Whitelaw (2000), Ang and Bekaert (2002a, 2002b), Ang and Chen (2002), Connolly, Stivers and Sun (2005) and Guidolin and Timmermann (2005a, 2006) report evidence of regimes in stock or bond returns. Therefore, in our study, we analyze the impact of such regime-switching models on optimal currency hedging. Closely related to our study are the works by Ang and Bekaert (2002a) and Guidolin and Timmermann (2005a). Both papers make use of regimeswitching models. Ang and Bekaert (2002) analyze the optimal investment strategy within a mean-variance framework. Concerning the modeling of regime switches, they do not consider a Bayesian updating rule to infer on state probabilities. Guidolin and Timmermann (2005b) assume preferences over the moments of wealth distribution. In addition, they explore the optimal asset allocation of an international portfolio with unhedged returns. They do not address the issue of optimal currency hedging. We use a more general CRRA (Constant Relative Risk Aversion) utility setting and we compare the non-Bayesian investor with the Bayesian one. We show that it really pays to go Bayes! Furthermore, we explicitly allow the investor to hedge his or her currency exposure. Whereas Guidolin and Timmermann (2005b) find that their model offers a rational explanation of the strong home bias observed in US investors’ asset allocation, our results contrast with their conclusion. While we find a slight decrease in foreign asset holdings for a strategy with unhedged returns, the strategy with optimal currency hedging substantially increases the exposure to foreign markets. Therefore, the home bias becomes even more puzzling. The plan of this chapter is as follows. In Section 2.2, we present the regimeswitching model and the optimization problem. Section 2.3 provides the estimation results for several model specifications. In Section 2.4, we provide

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INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

a discussion of several aspects of our results; in particular, we discuss the economic benefits of using regime-switching models and we analyze the optimal portfolio allocation. Section 2.5 concludes.

2.2 THE MODEL Regime-switching models consist of two generic processes, the state process st and the return process rt . The unobservable state process st determines which state is active at time t. For the state st , we assume a discrete firstorder Markov chain with S possible states or regimes. The constant transition probability for moving from state i to state j is denoted as pij = P{st+1 = j|st = i, st−1 = k, . . .} = P{st+1 = j|st = i},

for i, j = 1, . . ., S

and we collect all the pij ’s in the transition matrix P. For the N-dimensional return vector rt , we assume the state dependent dynamics drt = µ(st )dt + (st )dBt where Bt is a N × 1 dimensional Wiener process. Both the drift vector µ(st ) and the N × Ndimensional covariance matrix (st ) depend on the active regime. Therefore, the distribution of rt+1 conditional on the state st is a mixture of S normal distributions with probability density function f (rt+1 |st = i) =

S

pij f (rt+1 |st+1 = j)

j=1

We note that the regime-switching model defined above can account for skewed and fat-tailed returns. Furthermore, with pij > 1/S as a sufficient condition, we can also generate correlation breakdowns and volatility clusters. Both are often observed in the joint dynamics of international stock markets.

2.2.1 Portfolio selection with perfect knowledge of the active state We assume that the investor can invest in N assets, where the Nth asset is the risk-free asset. The investment horizon T is fixed. The investor has the possibility of rebalancing the asset allocation at the beginning of every period; for example, at times t = 0, . . ., T − 1. There are no transaction costs.

MARKUS LEIPPOLD AND FELIX MORGER

19

We further assume that the investor has a CRRA utility function defined over wealth, for example U(W) =

1 W 1−γ 1−γ

where we assume γ > 1 for the relative risk aversion coefficient. We start with the situation in which the investor has perfect knowledge of the active state. We denote by αt the vector of portfolio weights at time t. To maximize the investor’s terminal wealth, he/she has the following objective function: max E0 [U(WT )]

α0 ,...,αT−1

s.t.

Wt+1 = Wt (αTt exp (rt+1 ))

(2.1)

1 = αTt 1 where E0 [·] = E[·|F0 ]. To simplify notation, we write Wt+1 = Wt αTt exp (rt+1 ) = Wt Rt+1 (αt ) where Rt+1 is the portfolio’s gross return from time t to time t + 1. Using a dynamic programming approach, we can solve the optimization problem recursively. At each time step t, we have to maximize the indirect utility function J: J(W, r, s, θ, t) = max Et [Qt+1,T U(Wt+1 )] αt

given the parameter set θ = {µ(s), (s), P} and with the indirect utility Qt+1,T = Et+1 [(RT (α∗T−1 ) · . . . · Rt+2 (α∗t+1 ))1−γ ]

QT,T = 1

where α∗T−1 , . . .α∗t+1 are the optimal portfolio weights determined recursively. These optimal portfolio weights have to solve the corresponding first-order conditions (FOC) of the optimization problem in Equation (2.1). Given state st = i, we obtain the FOC of the investor’s allocation problem as .−γ

Et [Qt+1,T Rt+1 (αi,t )λt+1 |st = i] =

S

.−γ

pij Et [Qt+1,T Rt+1 (αi,t )λt+1 |st+1 = j] = 0

(2.2)

j=1

where αi,t = αt (st = i) and λi,t+1 is defined as the vector of excess returns over the risk-free rate, for example ⎛ 1 N )⎞ exp (ri,t+1 ) − exp (ri,t+1 ⎜ ⎟ .. λi,t+1 = ⎝ ⎠. . N−1 N ) exp (ri,t+1 ) − exp (ri,t+1

20

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

Whereas, for the case of one single regime with i.i.d. returns, the optimal asset allocation for a CRRA investor is constant over different time horizons (see, for example, Samuelson, 1969), the allocation in a multiple regime model depends on the prevailing regime st and on the investment horizon. This dependency arises because of the changing investment opportunity set induced by the regime switches.

2.2.2 Portfolio selection under hidden regime switches Since it is rather restrictive to assume that investors have perfect knowledge on the active state, we next assume that, at the moment they invest, they do not know which regime is active and they have to infer on the active regime using a specific updating procedure. For the updating procedure, we assume that the investor uses a Bayesian updating rule based on the filter developed by Hamilton (1989). To sketch out briefly the updating procedure, we denote the oneperiod-ahead forecast of investor beliefs as ut+1|t , for example, ut+1|t is the column vector of the probabilities P(st+1 = i|Ft ; θt ), and the set θt = {µ(s), (s), P} collects the time-t parameters of the regime-switching model. The optimal inference (posterior) and forecast (prior) on the active regime is found by iterating the equations uˆ t|t =

uˆ t|t−1 ◦ηt T 1 (uˆ t|t−1 ◦ηt )

(2.3)

and uˆ t+1|t = PT uˆ t|t

(2.4)

where ηt is a S × 1 vector of the multivariate normal densities determined by µ(st ) and (st ), 1 is the S × 1 unit vector, and by ‘◦’ we denote the elementby-element multiplication. Given a starting value u1|0 and a parameter set θ the algorithm defined by Equations (2.3) and (2.4) calculates for each time t the probability of a regime currently being active and also being active for the period ahead. To get parameter estimates, we maximize the likelihood function using a variate of the EM algorithm developed by Hamilton (1990). The log likelihood function L(θ) to be maximized is the sum of the denominator of Equation (2.3) over all t. We note that, with incomplete information on the current state, the indirect utility J is no longer a function of the active state st , but a function of the beliefs ut|t about the active state, where ut|t is the column vector of the probabilities P(st = i|Ft ; θt ). In particular, we have J(W, r, ut|t , θ, t) = maxαt Et (Qt+1,T U(X, Y))

MARKUS LEIPPOLD AND FELIX MORGER

21

We write the belief at time t + 1 given the filtration Ft+1 as vt+1|t+1 . For ease of notation, we drop the time indices from the beliefs ut|t and vt+1|t+1 whenever u and v are used as subscripts. Then, the FOC is given as .−γ

Et [Qt+1,T Rt+1 (αu,t )λt+1 |ut|t ] =

S i=1

ui,t|t

S j=1

1

.−γ

puv,j Et [Qv,t+1,T Rt+1 (αu,t )λt+1 |st+1 = j]dv = 0 (2.5)

pij 0

where puv,j = P{ut|t , vt+1|t+1 ; θt |st+1 = j}. By inspection, Equation (2.5) is a straightforward extension of the FOCs under full information on the active regime given in Equation (2.2). The expectation gives the FOCs for a given belief vt+1|t+1 , weights αu,t , and regime st+1 = j. The summation of the expectation over the different states j corresponds to the FOCs for the full information case. The summation over the beliefs ui,t|t and the integral over the combinations of beliefs vt+1|t+1 are because of the non-observability of the state process. With the state process being observable, ui,t|t equals 1 for the active regime and zero for all others. The integral over the combinations of beliefs vt+1|t+1 enters Equation (2.4), because the indirect utility Qv,t+1,T depends on the beliefs vt+1|t+1 , which are unknown at investment. The probability puv,j corresponds to the probability of the belief moving from ut|t to vt+1|t+1 , given that regime j is active at t+1. It assigns a weight to each expectation and its involved indirect utility. The probability puv,j can also be interpreted as the probability of occurrence of vt+1|t+1 given ut|t and regime st+1 = j. The optimal portfolio choice problem for a regime-switching model must be solved numerically. In the following empirical section, we apply a backward solution algorithm with a Monte Carlo simulation of size Z = 30,000. Furthermore, we have to discretize the state space of beliefs. For the model with two regimes, we have 3 grid points, and for the model with three regimes we have 6 grid points. With this parameterization the algorithm generates very accurate weights. Therefore, we do not consider more elaborated algorithms such as, for example, the algorithm proposed by Brandt et al. (2004) that is based on the Longstaff and Schwartz (2001) least-square Monte Carlo method.

2.3 ESTIMATION RESULTS 2.3.1 Data We take the perspective of a US investor, who allocates his/her wealth in risk-free assets and the US, UK and German stock markets. For the stock markets, we use MSCI (Morgan Stanby Capital International) country indices. We approximate the risk-free rate with the one-month Eurodollar rate. To

22

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

investigate the optimal currency hedging behavior of the investor, we let the investor take positions not only in the foreign market indices denominated in US dollars, but also in the foreign markets denominated in their local currency. The relative asset allocation between the unhedged and hedged indices determines the optimal hedging policy. The monthly data covers the period from December 1970 to June 2005, which results in 414 one-month excess returns over the risk-free rate. For the out-of-sample analysis, we use fifteen and a half years spanning the period January 1990 to June 2005. Table 2.1 presents the summary statistics for the excess returns. With the sole exception of the unhedged German index in the period prior to 1990, the index returns fail the Jarque–Bera test for normality, motivating the use of regime-switching models.

2.3.2 Specification test Using three model portfolios, we test for normality (Jarque–Bera – JB) and for the absence of serial correlation with three different likelihood ratio tests on the predictive density proposed by Berkowitz (2001). LR1;1 tests for serial correlation of lag one, LR2;1 for serial correlation of lag one and two, and LR2;2 for linear and squared serial correlation up to lag two. The latter is a test for omitted volatility dynamics. The results of the specification tests for the cases of two and three regimes are summarized in Table 2.2.1 We find that the Jarque–Bera statistic is reduced substantially by the use of regime-switching models. For example, we find that the Jarque–Bera statistics of the UK excess returns are above 2,000. In contrast, under the chosen specifications of the regime-switching models, the statistics for the UK are mainly below 20 during the period of the out-of-sample test. Hence, we are well advised to use a regime-switching model to account for the skewed and fat tails of stock market returns. In particular, for the subsequent analysis we shall use three regimes for both the unhedged and the optimal hedging strategy, and two regimes for the fully hedged strategy.

2.3.3 Parameter estimates as at June 2005 In this section, we present the parameter estimates for different model specifications. Table 2.3 presents the parameter estimates of the model specification with a single regime and unhedged indices. The mean excess returns are between 5 percent and 7.7 percent per year, and the volatilities are around 20 percent. The correlations are around 0.5. The highest correlation is between the UK and the US markets. All parameter estimates are significantly different from zero.

MARKUS LEIPPOLD AND FELIX MORGER

23

Table 2.1 Summary statistics, stock market returns, December 1970 – June 2005 GER, FH

GER, UH

UK, FH

UK, UH

Panel A: Summary statistics for June 2005 Moment statistics Mean 0.040 0.070

0.081

0.077

Std, dev. Skewness Kurtosis JB

US

0.050

0.198

0.216

0.208

0.230

0.154

−0.405

−0.214

1.367

1.337

−0.320

5.172

4.410

18.678

14.739

4.876

90.780

36.416

4319.407

2470.964

66.188

(0.000)

(0.000)

(0.000)

(0.000)

(0.000)

Correlation coefficients GER, FH 1.000 GER, UH

0.858

1.000

UK, FH

0.446

0.373

1.000

UK, UH

0.379

0.464

0.894

1.000

US

0.508

0.463

0.595

0.538

1.000

Panel B: Summary statistics for December 1989 Moment statistics Mean 0.035 0.088 0.108

0.093

0.032

Std, dev.

0.277

0.161 −0.228

Skewness Kurtosis JB

0.177

0.214

0.248

−0.180

0.009

1.459

1.338

5.138

3.716

16.296

12.305

5.545

42.827

4.440

1723.482

870.741

61.157

(0.000)

(0.109)

(0.000)

(0.000)

(0.000)

Correlation coefficients GER, FH 1.000 GER, UH

0.832

1.000

UK, FH

0.360

0.287

1.000

UK, UH

0.329

0.397

0.920

1.000

US

0.380

0.335

0.560

0.505

1.000

Notes: The table presents the summary statistics of the stock market returns in excess of the risk-free rate and includes the Jarque–Bera test. The market indices are MSCI country indices and the risk-free rate is the 1-month Eurodollar rate. The data are provided by Datastream. The statistics for Germany and the UK are given both as hedged (denoted as FH) and unhedged (denoted as UH) indices. Mean and standard deviation are annualized. Panel A gives the summary statistics for the entire sample. Panel B presents the numbers for the sub-sample from December 1970 until the start of the horse race (see section 2.2).

Table 2.4 presents the parameter estimates for a three-regime model specification with unhedged indices. Compared to the benchmark case in Table 2.3, the bear state exhibits negative drifts, slightly increased volatilities and correlations, and a relatively low persistence. Once the bear state is

24

Table 2.2 Specification tests, December 1989–December 2004 Test

1989.12

1992.6

1994.12

1997.6

1999.12

2002.6

2004.12

Panel A: Unhedged strategy 2 regimes JB 1* 2*

2**

2**

1*, 1**

1*, 1**

1**

LL-ratio

3*

1*

1*

2*

3*

1*

3*

3 regimes JB

1*

1**

1**

LL-ratio

2*

3*

3*

3*

3*

1*

2*,1**

Panel B: Fully hedged strategy 2 regimes JB 1*

1*

1*

1*

1**

2*

1*

2*

2*

3*

1*

3 regimes JB

1*

2*

1**

1**

LL-ratio

2*

2*

2*

1*, 1**

2*

3*

1*

Panel C: Optimally hedged strategy 2 regimes JB 2* 1* 2*

1*, 2**

3**

1*, 3**

5**

5*

4*, 1**

4*, 2**

2*, 4**

LL-ratio

LL-ratio

3*

4*

3 regimes JB

1**

1*

1*

4*

1*, 2**

LL-ratio

1*

4*, 1**

4*, 1**

4*, 1**

3*

2*

3*

Notes: The table presents an overview of the specification tests at intervals of 2.5 years. The entries in the table show the number of markets that failed the Jarque–Bera test and likelihood ratio test, respectively, at a significance level of 5% or 1%. Significance at the 5% level is denoted by * and at the 1% level by **. Panel A reports the results for the strategy without hedging, and Panel B results for the strategy with the hedged returns. Panel C presents the overview for the optimal hedging case.

Table 2.3 Parameters of the single regime strategy

Panel A: Drifts Mean excess return

GER

UK

USA

0.070 (0.008)

0.077 (0.008)

0.050 (0.006)

Panel B: Correlations and volatilities GER

0.216**

UK

0.464**

0.230**

US

0.463**

0.538**

0.154**

Notes: The table presents the parameter estimates for the international investor and single regime specification with the unhedged indices. Panel A gives the annualized mean excess returns. The values in brackets are the annualized standard deviations. The off-diagonal elements in Panel B report the correlations and the diagonal elements the annualized volatilities. Significance at the 0.05 level is represented by ** of the corresponding covariances.

25

MARKUS LEIPPOLD AND FELIX MORGER

Table 2.4 Parameter estimates for the three regimes specification with unhedged indices Panel A: Drifts GER −0.123 (0.104)

UK −0.024 (0.310)

US −0.197 (0.143)

Low correlation state

0.123 (0.013)

0.104 (0.025)

0.106 (0.018)

High correlation state

0.035 (0.023)

0.056 (0.015)

0.031 (0.016)

Bear state

Panel B: Correlations and volatilities GER Bear state GER 0.241**

UK

UK

0.537**

0.507**

US

0.651**

0.609**

Low correlation state GER

0.200**

UK

0.361**

0.191**

US

0.172**

0.448**

High correlation state GER

0.235**

US

0.240**

0.136**

UK

0.850**

0.136**

US

0.825**

0.765**

0.144**

Panel C: Transition probabilities Bear state Bear state 0.842 (0.208)

Low corr. 0.026 (0.005)

High corr. 0.000 (0.086)

Low correlation state

0.158 (0.363)

0.934 (0.036)

0.076 (0.181)

High correlation state

0.000

0.041

0.924

Notes: Panel A presents the annualized mean excess returns. The values in brackets are annualized standard deviations. The off-diagonal elements in Panel B report the correlations and the diagonal elements the annualized volatilities. Panel C shows the transition probabilities and in brackets their standard deviations. Significance at the .05 level is represented by ** of the corresponding covariances.

active, the probability that it will remain active for a year is only 14 percent. The other two states are both growth states, but with significantly different correlation structures. Table 2.5 reports the results for the model specification with two regimes and fully hedged currency risk. There is a clear distinction between the two regimes. The first regime is a bear state with low drifts, especially for Germany with a drift of −22.3 percent, high volatilities and very low persistence. The probability that it switches to the bull state within two months is close to 50 percent. The second regime is a bull state with high excess returns ranging between 8 percent and 10 percent, low volatilities, and high

26

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

Table 2.5 Parameter estimates for the three regimes specification with fully hedged currency risk Panel A: Drifts Bear state

GER −0.223 (0.005)

UK 0.043 (0.013)

US −0.097 (0.007)

0.101 (0.006)

0.088 (0.010)

0.082 (0.010)

Bull state

Panel B: Correlations and volatilities GER Recession state GER 0.298**

UK

UK

0.431**

0.403**

US

0.521**

0.640**

Growth state GER

0.167**

UK

0.496**

0.140**

US

0.482**

0.584**

Panel C: Transition probabilities Bear state Bear state 0.721 (0.007) Bull state

0.280

US

0.237**

0.130**

Bull state 0.053 (0.081) 0.947

Notes: Panel A presents the annualized mean excess returns. The values in brackets are annualized standard deviations. The off-diagonal elements in Panel B report the correlations and the diagonal elements the annualized volatilities. Panel C shows the transition probabilities and in brackets their standard deviations. Significance at the .05 level is represented by ** of the corresponding covariances.

persistence. Similar to the bear state, the correlations are close to the sample average. Finally, Table 2.6 presents the parameter estimates for the model specification with three regimes with both hedged and unhedged indices. As for the unhedged strategy, the three regimes can be referred to as bear state, low correlation state, and high correlation state. The low correlation state can be interpreted as a transitory state.

2.4 DISCUSSION With the empirical specification at hand, we next discuss some of the implications of regime switching on asset allocation. With regard to the numerical calculation of the optimal portfolio strategies, we use a Monte Carlo simulation with sample size Z = 30,000. Furthermore, for all calculations, we suppose that the investor has a risk aversion parameter equal to γ = 5.

Table 2.6 Parameter estimates for the three regimes specification with hedged and unhedged indices Panel A. Drifts Bear Low High

GER, FH −0.234 (0.073) 0.061 (0.016) 0.110 (0.026)

Panel B: Correlations and volatilities GER, FH Bear state GER, FH 0.282** GER, UH 0.923** UK, FH 0.432** UK, UH 0.386** US 0.480**

GER, UH −0.186 (0.070) 0.094 (0.006) 0.125 (0.025)

UK, FH 0.037 (0.025) 0.091 (0.007) 0.073 (0.015)

UK, UH 0.072 (0.026) 0.063 (0.009) 0.110 (0.015)

US −0.133 (0.028) 0.072 (0.006) 0.076 (0.016)

GER, UH

UK, FH

UK, UH

US

0.283** 0.456** 0.453** 0.555**

0.467** 0.987** 0.643**

0.480** 0.615**

0.261**

Low correlation GER, FH GER, UH UK, FH UK, UH US

0.158** 0.790** 0.344** 0.301** 0.291**

0.195** 0.199** 0.408** 0.216**

0.162** 0.805** 0.517**

0.199** 0.443**

0.134**

High correlation GER, FH GER, UH UK, FH UK, UH US

0.233** 0.920** 0.844** 0.724** 0.853**

0.227** 0.809** 0.849** 0.835**

0.130** 0.860** 0.807**

0.130** 0.771**

0.137**

Low corr. 0.032 (0.075) 0.929 (0.063) 0.039

High corr. 0.000 (0.035) 0.080 (0.098) 0.920

Panel C: Transition probabilities Bear state Bear 0.783 (0.003) Low 0.217 (0.019) High 0.000

27

Notes: Panel A presents the annualized mean excess returns. The values in brackets are annualized standard deviations. The off-diagonal elements in Panel B report the correlations and the diagonal elements the annualized volatilities. FH is short for fully hedged and UH for unhedged. Panel C shows the transition probabilities and in brackets their standard deviations. Significance at the 0.05 level is represented by ** of the corresponding covariances.

28

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

2.4.1 Economic importance of regimes The economic importance of regimes is a very relevant issue for the appraisal of regime-switching strategies. We measure this “importance” as utility costs that an investor bears, when he/she gives up the optimal strategy and follows instead a suboptimal one. More precisely, we are interested in the monetary compensation c, also called certainty equivalent compensation, that makes an investor with time horizon T indifferent between the suboptimal weights α− and the optimal weights α∗ . Formally, we must solve E0 [U(WT (α∗ )|W0 = 1)] = E0 [U(WT (α− )|W0 = 1 + c)] for c. Using the fact that CRRA utility preferences are homogeneous in W0 , and since E0 [U(WT )|W0 = 1] =

Q0,T 1−γ

we obtain c=

Q∗0,T Q− 0,T

1/(1−γ) −1

We report the certainty-equivalent compensations in Table 2.7 as cents per invested dollar. In Table 2.7, the reported cost can be attributed directly to international investment and regime-switching (Panel A) and to regimeswitching only (Panel B), respectively. Panel C documents the economic importance of regimes in a purely international setting. Panel A of Table 2.7 displays the economic cost of investing in a domestic US portfolio while disregarding the possibility of international diversification. Relative to the international investment strategy with one regime and unhedged indices, the costs (in terms of invested dollars) are only 0.53 percent per year for all time horizons. Hence the strategy with a single regime does not seem to offer substantial benefits, at least when only a small number of (highly correlated) markets are involved (as is the case for our analysis). However, turning to the regime-switching strategies in Panel A, we see that the economic costs for the three regime-switching strategies may be as high as 2.7 percent per year. Consequently, international investment combined with regime-switching does pay off – even when the number of potential markets is low and their correlation high. Also, we note that ignoring regimes tends to increase the annualized economic costs as the time horizon increases. This finding contrasts with Guidolin and Timmermann (2004), who explore the required compensation for buy-and-hold strategies, and find decreasing costs with a longer time horizon. Their result is

Table 2.7 The economic importance of regime-switching strategies 6 months Total

1 year

Annualized

2 years Total

Annualized

5 years Total

Annualized

10 years Total

Annualized

Panel A: International strategies versus domestic US strategy, 1 reg. Internat., 1 reg. 0.26 0.53 0.53 1.06

0.53

2.66

0.53

5.39

0.53

UH, 3 reg.

2.51

14.14

2.68

31.37

2.77

1.01

2.02

2.32

5.09

FH, 2 reg.

0.71

1.42

1.50

3.10

1.54

8.06

1.56

16.88

1.57

OH, 3 reg.

1.03

2.06

2.20

4.57

2.26

11.90

2.27

25.24

2.28

Panel B: RS strategies versus international unhedged portfolio, 1 reg. UH, 3 reg. 0.74 1.49 1.78 3.94

1.95

10.95

2.10

23.98

2.17

FH, 2 reg.

0.45

0.90

0.98

2.05

1.02

5.34

1.05

11.07

1.06

OH, 3 reg.

0.61

1.23

1.36

2.87

1.42

7.44

1.44

15.46

1.45

Panel C: States of OH versus the OH bear state Low corr. 1.06 2.13

1.33

1.46

0.73

1.50

0.30

1.50

0.15

High corr.

1.97

3.97

2.90

3.75

1.86

4.32

0.85

4.35

0.43

Uncond. state

1.06

2.13

1.51

1.83

0.91

2.02

0.40

2.03

0.20

2005/6

1.89

3.82

2.80

3.62

1.80

4.17

0.82

4.19

0.41

29

Notes: Panels A and B report the economic cost of investing suboptimally. The suboptimal strategies have the same investment opportunities as the optimal strategies. The economic cost of these three panels are calculated with the indirect utilities, Q∗0,T of the unconditional state probabilities. Panel C documents the importance of regimes for the case of the optimal hedging strategy. It gives the cost of being in the bear regime compared to a situation with another, economically more promising state belief. In all panels, the economic costs are given as cents per invested dollar required, making an investor indifferent between the optimal and the suboptimal weights. UH denotes unhedged; FH fully hedged; and OH optimal hedging.

30

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

as a result of the inability of the investor to react to market changes. Panel B gives the required compensation for an international investor to ignore regime-switching. With a horizon of ten years, the costs of ignoring regimeswitching average almost 17 percent, or 1.6 percent per year. Comparing Panel B with the regime-switching strategies of Panel A, we see that the economic gains are about 30 percent (on average) lower in Panel B than in Panel A. In Panel A, only 30 percent of the certainty equivalents can be attributed to international diversification, and 70 percent result from to taking into account the possibility of regime-switches. In Panel C, we present the results on the economic significance of the regimes for the optimal hedging strategy. The panel gives the cost of being in the bear state compared to a situation with another, economically more promising state belief. These costs are high in the short term and, in contrast to the other panels, decrease with longer investment horizons. They are high in the short term because the regimes have very different estimated parameters, and therefore very different expected returns. They decrease because, in the long term, the probability for a switch to another regime increases and the initial regime becomes less influential. There are almost no additional costs related to the initial regime after the first five years. The costs reported in Table 2.7 are not only economically, but also statistically significant. To test for statistical significance, we recall that, given some regularity conditions for the likelihood function (see, for example, Poirier, 1995), the asymptotic distribution of the MLE of θ is A θˆ −−→ N(0, J(θˆ )−1 )

where J(θˆ ) is the matrix of second derivatives of the log likelihood function with respect to the estimated parameters and observed returns – for example J(θˆ ) = −

T ∂2 log (rt ) t=1

∂θˆ i ∂θˆ i

, i = 1, . . ., k, j = 1, . . ., k

(·) is the normal distribution, and k is the number of parameters to be estimated. Therefore, to derive confidence intervals we first simulate Q = 200 parameter sets θˆ q from N(θˆ , J(θˆ )−1 ). Then the economic costs are calculated for each set of parameters θˆ q . Figure 2.1 plots the confidence intervals of the required compensation when investing in the one-regime international strategy rather than the three-regime strategy with unhedged indices. The lower bound of the 95 percent confidence interval lies above zero. Therefore, the null hypothesis, that ignorance of regime-switching does not cause any utility losses, is

31

MARKUS LEIPPOLD AND FELIX MORGER

Required compensation

0.07 0.06 0.05

Estimate Median 67% Cl 95% Cl

0.04 0.03 0.02 0.01 0

6

12 Investment horizon

18

24

Figure 2.1 Confidence intervals of economic cost estimates Notes: The figure shows the bootstrapped confidence intervals of the compensation required for ignoring the unhedged regime-switching strategy with the parameters estimated as per 2005/6. The economic costs are calculated for the indirect utilities of the unconditional state probabilities. In the bootstrap procedure, 200 parameter sets are drawn from N(θˆ ,J(θˆ )−1 ).

clearly rejected. For a one-year horizon, the required compensation can be as high as 3.5 percent with a median compensation of more than 1 percent. With long horizons, the non-realized wealth related to ignorance of regimeswitching is substantial.

2.4.2 Strategies in competition: horse race Most often, a model performs very well in sample, but fails out of sample. In this section, we present an out-of-sample test (horse race) for our regimeswitching strategies. This horse race spans the period from January 1990 to June 2005, or 186 real data returns. At the beginning of each month, we calculate the optimal asset allocation for the different strategies, and at the end of each month, we calculate the strategies’ performance in terms of cumulated wealth. To calculate the optimal portfolio allocation at the beginning of each month, the model parameters and beliefs are estimated with the data available to the investor. So, for example, the optimal weights for January 1990 are calculated based on the excess returns from January 1971 to December 1989. The horse race is therefore a truly out-of-sample comparison of different strategies.2

32

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

4

3.5

International, 1 reg UH, 3 reg FH, 2 reg OH, 3 reg

Cumulative wealth

3

2.5

2

1.5

1 1990

1995

2000

2005

Investment date

Figure 2.2 Cumulative wealth of the strategies Notes: The figure plots the cumulative wealth during the horse race of the regime-switching strategies and a benchmark strategy. The horse race is out-of-sample and spans the period from 1990/1 to 2005/6. The optimal asset allocations are calculated for an investor with a one-month investment horizon. UH is short for unhedged, FH for fully hedged, and OH for optimally hedged. Cumulative wealth and Sharpe ratio

Figure 2.2 plots the evolution of cumulative wealth for the four strategies under consideration. International strategy with a single regime serves as benchmark for the regime-switching strategies. With the exception of the first two years, the cumulative wealth of all regime-switching strategies is always above the benchmark. By the end of June 2005, the optimal hedging strategy performs best, outperforming the single-regime benchmark strategy by 37 percent. We also observe that the optimal hedging strategy recovers fast from the Russian crises in autumn 1998, suffers during the bursting of the Internet bubble, but recovers very well with strong performance during 2004/5. In comparison, the unhedged regime-switching strategy outperforms the benchmark by only 18 percent. Towards the end of the bursting of the

MARKUS LEIPPOLD AND FELIX MORGER

33

Internet bubble, the investor allocates most of the capital into the risk-free asset, but misses the beginning of the recovery period. The strategy with the fully hedged currency risk outperforms the benchmark by only 8 percent. Up to January 2000, its performance is similar to the unhedged regime-switching strategy. Relative to the unhedged strategy, which bears all the currency risk, it outperforms in the period from January 2000 to the early 2002, since the value of the dollar relative to the euro and pound increased. On the other hand, it underperforms towards the end of the horse race because it cannot take advantage of the depreciation of the dollar from early 2002 to the end of 2004. In 2005, it again outperforms because of a rise in the dollar. Table 2.8 presents the descriptive statistics of the horse race for the whole period and for two subperiods, split at the peak of the Internet bubble in January 2000. In comparison to Figure 2.2, we include additional strategies. First, we include a purely domestic single-regime strategy. Second, for comparability with Ang and Bekaert (2004), we introduce a naïve investor, who uses a regime-switching model with three possible states, but does not follow a Bayesian updating rule. Instead, he or she treats the one state with the largest probability of being active as the active state, with a probability of one. Panel A and B report the summary statistics (annualized) for the different strategies and for market indices, respectively. The numbers for the subperiods confirm that the high total performance of the US market is because of its outperformance in the 1990s. The standard deviations of the returns are low for the single-regime strategies, high for the market indices, and intermediate for regime-switching. The single regime strategies exhibit large Sharpe ratios. The largest Sharpe ratio is generated by US domestic strategy, which can be attributed to the outperformance of the US market. The Sharpe ratios of the regime-switching strategies are lower than those for the singleregime strategies. This is because of the low variance of the single-regime strategies. Finally, the regime-switching strategy with three regimes and a naïve investor has a lower mean return and higher return volatility than the Bayesian investor with the unhedged regime-switching strategy. Clearly, it pays to go Bayes. These conclusions for the whole period still hold when looking at the two subperiods (Table 2.8). We end this section with a word of caution. We have to be careful when ranking the different strategies using the Sharpe ratio. The Sharpe ratio would be the optimal measure when investors are only concerned about mean and volatility (or if asset returns are normally distributed). However, since we assume a CRRA utility, the investor also has concerns about higher moments of the portfolio strategy. Therefore, for the ranking of the different strategies, a comparison based on the economic costs as in Section 2.4.1 would be appropriate.

34

Table 2.8 Descriptive statistics on the horse race 1990.1–2005.6 Mean

Standard deviation

1990.1–1999.12

2000.1–2005.6

Sharpe ratio

Mean

Standard deviation

Sharpe ratio

Mean

Standard deviation

Sharpe ratio

Panel A: Strategies Domestic, 1 reg.

6.65

6.22

1.07

10.70

5.56

1.92

−0.35

6.90

−0.05

International UH, 1 reg.

6.16

8.45

0.73

10.46

8.03

1.30

−1.22

8.86

−0.14

UH, “naïve”, 3 reg.

6.60

11.84

0.56

13.08

12.79

1.02

−4.25

9.12

−0.47

UH, 3 reg.

7.28

11.17

0.65

13.15

12.05

1.09

−2.62

8.76

−0.30

FH, 2 reg.

6.77

11.24

0.60

13.52

11.10

1.22

−4.48

10.85

−0.41

OH, 3 reg.

8.33

11.79

0.71

14.05

10.60

1.33

−1.35

13.38

−0.10

Panel B: Markets GER, FH

6.44

22.22

0.29

14.48

20.02

0.72

−6.75

25.53

−0.26

GER, UH

6.76

21.84

0.31

12.85

19.18

0.67

−3.50

25.95

−0.13

UK, FH

8.42

14.46

0.58

14.25

14.34

0.99

−1.44

14.39

−0.10

UK, UH

9.16

15.27

0.60

14.24

15.37

0.93

0.49

14.91

0.03

10.69

14.55

0.73

19.01

13.40

1.42

−2.98

15.87

−0.19

US

Notes: We report the descriptive statistics of the horse race for the whole period and two subperiods. The descriptive statistics consist of the returns’ mean and standard deviation, and the Sharpe ratio (SR). The mean and standard deviation are annualized. Panel A gives the descriptive statistics of the strategies, and Panel B those of the market indices. FH denotes fully hedged and UH unhedged.

35

MARKUS LEIPPOLD AND FELIX MORGER

GER, FH

1 Weight

Weight

1

0.5

0 1990

1995

2000

0.5

0 1990

2005

GER, UH

Weight

Weight

2005

1995

2000

0.5

0 1990

2005

US

1995

2000

2005

Risk free asset

1

1 Weight

Weight

2000

1

0.5

0.5

0 1990

1995

UK, UH

1

0 1990

UK, FH

1995

2000

Investment date

2005

0.5

0 1990

1995

2000

2005

Investment date

Figure 2.3 Asset allocation of the optimal hedging strategy Notes: The figure plots the asset allocation of the optimal hedging strategy. The weights are derived for an investor with a one month horizon. The investor has a risk aversion of γ = 5. Optimal weights

It is instructive to take a look at the behavior of the optimal portfolio strategy through time. Figure 2.3 plots the asset allocation of the optimal hedging strategy. We observe that the weights have sharp peaks and temporarily drop to zero. These sudden moves are mainly because of changes in the state probability estimates. As we can see in Figure 2.4, beliefs about the active state change frequently and provoke the large variation in portfolio weights. Looking at the unhedged and hedged indices of one specific country, the optimization usually puts a lot of weight into the index with the higher historical sample mean and ignores the other index. This result is most probably because of the high correlation between the hedged and unhedged indices. Taking the example of Germany, it is better to invest in the unhedged index, which offers the higher expected drift, and to forgo the additional, but

36

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

pr(St 1) Probability

1

0.5

0 1990

1995

2005

pr(St 2)

1 Probability

2000

0.5

0 1990

1995

2000

2005

2000

2005

pr(St 3) Probability

1

0.5

0 1990

1995 Investment date

Figure 2.4 Changing beliefs during the horse-race Notes: The figure displays the state probabilities of the unhedged strategy. The plots give the probability that a state is active at a certain investment date. The state beliefs are estimated together with the parameter estimates by the EM algorithm. small diversification offered by the hedged index. However, the exception to this rule is investment into the unhedged UK index starting from July 2002. Here, the optimization picks up the depreciation of the dollar, which makes the pound returns more valuable for the US investor. To take advantage of the depreciation, it reduces its holdings in the UK local return index to zero and shifts into the unhedged UK index. The impressive performance at the end of the horse race is the reward.

Optimal hedge ratio

Figure 2.5 plots the optimal hedge ratios for the two foreign markets. The figure confirms the observations made above. The optimal currency hedge

MARKUS LEIPPOLD AND FELIX MORGER

37

GER 1 Hedge ratio

0.8 0.6 0.4 0.2 0 1990

1995

2000

2005

2000

2005

UK

1 Hedge ratio

0.8 0.6 0.4 0.2 0 1990

1995 Investment date

Figure 2.5 Optimal hedge ratios, Germany and UK, during the horse-race Notes: The figure plots the optimal hedge ratio of the foreign markets, Germany and UK, during the horse race. The hedge ratio is calculated as the allocation to the local currency index divided by the total investment to the foreign market. The weights are derived for an investor with a one month horizon. The investor has a risk aversion of γ = 5. No dot is plotted at investment dates where the hedged and unhedged indexes of the foreign markets do not receive any weight at all. ratio is time-varying, and the optimal hedging policy depends on two factors. The first is the trade-off between the diversification offered by the index with the lower expected drift, and the difference in the drifts of the hedged and unhedged indices. The second factor concerns the ex-ante identification of moves in the exchange rates that motivate a preference for the index with the lower drift. The prerequisite for this are parameter estimates for the regime-switching model that allow the identification of favorable moves in the exchange rates. As we observe in Figure 2.5, our model generates the optimal hedge ratios for the German and UK markets that are generally either zero or one. In contrast, in an equity-only international CAPM with regime-switching betas, Ang and Bekaert (2002) find optimal hedge ratios of 53 percent for the high volatility regime and 40 percent for the low volatility regime.

38

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

International, 1 reg. UH,3 reg. OH,3 reg.

1

Weight

0.8 0.6 0.4 0.2 0 1990

1995

2000

2005

Investment date

Figure 2.6 Optimal foreign investments Notes: The figure plots the foreign investments of two different regime-switching strategies and one single-regime strategy. The foreign investments are calculated as the total wealth allocated to foreign markets. The weights are derived for an investor with a one-month horizon. UH = unhedged; OH = optimal hedging.

2.4.3 Optimal foreign investment Figure 2.6 compares the optimal foreign investments of the regime-switching strategies with the international unhedged strategy with a single regime. The international strategy with a single regime starts with an asset allocation of more than 43 percent to foreign markets and reduces this number steadily to less than 30 percent at the end of the horse race. The reduction is caused by increasing drift estimates for the US market. The average international investment is 34 percent. The unhedged regime-switching strategy invests almost the same fraction into the foreign markets, namely 30 percent on average. However, compared to the benchmark strategy, the foreign investment is very volatile over time, moving between 0 percent and 76 percent. Therefore, the introduction of regimes makes the foreign investments strongly time dependent. With the exception of very few data points, the foreign investment of the optimal hedging strategy lies above the benchmark strategy, moving between 4 percent and 100 percent. The average allocation to foreign markets is 63 percent. This is almost twice the percentage of the foreign holdings induced by the single regime strategy. Hence, the introduction of optimal currency hedging increases the optimal allocation to foreign markets compared to the single regime international strategy and, as a backdrop to these results, the home bias question becomes even more of a puzzle.

MARKUS LEIPPOLD AND FELIX MORGER

39

2.5 CONCLUSION We investigated different regime-switching models for international asset allocation. Taking the perspective of a US investor, we applied regime-switching models to the US, UK and German stock markets. We show that, with a correctly specified model, the economic gains from explicitly modeling regime switches are significant and substantial. The cost of ignoring international regime-switching without currency hedging is 2.8 percent per year for a domestic US investor with a ten-year horizon. Of these costs, 70 percent can be attributed to regime-switching and 30 percent to international diversification. Concerning the hedging behavior, the optimal hedging strategy depends strongly on the conditional expected returns of the hedged and unhedged market indices. Most of the time, currency risk is either completely hedged, if the hedged index has the higher expected return, or not hedged at all. The ultimate test of the quality of our model specifications is the outof-sample horse race, where for different strategies we compare the wealth generated during a period spanning fifteen and a half years, from January 1990 to June 2005. Regime-switching strategies generate higher returns than the international and domestic benchmark strategy with a single regime, but at the cost of higher volatilities. The optimal hedging strategy outperforms the international strategy with a single regime by 37 percent in total and 2 percent per year, respectively. During the horse race, the optimal hedging strategy outperforms the strategy without currency hedging by an annualized 1.1 percent and the strategy with a full hedge by an annualized 1.6 percent. Finally, analyzing the size of foreign investments, we find that the optimal foreign investment under regime-switching is highly time-varying. The average foreign investment for the strategy without currency hedging is 30 percent, which is just below the 34 percent of foreign investments generated by the international strategy with a single regime. However, with optimal currency hedging, the investments in foreign markets increase to an average of 63 percent. Hence, under optimal currency hedging, the home bias becomes even more severe.

NOTES 1. A detailed econometric analysis can be obtained from the authors (email: [email protected]) 2. One could argue that our horse race is only pseudo out-of-sample, because the choice of the number of regimes is based on Table 12.2, which uses the whole sample. In real out-of-sample horse races, the specification test should be done at each investment date. This is certainly true. However, as Table 12.2 shows, the choice of regime would

40

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

be the same for each evaluated investment date. Because of this stability, it is legitimate to do the horse race without a specification test at each investment date. The horse race can be considered as a real out-of-sample test.

ACKNOWLEDGMENTS Part of this work was done when Markus Leippold was visiting professor at the Federal Reserve Bank of New York. The authors acknowledge the financial support of the Swiss National Science Foundation (NCCR FINRISK) and the University Research Priority Program “Finance and Financial Markets” at the University of Zurich.

REFERENCES Ang, A. and Bekaert, G. (2002a) “International Asset Allocation with Regime Shifts”, Review of Financial Studies, 15(4): 1137–87. Ang, A. and Bekaert, G. (2002b) “Regime Switches in Interest Rates”, Journal of Business and Economic Statistics, 20(2): 163–82. Ang, A. and Bekaert, G. (2004) “How Do Regimes Affect Asset Allocation?”, Financial Analysts Journal, 60(2): 86–99. Ang, A. and Chen, J. (2002) “Asymmetric Correlations of Equity Portfolios”, Journal of Financial Economics, 63(3): 443–94. Berkowitz, J. (2001) “Testing Density Forecasts with Applications to Risk Management”, Journal of Business and Economic Statistics, 19(4): 465–74. Black, F. (1989) “Universal Hedging: Optimizing Currency Risk and Reward in International Equity Portfolios”, Financial Analyst Journal, 45(4): 161–7. Brandt, M. W., Goyal, A., Santa-Clara, P. and Storud, J. (2004) “A Simulation Approach to Dynamic Portfolio Choice with an Application to Learning About Return Predictability”, Review of Financial Studies, 17(3): 831–73. Connolly, R., Stivers, C. and Sun, L. (2005) “Stock Market Uncertainty and the Stock–Bond Return Relation”, Journal of Financial and Quantitative Analysis, 40(1): 161–94. Dales, A. and Meese, R. (2001) “Strategic Currency Hedging”, Journal of Asset Management, 2(1): 9–21. Dempster, A. P., Laird, N. M. and Rubin, D. R. (1977) “Maximum Likelihood from Incomplete Data via the EM Algorithm”, Royal Statistical Society B, 39(1): 1–38. Froot, K. (1993) “Currency Hedging over Long Horizons”, NBER Working Paper No. W4355. Garcia, R. and Perron, P. (1996) “An Analysis of the Real Interest Rate under Regime Shifts”, The Review of Economics and Statistics, 78(1): 111–25. Glen, J. and Jorion, Ph. (1993) “Currency Hedging for International Portfolios”, Journal of Finance, 68(5): 1865–86. Gray, S. (1996) “Modeling the Conditional Distribution of Interest Rates as a RegimeSwitching Process”, Journal of Financial Economics, 42(1): 27–62. Guidolin, M. and Timmermann, A. (2004) “Strategic Asset Allocation and Consumption Decisions under Multivariate Regime Switching”, Working Paper, Federal Reserve Bank of St. Louis. Guidolin, M. and Timmermann, A. (2005a) “Economic Implications of Bull and Bear Regimes in UK Stock and Bond Returns”, Economic Journal, 115(500): 111–43.

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Guidolin, M. and Timmermann, A. (2005b) “International Asset Allocation under RegimeSwitching, Skew and Kurtosis Preferences”, Working Paper, Federal Reserve Bank of St. Louis. Guidolin, M. and Timmermann, A. (2006) “An Econometric Model of Nonlinear Dynamics in the Joint Distribution of Stock and Bond Returns”, Journal of Applied Econometrics, forthcoming. Hamilton, J. D. (1989) “A New Approach to the Economic Analysis of Non-Stationary Time Series and the Business Cycle”, Econometrica, 57(2): 357–84. Hamilton, J. D. (1990) “Analysis of Time Series Subject to Changes in Regime”, Journal of Econometrics, 45(1): 39–70. Longstaff, F. A. and Schwartz, E. (2001) “Valuing American Options by Simulation: A Simple Least-Squares Approach”, Review of Financial Studies, 14(1): 113–47. Perez-Quiros, G. and Timmermann, A. (2000) “Firm Size and Cyclical Variations in Stock Returns”, Journal of Finance, 55(3): 1229–62. Perold, A. and Schulman, E. (1988) “The Free Lunch in Currency Hedging: Implications for Investment Policy and Performance Standards”, Financial Analyst Journal, 44(3): 45–50. Poirier, D. (1995) Intermediate Statistics and Econometrics: A Comparative Approach, Cambridge, Mass.: MIT Press. Samuelson, P. (1969) “Lifetime Portfolio Selection by Dynamic Stochastic Programming”, Review of Economics and Statistics, 51(3): 239–46. Turner, C., Startz, R. and Nelson, C. (1989) “A Markov Model of Heteroskedasticity, Risk, and Learning in the Stock Market”, Journal of Financial Economics, 25(1): 3–22. Whitelaw, R. (2000) “Stock Market Risk and Return: An Equilibrium Approach”, Review of Financial Studies, 13(3): 521–47.

CHAPTER 3

The Determinants of Domestic and Foreign Biases: An Empirical Study Fathi Abid and Slah Bahloul

3.1 INTRODUCTION The international capital asset pricing model (ICAPM), based on traditional portfolio theory developed by Sharpe (1964) and Lintner (1965), suggests that, to maximize risk-adjusted returns, investors should hold a world market portfolio of risky assets. However, domestic assets are heavily weighted in investors’ portfolios even after the relaxing of capital control after 1980. For example, in 1997, 89.9 percent of US investors’ equity portfolios were domestic equities, while the size of the USA in world market capitalization was about 48.3 percent (Ahearne et al., 2004). The wide disparity between actual and recommended international equity portfolio weights constitutes the equity home bias, one of the unresolved puzzles in international finance literature.1 Various attempts have been made to explain the home asset bias. First explanations have focused on the institutional factor. The existence of equity home bias may be related to barriers to capital flow (Black, 1974: Stulz, 1981), hedging possibilities against domestic risk (Glassman and Riddick, 1996), and information asymmetries (Ahearne et al., 2004). Dissatisfaction with 42

FATHI ABID AND SLAH BAHLOUL

43

institutional explanations has led some authors to consider explanations based on investor behavior: optimism of investors about their domestic markets (French and Poterba, 1991), unfamiliarity with foreign market (Huberman, 2001), and subjective competence in the home market (Kilka and Weber, 2000). Recent explanations consider the problem of corporate governance and investor protection explains the home asset bias. The presence of controlling shareholders and the lack of investor protection led to low investment rates in foreign markets (Dahlquist et al., 2003). Explanations for the home bias seem not to be explored sufficiently to provide convincing arguments. Most studies on home bias use a singlefactor model, but international investing behavior seems to be determined by many factors (Faruqee et al., 2004). Besides, most of the previous studies are from the perspective of developed countries, in particular US investors, and neglect emergent countries’ points of view. The purpose of this chapter is not to add a new explanation but rather to examine the determinants of home and foreign equity biases for the period 2001–02. We start by considering different groups of factors that might intervene to explain equity home bias. These factors are economic development, capital controls, stock market development, information costs, investor behavior, familiarity, investor protection, and other variables. Following Chan et al. (2005), work on the determinants of home bias using mutual fund investors for the period 1999–2000, this chapter applies similar methodology to study the cross-border behavior of investors from various countries, including both developed and emerging countries, for a recent period (2001–02) and different investment strategies. We use the Coordinated Portfolio Investment Survey (CPIS) dataset from the International Monetary Fund (IMF) that lists the aggregate stockholdings of both individual and institutional investors. The effects of two other causes (information costs and investors’ behavior) on the home bias will be analyzed in addition to those considered by Chan et al. (2005). As did Chan et al. (2005), we distinguish between the domestic and foreign components of the home bias. The domestic bias reflects the extent to which investors overweigh the local market in their holdings, while the foreign bias reflects the extent to which investors underweigh or overweigh foreign markets. We also use, as an additional test, a measure of the home bias defined by Ahearne et al. (2004). Using the two-year data on equity holdings of thirty countries, we find that equity home bias is a feature in both developed and emerging markets. All of the thirty countries show a domestic bias. The fraction of domestic assets held by local investors is much larger than the world-market capitalization weight of the country. However, domestic bias varies greatly across countries. Venezuela, for example, has the highest domestic bias. Investors from the USA, the European bloc and developed countries have the lowest domestic bias.

44

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

Results show that the impact on domestic and foreign biases is asymmetric. Stock market development and information costs affect domestic bias the most, whereas information costs and familiarity variables contribute significantly to explaining foreign bias. Investor behavior, in contrast, has a significant effect on domestic bias but not on foreign bias. Results indicate that factors such as economic development, capital control and investor protection have smaller effects on these biases. The remainder of the chapter is organized as follows. Section 3.2 provides a theoretical framework for the domestic and foreign biases. In section 3.3, we present the descriptive statistics of investors’ holdings, domestic bias and foreign bias. Section 3.4 discusses the different causes of home bias. Section 3.5 presents and interprets the results, and section 3.6 presents a number of additional tests.

3.2 THEORETICAL FRAMEWORK OF DOMESTIC AND FOREIGN BIASES Chan et al. (2005) have used the theoretical framework developed by Cooper and Kaplanis (1986) to analyze domestic and foreign biases. Cooper and Kaplanis’s model assumes that a representative investor in country i acts as an expected return maximizer for a given level of variance Max(wi R − wi ci )

(3.1)

subject to wi Vwi = v wi I = 1 where wi is a column vector, the jth element of which is wij ; wij is the proportion of individual i’s total wealth invested in risky securities of country j; R is a column vector of pre-tax expected returns; ci is a column vector, the jth element of which is cij ; cij is the deadweight cost to investor i of holding securities in country j; v is the given constant variance; V is the variance/covariance matrix of the gross (pre-cost, pre-tax) returns of the risky securities; and I is a unity column vector. The Lagrangean of the above maximization problem is L = (wi R − wi ci ) − (h/2)(wi Vwi − v) − ki (wi I − 1)

(3.2)

where h and ki are Lagrange multipliers. The derivation of objective function with respect to wi equal to zero lead to R − ci − hVwi − ki I = 0

(3.3)

FATHI ABID AND SLAH BAHLOUL

45

Therefore the optimal portfolio for investor i is wi = (V −1 /h)(R − ci − ki I)

(3.4)

where

ki = I V −1 R − I V −1 ci − h /I V −1 I

Given the individual portfolio holdings, the aggregation leads to the world capital market equilibrium. The clearing condition for the model is pi wi = w∗ (3.5) where pi is the proportion of world wealth owned by country i; w∗ is a column vector, the ith element of which is wi∗ ; and wi∗ is the proportion of the world market capitalization in country i’s market. Using Equations (3.4) and (3.5), and defining z as the global minimumvariance portfolio (V −1 I/(I V −1 I)), Cooper and Kaplanis have obtained hV(wi − w∗ ) = pi ci − ci − z pi ci − ci I (3.6) If deadweight costs are zero (cij are equal to zero for all i and j), the righthand side of Equation (3.6) is zero, and each investor holds the world market portfolio. If deadweight cost of any country/investor pair is equal to c, then the portfolio holdings of each investor will deviate from the world market portfolio. To examine the deviation, Chan et al. (2005) have considered the simple case when the covariance matrix, V, is diagonal with all variances equal to s2 . The deviation of the portfolio weight of investor i in country j from the world market portfolio is given by hs2 (wii − wi∗ ) = −cii + bi + ai − d, i = j 2

hs

(wij − wj∗ )

= −cij + bj + ai − d, i = j

(3.7) (3.8)

where a i = z ci bj = pk ckj pi ci d = z ai can be interpreted as the weighted average deadweight cost for investor i, bj as the weighted marginal deadweight cost for investors investing in country j, and d as the world weighted average marginal deadweight cost. Equation (3.7) measures the extent to which domestic asset holdings of

46

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

investor i deviate from those of the world market portfolio, whereas, Equation (3.8) measures the extent to which investors of country i holdings in foreign market j deviate from the world market portfolio. Similar to Chan et al. (2005), we refer to the former as domestic bias (DBIASi ) and the latter as foreign bias (FBIASij ). From Equation (3.7), investor i overweights domestic country (DBIASi > 0) if the deadweight cost for this investor investing in his/her own country i (cii ) is considerably less than the weighted average deadweight cost for world investors (bi ), or if the weighted average deadweight cost he/she faces (ai ) is large enough to discouraged him/her from investing in foreign markets. Equation (3.8) shows that country j asset holdings by investor i depend on the difference between the deadweight costs for investor i investing in country j (cij ) and the weighted average deadweight cost for world investors (bj ). If cij is greater than bj , investor i underweights country j (FBIASij < 0). Then, the more important the deadweight cost for investor i investing in country j, the greater is the foreign bias (a more negative FBIASij ).

3.3 DATA AND PRELIMINARY STATISTICS 3.3.1 Data sources The cross-border equity data is taken from a survey of international portfolio holdings coordinated by the IMF for seventy countries for the end of December in both 2001 and 2002.2 Data on explanatory variables are not available for all countries; we explore data for only thirty investing and forty-three receiving countries. The data of capital market capitalization is from the international federation of stock exchanges (FIBV).

3.3.2 Statistics on investor holdings For each of the thirty investing countries, we calculate the percentage allocation of local investors in forty-three countries as follows: Wij =

MVij 43

(3.9)

MVij

j=1

where Wij is the share of country j in investor holdings of investing country i; and MV ij is the market value of country j s asset holdings by investors of investing country i.

FATHI ABID AND SLAH BAHLOUL

47

The weight of country j in the world market portfolio is defined as the portfolio of the forty-three countries included in the sample MVj∗ Wj∗ = (3.10) 43 ∗ MVj j=1

Wj∗

where is the share of country j in the world market portfolio; and MVj∗ is the market capitalization of country j; We compute Wij and Wj∗ in 2001 and 2002 separately, and then take an average of the two values. Table 3.1 presents the distribution of the average equity allocations (in percentages) of thirty investing countries’ investors in forty-three national markets across the world. The table shows that domestic bias is revealed in all countries in the sample. Across all of the thirty countries, the shares of investors’ holdings in the domestic market are much more important than the world market capitalization weight of the country. Venezuela has the highest percentage of investors’ holdings of domestic equities (99.64 percent) and the lowest share in the world market portfolio (0.02 percent). Austria has the lowest percentage of domestic asset holdings (50.12 percent), though its share in the world market portfolio is 0.12 percent. The same table shows that investors do underweight foreign markets in their asset holdings. Generally, the share of foreign asset holdings is by far smaller than the shares of foreign country in the world market portfolio. Yet, there are some exceptions. For example, the German and Belgian investors hold a proportion of 4.37 percent and 7.17 percent respectively, of French assets, while the share of the French market in the world market portfolio is only 3 percent. This may provide preliminary evidence that geographical proximity plays an important role in determining the extent to which investors’ overweight foreign markets. Table 3.2 presents the average share of domestic assets held by investors from different blocs: European, American, Asia/Pacific and African. It shows that European investors have the smallest percentage of domestic asset holdings among the four blocs. From Table 3.3 it can be seen that investors from developed markets have a lower percentage of domestic asset holdings compared to investors from emerging markets.

3.3.3 Statistics on domestic and foreign biases Chan et al. (2005) have used the theoretical framework developed by Cooper and Kaplanis (1986) to compute domestic and foreign biases. The domestic bias for a specific country j (DBIASj ) refers to the deviation of the proportion of country j’s investors in the local market from its world market capitalization weight. Therefore, DBIASj is defined as the log ratio of the

48

Table 3.1 Equity allocation for thirty countries, 2001 and 2002, percentages Panel A: First 15 countries Country

% WMP Argentina Austria Australia Belgium Brazil Canada Chile

Czech Denmark Finland France Germany Greece Hong Italy Kong Republic

Argentina

0.10

76.67

0.00

0.00

0.00

0.06

0.00

0.00

0.00

0.23

0.00

0.00

0.00

0.00

Australia

1.51

0.00

83.39

0.31

0.07

0.00

0.27

0.00

0.00

0.17

0.04

0.07

0.08

0.00

0.00 0.06 0.12 0.22

Austria

0.12

0.00

0.01

50.12

0.04

0.00

0.02

0.00

0.57

0.13

0.01

0.02

0.13

0.00

0.00 0.08

Belgium

0.74

0.00

0.03

0.80

74.12

0.00

0.04

0.00

1.18

0.26

0.06

0.97

0.21

0.01

0.00 0.14

Brazil

0.61

0.18

0.03

0.05

0.06

99.16

0.09

0.05

0.00

0.13

0.00

0.05

0.02

0.00

0.00 0.23

Canada

2.53

0.00

0.18

0.32

0.09

0.00

75.16

0.01

0.01

0.21

0.06

0.14

0.08

0.00

0.23 0.12

Chile

0.21

0.02

0.00

0.00

0.00

0.00

0.00

97.29

0.00

0.01

0.00

0.00

0.00

0.00

0.00 0.01

China P.R.

2.00

0.00

0.01

0.04

0.01

0.00

0.01

0.00

0.00

0.31

0.00

0.03

0.01

0.00

1.15 0.02

Czech Republic

0.05

0.00

0.00

0.17

0.03

0.00

0.00

0.00

94.53

0.03

0.00

0.00

0.01

0.00

0.00 0.01

Denmark

0.32

0.00

0.05

0.09

0.09

0.00

0.08

0.00

0.00

62.55

0.41

0.06

0.06

0.00

0.00 0.04

Egypt

0.10

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.01

0.00

0.00

0.00

0.00

0.00 0.00

Finland

0.65

0.09

0.08

0.92

0.63

0.00

0.20

0.00

0.06

1.01

79.46

0.66

1.24

0.02

0.01 0.48

France

3.00

0.05

0.65

3.40

7.17

0.03

1.03

0.01

0.39

2.49

1.65

73.87

4.37

0.11

0.09 2.65

Germany

3.44

0.07

0.42

11.58

1.76

0.02

0.59

0.04

0.75

1.54

1.00

3.01

73.56

0.09

0.06 2.19

Greece

0.30

0.00

0.01

0.08

0.03

0.00

0.01

0.00

0.00

0.03

0.01

0.01

0.02

98.71

0.00 0.03

Hong Kong

1.93

0.00

0.19

0.14

0.12

0.00

0.36

0.00

0.00

0.19

0.14

0.12

0.08

0.00

90.13

0.16

Hungary

1.02

0.00

0.00

0.81

0.01

0.00

0.00

0.00

0.11

0.06

0.00

0.01

0.01

0.01

0.00

0.02

India

0.47

0.00

0.02

0.01

0.01

0.00

0.02

0.00

0.00

0.06

0.00

0.01

0.02

0.00

0.00

0.04

Indonesia

0.11

0.00

0.00

0.02

0.01

0.00

0.01

0.07

0.00

0.01

0.00

0.01

0.00

0.00

0.00

0.03

Italy

1.99

0.15

0.20

0.86

0.96

0.01

0.35

0.00

0.16

1.04

0.43

1.67

1.36

0.02

0.01 77.53

Japan

8.61

0.00

1.06

1.81

0.68

0.00

1.68

0.00

0.00

1.93

0.72

1.08

0.76

0.01

0.44

1.63

Korea. Republic

0.82

0.00

0.05

0.15

0.05

0.00

0.25

0.00

0.00

0.49

0.04

0.09

0.11

0.01

0.31

0.21 0.03

Malaysia

0.27

0.00

0.01

0.03

0.01

0.00

0.01

0.00

0.00

0.07

0.00

0.02

0.01

0.00

0.14

Mexico

0.46

0.06

0.02

0.07

0.02

0.00

0.18

0.04

0.00

0.23

0.00

0.05

0.05

0.00

0.00

0.06

Netherlands

2.77

0.11

0.47

3.38

3.61

0.04

0.68

0.14

0.54

1.71

1.43

3.66

3.12

0.05

0.04

1.97

New Zealand

0.08

0.00

0.01

0.01

0.00

0.00

0.03

0.00

0.00

0.02

0.00

0.00

0.00

0.00

0.00

0.07

Norway

0.27

0.00

0.01

0.10

0.07

0.00

0.05

0.00

0.00

0.33

0.23

0.05

0.04

0.00

0.00

0.01

Pakistan

0.03

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.01

0.00

0.00

0.00

0.00

0.00

0.00

Philippines

0.08

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.01

0.00

0.00

0.00

0.00

0.01

0.00

Continued

49

50

Table 3.1 Continued Panel B: Second 15 countries Country

% WMP Argentina Austria Australia Belgium Brazil Canada Chile

Czech Denmark Finland France Germany Greece Hong Italy Kong Republic

Poland

0.11

0.00

0.00

0.20

0.02

0.00

0.00

0.00

0.05

0.01

0.00

0.00

0.01

0.00

0.00

0.02

Portugal

0.18

0.00

0.01

0.10

0.09

0.07

0.03

0.00

0.01

0.05

0.02

0.07

0.07

0.01

0.00

0.15

Russian Federation

0.41

0.00

0.01

0.24

0.03

0.00

0.02

0.00

0.04

0.10

0.09

0.02

0.08

0.01

0.00

0.05

Singapore

0.43

0.00

0.06

0.15

0.03

0.00

0.15

0.00

0.00

0.09

0.00

0.04

0.04

0.00

0.28

0.07

South Africa

0.66

0.00

0.05

0.06

0.05

0.00

0.06

0.00

0.00

0.06

0.00

0.04

0.04

0.00

0.00

0.04

Spain

1.85

1.08

0.14

0.84

1.11

0.20

0.34

0.00

0.04

0.73

0.29

1.93

1.23

0.02

0.01

0.74

Sweden

0.82

0.01

0.09

0.36

0.21

0.00

0.19

0.08

0.02

3.32

5.28

0.19

0.29

0.00

0.01

0.16

Switzerland

2.29

0.00

0.37

3.03

1.11

0.02

0.64

0.01

0.10

2.19

1.01

1.62

2.20

0.05

0.02

1.41

Taiwan

1.10

0.00

0.04

0.06

0.04

0.00

0.07

0.00

0.00

0.17

0.00

0.05

0.03

0.00

0.30

0.08

Thailand

0.16

0.00

0.01

0.06

0.06

0.00

0.02

0.00

0.00

0.07

0.00

0.02

0.03

0.00

0.15

0.03

Turkey

0.16

0.00

0.00

0.03

0.00

0.00

0.01

0.00

0.00

0.01

0.00

0.01

0.01

0.01

0.00

0.05

United Kingdom

7.79

0.17

1.57

5.61

2.96

0.06

2.74

0.04

0.30

5.98

3.20

3.95

4.25

0.37

4.73

2.80

49.44

21.32

10.74

13.99

4.61

0.30

14.58

2.20

1.11

11.96

4.43

6.39

6.35

0.46

1.74

6.36

0.02

0.00

0.00

0.00

0.00

0.01

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

United States Venezuela

Panel A: First 15 countries continued Country

% Japan Korea Malaysia Netherlands New Norway Portugal Singapore South Spain Sweden Switzerland UK WMP Zealand Africa

USA Venezuela

Argentina

0.10

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.04

0.00

0.02

0.02 0.00

0.00

Australia

1.51

0.16

0.00

0.05

0.34

5.28

0.84

0.00

0.77

0.06

0.00

0.25

0.13

0.76 0.28

0.00

Austria

0.12

0.00

0.00

0.00

0.06

0.00

0.09

0.09

0.00

0.00

0.00

0.06

0.13

0.03 0.01

0.00

Belgium

0.74

0.03

0.00

0.00

0.60

0.00

0.62

0.84

0.01

0.00

0.14

0.08

0.14

0.13 0.07

0.00

Brazil

0.61

0.01

0.00

0.00

0.06

0.00

0.05

0.26

0.00

0.00

0.10

0.03

0.04

0.15 0.14

0.00

Canada

2.53

0.14

0.01

0.01

0.27

0.40

0.43

0.02

0.15

0.02

0.01

0.22

0.37

0.21 0.59

0.00

Chile

0.21

0.00

0.00

0.00

0.05

0.00

0.00

0.00

0.00

0.00

0.06

0.00

0.00

0.02 0.01

0.00

China. P.R. 2.00

0.04

0.01

0.01

0.02

0.00

0.27

0.00

0.55

0.00

0.04

0.04

0.02

0.09 0.02

0.00

Czech Republic

0.05

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.01

0.01

0.01 0.00

0.00

Denmark

0.32

0.02

0.00

0.00

0.13

0.03

1.38

0.00

0.02

0.02

0.01

0.40

0.06

0.14 0.05

0.00 0.00

Egypt

0.10

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00 0.00

Finland

0.65

0.09

0.00

0.00

0.54

0.06

1.05

0.20

0.07

0.02

0.28

1.92

0.47

0.49 0.34

0.00

France

3.00

0.45

0.00

0.01

2.49

0.23

2.86

1.23

0.67

0.15

1.61

2.12

1.90

3.30 0.76

0.00

Germany

3.44

0.28

0.01

0.00

1.93

0.51

1.93

0.91

0.31

0.04

1.27

1.81

3.89

1.74 0.41

0.01 Continued

51

52

Table 3.1 Continued Panel B: Second 15 countries continued Country

% Japan Korea Malaysia Netherlands New Norway Portugal Singapore South Spain Sweden Switzerland UK USA Venezuela WMP Zealand Africa

Greece

0.30

0.01

0.00

0.00

0.04

0.00

0.08

0.00

0.00

0.01

0.01

0.02

0.03

0.05 0.02

0.00

Hong Kong

1.93

0.19

0.06

0.07

1.02

0.16

0.24

0.00

2.82

0.01

0.00

0.23

0.15

0.71 0.20

0.00

Hungary

1.02

0.00

0.00

0.00

0.01

0.00

0.01

0.00

0.00

0.00

0.00

0.03

0.01

0.03 0.01

0.00

India

0.47

0.00

0.00

0.01

0.12

0.00

0.00

0.03

0.25

0.00

0.05

0.01

0.01

0.10 0.06

0.00

Indonesia

0.11

0.00

0.01

0.02

0.01

0.00

0.00

0.00

0.86

0.00

0.00

0.00

0.01

0.04 0.02

0.00

Italy

1.99

0.16

0.00

0.00

1.00

0.08

0.85

0.76

0.06

0.03

0.83

0.57

0.57

0.91 0.24

0.00

Japan

8.61 90.51

0.10

0.01

1.43

2.23

4.05

0.14

1.56

0.14

0.50

1.97

1.14

2.87 1.28

0.00

Korea. Republic

0.82

0.02 99.31

0.01

0.16

0.09

0.36

0.00

1.07

0.00

0.00

0.17

0.14

0.46 0.24

0.00

Malaysia

0.27

0.02

0.05

99.23

0.05

0.00

0.01

0.00

4.40

0.00

0.00

0.04

0.02

0.09 0.02

0.00

Mexico

0.46

0.00

0.00

0.00

0.06

0.00

0.05

0.01

0.01

0.00

0.03

0.04

0.24

0.23 0.18

0.00

Netherlands 2.77

0.25

0.01

0.00

67.97

0.19

1.81

0.68

0.16

0.04

0.75

1.01

2.43

1.73 0.77

0.00

New Zealand

0.01

0.00

0.00

0.00

68.37

0.02

0.00

0.07

0.00

0.00

0.01

0.00

0.03 0.02

0.00

0.08

Norway

0.27

0.01

0.00

0.00

0.06

0.04

54.34

0.02

0.02

0.00

0.00

0.37

0.13

0.14 0.06

0.00

Pakistan

0.03

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00 0.00

0.00

Philippines

0.08

0.01

0.00

0.04

0.00

0.00

0.00

0.00

0.34

0.00

0.00

0.01

0.00

0.01 0.01

0.00

Poland

0.11

0.00

0.00

0.00

0.01

0.00

0.21

0.05

0.00

0.00

0.00

0.06

0.01

0.02 0.01

0.00

Portugal

0.18

0.01

0.00

0.00

0.06

0.00

0.11

87.49

0.01

0.00

0.26

0.03

0.03

0.13

0.02

0.00

Russian Federation

0.41

0.00

0.00

0.00

0.02

0.00

0.03

0.00

0.00

0.00

0.00

0.29

0.09

0.09

0.04

0.00

Singapore

0.43

0.04

0.00

0.38

0.11

0.07

0.20

0.00

74.08

0.01

0.00

0.09

0.04

0.29

0.17

0.00

South Africa

0.66

0.00

0.00

0.01

0.05

0.00

0.02

0.44

0.00

86.32

0.01

0.04

0.06

0.12

0.06

0.00

Spain

1.85

0.14

0.00

0.00

1.10

0.10

0.81

2.56

0.04

0.03

89.85

0.38

0.41

0.84

0.23

0.00

Sweden

0.82

0.07

0.00

0.00

0.34

0.13

2.95

0.06

0.06

0.04

0.05

64.65

0.36

0.66

0.13

0.01

Switzerland

2.29

0.32

0.00

0.00

1.60

0.19

1.81

0.27

0.26

0.10

0.34

1.83

75.75

1.55

0.58

0.00

Taiwan

1.10

0.02

0.00

0.01

0.09

0.00

0.28

0.00

1.03

0.00

0.00

0.12

0.08

0.35

0.14

0.00

Thailand

0.16

0.01

0.01

0.02

0.02

0.00

0.00

0.00

1.45

0.00

0.00

0.01

0.01

0.09

0.02

0.00

Turkey

0.16

0.00

0.00

0.00

0.01

0.00

0.01

0.00

0.00

0.00

0.02

0.00

0.04

0.11

0.01

0.00

United Kingdom

7.79

1.35

0.04

0.03

4.67

4.58

8.27

1.57

2.92

10.75

2.08

6.40

2.45

73.88

2.56

0.00

49.44

5.59

0.36

0.09

13.47

17.27

13.93

2.36

5.96

2.20

1.66

14.68

8.60

7.41

90.18

0.25

0.02

0.00

0.00

0.00

0.00

0.00

0.01

0.00

0.00

0.00

0.00

0.00

0.00

0.00

United States Venezuela

0.00 99.64

Notes: This table contains the distribution of 30 investing countries’ average investors’ allocations across 43 national markets for 2001 and 2002. The second column contains a country’s average stock market capitalization weight in the world market portfolio. Panel A – WMP: world market portfolio – Argentina, Australia, Austria, Belgium, Brazil, Canada, Chile, Czech Republic, Denmark, Finland, France, Germany, Greece, Hong Kong and Italy. Panel B – WMP: world market portfolio – Japan, Korea, Malaysia, Netherlands, New Zealand, Norway, Portugal Singapore, South Africa, Spain, Sweden, Switzerland, UK, USA, Venezuela.

53

54

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

Table 3.2 Average shares of domestic asset holdings by investors from different blocs American bloc

Asia/pacific bloc

African bloc

European bloc

89.68%

86.43%

86.32%

74.89%

Notes: American bloc: Argentina, Brazil, Canada, Chile, USA and Venezuela; Asia/Pacific: Australia, Korea, Hong Kong, Japan, Malaysia, New Zealand and Singapore; European bloc: Germany, Austria, Belgium, Denmark, Spain, Finland, France, Greece, Italy, Norway, Netherlands, Portugal, Czech Republic, UK, Sweden and Switzerland; African bloc: South Africa only.

Table 3.3 Shares of domestic asset holdings for developed and emerging countries Developed countries

Emerging countries

77.17%

93.24%

Notes: Developed countries: Germany, Australia, Austria, Belgium, Canada, Korea, Denmark, Spain, USA, Finland, France, Greece, Hong Kong, Italy, Japan, New Zealand, Norway, Netherlands, Portugal, UK, Singapore, Sweden, and Switzerland; Emerging countries: Argentina, Venezuela, Brazil, Chile, South Africa, Malaysia and Czech Republic.3

share of country j’s investors’ holdings in the domestic market (Wjj ) to the world market capitalization weight of country j (Wj∗ ): log

Wjj Wj∗

(3.11)

The foreign bias refers to the extent to which domestic investors underweight or overweight foreign markets in their asset holdings. The foreign bias (FBIASij ) is defined as log

Wij Wj∗

(3.12)

The average foreign bias of foreign investors in an investing country j is calculated by averaging FBIASj across all remaining countries. Table 3.4 exhibits the distribution of the domestic bias for domestic investors and the average foreign bias of foreign investors in an investing country, across thirty investing countries. Average values are computed for the sample periods 2001 and 2002. In general, we observe a significant cross-section variation in domestic bias and average foreign bias measures. The domestic bias fluctuates between 0.6 (USA) and 8.5 (Venezuela). The average foreign bias varies

FATHI ABID AND SLAH BAHLOUL

55

Table 3.4 Domestic bias of domestic investors and average foreign bias of foreign investors Country

Domestic bias

Average foreign bias

Argentina

6.71

−4.79

Australia

4.02

−3.82

Austria

6.07

−3.40

Belgium

4.61

−3.62

Brazil

5.09

−4.07

Canada

3.39

−4.05

Chile

6.15

−5.28

Czech Republic

7.59

−3.89

Denmark

5.27

−3.46

Finland

4.81

−2.24

France

3.21

−2.09

Germany

3.07

−2.44

Greece

5.80

−4.43

Hong Kong

3.85

−4.56

Italy

3.66

−3.05

Japan

2.35

−3.64

Korea

4.80

−3.99

Malaysia

6.14

−3.79

New Zealand

6.77

−4.19

Netherlands

3.20

−2.56

Norway

5.32

−3.58

Portugal

6.16

−2.95

Singapore

5.16

−3.82

South Africa

4.89

−4.71

Spain

3.88

−2.79

Sweden

4.37

−2.64

Switzerland

3.51

−2.69

UK

2.25

−1.97

USA

0.60

−2.64

Venezuela

8.50

−4.22

from −1.97 (UK) to −5.28 (Chile). The values of these measures are important relatively to those of Chan et al. (2005). We use aggregate cross-section data for institutional and individual investors. Chan et al. (2005) have used the data for mutual funds that are likely to invest more in foreign markets, so it seems acceptable to reach the end with different results.

56

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

Table 3.5 Domestic bias and average foreign bias Domestic bias

Average foreign bias

American bloc

5.07

−4.18

Asia/Pacific bloc

4.73

−3.97

European bloc

4.55

−2.99

African bloc

4.89

−4.71

Developed countries

4.21

−3.25

Developing countries

6.44

−4.39

The domestic and average foreign biases are calculated for investors from developed and developing countries as well as for investors from the American, European, Asia/Pacific and African blocs separately. Results are summarized in Table 3.5. The table shows that domestic bias and average foreign bias are less important for investors from the European bloc than for those from other regional blocs. Results corroborate once again the fact that European investors invest less in domestic assets. Developed countries also have a less important domestic bias and an average foreign bias compared to developing countries.

3.4 THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES There may be a large number of causes to explain home bias. We select several explanatory variables and regroup them into economic development, capital control, stock market development, information cost, familiarity, investor behavior, investor protection, and others. We calculate descriptive statistics of these variables for the years 2001 and 2002 separately. We find that an important number of these variables remain stable during this period. Consequently, we shall consider the descriptive statistics only for the year 2001, as shown in Table 3.6.

3.4.1 Economic development Chan et al. (2005) have found that the level of economic development of a country has a significant effect on the investment decisions of foreign investors. To study the impact of economic development on the home bias, we set a number of measures of economic development. We distinguish between gross domestic product (GDP) per capita in US dollars (GDPC); the real growth rate of gross domestic product (RGDP); the average of exports

Table 3.6 Summary statistics for the explanatory variables, 2001 Panel A: The first set of variables Country

Economic development GDP per capita US($)

Real GDP growth (%)

Trade volume (% of GDP)

Financial market development Foreign direct investment (% of GDP)

Transaction costs (basis points)

Stock market capitalization (% of GDP)

Capital control

Emerging market dummy

Capital flow Restrictions

Argentina

7,430

−4

17

0.01

70.8

0.12

1

5.8

Australia

18,995

4

35

0.01

51.0

1.02

0

6.1

Austria

23,603

1

77

0.03

45.6

0.13

0

8.1

Belgium

22,120

1

172

0.3

27.5

0.81

0

9.2

2,949

1

23

0.04

58.6

0.37

1

4.2

22,343

1

70

0.04

36.4

1.01

0

8.6

4,314

3

55

0.07

114.3

0.85

1

7.0

924

8

43

0.04

n.a.

0.45

1

2.7

5,593

3

122

0.10

72.9

0.16

1

7.0

29,713

1

61

0.06

41.3

0.53

0

9.0

1,511

4

17

0.01

n.a.

0.25

1

7.3

Finland

23,422

1

62

0.03

42.3

1.57

0

8.1

France

22,308

2

49

0.04

28.2

0.69

0

7.6

Germany

22,511

1

57

0.01

27.3

0.58

0

9.5

Greece

11,062

4

33

0.01

74.4

0.72

0

8.3

Hong Kong

24,213

0

241

0.15

53.2

3.11

0

9.6

5,088

4

124

0.05

103.8

0.20

1

8.8

India

463

5

19

0.01

44.4

0.23

1

2.0

Indonesia

676

3

62

−0.02

83.7

0.16

1

4.8

Brazil Canada Chile China Czech Republic Denmark Egypt

Hungary

57 Continued

Table 3.6 Continued Country

Italy Japan Korea, Republic of Malaysia Mexico Netherlands New Zealand Norway Pakistan Philippines Poland Portugal Russian Fed Singapore South Africa Spain Sweden Switzerland Taiwan Thailand Turkey UK USA Venezuela

58

Panel A: The first set of variables continued Economic development

Financial market development

Capital control

GDP per capita US($)

Real GDP growth (%)

Trade volume (% of GDP)

Foreign direct investment (% of GDP)

Transaction costs (basis points)

Stock market capitalization (% of GDP)

Emerging market dummy

Capital flow Restrictions

18,921 32,869 10,180 3,696 6,262 23,944 13,241 37,620 415 920 4,808 10,835 2,118 20,545 2,549 14,315 24,673 33,998 n.a. 1,888 2,119 24,211 35,118 5,123

2 0 4 0 0 1 3 2 3 3 4 2 5 −2 3 3 1 1 n.a. 2 −7 2 0 3

43 18 68 184 53 114 53 54 33 89 47 58 51 280 50 47 63 68 n.a. 110 50 42 19 36

0.01 0.00 0.01 0.01 0.04 0.13 0.04 0.01 0.01 0.01 0.03 0.05 0.01 0.13 0.06 0.05 0.06 0.04 n.a. 0.03 0.02 0.04 0.02 0.03

40.5 24.4 73.4 90.9 65.9 27.7 39.0 30.0 n.a. 113.2 n.a. 44.5 n.a. 74.0 88.5 39.4 29.3 41.5 59.7 87.4 45.3 46.6 28.5 102.7

0.48 0.54 0.40 1.35 0.20 1.81 0.35 0.41 0.08 0.29 0.14 0.42 0.25 1.36 1.29 0.80 1.08 2.15 n.a. 0.31 0.32 1.50 1.40 0.05

0 0 0 1 1 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 1 0 0 1

8.7 8.4 n.a. 3.7 5.1 9.5 8.9 7.8 0.8 4.6 3.8 7.6 3.2 7.6 4.1 6.9 7.2 9.1 7.6 4.3 5.8 9.1 8.2 7.8

Panel B: The second set of variables Country

Capital control

Other variables

Investor protection

Intensity of capital control

Rule of law

Accounting

Minority

Expropriation

Argentina

0.05

5.35

45

4

5.91

6

0

Australia

0.00

10

75

4

9.27

10

Austria

0.00

10

54

2

9.69

9.5

Belgium

0.00

10

61

0

9.63

9.5

Brazil

0.05

Canada

0.00

6.32 10

Efficiency

Legal system dummy

Lag 2-year return

Return correlation (average)

2.5

0.017

1

1.1

0.136

0

−12.0

0.004

0

−16.6

−0.028

54

3

7.62

5.75

0

22.1

0.126

74

4

9.67

9.25

1

19.7

0.202

Chile

0.11

7.02

52

3

7.5

7.25

0

13.4

0.056

China

0.59

n.a.

n.a.

n.a.

n.a.

n.a.

n.a.

29.4

−0.037

Czech Republic

0.02

n.a.

n.a.

n.a.

n.a.

n.a.

n.a.

1.3

0.162

Denmark

0.00

10

62

3

9.67

0

12.1

0.090 −0.029

Egypt

0.29

Finland

0.00

France

0.00

Germany Greece

4.17

10

24

2

6.3

0

23.1

77

2

9.67

10

0

n.a.

8.98

69

2

9.65

8

0

0.00

9.23

62

1

9.9

9

0

0.00

6.18

55

1

7.12

7

0

n.a.

n.a.

Hong Kong

0.00

8.22

69

4

8.29

10

1

20.8

0.169

Hungary

0.03

n.a.

n.a.

n.a.

n.a.

n.a.

n.a.

n.a.

India

0.59

4.17

57

2

7.75

8

1

20.0

0.144

Indonesia

0.14

3.98

65

2

7.16

2.5

0

14.0

0.056

10

6.5

n.a.

n.a.

5.9

0.051

1.5

0.145

59

Continued

Table 3.6 Continued Country

Italy Japan Korea, Republic of Malaysia Mexico Netherlands New Zealand Norway Pakistan Philippines Poland Portugal Russian Federation Singapore South Africa Spain Sweden Switzerland Taiwan Thailand Turkey UK USA Venezuela

60

Panel B: The second set of variables continued Capital control

Other variables

Investor protection

Intensity of capital control

Rule of law

Accounting

Minority

Expropriation

Efficiency

Legal system dummy

0.00 0.00 0.03 0.06 0.01 0.00 0.00 0.00 1.00 0.54 0.03 0.00 0.26 0.00 0.00 0.00 0.00 0.00 0.44 0.34 0.01 0.00 0.00 1.00

8.33 8.98 5.35 6.78 5.35 10 10 10 3.03 2.73 n.a. 7.8 n.a. 8.57 4.42 7.8 10 10 8.52 6.25 5.18 8.57 10 6.37

62 65 62 76 60 64 70 74 61 65 n.a. 36 n.a. 78 70 64 83 68 65 64 51 78 71 40

0 3 2 3 0 2 4 3 4 4 n.a. 2 n.a. 3 4 2 2 1 3 3 2 4 5 1

9.35 9.67 8.31 7.95 6.07 9.98 9.69 9.88 5.62 5.22 n.a. 8.9 n.a. 9.3 6.88 9.52 9.4 9.98 9.12 7.42 7 9.71 9.98 6.89

6.75 10 6 9 6 10 10 10 5 4.75 n.a. 5.5 n.a. 10 6 6.25 10 10 6.75 3.25 4 10 10 6.5

0 0 0 1 0 0 1 0 1 0 n.a. 0 n.a. 1 1 0 0 0 0 1 0 1 1 0

Lag 2-year return

−0.2 4.3 19.4 29.2 28.4 −2.9 n.a. −3.1 26.3 −23.9 n.a. −12.7 129.6 18.7 0.0 −13.3 18.5 −0.1 −9.4 −9.8 56.9 −4.2 4.8 43.8

Return correlation (average) 0.140 0.155 0.098 0.061 0.110 0.022 n.a. −0.062 0.084 0.066 n.a. 0.095 −0.003 0.153 n.a. 0.134 0.218 0.071 0.140 0.060 0.118 0.124 0.142 −0.002

Panel C: The third set of variables Country

Information costs

Investor behavior

Phone costs (By minute in $)

Investor behavior toward foreign market returns (basis points)

Distance (kilometers) (average)

Common language dummy (average)

Argentina

0.5743

−12

12,274

0.12

Australia

0.3881

−8

12,726

0.31

Austria

0.3323

−13

5,843

0.05

Belgium

0.3696

−28

5,970

0.12

Brazil

0.5861

41

11,417

0.02

Canada

0.2559

12

8,745

0.38

Chile

0.7472

−12

12,595

0.12

Czech Republic

0.6327

26

n.a.

0.00

Denmark

0.2706

−7

5,879

0.00

Finland

0.4442

N.A.

5,967

0.00

France

0.2997

−19

6,049

0.07

Germany

0.2417

−5

5 916

0.05

Greece

0.4996

N.A.

6 109

0.00

Hong Kong

0.2935

24

8 403

0.36

Italy

0.2991

Japan

0.5163

Korea, Republic of

0.3471

−2

Familiarity

6,044

0.00

12

8,809

0.05

20

8,324

0.05 Continued

61

62

Table 3.6 Continued Panel C: The third set of variables continued Country

Information costs

Investor behavior

Phone costs (by minute in $)

Investor behavior toward foreign market returns (basis points)

Familiarity Distance (kilometers) (average)

Common language dummy (average) 0.31

Malaysia

0.8926

5

8 846

Netherlands

0.3287

−19

5 956

0.05

New Zealand

0.2928

n.a.

13 996

0.31

Norway

0.3576

−25

5 997

0.00

Portugal

0.4850

−11

6 776

0.02

Singapore

0.3567

17

9 013

0.38

South Africa

0.3448

n.a.

9 536

0.31

Spain

0.3565

6 513

0.12

Sweden

0.2849

12

5 931

0.00

Switzerland

0.2556

−13

6 016

0.12

UK

0.3002

−9

6 093

0.31

USA

0.1560

−3

9 217

0.31

Venezuela

0.8008

−24

9 501

0.10

0.2

This table presents, summary statistics for each country, for eight groups of explanatory variables: (i) Economic development variables: GDP per capita, real GDP growth, trade volume (% of GDP) and foreign direct investment; (ii) Financial market development variables: transaction costs, stock market capitalization (% of GDP) and emerging market dummy variable; (iii) Capital control variables: capital flow restrictions and stock-holding restrictions; (iv) Investor protection variables: rule of law index, accounting standard index, minority investor protection index, risk of expropriation index and efficiency of judicial system index; (v) Other variables: past 2-year return and average return correlation; (vi) Information costs: average phone costs by minute; (vii) Investor behavior: average degree of pessimism toward foreign market return; (viii) Familiarity: average distance in kilometers and average common language dummy variable.4

FATHI ABID AND SLAH BAHLOUL

63

and imports scaled by GDP (TRADE); and foreign direct stock investment inward scaled by GDP (DI). All these variables are obtained from the World Development Indicators (WDI). Table 3.6 shows significant cross-sectional variation in the four measures of economic development. The country that has the highest value of GDPC is a developed country (Norway: US$37,620), while the most important value of RGDP is held by an emerging country (China: 8 percent). Belgium has the highest foreign direct stock investment inward relative to its GDP (DI), while the country that has the largest trade volume, as a percentage of GDP (TRADE) is Singapore.

3.4.2 Capital control Ahearne et al. (2004) suggest that, while capital controls have been greatly reduced in many countries, they still affect cross-border investment. In this chapter we measure capital control according to two parameters. The first is the intensity of capital control (RESTRICT) developed in Edison and Warnock (2003) and used by Ahearne et al. (2004). This measure is constructed by using International Finance Corporation (IFC) indexes. It equals one minus the ratio of the market capitalization of a country’s Investable (IFCI) and Global (IFCG) indexes.5 Restrictions vary greatly across developing countries. For industrial countries, the IFC does not publish investable indexes. We assume that, for these countries, Investable and Global indexes are identical. Table 3.6 shows that RESTRICT ranges between zero (in developed countries) to one (in Pakistan). The second variable (RFLOW) measures the restrictions of countries on capital flows. It is constructed by the Economic Freedom Network by assigning lower ratings to countries with more restrictions on foreign capital transactions and was used by Chan et al. (2005). Table 3.6 shows that RFLOW varies from 0.8 in Pakistan to 9.6 in Hong Kong.6 When a country imposes capital controls, this will stop, or at least discourage, foreign investors from holding stocks of companies in that country. When a country imposes higher capital control measures, the degree of foreign bias becomes higher (more negative FBIAS). Also, when a score on RFLOW is low (and the score on RESTRICT is important), domestic investors find it difficult to invest overseas, as it requires government approval. Then they will invest a large amount of their wealth in the domestic market and the domestic bias will consequently be important (more positive DBIAS).

3.4.3 Stock market development Chan et al. (2005) suggest that investors tend in general to invest more in developed stock markets. In fact, these markets present high liquidity and lower transaction costs. We measure the stock market development according to three distinct variables.

64

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

The first variable is the relative size of the stock market of each country, measured by the stock market capitalization as a percentage of the country’s GDP (SIZE). The value of SIZE varies from 0.05 percent in Venezuela to 3.11 percent in Hong Kong. The data on GDP are taken from the World Development Indicators (WDI) and data on market capitalization are from the International Federation of Stock Exchanges (FIBV). The second variable is the transaction costs associated with trading foreign securities (COST). The role of transaction costs in the explanation of the home bias has been neglected because of the existence of an important turnover rate of foreign assets compared to domestic assets (Tesar and Werner, 1995). However, Carmichael and Coen (2003) reveal, by using a simple OLG model of the world economy with transaction costs, that the introduction of very small transaction costs is sufficient to reproduce the large home bias observed in portfolios. We use the Elkins–McSherry Co. measure of transaction costs. The latter consists of three components: commissions, fees and market impact costs. This measure has been used by Ahearne et al. (2004) and Chan et al. (2005). Although the other explanatory variables are for the year 2001, we use the data for the year 1999.7 We assume that these transaction cost estimates do not change substantially, and we can utilize it in our analysis. We have transaction cost data for thirty-eight of the forty-three markets, ranging from 24.4 basis points for Japan to 114.3 for Chile. We also consider a dummy variable (DUMEMERG) that equals one for an emerging market, and zero otherwise. We expect that foreign investors will opt for investing in local markets which are large, developed and require low transaction costs. Foreign bias will be less important in these markets. The proportion of local asset holdings by domestic investors will be smaller and the domestic bias less important.

3.4.4 Information asymmetries Gehrig (1993) argues that one of the explanations for the home bias is that local investors spend too much on information about foreign markets. A high information cost discourages investors from investing abroad. Zhou (1998) shows that, with differential information, agents on average tilt their portfolio towards stocks about which they have better information. Portes and Rey (2005) use the volume of telephone call traffic as a proxy for information costs. To measure information costs, we shall consider the cost of international phone calls per minute from a country i to a country j.8 Table 3.6 shows that average values of information costs vary from US$0.1560 (USA) to US$0.8008 (Venezuela). High information costs between country i and country j make country i’s investors hold fewer assets in country j, so we expect an important FBIASij (more negative FBIAS).

FATHI ABID AND SLAH BAHLOUL

65

3.4.5 Investors’ behavior One explanation of the home bias is that investors in each country expect returns from their domestic equity market to be several hundred basis points higher than returns from foreign markets. French and Poterba (1991) show that investors may be relatively more optimistic about their home markets than are foreign investors. Strong and Xu (2003) find that fund mangers from the USA, UK, continental Europe and Japan show a significant relative optimism towards their equity markets. To measure an investor’s degree of optimism or pessimism towards a market, we use BEHAV, a variable that equals the difference between the market return implied by the actual portfolio holdings and the returns implied by an international value-weighted portfolio for each country. To determine this vector of return, we have used French and Poterba’s (1991) model: µ = λw∗

(3.13)

where µ is the vector of expected return; w∗ is optimal portfolio weights; is the covariance matrix; and λ is relative risk aversion. Average values of BEHAV vary from –28 basis points for Belgian investors to 41 basis points for Brazilian investors. In general, investors are pessimistic about foreign markets, which can help to explain why investors underweight foreign markets and overweight domestic markets.

3.4.6 Familiarity One explanation for the home bias is that investors may not be familiar with foreign markets. Huberman (2001) finds that “Shareholders of a Regional Bell Operating Company (RBOC) tend to live in the area which it serves, and an RBOC’s customers tend to hold its shares rather than other RBOCs’ equity”. Like Chan et al. (2005) and Sarkissan and Schill (2004), we use three proxies of familiarity9 variables for each pair of countries i and j. The first familiarity variable is common language. Data are obtained from the World Factbook 2001. For each pair of countries i (investing) and j, we construct a language dummy variable (DUMLANG) which equals one if i and j share the same language and zero otherwise. The second variable is geographical proximity (DISTANCE). Data are obtained from http://www.ksg.harvard.edu/people/sjwei. Average values of distance vary from 5,843 kilometers for Austria to 13,996 for New Zealand. The last variable is the amount of bilateral trades (TRADEB), with values ranging from 0 to 1. A value of 0.17 for TRADEB between the UK (investing) and the USA means that 17 percent of the total UK trade (imports

66

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

and exports) is with the USA. (Data are obtained from the United Nations Statistics Division Databases.) For the familiarity variables, we expect that investors from country i, who are more familiar with country j, through sharing a common language, being close to each other, or having larger bilateral trade volumes, will display less foreign bias (FBIASij ). Familiarity variables do affect domestic bias. Investors from a country that is isolated from the rest of the world will also hold a large proportion of domestic assets.

3.4.7 Investor protection La Porta et al. (1997) argue that capital markets are narrow in countries with poorer investor protection. Giannetti and Koskinen (2004) show that foreign investors are reluctant to invest in a country where expropriating minority shareholders is easy, while wealthy investors have an incentive to become controlling shareholders by investing a large proportion of their wealth in the stock market in a country with poor investor protection. Similarly to Chan et al. (2005), we use six measures of investor protection based on La Porta et al. (1997, 1998, 2000).10 The first measure is the rule of law index (LAW), elaborated by the International Country Risk Agency. It arranges countries on a scale ranging from zero to 10, with lower scores for countries with less respect for law and order. The index varies from 2.73 in the Philippines to 10 in twelve countries across the world. The second measure is the accounting standard index (ACC). This defines the amount and transparency of information available to investors. Table 3.6 shows that Sweden has the highest score (83) while the lowest is for Egypt (24). The third measure is the anti-director rights (MINORITY). It indicates the degree of protection for minority investors. Values vary from zero for Italy, Belgium and Mexico, to 5 for the USA. The fourth measure is the risk of expropriation index (EXPROP). This index is constructed by the International Country Risk Agency. It has a scale ranging from zero to 10, with lower scores for higher risk. This index varies from 5.22 for the Philippines to 9.98 for the USA, Switzerland and the Netherlands. The fifth measure is the efficiency of the judicial system (EFFICIENCY). This index is constructed by the Business International Corporation. Values range from 2.5 for Indonesia to 10 for fourteen developed countries. The sixth measure is a dummy variable that considers the type of legal system (DUMLEGAL). It equals 1 for common-law countries and zero otherwise. In fact, La Porta et al. (1997) have found that the French civil law countries have the weakest investor protection, particularly when compared to common-law countries.

FATHI ABID AND SLAH BAHLOUL

67

3.4.8 Other variables In addition to the above variables, we include several others that account for the home bias. The first variable is the two-year return (RET2). Bohn and Tesar (1996) find that US investors exhibit returns-chasing behavior, with a tendency to underweight countries whose stock markets have performed poorly. The second variable is the correlation between the returns of two countries (CORR). For each pair of countries, i and j, we compute the correlation coefficient using country returns in US dollars from 1999 to 2001. Data for indexes are from Yahoo Finance, and data for exchange rates are from the Oanda website. The Correlation coefficient is used as a proxy for the diversification potential between two countries.

3.5 THE EMPIRICAL ANALYSIS Following Chan et al.’s (2005) methodology, this section studies the causes of domestic and foreign biases. In all tests, we stack up all the observations on domestic-bias measure (DBIASj ) and regress them against each set of the explanatory variables; then we do the same for the foreign bias measures (FBIASij ). The dependent variables (DBIAS and FBIAS) are average values for the 2001–02 period, while, explanatory variables are for the year 2001.

3.5.1 Results concerning domestic bias The second column of Table 3.7 shows the predicted signs of the coefficients. The other columns contain estimates of explanatory variables for the eight categories separately and for all variables. Table 3.7 shows that stock market development variables have the most important explanatory power, with adjusted R2 of 58 percent, while familiarity variables exhibit a low adjusted R2 of 17 percent. For the market development variables, we find that DBIAS is negatively related to the size of the market (SIZE) and positively linked to transaction costs (COST). Large markets have a higher visibility and attract foreign investors, so domestic investors will invest less in large domestic markets. If transaction costs are very high, foreign investors will tend to invest less in local markets, and investments in domestic market by local investors will then be important. Information costs have a significant positive effect on DBIAS; the coefficient is 6.35 with a t-ratio of 4.67. If information costs in foreign market are very important, local investors opt for local markets, and then the domestic bias will be very important.

68

Table 3.7 Regression analysis of domestic bias Predicted sign

Economic development variables Coeff.

Constant

4.00

GDPC RGDP TRADE DI

− − − −

RFLOW RESTRICT

− +

SIZE COST DUMEMERG

− + +

INFOR

+

BEHAV

+

DUMLANG DIST

− +

LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL

− − − − − −

RET2 CORR

− +

Adjusted R2

−0.001 0.03 0.00 −0.97

Capital control

Stock market development

Information costs

Investor behavior

Familiarity

Investor protection

Other variables

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

8.43

7.10

5.16

−5.34

−1.82

2.10

3.47

3.82

9.03

−21.03

−2.10

15.87

6.30

5.36

10.51

28.71

1.83

0.00 −0.01 0.01 −4.63

−0.05 −0.06 1.12 −0.94

0.01 1.14

0.19 0.51

−0.77 2.40 −2.00

−1.97 2.50 −0.87

4.79

1.61

2.72

1.35

2.79 −3.27

0.52 −2.37

1.55 0.01 0.29 −1.94 −0.35 −1.62

2.37 0.23 1.09 −3.29 −0.80 −1.16

−0.02 1.14

−0.56 0.21

−4.15 0.16 0.71 −0.19 −0.34 4.17

−1.97 2.77 −1.20 2.94 −0.70

−3.62 3.55 −0.85 6.35

4.67 4.19

2.61 −5.14 2.94

−2.17 2.60 0.57 −0.03 −0.32 −1.48 −0.01 0.36

1.21 −0.89 −1.19 −2.74 −0.03 0.42 0.05 −11.55

0.35

All variables

0.26

0.58

0.42

0.19

0.17

0.38

0.23

2.17 −2.65

t-stat

0.86

Notes: DBIAS j : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flow restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: average common language dummy variables; DIST : average of log geographical distances; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET2: past 2-year return; CORR: average of return correlations. * Bold numbers, indicate t-stat is significant.

FATHI ABID AND SLAH BAHLOUL

69

Concerning the investor-protection variables, only the index of risk expropriation is statistically significant at the 5 percent level with a coefficient reaching –1.48. Expropriation risk will have less of an impact on the decisions of local investors to invest in local markets than on foreign investors. As a result, when the expropriation risk is small (lager EXPROP), this will attract relatively more foreign investments to local markets. In this case, the domestic bias will be lower. Furthermore, the result of economic development variables shows that only GDP per capita (GDPC) has a significant negative coefficient. GDP per capita will reflect the development of a country and its financial markets. If GDPC is positive, a country will attract more foreign investors and domestic bias will consequently be less important. The two control variables (RFLOW) and (RESTRICT) are significant at the 5 percent level. The coefficient of RFLOW is positive. In fact, the lower the value of RFLOW, the more important are the restrictions facing the domestic investors in foreign markets and foreign investors in local markets. Then domestic investors will invest more in domestic markets and DBIAS will be important. The relation between DBIAS and RESTRICT is positive and this implies that if the restrictions on foreign investment are important, local investors will invest more in the domestic market. Investors’ behavior also has an impact on domestic bias. Its coefficient is 4.19 and the t-ratio 2.61. The more optimistic investors are about the local market, the more they invest in it; hence the importance of DBIAS. Common language (DUMLAN) and geographical proximity (DIST), which are proxies for familiarity, both have a significant impact on domestic bias. Countries that have the same language as a large number of countries in the world tend to have a smaller domestic bias. Countries that are farther away from the rest of the world have an important domestic bias. When, all explanatory variables are estimated jointly, results show significant coefficients generally for financial market development and for two variables of investor protection (LAW and EXPROP). Other variables, such as a common language, are not significant. Our findings corroborate those of Chan et al. (2005) especially regarding the importance of stock market development in the explanation of the domestic bias.

3.5.2 Results concerning foreign bias Table 3.8 shows regression results for the foreign bias measure, FBIASij . We introduce an additional independent variable, DBIASi , which controls the impact of domestic bias on foreign bias. If investors invest more in a domestic market, the proportion they could invest in other markets will be considerably lower. Then, if the domestic bias is important (DBIASi

Predicted sign

Economic development variables

Capital control

70

Table 3.8 Regression analysis of foreign bias Stock market development

Information costs

Investor behavior

Familiarity

Investor protection

Other variables

All variables

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff. t-stat

Coeff.

t-stat

−2.31 −0.51

−6.72 −12.2

1.01 −0.57

1.03 −13.0

−1.00 −0.18

−5.03 −4.06

−1.27 −0.52

−5.47 −10.9

8.46 −0.43

13.86 −10.6

−5.89 −0.55

−8.58 −12.8

−1.21 −5.05 −0.50 −10.7

15.32 −0.08

6.09 −1.48

0.00 0.05 0.00 −4.42

1.02 0.91 1.58 −2.12

−0.25 1.21

−2.57 2.21

0.30 −0.80 0.43

1.71 −1.87 0.77

Constant DBIAS

−

−2.15 −0.52

−8.36 −12.5

GDPC RGDP TRADE DI

+ + + +

0.001 −0.02 0.00 −0.99

9.12 −0.77 0.16 −0.62

RFLOW RESTRICT

+ −

SIZE COST DUMEMERG

+ − −

INFOR

−

BEHAV

+

DUMLANG TRADEB DIST

+ + −

LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL

+ + + + + +

RET2 CORR Adjusted R2

+ −

0.16 −1.31

4.23 −4.04 0.06 −0.42 −0.88

0.44 −1.55 −3.24 −4.86

−14.84 −2.83

−1.88 0.54 3.71 −1.21

2.89 2.80 −17.50

0.15

0.18

0.24

0.11

0.35

−9.96

−9.09

−4.78

0.86 4.26 −0.75 −0.54 −1.22 −15.98 0.04 0.03 −0.09 0.44 −0.11 −0.38

0.17

−4.92

0.20

0.40 4.03 −1.21 3.24 −1.91 −1.75 −0.02 −5.11 0.85 2.42 0.14

0.00 −0.01 −0.12 0.01 −0.21 0.59

−0.02 −0.80 −1.48 0.06 −3.13 2.21

−0.01 1.19 0.53

−1.49 2.70

Notes: FBIAS ij : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: common language dummy variables between two countries; DIST : log geographical distances between two countries; TRADE: bilateral trade volume between two countries; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET 2: past 2-year return; CORR: return correlations between two countries; DBIAS i : control variable.

FATHI ABID AND SLAH BAHLOUL

71

important), foreign bias will be important (more negative FBIASij ). The coefficients of DBIASi are negative and significant for almost all specification models. Among the eight different categories of explanatory variables, familiarity has the greatest influence on foreign bias. All familiarity variables are statistically significant at the 5 percent level. Foreign investors tend to invest in a country that is geographically close to them, that enjoys a large bilateral trade volume, and with which they share a language. The more familiar investors are with a foreign market, the more they invest abroad and the less the foreign bias (FBIAS important). Moreover, investors are less willing to invest in a foreign market that require a high information cost. The coefficient of INFOR is –4.86 and the t-ratio equals –14.84. Results concerning the investor protection variables show that the two variables ACC and EXPROP have a significant effect on foreign asset holdings. Foreign investors prefer markets with more transparency of information (ACC important) and a low expropriation risk. The more investors’ rights are maintained, the smaller the foreign bias. Economic market development (DUMEMERG) and financial market development (GDPC) have a significant impact on foreign bias. Foreign investors tend to invest more in a developed market (DUMEMERG = 0) and in a country with an important GDP per capita. In other words, foreign investors prefer large and developed markets with high levels of transparency. Capital control variables have a strong impact on foreign bias. The coefficient on RFLOW and RESTRICT are, respectively, 0.16 with a t-ratio of 4.23 and –1.31 with a t-ratio of –4.04. This result suggests that a country with fewer restrictions on capital flows (RFLOW important) or/and on foreign asset holding (RESTRICT low) attract more foreign investors, then foreign bias in this country will be less important (FBIAS important). When we regress the foreign bias on all variables, some of the coefficients are no longer statistically significant at the conventional level. Information costs and familiarity variables (except TRADEB) remain statistically significant; yet investor protection variables are insignificant. These results corroborate those found by Chan et al. (2005), in particular regarding the importance effect of familiarity on the foreign bias.

3.6 ADDITIONAL TESTS 3.6.1 Results concerning home bias In addition to domestic and foreign bias measures, we have used another measure of home bias developed by Ahearne et al. (2004) to confirm the preceding results.

72

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

The home bias of country i’s investors against country j (HBIASij ) is share of country j in the portfolio of country i’s investors HBIASij = 1 − (3.14) share of country j in the world market portfolio Regression results of the home bias on eight categories of variables are presented in Table 3.9. Table 3.9 shows that familiarity variables are the most important determinants of home bias. In fact, they present the most important adjusted R2 (0.13). Coefficients are −0.21 for common language (DUMLANG), −1.63 for bilateral trade volume (TRADEB) and 0.31 for geographic proximity (DIST), with t-ratios, respectively, of −2.64, −2.84 and 10.52. Familiarity variables seem to contribute similarly in explaining the foreign bias. Among the remaining factors, only four variables are statistically significant at the conventional level. The first variable is information costs, with a coefficient of 0.52 and a t-ratio of 4.49. Then, if foreign investors are less informed about the local market, they will invest less and the home bias will be very important. The second variable is related to investor protection (ACC). Foreign investors prefer markets with more information transparency (ACC important). The third variable is the restriction on asset holdings by foreign investors (RESTRICT). If foreign investors have this constraint, they will invest less abroad, and home bias will be very important. The last variable is trade volume scaled by GDP (TRADE). Its coefficient is −0.02, with a t-ratio of −2.84. Foreign investors tend to hold assets in a country with very important trade volume scaled by GDP. When all the explanatory variables are estimated jointly, apart from correlation coefficient (CORR), only familiarity variables present a statistically significant coefficient that confirms the hypothesis. Results corroborate those of Ahearne et al. (2004), especially for the impact of information costs on home bias.

3.6.2 Domestic, foreign and home biases according to a world float portfolio Dahlquist et al. (2003) show that the prevalence of closely held firms in most countries helps to explain why these countries exhibit a home bias in equity holdings. Based on their estimates of the percentage of closely held market capitalization, we construct a world float portfolio with country weights based on the free-floating shares available to investors. We calculate the float adjusted domestic bias (DBIAS_FLOAT), foreign bias (FBIAS_FLOAT) and home bias (HBIAS_FLOAT).

Table 3.9 Regression analysis of home bias Predicted sign

Economic development variables Coeff.

Constant GDPC RGDP TRADE DI

− − − −

RFLOW RESTRICT

− +

SIZE COST DUMEMERG

− + +

INFOR

+

BEHAV

−

DUMLANG TRADEB DIST

− − +

LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL

− − − − − −

RET2 CORR

− +

Adjusted R2

Capital control

Stock market development

t-stat

Coeff.

t-stat

Coeff.

t-stat

0.90

13.28

0.71

6.39

0.95

2.28

−0.00 −0.00 −0.02 1.18

−1.75 −0.04 −2.84 1.83 0.00 0.26

Information costs

Investor behavior

Familiarity

Investor protection

Other variables

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

0.55

9.92

0.75

22.50

−1.86

−7.11

1.24

4.65

0.72

18.35

−2.03

−1.38

0.00 −0.01 0.01 1.91

0.93 −0.32 −1.41 1.57

0.09 0.10

1.61 0.31

−0.08 −0.07 −0.05

−0.82 −0.30 −0.17

0.09 2.04 −0.03 −0.06 0.10

−0.55 −0.51 0.81 0.52

4.49 −0.65

−1.06 −0.21 −1.63 0.31

−2.64 −2.84 10.52 −0.02 −0.01 0.04 0.00 0.01 0.03

−0.40 −2.70 1.24 0.04 0.22 0.38 0.00 −1.01

0.01

0.01

0.01

All variables

0.04

0.01

0.13

0.01

0.00

1.46 −0.10

0.25

1.15

−1.92

−1.73

−0.30 −2.64 0.34

−2.52 −3.24 7.65

−0.11 0.00 0.00 0.04 0.00 0.11

−1.58 −0.40 0.08 0.38 −0.03 0.71

0.00 0.61

−0.29 2.39

0.15

73

Notes: HBIAS ij : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: common language dummy variables between two countries; DIST : log geographical distances between two countries; TRADE: bilateral trade volume between two countries; LAW : rule of law index: ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET 2: past 2-year return; CORR: return correlations between two countries.

74

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

Results concerning the float-adjusted domestic bias

Table 3.10 presents the result of the DBIAS_FLOAT estimation. The results of the float-adjusted domestic bias are almost the same as those of the unadjusted domestic bias. Variables that have a significant impact on domestic bias also have a significant effect on DBIAS_FLOAT. Moreover, the predictive power is generally more important for DBIAS_FLOAT than for DBIAS. The results confirm those of Dahlquist et al. (2003) for the impact of corporate governance structure on the home asset bias.

Results concerning the float-adjusted foreign bias

Table 3.11 shows the result of the FBIAS_FLOAT estimation for the eight categories of explanatory variables. The foreign bias calculated on the basis of the world float portfolio seems to be influenced by the same characteristics as the foreign bias. Except bilateral trade volume (TRADEB), all significant variables for the foreign bias are statistically significant at the 5 percent level for the foreign bias float.

Results for the float-adjusted home bias

Results found for the float-adjusted home bias are qualitatively the same as those of the home bias. Information costs and familiarity variables remain statistically significant for the home bias float. See Table 3.12.

3.7 CONCLUSION This chapter presented an analytical study of the bilateral asset holdings of investors from thirty investing countries in forty-three receiving countries. Similarly to Chan et al. (2005), we distinguish between domestic bias (domestic investors overweighting the local markets) and foreign bias (foreign investors under-or overweighting the overseas markets). We find that home bias is a large phenomenon for both developed and developing nations. The results show that stock market development and information costs have an important impact on domestic bias, while information costs and familiarity variables have an important effect on foreign bias. Economic development, capital control and investor protection variables have only a small effect on theses biases. However, investor behavior has a significant impact only on the domestic bias. Additional tests show that, only information costs and familiarity variables have an important effect on home bias. Investment behavior appears to be determined by multiple-factor models.

Table 3.10 Regression analysis of domestic bias based on the world float portfolio Economic development variables Coeff. Constant GDPC RGDP TRADE DI

Capital control

Stock market development

Information costs

Investor behavior

Familiarity

Other variables

All variables

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

7.09

8.34

7.91

4.84

−5.68

−1.68

2.06

2.91

3.99

8.47

−26.95

−2.38

18.15

6.35

5.73

9.72

25.75

2.08

0.01 0.05 0.01 −0.86

−4.49 0.28 0.76 −0.15

0.00 0.04 0.02 −7.14

−0.15 0.23 2.16 −1.18

0.15 1.00

0.60 0.57

−1.04 2.29 −1.33

−3.32 3.03 −0.73

−0.39 4.56

RFLOW RESTRICT

−1.98 2.54 −1.37 3.15 −0.46

SIZE COST DUMEMERG

−3.59 3.30 −0.49 7.41

INFOR

4.67 5.12

BEHAV

2.86 −5.76 3.65

DUMLANG DIST

−2.15 2.85 0.62 −0.04 −0.39 −1.68 −0.01 0.60

LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL RET2 CORR Adjusted R2

Investor protection

0.35

0.24

0.59

0.42

0.22

0.20

0.40

1.16 −0.95 −1.28 −2.74 −0.03 0.63 0.05 −12.34 0.22

2.15 −2.45

t-stat

4.95

2.11

3.80

2.39

5.03 −3.13

1.20 −2.29

1.71 0.02 0.34 −2.02 −0.40 −2.40

3.34 0.46 1.64 −4.36 −1.17 −2.18

−0.03 2.09 0.93

−0.99 0.49

75

Notes: DBIAS_FLOAT j : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: average common language dummy variables; DIST : average of log geographical distances; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET2: past 2-year return; CORR: average of return correlations.

Table 3.11 Regression analysis of foreign bias based on the world float portfolio 76

Economic development variables

Capital control

Stock market development

Information costs

Investor behavior

Familiarity

Investor protection

Other variables

All variables

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Constant DBIASFLOAT

−1.76 −0.43

−7.72 −13.6

−1.78 −0.42

−5.61 −13.5

0.46 −0.46

0.48 −14.0

−0.59 −0.22

−3.49 −6.6

−1.19 −0.43

−6.23 −11.8

8.12 −0.35

13.74 −11.6

−4.61 −0.44

−6.90 −13.8

−1.13 −0.42

−5.68 −11.7

13.34 −0.09

5.33 −2.1

GDPC RGDP TRADE DI

0.00 −0.03 0.00 −0.74

6.02 −0.89 0.47 −0.47

0.00 0.04 0.01 −4.45

0.54 0.76 2.66 −2.15

−0.20 1.29

−2.11 2.37

0.17 −0.76 0.93

0.95 −1.79 1.68

−4.78

−10.23

−9.16

−4.84

0.88 −0.95 −1.21

4.37 −0.68 −15.90

−0.01 −0.01 −0.02 0.23 −0.26 0.30

−0.04 −0.86 −0.25 1.23 −3.88 1.13

−0.01 1.19

−0.92 2.72

RFLOW RESTRICT

0.10 −1.16

2.66 −3.64 −0.08 −0.26 −0.69

SIZE COST DUMEMERG

−0.59 −0.98 −2.56 −4.16

INFOR

−13.13 −2.21

BEHAV

−1.50 0.51 2.41 −1.14

DUMLANG TRADEB DIST

2.82 1.85 −16.88 0.09 0.03 −0.10 0.31 −0.17 −0.19

LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL

−0.01 0.68

RET2 CORR Adjusted R2

0.96 3.75 −1.38 2.37 −2.99 −0.91

0.15

0.15

0.17

0.23

0.13

0.35

0.19

0.15

−3.77 1.99

0.53

Notes: FBIAS_FLOAT ij : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: common language dummy variables between two countries; DIST : log geographical distances between two countries; TRADE: bilateral trade volume between two countries; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET2: past 2-year return; CORR: return correlations between two countries; FLOAT _DBIAS j : control variable.

Table 3.12 Regression analysis of home bias based on the world float portfolio Economic development variables Coeff.

t-stat

Constant

0.78

7.02

GDPC RGDP TRADE DI

0.00 0.00 0.00 2.07

0.01 −0.04 −3.25 1.95

RFLOW RESTRICT

Capital control

Stock market development

Coeff.

t-stat

Coeff.

t-stat

0.42

2.26

1.21

1.76

0.02 0.42

Investor behavior

Familiarity

Investor protection

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

0.37

4.01

0.62

10.92

−3.05

−7.06

1.04

2.39

0.66

INFOR

t-stat

Coeff.

t-stat

0.58

8.64

−3.14

−1.27

0.00 −0.03 −0.01 3.34

0.65 −0.55 −1.96 1.63

0.12 0.29

1.22 0.53

−0.04 −0.06 −0.40

−0.22 −0.14 −0.73

3.44 −0.93

−0.90 −0.35 −1.84 0.44

DUMLANG TRADEB DIST

−2.64 −1.97 8.98 −0.03 −0.01 0.06 0.01 0.04 −0.05

LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL

−0.51 −2.09 1.20 0.11 1.06 −0.37 0.00 0.11

RET2 CORR 0.01

0.01

0.01

0.02

0.01

0.10

0.01

All variables

Coeff.

0.20 −0.86 0.43

BEHAV

Adjusted R2

Other variables

0.99 1.96 0.02 −0.17 0.09

SIZE COST DUMEMERG

Information costs

0.00

0.98 0.46

0.28

0.78

−2.73

−1.45

−0.49 −3.70 0.51

−2.46 −2.69 6.78

−0.16 0.00 −0.02 0.00 0.03 0.15

−1.38 −0.21 −0.24 0.03 0.52 0.56

0.00 0.91

−0.37 2.12

0.12

77

Notes: HBIAS_FLOAT ij : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: common language dummy variables between two countries; DIST : log geographical distances between two countries; TRADE: bilateral trade volume between two countries; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET2: past 2-year return; CORR: return correlations between two countries.

78

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

Our results corroborate those of Chan (2005), namely about the effect of familiarity variables on domestic and foreign biases as well as the conclusion reached by Ahearne et al. (2004) as to the impact of information costs on the home bias.

NOTES 1. See Lewis (1999) and Karolyi and Stulz (2003) for a survey of literature on the home bias. 2. http://www.imf.org/external/np/sta/pi/datarsl.htm. 3. http://en.wikipedia.org/wki/developed_country. 4. For variable groups (vi), (vii) and (viii), average values are calculated for each investing country face other receiving countries. Average value of TRADEB (see page 000) is not presented with other familiarity variables because it is equal to 1/43 for each observation. 5. We thank Jack Glen for providing us with data. 6. Source: http://www.freetheword.com. 7. The data for the year 1999 are used by Bartram and Dufey (2001). 8. http://www.phone-rate-calculator.com. 9. In this chapter we use the term “familiarity” broadly to capture the effects of both asymmetric information and psychological factors. 10. Data are used by Chan et al. (2005).

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French, K. and Poterba, J. (1991) “Investor Diversification and International Equity Markets”, American Economic Review, 81(2): 222–26. Gehrig, T. (1993) “An Information Based Explanation of the Domestic Bias in International Equity Investment”, Scandinavian Journal of Economics, 95(1): 97–109. Giannetti, M. and Koskinen, Y. (2004) “Investor Protection and the Demand for Equity”, SSE/EFI Working Paper, Series in Economics and Finance, n.526. Glassman, D. A. and Riddick, L. A. (1996) “Why Empirical International Portfolio Models Fail: Evidence that Model Misspecification Creates Home Bias”, Journal of International Money and Finance, 15(2): 275–312. Huberman, G. (2001) “Familiarity Breeds Investment”, Review of Financial Studies, 14(3): 659–80. Karolyi, A. and Stulz, R. (2003) “Are Financial Assets Priced Locally or Globally?”, in G. Constantinides, M. Harris, and R. M. Stulz (eds), Handbook of the Economics of Finance, (Amsterdam: North-Holland). Kilka, M. and Weber, M. (2000) “Home Bias in International Stock Return Expectations”, Journal of Psychology and Financial Markets, 1(3–4): 176–92. La Porta, R., Lopez-De-Silanes, F., Shleifer, A. and Vishny, R. (1997) “Legal Determinants of External Finance”, Journal of Finance, 52(3): 1131–51. La Porta, R., Lopez-De-Silanes, F., Shleifer, A. and Vishny, R. (1998) “Law and Finance”, Journal of Political Economy, 106(6): 1113–55. La Porta, R., Lopez-De-Silanes, F., Shleifer, A. and Vishny, R. (2000) “Investor Protection and Corporate Governance”, Journal of Financial Economic, 58(1–2): 3–27. Lewis, K. (1999) “Trying to Explain Home Bias in Equities and Consumption”, Journal of Economic Literature, 37(2): 571–608. Lintner, J. (1965) “The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolio and Capital Budgets”, Review of Economics and Statistics, 47(1): 13–37. Portes, R. and Rey, H. (2005) “The Determinants of Cross-Border Equity Flows”, Journal of International Economics, 65(2): 269–96. Sarkissan, S. and Schill, M. (2004) “The Overseas Listing Decision: New Evidence of Proximity Preference”, Review of Financial Studies, 17(3): 769–809. Sharpe, W. F. (1964) “Capital Asset Prices: A Theory of Market Equilibrium under the Condition of Risk”, Journal of Finance, 19(3): 425–42. Strong, N. and Xu, X. (2003) “Understanding the Equity Home Bias: Evidence from Survey Data”, Review of Economics and Statistics, 85(2): 307–12. Stulz, R. M. (1981) “On the Effect of Barriers to International Investment”, Journal of Finance, 36(4): 923–34. Tesar, L. and Werner, I. (1995) “Home Bias and High Turnover”, Journal of International Money and Finance, 14(4): 467–92. Zhou, C. (1998) “Dynamic Portfolio Choice and Asset Pricing with Differential Information”, Journal of Economic Dynamic and Control, 22(7): 1027–51.

CHAPTER 4

The Critical Line Algorithm for UPM–LPM Parametric General Asset Allocation Problem with Allocation Boundaries and Linear Constraints Denisa Cumova, David Moreno and David Nawrocki

4.1 INTRODUCTION Assume that there are information costs and asymmetric information in the marketplace. These conditions have been associated with segmented markets, with investors having a preferred habitat. Therefore, investors will be searching for local minima and maxima in their preferred habitat rather than a global market optimization based on equilibrium market asset pricing. Investors will have unique utility functions depending on their preferred habitat, and attempt to maximize their utility through a localized solution. This idea is not new as it traces back to Simon (1955), who proposed satisficing investors who restrict their searches to localized searches with rationality bounded by the area of the search. Cyert and March (1963) followed with their behavioral theory which suggests that decision-makers 80

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break complex problems into a sequence of simpler problems, which they solve sequentially using local rationality. This is a key component of the Kahneman and Tversky (1979) prospect theory, where investors set up multiple mental accounts and optimize each account. Under these conditions, the traditional Markowitz (1959) portfolio theory may be used to optimize within a preferred habitat or mental account. This leads to the question of whether the quadratic utility function of traditional mean-variance analysis is the appropriate utility model for portfolio analysis. There is much evidence indicating that investors are more sensitive to losses than gains.1 This introduces a discontinuous change in the shape of the investor’s utility function at some target return and plays a role in prospect theory, developed by Kahneman and Tversky (1979) and Tversky and Kahneman (1991). However, there is evidence that investors are not risk-averse throughout the range of returns, and will exhibit risk-seeking behavior in special situations. Friedman and Savage (1948) and Markowitz (1952) argue that willingness to purchase both insurance and lottery tickets implies reverse S-shaped (both concave and convex) utility functions. A reverse S-shaped utility function provides an explanation for investors engaging in risk-averse behavior for losses and risk-seeking behavior for gains. Fishburn (1977) proposed the lower partial moment (LPM) a, τ model to explain risk-seeking and risk-averse behavior below a target return (τ). Investor behavior is explained through a coefficient (a) as a < 1 is risk-seeking behavior and a > 1 is risk-averse behavior, thus the LPM (a, τ) model. The LPM (a, τ) model proved to be a very useful risk measure because of its flexibility in capturing investor behavior (Nawrocki, 1999). However, it was not immune to criticism. Kaplan and Siegel (1994a, 1994b) zeroed in on its characteristic of a linear utility function above the target return, which assumes that the investor is risk-neutral to all above-target returns. A recent paper by Post and van Vliet (2002) found evidence that, while investors are risk-averse to below-target returns, they are risk-seeking above the target return. In order to apply more realistic behavior to above-target returns, Sortino et al. (1999) proposed a performance measure, the upper partial moment/lower partial moment (UPM/LPM) ratio. Given the potential usefulness of the UPM/LPM model, we develop a critical line UPM/LPM portfolio optimization algorithm (CLA) with bounded investment constraints for investors to generate optimal solutions within separate mental accounts or preferred habitats. This algorithm is important as it allows us to study the behavior of UPM/LPM portfolios in greater detail. Section 4.2 of the chapter derives the UPM/LPM CLA and provides a proof that it is consistent with Kuhn–Tucker conditions. Section 4.3 presents a short empirical test to demonstrate that the resulting algorithm does work relative to the traditional mean-variance (EV) algorithm, and section 4.4 offers a summary and conclusions.

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4.2 THE UPSIDE POTENTIAL–DOWNSIDE RISK PORTFOLIO MODEL The UPM is also known as the upside potential, and the LPM is a family of downside risk measures. Therefore, the upside potential–downside risk (UPM/LPM) model may be formulated as follows: Maximize E(UPMp ) =

K

pt [max{0, E(Rpt ) − τ}]c

t=1

Minimize E(LPMp ) =

K

(4.1)

pt [max{0, τ − E(Rpt )}]a

t=1

subject to n

wi = 1

(4.2)

i=1

Although Markowitz (1959) developed the CLA for mean-variance optimization, this algorithm is not exclusive to the mean/variance problem; it can also be constructed for other risk–return portfolio models. In the next section, we derive the CLA for general asset allocation problem with bounds and linear equality constraints for the UPM–LPM portfolio model. Let r = (r1 , r2 , . . . rn )T be a vector of asset returns and x = (x1 , x2 , . . . xn ) a related vector of investment. It is assumed that x varies in the compact and convex set S. General asset allocation problems in the UPM–LPM framework can be formulated as looking for a legitimate investment vector x = (x1 , x2 , . . . xn ) with minimal downside risk for a given portfolio upper partial moment (b0 ). The investment vector x is legitimate whenever it fulfills the constraints. Therefore, the general optimization problem may be stated as Select x = (x1 , x2 , . . . . xn ) for which min E(LPMp ) =

n n

xi xj CLPMij

i=1 j=1

=

n i=1

xi2 LPMi +

n i =j

xi xj CLPMij

(4.3)

DENISA CUMOVA, DAVID MORENO AND DAVID NAWROCKI

83

subject to n n

E(UPMp ) =

xi xj CUPMij =

i=1 j=1

n i=1

xi2 UPMi +

a11 · x1 + a12 · x2 + · · · + a1n · xn = b1 a21 · x1 + a22 · x2 + · · · + a2n · xn .. .

n i=j

xi xj CUPMij

= b2 .. .

am1 · x1 + am2 · x2 + · · · + amn · xn = bm 0 ≤ x ≤ 1 ∀i = 1 . . . n where LPMi =

T

pt · [Max{0; (τ − rit )}]a

t=1

CLPMij =

T

pt · [Max{0; (τ − rit )}]a−1 (τ−rjt )

t=1

UPMi =

T

pt · [Max{0; (rit − τ)}]c

(4.4)

t=1

CUPMij =

T

pt · [Max{0, (rit − τ)}]c−1 (rjt − τ)

t=1

LPMi = CLPMij

for

∀i = j

UPMi = CUPMij

for

∀i = j

In matrix form, this optimization problem can be formulated as (a) min x

LPMp = xT · L · x

(b) UPMp = xT · U · x (c) A · x = b (d) 0 ≤ x with x = (x1 , x2 , . . . xn )T ⎞ ⎛ CLPM11 · · · CLPM1n ⎟ ⎜ .. .. .. L=⎝ ⎠ . . . CLPMn1 · · · CLPMnn

(4.5)

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CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES

⎞ CUPM11 · · · CUPM1n ⎟ ⎜ .. .. .. U=⎝ ⎠ . . . CUPMn1 · · · CUPMnn ⎛

⎛

⎞ a11 · · · a1n ⎜ ⎟ A = ⎝ ... . . . ... ⎠ am1 · · · amn ⎛

⎞ b1 ⎜ ⎟ b = ⎝ ... ⎠ bm First, we have to prove whether this formulation is convex, as the Kuhn– Tucker conditions for finding a global optimum can only be applied in this case. The objective function E(LPMp ) is convex for all a ≥ 1 in bounded x ∈ S if and only if it is positive semi-definite for all bounded x ∈ S. The E(LPMp ) is a positive variable for all bounded x by a ≥ 1, which implies that E(LPMp ) (and the associated matrix L) is positive semi-definite. E(LPMp ) ≥ 02 for all bounded x by a ≥ 1 assures construction of LPM3 taking positive value or zero. Thus, we can state that the objective function E(LPMp ) is convex for all a ≥ 1 in bounded x ∈ S. With the exception of E(UPMp ) all constraints are linear, and hence convex. As the quadratic function E(UPMp ) ≥ 04 is bounded for all x by c ≥ 1, convexity is assured by the formulation of UPM taking positive values or zero. Therefore, E(UPMp ) (and the associated matrix U) is positive semi-definite, and therefore convex. Kuhn–Tucker conditions are based on Lagrangian multipliers. In this case, the Lagrangian function is5

1 1 T x · U · x − UPMp L = xT · L · x − λ · (A · X − b) − λu · 2 2 where λ = {λ1 , λ2 , . . . , λm } ∈ m and λu denote Lagrangian multipliers for constraints (Equations 4.5b and 4.5c). Using the matrix form, the Kuhn– Tucker conditions are now constructed.

4.2.1 Kuhn–Tucker conditions Equation (4.5) represents a convex quadratic minimization problem with convex constraints.6 The necessary and sufficient conditions for x to be a global optimum is that x fulfills the Kuhn–Tucker conditions.7

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Let η represent the vector of partial derivatives of the Lagrangian L with respect to the n decision variables xi (0 ≤ i ≤ 1):

∂L ∂L ∂L , ,..., η = (η1 , η2 , . . . , ηn ) = ∂x1 ∂x2 ∂xn The Kuhn–Tucker conditions for (4.5) are formulated as (a) η = L · x − λ · A − λu · (U · x) ≥ 0 (b) x ≥ 0, λ ≥ 0 (c)

∀ η · x = 0 ⇔ (η > 0 ∧ x = 0)

1≤i≤n

and

(η = 0 ∧ x > 0)

(4.6)

(d) A · x = b (e) xT · U · x = UPMp Similar to Markowitz’s CLA condition (4.6c) implies that the partial derivative ηi of L with respect to xi equals zero if and only if xi is greater than zero – that is, if asset i is included in the base solution. This is the optimality condition. When xi equals zero, then the respective partial derivative ηi is positive. Equations (4.6a) and (4.6d) can be summarized, then Equation (4.6) can be rearranged in Equation (4.7), where it is assumed that all partial derivatives are equal to zero:

X U 0 X 0 L A · − λu · = (a) A 0 −λ 0 0 −λ b (b) x ≥ 0, λ ≥ 0 (c)

∀

1≤i≤n

η > 0 ⇔ x = 0 and η = 0 ⇔ x > 0

(4.7)

(d) η = L · x − λ · A − λu · (U · x) ≥ 0 (e) xT · U · x = UPMp Equation (4.7d) is added to ascertain that (4.6a) remains satisfied. Each xi > 0 represents a base or “IN” variable. Each xi = 0 is a non-base or “OUT” variable. The first-order condition (Lagrangean function equals zero) remains appropriate for the variables that are in the solution, as their bounds are not binding and hence could have been omitted entirely (at least for the risk tolerance being examined). Assets xi = 0 will not have any impact on portfolio expected LPM, and therefore, a particular IN-set in the L matrix can replace its ith row by identity vector ei , for every i, which is not in the IN-set. Vector ei has a 1 in its ith position and zero in other positions. Let this matrix be L. Let U be the U matrix with its ith row replaced by 0 vector, for every i,

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CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES

which is not in the IN-set. Next, the A matrix is defined as A matrix with the ith row replaced by a 0 vector, for every i not in the IN-set. In vector 0 , the ith zero position should be replaced with a downside boundary b for x in the OUT-position. But this downside boundary is zero (xi = 0), so

0 the vector does not change. Thus, it holds true that b

0 x x U 0 L A − λ = · · u b −λ −λ 0 0 A 0

L A

A 0

− λu

L − λu · U A

A 0

U 0

0 0

0 x = · b −λ

(4.8)

0 x = · b −λ

For example, if we have three assets and the second is IN, the first and the third are OUT, and the A · X = b constraint is simply x1 + x2 + x3 = 1, we shall have ⎞ ⎡⎛ 1 0 0 0 ⎢⎜ CLPM21 CLPM22 CLPM23 1 ⎟ ⎟ ⎢⎜ ⎣⎝ 0 0 1 0⎠ 1 1 1 0 ⎛ ⎞⎤ ⎛ ⎞ ⎛ ⎞ 0 0 0 0 x1 0 ⎜ CUPM21 CUPM22 CUPM23 0 ⎟⎥ ⎜ x2 ⎟ ⎜ 0 ⎟ ⎟⎥ · ⎜ ⎟=⎜ ⎟ − λu ⎜ ⎝ 0 0 0 0 ⎠⎦ ⎝ x3 ⎠ ⎝ 0 ⎠ 0 0 0 0 −λ 1 The matrix with the portfolio fractions emerges from

x −λ

=

L − λu · U A

A 0

−1

0 · b

Rearranging the inverse matrix gives

−1 0 0 A x = · −1 −1 −1 b −λ (A ) −(A ) · (L − λu · U) · (A ) After multiplying, we obtain

−1 A ·b x = −1 −1 −λ −(A ) · (L − λu · U) · (A ) · b

(4.9)

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87

It would be useful to divide this equation with and without λu :

−1 0 x A ·b = + λu −1 −1 −1 −1 −λ (A ) · U · (A ) · b −(A ) · L · (A ) · b Then, we can simply write

x = + λu · θ −λ

(4.10)

In addition to Equation (4.10), Equation (4.7d) has to be satisfied, thus

x η = ((L − λu · U) A ) · ≥0 −λ

x Substituting matrix gives −λ η = ((L − λu · U) A ) ·

−(A

−1 )

· L · (A

−1

A

−1

·b

) · b + λu · (A

−1 )

· U · (A

−1

)·b

≥0

Again, it would be useful to divide this equation with and without λu , and then simplify the denotation η=L·A

−1

· b − λu · U · A

+ λu · A · (A

−1

−1

) · U · (A

· b − A · (A

−1

)≥0

−1

) · L · (A

−1

)·b (4.11)

η = ϑ + λu · ϑ ≥ 0 where Equation (4.8) forces η = 0 + λu · 0

for

i ∈ IN-set.

Equations (4.10) and (4.11) provide the Kuhn–Tucker conditions (6a, 6b, 6d) expressed as linear functions of λu in the same way as the CLA for M–V. These linear functions are known as “critical lines”. Thus it is easy to compute efficient segments and corner portfolios in similar way as with the CLA for M–V. Portfolios with the same structure are defined as portfolios with the same i-assets in the “IN”-set. As we change the value of the λu -parameter expressing risk tolerance, the portfolio structure does not change with the value of λu , but remains the same for a certain interval. Portfolios, where the portfolio structure changes, are called “corner” portfolios by Markowitz (1959). These corner portfolios divide the λu (efficient frontier) into piecewise intervals, for which piecewise critical lines will be calculated. Hence, the efficient frontier is said to be segmented.

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CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES

4.2.2 Efficient segments on the efficient frontier Each efficient segment is determined by the interval λlow ≤ λE ≤ λhi , with which Equation (4.10) and (4.11) are satisfied. For any critical line, the interval boundaries are defined as λhi = min(λc , λd ) λlow = max(λa , λb , 0) where

λa =

βi >0

−∞

λb =

δi >0

λc =

for

i = 1, · · · , n

if υi ≤ 0

for

i = 1, · · · , n

for

i = 1, · · · , n

for

i = 1, · · · , n

max −i /θi βi 9%

9% to 3%

3% to −3%

−3% to −9%

0 r>1

0.133768 0.018052

56.15137∗∗ 6.321137

AOI Day Close, SPI Night Open

r=0 r≤1

r>0 r>1

0.15986 0.017473

66.55955∗∗ 6.116847

SPI Day Open, AOI Day Open

r=0 r≤1

r>0 r>1

0.149069 0.017637

62.18897∗∗ 6.174793

AOI Day Close, SPI Day Close

r=0 r≤1

r>0 r>1

0.162063 0.017576

67.50692∗∗ 6.153183

AOI Day Close, SPI Day Open

r=0 r≤1

r>0 r>1

0.132571 0.01729

55.40282∗∗ 6.051996

Note: ∗∗ denotes rejection at 1% significant level.

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LIQUIDITY AND MARKET EFFICIENCY

Table 9.9 Johansen cointegration test log results for the post-automation period Post-automation variables

H0

H1

Eigenvalues

Likelihood ratio

SPI Night Close, AOI Day Open

r=0 r≤1

r>0 r>1

0.171061 0.008075

66.54364** 2.756496

AOI Day Close, SPI Night Open

r=0 r≤1

r>0 r>1

0.102473 0.010521

40.47313** 3.60673

SPI Day Open, AOI Day Open

r=0 r≤1

r>0 r>1

0.177129 0.09194

69.62966** 3.149613

AOI Day Close, SPI Day Close

r=0 r≤1

r>0 r>1

0.076209 0.007512

29.6022** 2.571375

AOI Day Close, SPI Day Open

r=0 r≤1

r>0 r>1

0.087967 0.00663

33.56882** 2.261839

Note: ** denotes rejection at 1% significant level.

1999) that the behavior of index prices is consistent with the semi-strong form of the efficient market hypothesis. The results shown in Tables 9.8 and 9.9 provide evidence that the spot and futures prices are in a cointegrated equilibrium relationship. However, no previous empirical evidence has been provided to support the findings in this study for night open and close prices for a cointegrated relationship. Overall, results using a cointegration approach provide evidence supportive of a semi-strong form of the efficient market hypothesis.

Granger causality tests

As discussed earlier, the price discovery relationship is centered around the lead–lag behavior of the ASX All Ordinaries Index and the SFE SPI futures contracts. The causal relationship was examined using the Granger causality test to detect the direction of information flow as reflected in price change. The results for the Granger causality performed on the day-traded indexes are seen in Panel A in Table 9.10, and the night-traded indexes (SPI) are shown in Panel B. The causal relationships are tested on the open and close prices for the SPI futures and the All Ordinaries Index. As the unit root test results shown in Table 9.3 show that the variables are integrated of order one, I (1), the first differenced variables are used in the causality tests. It is observed from Panel A in Table 9.10 that the day traded All Ordinaries open value (day AOI open value) does cause the share price index (day open value), while the share price index (day close value) does cause the All Ordinaries Index (day close value). Causality seems to run both ways

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Table 9.10 Granger causality test results Null Hypothesis

Panel A: Day traded series SPI Day Open, SPI day open does not Granger AOI Open cause AOI open AOI open does not Granger cause SPI day open AOI Close, SPI Day Close

AOI close does not Granger cause SPI day close SPI day close does not Granger cause index close

Panel B: Night traded series AOI Close, AOI close does not Granger SPI Night Open cause SPI night open SPI night open does not Granger cause AOI close SPI Night Close, SPI night close does not Granger AOI Open cause AOI open AOI open does not Granger cause SPI night close AOI Close, AOI close does not Granger SPI Night Close cause SPI night close SPI night close does not Granger cause AOI close

Pre-automation Post-automation F-stat F-stat 44.6516∗∗∗ 0.13576

4.30455∗∗∗ 0.36373

1.94637 0.37811

2.48783∗ 1.58893

1402.57∗∗∗ 0.72121

0.59129 59.8435∗∗∗

33.2951∗∗∗ 0.82011

123.388*** 1.09953

5.46812∗∗∗ 0.38735

22.0058∗∗∗ 1.99863

Note: ∗∗∗ (∗ ) denotes significance at the 1% (10%) level.

(bi-directional causality) from/to the All Ordinaries Index (AOI) to/from share price index futures in the day trades. The night trades of series are shown in Panel B of Table 9.10. The share price index (SPI) night close causes the All Ordinaries Index day close value in the post-automation period, whereas the All Ordinaries Index (AOI) day close causes the share price index (SPI) night close in the pre-automation period. Thus causality seems to run both ways (bi-directional causality) from/to the All Ordinaries Index (AOI) to/from night traded share price index futures in the day trades in the night traded period. Comparing the pre-automation and the post-automation periods, the results suggest that, since automation, the structural alignment between the night traded futures and the day traded futures seems to have created a synergy from the 24-hour trading of the SPI futures contract. This can be seen in Table 9.10 (Panel B), with the shift to the SPI night open causing the AOI close in the “post-” period. The results broadly show bi-directional results over the 24-hour period before and after the introduction of electronic trading at the SFE. Thus Research Proposition 4 is supported in that

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LIQUIDITY AND MARKET EFFICIENCY

the lead–lag relationship remains unchanged (still bi-directional) after the start of electronic trading at the SFE.

9.6 CONCLUSION This study examined whether the Sydney Futures Exchange has benefited from the introduction of electronic trading. Under the first research proposition, empirical tests were carried out on the liquidity of the Sydney Futures Exchange with the analysis of the “at the money” (ATM) SPI call options, and SPI futures contracts. This provided an insight to the change of liquidity between the two derivative markets. By classifying the option contracts by date to expiry and the closeness of the strike price to the spot price, the ATM volume was analyzed along with the SPI futures against the All Ordinaries Index (AOI). The tests were run by classifying the volumes into size groups, following the assumptions of the mixture of distributions hypotheses so as to provide relative levels of market volatility against which to compare the liquidity ratios. The results show that the “at the money” SPI options were more liquid in times of high volatility after the SFE became automated. The SPI futures are less liquid in times of medium to low market volatility. This overall result supports Research Proposition 1, that the SFE’s liquidity has changed with the introduction of electronic trading. Therefore it can be concluded that the liquidity of the Sydney Futures Exchange seems to have increased the operational efficiency within the SPI call options market, while there seems to have been a decline in the operational efficiency of SPI futures market. The importance of the analysis of liquidity in this study is that it was able to account for times of high volatility, such as the technology crash at the beginning of 2000. This was shown clearly by segmenting the option and futures market responses to the differing levels of market volatilities. The examination of the price discovery process was incorporated into the last three research propositions. Research Propositions 2 and 3 were used to test semi-strong form market efficiency. Under this assumption, the trading prices in the Australian Stock Exchange (ASX) and Sydney Futures Exchange (SFE) should have a long-run cointegrating relationship. The results confirm the belief that ASX and the SFE are semi-strong efficient. The existence of cointegration between the two markets before and after the introduction of electronic trading supported the semi-strong market efficiency. The presence of a bi-directional lead–lag relationship between the SPI futures price and the All Ordinaries Index price before and after the introduction of electronic trading supported the fourth research proposition. The automation of the SFE did alter the price discovery process. First, it appeared to synergize the night traded market. This is most probably a

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179

result from the SFE day and night structural alignment, of both instruments being traded on the same system. However, market leads and lags were bi-directional both before and after automation. This suggests that the electronic trading structure does not greatly enhance the price discovery price of the SFE. If it did, this would be observed in the SPI futures leading the AOI. Therefore, it can be concluded that a change in the liquidity is evident in the SPI futures and SPI call option contracts, but the price discovery process does not appear to have been enhanced by the automation of the Sydney Futures Exchange in the early stages up to August 2000.

APPENDIX Ratio variables Ratio_A =

Day_ATM_Volume Day_Index_Volume

Ratio_C =

Day_Futures_Volume Day_Index_Volume

Ratio_D =

Night_Futures_Volume Day_Index_Volume

Ratio_E =

Day_ATM_Volume Day_Futures_Volume

Day

= Traded during the day trading session

Night = Traded during the night trading session Futures = Share price index futures contract ATM

= “At the money” SPI call option contracts

Index = All Ordinaries Index

NOTES 1. Massimb and Phelps (1994) defined operational efficiency as the market’s ability to lower the costs of trading, and its execution speed of orders between buyer and seller. See Ates and Wang (2005) for the latest evidence on operational efficiency in the US futures market. 2. Informational efficiency means that all traders have equal access to all public information, and that the information is quickly reflected in trading prices (Tsang, 1999). 3. Overstatement of the lead–lag relationship will be discussed later in the chapter.

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LIQUIDITY AND MARKET EFFICIENCY

4. Fama (1970) has identified three levels of market efficiency. The semi-strong form of the “efficient market hypothesis” states that prices reflect all publicly available information. 5. Market efficiency is discussed in detail in the following section. 6. Cost-of-carry is the cost involved in storing an asset and the interest lost on funds tied up therein. 7. A moneyness portfolio denotes the categorization of strike prices relative to the spot price, so as to create portfolios reflecting the moneyness at that point in time (Rubinstein, 1994). 8. A brief discussion on the mixture of distribution hypothesis was given earlier. 9. Variables definitions are given in the footnotes to Table 9.4.

REFERENCES Ackert, L. F. and Racine, M. D. (1999) “Stochastic Trends and Cointegration in the Market for Equities”, Journal of Economics and Business, 51(2): 133–43. Anthony, J. H. (1988) “The Interrelation of Stock and Option Market Trading Volume Data”, Journal of Finance, 43(4): 949–64. Ates, A. and Wang, G. H. K. (2005) “Information Transmission in Electronic versus OpenOutcry Trading Systems: An Analysis of U.S. Equity Index Futures Markets”, Journal of Futures Markets, July, 25(7): 679–715. Australian Stock Exchange (2000) website address: www.asx.com.au/B1400.htm (accessed April 24, 2000). Bortoli, L., Gareth, A. and Jarnecic, E. (2004) “Differences in the Cost of Trade Execution Services on Floor-Based and Electronic Futures Markets”, Journal of Financial Services Research, August, 26(1): 73–87. Brailsford, T. (1996) “The Empirical Relationship between Trading Volume, Returns and Volatility”, Accounting and Finance, 36(1): 89–111. Brenner, K. and Kroner, K. (1995) “Arbitrage, Cointegration, and Testing the Unbiasedness Hypothesis in Financial Markets”, Journal of Financial and Quantitative Analysis, 30(1): 23–42. Brooks, C., Garrett, I. and Hinich, M. J. (1999) “An Alternative Approach to Investigating Lead–Lag Relationships between Stock Index and Stock Index Futures Markets”, Applied Financial Economics, 9(6): 605–13. Clark, P. K. (1973) “A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices”, Econometrica, 41(1): 135–55. Copeland, L., Lam, K. and Jones, S.-A. (1976) “The Index Futures Markets: Is Screen Trading more Efficient?”, Journal of Futures Markets, 24(4): 337–57. Copeland, T. E. (1976) “A Model of Asset Trading under the Assumption of Sequential Informational Arrival”, Journal of Finance, 31(5): 1149–68. Cornell, B. (1981) “The Relationship between Volume and Price Variability in Futures Markets”, Journal of Futures Markets, 1(2): 303–16. Dickey, D. A. and Fuller, W. (1979) “Distribution of the Estimators for Auto Regression Time Series with a Unit Root”, Journal of the American Statistical Association, 74(2): 427–31. Domowitz, G. (1993) “A Taxonomy of Automated Trade Execution Systems”, Journal of International Money and Finance, 12(3): 607–31. Engle, R. F. and Granger, C. W. J. (1987) “Co-integration and Error Correction: Representation, Estimation, and Testing”, Econometrica, 55(2): 251–76.

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Epps, T. W. and Epps, M. L. (1976) “The Stochastic Dependence of Security Price Change and Transaction Volumes: Implications for the Mixture-of-Distributions Hypothesis”, Econometrica, 44(2): 305–21. Fama, E. F. (1970) “Efficient Capital Markets: A Review of Theory and Empirical Work”, Journal of Finance, 25(2): 383–417. Fama, E. F. (1991) “Efficient Capital Markets”, Journal of Finance, 46(5): 1575–617. Franke, G. and Hess, D. (1995) “Anonymous Electronic Trading Versus Floor Trading”, Working Paper, Series II, no. 285, Universitat Konstanz. Freund, W. C., Larrain, M. and Pagano, M. S. (1997) “Market Efficiency Before and After the Introduction of Electronic Trading at the Toronto Stock Exchange”, Review of Financial Economics, 6(1): 29–56. Frino, A., Bortoli, L., and Jarnecic, E. (2004), “Differences in the Cost of Trade Execution Serouson Floor-based and Electronic Future, Markets”, Journal of Financial Services Research, 26(1): 73–87. Frino, A. I. and Jarnecic, E. (2000) “An Empirical Analysis of the Supply of Liquidity by Locals in Futures Markets: Evidence from the Sydney Futures Exchange”, Pacific Basin Finance Journal, 8(3–4): 443–56. Frino, A., McInish, T. and Toner, M. (1998) “The Liquidity of Automated Exchanges: New Evidence from the German Bund Futures”, Journal of International Financial Markets, Institutions and Money, 8(3–4): 225–41. Granger, C. W. J. (1969) “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods”, Econometrica, 37(3): 424–38. Granger, C. W. J. (1981) “Some Properties of Time Series Data and Their Use in Econometric Model Specification”, Journal of Econometrics, 16(1): 121–30. Groenewold, N. (1997) “Share Market Efficiency: Tests Using Daily Data for Australia and New Zealand”, Applied Financial Economics, 7(6): 645–57. Grünbichler, A., Longstaff, F. A. and Schwartz, E. S. (1994) “Electronic Screen Trading and the Transmission of Information: An Empirical Examination”, Journal of Financial Intermediation, 3(2): 166–87. Harris, L. (1982) “Transaction Data Tests of the Mixture of Distributions Hypothesis”, Journal of Financial and Quantitative Analysis, 22(2): 127–41. Hemler, M. L. and Longstaff. F. (1991) “General Equilibrium Stock Index Futures Prices: Theory and Empirical Evidence”, Journal of Financial and Quantitative Analysis, 26(3): 287–308. Hinich, M. (1996) “Testing for Dependence in the Input to a Linear Time Series Model”, Journal of Nonparametric Statistics, 6(3): 205–21. Hurst, H. E. (1951) “The Long-term Storage Capacity of Reservoirs”, Transactions of the American Society of Civil Engineers, 116(3): 143–52. Jarnecic, E. (1999) “Trading Volume Relations between the ASX and ASX Options Market: Implication of Microstructure”, Australian Journal of Management, 24(1): 77–91. Johansen, S. (1991) “Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Autoregressive Models”, Econometrica, 59(6): 1551–80. Johansen, S. (1995) “Likelihood-based Inference in Cointegrated Vectors Autoregressive Models”, (Oxford University Press). Johansen, S. and Juselius, K. (1990) “Maximum Likelihood Estimation and Inference on Cointegration – with Application to the Demand for Money”, Oxford Bulletin of Economics and Statistics, 52(2): 169–210. Karpoff, J. M. (1987) “The Relation between Price Change and Trading Volume: A Survey”, Journal of Financial and Quantitative Analysis, 22(1): 109–26. Kempf, A. and Korn, O. (1998) “Trading System and Market Integration”, Journal of Financial Intermediation, 7(3): 220–39.

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Kofman, P. and Moser, J. (1997) “Spread Information Flows and Transparency Across Trading Systems”, Applied Financial Economics, 7(3): 281–94. MacKinnon, J. G. (1991) “Critical Values for Cointegration Tests”, in R. F. Engle, and C. W. J. Granger (eds), Modelling Long-run Economic Relationships (London: Oxford Publishing Company). Mananyi, A. and Struthers, J. J. (1997) “Cocoa Market Efficiency: A Cointegration Approach”, Journal of Economic Studies, 24(3): 141–51. Manaster, S. and Mann, S. C. (1998) “Life in the Pits: Competitive Market Making and Inventory Control”, Review of Financial Studies, 36(1): 51–64. Martell, T. F. and Wolf, A. S. (1985) “Determinants of Trading Volume in Futures Markets”, Working paper, Centre for the Study of Futures Markets, Columbia Business School, USA. Massimb, M. and Phelps, B. (1994) “Electronic Trading, Market Structure and Liquidity”, Financial Analysts Journal, 50(1): 39–50. O’Connor, S. M. (1993) “The Development of Financial Derivatives Markets: The Canadian Experience”, Technical Report, No. 62 (Ottawa: Bank of Canada). O’Hara, M. (1995) Market Microstructure Theory (Cambridge, Basil Blackwell). Perron, P. (1990) “Testing for a Unit Root in a Time Series with a Changing Mean”, Journal of Business and Economic Statistics, 8(1): 153–62. Peters, E. (1992) “R/S Analysis Using Logarithmic Returns: A Technical Note”, Financial Analysts Journal, 48(6): 81–2. Phillips, C. C. and Perron, P. (1988) “Testing for a Unit Root in Time Series Regression”, Biometrika, 8(2): 153–62. Pirrong, C. (1996) “Market Liquidity and Depth on Computerized and Open Outcry Trading Systems: A Comparison of DTB and LIFFE Bund Contracts”, Journal of Futures Markets, 16(5): 519–43. Ragunathan, V. and Peker, A. (1997) “Price Variability, Trading Volume and Market Depth: Evidence from the Australian Futures Market”, Applied Financial Economics, 7(5): 447–54. Richardson, M. and Smith, T. (1994) “A Direct Test of the Mixture of Distributions Hypothesis: Measuring the Daily Flow of Information”, Journal of Financial and Quantitative Analysis, 29(1): 101–16. Rubinstein, M. (1994) “Implied Binomial Trees”, Journal of Finance, 49(3): 771–818. Shyy, G. and Lee, J. (1996) “Price Transmission and Information Asymmetry in Bund Futures Markets: LIFFE vs. DTB”, Journal of Futures Markets, 15(1): 437–55. Sydney Futures Exchange (SFE) (1999) Full Electronic Trading: A Guide for Customers (Sydney, Australia: SFE). Tauchen, G. and Pitts, M. (1983) “The Price Variability–Volume Relationship on Speculative Markets”, Econometrica, 51(2): 485–505. Taylor, M. P. and Sarno, L. (1997) “Capital Flows to Developing Countries: Log and Short-term Determinants”, World Bank Economic Review, 11(3): 183–212. Tsang, R. (1999) “Open Outcry and Electronic Trading in Futures Exchanges”, Bank of Canada Review, Spring, 21–39. Turkington, J. and Walsh, D. (1999) “Price Discovery and Causality in the Australian Share Price Index Futures Market”, Australian Journal of Management, 24(2): 97–113.

C H A P T E R 10

How Does Systematic Risk Impact Stocks? A Study of the French Financial Market Hayette Gatfaoui

10.1 INTRODUCTION Systematic risk is known to affect the market prices of traded financial assets (Stulz, 1999a, 1999b, 1999c). Indeed, the capital asset pricing model (CAPM) theory argues that each financial asset bears an undiversifiable risk known as systematic or market risk, as introduced by Sharpe (1963, 1964, 1970) and Treynor (1961) among others.1 Such a risk can be estimated through a well-diversified portfolio so far as this portfolio presents as low as possible an idiosyncratic risk (French and Poterba, 1991). Recent literature focuses mainly on a sound assessment of the influence of systematic risk on financial assets, along with the beta coefficient in a CAPM framework. Koutmos and Knif (2002) estimate the influence of systematic risk while employing timevarying distributions (for example, conditional distributions depending on past innovations). Using market stock indices of the financial markets under consideration, they find that financial assets’ betas are stationary meanreverting processes with an average degree of persistence equal to four days. Gençay, Selçuk and Whitcher (2003) use wavelet techniques to assess the influence of systematic risk on any asset, or equivalently to compute its beta 183

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in a CAPM model. These authors use the S&P 500 index as a systematic risk benchmark. Therefore, common practice resorts to available stock indices as proxies for a well-diversified market portfolio, and pays little attention to the sound assessment of systematic risk itself.2 However, a study by Campbell et al. (2001) shows that the number of stocks in such an index has to be high enough to offset idiosyncratic risk. They find that the number of assets required to create a well-diversified portfolio has grown over time. Therefore, using market indices with an insufficient number of stocks (for example, small stock indices) may be an inaccurate and even wrong benchmark for systematic risk. Indeed, a market stock index represents a sub-set of the whole range of financial assets that should enter the composition of an actual market portfolio, according to the critique of Roll (1977). This author underlines that the actual market portfolio is non-identifiable, since a market portfolio should be composed of stocks, bonds, real estate and human capital assets, among others. However, Campbell et al. (2001) show that market volatility (which is that part of the global volatility related to market factor, and specifically their market factor proxy) tends to drive global volatility. Therefore, in this chapter we address the question of how to find a proxy for the market factor, such as the systematic risk factor, in markets where only small stock indices are available, and where options on such indices are traded. It is a hard task, since the undiversifiable risk is not directly observable and can only be estimated. Hence, lacking a portfolio diversified enough to represent the market factor accurately, we attempt to infer the fair level of market risk factor from only observed available stock indices and related European call prices. The chapter is organized as follows. Section 10.2 introduces the assumptions and theoretical framework aimed at finding a proxy for the systematic risk factor. Section 10.3 employs an empirical application of such a framework, focusing on the French financial market and its CAC40 stock index. Section 10.4 studies the impact of the implied market factor on a pool of French stocks. The impact of systematic risk is analyzed through a twostep methodology, namely a correlation study and a Granger causality test. For further investigation, section 10.5 attempts to test for a non-linear relationship between both prices and returns of the implied market factor, and French financial assets. This study is realized in two stages: a linear regression analysis and a volatility analysis. The linear regression analysis considers first simple regressions of returns, and then Jensen-type (1968, 1969) regressions. The volatility study considers weekly rolling volatilities of asset returns. Section 10.6 attempts to draw some conclusions while comparing our implied market factor with other available market stock indices. A two-step study is undertaken, considering first the explanatory power corresponding to each stock index. The empirical weekly forecasting performance underlying each available market proxy is then assessed, employing the average absolute relative error as a performance measure. Finally, the

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study ends with concluding remarks and suggestions for future research in section 10.7.

10.2 THEORETICAL FRAMEWORK In this section, we introduce our assumptions and the related theoretical framework allowing the induction of the market factor.

10.2.1 Valuation setting We assume that any small stock index is a non-perfect proxy of the systematic risk factor. Specifically, we suppose that a small stock index is a disturbed observation of the market factor.

Assumptions. Any small stock index It , at current time t, depends on market factor Xt such that It = t Xt

(10.1)

where t represents a (strictly) positive determinist scale factor that is time∗ . Moreover, is a continuous and derivable varying and bounded on R+ t function of time. We assume implicity that any small stock index is diversified so as to exhibit a sufficiently low level of idiosyncratic risk. Therefore, the scale factor encompasses this. This parameter is not purely, or mainly, driven by an idiosyncratic component. Hence the scale parameter can encompass many effects/factors such as liquidity phenomena, and short-term shocks resulting from some announcement effects or specific events occurring in the financial market. Further, all the assumptions of the Black and Scholes (1973) option valuation framework are supposed to hold. To sum up, trading is continuous; there are no dividend payments, no transaction costs and no taxes. Moreover, there is no arbitrage opportunity and a constant spot risk-free interest rate r prevails in the complete market.3 We also assume that the market fact tor follows a geometric Brownian motion such as dX Xt = µ dt + σ dWt where t is the current date; µ and σ are constant drift and volatility parameters of the systematic factor’s instantaneous rate of return;4 Wt is a standard Brownian motion under the historical probability.

Dynamic of the stock index. Applying Ito’s lemma in the risk-neutral universe and on-time sub-set [t, T], the stock index dynamic writes under risk neutral probability σ2 1 ∂t d ln (It ) = + r− dt + σ dWt∗ (10.2) t ∂t 2

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σ2 ∗ ∗ T (T − t) + σ W which rewrites IT = It exp r − T−t , where (Wt ) is a 2 t standard Brownian motion. Of course, we could estimate t and T while building a well-diversified portfolio. Such a portfolio should be a good proxy of market factor so that the market is complete. However, along with Roll’s critique and Campbell et al. (2001), we address the question of how to proxy the market factor from a small-stock index, which is an imperfect proxy of market factor. Hence, we consider the prices of options on a small index. Indeed, observed index prices and call market prices will give information about both scale and market factors.

10.2.2 Option pricing We introduce a call pricing formula for European calls on the small-stock index I.

Call’s dynamic in a no-dividend framework. We consider a European call on stock index I whose strike price and expiring date are, respectively, K and T. At maturity, such a call is valued C(T, IT ) = max(0, IT − K) = (IT − K)+ . Like Black and Scholes (1973), we apply the no-opportunity arbitrage valuation principle, which states that the current value of any contingent claim is equal to the discount expected value of its future cash flows under risk neutral probability. Then, our European call Q Q price writes C(t, It ) = Et e−r(T−t) (IT − K)+ , where Et [.] is the expectation operator under risk neutral probability Q, conditional on the information set Ft = σ{Ws ,0 ≤ s ≤ t} available at current date t. Therefore, from Equations (10.1) and (10.2) of the stock index, the pricing formula for a European call on stock index I at current date t reads: C(t, It ) ≡ C(T − t, K, It , r, t , T , σ) =

where N(.)

T It N(d1 ) − K e−r(T−t) N(d2 ) t (10.3)

is the cumulative distribution function of the stan

I σ2 ln T + ln Kt + r + 2 (T − t) √ t √ ; d = d − σ T −t= dard normal law; d1 = 2 1 σ T −t 2 I σ ln T + ln Kt + r − 2 (T − t) t √ . If we assume that the small-stock index is a σ T −t

perfect proxy of market factor, we get the classical Black and Scholes (1973) option pricing formula, since we have t = T = 1 for each date t < T. Therefore, introducing a disturbance in our modifies the classical Black setting

T and Scholes formula through ratio t . However, assuming a Black and Scholes (1973) setting to value a call on a stock index is inappropriate in so

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far as the no-dividend assumption is unrealistic. That is why we adapt the previous formula to account for a stock index comprising dividend-paying equities.

Call’s dynamic in a dividend framework. Since most of the stocks that constitute financial indices pay dividends, we assume that index I pays a dividend at a continuous annualized rate q (see Merton, 1973; Black, 1975). Therefore, under the Black and Scholes’ world and dividend-paying assumptions, the current price of underlying It has to be replaced with It e−q(T−t) . Then, adjusting the European call pricing formula in Equation (10.3) to become a dividend-paying framework, a European call on a dividend-paying stock index I is valued as C(T − t, K, It , r, t , T , σ) =

T It e−q(T−t) N(d1 ) − K e−r(T−t) N(d2 ) t (10.4)

where N (.) is the cumulative distribution function of the standard

I e−q(T − t) σ2 ln ( T ) + ln t K + r + 2 (T − t) √ t √ ; d2 = d1 − σ T − t = normal law; d1 = σ T − t −q(T−t) I e σ2 ln T + ln t K + r − 2 (T − t) t √ . In European call formula Equations σ T −t

(10.3) or (10.4), all parameters are known except the scale parameter at instants t and T (for example, t and T ), and volatility parameter σ. Therefore we shall use our knowledge about observed index prices and market prices of European index calls to extract information about the scale parameter and volatility parameter σ. Such a process will give information about the market factor itself.

10.3 EMPIRICAL STUDY We apply our European call pricing here to the French stock market and its CAC40 stock index.

10.3.1 Data We use Bloomberg daily closing data from January 2, 2002 to March 19, 2002, a total of 55 observations by series. We observe one-month r1M , twomonth r2M and three-month r3M risk-free interest rates, and consider market prices of the CAC40 French stock index. This index is composed of the forty most liquid and representative stocks listed on the French financial market, and pays a continuous annualized dividend rate q. The CAC40 INDEX is a weighted stock index whose weights are proportional to each of its forty

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Table 10.1 Index information Index

q(%)

nb K

Spread

CAC40

2.2650

3

4238.99–4682.79

Table 10.2 Features of CAC40 INDEX calls Call name

Strike price (€ )

CAC 3/02 C4000

4,000

CAC 3/02 C4500

4,500

CAC 3/02 C5000

5,000

stocks’ capitalization. We also obtain closing prices of three European calls on CAC40 while considering option contracts of the continuous listing class. These calls are traded on the French options market called MONEP (Marché des Options Négociables de Paris). Let q, nbK and spread be, respectively, the dividend rate, the number of different strike prices of CAC40 INDEX calls, and the variation bounds of the index value (that is, lowest–highest in euros) over the studied time period (see Table 10.1). European calls on the CAC40 INDEX, maturing on March 27, 2002, exhibit the features shown in Table 10.2. Over our time horizon, time to maturity of calls falls from 84 calendar days to 8 calendar days (6 working days). Part of these data will help us to compute the risk-free interest rate, which must be defined. Given our European call pricing formula, we compute the risk-free rate as a function of time to maturity. We choose a quadratic interpolation method to infer our short-term risk-free rate from the one-, two- and three-month term risk free rates. Let r(t, T) be the risk free rate at current time t for time horizon T. This rate is then described by relation r(t, T) = a(T − t)2 + b(T − t) + c with a = 72[r1M (t) − 2r2M (t) + r3M (t)], b = 12[r2M (t) − r1M (t) − (a/48)] = −30r1M (t) + 48r2M (t) − 18r3M (t) and c = r1M (t) − (a/144) − (b/12) = 3r1M (t) − 3r2M (t) + r3M (t). This method gives a good risk-free rate proxy, given that European calls’ time to maturity (for example, (T − t)) is, at most, three calendar months.

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This proxy is employed to infer the fair value of the market risk factor from the CAC40 INDEX and related European call prices. Incidentally, the lack of control of the CAC40 INDEX weights introduces size effects into this index, among others. Therefore, an important role is played by some specific effects that are peculiar to any given highly-capitalized firm belonging to the CAC40 INDEX. In this light, the market benchmark role of CAC40 is strongly compromised. Dow Jones STOXX market indices bypass such a bias by applying some weight constraint when a given stock’s weight exceeds some specific threshold among the indices under consideration (high free-floating market capitalization).

10.3.2 Induction of systematic risk We explain how to estimate the level of market factor from market prices of a small stock index and the closing prices of European calls on such an index. From Equation (10.4), the estimation of the market factor’s level requires the estimation of the scale parameter at instants t and T (that is, t and T ), and volatility parameter σ (the volatility of the market factor’s instantaneous rate of return). As we observe market prices of the CAC40 INDEX (the small-stock index) and closing prices of related European calls, one solution consists of inverting Equation (10.4) relative to the scale parameter at times t and T, and the volatility parameter. We estimate these implied parameters while minimizing the sum of squared valuation errors at each fixed date t as ⎧ ⎫ nbK ⎨

2 ⎬ Min CObs T − t, Kj , It − C T − t, Kj , It , r, t , T , σ ⎭ t , T , σ ⎩ j=1

where Kj ∈ {4000, 4500, 5000}, and CObs (T − t, Kj , It ) are the European call’s market price. We solve this non-linear minimization problem numerically with a quasi-Newton method, and a Davidon–Fletcher–Powell type of algorithm. First, we get T = 2.3050 and XT = 2033.8482. Second, results allow plotting the implied values of t and σ against time to maturity. Whaley, (1982) argues that valuation errors do not necessarily depend on options’ moneyness, hence we draw plots according to time rather than moneyness. The implied volatility parameter σ is time-varying with a quadratic trend (a “smirk” type trend). Moreover, implied time series t and σ exhibit the statistical profiles shown in Table 10.3. We then observe a non-normal behavior for t and σ, namely leptokurtic distributions. Specifically, the volatility of the systematic risk factor should be modeled by a non-normal stochastic process or time-varying series. This stylized fact is known as the Black and Scholes volatility bias characterizing non-normal observed market asset returns. Knowing the market trend, we can now characterize the impact of systematic risk on the French financial market.

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0.2350

2.305 2.3

0.2200

2.295

0.2150

2.29

0.2100

2.285

0.2050

2.28

0.2000

2.275

0.1950

2.27

0.1900

2.265

0.1850

2.26

84 80 78 76 72 70 66 64 62 58 56 52 50 48 44 42 38 36 34 30 28 24 22 20 16 14 10 8

Volatility level

0.2250

2.31 Implied volatility Implied lambda

Scale factor’s level

0.2300

Time to maturity (days)

Figure 10.1 Daily implied scale factor and market factor volatility

Table 10.3 Descriptive statistics t Mean

2.2881

σ 0.2069

Xt 1952.2699

Standard deviation

0.0086

0.0069

52.0441

Skewness

0.2144

1.1208

−0.1669

−1.0728

2.9231

−0.8446

3.0589

31.0976

1.8901

Excess Kurtosis Jarque-Bera Statistic

10.4 THE IMPACT OF SYSTEMATIC RISK Given our market factor’s estimation, we try to quantify its impact on prices of French stocks. Our primary econometric study is composed of a correlation study and a Granger causality test.

10.4.1 Correlation We study correlations between implied market factor and, on the one hand, French stock indices (CAC40, SBF120 and SBF250), and on the other ten French stocks: Air Liquide, Danone, L’Oréal, Renault, Schneider, Société Générale, Thomson, Totalfina Elf, Valéo and Vivendi (see Table 10.4). Most of commonly used descriptive statistics are valid only under the strong assumption of an elliptical distribution. When this is not the case, statistics

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Table 10.4 Correlation of assets with the implied market factor Asset return

Correlation coefficient

Asset return

Correlation coefficient

SBF120

0.9959

Valéo

0.5288

SBF250

0.9952

Société Générale

0.7078

CAC40

0.9967

L’Oréal

0.6775

Air Liquide

0.4667

Renault

0.5365

Danone

0.2002

Schneider

0.4736

Vivendi

0.7982

Thomson

0.5329

Totalfina Elf

0.6569

are false. Indeed, this point fits some of the current questions considered by the Basel Committee. Szego (2002) and Artzner et al. (1999, 2000), highlight the coherency problem of risk measures such as linear correlation or covariance. Such risk measures are valid only for, at least, stationary distributions when not elliptical. Specifically, leptokurtic distributions violate one main property ensuring risk measures’ coherency, namely the sub-additivity principle. Following this concern, we compute correlations between the return of the implied market factor and returns of French stocks. Returns of both series are stationary over the time period studied. We then study the link between evolutions of both the systematic risk factor’s return and French asset returns. The average correlation of our three stock indices is 0.9959. The implied market factor is highly correlated with stocks, whose correlation coefficients range from 0.2002 for Danone to 0.7982 for Vivendi. In the rest of the chapter, we study the dependency between systematic risk and French stocks.

10.4.2 Causality Any causality study needs a vector autoregressive (VAR) specification as a starting point. We first introduce our VAR specification and then apply a Granger causality test. VAR specification

We look for a link between the implied market risk’s return RX and French stock or index returns RS . Hence, we consider VAR representations linking RX to RS with S ∈ {SBF120,5 SBF250, CAC40, Air Liquide, Danone, L’Oréal, Renault, Schneider, Société Générale, Thomson, Totalfina Elf, Valéo and Vivendi}. A VAR model allows us to test for a statistical relation between variables. Moreover, any VAR process parameters have to be estimated for

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stationary time series such as our asset returns. The related bidimensional VAR with p lags, called VAR(p), writes Yt = A0 + A1 Yt−1 + A2 Yt−2 + · · · + Ap Yt−p + εt where Yt = [RXt RSt ] is the vector of variables; A0 = [a01 a02 ] is a vector of j

constant parameters; Ap = [aip ] 1 ≤ i,j ≤ 2 is the coefficient matrix for lag p; and εt = [ε1t ε2t ] is the vector of innovations that is assumed to follow a normal law. In practice, disturbances may be correlated contemporaneously with each other, without being correlated with, on the one hand, their own lagged values, and on the other, all the lagged values of the variables. When disturbances (εt ) are correlated, the variation of one error component has an impact on the other components – variables have a synchronous influence on each other. A causality analysis allows then to study the kind of influence variables have on each other. Moreover, the optimal lag is determined while minimizing Akaike and Schwarz information criteria. We investigate optimal lags of one to five days while looking for a weekly influence at most, as compared to the four days of persistence documented by Koutmos and Knif (2002) for beta estimates. The maximum likelihood method then gives an optimal lag p of one. Such a first-order relationship between asset returns may result from asynchronous trading in the financial markets or asset prices’ speed of adjustment to new private/public information. Indeed, large firms’ asset prices integrate information more easily and quickly than small ones’ asset prices, since large firms’ assets are usually more liquid: as large firms’ assets are usually traded more frequently than small firms’ assets, their prices adjust more quickly to the arrival of new information. Moreover, McKenzie and Faff (2003) show that trading volumes and market returns determine time-varying autocorrelations of asset returns. This setting leads to the results shown in Table 10.5. a01 a111 a211 In each column, the coefficients of returns are displayed as 0 1 a2 a21 a221 with their related Student statistics between brackets under each coefficient. Moreover, the R2 statistic related to the estimation of each univariate relation is displayed in percent as: [R2 (RXt ) R2 (RSt )] . Recall that we have the next VAR(1) bivariate specification:

RXt RS t

=

a01 a02

+

a111 a211 a121 a221

RXt−1 RSt−1

+

ε1t ε2t

(10.5)

Our VAR(1) specification does not exhibit any influence between the implied market factor’s return and returns of French indices. As a rough guide, we also compute the statistics and coefficients related to our ten stocks’ VAR(1) specification (see Table 10.6).

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Table 10.5 VAR results for stock indices Index return CAC40

SBF120

SBF250

R 2 (%)

Coefficients −0.0546

−1.1382

1.0206

(−0.3488)

(−0.6794)

(0.6428)

1.2644

−0.0219

−0.6581

0.5539

0.7644

(−0.1327)

(−0.3721)

(0.3305)

−0.0487

−0.4722

0.4102

(−0.2888)

(−0.3079)

(0.2666)

0.5899 0.4677

−0.0053

−0.4560

0.4037

(−0.0317)

(−0.2995)

(0.2643)

−0.0609

−0.5750

0.5389

(−0.3473)

(−0.4061)

(0.3619)

0.7086

−0.0030

−0.5750

0.4852

0.4455

(−0.0178)

(−0.3738)

(0.3441)

At the 5 percent level of Student test, Air Liquide and Renault stocks impact implied a systematic risk factor, while Société Générale stock influences implied a systematic risk factor at the 10 percent level. We further investigate these results through a causality test. Granger causality test

A natural application of VAR modeling is a causality test. Granger (1969) defines causality as follows: RXt is said to be the cause of RSt when taking into account the information set associated to RXt helps to improve predictions of RSt . Analyzing causality of RXt towards RSt is equivalent to realizing a test with constraints on the coefficients of RXt in its VAR representation (Equation (10.5)) (a restricted VAR specification for RXt , also known as RVAR). Specifically, consider assumption H0 : a121 = a221 = 0. If we accept H0 , then RXt does not cause RSt . To test assumption H0 , we compare the unrestricted VAR (for example, UVAR, in Equation 10.5) with the VAR specification restricted to H0 (RVAR). The related test statistic is the likelihood ratio L = (n − c) ln{|RVAR |/|UVAR |} where n is the number of observations; c is the numberof estimated coefficients in each univariate relation of the UVAR model; RVAR , UVAR are the covariance matrices of restricted and unrestricted VAR models, respectively; |A| represents the determinant of matrix A. In this case, L is assumed to follow a chi-square law with two degrees of freedom (for example, χ2 (2)). Therefore, we reject H0 assumption for a given test level α if L is greater than the critical value of the χ2 (2) law 2 for level α (for example, L > χcritical (2); see Hamilton, 1994).

194

Table 10.6 VAR results for stocks Stock return Air Liquide

Danone

Vivendi

Totalfina Elf

Valéo

R 2 (%)

Coefficients −0.0588 (−0.4002) 0.1104 (0.5906)

−0.2090 (−1.4016) −0.2699 (−1.4223)

0.2525 (2.0454) 0.0547 (0.3481)

8.1354 4.1368

−0.0352 (−0.2366) 0.0329 (0.2584)

−1.1071 (−0.7840) 0.1050 (0.8981)

0.2532 (1.5238) −0.1050 (−0.7379)

4.8662 2.2082

−0.0078 (−0.0477) −0.5975 (−1.5866)

−0.1317 (−0.5784) 0.0990 (0.1881)

0.0367 (0.3665) 0.0317 (0.1371)

0.7152 0.5276

−0.0326 (−0.2106) 0.1860 (1.1717)

−0.0787 (−0.4343) −0.0572 (−0.3077)

0.0208 (0.1160) −0.1560 (−0.8388)

0.4753 3.9099

−0.0281 (−0.1817) 0.2474 (1.0045)

−0.0611 (−0.3786) 0.0482 (0.1875)

−0.0047 (−0.0455) 0.0080 (0.0485)

0.4527 0.1308

Stock return Société Générale

L’Oréal

Renault

Schneider

Thomson

R 2 (%)

Coefficients −0.0808 (−0.5361) 0.1979 (0.7984)

−0.3013 (−1.5920) 0.1241 (0.3985)

0.2120 (1.7529) −0.0068 (−0.0341)

6.2121 0.5678

−0.0032 (−0.0209) 0.1839 (1.0659)

0.0924 (0.5057) 0.0766 (0.3671)

−0.1809 (−1.2728) −0.5131 (−3.1637)

3.5726 24.0139

−0.1423 (−0.9186) 0.4518 (1.6360)

−0.2426 (−1.5658) 0.0960 (0.3477)

0.1884 (2.1443) 0.1453 (0.9278)

8.8326 3.5828

−0.0233 (−0.1521) 0.1484 (0.5015)

−0.0384 (−0.2468) 0.4611 (1.5319)

−0.0278 (−0.3571) −0.1657 (−1.1000)

0.7017 4.8022

−0.0295 (−0.1934) 0.0393 (0.1082)

−0.0671 (−0.4149) 0.0559 (0.1451)

0.0017 (0.0250) −0.0438 (−0.2698)

0.4498 0.1454

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Table 10.7 Granger statistics for indices and stocks Asset return

L

Probability

CAC40

0.1385 0.4132

0.7114 0.5233

SBF120

0.0897 0.0711

SBF250

Stock return

L

Probability

Valéo

0.0352 0.0021

0.8520 0.9639

0.7658 0.7909

Société Générale

0.1588 3.0727

0.6920 0.0858

0.1397 0.1310

0.7101 0.7190

L’Oréal

0.1348 1.6199

0.7151 0.2090

Air Liquide

2.0229 4.1838

0.1612 0.0461

Renault

0.1209 4.5982

0.7296 0.0369

Danone

0.8066 2.3218

0.3734 0.1339

Schneider

2.3468 0.1275

0.1318 0.7226

Vivendi

0.0354 0.1343

0.8515 0.7156

Thomson

0.0211 0.0006

0.8852 0.9801

Totalfina Elf

0.0947 0.0135

0.7596 0.9081

Note: bold indicates a chi-squares results.

Studying relationships between an implied systematic risk factor’s return and French stock returns, we tested two assumptions, namely: “H0 : RXt does not Granger cause RSt ” and “H0∗ : RSt does not Granger cause RXt ”, and obtained the results shown in Table 10.7. For each asset, the first and second lines correspond to the results of H0 and H0∗ assumptions, respectively. At the 15 percent level, Air Liquide and Renault returns cause the implied market factor’s return (RXt ). Enlarging our test level to 40 percent, Société Générale also causes the implied market factor’s return (RXt ). Our study therefore shows a smaller impact of the implied market factor on French assets than was expected. Our results’ weakness may come from the small sample size used. For further investigation, we look for contemporaneous links between variables without lag consideration. Specifically, we test for a non-linear influence of the implied market factor’s price on the prices of French stocks and indices.

10.5 FURTHER INVESTIGATION We attempt to exhibit non-linear dependence and “quadratic” causality between implied market factor and French stocks. Non-linearity is captured through the study of returns. We proceed in two steps: a regression analysis of asset returns and a volatility analysis of these daily returns.

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10.5.1 Simple regression Focusing on a non-linear link between the price of the implied market factor and the price of an asset is equivalent to regressing this asset return on the return of the implied market factor. Specifically, we look for the following kind of relationship β

S t = αS X t S

(10.6)

where αS and βS are constant terms, and St ∈ {SBF120, SBF250, CAC40, Air Liquide, Danone, L’Oréal, Renault, Schneider, Société Générale, Thomson, Totalfina Elf, Valéo Moreover, we make the approximation that Vivendi}.

t−1 t RSt = StS−S for each time t ranging from 2 to 55, and rewrite ≈ ln SSt−1 t−1 Equation (10.6) as a logarithm variation between times (t − 1) and t

RSt = βS RXt

(10.7)

for t ∈ {2, … , 55}. Consequently, the non-linear link between Xt and St is equivalent to a linear regression of St return (RSt ) on Xt return (RXt ). Such a study is practical, given that returns are stationary variables here. Moreover, our methodology consists of applying a single-index model that translates into a one-factor model that is close to CAPM (see Tables 10.8 and 10.9)6 . Regressions of French asset returns on the return of the implied market factor are all significant at the 1 percent level, apart from Danone stock’s

Table 10.8 Regression results for stock indices β

Student t

R 2 (%)

CAC40

1.0514

86.3097

99.2930

SBF120

0.9925

73.1893

99.0179

SBF250

0.9460

64.8178

98.7496

Index return

Table 10.9 Regression results for stocks β

Student t

R 2 (%)

Air Liquide

0.5650

3.8326

21.2243

Danone

0.1645

1.4883

Vivendi

1.8302

9.0541

Totalfina Elf

0.6694

6.2260

Valéo

0.8384

4.4587

Stock return

β

Student t

R 2 (%)

Société Générale

1.1088

7.1866

48.4465

3.9693

L’Oréal

0.8737

6.6437

44.7481

58.9820

Renault

0.9527

4.3560

18.8525

40.7725

Schneider

0.9480

3.9014

21.6706

25.0821

Thomson

1.2628

4.5825

28.2329

Stock return

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regression. Among available French stock indices, the highest explanatory power is reached for CAC40 (for example, R2 (CAC40) = 99.2930%), whereas the highest explanatory power among French stocks is reached for Vivendi stock (for example, R2 (ex) = 58.8820%). Therefore, the implied market factor has an important influence, in terms of explaining daily returns, on all our financial assets apart from Danone. Such a pattern indicates that the residual risk factor (idiosyncratic risk factor) additional to the systematic risk factor explains the main part of Danone stock’s evolution. We also tested for the assumption “H0 : βS = 1” in Equation (10.7) for stocks. We found that βS has a significant unit value only for Valéo, Société Générale, L’Oréal, Renault Schneider and Thomson stocks. Therefore, these six assets are driven purely by market trends as represented by the implied market factor. Moreover, Air Liquide’s, Danone’s, and Totalfina Elf’s, stock returns absorb the influence of the implied market factor’s return, whereas Vivendi’s stock return amplifies such an impact. Finally, our ten stocks are globally market-driven, since their returns exhibit a positive link with that of the implied market factor. Such a finding is coherent with the work of Campbell et al. (2001). Brailsford and Faff (1997) found poor support for CAPM when studying Australian daily stock returns. Moreover, in a daily stock return setting, Koutmos and Knif (2002) show that the simple regression model works well for systematic risk measurement purposes (estimating the beta coefficient). However, a dynamic model with time-varying parameters is better for forecast purpose (forecasting efficient conditional beta estimates). Our main goal is to assess the impact of systematic risk on French stocks rather than value the validity and performance of CAPM (that is, to assess the mean-variance efficiency of our market proxy). Explanations about validity and performance of CAPM are proposed by Roll (1977) and Campbell et al. (1997), among others. Jagannathan and Wang (1996) also propose a good performance study. However, given the closeness of our single-index model to CAPM, we further investigate some linear dependency between returns of our implied market factor and French stocks. For this purpose, we employ the one-factor model of Jensen (1968, 1969) for each time t ranging from 2 to 55, namely RSt − r1M (t) = αS + βS (RXt − r1M (t)) + εt , where RXt is the return of the implied market factor X at time t; RSt is the return of stock S at time t; r1M (t) is the one-month French risk-free rate; βS is the sensitivity of stock S to the implied market factor X; αS is a constant term of regression; εt is a random normal error with zero expectation and constant variance; RSt − r1M (t) and RXt − r1M (t) are, respectively, stock S and implied market factor X market-risk premia. Jensen’s methodology allows the assessment of a risk-adjusted performance, the relevant risk measure being the beta of Sharpe (1963). The alpha coefficient of the previous regression is known as Jensen’s alpha and represents the abnormal return or excess return of a given stock relative to its CAPM return if this model were valid. In fact, alpha is the

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Table 10.10 Jensen’s regression results for stocks Stock return

α

Student t α

β

Student t β

R 2 (%)

CAC40

0.1945

4.6472

1.0511

88.2375

99.3366

SBF120

0.0191

0.4367

0.9922

79.8204

99.1905

SBF250

−0.1257

−2.7697

0.9457

73.1643

99.0379

Air Liquide

−1.3580

−2.6126

0.5640

3.8116

21.8380

Danone

−2.7691

−7.0750

0.1663

1.4924

4.1072

Vivendi

−0.2698

−0.5857

0.8737

6.6611

46.0417

Totalfina Elf

0.4514

0.6223

0.9511

4.6061

28.9774

Valéo

0.0167

0.0195

0.9475

3.8880

22.5227

Société Générale

0.5767

1.0732

1.1062

7.2304

50.1336

L’Oréal

0.9755

1.0011

1.2599

4.5414

28.3987

Renault

−0.9362

−2.5121

0.6680

6.2962

43.2574

Schneider

−0.2530

−0.3877

0.8357

4.4978

28.0081

Thomson

2.2339

3.3201

1.8333

9.5707

63.7881

non-equilibrium return that the stock brings in over the studied time horizon (see Table 10.10). Jensen’s alpha is significant for SBF250, CAC40, Air Liquide, Danone, Renault and Thomson stock returns. And the alpha is negative (that is, an abnormal return leading to a loss in value for an investment in the considered stock) for SBF250, Air Liquide, Danone, Vivendi, Renault and Schneider returns. Moreover, the beta coefficient is significant for all indices and stocks apart from Danone’s stock return, whose beta is close to zero. Hence, Danone’s stock evolution is uncorrelated or extremely low-correlated with the market, which means that this stock is low or not sensitive to market evolution. Put differently, Danone and Thomson stocks exhibit the lowest and highest beta coefficients (systematic risk), respectively, whereas beta estimates of CAC40, Société Générale, L’Oréal and Thomson returns lie above unity (for example, amplify the market effect). Finally, the explanatory power of our regressions is globally good in so far as CAC40 and Thomson stocks exhibit the highest explanatory power among indices and stocks, respectively (R2 (CAC40) = 99.3366% and R2 (Thomson) = 63.7881%). In contrast, Danone stock exhibits the lowest explanatory power (for example, R2 (Danone) = 4.1072%). Consequently, the implied market factor generally has a strong impact and influence in explaining stock return evolutions. Such an influence is nevertheless insufficient to explain the whole evolution of assets given both the limited explanatory power of regressions and the significance of Jensen’s alpha. Such a pattern can be explained by firm-specific features (for example, size effect) that are left aside while considering only systematic risk’s impact on French stocks (see Fama and French, 1992, 1993; Berk, 1995). This point is emphasized with

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Danone stock (Danone) whose evolution is not explained by the overall latent systematic risk factor prevailing in the French market. However, stronger evidence concerning the influence of the implied market factor on French stocks can be found while further investigating non-linear dependency between asset returns.

10.5.2 Volatility impact Investigating non-linear relationships between returns, we study the influence of the implied market factor on the volatility of our French assets. Indeed, linear causality analysis is unable to account for non-linear dependency between financial assets (for example, implied market factor and French stocks). Non-linear phenomena describing both financial markets and the underlying dynamics of the various assets composing such markets have been widely documented in the financial literature. Mele (1998) explains different kinds of non-linear dynamics, volatility and equilibrium that may describe a financial market. The simple existence of conditional heteroskedasticity in asset prices already describes some non-linear patterns in financial markets (Gourieroux and Jasiak, 2001). Put differently, exhibiting links between asset volatilities is a means of accounting for the non-linear features and effects that prevail between assets in markets. To this end, testing for a quadratic dependency between returns, we consider the weekly rolling volatilities of assets. As one calendar week represents five working days (a financial week), the weekly rolling volatility of return RSt at date t is written as σ(RSt ) = t t 1 2 with R = 1 (R − R ) (RSi ) for t ∈ {6, . . . , 55}. We analyze the S S S i t t 5 5 i=t−4

i=t−4

impact of the volatility of the implied market factor while considering the following first differences regressions: (10.8) σ RSt = aS σ RXt + ηt where ∀t ∈ {7, . . . , 55}, ∀ Xt , σ(RXt ) = σ(RXt ) − σ(RXt−1 ); aS is a constant coefficient; ηt is a “normal” disturbance; St ∈ {SBF120, SBF250, CAC40, Air Liquide, Danone, L’Oréal, Renault, Schneider, Société Générale, Thomson, Totalfina Elf, Valéo and Vivendi}. Results for first difference regressions (Equation (10.8)) of the weekly rolling volatilities of French assets on the weekly rolling volatility of the implied market factor are listed in Tables 10.11 and 10.12. Volatility regressions in Equation (10.8) are significant at a 1 percent level for CAC40, Vivendi, Totalfina Elf, Valéo, Société Générale, L’Oréal, Renault, SBF120 and SBF250. Among French indices, SBF250 presents the highest explanatory power (R2 (SBF250) = 97.1677%) whereas Vivendi exhibits the highest explanatory power (R2 (Vivendi) = 52.1625%) among stocks. Results

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Table 10.11 Volatility regression results for stock indices a

Student t

R 2 (%)

CAC40

1.0329

40.2860

97.1143

SBF120

0.9614

39.3774

96.9823

SBF250

0.9349

40.6838

97.1677

Index return

Table 10.12 Volatility regression results for stocks Stock return

a

Student t R 2 (%)

Stock return

a

Student t R 2 (%)

2.2470 Société Générale

0.8547

2.8667 14.4792

0.2888

0.2015 L’Oréal

0.4608

4.0223 23.6650

7.2466

52.1625 Renault

0.8075

2.7771 13.8271

0.3720

2.8797

14.7152 Schneider

−0.0938 −0.3519

0.3325

0.6732

2.7991

14.0162 Thomson

−0.5603 −1.3889

3.7211

Air Liquide

0.1634

1.1522

Danone

0.0402

Vivendi

1.7898

Totalfina Elf Valéo

suggest that the implied market factor has a strong influence on the weekly rolling volatilities of CAC40, L’Oréal, Renault, SBF120, SBF250, Société Générale, Totalfina Elf, Valéo, and finally Vivendi assets. However, from the explanatory power of regressions, the implied market factor fails to explain the whole evolution of assets. As shown by Campbell et al. (2001) and Goyal and Santa-Clara (2003), idiosyncratic risk should be the additional factor explaining the part of stock returns that is unexplained by systematic risk factor. Results require a global remark drawing a comparison between the global market information embedded in both the CAC40 stock index and its filtered counterpart, as represented by implied market factor X. The results we get while using the CAC40 stock index as a market proxy instead of an implied market factor give preliminary insights. Indeed, as the correlation between returns of the CAC40 stock index and implied market factor is 99.6666 percent, why should the CAC40 stock index not be the market? Given our results, the average (that is, arithmetic mean over time horizon) price of the CAC40 Index is 2.2881 times the average price of the implied market factor, with an average scale factor of 2.2883. Stated differently, the CAC40 INDEX average return is 3.8164 times the implied market factor’s average return. However, we also notice that the average ratio of the CAC40 INDEX return to the implied market factor’s return is 0.9636. These preliminary statistics suggest that CAC40 is a good market proxy if we assume that the implied market factor is the actual market portfolio. However, we cannot draw such

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a conclusion when considering the CAC40 INDEX as a market proxy. Moreover, being interested only in a view of the level of market return, CAC40, as well as the SBF120 and SBF250 indices, seem to be a convenient approximation of market return (as represented by implied market factor). Recall that both SBF120 and SBF250 indices are correlated almost as highly with the implied market factor as the CAC40 stock index. Nevertheless, for forecast purpose, this viewpoint changes greatly. Such results are introduced and summarized in the next section, which makes a comparison between our four different benchmarks.

10.6 MARKET BENCHMARK COMPARISON To answer the question about choosing between implied market factor and the CAC40 INDEX, we compare the results we get when employing successively Xt , CAC40, SBF250 and SBF120 indices as market benchmarks. First, we summarize the results obtained for our four different benchmarks, then we make a study of forecasting performance.

10.6.1 Basic empirical study We have estimated the three previous types of regressions that we called simple, Jensen and volatility regressions. We summarize the results in this section, displaying only relevant results to save space. We use two criteria to discriminate between market benchmarks. First, we consider the explanatory power of related regressions, and second, how close beta estimates of such regressions lie relative to the beta estimates we get for implied market factor X. In this way, we observe the impact of the bias, which comes from the fact that studied stocks are part of our three French stock indices. Recall that the Granger causality test allows for the classifying of benchmark-based relations with decreasing value of significance as Xt , SBF250, SBF120 and the CAC40 INDEX. We display the results for the explanatory power of our three types of regressions in Table 10.13. The first, second and third lines of each asset correspond to the simple, Jensen and volatility regressions, respectively. As regards simple regressions, SBF250-based regressions exhibit the highest explanatory power for 50 percent of stocks (Valéo, Société Générale, Schneider, Renault and Thomson returns) whereas CAC40-based regressions exhibit the highest explanatory power for 30 percent of stocks (Air Liquide, Totalfina Elf and L’Oréal returns). In a less powerful way, SBF120-based regressions exhibit the highest explanatory power for 10 percent of stocks (Danone returns) analogously to Xt -based regressions (Vivendi returns). As regards Jensen-type regressions, considering the proportion of stocks

202

Table 10.13 Explanatory power of regressions (percentages) Stock

Xt

CAC40

SBF120

SBF250

Stock

21.2243 21.8380 2.2470

21.4424 21.9272 4.0148

20.1345 20.5040 3.1089

20.0694 20.3780 2.7512

Danone

3.9693 4.1072 0.2015

4.1214 4.2446 0.0354

4.1579 4.2681 0.1733

Vivendi

58.9820 46.0416 52.1625

57.6816 46.8405 47.8086

Totalfina Elf

40.7725 28.9774 14.7152

Valéo

25.0821 22.5227 14.0162

Air Liquide

Xt

CAC40

SBF120

SBF250

Société Générale

48.4465 50.1336 14.4792

50.4244 51.7562 17.1930

50.2269 51.2400 15.8954

50.4256 51.2706 16.1946

4.0683 4.1740 0.0841

L’Oréal

44.7481 28.3987 23.6650

45.8329 27.9946 26.9866

44.8268 29.3573 25.4735

44.0374 29.5735 23.1302

56.7106 45.6010 47.8346

56.4111 44.7000 49.3285

Renault

18.8525 43.2574 13.8271

18.8192 44.9391 14.9337

18.9169 42.8588 14.1422

19.4062 42.1550 13.6872

42.8574 28.2652 17.3273

41.1395 27.7130 15.9644

40.6282 27.8279 14.6540

Schneider

21.6706 28.0081 0.3325

22.2030 28.9954 0.3286

23.7321 30.7258 0.1534

24.1688 31.3020 0.0724

26.4305 22.8926 14.3045

28.5368 24.2689 14.3560

29.3201 24.6283 15.0222

Thomson

28.2329 63.7881 3.7211

27.9002 63.2092 3.4569

29.3254 63.0331 2.6751

29.5660 63.2194 3.1387

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that are best explained (highest explanatory power) through our four market benchmarks leads to classifying benchmarks with a decreasing proportion’s of value as CAC40 (40 percent of stocks, namely Air Liquide, Vivendi, Société Générale and Renault returns); SBF250 (30 percent of stocks, namely Valéo, L’Oréal and Schneider returns); Xt (20 percent of stocks, namely Totalfina Elf, and Thomson returns); and SBF120 (10 percent of stocks, namely Danone returns). As regards the explanatory power of volatility-based regressions, comparing the proportion of best-explained stocks through our different market proxies allows us to order results with decreasing proportion’s of value as CAC40 (50 percent of stocks, namely Air Liquide, Totalfina Elf, Société Générale, L’Oréal and Renault returns); Xt (40 percent of stocks, namely Danone, Vivendi and Thomson returns); SBF250 (for example, 10 percent of stocks, namely Valéo’s return); and SBF120. However, explanatory power-based results are probably upward-biased because of the weights of studied stocks that belong to available French stock indices (for example, important free-floating market capitalization weights in indices). To bypass such a bias, we consider how close the beta estimates of our different benchmark-based regressions are to the beta estimates of Xt -based regressions. For an overview, we display related beta estimates in Table 10.14. For each stock return, the first, second and third levels refer to simple, Jensen-type and volatility regressions, respectively. As regards simple regressions, SBF120-based regressions exhibit the closest estimates for Danone, Vivendi, Totalfina Elf, Société Générale, L’Oréal, Renault and Thomson stocks (70 percent of stocks) while CAC40- and SBF250-based regressions exhibit the closest estimates for Valéo and Schneider stocks (20 percent of stocks), and Air Liquide stock (10 percent of stocks), respectively. As regards Jensen-type regressions, SBF120-based regressions exhibit the closest beta estimates for Danone, Vivendi, Valéo, Totalfina Elf, Société Générale, L’Oréal, Renault and Thomson stocks (80 percent of stocks) while SBF250- and CAC40-based regressions exhibit the closest estimates for Air Liquide (10 percent of stocks) and Schneider stocks (10 percent of stocks), respectively. As regards volatility regressions, CAC40-based regressions exhibit the closest beta estimates for Danone, Totalfina Elf, Valéo, Société Générale, L’Oréal, Renault and Schneider stocks (70 percent of stocks) while SBF250- and SBF120-based regressions exhibit the closest estimates for Air Liquide and Thomson stocks (20 percent of stocks), and Vivendi-stock (10 percent of stock), respectively. Hence, given the closeness to the beta estimates of Xt -based regressions, the SBF120 index seems to be the best proxy for implied market factor for both simple and Jensen-type regressions (best for 80 percent, on average). Therefore, assuming that Xt is the actual market portfolio and given that the correlation between returns of Xt and SBF120 is 0.9959, employing SBF120 as a market proxy follows the findings of both Kandel and Staumbaugh (1987) and Shanken (1987). But differently, the CAC40 INDEX seems to be the best proxy of the implied market factor for

204

Table 10.14 Beta estimates for three types of regressions (1) Stock

Xt

CAC40

SBF120

SBF250

Stock

Air Liquide

0.5650 0.5640 0.1634*

0.5381 0.5359 0.2005*

0.5521 0.5486 0.1922*

0.5756 0.5734 0.1876*

Société Générale

Danone

0.1645*

0.1588*

0.1688*

0.1749*

L’Oréal

0.1663* 0.0402*

0.1603* 0.0448*

0.1702* 0.0605*

0.1764* 0.0543*

Vivendi

1.8302 0.8737 1.7898

1.7167 0.8356 1.6351

1.8017 0.8728 1.7560

1.8831 0.9059 1.8353

Renault

Valéo

0.6694 0.9511 0.3720

0.6495 0.8907 0.3851

0.6739 0.9337 0.3969

0.7020 0.9808 0.3914

Totalfina Elf

0.8384 0.9475 0.6732

0.8135 0.9058 0.6489

0.8907 0.9873 0.6979

0.9448 1.0427 0.7348

Note: *Non-significant estimates at a 5% test level.

Xt

CAC40

SBF120

SBF250

1.1088 1.1062 0.8547

1.0714 1.0658 0.8879

1.1312 1.1227 0.9170

1.1875 1.1773 0.9526

0.8737

0.8378

0.8767

0.9107

1.2599 0.4608

1.1861 0.4672

1.2858 0.4883

1.3530 0.4808

0.9527 0.6680 0.8075

0.9024 0.6457 0.8006

0.9562 0.6675 0.8365

1.0102 0.6940 0.8471

Schneider

0.9480 0.8357 −0.0938*

0.9091 0.8063 −0.0892*

0.9930 0.8786 −0.0610*

1.0497 0.9297 −0.0391*

Thomson

1.2628 1.8333 −0.5603*

1.1898 1.7305 −0.5161*

1.2901 1.8294 −0.4903*

1.3572 1.9207 −0.5445*

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volatility regressions. We investigate such preliminary results through a performance study.

10.6.2 Forecasting performance We attempt to discriminate between our four market proxies while considering their weekly forecasting performance in our three types of regressions. For this purpose, we first estimate regressions on the initial time horizon less one week of data (five observations). Then, we forecast corresponding returns or volatilities on the remaining week. Finally, we assess the related performance while computing the related average absolute relative error. Hence, we can assess the realized forecasting error relative to the actual level of return or volatility. Simple, Jensen-type and volatility regressions are successively estimated on t ∈ {2, … , 50}, {2, … , 50} and {7, … , 50} time horizons, respectively. We drop the last week of data for t ∈ {51, … , 55}. We display in Table 10.15 the beta estimates for our three types of regressions while employing successively our four market proxies. For each stock return, the first, second and third lines refer to simple, Jensen-type and volatility regressions, respectively. With regard to the closeness of our beta estimates to Xt -based beta estimates, a similar conclusion to that in the previous sub-section applies. On average (for 75 percent of stocks), the SBF120 index represents the best proxy of implied market factor for both simple and Jensen-type regressions whereas the CAC40 INDEX is the best proxy of Xt for volatility regressions. For the second part of our study, we first use previous regression estimates to forecast related returns and volatilities in the last week of our initial time horizon. Then, to assess the weekly forecasting performance of our benchmark-based regressions, we compute the corresponding average relative absolute error (average normalized absolute error). Such a performance measure allows us to highlight the percentage of forecasting errors relative to the actual level of both returns and volatilities under consideration. For this purpose, we compute respective forecasting errors eS of stock S during the last week of data as 55 ∗ ˆ St R − R 1 S eS = t ∗ RS 5 t t=51 ˆ St − rˆ1M (t) 55 R∗ − r1M (t) − R S 1 t eS = ∗ 5 RSt − r1M (t) t=51 and

55 ∗ R σ − σ ˆ RSt 1 S t eS = ∗ 5 σ RSt t=51

206

Table 10.15 Beta estimates for three types of regressions (2) Xt

CAC40

SBF120

SBF250

Stock

Air Liquide

0.5789 0.5808

0.5520 0.5522

0.5664 0.5654

0.5911 0.5895

Société Générale

0.1397*

0.1748*

0.1740*

0.1714*

Danone

0.1644

0.1582

0.1630*

0.1672*

Stock

Vivendi

Valéo

Totalfina Elf

L’Oréal

Xt

CAC40

SBF120

SBF250

1.1122 1.1170

1.0748 1.0750

1.1314 1.1282

1.1878 1.1827

0.8820

0.9159

0.9537

0.9977

0.8533

0.8187

0.8569

0.8910

0.1654

0.1596

0.1650*

0.1696*

1.2302

1.1586

1.2588

1.3232

0.0200

0.0288

0.0465*

0.0363*

0.4747

0.4778

0.5062

0.5036

1.8708

1.7585

1.8422

1.9279

0.9135

0.8667

0.9235

0.9751

0.8577

0.8201

0.8565

0.8898

0.6808

0.6582

0.6842

0.7133

1.8490

1.6884

1.8192

1.9103

0.7195

0.7232

0.7699

0.7714

Renault

0.6756

0.6574

0.6868

0.7178

0.9949

0.9546

1.0407

1.0998

0.9291

0.8714

0.9200

0.9667

0.8648

0.8373

0.9086

0.9622

0.3930

0.4014

0.4228

0.4228

−0.1106*

−0.1105*

−0.1098*

−0.0954*

0.8592

0.8370

0.9125

0.9686

1.2321

1.1606

1.2607

1.3247

1.0017

0.9567

1.0395

1.0969

1.8570

1.7554

1.8495

1.9424

0.7090

0.6761

0.7209

0.7621

−0.6688*

−0.6184*

−0.6215*

−0.6960*

Note: *Non-significant estimates at a 5% test level.

Schneider

Thomson

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207

for simple, Jensen-type and volatility regressions. RS∗t , RS∗t − r1M (t) and σ ∗ (RSt ) are, respectively, observed market values of stock return, market risk premium and the first difference of weekly rolling volatility of stock St , while Rˆ St , (Rˆ St − rˆ1M (t)) and σ(R ˆ St ) are corresponding respective regression estimates of asset return, market risk premium and first-order weekly rolling volatility of asset St . We display our results in Table 10.16, where the first, second and third lines of each stock refer to simple, Jensen-type and volatility regressions, respectively. With regard to simple regressions, we classify benchmark-based regressions as an increasing function of the average relative absolute error, and get Xt implied factor, SBF250, SBF120 and CAC40 indexes. Indeed, Xt -based regressions exhibit the lowest absolute errors for Air Liquide, Totalfina Elf, Renault, and Schneider returns (40 percent of the stocks) while SBF250 and SBF120-based regressions exhibit the lowest ones for Danone, Vivendi and Thomson returns (30 percent of stocks), and Valéo, Société Générale, and L’Oréal returns (30 percent of stocks), respectively. Moreover, the mean of average absolute relative errors over all stocks for SBF250-based regressions is 130.8780 percent, and lies below the mean of SBF120-based regressions at is 131.2630 percent. On average, for 50 percent of cases, Xt average relative absolute errors are below the observed absolute errors of other stock indexbased simple regressions. With regard to Jensen-type regressions, ordering benchmark-based regressions with increasing value of average relative absolute errors results in Xt implied factor, SBF250, SBF120 and CAC40 indexes. In particular, Xt -based regressions exhibit the lowest average relative absolute errors for Totalfina Elf, Valéo, L’Oréal, Renault and Schneider returns (50 percent of the stocks), while SBF250 and SBF120-based regressions exhibit the lowest errors for Air Liquide, Danone, Société Générale and Thomson returns (40 percent of stocks), and Vivendi returns (10 percent of stocks), respectively. On average for 63.3334 percent of cases, Xt -based regressions exhibit average relative absolute errors that lie below the ones observed for the other stock index-based Jensen regressions. With regard to volatility regressions, ordering benchmark-based regressions with increasing value of average relative absolute errors results in Xt implied factor, SBF250, SBF120 and CAC40 indexes. Specifically, Xt -based regressions exhibit the lowest average relative absolute errors for Vivendi, Totalfina Elf, Société Générale and Renault returns (40 percent of stocks) while SBF250-based regressions exhibit the lowest for Air Liquide, Valéo and Schneider returns (30 percent of stocks). CAC40-based regressions exhibit lowest average relative absolute errors for L’Oréal and Thomson returns (20 percent of stocks), whereas SBF120-based regressions exhibit the lowest ones for Danone returns (10 percent of stocks). On average, for 50 percent of cases, Xt -based average relative absolute errors are lower than other benchmark-based average relative absolute errors.

208

Table 10.16 Average absolute relative errors for three types of regressions (percentages) Stock

Xt

CAC40

SBF120

SBF250

Air Liquide

190.2920 34.1998 93.6811

198.2960 34.4080 91.3792

199.7410 34.5058 89.5556

202.1020 34.1037 89.0935

Danone

108.7540 40.9120 100.5850

108.2980 40.8938 100.3970

106.8060 40.5194 97.5889

Vivendi

198.2090 21.3727 170.7980

203.5010 21.4338 181.0560

98.5802 75.7842 98.5722

Valéo

Mean error

Totalfina Elf

Stock

Xt

CAC40

SBF120

SBF250

Société Générale

156.1130 35.9794 156.3100

156.9040 34.9911 163.5170

145.9650 32.6395 165.2350

147.3510 32.2085 174.7310

105.9940 40.3935 97.6414

L’Oréal

58.8281 206.2060 228.1930

58.7250 208.0390 206.3310

56.9955 209.0130 242.5520

58.6216 206.6140 268.5740

179.0770 20.8868 285.0580

167.8610 21.3040 324.9240

Renault

118.5050 24.4820 76.1971

119.2220 24.7314 77.8722

123.4980 26.0944 101.8760

123.9320 26.5305 104.7220

99.4735 77.2388 103.1750

105.3530 78.5040 154.4290

107.1970 78.5563 187.6690

Schneider

178.6000 117.4490 134.1100

180.7920 119.7430 132.5430

200.0930 118.7390 127.7270

204.9850 119.3130 126.0830

119.8940 33.9410 124.9710

121.7060 34.1834 138.8180

118.5630 34.1897 122.2820

118.5890 33.9615 121.3980

Thomson

87.5991 93.7890 235.0740

84.4210 96.5114 91.5840

76.5346 90.8615 113.5140

72.1501 90.3767 164.9720

131.5370 68.4115 141.8490

133.1340 69.2174 128.6670

131.2630 68.5953 149.9820

130.8780 68.3362 165.9810

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The extremely good performance exhibited by the explanatory powers of some index-based regressions probably results from an upward bias. Such a bias comes from the non-negligible weights assigned to studied stocks from which French market indices are composed. Also, beta estimate-based analysis indicates globally that, among observed market proxies, SBF120based regressions exhibit the closest beta estimates relative to the ones we get for Xt -based regressions. On the other hand, CAC40-based regressions are far behind SBF120-based ones when considering linear relations between asset returns. Hence, in the prospect of an assessment of the impact of systematic risk on French stocks, the CAC40 INDEX represents a much less accurate approximation of market return than the implied market factor (fewer significant relations between returns). Finally, our forecasting performance study based on average relative absolute errors suggests that the Xt benchmark (implied market factor) is, on average, a more powerful proxy for systematic risk factor than our three French stock indices. Specifically, our implied market factor has a far more powerful forecasting performance than the CAC40 stock index. Such a viewpoint is sustained by the results of Jensen-type regressions when we employ the CAC40 INDEX as a market proxy (rather than implied market factor). Regarding the negative estimated abnormal returns (negative alpha coefficients): these have a higher absolute value for the CAC40 benchmark than for the implied market factor. Hence, the CAC40 benchmark leads to higher abnormal losses. On the other hand, regarding positive estimated alpha coefficients, abnormal returns brought in by French stocks are lower for the CAC40 benchmark than for the implied market factor. Moreover, there is a difference of sign for Jensen’s alpha of the stock Valéo. In conclusion, assuming that implied market factor X is an accurate proxy of the actual market portfolio, the CAC40 INDEX leads to an underestimation of positive abnormal returns and an overestimation of losses or negative abnormal returns brought in by French stocks. Consequently, CAC40 stock index does not represent the market, though it has been employed to infer the level of implied market risk factor (through some non-linear filtering methodology).

10.7 CONCLUSION Considering the wide literature about systematic risk initiated by Sharpe (1963) and the debate initiated by the famous critique by Roll (1977), we addressed the problem of finding a good market risk proxy when considering a small stock index with traded options on such an index. We proceeded in five steps: a theoretical framework; an empirical application of this setting; two empirical studies assessing impact of the implied systematic risk

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on French financial assets; and a critical study (forecasting performance analysis of available market benchmarks). First, our theoretical setting assumed that the small stock index was a disturbed observation of the actual market factor. This stock index depends on the market factor through a scale parameter, which is a continuous function of time. We further assumed that the market factor follows a geometric Brownian motion. Then, we induced an analytical formula pricing European calls on the stock index. All the parameters of our closed form formula are known apart from the scale parameter at times t and T, and the volatility of the market factor. Second, inverting our European call pricing formula given observed market prices of European index calls, we calculated the values of the scale factor at dates t and T, and the volatility parameter. These estimates allowed the computation of the implied market factor’s level from stock index prices. We applied this empirical study to the French financial market, and considered its CAC40 stock index. Results showed that the implied volatility parameter is time-varying, and the distributions of both volatility and market factor are leptokurtic. Third, we attempted to assess the implied market factor’s impact on a basket of French stocks and indices. We studied correlations between the implied market factor’s return and French asset returns. Results are poor in so far as our VAR study and the Granger causality test only show the strong influence of Air Liquide and Renault daily stock returns on the implied market factor’s return. Fourth, we investigated a non-linear relationship between French asset prices and the level of implied market factor. This led to the study of linear regressions of French asset returns on the implied market factor’s return. Our linear framework assumes that residual risk, which we assimilated to idiosyncratic risk, is normally distributed, with zero mean and constant variance. Results obtained are fruitful in that the implied market factor’s return appears to have a strong influence on French asset returns, apart from on Danone stock. Indeed, regressions exhibit high explanatory power. Further, we also estimated first differences regressions of French assets’ weekly rolling volatilities on the weekly rolling volatility of implied market factor. The implied market factor exhibits a strong link with CAC40, L’Oréal, Renault, SBF120, SBF250, Société Générale, Totalfina Elf, Valéo and Vivendi assets. However, it fails in explaining the whole evolution of assets, probably because idiosyncratic risk plays an important role. Indeed, such a risk factor can explain that part of assets’ evolution which remains unexplained by systematic risk factors, as suggested by Campbell et al. (2001) and Goyal and Santa-Clara (2003). Finally, we attempted to discriminate between implied market factor and French stock indices as a market proxy. Specifically, our forecasting performance study examines the highest correlation observed between both the CAC40 INDEX and implied market factor returns. Namely, the CAC40 INDEX can be a useful proxy to estimate current level of market return, whereas implied market factor is a more convincing market benchmark for

HAYETTE GATFAOUI

211

forecast purpose and performance measure (systematic risk level), along with the CAPM setting. Suggested improvements are, first, the lengthening of the time period. A larger sample could give stronger and more significant results. Second, building a diversified portfolio (which replicates market factors accurately) would give a systematic risk benchmark to be compared with implied market factor. Prior to this, we should address what the optimal number of stocks and optimal composition of a well-diversified portfolio should be to achieve a sound and standardized assessment of systematic risk. Third, as in CAPM theory, firm-specific risk or unsystematic risk was not considered to explain realized stock returns. Our single factor framework ignores the part of any return’s global variance that is a result of firm-specific patterns. However, firm-specific factors are important explanatory variables for asset returns, as shown by Fama and French (1992, 1993) and Berk (1995). Hence, given Roll’s critique and advice in favor of multi-index models, future research should apply at least a two-factor model accounting for both systematic and idiosyncratic risk factors.

NOTES 1. Improved versions of CAPM are also given by Mossin (1966), Lintner (1965, 1969) and Black (1972). Dynamic versions of CAPM are also proposed, along with intertemporal models like that in Merton (1974). 2. Milevsky (2002) studies two dimensions of diversification, namely the number of stocks in a portfolio, and the time horizon for investment. The author discusses the benefits of the number of stocks diversification versus time horizon diversification. 3. Under completeness, financial asset prices can be reached (for example, each market variable is observable or has a proxy). 4. Drift and volatility parameters must satisfy the Lipschitz conditions, which ensure the existence and uniqueness of the solution to the stochastic differential equation satisfied by the market factor’s dynamic (given a starting value). 5. The SBF120 index is a weighted stock index composed of the forty values of CAC40 INDEX and another eighty most liquid French stocks. The SBF250 index is a weighted index composed of the 120 stocks of the SBF120, and 130 stocks selected for their importance and sector representativity. There is no traded option on such indices in the MONEP. 6. We also performed regressions with a constant term. Unfortunately, the constant coefficient does not generally appear to be significant, and its estimated value is very different from the levels of the one-, two- or three-month French risk-free rates observed in the market. We notice that constant term of regression =(1 – β) rf 55 1 for rf ∈ {¯r1M , r¯2M , r¯3M } with r¯iM = 54 riM (t) whatever i = 1, 2, 3. t=2

ACKNOWLEDGMENTS I would like to thank participants and referees at the MODSIM conference (Townsville, Australia, July 2003) and Professor Peter Verhoeven (Curtin University of Technology)

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for their helpful comments. I am also grateful to Professor Pascal St-Amour (HEC Montreal) and attendees of the EFMA annual meeting (Basle, Switzerland, July 2004) for their interesting remarks and suggestions. Finally, I thank participants at the 17th IAE National Days (Lyon, France, September 2004) and Deloitte Risk Management Conference (Antwerp, Belgium, May 2005) for their interesting comments. The usual disclaimer applies.

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C H A P T E R 11

Matrix Elliptical Contoured Distributions versus a Stable Model: Application to Daily Stock Returns of Eight Stock Markets Taras Bodnar and Wolfgang Schmid

11.1 INTRODUCTION The assumptions of independency and normality are not appropriate in many situations of practical interest, especially in modeling financial data from emerging markets. It was pointed out in numerous studies that daily financial data is heavily tail distributed (Blattberg and Gonedes, 1974; Fama, 1976; Engle, 1982; Bollerslev, 1986; Nelson, 1991; Rachev and Mittnik, 2000). These studies proposed to pick up the assumptions of t-distribution, symmetric stable distribution, or the autoregressive conditional heteroskedasticity (ARCH) process instead of normality. In this chapter, the much weaker assumption of matrix ellipticalcontoured distribution is imposed on the asset returns. This family covers a wide range of distributions – for example, the matrix normal distribution, the matrix mixture of normal distribution, the matrix t-distribution and 214

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215

the matrix symmetric stable distribution. Elliptical distributions, whose contours of equal densities have the same elliptical shape as the normal distribution, provide an attractive alternative to the multivariate stable law. These distributions have been already discussed in financial literature (Chamberlain, 1983; Owen and Rabinovitch, 1983; Zhou, 1993; Berk, 1997; Bodnar and Schmid, 2003, 2004). For instance, Owen and Rabinovitch (1983) showed that Tobin’s separation theorem, Bawa’s rules of ordering certain prospects can be extended to elliptically contoured distributions. While Chamberlain (1983) showed that elliptical distributions imply mean– variance utility functions, Berk (1997) argued that one of the necessary conditions for the capital asset pricing model (CAPM) is an elliptical distribution for the asset returns. Furthermore, Zhou (1993) extended findings of Gibbons et al. (1989) by applying their test for the validity of the CAPM to elliptically distributed returns. The first paper dealing with the application of matrix elliptically contoured distributions in finance, however, seems to be Bodnar and Schmid (2003). They introduced a test for the global minimum variance. It is analyzed whether the lowest risk is larger than a given benchmark value or not. The aim of the present study is to derive the statistical procedures for testing the elliptical symmetry of multivariate sample, for example, matrix ellipticity. While there are several procedures for testing the multivariate elliptical and spherical symmetry under independency assumptions (Beran, 1979; Baringhaus, 1991; Fang et al., 1993; Heathcote, et al., 1995; Koltchinskii and Li, 1998; Manzotti et al., 2002; Zhu and Neuhaus, 2003), the matrix elliptical symmetry was not treated in literature up to now. Furthermore, only the limiting distributions of above-mentioned statistics were derived in the studies. Conversely, our approaches are based on the small sample tests. Empirically we show that daily returns of seven developed countries follow a matrix elliptical distribution. This result is in line with the findings of Andersen et al. (2001), who, using the distributional properties of realized volatility, argue that daily returns can be well approximated by the mixture of normal distributions (the partial case of elliptical family). The remainder of the chapter is organized as follows. The main results are presented in section 11.2. Under the null hypothesis of matrix elliptical symmetry, the finite sample distributions of the proposed statistics are derived in all cases when the type of elliptical symmetry, location vector and scale matrix are known or unknown. The test’s powers are considered in section 11.3. In section 11.4 we implement our findings empirically by considering daily returns of seven developed stock markets. We show that the null hypothesis of the matrix ellipticity cannot be rejected for this data. Furthermore, as the test power for the symmetric stable distribution is always very high, a practitioner should be very careful with modeling daily financial data by using the stable law. Final remarks and conclusions are presented in Section 11.5. All proofs are given in the Appendix.

216

MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS

11.2 SMALL SAMPLE TESTS Before presenting the main results of the study, we introduce the family of the matrix elliptical contoured distributions and briefly discuss their main properties. Following Gupta and Varga (1993) a random matrix X of dimensions k × n is said to have a matrix variate elliptically contoured distribution if its characteristic function has the form

(T) = exp(tr(iT M)) (tr(T T)) where T and M are k × n matrices, is a k × k positive semidefinite matrix, and : [0, ∞) → R. The symbol tr stands for the trace of a matrix. This family of distributions we denote by Ek,n (M, , ). If X = (X1 , … ,Xn ) ∼ Ek,n (M, , ) and if its second moments exist then it holds with M = (µ1 , … ,µn ) that the random vectors X1 , … , Xn are uncorrelated and Xi ∼ Ek,1 (µi , , ) (see Gupta and Varga, 1993, theorem 2.4.1, corollaries 2.4.1.1 and 2.4.1.2, theorem 2.3.2). Thus the columns of X follow a vector elliptically contoured distribution. It holds that E(X) = M and = Cov(Xi ) = −2 (0) (see Fang and Zhang, 1990, theorem 2.6.5). Note that the columns are independent if X follows a matrix variate normal distribution, for example, if is taken as being equal to exp(−x/2) (Gupta and Varga, 1993, theorem 2.1.5). Assuming X to be absolutely continuous, it follows that X ∼ Ek,n (M, , ) if and only if the density of X has the form f (X) = det()−n/2 h(tr((X − M) −1 (X − M)), where h and determine each other for specified k and n (see Gupta and Varga, 1993, theorem 2.2.1). The matrix elliptically contoured distributions possess several desirable properties that have been observed for financial assets. It presents an extension of the assumption of an independent normal sample. First, the returns must not be independent, and, second, they may have heavy tails. The aim of our study is to compare the ability of the matrix elliptical and symmetric stable distributions to explain the stochastic behavior of daily asset returns. For these purposes, we derive statistical procedures for testing the matrix elliptical symmetry of a multivariate sample. The cases of known and unknown nature of elliptical symmetry, scale parameters and location vector are considered. Let us denote the set of distribution functions with the known location vector, the known scale matrix, and the known characteristic function by µ,, = {F : F ∈ Ek,n (µ × 1, , )}

(11.1)

If some parameters are not precisely known we put points on the corresponding places in Equation (11.1). For example, when the location vector µ is unknown, we put .,, = {F : F ∈ Ek,n (µ × 1, , ), µ ∈ Rk }

(11.2)

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In a similar way, the testing hypotheses are denoted. When the location vector is unknown we obtain H0,.,, : X ∼ F ∈ .,,

against H1,.,, : X ∼ F ∈ .,,

(11.3)

11.2.1 Known type of elliptical symmetry (known characteristic function) In this section we derive several procedures to test the ellipticity of the sample with the precise definition of the type of elliptical symmetry, for example, to test if X ∼ Ek,n (µ × 1 , , ) with the known characteristic function . First, we consider the case of the known scale matrix and unknown location vector µ. The test hypothesis is H0,.,, : X ∼ F ∈ .,,

against H1,.,, : X ∼ F ∈ .,, .

(11.4)

Let us denote the following random variable ˆ τ τ Q1 = τ τ

(11.5)

where τ is a nonzero vector of constants and is estimated by ˆ =

n 1 1 1 (Xt − X)(Xt − X) = X(I − 11 )X n−1 n−1 n

(11.6)

t=1

The stochastic representation of the random matrix X is essential for deriving the distribution of Q1 . Let be positive definite. It holds that X ∼ Ek,n (M, , ) if and only if X has the same distribution as M + R 1/2 U, where U is a k × n random matrix and vec (U ) is uniformly distributed on the unit sphere in Rkn , R is a non-negative random variable, and R and U are independent (see Gupta and Varga, 1993, theorem 2.5.2). The expression M + R 1/2 U is a stochastic representation of X, that is, it holds that X ≈ M + R 1/2 U. The symbol A ≈ B says that the two random variables A and B have the same distribution. The variable R is called the generating variable of X. The distribution of R2 is equal to the distribution of ni=1 (Xi − µi ) −1 (Xi − µi ). If X is absolutely continuous, then R is also absolutely continuous and its density is fR (r) =

2πnk/2 nk−1 r h(r2 ) nk 2

(11.7)

for r ≥ 0 (see Gupta and Varga, 1993, theorem 2.5.5). Note that for the matrix 2 ∼ χ2 . The index N refers to the normal variate normal distribution RN nk distribution.

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MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS

Lemma 1. Let X = (X1 . . . Xn ) ∼ Ek,n (M,,) and n > k. Let be positive definite and suppose that X is absolutely continuous. Then it holds that (n − 1) Q1 has a stochastic representation R2 b, for example, (n − 1)Q1 ≈ R2 b with R being the generating variable of X, nk−n+1 b ∼ B( n−1 ) (Beta distribution), and the random variables R and b are independent. 2 2 ,

The proof of the lemma is given in the Appendix. From the result of Lemma 1, the moment sequence of random variable Q is calculated mi = E(Qi ) =

k E(R∗2i ) (i + n−1 2 ) ( 2 ) k (n − 1)i ( n−1 2 ) (i + 2 )

(11.8)

where R∗ is the generating variable of X1 . Finally, from Lemma 1 we obtain Theorem 1 Let X = (X1 , . . . , Xn ) ∼ Ek,n (µ × 1 , , ) and n > k. Then it follows that under the null hypothesis H0,.,, the test statistic T1 = (n−1)Q1 has the same distribution as R˜ 1 , where R˜ 1 is the generating variable of En−1,1 (.,., ).

In case of the known scale matrix and the known location vector µ we consider the following random variable Q2 =

ˆ ˜ τ τ τ τ

with

n ˆ˜ = 1 (X − µ)(X − µ) t t n

(11.9)

t=1

The test hypothesis is given by H0,µ,, : X ∼ F ∈ µ,,

against

H1,µ,, : X ∼ F ∈ µ,,

ˆ˜ has a Wishart distriFrom the result of Lemma 1 and the fact that n bution with n degrees of freedom and the parameter matrix , namely, ˆ˜ ∼ W (n, ), it follows n k Theorem 2 Let X = (X, . . . Xn ) ∼ Ek,n (µ × 1 , , ) and n > k. Then it follows that under the null hypothesis H0,µ,, the test statistic T2 = nQ2 has the same distribution as R˜ 2 , where R˜ 2 is the generating variable of En,1 (.,.,).

11.2.2 Unknown type of elliptical symmetry (unknown characteristic function) In contrast to the distributional properties of the proposed statistics in section 11.2.1, the test statistics presented in this section are distributional free within the class of matrix elliptical contoured distributions. Their distributions, in general, are presented by central F-distributions with some degrees of freedom. These results make them available to be applied without specifying the concrete type of elliptical symmetry. Again, the cases with known

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and unknown location vectors are treated separately. We omit the moments requirements imposed on the asset returns. The proposed approaches can be applied to rather heavy-tailed distributions, even which do not possess the first and higher moments. Let X = (X(1) , X(2) ), where X(1) = (X1 , . . . , Xn1 ) and X(2) = (X1+n1 , . . . , Xn ) with n2 = n − n1 . First, we treat the case of unknown location vector µ. When the scale matrix is unknown one has to estimate it by previous observations. Then based on the first n1 observations we estimate ˆ 1) = (n

n1 1 1 1 (Xt − X¯ (1) )(Xt − X¯ (1) ) = X(1) (I − 11 )X(1) n1 − 1 n1 − 1 n1 t=1

Using the rest of the n2 observations we obtain ˆ 2) = (n

1 n2 − 1

n

(Xt − X¯ (2) )(Xt − X¯ (2) ) =

t=n1 +1

1 1 X(2) (I − 11 )X(2) n2 − 1 n2

Finally, to test the null hypothesis H0,.,.,. : X ∼ F ∈ .,.,.

against

H1,.,.,. : X ∼ F ∈ .,.,.

(11.10)

we use the result of the following theorem Theorem 3 Let X = (X1 . . . Xn ) ∼ Ek,n (µ × 1 , , ) and n > k. Then it follows that ˆ 2 )τ/τ (n ˆ 1 )τ has a central under the null hypothesis H0 ,.,.,. the test statistic T3 = τ (n F-distribution with n2 − 1 and n1 − 1 degrees of freedom.

The proof of the theorem is given in the Appendix. If the location vector µ is known, a similar statistic as above is considered. However, the estimators of the scale parameters are given by ˆ˜ (n 1) =

n1 1 (Xt − µ)(Xt − µ) n1 − 1

ˆ˜ and (n 2)

t=1

n 1 = (Xt − µ)(Xt − µ) n2 − 1 t=n1 +1

correspondingly. From Muirhead (1982, theorem 3.2.8) and Fang and Zhang (1990, theorem 5.1.1) we obtain Let X = (X1 , . . . , Xn ) ∼ Ek,n (µ × 1 , , ) and n > k. Then it follows that ˆ ˆ˜ ˜ 2 )τ/τ (n under the null hypothesis H0,µ,.,. the test statistic T4 = τ (n 1 )τ has a central F-distribution with n2 and n1 degrees of freedom. Theorem 4

Note that the distributions of T1 , T2 , T3 and T4 statistics do not depend on τ.

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MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS

11.2.3 Further statistics If the type of ellipticity is unknown, additionally, the following four statistics are considered: −1 ˆ τ −1 τ n − k ˆ −1 ˆ )−1 − −1 T5 = (LR()L (τ τ) −1 p τ τ ˆ τ τ −1 ˆ τ −1 τ × − −1 τ τ ˆ −1 τ τ ˆ˜ −1 τ −1 τ n − k ˆ −1 ˆ˜ )−1 ˜ τ) − −1 (τ (LR()L T6 = ˆ˜ −1 τ p τ τ τ ˆ˜ −1 τ −1 τ × − −1 ˆ −1 τ τ ˜ τ τ −1 ˆ (n2 )τ n − k ˆ −1 T7 = (τ (n2 )τ) (n2 ) − p ˆ −1 τ τ −1 ˆ (n2 )τ −1 ˆ 2 ))L ) × (LR((n − ˆ −1 (n2 )τ τ

ˆ −1 (n1 )τ

ˆ −1 (n1 )τ τ

ˆ −1 (n1 )τ

ˆ −1 (n1 )τ τ

ˆ˜ −1 (n )τ ˆ˜ −1 (n )τ n − k ˆ −1 2 1 ˜ (n2 )τ) T8 = − (τ ˆ −1 ˆ −1 p ˜ ˜ τ (n2 )τ τ (n1 )τ ˆ˜ −1 × (LR((n 2 ))L )

ˆ˜ −1 (n )τ ˆ˜ −1 (n )τ 2 1 − ˆ −1 ˆ −1 ˜ (n2 )τ ˜ (n1 )τ τ τ

where R(A) = A−1 − A−1 ττ A−1 /τ A−1 τ. The statistic T5 is used to test the null hypothesis H0,.,,. : X ∼ .,,. , while the statistic T6 corresponds to the hypothesis H0,µ,,. : X ∼ µ,,. , the statistic T7 to H0,.,.,. : X ∼ .,.,. , and the statistic T8 to H0,µ,.,. : X ∼ µ,.,. . L is a p × k matrix of constants, p ≤ k − 1, such that (L,τ) is of full rank p + 1. The distributions of these statistics have already been derived in Bodnar (2004) and Bodnar and Schmid (2004). They do not depend on the type of elliptical symmetry within the class of matrix elliptical contoured. The critical values of the T7 - and T8 statistics can be obtained by numerical integration from the mathematical software package Mathematica. They are twice as large as the corresponding ones of central F-distributions (T5 - and T6 statistics).

Power

TARAS BODNAR AND WOLFGANG SCHMID

0.40 0.39 0.38 0.37 0.36 0.35 0.34 0.33 0.32 0.31 0.30 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.13

221

T4 T8

0

1

2

3

4

5

6

7

8

9

10

Degrees of freedom

Figure 11.1 Power functions of the T4 and T8 tests for testing sample elliptical symmetry against independent multivariate t-distribution with different degrees of freedom

11.3 ANALYSIS OF THE POWER FUNCTIONS In this section we deal with the power functions of the proposed tests. Note that the rejection of the null hypothesis may be caused by a change in the covariance matrix or misspecification of the underlying distribution. As the first case is not of interest to us, we fix the covariance matrix of the process. Then the rejection of the null hypothesis is the result of an incorrect specification. Since a huge number of alternative hypotheses can be modeled, we do not consider all of them. Furthermore, because of the analytical difficulties of deriving the distributional function under the alternative hypothesis H1 to calculate the power of the test, we apply a Monte Carlo study. Several situations are modeled by drawing a sample of an independent multivariate t-distribution in the first case and an independent symmetric multivariate stable distribution in the second one. The location vector is chosen to be 0, and the scale matrix is identical. This choice is not restrictive, as neither of the T4 and T8 statistics depend on τ, and T8 is independent of L. In all cases, 104 seven-dimensional vectors of the corresponding distributions are drawn. The procedures for generating a multivariate symmetric stable distribution and a multivariate t-distribution are discussed in the Appendix. The powers of the T4 and T8 tests are shown in Figure 11.1 for multivariate t-distributions with different degrees of freedom. The powers of both tests decrease as the degrees of freedom increase. It is not surprising that the t-distribution converges to the normal when the degree of freedom tends

Power

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MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS

0.66 0.64 0.62 0.60 0.58 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 0.30 0.28

T8

T4

0.6

0.8

1.0

1.2 1.4 Stability index

1.6

1.8

2.0

Figure 11.2 Power function of the T4 and T8 tests for testing sample elliptical symmetry against independent symmetrical multivariate stable distribution with different stability indices to infinity. Finally, the power functions of the T4 and T8 statistics are almost the same for the different degrees of freedom. Figure 11.2 contains the power functions of the T4 and T8 tests for independent symmetric stable distributions with different stability indices. The power of a T8 test is always higher than the power of T4 . When the stability index is around one, the power is over 0.6. For larger values of stability indices this probability decreases. It is around 35 percent when the stability index values are around 1.65, the recommended value for describing daily data (Blattberg and Gonedes, 1974). We make use of these results in the next section, when an empirical example of the daily returns of seven developed stock markets is discussed.

11.4 EMPIRICAL STUDY In this section, the results of the empirical study are presented. Because the location parameter, the scale matrix and the type of elliptical symmetry are usually unknown in a practical situation, we make use of the T8 statistic for testing the null hypothesis that daily returns follow a matrix ellipticalcontoured distribution. It is seen how the finite sample properties of these statistics can be used. We consider the daily price data from Morgan Stanley Capital International for the equity markets of seven developed countries (France, Germany, Italy, Japan, Spain, the UK and the USA) for the period January 1, 1994 to December 31, 2000. We group our data set by half-year

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Table 11.1 The 5% and 10% critical values for the T8 statistic depending on the sample sizes n1 , n2 (k = 1) 0.05

α n1 \n2

63

0.1

0.05

63

64

0.1 64

0.05

0.1

65

65

63

8.195

5.695

7.695

5.495

7.265

5.287

64

8.815

5.927

8.190

5.691

7.689

5.498

65

9.581

6.192

8.792

5.927

8.176

5.684

Table 11.2 Value of the T8 statistic for different linear restrictions Year\Test

T 8, l 1

T 8, l 2

T 8, l 3

T 8, l 5

T 8, l 6

1994, I

2.608

1.283

5.175

1994, II

0.106

3.781

0.271

0.840

6.629

0.418

5.597

0.008

1995, I

2.616

1.702

0.168

1995, II

3.869

0.128

2.482

4.285

0.123

0.650

0.014

2.981

5.399

0.0

5.077

1996, I

3.552

3.827

1996, II

2.883

5.263

4.098

2.190

0.137

0.667

0.305

8.092

1997, I

0.001

3.569

4.247

0.025

0.170

0.761

1.146

1997, II

0.084

1.034

1998, I

0.270

1.808

0.255

1998, II

0.075

0.899

1999, I

0.108

1999, II

2.756

2000, I 2000, II

0.599

T 8, l 4 11.50

13.79 3.209

0.809 0.005

0.017

3.436

2.668

0.159

1.726

4.466

2.224

5.387

0.786

3.592

0.152

0.021

3.736

0.027

0.206

2.605

1.968

1.149

1.629

0.136

1.743

0.734

0.518

0.019

0.637

0.142

2.176

0.675

2.753

8.155

2.568

3.298

0.314

25.63

20.73

4.815 11.93

T 8, l 7

11.57

0.673 4.682

11.18

data. Furthermore, each group is additionally partitioned into two subsamples of three-month data sets. The first sub-sample is used to calculate ˆ 1 ) and the second for (n ˆ 2 ). (n In Table 11.1, the 5 percent and 10 percent critical values of the T8 statistics for different sample sizes, n1 , n2 ∈ 63,64,65 are presented. These change significantly with a change of sample size. However, the critical values are almost the same on the diagonals that are parallel to the main diagonal of the table. The values of the T8 statistic are presented in Table 11.2. They are calculated for different linear restrictions, for example, l1 = (1, 0, 0, 0, 0, 0, 0), . . . , l7 = (0, 0, 0, 0, 0, 0, 1). The values of the statistics that are greater than the 5 percent critical value are indicated in bold type. The null hypothesis of matrix elliptical symmetry is rejected in null cases out of fourteen for the T8

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MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS

statistic with the l1 and l2 linear restrictions. In other cases there are only two rejections per a column. In general we observe seven rejections out of ninety-eight for the 5 percent level of significance, and ten rejections out of ninety-eight for the 10 percent level. Keeping in mind that the rejection of the null hypothesis can also be effected by changes in the covariance structure of the stock returns process, we are unable to reject the matrix ellipticity of the considered returns. From the other side, using the results of section 11.3 (the test power is very high for the multivariate symmetric stable distribution), one should be very careful with the assumption that daily stock returns follow a multivariate stable law.

11.5 CONCLUSION In this chapter, several statistics for testing the null hypothesis of a matrix elliptical-contoured distribution are proposed. The finite sample properties are derived in all cases of known and unknown types of elliptical symmetry, scale matrix and location vector. The T1 - and T2 statistics do not possess the invariance property with respect to matrix elliptical-contoured distributions and their null distributions are specified by the corresponding generating variables. From the other side, the statistics from T3 to T8 are distributionally free. Their stochastic properties, apart from T7 and T8 , are based on the central F-distribution with some degrees of freedom. The control limits of the T7 - and T8 statistics can be obtained by numerical calculations in the software package Mathematica. We applied the T8 statistic in a situation of practical interest by considering the daily stock returns of seven developed stock markets. The null hypothesis of the matrix elliptical symmetry is rejected ten times out of ninety-eight for the 10 percent level of significance. In addition, the rejection of the null hypothesis may be caused by changes in the covariance structure of the underlying distribution. Keeping everything together, we conclude that the results of the empirical study provide support to model the daily data by matrix elliptical-countered distributions. They are in line with the suggestions of Andersen et al. (2001) and Andersen et al. (2004), who argued that daily returns normalized by the realized volatility can be well approximated by normal distribution. Furthermore, researchers should be very careful with the application of the multivariate symmetric stable law in modeling daily data. Instead, the assumption of a matrix elliptical-contoured distribution should be maintained.

APPENDIX In this section, the proofs of Lemma 1 and Theorem 3 are given. Furthermore, we deal with the problem of generating independent multivariate t- and symmetrical stable distributed random vectors.

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We say that a characteristic function belongs to the class (U) if it can be equal to another characteristic function in the neighborhood of zero without being identical to it. Correspondingly, a characteristic function belongs to the class (U) if it does not belong to (U). We denote a set of orthogonal matrices of the order k by O(k).

Proof of Lemma 1 We have it that Q ≈ R2 τ

(1/2 U(I − 11 /n)U 1/2 )τ τ τ

= R2 Q∗ . A similar presentation is obtained

2 Q . The index N is used to when X is matrix normally distributed; for example, QN ≈ RN ∗ 2 ∼ χ2 , R2 Q ∼ χ2 , it follows from Fang and Zhang indicate the normal case. Because RN N ∗ n−1 nk 1 nk − n + 1 2 Q ≈ R2 b . Furthermore, we ) exists such that RN (1990, p. 59) that b∗ ∼ B( n − ∗ N ∗ 2 , 2 2 > 0) = 1 and P(b > 0) = 1. have it that P(RN ∗ ˆ is positive definite with From the assumption of the lemma, it follows that probability 1 (see Muirhead, 1982, theorem 3.1.4). Hence, Q is greater than 0 with probability 1, and therefore, P(Q∗ > 0) = 1. From the above consideration, it follows that the density of ln b∗ is

fln b∗ (t) =

n−1

et 2 (1 − et ) 0,

nk−n+1 −1 2

,

if if

t≤0 t>0

∞ Hence, for any positive r it holds 0 ert fln b∗ (t)dt = 0 < ∞. Using the results of Fang and Zhang (1990), it follows that characteristic function φln b∗ ∈ (U). Thus, from the property of the operation ≈ (see Fang and Zhang, 1990, p. 38) it follows that Q∗ ≈ b∗ . As a 1 nk − n + 1 result we obtain Q∗ (R2 ) ≈ R2 Q∗ ≈ R2 b∗ , where b∗ ∼ B( n − ) and R2 and b∗ are 2 , 2 independently distributed.

Proof of Theorem 3 Let us consider T3 =

ˆ 2 )τ ˆ 2 )τ τ τ τ (n τ (n = ˆ 1 )τ ˆ 1 )τ τ τ τ (n τ (n

The rest of the proof follows from Muirhead (1982, theorem 3.2.8) and Fang and Zhang (1990, theorem 5.1.1).

Drawing samples from multivariate t- and symmetric stable distributions The way of generating samples of independent multivariate t-distributions and symmetric stable distributions follow immediately from their stochastic representations. From Fang et al. (1990, p. 85) we obtain that the k-dimensional t-distributed random vector Y is equal to Y = Z/

χ n

(11.11)

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MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS

where Z has a k variate normal distribution, χ has a chi-squared distribution with n degrees of freedom, and Z and χ are independently distributed. The stochastic representation of the k-dimensional stable random vector with the index of stability equal to α, 0 < α < 2 is given by S=

√ AZ

(11.12)

where Z has a k variate normal distribution, A has an univariate α/2-stable distribution with skewness parameter equal to 1, the location parameter 0, and the scale parameter (cos(πα/4))2/α , and Z and A are independently distributed (see Samorodnitsky and Taqqu, 1994, p. 77). Following Kantner (1975), the stochastic representation of A is

A = cos (πα/4)

1/α

sin ((1 − α/2)θ) sin (αθ/2)α/(2−α) sin (θ)2/(2−α) W

(2−α)/α (11.13)

where θ is uniform on (0,π),W has a standard exponential distribution, and θ and W are independently distributed.

REFERENCES Andersen, T. G., Bollerslev, T. and Diebold, F. X. (2004) “Parametric and Nonparametric Measurements of Volatility”, in Y. Aït-Sahalia and L. P. Hansen (eds), Handbook of Financial Econometrics, (Amsterdam: North-Holland). Andersen, T. G., Bollerslev, T., Diebold, F. X. and Ebens, H. (2001) “The Distribution of Realized Exchange Rate Volatility”, Journal of the American Statistical Association, 96(453): 42–55. Baringhaus, L. (1991) “Testing for Spherical Symmetry of a Multivariate Distribution”, The Annals of Statistics, 19(2): 899–917. Beran, R. (1979) “Testing Ellipsoidal Symmetry of a Multivariate Density”, The Annals of Statistics, 7(1): 150–62. Berk, J. B. (1997) “Necessary Conditions for the CAPM”, Journal of Economic Theory, 73(1): 245–57. Blattberg, R. C. and Gonedes, N. J. (1974) “A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices”, The Journal of Business, 47(2): 244–80. Bodnar, T. (2004) “Optimal Portfolios in an Elliptical Model – Statistical Analysis and Tests for Efficiency”, Ph.D. thesis, Europa University Viadrina, Frankfurt (Oder), Germany. Bodnar, T. and Schmid, W. (2003) “The Distribution of the Global Minimum Variance Estimator in Elliptical Models”, EUV Working paper 22. Bodnar, T. and Schmid, W. (2004) “A Test for the Weights of the Global Minimum Variance Portfolio in an Elliptical Model”, EUV Working paper 2. Bollerslev, T. (1986) “Generalized Autoregressive Conditional Heteroskedasticity”, Journal of Econometrics, 31(3): 307–27. Chamberlain, G. A. (1983) “A Characterization of the Distributions that Imply MeanVariance Utility Functions”, Journal of Economic Theory, 29(1): 185–201. Engle, R. F. (1982) “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation”, Econometrica, 50(4): 987–1008. Fang, K. T. and Zhang, Y. T. (1990) Generalized Multivariate Analysis (Berlin: Springer-Verlag Beijing: Science Press). Fang, K. T., Kotz, S. and Ng, K. W. (1990) Symmetric Multivariate and Related Distributions (London: Chapman & Hall).

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227

Fang, K. T., Zhu, L. X. and Bentler, P. M. (1993) “A Necessary Test of Goodness of Fit for Sphericity”, Journal of Multivariate Analysis, 45(1): 34–55. Fama, E. F. (1965) “The Behavior of Stock Market Prices”, Journal of Business, 38(1): 34–105. Fama, E. F. (1976) Foundations of Finance (New York: Basic Books). Gibbons, M. R., Ross, S. A. and Shanken, J. (1989) “A Test of the Efficiency of a Given Portfolio”, Econometrica, 57(5): 1121–52. Gupta, A. K. and Varga, T. (1993) Elliptically Contoured Models in Statistics (Dondrecht: Kluwer Academic). de Haan, L. and Rachev, S. T. (1989) “Estimates of the Rate of Convergence for Max-Stable Processes”, The Annals of Probability, 17(2): 651–77. Heathcote, C. R., Cheng, B. and Rachev, S. T. (1995) “Testing Multivariate Symmetry”, Journal of Multivariate Analysis, 54(2): 91–112. Kantner, M. (1975) “Stable Densities under Change of Scale and Total Variation Inequalities”, The Annals of Probability, 3(4): 697–707. Koltchinskii, V. I. and Li, L. (1998) “Testing for Spherical Symmetry of a Multivariate Distribution”, Journal of Multivariate Analysis, 65(2): 228–44. Manzotti, A., Perez, F. J. and Quiroz, A. J. (2002) “A Statistic for Testing the Null Hypothesis of Elliptical Symmetry”, Journal of Multivariate Analysis, 81(2): 274–85. Muirhead, R. J. (1982) Aspects of Multivariate Statistical Theory (New York: John Wiley). Nelson, D. (1991) “Conditional Heteroscedasticity in Stock Returns: A New Approach”, Econometrica, 59(2): 347–70. Owen, J. and Rabinovitch, R. (1983) “On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice”, The Journal of Finance, 38(3): 745–52. Rachev, S. T. and Mittnik, S. (2000) Stable Paretian Models in Finance. (New York: John Wiley). Samorodnitsky, G. and Taqqu, M. S. (1994) Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance (New York/London: Chapman & Hall). Tu, J. and Zhou, G. (2004) “Data-generating Process Uncertainty: What Difference Does It Make in Portfolio Decisions?”, Journal of Financial Economics, 72(2): 385–421. Zhou, G. (1993) “Asset-pricing Tests under Alternative Distributions”, The Journal of Finance, 48(5): 1927–42. Zhu, L. X. and Neuhaus, G. (2003) “Conditional Tests for Elliptical Symmetry”, Journal of Multivariate Analysis, 84(2): 284–98.

C H A P T E R 12

The Modified Sharpe Ratio Applied to Canadian Hedge Funds Greg N. Gregoriou

12.1 INTRODUCTION The assessment of portfolio performance is fundamental for both investors and fund managers, and this applies also to Canadian hedge funds. Traditional portfolio measures present some limitations when applied to hedge funds. For example, the Sharpe ratio uses the excess reward per unit of risk as a measure of performance, with risk represented by the standard deviation. The mean-variance approach to the portfolio selection problem developed by Markowitz (1952) has frequently been the subject of undue criticism because of its utilization of variance as a measure of risk exposure when examining the non-normal returns of hedge funds. The value-at-risk (VaR) measure for financial risk has recently grown to be accepted as a traditional measure in investment firms, large banks and pension funds. As a result of the recurring frequency of down-markets since the collapse of Long-Term Capital Management (LTCM) in August 1998, VaR has played a paramount role as a risk management tool, and is considered to be a mainstream technique for estimating and conveying the exposure a hedge fund has to market risk. With the wide acceptance of VaR, and specifically, of modified VaR as a relevant risk-management tool, a more suitable portfolio performance measure for hedge funds can be formulated in terms of a modified Sharpe ratio.1 228

GREG N. GREGORIOU

229

Using the traditional Sharpe ratio to rank hedge funds will underestimate the tail risk, and then overestimate performance. Therefore, the further the distribution is from the normal, the greater the risk of underestimation. In this chapter, we rank nine funds according to the Sharpe and modified Sharpe ratios. Our results indicate that the modified Sharpe is lower and more accurate when examining non-normal returns.

12.2 LITERATURE REVIEW Many hedge funds produce statistical reports for clients using the traditional Sharpe ratio, which can be misleading because funds will have a tendency to look better in terms of risk-adjusted returns. The drawback of using a traditional Sharpe ratio is that no distinction is made between upside and downside risk, but rather a fund is penalized for upside risk as much as downside risk and does not differentiate irregular losses compared to repeated losses. VaR has emerged progressively in the finance literature as a prevailing measure of risk. However, its simple version also presents some limitations because of the skewed returns hedge funds possess. Methods of measuring VaR, such as the delta-normal method developed by Jorion (2000), are straightforward and simple to apply. However, the formula has its drawbacks, since the assumptions of normality of the distributions are violated largely because of the use of short-selling and derivatives strategies, such as futures and options, frequently used by hedge funds. Current methods have been proposed to properly assess the VaR for nonnormal returns as developed by Rockafellar and Uryasev (2001) using a conditional VaR for general loss distributions, while Agarwal and Naik (2004) construct a mean conditional VaR demonstrating that mean-variance analysis also underestimates tail risk. Furthermore, Favre and Galeano (2002) also developed a technique to properly assess funds with non-normal distributions. The authors demonstrate that the modified VaR (MvaR) can significantly improve the accuracy of the traditional VaR. The difference between the modified VaR and the traditional VaR is that the latter considers only mean and standard deviation, while the former takes into account higher moments such as skewness and kurtosis. In addition, it is possible to reduce the probability of large negative returns by at least 15 percent (Favre and Singer, 2002). The modified VaR allows us to define a modified Sharpe ratio, which is more suitable for hedge funds. For example, when two portfolios have the same mean and standard deviation, in essence they may be different because of extreme losses. Therefore an advantage exists when using the modified VaR measure and modified Sharpe ratio.

230

THE MODIFIED SHARPE RATIO AND CANADIAN HEDGE FUNDS

12.3 DATA AND METHODOLOGY The dataset we use contains hedge fund returns for fifty funds in Canada. However, the majority of the funds commenced operations in 2001 and have been discarded because of the small number of data points available at the time of writing. Only nine live Canadian hedge funds reporting monthly performance figures spanning the period January 1998–December 2002 have been investigated. We obtain data from Beck and Nagy (2003). This period contains the extreme market event of August 1998 as well as the September 11, 2001 terrorist attacks. We use the Extreme Metrics software and assume a risk-free rate of 0 percent to compute the results, using a 95 percent VaR probability. This means that the investor is able to borrow and reinvest in the market portfolio at zero cost in order to move along the capital market line. This assumption simplifies the ranking of assets, especially when some of them have an average return below the risk-free rate, which yields a negative Sharpe or modified Sharpe ratio. The difference between the traditional and modified Sharpe ratios is that, in the latter, the standard deviation is replaced by the modified VaR (at 95 percent) in the denominator. The traditional Sharpe ratio is generally defined as the excess return per unit of standard deviation, as represented by the following equation: Sharpe ratio =

R p − RF σ

(12.1)

where RP = return of the portfolio; RF = risk-free rate; and σ = standard deviation. Since Equation (12.1) presents several limitations for non-normal distribution, a modified Sharpe ratio can be defined in term of modified VaR, as follows: Modified Sharpe ratio =

Rp − RF , MVaR

(12.2)

with 1 1 MVaR = W[µ − {zc + (zc2 − 1)S + (zc3 − 3zc )K 6 24 1 − (2zc3 − 5zc )S2 }σ] 36

(12.3)

where RP = return of the portfolio; RF = risk-free rate; σ = standard deviation; Zc = is the critical value for probability (1 − α) − 1.96 for a 95 percent probability; S = skewness; and K = excess kurtosis. The detailed derivation of the formula for modified VaR is beyond the scope of this chapter. Readers are guided to Favre and Galeano (2002) for a more detailed explanation.

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231

12.4 EMPIRICAL RESULTS 12.4.1 Descriptive statistics Table 12.1 displays monthly statistics on mean return, standard deviation, skewness, excess kurtosis, normal and modified VaR, Jarque–Bera statistic and compounded returns of the hedge funds during the examination period. The average of the compounded returns and mean monthly returns are greatest in the highest group and least in the lowest group – an expected finding. In addition, we find that positive skewness is more pronounced in the lowest group, yielding more positive monthly returns, whereas the top group has the least average positive skewness. A likely explanation is that smaller hedge funds can better control skewness in negative extreme market events, and on average will have less negative monthly returns. The lowest group (see Table 12.1, Panel C) has the highest volatility and lowest returns, which could be attributed to hedge funds taking on more risk to achieve greater returns while increasing assets under management.

12.4.2 Performance discussion Market risk and performance results are also presented in Table 12.1. First we observe that the middle group has, in absolute value, the lowest normal and modified VaR, so is less exposed to extreme market losses. Furthermore, we find that the non-normality when skewness and kurtosis are considered simultaneously using the Jarque–Bera tend to be the largest for small hedge funds. With regard to performance, we notice that the lowest group has the lowest traditional Sharpe and modified Sharpe ratios. It appears that medium-sized hedge funds can do a better job in controlling riskadjusted performance than either small or large funds. Since medium-sized hedge funds receive a greater inflow of money than small funds, they can alter their allocation more frequently. However, there exists a huge difference of assets under management between large and medium funds. When receiving a vast inflow of capital, large hedge funds could be overwhelmed and might experience trouble in producing superior risk-adjusted returns than medium-sized hedge funds. Capacity constraints may exist, since the Toronto Stock Exchange is relatively small compared to the US markets, and trading securities may further restrict large Canadian hedge funds, thus making trading sporadic, especially when leverage and short-selling is involved. Smaller hedge funds with fewer assets might have no choice but to hold their portfolio for a long period of time, irrespective of changing economic conditions.

Fund name

232

Table 12.1 Descriptive statistics of Canadian hedge funds, 1998–2002 Assets Mean Std. dev. Skewness Excess Modified Normal Traditional Modified Jarque–Bera Compound (millions (%) (%) kurtosis VaR VaR Sharpe sharpe statistic return $) 99% 99% ratio ratio (%)

Panel A: Sub-sample 1: Top 3 funds Arrow Clocktower

325

1.4

3.5

0.2

0.00

−6.3

−6.8

0.18

0.16

Goodwood Fund

200

1.6

4.6

0.4

0.5

BPI Global Opportunities

195

1.5

5.9

0.9

1.1

−8.2

−9.1

0.15

−8.3

−12.3

0.14

Average

240

1.5

4.67

0.5

0.53

−7.60

−9.4

−3.9

0.26

127.73

0.13

1.92

137.07

0.09

10.88

119.97

0.16

0.13

4.35

128.26

−4.6

0.14

0.12

0.97

65.10

Panel B: Sub-sample 2: Middle 3 funds −0.4

Horizons Mondiale

125

0.9

2.3

0.2

Horizons Univest 2

107

0.9

0.7

0.2

1.2

−0.8

−0.9

0.75

0.66

3.74

74.44

82

1.8

5.7

0.00

3.4

−11.5

−16.2

0.13

0.09

28.57

162.52

104.67

1.2

2.9

0.13

1.4

5.4

7.2

0.34

0.29

11.09

100.69

0.10

0.07

24.79

112.03

Vertex Average

Panel C: Sub-sample 3: Bottom 3 funds Friedberg TT Equity Hedge

6

1.6

8.0

1.1

2.3

−11.9

−17.1

Horizons Strategic

3

1.8

7.0

3.2

18.4

−0.1

−14.5

0.10

0.09

948.02

77.22

Hillsdale Market Neutral ($US)

2

0.2

4.2

0.4

1.8

−9.6

−9.7

−0.02

−0.02

10.18

6.18

Average

3.67

1.2

6.4

1.57

7.5

−7.2

−13.76

0.06

0.05

327.66

65.14

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When we compare the results between the traditional and the modified Sharpe ratios, we find that the traditional Sharpe ratio is higher, confirming that tail risk is underestimated.

12.5 CONCLUSION It is of critical importance to understand that complications will arise when a traditional measure of risk-adjusted performance, such as the traditional Sharpe ratio, is used to investigate fat tails and non-normal returns of hedge funds. Institutional investors must use the modified Sharpe ratio to measure the risk-adjusted returns correctly; and the modified VaR is recommended to measure extreme negative returns because the normal VaR only considers the first two moments of a distribution, namely mean and standard deviation. The modified VaR, however, takes into consideration the third and fourth moments of a distribution – skewness and kurtosis. Using both the modified Sharpe and modified VaR will enable investors to obtain a more accurate picture without any bias. Furthermore, the modified VaR is lower than the normal VaR because of negative skewness in hedge fund returns and the small excess positive kurtosis. The statistics we have presented can be applied to all hedge fund and commodity trading adviser (CTA) classifications to evaluate non-normal returns. We believe many institutional investors wanting to add hedge funds and funds of hedge funds to traditional stock and bond portfolios must request additional and more appropriate statistics such as the modified Sharpe ratio in analyzing the returns of hedge funds.

NOTE 1. The standard VaR, which assumes normality and uses the traditional standard deviation measure, looks only at the tail of the distribution of extreme events. This is common when examining mutual funds, but when applying this technique to funds of hedge funds, difficulties arise because of the non-normality of returns (Favre and Galeano, 2002). The modified VaR takes into consideration the mean, standard deviation, skewness and kurtosis to evaluate correctly the risk-adjusted returns of funds of hedge funds. Computing the risk of a traditional investment portfolio consisting of 50% stocks and 50% bonds with the traditional standard deviation measure could underestimate the risk by as much as 35% (Favre and Singer, 2002).

REFERENCES Agarwal, V. and Naik, N. (2004) “Risks and Portfolio Decisions Involving Hedge Funds”, Review of Financial Studies. 17(1): 63–98. Beck, P. and Nagy, M. (2003) Hedge Funds for Canadians (Toronto: John Wiley).

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Favre, L. and Galeano, J. A. (2002) “Mean-Modified Value-at-Risk with Hedge Funds”, Journal of Alternative Investments, 5(2): 21–5. Favre, L. and Singer, A. (2002) “The Difficulties in Measuring the Benefits of Hedge Funds”, Journal of Alternative Investments, 5(1): 31–42. Jorion, P. (2000) Value at Risk (New York: McGraw-Hill). Markowitz, H. (1952) “Portfolio Selection”, Journal of Finance, 77(1): 77–91. Rockafellar, R. T. and Uryasev, S. (2001) “Conditional Value-at-Risk for General Loss Distributions”, Research Report, ISE Dept, University of Florida.

Index

ABN Amro 1 accounting standard index 59–60, 66, 67–74, 75–7 Ackert, L.F. 157, 164, 174, 175 African bloc 47, 54, 56 Agarwal, V. 229 Ahearne, A. 42, 43, 63, 64, 71–2, 78 Air Liquide 190–209, 210 alpha art market 9–10 Jensen’s alpha 197–8 American bloc 47, 54, 56 Amin, G. 140 analysis of variance (ANOVA) 161, 162, 167–73 Andersen, T.G. 215, 224 Ang, A. 17, 33, 37 Anthony, J.H. 155 anti-director rights 59–60, 66, 67–74, 75–7 arbitrage pricing theory (APT) 114–15, 116–17 Argentina 104 art 1–15 art market 3–4; defining a bubble in 5–6 data 6–8 empirical studies 4–5 methodology 9–10 results 10–11 Art Market Research (AMR) database 6 Artzner, P. 191 Asia/Pacific bloc 47, 54, 56 Asian currency crisis 96–102 at the money (ATM) option contracts 160, 167, 168–9, 172 Ates, A. 153

auction houses 3 Augmented Dickey–Fuller (ADF) test 163, 174 Australian Statex Actuaries Price Index 157 Australian Stock Exchange (ASX) 151–2 All Ordinaries Index 154, 157, 158, 159, 161, 165–7, 168, 169, 175, 176–8 lead–lag relationship with SFE 155, 158–9, 164–5, 176–8 multifactor model 114–36; cross-sectional regressions 124–7; data 119–27; data analysis and results 127–32; parsimonious model 119, 130–2; portfolio characteristics 124; returns to be explained 123–4 Austria 47 automation 151–82 analysis of results 165–78; ANOVA results 167–73; descriptive statistics 165–7; price discovery analysis 174–8 sample design 159–65; cointegration 163–5; data sources 159–60; methodology 160; model and statistical procedures 160–3 Baig, T. 97, 104 Barbone, L. 96, 104 Bauer, R. 5 Baumol, W. 4 bear state 23–6, 27 Beck, P. 230 behavioral anomalies

12–13 235

236

INDEX

behavioral theory 80–1 Bekaert, G. 17, 33, 37 Belgium 47, 63 benchmarks benchmark assets in efficiency gain/loss methodology 140–1; return distribution 144 market benchmarks for French stocks 201–9; forecasting performance 205–9 Benjamin, W. 12 bequests 13 Berk, J.B. 211, 215 Berkowitz, J. 22 beta 197–8, 203–5, 206 bilateral trades 65–6, 67–74, 75–7 Black, F. 17, 185, 186 Black and Scholes volatility bias 189 Blattberg, R.C. 222 blocs 47, 54, 56 Bodnar, T. 215, 220 Bohn, H. 67 bond market–stock market linkages 103–15 book-to-market value 118–19, 121–2, 123–4, 125, 126–32, 133 Bortoli, L. 153 bounded rationality 80 Brady bonds 103–13 stripped-yield spreads 104, 105, 107, 108, 109–11 Brailsford, T. 154, 197 Brandt, M.W. 21 Brazilian bond market–stock market linkages 103–13 Brooks, C. 158, 159 Brownian motion 185–6 bubbles 11–14 art market 1–2, 5, 8, 10–11; defining a bubble in the art market 5–6 Internet 32–3 real estate 6 “bubbliness”, degree of 13–14 CAC40 index 187–209, 210–11 call pricing formula 186–7 Campbell, J.Y. 184, 186, 197, 200, 210 Campbell, R.A. 4, 8 Canadian hedge funds 228–34 capital asset pricing model (CAPM) 114, 115–16, 183, 197, 215 international (ICAPM) 42 capital controls 43, 44, 57–8, 59–60, 63, 67–74, 75–7

capital flow restrictions 57–8, 63, 67–74, 75–7 Carmichael, B. 64 Case, K.E. 6 causality analysis impact of systematic risk on French stocks 191–5; Granger causality test 193–5; VAR specification 191–3 lead–lag relationship 164–5, 176–8 Central and Eastern European investment funds 143–50 certainty-equivalent compensation 28–30 Chamberlain, G.A. 215 Chan, K. 43, 44, 45, 46, 47, 55, 56, 63, 64, 65, 67, 69, 74, 78, 127 Chanel, O. 4 Chen, J. 17 Chen, N.F. 116, 117, 118, 119, 125, 126, 127, 130, 132 Cheng, C.S. 119 China 63 Chordia, T. 105 Clare, A.D. 127, 130 Clark, P.K. 154 cocoa prices 156–7 Coen, A. 64 cointegration 156–8, 163–5, 175–8 Johansen test 163–4, 175–6 lead–lag relationship 155, 158–9, 164–5, 176–8 unit root tests 163, 174 collateral, art as 8 compensation, certainty-equivalent 28–31 Connor, G. 117 contagion 104 currency crises and portfolio selection 96–102 Cooper, I. 44–5, 47 Coordinated Portfolio Investment Survey (CPIS) dataset 43 Copeland, L. 154, 156 corner portfolios 87, 91 correlation studies correlation coefficient and home bias 67–74, 75–7 impact of systematic risk on French stocks 190–1 peer group analysis 148–9 stock market returns 97, 99–100, 101 cost-of-carry 158, 180 creditworthiness 103–4

INDEX

critical line UPM/LPM portfolio optimization algorithm (CLA) 80–95 derivation 82–4 efficient segments on the efficient frontier 88; adjacent efficient segments 89–92 empirical example 92–4 Kuhn–Tucker conditions 84–7 cross-sectional regressions 124–32 CRR (Chen, Roll and Ross) macroeconomic variables 122–3, 126, 127–32, 133 cumulative wealth 32–4 currency crises 96–102, 104 portfolio performance 100–1 stock market average rates of return and average volatility 97–9, 101 stock market correlations 97, 99–100, 101 currency hedging 16–41 economic importance of regimes 28–31 estimation results for regime-switching models 21–6, 27 optimal foreign investment 38, 39 out-of-sample test for regime-switching strategies 31–7, 39 Cyert, R.M. 80 Dahlquist, M. 43, 72 daily financial data 214 matrix elliptical contoured distributions 222–4 Dales, A. 17 Danone 190–209, 210 DAX 157 DeBondt, W.F.M. 117 default probabilities 104 depreciation of the dollar 33, 36 descriptive statistics automation of SFE 161, 165–7 Canadian hedge funds 231, 232 developed countries 47, 54, 56 Diacogiannis, G.P. 116 Diamandis, P. 116 disposition effect 13 distributional price 139 dividend-paying framework 187 dollar, depreciation of the 33, 36 domestic bias 42–79 data sources 46

237

determinants of 56–67; capital control 57–8, 59–60, 63; economic development 56–63; familiarity 61–2, 65–6; information costs 61–2, 64; investor protection 59–60, 66; investors’ behavior 61–2, 65; other variables 67; stock market development 57–8, 63–4 results of empirical analysis 67–9 statistics on 47–56 statistics on investor holdings 46–53, 54 theoretical framework 44–6 world float portfolio 72–4, 75 double-lognormal (DLN) framework 138, 142–3, 144, 150 Dow Jones STOXX market indices 189 downside risk 229 time-varying 1–15 Durbin–Watson (DW) test 130, 131 Dybvig, P.H. 139 East Asian stock markets 96–102 economic development 43, 44, 56–63, 67–77 economic importance of regimes 28–31 Ederington, L.H. 143 Edison, H. 63 efficiency gain/loss 138, 140–3, 149–50 benchmark 141–2 definition 140–1 higher moment performance characteristics 145–7 underlying 142–3 efficient frontiers 87–94 efficient segments on 88; adjacent efficient segments 89–92 mean-variance and UPM/LPM models 92–4 efficient market hypothesis (EMH) 114 semi-strong form 156, 175–6, 180 Eichenberger, R. 12, 13 Eichengreen, B. 96 electronic trading see automation elliptical distributions see matrix elliptical contoured distributions emerging markets Brady bonds see Brady bonds domestic and foreign biases 47, 54, 56, 64, 67–77 performance evaluation 137–50

238

INDEX

endowment effect 12 Engle, R.F. 163 Epps, M.L. 154 Epps, T.W. 154 Erb, C.B. 96–7 European bloc 47, 54, 56 expected inflation, change in 122–3, 126, 127–32, 133 expropriation, risk of 59–60, 66, 67–77 extreme value theory (EVT) 9 Faff, R.W. 118, 119, 192, 197 Fama, E.F. 115, 117–19, 121, 123, 125, 126, 127, 130, 132, 156, 158, 180, 211 familiarity 43, 44, 61–2, 65–78 Fang, K.T. 219, 225 far in the money (FITM) option contracts 160, 168–9 Faruquee, H. 43 Favre, L. 229, 230 firm-attribute factors 211 multifactor models 117, 118–19; ASX 121–32, 133 Fishburn, P.C. 81 Fleming, J. 97, 105 foreign bias 42–79 data sources 46 determinants of 56–67; capital control 57–60, 63; economic development 56–63; familiarity 61–2, 65–6; information costs 61–2, 64; investor protection 59–60, 66; investors’ behavior 61–2, 65; other variables 67; stock market development 57–8, 63–4 results of empirical analysis 69–71 statistics on 47–56 statistics on investor holdings 46–53, 54 theoretical framework 44–6 world float portfolio 72–4, 76 foreign direct investment (FDI) 56–63, 67–77 Forni, L. 96, 104 France 47 impact of systematic risk on stocks in French financial market 183–213 Fraser, P. 116, 118, 119 French, K.R. 43, 65, 115, 117–19, 121, 123, 125, 126, 127, 130, 132, 183, 211 Freund, W.C. 156, 161, 163 Frey, B.S. 12, 13 Friedman, M. 81

Friend, I. 116 Frino, A. 154 Froot, K. 16 Fund of Art Funds 1 futures see Sydney Futures Exchange (SFE) gamma estimates 9, 10–11, 13 Galeano, J.A. 229, 230 Garcia, R. 17 GDP per capita 56–63, 67–77 Gehrig, T. 64 Gençay, R. 183–4 geographical proximity 47, 61–2, 65, 67–74, 75–7 Germany 47 currency hedging and regime switching 21–6, 27, 33–6; optimal hedge ratio 36–7 DAX 157 Giannetti, M. 66 Gibbons, M.R. 215 Glassman, D.A. 42 Glen, J. 17 Glosten, L.R. 140 Goetzmann, W.N. 4 Goldfajn, I. 97, 104 Gonedes, N.J. 222 Gourieroux, C. 199 Goyal, A. 200, 210 Granger, C.W.J. 163 Granger causality test automation of SFE and lead–lag relationship 164–5, 176–8 systematic risk and French stocks 193–5 Gray, S. 17 Groenewold, N. 116, 118, 119, 157, 174, 175 Grünbichler, A. 158 Guidolin, M. 17, 28 Gupta, A.K. 216, 217 Halliwell, J. 118, 119, 123 Hamilton, J.D. 20 Hartmann, P. 96–7 Harvey, C.R. 96–7 He, J. 119, 131–2 hedge funds, Canadian 228–34 hedging, currency see currency hedging; optimal currency hedging hidden regime switches 20–1 high correlation state 25–6, 27

INDEX

higher moment performance analysis 138–40, 145–7 portfolio replication 139–40 rationale 139 role of higher moments 138–9 Hill, B. 9 home bias 42–79 Ahearne measure 71–2, 73 causes 56–67 data and preliminary statistics 44–56 empirical analysis 67–71 theoretical framework of domestic and foreign biases 44–6 world float portfolio 72–4, 77 Hong Kong 97–100 horse race (out-of-sample test) 31–7, 39 house prices 6 Huberman, G. 43, 65 Huisman, R. 9 Hungarian investment funds 143–50 in the money (ITM) option contracts 160, 168–9 Indonesia 97–100 industrial production growth rate, unexpected change in 122–3, 126, 127–32, 133 inflation change in expected 122–3, 126, 127–32, 133 unexpected inflation rate 122–3, 126, 127–32, 133 information costs 43, 44, 61–2, 64, 67–78 information flow 154, 155, 164–5 informational efficiency 152, 179 intensity of capital control 63, 67–74, 75–7 interest rates risk-free 188–9 unexpected change in term structure 122–3, 126, 127–32, 133 international capital asset pricing model (ICAPM) 42 International Finance Corporation (IFC) 63 Internet bubble 32–3 investor behavior domestic and foreign biases 43, 44, 61–2, 65, 67–77 UPM/LPM critical line algorithm 80–95

investor protection 67–77 Izvorski, I. 104

239

43, 44, 59–60, 66,

Jagannathan, R. 140, 197 Japan 5, 8, 97–100 Jarnecic, E. 155 Jarque–Bera test statistic 22, 24, 145, 231, 232 Jasiak, J. 199 Jegadeesh, N. 117 Jensen, C.M. 184, 197 Jensen-type regressions 197–8, 201–3, 204, 207, 208 Johansen, S. 164, 175 Johansen cointegration test 163–4, 175–6 Jorion, P. 17, 229 judicial system efficiency 59–60, 66, 67–77 Juselius, K. 164, 175 Kahneman, D. 12, 13, 81 Kantner, M. 226 Kaplan, P.D. 81 Kaplanis, E. 44–5, 47 Karolyi, G.A. 96 Karpoff, J.M. 154 Kat, H.M. 139, 140 Kelly, J.M. 105 Kempf, A. 157 Kilka, M. 43 Knif, J. 183, 192, 197 known characteristic function 216, 217–18 known location vector 216, 218, 219 Kofman, P. 152, 158 Korea 97–100 Korn, O. 157 Koskinen, Y. 66 Koutmos, G. 183, 192, 197 Kuhn–Tucker conditions 84–7 kurtosis 229, 231, 232, 233 La Porta, R. 66 Lagrange multiplier tests 130, 131 language, common 61–2, 65, 67–77 law, rule of 59–60, 66, 67–77 lead–lag relationship 155, 158–9, 164–5, 176–8 Lee, J. 153 legal system, type of 66, 67–77 leptokurtic distributions 189, 191

240

INDEX

likelihood ratio tests 22, 24 Lintner, J. 42, 115 liquidity 151–82 options data volume as a proxy for 154–9; price discovery and operational efficiency of a market structure 156–9 liquidity ratios 162, 165–7, 168, 169 and market volatility 169–71 London Futures and Options Exchange 156–7 Long-Term Capital Management (LTCM) 228 Longstaff, F.A. 21 L’Oréal 190–209 loss aversion 12 low correlation state 25–6, 27 lower partial moment (LPM) model 81 see also upside potential–downside risk portfolio model MacBeth, J.D. 119, 125 MacKinlay, C. 115 MacKinnon, J.G. 174 macroeconomic variables 117, 118–19 multifactor model for ASX 122–3, 126, 127–32, 133 Malaysia 97–100 Mananyi, A. 156–7 March, J.G. 80 market efficiency 156 market factor 185–7, 189–90 impact of systematic risk on French stocks 190–211 market return index 126, 127–32, 133 market structure 151–82 dynamics of a changing market structure 152–3 price discovery and operational efficiency of 156–9 Markowitz, H. 81, 82, 87, 88, 228 Massimb, M. 152–3 Masson, P. 96 matrix elliptical contoured distributions 214–27 analysis of the power functions 221–2 empirical study 222–4 small sample tests 216–20; further statistics 220; known type of elliptical symmetry 217–18; unknown type of elliptical symmetry 218–19

McKenzie, M.D. 192 mean–variance analysis 81, 228 critical line UPM/LMP model and 92–4 Meese, R. 17 Mei, J. 4 Mele, A. 199 Merton, R.C. 130 Mexico bond market–stock market linkages 103–13 currency crisis 96, 104 Min, H. 103, 112 mixture of distributions (MDH) hypothesis 154, 161, 170 modified Sharpe ratio 228–34 modified VaR (MvaR) 228, 229, 230, 231, 232, 233 Mody, A. 96 MONEP (Marché des Options Négociables de Paris) 188 moneyness portfolios 154–5, 159–60, 161, 167, 168–9 monsoonal effect 96 Moser, J. 152, 158 Moses, M. 4 Mossin, J. 115 Muirhead, R.J. 219, 225 multifactor arbitrage pricing theory (APT) 114–17 multifactor models (MFM) 114–36 data 119–27; cross–sectional regressions 124–7; CRR macroeconomic variables 122–3; explanatory returns 119–22; portfolio characteristics 124; returns to be explained 123–4 data analysis and results 127–32 existing evidence 115–18 parsimonious model for ASX 119, 130–2 multivariate t-distributions 221–2, 225–6 Nagy, M. 230 Naik, N. 229 Nawrocki, D. 81 New Zealand Gross Index 157 Ng, L.K. 119, 131–2 no-opportunity arbitrage valuation principle 186 non-linearity 195–201 Norway 63 NZSE-40 Index 157

INDEX

omission bias 13 open outcry 152, 153 operational efficiency 152, 153, 179 of a market structure 156–9 opportunity cost effect 12 optimal currency hedging 16–41 certainty equivalent compensation 28–31 estimation results for regime-switching models 21–6, 27 horse race for regime-switching strategies 31–7, 39 optimal foreign investment 38, 39 optimal hedge ratios 36–7 optimal weights 35–6 options benchmark assets 141–2; return distribution 144 option moneyness portfolios 154–5, 159–60, 161, 167, 168–9 options data volume as a proxy for liquidity 154–9 portfolio replication 139–40 pricing 186–7; dividend framework 187; no-dividend framework 186–7 out-of-sample test (horse race) 31–7, 39 out of the money (OTM) option contracts 160, 168–9 Owen, J. 215 ownership effect 12 Pacific/Asia bloc 47, 54, 56 Palaro, H.P. 139 parsimonious multifactor model 119, 130–2 payoff distribution pricing model 139 peer group analysis 148–9 Perez-Quiros, G. 17 perfect knowledge 18–20 performance Canadian hedge funds 231–3 forecasting 205–9 performance evaluation 137–50 efficiency gain/loss methodology 140–3 higher moment performance analysis 138–40 testing results 143–9; basic performance characteristics 145; data for the analysis 143–4; higher moment performance

241

characteristics 145–7; peer group analysis 148–9; return distribution of the benchmark asset 144 Perold, A. 16 Perron, P. 17 Phelps, B. 152–3 Philippines 97–100, 104 Phillip–Perron (PP) test 163, 174 phone call costs 61–2, 64, 67–77 Pirrong, C. 153 Pitts, M. 154 Portes, R. 64 portfolio replication 139–40, 145–7, 150 Post, T. 81 Poterba, J. 43, 65, 183 power functions 221–2 price art price indices 6–8 and trading volume 154 price discovery analysis 156–9, 174–9 prospect theory 81 Rabinovitch, R. 215 Racine, M.D. 157, 164, 174, 175 ratio analysis 162, 172–3, 179 real-estate bubble 6 real GDP growth rate 56–63, 67–77 regime switching 16–41 economic importance of regimes 28–31 estimation results 21–6, 27; data 21–2, 23; parameter estimates 22–6; specification test 22, 24 implications on asset allocation 26–38 model 18–21; portfolio selection under hidden regime switches 20–1; portfolio selection with perfect knowledge of the active state 18–20 optimal foreign investment 38, 39 strategies in competition 31–7, 39; cumulative wealth and Sharpe ratio 32–4; optimal hedge ratio 36–7; optimal weights 35–6 regression analysis cross-sectional regressions 124–32 impact of systematic risk on French stocks 196–209 liquidity and automation of SFE 162–3, 172–3 Reinganum, M.R. 115, 116, 119

242

INDEX

Renault 190–209, 210 replication, portfolio 139–40, 145–7, 150 return correlations 96 East Asian economies 97, 99–100, 101 returns Australian stock market 123–4, 127–32 bond and stock market linkages 105–12 East Asian stock markets average rates of return 97, 97–9, 101 return distribution of benchmark asset for Hungarian investment funds 144 reverse S-shaped utility functions 81 Rey, H. 64 Richardson, M. 161 Riddick, L.A. 42 risk downside see downside risk upside 229 upside potential–downside risk portfolio model 80–95 risk aversion 81, 92, 93, 94 risk-free interest rate 188–9 risk premiums, unexpected change in 122–3, 126, 127–32, 133 risk-seeking behavior 81 Rockafellar, R.T. 229 Roll, R. 114, 116, 184, 197, 209 Ross, S.A. 114, 116 Rubinstein, M. 154, 160, 161 rule of law 59–60, 66, 67–74, 75–7 S&P 500 index 157 Samorodnitsky, G. 226 Samuelson, W. 12 Santa-Clara, P.P. 200, 210 Sarkisson, S. 65 Sarno, L. 164 Savage, L.J. 81 SBF120 index 190–209, 210 SBF250 index 190–209, 210 scale factor 185–7, 189–90 Schill, M. 65 Schmid, W. 215, 220 Schneider 190–209 Scholes, M. 185, 186 Schulman, E. 16 Schwartz, E. 21 Selçuk, F. 183–4 self-deception theory 13

Sharpe, W.F. 42, 115, 183, 197, 209 Sharpe ratio 138, 228, 229, 230, 231, 232, 233 CEE investment funds 147, 150 modified 228–34 regime switching and optimal currency hedging 32–4 Shefrin, H. 13 short selling 145–7 Shyy, G. 153 Siegel, J.J. 5 Siegel, L.B. 81 Simon, H.A. 80 simple regression analysis 196–9, 201, 202, 203, 204, 206, 207, 208 Singapore 63 Singer, A. 229 size 118–19, 121–33 skewness 229, 231, 232, 233 Smith, T. 161 Société Générale 190–209, 210 Solnik, B. 117 Sortino, F. 81 spillover effect 96 SPI futures/All Ordinaries Index ratio 162–3, 172–3 Statex Actuaries Accumulation Index 157 statistical multifactor models 117 Statman, M. 13 status quo bias 12 Stiglitz, J.E. 5 stock index dynamic 185–6 stock market capitalization 57–8, 64, 67–77 stock market development 43, 44, 57–8, 63–4, 67–77 stock markets linkages to bond markets 103–13 currency crises, contagion and portfolio selection 96–102 Strong, N. 65 Struthers, J.J. 156–7 Stulz, R.M. 96, 183 sunk cost effect 12–13 Sydney Futures Exchange (SFE) 151–82 lead–lag relationship with ASX 155, 158–9, 164–5, 176–8 Share Price Index (SPI) 157, 158, 159, 161, 165–7, 175, 176–8 SPI futures/All Ordinaries Index ratio 162–3, 172–3

INDEX

symmetric stable distributions 221, 222, 225–6 systematic risk 183–213 empirical study 187–90; data 187–9; induction of systematic risk 189–90 impact 190–5; causality 191–3; correlation 190–1; Granger causality test 193–5 market benchmark comparison 201–9; basic empirical study 201–5; forecasting performance 205–9 non-linearity 195–201; simple regression analysis of asset returns 196–9; volatility impact 199–201 theoretical framework 185–7; option pricing 186–7; valuation setting 185–6 Szego, G. 191 tail index estimator 8, 9–12 Taiwan 97–100 Taqqu, M.S. 226 Tauchen, G. 154 Taylor, M.P. 164 term structure, unexpected change in 122–3, 126, 127–32, 133 Tesar, L. 64, 67 Thaler, R.H. 12, 13, 117 Thomas, S.H. 127, 130 Thomson 190–209 time series properties 174–5 time-varying downside risk 1–15 Timmerman, A. 17, 28 Titman, S. 117 Toronto Stock Exchange (TSE) 156, 231 Totalfina Elf 190–209, 210 trade, scaled by GDP 56–63, 67–77 trading volume 154, 155, 161, 165–7 market volatility and 169–71 transaction costs 64, 67–77 Treynor, J. 183 Turkington, J. 157, 158 Turner, C. 17 Tversky, A. 12, 81 two-year return 67–77 underlying distribution 142–3 unexpected inflation rate 122–3, 126, 127–32, 133 uniqueness of art works 3

243

unit root tests 163, 174 United Kingdom art market 6–8, 10 currency hedging and regime switching 21–6, 27, 33–6; optimal hedge ratio 36–7 United States art market 6–8, 10 regime switching and currency hedging 21–7, 33–6 stock market levels and returns 107 unknown characteristic function 218–19 unknown location vector 216–19 upper partial moment/lower partial moment (UPM/LPM) ratio 81 see also upside potential–downside risk portfolio model upside potential–downside risk portfolio model 80–95 efficient segments on the efficient frontier 88; adjacent efficient segments 89–92 empirical example 92–4 Kuhn–Tucker conditions 84–7 upside risk 229 upward bias 209 Uryasev, S. 229 Valéo 190–209, 210 valuation 185–6 value, art and 3–4 value-at-risk (VaR) 228, 229, 231, 232, 233 modified (MvaR) 228, 229, 230, 231, 232, 233 Van Vliet, P. 81 Varga, T. 216, 217 variance decompositions 109, 110, 111 vector autoregressive (VAR) models 164 Brady bonds 105–12 systematic risk and French stocks 191–3, 194 Venezuela 43, 47, 104 Vivendi 190–209, 210 volatility Asian currency crisis 97–9, 101 automation of SFE 161, 169–71 Black and Scholes volatility bias 189

244

INDEX

volatility continued impact of systematic risk on French stocks 199–205, 207, 208 volatility parameter 185, 189–90

Whaley, R. 189 Whitcher, B. 183-4 White, H. 127 world float portfolio

Walsh, D. 157, 158 Wang, G.H.K. 153 Wang, Z. 197 Warnock, F. 63 Weber, M. 43 Werner, I. 64

Xu, X.

65

Zeckhauser, R. 12 Zhang, Y.T. 219, 225 Zhou, C. 64 Zhou, G. 215

72–7

Greg N. Gregoriou

ASSET ALLOCATION AND INTERNATIONAL INVESTMENTS

Also edited by Greg N. Gregoriou ADVANCES IN RISK MANAGEMENT DIVERSIFICATION AND PORTFOLIO MANAGEMENT OF MUTUAL FUNDS PERFORMANCE OF MUTUAL FUNDS

Asset Allocation and International Investments

Edited by GREG N. GREGORIOU

Selection and editorial matter © Greg N. Gregoriou 2007 Individual chapters © contributors 2007 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1T 4LP. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The authors have asserted their rights to be identified as the authors of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2007 by PALGRAVE MACMILLAN Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N.Y. 10010 Companies and representatives throughout the world PALGRAVE MACMILLAN is the global academic imprint of the Palgrave Macmillan division of St. Martin’s Press, LLC and of Palgrave Macmillan Ltd. Macmillan® is a registered trademark in the United States, United Kingdom and other countries. Palgrave is a registered trademark in the European Union and other countries. ISBN-13: 978–0–230–01917–1 ISBN-10: 0–230–01917–X This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. A catalogue record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Asset allocation and international investments / edited by Gerg N. Gregoriou. p.cm. — (Finance and capital markets) Includes bibliographical references and index. ISBN 0–230–01917–X 1. Asset allocation. 2. Investments, Foreign. 3. Globalization—Economic aspects. 4. Portfolio management. I. Gregoriou, Greg N., 1956– II. Series. HG4529.5.A83 2006 332.67’3—dc22 2006045369 10 9 8 7 6 5 4 3 2 1 16 15 14 13 12 11 10 09 08 07 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham and Eastbourne

To my mother Evangelia and in loving memory of my father Nicholas

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Contents

Acknowledgments

xi

Notes on the Contributors

xii

Introduction

xvii

1 Time-Varying Downside Risk: An Application to the Art Market

1

Rachel Campbell and Roman Kräussl 1.1 Introduction 1.2 Art as an investment 1.3 Previous empirical studies 1.4 Empirical analysis 1.5 Data 1.6 Methodology 1.7 Results 1.8 Discussion 1.9 Conclusion

2 International Stock Portfolios and Optimal Currency Hedging with Regime Switching

1 3 4 5 6 9 10 11 13

16

Markus Leippold and Felix Morger 2.1 2.2 2.3 2.4 2.5

Introduction The model Estimation results Discussion Conclusion

16 18 21 26 39 vii

viii

CONTENTS

3 The Determinants of Domestic and Foreign Biases: An Empirical Study Fathi Abid and Slah Bahloul 3.1 Introduction 3.2 Theoretical framework of domestic and foreign biases 3.3 Data and preliminary statistics 3.4 The determinants of domestic and foreign biases 3.5 The empirical analysis 3.6 Additional tests 3.7 Conclusion

4 The Critical Line Algorithm for UPM–LPM Parametric General Asset Allocation Problem with Allocation Boundaries and Linear Constraints

42 42 44 46 56 67 71 74

80

Denisa Cumova, David Moreno and David Nawrocki 4.1 Introduction 4.2 The upside potential–downside risk portfolio model 4.3 An empirical example 4.4 Conclusion

5 Currency Crises, Contagion and Portfolio Selection

80 82 92 94

96

Arindam Bandopadhyaya and Sushmita Nagarajan 5.1 5.2 5.3 5.4 5.5

Introduction Stock market average rates of return and average volatility Stock market correlations Portfolio performance Conclusion

6 Bond and Stock Market Linkages: The Case of Mexico and Brazil

96 97 99 100 101

103

Arindam Bandopadhyaya 6.1 6.2 6.3 6.4

Introduction The estimation equations and data Results Conclusion

7 The Australian Stock Market: An Empirical Investigation

103 105 109 112

114

Adeline Chan and J. Wickramanayake 7.1 7.2 7.3

Introduction Existing evidence Hypothesis

114 115 118

CONTENTS

7.4 7.5 7.6

The data Data analysis and results Conclusion

ix

119 127 132

8 The Price of Efficiency – So, What Do You Think About Emerging Markets? 137 Zsolt Berényi 8.1 8.2 8.3 8.4 8.5

Introduction Higher moment performance analysis – the theory The efficiency gain/loss methodology Testing results Conclusion

9 Liquidity and Market Efficiency Before and After the Introduction of Electronic Trading at the Sydney Futures Exchange

137 138 140 143 149

151

Mark Burgess and J. Wickramanayake 9.1 Introduction 9.2 Review of the literature 9.3 Options data volume as a proxy for liquidity 9.4 Sample design 9.5 Analysis of results 9.6 Conclusion

10 How Does Systematic Risk Impact Stocks? A Study of the French Financial Market

151 152 154 159 165 178

183

Hayette Gatfaoui 10.1 10.2 10.3 10.4 10.5 10.6 10.7

Introduction Theoretical framework Empirical study The impact of systematic risk Further investigation Market benchmark comparison Conclusion

11 Matrix Elliptical Contoured Distributions versus a Stable Model: Application to Daily Stock Returns of Eight Stock Markets

183 185 187 190 195 201 209

214

Taras Bodnar and Wolfgang Schmid 11.1 Introduction 11.2 Small sample tests 11.3 Analysis of the power functions 11.4 Empirical study 11.5 Conclusion

214 216 221 222 224

x

CONTENTS

12 The Modified Sharpe Ratio Applied to Canadian Hedge Funds

228

Greg N. Gregoriou 12.1 12.2 12.3 12.4 12.5 Index

Introduction Literature review Data and methodology Empirical results Conclusion

228 229 230 231 233 235

Acknowledgments

I would like to thank Stephen Rutt, Publishing Director, and Alexandra Dawe, Assistant Editor, at Palgrave Macmillan for their suggestions, efficiency and helpful comments throughout the production process, as well as Keith Povey (with Elaine Towns) for copy-editing and editorial supervision of the highest order. In addition, I would like to thank the numerous anonymous referees in the US and Europe during the review and selection process of the articles proposed for this volume.

xi

Notes on the Contributors

The Editor Greg N. Gregoriou is Associate Professor of Finance and coordinator of faculty research in the School of Business and Economics at the State University of New York (Plattsburgh). He obtained his PhD (Finance) from the University of Quebec at Montreal and is the hedge fund editor for the peer-reviewed journal Derivatives Use, Trading and Regulation, published by Palgrave Macmillan, based in the UK. He has authored over fifty articles on hedge funds, and managed futures in various US and UK peer-reviewed publications, including Journal of Portfolio Management, Journal of Futures Markets, European Journal of Finance, Journal of Asset Management, European Journal of Operational Research and Annals of Operations Research. He has published four books with John Wiley and Sons Inc. and four with Elsevier.

The Contributors Fathi Abid is a Professor of Finance. He is Director of the research team MODESFI specializing in financial modeling and financial strategy. He lectures frequently on financial market theory and has taught investment and portfolio management at Tunisian and European universities. He has written and co-authored numerous articles in national and international scientific journals, books and proceedings. xii

NOTES ON THE CONTRIBUTORS

xiii

Arindam Bandopadhyaya is the Chairman and Associate Professor of Finance in the Accounting and Finance Department at UMass Boston, USA. He is also the Director of the College of Management’s Financial Services Forum. A recipient of the Dean’s Award for Distinguished Research, Dr Bandopadhyaya has published in journals such as the Journal of International Money and Finance, Journal of Empirical Finance, Journal of Banking and Finance and Review of Economics and Statistics. He has presented his work at national and international conferences such as those of the Financial Management Association, European Finance Association and European Economic Association. He has presented research reports of the Financial Services Form at the Boston Stock Exchange and the Federal Reserve Bank of Boston. Dr Bandopadhyaya teaches corporate finance, international finance and managerial economics. He has received teaching awards from the College of Management, including the Professor of the Year Award and the Betty Diener Award for Teaching Excellence. Slah Bahloul is an Assistant Professor of Finance at Higher School of Business Administration in Sfax, Tunisia. He is a research assistant in the MODESFI team and has taught international finance and financial decision-making. Zsolt Berényi holds an MSc in Economics from the University of Economic Sciences in Budapest, and a PhD in Finance from the University of Munich. His main interests lie in the risk and performance evaluation of alternative investments: hedge funds, CTAs and credit funds. After working for many years for the Deutsche Bank, HypoVereinsbank and KPMG at various locations throughout Europe, Zsolt now leads an independent consultancy in Budapest, Hungary. Taras Bodnar studied Mathematics at the Lviv National University, Ukraine from 1996 to 2001. He received a PhD in Economics in 2004 from the European University Viadrina, Frankfurt (Oder), Germany. Currently, he is a research assistant at the Department of Statistics, European University Viadrina. His fields of interest are quantitative methods in finance, nonstationary time series, elliptical distributions and econometric applications. Mark Burgess currently works in the financial services industry in Australia. He has a Bachelor of Business (Honors) degree from Monash University, Australia. Rachel Campbell completed her PhD on Risk Management in International Financial Markets at Erasmus University, Rotterdam, The Netherlands in 2001. She currently works at the University of Maastricht as an Assistant Professor of Finance. Her work has been published in a number of leading

xiv

NOTES ON THE CONTRIBUTORS

journals, including the Journal of International Money and Finance, Journal of Banking and Finance, Financial Analysts Journal, Journal of Portfolio Management, Journal of Risk, and Derivatives Weekly. She teaches with Euromoney Financial Training on Art Investment, and works as an Independent Economic Adviser for the Fine Art Fund in London, and for Fine Art Wealth Management, UK. Adeline Chan currently works in the financial services industry in Singapore. She has a Bachelor of Business (Honors) degree from Monash University, Australia. Denisa Cumova works in the fund management group at the Berenberg Bank in Hamburg, Germany. She received her PhD in Finance from the University of Technology, Chemnitz, Germany. Hayette Gatfaoui gained a PhD in “Default Risk Valuation of Financial Assets” University Paris 1 in 2000. He taught for five years at the University Paris 1 (Pantheon-Sorbonne) France, and is now an Associate Professor at Rouen Graduate School of Management, France. He is a specialist in applied mathematics (holding a Master’s degree in stochastic modeling for finance and economics). He is currently advising financial firms about risk measurement and risk management topics for asset management, and for credit risk management purposes. Dr Gatfaoui is also a referee for the International Journal of Theoretical and Applied Finance (IJTAF). His current research areas concern risk typology in financial markets, quantitative finance and risk analysis. Roman Kräussl obtained a first-class honors Master’s degree in Economics with a specialization in Financial Econometrics from the University of Bielefeld, Germany, in 1998. He completed his PhD on the Role of Credit Rating Agencies in International Financial Markets at Johann Wolfgang Goethe University, Frankfurtam Main, Germany, in 2002. As the Head of Quantitative Research at Cognitrend GmbH, he was closely involved with the financial industry. Currently he is Assistant Professor of Finance at Vrije Universiteit Amsterdam, The Netherlands and research fellow with the Centre for Financial Studies, Frankfurtam Main. Markus Leippold is Assistant Professor of Finance at the Swiss Banking Institute of the University of Zurich, Switzerland. Prior to moving back to academia he worked for Sungard, Trading and Risk Management Systems, and the Zurich Cantonal Bank. His main research interests are term structure modeling, asset pricing and risk management. He obtained his PhD in financial economics from the University of St. Gallen, Switzerland, in 1999. During his PhD studies, he was a research fellow at the Stern School of Business in New York. He has published in several journals, such as the

NOTES ON THE CONTRIBUTORS

xv

Journal of Financial and Quantitative Analysis, Journal of Economic Dynamics and Control, Journal of Banking and Finance, Review of Derivative Research, Journal of Risk, and Review of Finance. In 2003, he and his colleagues received an award from the German Finance Association for their paper on the equilibrium impacts of value-at-risk regulation, and an achievement award from RISK for their paper on operational risk. In 2004, their research paper on credit contagion won the STOXX Gold Award at the annual conference of the European Financial Management Association. David Moreno holds a PhD degree in Economics from the Universidad Carlos III, Madrid, Spain, and a BSc degree in Mathematics from the Universidad Complutense, Madrid. He is currently Assistant Professor of Financial Economics and Accounting at Universidad Pompeu Fabra, Barcelona and Co-Director of the Master’s Program in Finance. He has previously held teaching and research positions at the Financial Option Research Centre (Warwick Business School, UK), Universidad Carlos III de Madrid, and at the IESE Business School, Barcelona, Spain. His research interests focus on finance in continuous time, with special emphasis on derivatives markets, financial engineering applications, pricing of derivatives, empirical analysis of different pricing models, portfolio management and term structure models. His research has been published in a number of academic journals including Review of Derivatives Research and Journal of Futures Markets, as well as in professional volumes. He has presented his work at a number of international conferences and has given invited talks at many academic and nonacademic institutions. He is associate editor of Revista de Economía Financiera and a member of GARP (the Global Association of Risk Professionals). Felix Morger is a fourth-year PhD student at the Swiss Banking Institute of the University of Zurich, Switzerland. The main part of his thesis is concerned with the theoretical and empirical aspects of Bayesian learning models with Markov switching and their application to asset allocation. Prior to his PhD studies, he worked as a consultant in pension funds. Sushmita Nagarajan is a Senior Associate in the Structured Finance Group at Moody’s Investor Service, New York. Her areas of expertise are rating and monitoring various types of structured derivative products using Moody’s rating methodologies. She also provides quantitative analysis and research surrounding complex derivative products such as asset-backed commercial paper structures. Prior to joining Moody’s she was an intern at State Street Research and Management as a Fixed Income Research Analyst with emphasis on Collateralized Debt Obligations. She graduated summa-cumlaude with a MSc degree in Finance from Boston College, and has an MBA in Finance from Jawaharlal Nehru Technological University, India.

xvi

NOTES ON THE CONTRIBUTORS

David Nawrocki is the Katherine M. and Richard J. Salisbury Jr. Professor of Finance at Villanova University, Villanova, Pa., USA. He is a registered investment adviser and is the director of the Institute for Research in Advanced Financial Technology (IRAFT) at Villanova. Nawrocki’s research includes work on financial market theory, downside-risk measures, systems theory, portfolio theory, and business cycles. He received his PhD in Finance from the Pennsylvania State University, USA. Wolfgang Schmid is a Full Professor at the European University in Frankfurt (Oder), Germany. He received a PhD in Mathematics in 1984 at the University of Ulm, Germany. His fields of major statistical activities are quantitative methods in finance, statistical process control and econometric applications. J. Wickramanayake obtained his PhD in 1994 from La Trobe University, Australia. He completed his Master’s degree at Williams College, Williamstown, Ma., USA in 1982, and did postgraduate studies in the Netherlands in 1978. He has been a member of the Financial Services Institute of Australasia for over ten years. Dr Wickramanayake has more than twenty years’ experience as a financial analyst at a central bank. Currently, he teaches finance at both undergraduate and postgraduate levels at Monash University, Australia. Dr Wickramanayake’s research interests involve banking, financial markets, mergers/acquisitions, bankruptcy and business failures, fund management, superannuation and pension finance.

Introduction

Chapter 1 deals with the economic downturn during 2000 which left many investors with burnt fingers and weary of investing in equities. There has been a continued search for alternative asset classes to fulfill the need for preserving returns while not taking on too high a risk. One such innovative alternative is investing in art as an alternative to stocks, bonds and real estate. This chapter analyses in a detailed empirical study how the risk during the art market bubble increased dramatically before the collapse of the market in the early 1990s. Understanding how deviations from normality in the form of extreme market returns link to the creation of a bubble in asset prices is crucial to our understanding of risk-and-return relationships. Chapter 2 presents a model for strategic asset allocation and currency hedging for an international investor, where the returns on stock indices follow a Gaussian regime-switching model. The authors study a Bayesian investor, who has only partial information on the current regime switching model being active, but updates the investor’s beliefs over time. The results indicate that engaging in optimal currency hedging significantly improves the risk and return characteristics of the Bayesian investor. Chapter 3 describes an empirical study of the determinant factors of domestic and foreign home biases. Using the equity holdings of thirty countries, the authors find that a severe equity home bias exists for both developed and emerging markets. Stock market development, information costs and familiarity factors are found to contribute the most to explaining foreign bias, whereas investor’s behavior has a significant effect on domestic bias. Chapter 4 discusses how human beings have always engaged in different behavior above and below a target rate of return. As a result, reverse S-shaped utility functions have been utilized to describe this human investment behavior, ever since Friedman and Savage (1948) and Markowitz xvii

xviii

INTRODUCTION

(1952). Fishburn (1977) made this approach operational with the lower partial moment, LPM(a, t), model, which detailed risk-seeking and risk-averse behavior below a minimum target return. However, the Fishburn utility measures have attracted criticism, since they assume a linear utility (risk neutral) above the target return. Recently, the upper partial moment/lower partial moment (UPM/LPM) has been put forward as a solution to this problem. This chapter develops a UPM/LPM critical line algorithm that allows this model to be operational. Chapter 5 examines the characteristics of domestic and international portfolios from the perspective of a US investor in Asian emerging markets during a period where the economies have suffered a currency crisis. Among various portfolios constructed, a purely international portfolio posts superior performance compared to a purely domestic one or a combination of domestic and international portfolios in the post-crisis period. Chapter 6 investigates the Brady bond markets of the two largest LatinAmerican economies – Mexico and Brazil. Results indicate that, for the very near future, the yield in each market is determined primarily by past yields in the respective markets. However, over a longer-term horizon, the interrelationships between the bond markets and the stock markets of the two countries become increasingly important. Chapter 7 provides an evaluation and comparison between the explanatory power of the macroeconomic model of Chen et al. (1986) and the three-factor model of Fama and French (1993) in explaining the variation in returns in the Australian equity market for the decade of the 1990s. The empirical results show that firm attributes (Fama and French, 1993) alone are insufficient to explain returns and macroeconomic variables (Chen et al., 1986) can be combined in a better multifactor model to explain the variation in returns. Chapter 8 evaluates inter-market investment efficiency, which may be a complicated task, especially across investment forms with widely differing return characteristics. This chapter offers some new ideas on how to evaluate such investments, using the example of emerging markets. The authors show that replicating the expected return distribution using options, the efficiency of any investment portfolio – for example, not just “emerging market” or “equity” – can be assessed and compared. Chapter 9 examines whether the Sydney Futures Exchange (SFE) in Australia has benefited from the introduction of electronic trading on November 15, 1999. Empirical results in this study show that during the early stage, up to the beginning of August 2000 that the money SPI options were more liquid at times of high volatility after the automation of the SFE. However, the SPI futures were less liquid at times of medium to low market volatility after this event. The authors also found a cointegrating relationship between the Australian Stock Exchange (ASX) and the derivative market (SFE) before

INTRODUCTION

xix

and after the introduction of electronic trading supporting the semi-strong market efficiency hypothesis. Chapter 10 discusses how many researchers have focused on the common latent component underlying the evolution of stock returns. The authors propose to infer such an unobserved common component while employing the well known Black and Scholes (1973) option pricing formula. Their study is based on the assumption that any small stock market index is a distorted representative of such a latent component. Once this systematic risk factor is exhibited, the authors attempt to assess its impact on a basket of French stock returns. Chapter 11 explores the assumptions of independency and normality which are not appropriate in many situations of practical interest, especially for the data sets from emerging markets. The authors propose to make use of matrix elliptical distribution instead of the normal distribution. Empirically, they show that the assumptions of the elliptical symmetry cannot be rejected for daily returns. Chapter 12 applies the modified Sharpe ratio to a small sample of Canadian hedge funds. Many investors today use the traditional Sharpe ratio to measure risk-adjusted performance, but the proposed modified VaR Sharpe ratio is a superior and more precise method that can deal with the skewed/non-normal returns that hedge fund possess. The results show that the modified Sharpe ratio is more precise when examining non-normal returns.

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CHAPTER 1

Time-Varying Downside Risk: An Application to the Art Market1 Rachel Campbell and Roman Kräussl

1.1 INTRODUCTION The economic downturn during 2000 left many investors with burnt fingers and weary of investing in equities. Since then, there has been a search for alternative asset classes to fulfill the need to preserve returns, while not involving too high a risk. Arising from the media’s continued concern about a potential bubble in the housing market, many investors are showing an increasing interest in alternative asset classes that are not so highly correlated with equities, and provide hedging potential as part of a diversified portfolio of investments. One such innovative alternative asset class to stocks, bonds and real estate is art, which is seen increasingly as not merely items with aesthetic value, but also as attractive investments with a potential capital gain. The planned launch of a Fund of Art Funds by ABN Amro in 2005, aiming to channel money into some existing (and some yet to be launched) independent art funds, serves to highlight this point. It is a well-known fact that investment in art is influenced strongly by income and other fundamental economic factors. The effect on the economy from a collapse in the art market depends on the contagious impact of the art market on the rest of the financial system, predominately through the banking system. Thus, what is the impact of a negative shock in the art market on the overall economy? We argue that the extent to which real effects are likely to occur from a bubble in the art market is likely to 1

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be significantly less because of the type of investment that is made in the art market. There are two main reasons for this. First, as art is a luxury good, investors tend to invest money into the art market that would not necessarily be invested elsewhere in other asset classes beyond holding it as surplus cash. Second, the initial wealth levels of investors typically investing in art markets is higher, and therefore less at the mercy of the banking system, as the banks are unlikely to let such investors become insolvent. We argue that the likelihood of falling prices is only liable to affect the general economy to the extent that the losses made might reduce liquidity in financial markets. Even though booms in other markets, such as in real estate, may lead to a collapse in the initial market followed by a collapse in the banking sector, this is much less likely to be the case in the art market. Although the real effects from a collapse in the art market may be significantly less than in other financial markets, the development of bubbles in the art market is likely to be significantly greater. The rate at which prices in the art market are driven by taste and fashion, predominately via the media, is much greater than in other financial markets, where “value” is a greater function of market fundamentals. The development of a large bubble in the general price of all works of art was well documented in the early 1990s for most classes of art. Indeed, it would appear that there was a severe deviation away from the fundamental valuation of art pieces during this period. This provides an extremely interesting and unique data series with which to analyze the risk to the investor around the period of the bubble’s development. We focus on time-varying downside risk in relation to theory from behavioral finance. This, given our knowledge of the literature, is an area of research that has not been undertaken before. In this chapter we analyze the art market using a measure for time variance in the downside risk, which reflects “bubbliness” in the market. This estimate measures the changing probability of large movements occurring in the return distribution of the historical time series of art price data. Taking such an approach and using techniques developed in extreme value theory (EVT), we are able to provide some new insight into the creation and measurement of risk during times of the development of bubbles in financial markets. We focus on a particularly interesting case: the art market. This market is highly media- and taste-driven, is illiquid and lacks transparency, and thus offers an ideal application in which to observe downside risk with prices that may deviate significantly from fundamental values. This chapter is organized as follows. Section 1.2 briefly surveys the economic literature concerning art as an investment; we explore the financial aspects of art investing by emphasizing similarities and differences among financial assets. Section 1.3 discusses the data and the methodology, and presents the empirical results. Section 1.4 presents some behavioral

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explanations for our results. Section 1.5 concludes and presents an outlook for future research.

1.2 ART AS AN INVESTMENT 1.2.1 The art market in general Financial assets tend to be very liquid, allowing for diversification benefits, and thus reduce risk. Additionally, they are relatively transparent. Most financial assets can be selected on the basis of a fairly small set of objective criteria. Fundamentals do exist and can be analyzed with standard finance tools. Such financial markets are characterized by a large number of individual buyers and sellers, transaction costs are low, and trades in perfectly (or nearly) identical assets are repeated millions of times daily in various exchanges. It goes without saying that, the first impression of the art markets is that they differ significantly from other types of financial markets. Most art markets would appear to be characterized by product heterogeneity, illiquidity, behavioral anomalies, market segmentation, information asymmetries, and almost monopolistic price setting. Moreover, there is no doubt that a substantial amount of the return from art investment is derived not from classical financial returns but rather from intrinsic aesthetic qualities through art as a consumption good. Art works are not liquid assets, and transaction costs are high. Short selling is not possible, and supply is rather inelastic in the short term. There are unavoidable delays between an owner’s decision to sell and the actual sale, since it takes about three to six months to “market a work” – that is, to have it accepted by the auction house, take photographs and print and distribute the catalogue, publish advertisements for the coming auction and so on. Investing in art typically requires substantial knowledge of art and the art market in general, and often a significant amount of capital to acquire a work of a well-known artist. Moreover, the art market is highly segmented and dominated by a few large auction houses. These auction houses, such as Sotheby’s and Christie’s, are used by a restricted number of buyers, mostly wealthy collectors, public museums or private foundations. Informational asymmetries are essential features of these markets. Furthermore, art sells only occasionally. Art objects are created by individuals. Accordingly, there is only a single, unique piece of original work available. This is an extreme case of a heterogeneous commodity. Therefore, financial risk in the art market is related to specific material risk factors associated with the unique physical nature of art works such as theft, fire, water damage, or the possible reattribution to another (less famous) artist. Moreover, the value of an art object is

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determined by a complex and subjective set of beliefs about past, present and future prices. Art has little intrinsic value; its appeal is ultimately dependent on individual tastes and fashion, which can change over time. The future sales price of a piece of art depends on both the number of people who wish to buy the piece when it is put up for sale and the (available) wealth of the individuals or institutions who desire it at that time. The most distinctive difference between financial markets and the art market is that the individual investor’s expected return from investing in art consists not only of a rise in price. It also involves the psychic return from art works through their aesthetic qualities. Most empirical studies have been unable so far to quantify these psychic returns associated with art as a consumption good. Recognizing art as a consumption good helps in part to explain behavioral anomalies less well-known in modern financial markets.

1.3 PREVIOUS EMPIRICAL STUDIES In recent years, an extensive literature has arisen based on calculating the returns on art investments. Starting with Baumol (1986), these include, among others, empirical studies by Goetzmann (1993), Chanel (1995), Mei and Moses (2002) and Campbell (2005). Baumol (1986) and Goetzmann (1993) tend to concur that art is dominated as an investment product by stocks, bonds and real estate. Goetzmann (1993) finds a positive relationship between art investments and the stock market over shorter time periods. He argues that the high and significant positive correlation clearly makes art investment a poor instrument for the purposes of portfolio diversification. Goetzmann (1993) also finds evidence of a significant relationship between aggregate financial wealth and the demand for art. He concludes that this empirical finding is sufficient evidence that the demand for art increases with the wealth of art collectors since, in the twentieth century, art prices tended to follow stock market trends. Chanel (1995) follows this argumentation and concludes that financial markets react quickly to shocks in the economy. Profits generated on financial markets may be invested in art, so that developments in stock markets may be considered as leading indicators for returns in the art markets. Mei and Moses (2002) take a somewhat different view. They argue that a diversified portfolio of works of art play a more important role in portfolio diversification. They base their conclusions on their empirical finding that their art price index has lower volatility and a much lower correlation with other asset classes than was discovered in earlier research. Campbell (2005) focuses on the extent of downside risk, which is less for the art market during periods in which the stock market performs badly. This is highly likely to be driven by issues relating to theories from behavioral finance, caused not only by the low liquidity on the art market, but also to investors being

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anxious not to sell off art works representing a symbol of their reputation and status during falling financial markets. Thus, maintaining art investment remains strong during periods of economic downturn. This helps to drive the hedging connotation of art in the portfolio. The cyclicality of the art and equity markets has been documented in a recent working paper by Bauer et al. (2005), who show that art investments perform well at times when other asset classes are performing badly. Despite this strength during downturns, bubbles are also evident in the art market. The famous bubble in the 1990s occurred because of the excessive demand for works of art by the Japanese. What happens to risk-and-return characteristics during these periods? Is risk time-varying during the expansion of the bubble? Could we have seen the extent to which a bubble was developing in the market? Before answering these questions, a number of preliminary queries need to be addressed on the definition and possible estimation of a bubble.

1.4 EMPIRICAL ANALYSIS 1.4.1 How to define a bubble in the art market? What is a bubble in financial markets? How do we define bubbles ex ante or even ex post? A financial market bubble may be defined loosely as a sharp increase in the price of an asset in a continuous process, with the initial rise generating investors’ expectations of further future increases and thereby attracting new buyers. These buyers are generally speculators, interested in profits from trading in the asset rather than the asset’s earning capacity. Such a definition implies that a high and increasing price is not justified and is fed by momentum investors who buy with the sole purpose of selling quickly to other investors at a higher price. In recent years, economists have tried to give additional substance to the definition of a financial market bubble by linking asset price movements to fundamentals. Fundamentals refer to those economic factors that together determine the price of any asset, such as cash flows and discount rates. For example, Stiglitz (1990, p. 13) defines a bubble in financial markets in the following way: “If the reason that the price is high today is only that the selling price will be high tomorrow – when fundamental factors do not seem to justify such a price – then a bubble exists”. In this context, Siegel (2003) argues that one cannot identify any asset price bubble immediately, because one has to wait a sufficient length of time to determine whether the previous asset prices can be justified by the asset’s subsequent cash flows. Unfortunately, it is not that easy to find an operational definition of a bubble in the art market. If a bubble is defined only as excess changes in prices that are not captured by underlying economic fundamentals, then

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the extent to which bubbles occur in the art market is only by definition. Although some mass media treat the rapid rate of increase in some pieces of art as de facto evidence of a bubble in the art market, our understanding dictates that such rises alone are necessary but not sufficient evidence. Additional evidence is needed that relates current art prices to their fundamental determinants. It is vital to keep in mind that the only constraint limiting the price of a particular piece of art is the wealth of the agents willing to pay it. Economists have identified a number of transmission mechanisms where fluctuations in housing prices can have an effect on the overall economy. One potential effect of a severe home price decline could result from a consumption wealth effect. Although the magnitude of this effect remains controversial in some quarters, a number of empirical studies find significant wealth effects from real estate assets. For example, Case et al. (2001) show that if the magnitude of the wealth effect from housing is around 5 percent, then a severe decline would lead to reduction in consumption of roundabout US$150bn, which is about 2 percent of total personal consumption expenditures. Many analysts argue that the recent increase in home prices is symptomatic of a real-estate bubble that will burst eventually, just as the stock market bubble did in 2000. This would imply the erasing of a significant amount of household wealth. They add that such a decline of disposable income would have sharp adverse macroeconomic effects, as already indebted consumers reduce spending even further to improve their weakened financial situation. Despite the (technical and conceptual) difficulties of defining bubbles in art markets, we believe that the likelihood of falling prices will affect the general economy only to the extent that the losses made may reduce liquidity in financial markets.

1.5 DATA In order to look more specifically at both bull and bear markets, we use almost thirty years’ of monthly data, from January 1976 to December 2004, from the Art Market Research (AMR) database. AMR uses over 800 auction houses to collect sales data for hundreds of individual artists worldwide. This is the most comprehensive data set available for looking at performance during market extremes, since it is available as a monthly index. Indices are constructed for individual artists using the average prices of his or her paintings obtained in the market. A national index is constructed, comprised of a number of chosen artists for each country. Art indices for the USA and the UK art markets are used, covering a large number of artists over several movements and periods in the art scene, as well as a general index which covers the markets that dominate the global market for art.2 Log returns are

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computed for the art indices over the period January 1976–December 2004 (see Table 1.1). For an art fund, the only dividend received is the pleasure of the painting for the portfolio manager, unless the art fund allows its investors to rent or borrow some of its paintings for display in their own homes, or the works of art are rented out to museums or art collectors on loan, thus providing an additional income stream. The indices do not cover sales by private dealers, or works of art that are bought in – that is, pieces put on the block but not selling; however, these represent a highly significant part of the global art market3 . Figure 1.1 displays the development of the average price indices

Table 1.1 Summary statistics: monthly log return data, January 1976– December 2004 ART 100 Annual average return

US 100

UK 100

5.27%

8.26%

5.12%

17.11%

15.86%

11.10%

Average

0.026

0.041

0.026

Standard deviation

0.121

0.112

0.078

−0.837

−0.817

−0.097

1.694

1.029

−1.083

Annual average standard deviation

Skewness Kurtosis

12,000 10,000

General US British

8000 6000 4000

0

1976 1977/03 1978/06 1979/09 1980/12 1982/03 1983/06 1984/09 1985/12 1987/03 1988/06 1989/09 1990/12 1992/03 1993/06 1994/09 1995/12 1997/03 1998/06 1999/09 2000/12 2002/03 2003/06 2004/09

2000

Figure 1.1 Art indices: International art performance, January 1976–December 2004 Note: The average price indices from AMR for the General Art Market (ART 100): the top US artists (US 100) and the top British artists (UK 100).

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for the general art market, the top 100 US artists and the top 100 UK artists since the later 1970s. From Figure 1.1, it would appear that there was an enormous bubble in the art market during the early 1990s. This has been documented in the literature and is usually considered to have arisen from the effect of the expanding Japanese economy, with wealth flowing directly from this boom into the luxury art market. From the graph, it would appear that prices returned to their fundamental values in the two years following the bubble in 1990. Figure 1.1 also indicates a Japanese phenomenon after the collapse of the Japanese economy, with money flowing directly out of the art market during this period. Interestingly, the causality has been documented between these two markets, as well as between other equity markets and the art market, by Campbell (2005). The restrictive supply of art, recently cited as the reason for increasing prices, and hence returns being made in art investment, is also a driving factor behind the occurrence of bubbles in the art market. More recent developments using art as collateral for credit loans only serve to lengthen the extent and duration of the resulting price rises and the size of the financial bubble. This optimism is exacerbated by the tendency of investors and banks towards myopic disaster behavior. The highly leveraged positions of banks, holding collateral consisting of such opaque assets as real estate and artworks result in downside risk from the real estate and art markets being shifted to the banking sector. This effect is exaggerated by the feeling of wealth created by increases in property prices and art prices feeding on each other. This can have severe implications for the banking sector and the macroeconomy, so this is one reason to evaluate carefully how much of these assets act as collateral on the balance sheet. The extent to which a bubble is able to develop depends on the upward pressure on movements in prices, notably through such mechanisms as greater media coverage, so that a large response to large price changes occurs. The development of a bubble through the occurrence of large price movements should be reflected by greater conditional volatility in financial markets. Movements which occur with a greater probability than conditional volatility would suggest, can be accounted for by the use of a tail index. This is not a new methodology, but its application to the measurement of speculative asset-pricing bubbles, is, to our knowledge, new. Before we observe the relationship empirically between estimates for the probability of larger than conditionally normal movements in prices occurring during the period of development of a speculative bubble, we shall first outline the methodology for estimating a tail index. It is the conditional estimate of the tail index that we estimate using rolling observations, to see how the probability of larger than “normal” movements in prices change over the development and bursting of the bubble in the art market.

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1.6 METHODOLOGY In order to analyze the extent to which the market moves away from fundamental values through larger than “normal” probabilities occurring in large movements of the return distribution, we shall apply below the Hill’s (1975) tail index estimator, which was further extended by Huisman et al. (1997). We use EVT to provide us with estimates of tail indices. EVT looks specifically at the distribution of the returns in the tails, and the tail fatness of the distribution is reflected by the tail index. This concept was first introduced by Hill (1975), and measures the speed with which the distribution’s tail approaches zero. The fatter the tail, the slower the speed and the lower the tail index given. An important feature about the tail index is that it equals the number of existing moments for the distribution. A tail index estimate equal to 2 therefore reveals that both the first and second moments exist, in that case the mean and the variance; however, higher moments will be infinite. By definition, the tail index for normal distribution equals infinity, since in that case, all moments exist. Since the number of degrees of freedom reflects the number of existing moments, the tail index can thus be used as a parameter for the number of degrees of freedom to parameterize the student-t distribution. To obtain tail index estimates, we use a modified version of the Hill estimator, developed by Huisman et al. (1997). Their estimator has been modified to account for the bias in the Hill estimator, with the additional advantage of producing almost unbiased estimates in relatively small samples. Specifying k as the number of tail observations, and ordering their absolute values as an increasing function of size, we obtain the tail estimator proposed by Hill. This is denoted by γ, which is the inverse of α: 1 ln(xn−j+1 ) − ln(xn−k ) k k

γ(k) =

(1.1)

j=1

Following the methodology of Huisman et al., we can use a modified version of the Hill estimator to correct for the bias in small samples. The bias in the Hill estimator stems from the fact that it is a function of the sample size. A bias-corrected tail index is therefore obtained by observing the bias of the Hill estimator as the number of tail observations increases up to κ, whereby κ is equal to half of the sample size γ(k) = β0 + β1 k + ε(k),

k = 1, . . . , κ

(1.2)

The optimal estimate for the tail index is the intercept β0 , while the α estimate is the inverse of this estimate. This is the estimate of the tail index that we use to estimate rolling estimations of the degree with which larger

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Table 1.2 Alpha estimates for the All Art Index, January 1976–December 2004 Log returns Alpha

Gamma

SE

Kappa

All Art Index

Both Left Right

1 3.26994 3.18242 3.46838

0.305816 0.314226 0.288319

0.053088 0.081208 0.069766

1 3.26994 3.18242 3.46838

US Art Index

Both Left Right

2 3.87852 3.99025 3.13632

0.25783 0.250611 0.318845

0.044758 0.062053 0.079892

2 3.87852 3.99025 3.13632

UK Art Index

Both Left Right

3 9.60735 9.35147 10.4014

0.104087 0.106935 0.096141

0.018069 0.026795 0.023805

3 9.60735 9.35147 10.4014

Note: This table provides the alpha estimates using the Huisman et al. (1997) estimator for the All Art Index from Art Market Research, using monthly data.

than “conditionally normal” returns occur in the historical distribution of returns over time.

1.7 RESULTS Table 1.2 provides the alpha estimates using the Huisman et al.’s estimator over the period January 1976 to December 2004. We first look at the alpha estimates for the whole sample. We see that there is indeed deviation from the assumption of “normality”, since the alpha estimates are between 2 and 3. This would imply that the tail index is able to capture some of the additional movement occurring in returns beyond that of the assumption of normality, captured by volatility alone. There is a move away from fundamental distribution over time. In Figure 1.2, the inverse alpha estimates (gammas) using the previous eight years’ sample of monthly data are plotted next to the actual monthly returns. We see that, the more the returns fluctuate, the higher the inverse alpha estimate and the greater the movement away from fundamental values. Indeed, the correlation between volatility and alpha is −0.42, which is highly significant at the 95 percent confidence level. It has been shown that this measure increases during periods of instability.4 Therefore, we would expect that the use of the gamma estimate is a good indicator for a movement away from fundamental values, and the development of an asset pricing bubble.

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General art index gamma Art returns General art index

/0 2

/0 2

04 20

/0 2

03 20

/0 2

02 20

/0 2

01 20

/0 2

00 20

/0 2

99 19

/0 2

98 19

97 19

96 19

95 19

94 19

93 19

91

92 19

19

90 19

89 19

87

88 19

19

02

86

19

19 85 /

84 19

⫺0.1

/0 2

1

/0 2

0

/0 2

2 /0 2

0.1

/0 2

3

/0 2

0.2

/0 2

4

/0 2

0.3

/0 2

5

/0 2

0.4

/0 2

6

Indices value

7

0.5

/0 2

Gamma estimates

0.6

0

Figure 1.2 Art returns and time-varying downside risk, January 1976–December 2004 Notes: Art Index. Returns and Gamma Estimates Monthly Data 96 Rolling Observation for Gamma Estimates for left tail of the Art Index using 96 observations to estimate the downside risk. Based on the data for the art market, we use the eight years’ monthly data available from 1976 to 1984 – a total of 96 observations – to calculate the conditional gamma estimate for the distribution. Obviously, the other moments of the distribution, the mean and the standard deviation are able to change conditionally over time, so that the gamma estimate is able to capture the extent of larger than conditionally normal movements occurring in the return distribution. The results are shown in Figure 1.2. There is an extremely high correlation between the bubble occurring in 1990 and the high values obtained from the tail index estimator over the period of the bubble. The gamma estimates converge to their average values after the bubble bursts in 1991, and maintain a value around the average value over the rest of the sample until the current period.

1.8 DISCUSSION It would appear that the phenomenon of the bubble developing in the art market may be captured through the use of the tail index estimator, which captures the probability of larger than conditionally normal movements in large returns occurring over the return distribution over time. The analysis so far has only been applied to the art market, but there is no reason why

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the methodology may not be used for other financial markets in which it is thought that bubbles have occurred, or are thought to be present. Indeed, the use of a relatively small sample of observations to analyze the tail index estimator provides a robust estimator, which can be applied conditionally over the historical time series of returns. A further issue is that it is not possible to test strictly for efficiency in the art market. We have discussed so far possible reasons for inefficiencies of the art market – for example, information asymmetries. But there are good reasons why particular behavioral anomalies are even larger and more widespread in the art market compared to the financial markets. Many private collectors are not profit-oriented and are particularly prone to the behavioral anomalies that arise from leaving endowments, opportunity costs and sunk cost effects. Circumstantial evidence suggests that private collectors are strongly subject to the endowment effect, which implies that they value an art object owned to a greater extent than one not owned. The result is that people often demand much more to give up an object than they would be willing to pay to acquire it (see Thaler, 1980). This is what Samuelson and Zeckhauser (1988) call a status quo bias; that is, the preference for the current state that biases someone against both buying or selling an object. These anomalies are manifestations of an asymmetry of value that Tversky and Kahneman (1991) call “loss aversion”. Loss aversion means that the disutility of selling an object is greater than the utility associated with buying it. Loss aversion also explains why there is no market for renting art objects. Frey and Eichenberger (1995) argue that the consumption benefits of viewing art should be revealed in the rental fees for art objects. The consumer would pay a fee for enjoying art while being unaffected by price changes in the art market. The reason why such market-revealing pure psychic benefits from art do not exist must be sought in property rights and a corresponding ownership effect. While the decision to buy art might be based on financial calculations, the desire to possess a beautiful and internationally famous work that will impress friends and clients unquestionably adds to the attraction. The owner of a work of art has a monopoly over that specific object, while other assets may be held by many individuals. The major difference between investing in art and in common financial assets is that art is tangible and is associated with a given lifestyle. This implies that an art object yields additional benefits if it is owned and not just rented, because the art object’s aura is also appropriated (Benjamin, 1963). Apart from the endowment effect and its corresponding ownership effect, there is also the opportunity cost effect. This implies that many collectors isolate themselves from considering the returns of alternative uses for their investments. A third behavioral anomaly that plays a large part in the art market is the sunk cost effect. This describes the tendency to be excessively attached to activities (things) for which one has expended resources resulting

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from past efforts at building up a (specific) art collection. Additionally, the self-deception theory suggests that the tendency to adjust attitudes to match past actions is a mechanism designed to persuade the individual that he or she is a skillful decision-maker. Are art investors reluctant to realize their losses? Or are investors extremely reluctant to realize their losses in art? Mental accounting is a kind of narrow framing that involves keeping track of gains and losses related to decisions in separate mental accounts. Thaler (1985) argues that individuals reexamine each account only intermittently when it is action-relevant. Mental accounting may explain the disposition effect (Shefrin and Statman, 1985) – that is, the excessive propensity to hold on to assets that have declined in value and to sell the winners. Such a mechanism may even be side-tracked when the individual avoids recognizing losses. Self-deception theory reinforces this argument, since a loss is an indicator of poor decision-making, and a self-deceiver maintains self-esteem by avoiding the recognition of this. Regret avoidance may also reflect a self-deception mechanism designed to protect self-esteem about poor decision ability. Kahneman et al. (1991) show that regret is stronger for individual decisions that involve action rather than passivity. This effect is also known as the “omission bias”. A bequest aspect is also highly relevant. Gifts from parents to their children, or inheritances of family members in the form of art objects are valued more highly by the owner than they would be purely for their monetary value. Frey and Eichenberger (1995) argue that, by selling the object, the owners are transferring with it part of their own “nature”.

1.9 CONCLUSION Using a unique set of data with which to observe and quantify the extent of a bubble in the art market, we have been able to gain a greater insight into the nature of bubbles with respect to the larger than “normal” movements that appear to occur during the build-up and breakdown of financial bubbles. More detailed analysis with regard to return distribution will no doubt enable a richer analysis of the make-up of the asset bubbles, and will be extremely interesting avenues for further research. By defining the degree of “bubbliness” in a market as the degree to which large movements are more likely to occur, the gamma of the distribution of historical returns can be estimated conditionally over time. We see that there is an extremely high correlation between the size of the gamma estimates and prices during the period of the bubble. The larger the gamma estimate, the greater the probability of more extreme movements in the return distribution. This should indeed be constant over time. However, we see that the correlation of the gamma estimates increases during the period of the bubble, and is thereafter fairly constant. We therefore premise that the bubble

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can be defined ex post from a larger probability occurring in the tails of the distribution, observed conditionally over time – from rolling observations used to estimate the degree of “bubbliness” in the market. Although the results presented here are preliminary in nature, they provide an extremely innovative and interesting avenue for further research into the notion of bubbles in financial markets. The use of the art market, which represents a market in which deviations from fundamental values are much more likely, provides a particularly interesting market with which to observe such measures. There are many further areas that may need to be addressed before any definite conclusions can be drawn. For example, the use of this measure on alternative asset classes in which bubbles have been observed. The “dot.com” mania and real estate markets in particular. Although the results are in a preliminary form, they should help to generate further discussion and insight into the determination and measurement of bubbles in financial markets.

NOTES 1. All errors are the responsibility of the authors. Many thanks to participants at the conference on “Art: An Alternative Asset Class” at Sotheby’s, London, for their valuable comments. 2. Figures on the exact numbers of artists per index are available from the authors on request, or from AMR. 3. In a similar manner, the S&P 500 only represents a segment of the whole market for US equities. 4. See Pownall and Koedijk (1999) for an analysis of the Asian financial crises of 1997–8, with the use of the same methodology.

REFERENCES Bauer, R., Campbell, R. A. and Dil, N. (2005) “Art Diversification over the Business Cycle: The Case for the UK”, Working Paper, Maastricht University. Baumol, W. (1986) “Unnatural Value: or Art Investment as a Floating Crap Game”, American Economic Review, 76(2): 10–14. Benjamin, W. (1963) Das Kunstwerk im Zeitalter seiner technischen Reproduzierbarkeit, 4th edn (Frankfurt am Main: Edition Suhrkamp). Campbell, R. A. (2005) “Art as an Alternative Asset Class”, Working Paper, Maastricht University. Case, K. E., Quigley, J. M. and Shiller, R. J. (2001) “Comparing Wealth Effects: The Stock Market versus the Housing Market”, Advances in Macroeconomics, 5(1): 1–32. Chanel, O. (1995) “Is Art Market Behavior Predictable?”, European Economic Review, 39(3–4): 519–27. Frey, B. S. and Eichenberger, R. (1995) “On the Rate of Return in the Art Market: Survey and Evaluation”, European Economic Review, 39(3–4): 528–37. Goetzmann, W. N. (1993) “Accounting for Taste: Art and the Financial Markets over Three Centuries”, American Economic Review, 83(5): 1370–6.

RACHEL CAMPBELL AND ROMAN KRÄUSSL

15

Hill, B. (1975) “A Simple General Approach to Inference about the Tail of a Distribution”, Annals of Mathematical Statistics, 3(5): 1163–74. Huisman, R., Koedijk, K. G., Kool, C. and Palm, F. (1997) “Fat Tails in Small Samples”, Working Paper, Erasmus University, Rotterdam. Kahneman, D., Knetsch, J. and Thaler, R. (1991) “The Endowment Effect, Loss Aversion, and Status Quo Bias”, Journal of Economic Perspectives, 5(1): 193–206. Mei, J. and Moses, M. (2002) “Art as an Investment and the Underperformance of Masterpieces”, American Economic Review, 92(5): 1656–68. Pownall, R. A. and Koedijk, K. G. (1999) “Capturing Downside Risk in Financial Markets: the Case of the Asian Crisis”, Journal of International Money and Finance, 18(6): 853–70. Samuelson, W. and Zeckhauser, R. (1988) “Status Quo Bias in Decision Making”, Journal of Risk and Uncertainty, 1(1): 7–59. Shefrin, H. and Statman, M. (1985) “The Disposition to Sell Winners Too Early and Ride Losers Too Long: Theory and Evidence”, Journal of Finance, 40(3): 777–90. Siegel, J. J. (2003) “What Is an Asset Price Bubble? An Operational Definition”, European Financial Management, 9(1): 11–24. Stiglitz, J. E. (1990) “Symposium on Bubbles”, Journal of Economic Perspectives, 4(2): 13–18. Thaler, R. H. (1980) “Toward a Positive Theory of Consumer Choice”, Journal of Economic Behavior and Organization, 1(1): 39–60. Thaler, R. H. (1985) “Mental Accounting and Consumer Choice”, Marketing Science, 4(3): 199–214. Tversky, A. and Kahneman, D. (1991) “Loss Aversion in Riskless Choice: A ReferenceDependent Model”, Quarterly Journal of Economics, 106(4): 1039–61.

CHAPTER 2

International Stock Portfolios and Optimal Currency Hedging with Regime Switching Markus Leippold and Felix Morger

2.1 INTRODUCTION Despite the vast literature on optimal currency hedging, there still is considerable disagreement about how international investors should hedge their currency risk. One argument is that investors should fully hedge, since exchange-rate changes in excess of the forward discount rate average out. Therefore, hedging decreases the risk of foreign investment, but does not reduce its expected returns. In the words of Perold and Schulman (1988), currency hedging is a free lunch. However, there is a large branch of literature that does not agree with this viewpoint. As an early example, Froot (1993) argues that the free-lunch argument does not hold in the long run. If exchange rates and asset prices display mean reversion, the optimal hedging policy becomes time-varying. In particular, real exchange rates revert to their means according to the theory of purchasing power parity, and investors should maintain an unhedged foreign currency position. Therefore, for an investor with a long investment horizon, it becomes optimal not to hedge at all. Froot argues that real-exchange rates may deviate from their theoretical fair value over shorter horizons, and currency hedging in this context may become beneficial. As a compromise between these two extreme viewpoints, Black (1989) argues that, using Siegel’s paradox, there is a constant universal hedge 16

MARKUS LEIPPOLD AND FELIX MORGER

17

ratio between zero and one. However, Black has to impose some strong assumptions and because of the time-period sensitivity and significant variability and volatility of input parameters in the optimal hedge ratio, there is a significant dispersion in what constitutes the optimal constant hedge ratio. In contrast, the evidence of Glen and Jorion (1993), who analyze the performance of mean-variance efficient stock and bond portfolios from the G5 countries when hedging the associated currency risk with currency forwards, shows that there is a substantial improvement when using conditional time-varying hedging strategies. It is beyond the scope of this chapter to provide a full account of the existing literature on currency hedging, but we refer, for example, to the recent contribution by Dales and Meese (2001) for an overview. We note that most of the literature builds on simplifying assumptions on the dynamics of the underlying returns. Indeed, there is now ample empirical evidence against the normal distribution for return dynamics and a lot of statistical justification for so-called regime-switching models. For example, Turner et al. (1989), Garcia and Perron (1996), Gray (1996), Perez-Quiros and Timmermann (2000), Whitelaw (2000), Ang and Bekaert (2002a, 2002b), Ang and Chen (2002), Connolly, Stivers and Sun (2005) and Guidolin and Timmermann (2005a, 2006) report evidence of regimes in stock or bond returns. Therefore, in our study, we analyze the impact of such regime-switching models on optimal currency hedging. Closely related to our study are the works by Ang and Bekaert (2002a) and Guidolin and Timmermann (2005a). Both papers make use of regimeswitching models. Ang and Bekaert (2002) analyze the optimal investment strategy within a mean-variance framework. Concerning the modeling of regime switches, they do not consider a Bayesian updating rule to infer on state probabilities. Guidolin and Timmermann (2005b) assume preferences over the moments of wealth distribution. In addition, they explore the optimal asset allocation of an international portfolio with unhedged returns. They do not address the issue of optimal currency hedging. We use a more general CRRA (Constant Relative Risk Aversion) utility setting and we compare the non-Bayesian investor with the Bayesian one. We show that it really pays to go Bayes! Furthermore, we explicitly allow the investor to hedge his or her currency exposure. Whereas Guidolin and Timmermann (2005b) find that their model offers a rational explanation of the strong home bias observed in US investors’ asset allocation, our results contrast with their conclusion. While we find a slight decrease in foreign asset holdings for a strategy with unhedged returns, the strategy with optimal currency hedging substantially increases the exposure to foreign markets. Therefore, the home bias becomes even more puzzling. The plan of this chapter is as follows. In Section 2.2, we present the regimeswitching model and the optimization problem. Section 2.3 provides the estimation results for several model specifications. In Section 2.4, we provide

18

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

a discussion of several aspects of our results; in particular, we discuss the economic benefits of using regime-switching models and we analyze the optimal portfolio allocation. Section 2.5 concludes.

2.2 THE MODEL Regime-switching models consist of two generic processes, the state process st and the return process rt . The unobservable state process st determines which state is active at time t. For the state st , we assume a discrete firstorder Markov chain with S possible states or regimes. The constant transition probability for moving from state i to state j is denoted as pij = P{st+1 = j|st = i, st−1 = k, . . .} = P{st+1 = j|st = i},

for i, j = 1, . . ., S

and we collect all the pij ’s in the transition matrix P. For the N-dimensional return vector rt , we assume the state dependent dynamics drt = µ(st )dt + (st )dBt where Bt is a N × 1 dimensional Wiener process. Both the drift vector µ(st ) and the N × Ndimensional covariance matrix (st ) depend on the active regime. Therefore, the distribution of rt+1 conditional on the state st is a mixture of S normal distributions with probability density function f (rt+1 |st = i) =

S

pij f (rt+1 |st+1 = j)

j=1

We note that the regime-switching model defined above can account for skewed and fat-tailed returns. Furthermore, with pij > 1/S as a sufficient condition, we can also generate correlation breakdowns and volatility clusters. Both are often observed in the joint dynamics of international stock markets.

2.2.1 Portfolio selection with perfect knowledge of the active state We assume that the investor can invest in N assets, where the Nth asset is the risk-free asset. The investment horizon T is fixed. The investor has the possibility of rebalancing the asset allocation at the beginning of every period; for example, at times t = 0, . . ., T − 1. There are no transaction costs.

MARKUS LEIPPOLD AND FELIX MORGER

19

We further assume that the investor has a CRRA utility function defined over wealth, for example U(W) =

1 W 1−γ 1−γ

where we assume γ > 1 for the relative risk aversion coefficient. We start with the situation in which the investor has perfect knowledge of the active state. We denote by αt the vector of portfolio weights at time t. To maximize the investor’s terminal wealth, he/she has the following objective function: max E0 [U(WT )]

α0 ,...,αT−1

s.t.

Wt+1 = Wt (αTt exp (rt+1 ))

(2.1)

1 = αTt 1 where E0 [·] = E[·|F0 ]. To simplify notation, we write Wt+1 = Wt αTt exp (rt+1 ) = Wt Rt+1 (αt ) where Rt+1 is the portfolio’s gross return from time t to time t + 1. Using a dynamic programming approach, we can solve the optimization problem recursively. At each time step t, we have to maximize the indirect utility function J: J(W, r, s, θ, t) = max Et [Qt+1,T U(Wt+1 )] αt

given the parameter set θ = {µ(s), (s), P} and with the indirect utility Qt+1,T = Et+1 [(RT (α∗T−1 ) · . . . · Rt+2 (α∗t+1 ))1−γ ]

QT,T = 1

where α∗T−1 , . . .α∗t+1 are the optimal portfolio weights determined recursively. These optimal portfolio weights have to solve the corresponding first-order conditions (FOC) of the optimization problem in Equation (2.1). Given state st = i, we obtain the FOC of the investor’s allocation problem as .−γ

Et [Qt+1,T Rt+1 (αi,t )λt+1 |st = i] =

S

.−γ

pij Et [Qt+1,T Rt+1 (αi,t )λt+1 |st+1 = j] = 0

(2.2)

j=1

where αi,t = αt (st = i) and λi,t+1 is defined as the vector of excess returns over the risk-free rate, for example ⎛ 1 N )⎞ exp (ri,t+1 ) − exp (ri,t+1 ⎜ ⎟ .. λi,t+1 = ⎝ ⎠. . N−1 N ) exp (ri,t+1 ) − exp (ri,t+1

20

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

Whereas, for the case of one single regime with i.i.d. returns, the optimal asset allocation for a CRRA investor is constant over different time horizons (see, for example, Samuelson, 1969), the allocation in a multiple regime model depends on the prevailing regime st and on the investment horizon. This dependency arises because of the changing investment opportunity set induced by the regime switches.

2.2.2 Portfolio selection under hidden regime switches Since it is rather restrictive to assume that investors have perfect knowledge on the active state, we next assume that, at the moment they invest, they do not know which regime is active and they have to infer on the active regime using a specific updating procedure. For the updating procedure, we assume that the investor uses a Bayesian updating rule based on the filter developed by Hamilton (1989). To sketch out briefly the updating procedure, we denote the oneperiod-ahead forecast of investor beliefs as ut+1|t , for example, ut+1|t is the column vector of the probabilities P(st+1 = i|Ft ; θt ), and the set θt = {µ(s), (s), P} collects the time-t parameters of the regime-switching model. The optimal inference (posterior) and forecast (prior) on the active regime is found by iterating the equations uˆ t|t =

uˆ t|t−1 ◦ηt T 1 (uˆ t|t−1 ◦ηt )

(2.3)

and uˆ t+1|t = PT uˆ t|t

(2.4)

where ηt is a S × 1 vector of the multivariate normal densities determined by µ(st ) and (st ), 1 is the S × 1 unit vector, and by ‘◦’ we denote the elementby-element multiplication. Given a starting value u1|0 and a parameter set θ the algorithm defined by Equations (2.3) and (2.4) calculates for each time t the probability of a regime currently being active and also being active for the period ahead. To get parameter estimates, we maximize the likelihood function using a variate of the EM algorithm developed by Hamilton (1990). The log likelihood function L(θ) to be maximized is the sum of the denominator of Equation (2.3) over all t. We note that, with incomplete information on the current state, the indirect utility J is no longer a function of the active state st , but a function of the beliefs ut|t about the active state, where ut|t is the column vector of the probabilities P(st = i|Ft ; θt ). In particular, we have J(W, r, ut|t , θ, t) = maxαt Et (Qt+1,T U(X, Y))

MARKUS LEIPPOLD AND FELIX MORGER

21

We write the belief at time t + 1 given the filtration Ft+1 as vt+1|t+1 . For ease of notation, we drop the time indices from the beliefs ut|t and vt+1|t+1 whenever u and v are used as subscripts. Then, the FOC is given as .−γ

Et [Qt+1,T Rt+1 (αu,t )λt+1 |ut|t ] =

S i=1

ui,t|t

S j=1

1

.−γ

puv,j Et [Qv,t+1,T Rt+1 (αu,t )λt+1 |st+1 = j]dv = 0 (2.5)

pij 0

where puv,j = P{ut|t , vt+1|t+1 ; θt |st+1 = j}. By inspection, Equation (2.5) is a straightforward extension of the FOCs under full information on the active regime given in Equation (2.2). The expectation gives the FOCs for a given belief vt+1|t+1 , weights αu,t , and regime st+1 = j. The summation of the expectation over the different states j corresponds to the FOCs for the full information case. The summation over the beliefs ui,t|t and the integral over the combinations of beliefs vt+1|t+1 are because of the non-observability of the state process. With the state process being observable, ui,t|t equals 1 for the active regime and zero for all others. The integral over the combinations of beliefs vt+1|t+1 enters Equation (2.4), because the indirect utility Qv,t+1,T depends on the beliefs vt+1|t+1 , which are unknown at investment. The probability puv,j corresponds to the probability of the belief moving from ut|t to vt+1|t+1 , given that regime j is active at t+1. It assigns a weight to each expectation and its involved indirect utility. The probability puv,j can also be interpreted as the probability of occurrence of vt+1|t+1 given ut|t and regime st+1 = j. The optimal portfolio choice problem for a regime-switching model must be solved numerically. In the following empirical section, we apply a backward solution algorithm with a Monte Carlo simulation of size Z = 30,000. Furthermore, we have to discretize the state space of beliefs. For the model with two regimes, we have 3 grid points, and for the model with three regimes we have 6 grid points. With this parameterization the algorithm generates very accurate weights. Therefore, we do not consider more elaborated algorithms such as, for example, the algorithm proposed by Brandt et al. (2004) that is based on the Longstaff and Schwartz (2001) least-square Monte Carlo method.

2.3 ESTIMATION RESULTS 2.3.1 Data We take the perspective of a US investor, who allocates his/her wealth in risk-free assets and the US, UK and German stock markets. For the stock markets, we use MSCI (Morgan Stanby Capital International) country indices. We approximate the risk-free rate with the one-month Eurodollar rate. To

22

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

investigate the optimal currency hedging behavior of the investor, we let the investor take positions not only in the foreign market indices denominated in US dollars, but also in the foreign markets denominated in their local currency. The relative asset allocation between the unhedged and hedged indices determines the optimal hedging policy. The monthly data covers the period from December 1970 to June 2005, which results in 414 one-month excess returns over the risk-free rate. For the out-of-sample analysis, we use fifteen and a half years spanning the period January 1990 to June 2005. Table 2.1 presents the summary statistics for the excess returns. With the sole exception of the unhedged German index in the period prior to 1990, the index returns fail the Jarque–Bera test for normality, motivating the use of regime-switching models.

2.3.2 Specification test Using three model portfolios, we test for normality (Jarque–Bera – JB) and for the absence of serial correlation with three different likelihood ratio tests on the predictive density proposed by Berkowitz (2001). LR1;1 tests for serial correlation of lag one, LR2;1 for serial correlation of lag one and two, and LR2;2 for linear and squared serial correlation up to lag two. The latter is a test for omitted volatility dynamics. The results of the specification tests for the cases of two and three regimes are summarized in Table 2.2.1 We find that the Jarque–Bera statistic is reduced substantially by the use of regime-switching models. For example, we find that the Jarque–Bera statistics of the UK excess returns are above 2,000. In contrast, under the chosen specifications of the regime-switching models, the statistics for the UK are mainly below 20 during the period of the out-of-sample test. Hence, we are well advised to use a regime-switching model to account for the skewed and fat tails of stock market returns. In particular, for the subsequent analysis we shall use three regimes for both the unhedged and the optimal hedging strategy, and two regimes for the fully hedged strategy.

2.3.3 Parameter estimates as at June 2005 In this section, we present the parameter estimates for different model specifications. Table 2.3 presents the parameter estimates of the model specification with a single regime and unhedged indices. The mean excess returns are between 5 percent and 7.7 percent per year, and the volatilities are around 20 percent. The correlations are around 0.5. The highest correlation is between the UK and the US markets. All parameter estimates are significantly different from zero.

MARKUS LEIPPOLD AND FELIX MORGER

23

Table 2.1 Summary statistics, stock market returns, December 1970 – June 2005 GER, FH

GER, UH

UK, FH

UK, UH

Panel A: Summary statistics for June 2005 Moment statistics Mean 0.040 0.070

0.081

0.077

Std, dev. Skewness Kurtosis JB

US

0.050

0.198

0.216

0.208

0.230

0.154

−0.405

−0.214

1.367

1.337

−0.320

5.172

4.410

18.678

14.739

4.876

90.780

36.416

4319.407

2470.964

66.188

(0.000)

(0.000)

(0.000)

(0.000)

(0.000)

Correlation coefficients GER, FH 1.000 GER, UH

0.858

1.000

UK, FH

0.446

0.373

1.000

UK, UH

0.379

0.464

0.894

1.000

US

0.508

0.463

0.595

0.538

1.000

Panel B: Summary statistics for December 1989 Moment statistics Mean 0.035 0.088 0.108

0.093

0.032

Std, dev.

0.277

0.161 −0.228

Skewness Kurtosis JB

0.177

0.214

0.248

−0.180

0.009

1.459

1.338

5.138

3.716

16.296

12.305

5.545

42.827

4.440

1723.482

870.741

61.157

(0.000)

(0.109)

(0.000)

(0.000)

(0.000)

Correlation coefficients GER, FH 1.000 GER, UH

0.832

1.000

UK, FH

0.360

0.287

1.000

UK, UH

0.329

0.397

0.920

1.000

US

0.380

0.335

0.560

0.505

1.000

Notes: The table presents the summary statistics of the stock market returns in excess of the risk-free rate and includes the Jarque–Bera test. The market indices are MSCI country indices and the risk-free rate is the 1-month Eurodollar rate. The data are provided by Datastream. The statistics for Germany and the UK are given both as hedged (denoted as FH) and unhedged (denoted as UH) indices. Mean and standard deviation are annualized. Panel A gives the summary statistics for the entire sample. Panel B presents the numbers for the sub-sample from December 1970 until the start of the horse race (see section 2.2).

Table 2.4 presents the parameter estimates for a three-regime model specification with unhedged indices. Compared to the benchmark case in Table 2.3, the bear state exhibits negative drifts, slightly increased volatilities and correlations, and a relatively low persistence. Once the bear state is

24

Table 2.2 Specification tests, December 1989–December 2004 Test

1989.12

1992.6

1994.12

1997.6

1999.12

2002.6

2004.12

Panel A: Unhedged strategy 2 regimes JB 1* 2*

2**

2**

1*, 1**

1*, 1**

1**

LL-ratio

3*

1*

1*

2*

3*

1*

3*

3 regimes JB

1*

1**

1**

LL-ratio

2*

3*

3*

3*

3*

1*

2*,1**

Panel B: Fully hedged strategy 2 regimes JB 1*

1*

1*

1*

1**

2*

1*

2*

2*

3*

1*

3 regimes JB

1*

2*

1**

1**

LL-ratio

2*

2*

2*

1*, 1**

2*

3*

1*

Panel C: Optimally hedged strategy 2 regimes JB 2* 1* 2*

1*, 2**

3**

1*, 3**

5**

5*

4*, 1**

4*, 2**

2*, 4**

LL-ratio

LL-ratio

3*

4*

3 regimes JB

1**

1*

1*

4*

1*, 2**

LL-ratio

1*

4*, 1**

4*, 1**

4*, 1**

3*

2*

3*

Notes: The table presents an overview of the specification tests at intervals of 2.5 years. The entries in the table show the number of markets that failed the Jarque–Bera test and likelihood ratio test, respectively, at a significance level of 5% or 1%. Significance at the 5% level is denoted by * and at the 1% level by **. Panel A reports the results for the strategy without hedging, and Panel B results for the strategy with the hedged returns. Panel C presents the overview for the optimal hedging case.

Table 2.3 Parameters of the single regime strategy

Panel A: Drifts Mean excess return

GER

UK

USA

0.070 (0.008)

0.077 (0.008)

0.050 (0.006)

Panel B: Correlations and volatilities GER

0.216**

UK

0.464**

0.230**

US

0.463**

0.538**

0.154**

Notes: The table presents the parameter estimates for the international investor and single regime specification with the unhedged indices. Panel A gives the annualized mean excess returns. The values in brackets are the annualized standard deviations. The off-diagonal elements in Panel B report the correlations and the diagonal elements the annualized volatilities. Significance at the 0.05 level is represented by ** of the corresponding covariances.

25

MARKUS LEIPPOLD AND FELIX MORGER

Table 2.4 Parameter estimates for the three regimes specification with unhedged indices Panel A: Drifts GER −0.123 (0.104)

UK −0.024 (0.310)

US −0.197 (0.143)

Low correlation state

0.123 (0.013)

0.104 (0.025)

0.106 (0.018)

High correlation state

0.035 (0.023)

0.056 (0.015)

0.031 (0.016)

Bear state

Panel B: Correlations and volatilities GER Bear state GER 0.241**

UK

UK

0.537**

0.507**

US

0.651**

0.609**

Low correlation state GER

0.200**

UK

0.361**

0.191**

US

0.172**

0.448**

High correlation state GER

0.235**

US

0.240**

0.136**

UK

0.850**

0.136**

US

0.825**

0.765**

0.144**

Panel C: Transition probabilities Bear state Bear state 0.842 (0.208)

Low corr. 0.026 (0.005)

High corr. 0.000 (0.086)

Low correlation state

0.158 (0.363)

0.934 (0.036)

0.076 (0.181)

High correlation state

0.000

0.041

0.924

Notes: Panel A presents the annualized mean excess returns. The values in brackets are annualized standard deviations. The off-diagonal elements in Panel B report the correlations and the diagonal elements the annualized volatilities. Panel C shows the transition probabilities and in brackets their standard deviations. Significance at the .05 level is represented by ** of the corresponding covariances.

active, the probability that it will remain active for a year is only 14 percent. The other two states are both growth states, but with significantly different correlation structures. Table 2.5 reports the results for the model specification with two regimes and fully hedged currency risk. There is a clear distinction between the two regimes. The first regime is a bear state with low drifts, especially for Germany with a drift of −22.3 percent, high volatilities and very low persistence. The probability that it switches to the bull state within two months is close to 50 percent. The second regime is a bull state with high excess returns ranging between 8 percent and 10 percent, low volatilities, and high

26

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

Table 2.5 Parameter estimates for the three regimes specification with fully hedged currency risk Panel A: Drifts Bear state

GER −0.223 (0.005)

UK 0.043 (0.013)

US −0.097 (0.007)

0.101 (0.006)

0.088 (0.010)

0.082 (0.010)

Bull state

Panel B: Correlations and volatilities GER Recession state GER 0.298**

UK

UK

0.431**

0.403**

US

0.521**

0.640**

Growth state GER

0.167**

UK

0.496**

0.140**

US

0.482**

0.584**

Panel C: Transition probabilities Bear state Bear state 0.721 (0.007) Bull state

0.280

US

0.237**

0.130**

Bull state 0.053 (0.081) 0.947

Notes: Panel A presents the annualized mean excess returns. The values in brackets are annualized standard deviations. The off-diagonal elements in Panel B report the correlations and the diagonal elements the annualized volatilities. Panel C shows the transition probabilities and in brackets their standard deviations. Significance at the .05 level is represented by ** of the corresponding covariances.

persistence. Similar to the bear state, the correlations are close to the sample average. Finally, Table 2.6 presents the parameter estimates for the model specification with three regimes with both hedged and unhedged indices. As for the unhedged strategy, the three regimes can be referred to as bear state, low correlation state, and high correlation state. The low correlation state can be interpreted as a transitory state.

2.4 DISCUSSION With the empirical specification at hand, we next discuss some of the implications of regime switching on asset allocation. With regard to the numerical calculation of the optimal portfolio strategies, we use a Monte Carlo simulation with sample size Z = 30,000. Furthermore, for all calculations, we suppose that the investor has a risk aversion parameter equal to γ = 5.

Table 2.6 Parameter estimates for the three regimes specification with hedged and unhedged indices Panel A. Drifts Bear Low High

GER, FH −0.234 (0.073) 0.061 (0.016) 0.110 (0.026)

Panel B: Correlations and volatilities GER, FH Bear state GER, FH 0.282** GER, UH 0.923** UK, FH 0.432** UK, UH 0.386** US 0.480**

GER, UH −0.186 (0.070) 0.094 (0.006) 0.125 (0.025)

UK, FH 0.037 (0.025) 0.091 (0.007) 0.073 (0.015)

UK, UH 0.072 (0.026) 0.063 (0.009) 0.110 (0.015)

US −0.133 (0.028) 0.072 (0.006) 0.076 (0.016)

GER, UH

UK, FH

UK, UH

US

0.283** 0.456** 0.453** 0.555**

0.467** 0.987** 0.643**

0.480** 0.615**

0.261**

Low correlation GER, FH GER, UH UK, FH UK, UH US

0.158** 0.790** 0.344** 0.301** 0.291**

0.195** 0.199** 0.408** 0.216**

0.162** 0.805** 0.517**

0.199** 0.443**

0.134**

High correlation GER, FH GER, UH UK, FH UK, UH US

0.233** 0.920** 0.844** 0.724** 0.853**

0.227** 0.809** 0.849** 0.835**

0.130** 0.860** 0.807**

0.130** 0.771**

0.137**

Low corr. 0.032 (0.075) 0.929 (0.063) 0.039

High corr. 0.000 (0.035) 0.080 (0.098) 0.920

Panel C: Transition probabilities Bear state Bear 0.783 (0.003) Low 0.217 (0.019) High 0.000

27

Notes: Panel A presents the annualized mean excess returns. The values in brackets are annualized standard deviations. The off-diagonal elements in Panel B report the correlations and the diagonal elements the annualized volatilities. FH is short for fully hedged and UH for unhedged. Panel C shows the transition probabilities and in brackets their standard deviations. Significance at the 0.05 level is represented by ** of the corresponding covariances.

28

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

2.4.1 Economic importance of regimes The economic importance of regimes is a very relevant issue for the appraisal of regime-switching strategies. We measure this “importance” as utility costs that an investor bears, when he/she gives up the optimal strategy and follows instead a suboptimal one. More precisely, we are interested in the monetary compensation c, also called certainty equivalent compensation, that makes an investor with time horizon T indifferent between the suboptimal weights α− and the optimal weights α∗ . Formally, we must solve E0 [U(WT (α∗ )|W0 = 1)] = E0 [U(WT (α− )|W0 = 1 + c)] for c. Using the fact that CRRA utility preferences are homogeneous in W0 , and since E0 [U(WT )|W0 = 1] =

Q0,T 1−γ

we obtain c=

Q∗0,T Q− 0,T

1/(1−γ) −1

We report the certainty-equivalent compensations in Table 2.7 as cents per invested dollar. In Table 2.7, the reported cost can be attributed directly to international investment and regime-switching (Panel A) and to regimeswitching only (Panel B), respectively. Panel C documents the economic importance of regimes in a purely international setting. Panel A of Table 2.7 displays the economic cost of investing in a domestic US portfolio while disregarding the possibility of international diversification. Relative to the international investment strategy with one regime and unhedged indices, the costs (in terms of invested dollars) are only 0.53 percent per year for all time horizons. Hence the strategy with a single regime does not seem to offer substantial benefits, at least when only a small number of (highly correlated) markets are involved (as is the case for our analysis). However, turning to the regime-switching strategies in Panel A, we see that the economic costs for the three regime-switching strategies may be as high as 2.7 percent per year. Consequently, international investment combined with regime-switching does pay off – even when the number of potential markets is low and their correlation high. Also, we note that ignoring regimes tends to increase the annualized economic costs as the time horizon increases. This finding contrasts with Guidolin and Timmermann (2004), who explore the required compensation for buy-and-hold strategies, and find decreasing costs with a longer time horizon. Their result is

Table 2.7 The economic importance of regime-switching strategies 6 months Total

1 year

Annualized

2 years Total

Annualized

5 years Total

Annualized

10 years Total

Annualized

Panel A: International strategies versus domestic US strategy, 1 reg. Internat., 1 reg. 0.26 0.53 0.53 1.06

0.53

2.66

0.53

5.39

0.53

UH, 3 reg.

2.51

14.14

2.68

31.37

2.77

1.01

2.02

2.32

5.09

FH, 2 reg.

0.71

1.42

1.50

3.10

1.54

8.06

1.56

16.88

1.57

OH, 3 reg.

1.03

2.06

2.20

4.57

2.26

11.90

2.27

25.24

2.28

Panel B: RS strategies versus international unhedged portfolio, 1 reg. UH, 3 reg. 0.74 1.49 1.78 3.94

1.95

10.95

2.10

23.98

2.17

FH, 2 reg.

0.45

0.90

0.98

2.05

1.02

5.34

1.05

11.07

1.06

OH, 3 reg.

0.61

1.23

1.36

2.87

1.42

7.44

1.44

15.46

1.45

Panel C: States of OH versus the OH bear state Low corr. 1.06 2.13

1.33

1.46

0.73

1.50

0.30

1.50

0.15

High corr.

1.97

3.97

2.90

3.75

1.86

4.32

0.85

4.35

0.43

Uncond. state

1.06

2.13

1.51

1.83

0.91

2.02

0.40

2.03

0.20

2005/6

1.89

3.82

2.80

3.62

1.80

4.17

0.82

4.19

0.41

29

Notes: Panels A and B report the economic cost of investing suboptimally. The suboptimal strategies have the same investment opportunities as the optimal strategies. The economic cost of these three panels are calculated with the indirect utilities, Q∗0,T of the unconditional state probabilities. Panel C documents the importance of regimes for the case of the optimal hedging strategy. It gives the cost of being in the bear regime compared to a situation with another, economically more promising state belief. In all panels, the economic costs are given as cents per invested dollar required, making an investor indifferent between the optimal and the suboptimal weights. UH denotes unhedged; FH fully hedged; and OH optimal hedging.

30

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

as a result of the inability of the investor to react to market changes. Panel B gives the required compensation for an international investor to ignore regime-switching. With a horizon of ten years, the costs of ignoring regimeswitching average almost 17 percent, or 1.6 percent per year. Comparing Panel B with the regime-switching strategies of Panel A, we see that the economic gains are about 30 percent (on average) lower in Panel B than in Panel A. In Panel A, only 30 percent of the certainty equivalents can be attributed to international diversification, and 70 percent result from to taking into account the possibility of regime-switches. In Panel C, we present the results on the economic significance of the regimes for the optimal hedging strategy. The panel gives the cost of being in the bear state compared to a situation with another, economically more promising state belief. These costs are high in the short term and, in contrast to the other panels, decrease with longer investment horizons. They are high in the short term because the regimes have very different estimated parameters, and therefore very different expected returns. They decrease because, in the long term, the probability for a switch to another regime increases and the initial regime becomes less influential. There are almost no additional costs related to the initial regime after the first five years. The costs reported in Table 2.7 are not only economically, but also statistically significant. To test for statistical significance, we recall that, given some regularity conditions for the likelihood function (see, for example, Poirier, 1995), the asymptotic distribution of the MLE of θ is A θˆ −−→ N(0, J(θˆ )−1 )

where J(θˆ ) is the matrix of second derivatives of the log likelihood function with respect to the estimated parameters and observed returns – for example J(θˆ ) = −

T ∂2 log (rt ) t=1

∂θˆ i ∂θˆ i

, i = 1, . . ., k, j = 1, . . ., k

(·) is the normal distribution, and k is the number of parameters to be estimated. Therefore, to derive confidence intervals we first simulate Q = 200 parameter sets θˆ q from N(θˆ , J(θˆ )−1 ). Then the economic costs are calculated for each set of parameters θˆ q . Figure 2.1 plots the confidence intervals of the required compensation when investing in the one-regime international strategy rather than the three-regime strategy with unhedged indices. The lower bound of the 95 percent confidence interval lies above zero. Therefore, the null hypothesis, that ignorance of regime-switching does not cause any utility losses, is

31

MARKUS LEIPPOLD AND FELIX MORGER

Required compensation

0.07 0.06 0.05

Estimate Median 67% Cl 95% Cl

0.04 0.03 0.02 0.01 0

6

12 Investment horizon

18

24

Figure 2.1 Confidence intervals of economic cost estimates Notes: The figure shows the bootstrapped confidence intervals of the compensation required for ignoring the unhedged regime-switching strategy with the parameters estimated as per 2005/6. The economic costs are calculated for the indirect utilities of the unconditional state probabilities. In the bootstrap procedure, 200 parameter sets are drawn from N(θˆ ,J(θˆ )−1 ).

clearly rejected. For a one-year horizon, the required compensation can be as high as 3.5 percent with a median compensation of more than 1 percent. With long horizons, the non-realized wealth related to ignorance of regimeswitching is substantial.

2.4.2 Strategies in competition: horse race Most often, a model performs very well in sample, but fails out of sample. In this section, we present an out-of-sample test (horse race) for our regimeswitching strategies. This horse race spans the period from January 1990 to June 2005, or 186 real data returns. At the beginning of each month, we calculate the optimal asset allocation for the different strategies, and at the end of each month, we calculate the strategies’ performance in terms of cumulated wealth. To calculate the optimal portfolio allocation at the beginning of each month, the model parameters and beliefs are estimated with the data available to the investor. So, for example, the optimal weights for January 1990 are calculated based on the excess returns from January 1971 to December 1989. The horse race is therefore a truly out-of-sample comparison of different strategies.2

32

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

4

3.5

International, 1 reg UH, 3 reg FH, 2 reg OH, 3 reg

Cumulative wealth

3

2.5

2

1.5

1 1990

1995

2000

2005

Investment date

Figure 2.2 Cumulative wealth of the strategies Notes: The figure plots the cumulative wealth during the horse race of the regime-switching strategies and a benchmark strategy. The horse race is out-of-sample and spans the period from 1990/1 to 2005/6. The optimal asset allocations are calculated for an investor with a one-month investment horizon. UH is short for unhedged, FH for fully hedged, and OH for optimally hedged. Cumulative wealth and Sharpe ratio

Figure 2.2 plots the evolution of cumulative wealth for the four strategies under consideration. International strategy with a single regime serves as benchmark for the regime-switching strategies. With the exception of the first two years, the cumulative wealth of all regime-switching strategies is always above the benchmark. By the end of June 2005, the optimal hedging strategy performs best, outperforming the single-regime benchmark strategy by 37 percent. We also observe that the optimal hedging strategy recovers fast from the Russian crises in autumn 1998, suffers during the bursting of the Internet bubble, but recovers very well with strong performance during 2004/5. In comparison, the unhedged regime-switching strategy outperforms the benchmark by only 18 percent. Towards the end of the bursting of the

MARKUS LEIPPOLD AND FELIX MORGER

33

Internet bubble, the investor allocates most of the capital into the risk-free asset, but misses the beginning of the recovery period. The strategy with the fully hedged currency risk outperforms the benchmark by only 8 percent. Up to January 2000, its performance is similar to the unhedged regime-switching strategy. Relative to the unhedged strategy, which bears all the currency risk, it outperforms in the period from January 2000 to the early 2002, since the value of the dollar relative to the euro and pound increased. On the other hand, it underperforms towards the end of the horse race because it cannot take advantage of the depreciation of the dollar from early 2002 to the end of 2004. In 2005, it again outperforms because of a rise in the dollar. Table 2.8 presents the descriptive statistics of the horse race for the whole period and for two subperiods, split at the peak of the Internet bubble in January 2000. In comparison to Figure 2.2, we include additional strategies. First, we include a purely domestic single-regime strategy. Second, for comparability with Ang and Bekaert (2004), we introduce a naïve investor, who uses a regime-switching model with three possible states, but does not follow a Bayesian updating rule. Instead, he or she treats the one state with the largest probability of being active as the active state, with a probability of one. Panel A and B report the summary statistics (annualized) for the different strategies and for market indices, respectively. The numbers for the subperiods confirm that the high total performance of the US market is because of its outperformance in the 1990s. The standard deviations of the returns are low for the single-regime strategies, high for the market indices, and intermediate for regime-switching. The single regime strategies exhibit large Sharpe ratios. The largest Sharpe ratio is generated by US domestic strategy, which can be attributed to the outperformance of the US market. The Sharpe ratios of the regime-switching strategies are lower than those for the singleregime strategies. This is because of the low variance of the single-regime strategies. Finally, the regime-switching strategy with three regimes and a naïve investor has a lower mean return and higher return volatility than the Bayesian investor with the unhedged regime-switching strategy. Clearly, it pays to go Bayes. These conclusions for the whole period still hold when looking at the two subperiods (Table 2.8). We end this section with a word of caution. We have to be careful when ranking the different strategies using the Sharpe ratio. The Sharpe ratio would be the optimal measure when investors are only concerned about mean and volatility (or if asset returns are normally distributed). However, since we assume a CRRA utility, the investor also has concerns about higher moments of the portfolio strategy. Therefore, for the ranking of the different strategies, a comparison based on the economic costs as in Section 2.4.1 would be appropriate.

34

Table 2.8 Descriptive statistics on the horse race 1990.1–2005.6 Mean

Standard deviation

1990.1–1999.12

2000.1–2005.6

Sharpe ratio

Mean

Standard deviation

Sharpe ratio

Mean

Standard deviation

Sharpe ratio

Panel A: Strategies Domestic, 1 reg.

6.65

6.22

1.07

10.70

5.56

1.92

−0.35

6.90

−0.05

International UH, 1 reg.

6.16

8.45

0.73

10.46

8.03

1.30

−1.22

8.86

−0.14

UH, “naïve”, 3 reg.

6.60

11.84

0.56

13.08

12.79

1.02

−4.25

9.12

−0.47

UH, 3 reg.

7.28

11.17

0.65

13.15

12.05

1.09

−2.62

8.76

−0.30

FH, 2 reg.

6.77

11.24

0.60

13.52

11.10

1.22

−4.48

10.85

−0.41

OH, 3 reg.

8.33

11.79

0.71

14.05

10.60

1.33

−1.35

13.38

−0.10

Panel B: Markets GER, FH

6.44

22.22

0.29

14.48

20.02

0.72

−6.75

25.53

−0.26

GER, UH

6.76

21.84

0.31

12.85

19.18

0.67

−3.50

25.95

−0.13

UK, FH

8.42

14.46

0.58

14.25

14.34

0.99

−1.44

14.39

−0.10

UK, UH

9.16

15.27

0.60

14.24

15.37

0.93

0.49

14.91

0.03

10.69

14.55

0.73

19.01

13.40

1.42

−2.98

15.87

−0.19

US

Notes: We report the descriptive statistics of the horse race for the whole period and two subperiods. The descriptive statistics consist of the returns’ mean and standard deviation, and the Sharpe ratio (SR). The mean and standard deviation are annualized. Panel A gives the descriptive statistics of the strategies, and Panel B those of the market indices. FH denotes fully hedged and UH unhedged.

35

MARKUS LEIPPOLD AND FELIX MORGER

GER, FH

1 Weight

Weight

1

0.5

0 1990

1995

2000

0.5

0 1990

2005

GER, UH

Weight

Weight

2005

1995

2000

0.5

0 1990

2005

US

1995

2000

2005

Risk free asset

1

1 Weight

Weight

2000

1

0.5

0.5

0 1990

1995

UK, UH

1

0 1990

UK, FH

1995

2000

Investment date

2005

0.5

0 1990

1995

2000

2005

Investment date

Figure 2.3 Asset allocation of the optimal hedging strategy Notes: The figure plots the asset allocation of the optimal hedging strategy. The weights are derived for an investor with a one month horizon. The investor has a risk aversion of γ = 5. Optimal weights

It is instructive to take a look at the behavior of the optimal portfolio strategy through time. Figure 2.3 plots the asset allocation of the optimal hedging strategy. We observe that the weights have sharp peaks and temporarily drop to zero. These sudden moves are mainly because of changes in the state probability estimates. As we can see in Figure 2.4, beliefs about the active state change frequently and provoke the large variation in portfolio weights. Looking at the unhedged and hedged indices of one specific country, the optimization usually puts a lot of weight into the index with the higher historical sample mean and ignores the other index. This result is most probably because of the high correlation between the hedged and unhedged indices. Taking the example of Germany, it is better to invest in the unhedged index, which offers the higher expected drift, and to forgo the additional, but

36

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

pr(St 1) Probability

1

0.5

0 1990

1995

2005

pr(St 2)

1 Probability

2000

0.5

0 1990

1995

2000

2005

2000

2005

pr(St 3) Probability

1

0.5

0 1990

1995 Investment date

Figure 2.4 Changing beliefs during the horse-race Notes: The figure displays the state probabilities of the unhedged strategy. The plots give the probability that a state is active at a certain investment date. The state beliefs are estimated together with the parameter estimates by the EM algorithm. small diversification offered by the hedged index. However, the exception to this rule is investment into the unhedged UK index starting from July 2002. Here, the optimization picks up the depreciation of the dollar, which makes the pound returns more valuable for the US investor. To take advantage of the depreciation, it reduces its holdings in the UK local return index to zero and shifts into the unhedged UK index. The impressive performance at the end of the horse race is the reward.

Optimal hedge ratio

Figure 2.5 plots the optimal hedge ratios for the two foreign markets. The figure confirms the observations made above. The optimal currency hedge

MARKUS LEIPPOLD AND FELIX MORGER

37

GER 1 Hedge ratio

0.8 0.6 0.4 0.2 0 1990

1995

2000

2005

2000

2005

UK

1 Hedge ratio

0.8 0.6 0.4 0.2 0 1990

1995 Investment date

Figure 2.5 Optimal hedge ratios, Germany and UK, during the horse-race Notes: The figure plots the optimal hedge ratio of the foreign markets, Germany and UK, during the horse race. The hedge ratio is calculated as the allocation to the local currency index divided by the total investment to the foreign market. The weights are derived for an investor with a one month horizon. The investor has a risk aversion of γ = 5. No dot is plotted at investment dates where the hedged and unhedged indexes of the foreign markets do not receive any weight at all. ratio is time-varying, and the optimal hedging policy depends on two factors. The first is the trade-off between the diversification offered by the index with the lower expected drift, and the difference in the drifts of the hedged and unhedged indices. The second factor concerns the ex-ante identification of moves in the exchange rates that motivate a preference for the index with the lower drift. The prerequisite for this are parameter estimates for the regime-switching model that allow the identification of favorable moves in the exchange rates. As we observe in Figure 2.5, our model generates the optimal hedge ratios for the German and UK markets that are generally either zero or one. In contrast, in an equity-only international CAPM with regime-switching betas, Ang and Bekaert (2002) find optimal hedge ratios of 53 percent for the high volatility regime and 40 percent for the low volatility regime.

38

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

International, 1 reg. UH,3 reg. OH,3 reg.

1

Weight

0.8 0.6 0.4 0.2 0 1990

1995

2000

2005

Investment date

Figure 2.6 Optimal foreign investments Notes: The figure plots the foreign investments of two different regime-switching strategies and one single-regime strategy. The foreign investments are calculated as the total wealth allocated to foreign markets. The weights are derived for an investor with a one-month horizon. UH = unhedged; OH = optimal hedging.

2.4.3 Optimal foreign investment Figure 2.6 compares the optimal foreign investments of the regime-switching strategies with the international unhedged strategy with a single regime. The international strategy with a single regime starts with an asset allocation of more than 43 percent to foreign markets and reduces this number steadily to less than 30 percent at the end of the horse race. The reduction is caused by increasing drift estimates for the US market. The average international investment is 34 percent. The unhedged regime-switching strategy invests almost the same fraction into the foreign markets, namely 30 percent on average. However, compared to the benchmark strategy, the foreign investment is very volatile over time, moving between 0 percent and 76 percent. Therefore, the introduction of regimes makes the foreign investments strongly time dependent. With the exception of very few data points, the foreign investment of the optimal hedging strategy lies above the benchmark strategy, moving between 4 percent and 100 percent. The average allocation to foreign markets is 63 percent. This is almost twice the percentage of the foreign holdings induced by the single regime strategy. Hence, the introduction of optimal currency hedging increases the optimal allocation to foreign markets compared to the single regime international strategy and, as a backdrop to these results, the home bias question becomes even more of a puzzle.

MARKUS LEIPPOLD AND FELIX MORGER

39

2.5 CONCLUSION We investigated different regime-switching models for international asset allocation. Taking the perspective of a US investor, we applied regime-switching models to the US, UK and German stock markets. We show that, with a correctly specified model, the economic gains from explicitly modeling regime switches are significant and substantial. The cost of ignoring international regime-switching without currency hedging is 2.8 percent per year for a domestic US investor with a ten-year horizon. Of these costs, 70 percent can be attributed to regime-switching and 30 percent to international diversification. Concerning the hedging behavior, the optimal hedging strategy depends strongly on the conditional expected returns of the hedged and unhedged market indices. Most of the time, currency risk is either completely hedged, if the hedged index has the higher expected return, or not hedged at all. The ultimate test of the quality of our model specifications is the outof-sample horse race, where for different strategies we compare the wealth generated during a period spanning fifteen and a half years, from January 1990 to June 2005. Regime-switching strategies generate higher returns than the international and domestic benchmark strategy with a single regime, but at the cost of higher volatilities. The optimal hedging strategy outperforms the international strategy with a single regime by 37 percent in total and 2 percent per year, respectively. During the horse race, the optimal hedging strategy outperforms the strategy without currency hedging by an annualized 1.1 percent and the strategy with a full hedge by an annualized 1.6 percent. Finally, analyzing the size of foreign investments, we find that the optimal foreign investment under regime-switching is highly time-varying. The average foreign investment for the strategy without currency hedging is 30 percent, which is just below the 34 percent of foreign investments generated by the international strategy with a single regime. However, with optimal currency hedging, the investments in foreign markets increase to an average of 63 percent. Hence, under optimal currency hedging, the home bias becomes even more severe.

NOTES 1. A detailed econometric analysis can be obtained from the authors (email: [email protected]) 2. One could argue that our horse race is only pseudo out-of-sample, because the choice of the number of regimes is based on Table 12.2, which uses the whole sample. In real out-of-sample horse races, the specification test should be done at each investment date. This is certainly true. However, as Table 12.2 shows, the choice of regime would

40

INTERNATIONAL STOCK PORTFOLIOS AND OPTIMAL CURRENCY HEDGING

be the same for each evaluated investment date. Because of this stability, it is legitimate to do the horse race without a specification test at each investment date. The horse race can be considered as a real out-of-sample test.

ACKNOWLEDGMENTS Part of this work was done when Markus Leippold was visiting professor at the Federal Reserve Bank of New York. The authors acknowledge the financial support of the Swiss National Science Foundation (NCCR FINRISK) and the University Research Priority Program “Finance and Financial Markets” at the University of Zurich.

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Guidolin, M. and Timmermann, A. (2005b) “International Asset Allocation under RegimeSwitching, Skew and Kurtosis Preferences”, Working Paper, Federal Reserve Bank of St. Louis. Guidolin, M. and Timmermann, A. (2006) “An Econometric Model of Nonlinear Dynamics in the Joint Distribution of Stock and Bond Returns”, Journal of Applied Econometrics, forthcoming. Hamilton, J. D. (1989) “A New Approach to the Economic Analysis of Non-Stationary Time Series and the Business Cycle”, Econometrica, 57(2): 357–84. Hamilton, J. D. (1990) “Analysis of Time Series Subject to Changes in Regime”, Journal of Econometrics, 45(1): 39–70. Longstaff, F. A. and Schwartz, E. (2001) “Valuing American Options by Simulation: A Simple Least-Squares Approach”, Review of Financial Studies, 14(1): 113–47. Perez-Quiros, G. and Timmermann, A. (2000) “Firm Size and Cyclical Variations in Stock Returns”, Journal of Finance, 55(3): 1229–62. Perold, A. and Schulman, E. (1988) “The Free Lunch in Currency Hedging: Implications for Investment Policy and Performance Standards”, Financial Analyst Journal, 44(3): 45–50. Poirier, D. (1995) Intermediate Statistics and Econometrics: A Comparative Approach, Cambridge, Mass.: MIT Press. Samuelson, P. (1969) “Lifetime Portfolio Selection by Dynamic Stochastic Programming”, Review of Economics and Statistics, 51(3): 239–46. Turner, C., Startz, R. and Nelson, C. (1989) “A Markov Model of Heteroskedasticity, Risk, and Learning in the Stock Market”, Journal of Financial Economics, 25(1): 3–22. Whitelaw, R. (2000) “Stock Market Risk and Return: An Equilibrium Approach”, Review of Financial Studies, 13(3): 521–47.

CHAPTER 3

The Determinants of Domestic and Foreign Biases: An Empirical Study Fathi Abid and Slah Bahloul

3.1 INTRODUCTION The international capital asset pricing model (ICAPM), based on traditional portfolio theory developed by Sharpe (1964) and Lintner (1965), suggests that, to maximize risk-adjusted returns, investors should hold a world market portfolio of risky assets. However, domestic assets are heavily weighted in investors’ portfolios even after the relaxing of capital control after 1980. For example, in 1997, 89.9 percent of US investors’ equity portfolios were domestic equities, while the size of the USA in world market capitalization was about 48.3 percent (Ahearne et al., 2004). The wide disparity between actual and recommended international equity portfolio weights constitutes the equity home bias, one of the unresolved puzzles in international finance literature.1 Various attempts have been made to explain the home asset bias. First explanations have focused on the institutional factor. The existence of equity home bias may be related to barriers to capital flow (Black, 1974: Stulz, 1981), hedging possibilities against domestic risk (Glassman and Riddick, 1996), and information asymmetries (Ahearne et al., 2004). Dissatisfaction with 42

FATHI ABID AND SLAH BAHLOUL

43

institutional explanations has led some authors to consider explanations based on investor behavior: optimism of investors about their domestic markets (French and Poterba, 1991), unfamiliarity with foreign market (Huberman, 2001), and subjective competence in the home market (Kilka and Weber, 2000). Recent explanations consider the problem of corporate governance and investor protection explains the home asset bias. The presence of controlling shareholders and the lack of investor protection led to low investment rates in foreign markets (Dahlquist et al., 2003). Explanations for the home bias seem not to be explored sufficiently to provide convincing arguments. Most studies on home bias use a singlefactor model, but international investing behavior seems to be determined by many factors (Faruqee et al., 2004). Besides, most of the previous studies are from the perspective of developed countries, in particular US investors, and neglect emergent countries’ points of view. The purpose of this chapter is not to add a new explanation but rather to examine the determinants of home and foreign equity biases for the period 2001–02. We start by considering different groups of factors that might intervene to explain equity home bias. These factors are economic development, capital controls, stock market development, information costs, investor behavior, familiarity, investor protection, and other variables. Following Chan et al. (2005), work on the determinants of home bias using mutual fund investors for the period 1999–2000, this chapter applies similar methodology to study the cross-border behavior of investors from various countries, including both developed and emerging countries, for a recent period (2001–02) and different investment strategies. We use the Coordinated Portfolio Investment Survey (CPIS) dataset from the International Monetary Fund (IMF) that lists the aggregate stockholdings of both individual and institutional investors. The effects of two other causes (information costs and investors’ behavior) on the home bias will be analyzed in addition to those considered by Chan et al. (2005). As did Chan et al. (2005), we distinguish between the domestic and foreign components of the home bias. The domestic bias reflects the extent to which investors overweigh the local market in their holdings, while the foreign bias reflects the extent to which investors underweigh or overweigh foreign markets. We also use, as an additional test, a measure of the home bias defined by Ahearne et al. (2004). Using the two-year data on equity holdings of thirty countries, we find that equity home bias is a feature in both developed and emerging markets. All of the thirty countries show a domestic bias. The fraction of domestic assets held by local investors is much larger than the world-market capitalization weight of the country. However, domestic bias varies greatly across countries. Venezuela, for example, has the highest domestic bias. Investors from the USA, the European bloc and developed countries have the lowest domestic bias.

44

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

Results show that the impact on domestic and foreign biases is asymmetric. Stock market development and information costs affect domestic bias the most, whereas information costs and familiarity variables contribute significantly to explaining foreign bias. Investor behavior, in contrast, has a significant effect on domestic bias but not on foreign bias. Results indicate that factors such as economic development, capital control and investor protection have smaller effects on these biases. The remainder of the chapter is organized as follows. Section 3.2 provides a theoretical framework for the domestic and foreign biases. In section 3.3, we present the descriptive statistics of investors’ holdings, domestic bias and foreign bias. Section 3.4 discusses the different causes of home bias. Section 3.5 presents and interprets the results, and section 3.6 presents a number of additional tests.

3.2 THEORETICAL FRAMEWORK OF DOMESTIC AND FOREIGN BIASES Chan et al. (2005) have used the theoretical framework developed by Cooper and Kaplanis (1986) to analyze domestic and foreign biases. Cooper and Kaplanis’s model assumes that a representative investor in country i acts as an expected return maximizer for a given level of variance Max(wi R − wi ci )

(3.1)

subject to wi Vwi = v wi I = 1 where wi is a column vector, the jth element of which is wij ; wij is the proportion of individual i’s total wealth invested in risky securities of country j; R is a column vector of pre-tax expected returns; ci is a column vector, the jth element of which is cij ; cij is the deadweight cost to investor i of holding securities in country j; v is the given constant variance; V is the variance/covariance matrix of the gross (pre-cost, pre-tax) returns of the risky securities; and I is a unity column vector. The Lagrangean of the above maximization problem is L = (wi R − wi ci ) − (h/2)(wi Vwi − v) − ki (wi I − 1)

(3.2)

where h and ki are Lagrange multipliers. The derivation of objective function with respect to wi equal to zero lead to R − ci − hVwi − ki I = 0

(3.3)

FATHI ABID AND SLAH BAHLOUL

45

Therefore the optimal portfolio for investor i is wi = (V −1 /h)(R − ci − ki I)

(3.4)

where

ki = I V −1 R − I V −1 ci − h /I V −1 I

Given the individual portfolio holdings, the aggregation leads to the world capital market equilibrium. The clearing condition for the model is pi wi = w∗ (3.5) where pi is the proportion of world wealth owned by country i; w∗ is a column vector, the ith element of which is wi∗ ; and wi∗ is the proportion of the world market capitalization in country i’s market. Using Equations (3.4) and (3.5), and defining z as the global minimumvariance portfolio (V −1 I/(I V −1 I)), Cooper and Kaplanis have obtained hV(wi − w∗ ) = pi ci − ci − z pi ci − ci I (3.6) If deadweight costs are zero (cij are equal to zero for all i and j), the righthand side of Equation (3.6) is zero, and each investor holds the world market portfolio. If deadweight cost of any country/investor pair is equal to c, then the portfolio holdings of each investor will deviate from the world market portfolio. To examine the deviation, Chan et al. (2005) have considered the simple case when the covariance matrix, V, is diagonal with all variances equal to s2 . The deviation of the portfolio weight of investor i in country j from the world market portfolio is given by hs2 (wii − wi∗ ) = −cii + bi + ai − d, i = j 2

hs

(wij − wj∗ )

= −cij + bj + ai − d, i = j

(3.7) (3.8)

where a i = z ci bj = pk ckj pi ci d = z ai can be interpreted as the weighted average deadweight cost for investor i, bj as the weighted marginal deadweight cost for investors investing in country j, and d as the world weighted average marginal deadweight cost. Equation (3.7) measures the extent to which domestic asset holdings of

46

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

investor i deviate from those of the world market portfolio, whereas, Equation (3.8) measures the extent to which investors of country i holdings in foreign market j deviate from the world market portfolio. Similar to Chan et al. (2005), we refer to the former as domestic bias (DBIASi ) and the latter as foreign bias (FBIASij ). From Equation (3.7), investor i overweights domestic country (DBIASi > 0) if the deadweight cost for this investor investing in his/her own country i (cii ) is considerably less than the weighted average deadweight cost for world investors (bi ), or if the weighted average deadweight cost he/she faces (ai ) is large enough to discouraged him/her from investing in foreign markets. Equation (3.8) shows that country j asset holdings by investor i depend on the difference between the deadweight costs for investor i investing in country j (cij ) and the weighted average deadweight cost for world investors (bj ). If cij is greater than bj , investor i underweights country j (FBIASij < 0). Then, the more important the deadweight cost for investor i investing in country j, the greater is the foreign bias (a more negative FBIASij ).

3.3 DATA AND PRELIMINARY STATISTICS 3.3.1 Data sources The cross-border equity data is taken from a survey of international portfolio holdings coordinated by the IMF for seventy countries for the end of December in both 2001 and 2002.2 Data on explanatory variables are not available for all countries; we explore data for only thirty investing and forty-three receiving countries. The data of capital market capitalization is from the international federation of stock exchanges (FIBV).

3.3.2 Statistics on investor holdings For each of the thirty investing countries, we calculate the percentage allocation of local investors in forty-three countries as follows: Wij =

MVij 43

(3.9)

MVij

j=1

where Wij is the share of country j in investor holdings of investing country i; and MV ij is the market value of country j s asset holdings by investors of investing country i.

FATHI ABID AND SLAH BAHLOUL

47

The weight of country j in the world market portfolio is defined as the portfolio of the forty-three countries included in the sample MVj∗ Wj∗ = (3.10) 43 ∗ MVj j=1

Wj∗

where is the share of country j in the world market portfolio; and MVj∗ is the market capitalization of country j; We compute Wij and Wj∗ in 2001 and 2002 separately, and then take an average of the two values. Table 3.1 presents the distribution of the average equity allocations (in percentages) of thirty investing countries’ investors in forty-three national markets across the world. The table shows that domestic bias is revealed in all countries in the sample. Across all of the thirty countries, the shares of investors’ holdings in the domestic market are much more important than the world market capitalization weight of the country. Venezuela has the highest percentage of investors’ holdings of domestic equities (99.64 percent) and the lowest share in the world market portfolio (0.02 percent). Austria has the lowest percentage of domestic asset holdings (50.12 percent), though its share in the world market portfolio is 0.12 percent. The same table shows that investors do underweight foreign markets in their asset holdings. Generally, the share of foreign asset holdings is by far smaller than the shares of foreign country in the world market portfolio. Yet, there are some exceptions. For example, the German and Belgian investors hold a proportion of 4.37 percent and 7.17 percent respectively, of French assets, while the share of the French market in the world market portfolio is only 3 percent. This may provide preliminary evidence that geographical proximity plays an important role in determining the extent to which investors’ overweight foreign markets. Table 3.2 presents the average share of domestic assets held by investors from different blocs: European, American, Asia/Pacific and African. It shows that European investors have the smallest percentage of domestic asset holdings among the four blocs. From Table 3.3 it can be seen that investors from developed markets have a lower percentage of domestic asset holdings compared to investors from emerging markets.

3.3.3 Statistics on domestic and foreign biases Chan et al. (2005) have used the theoretical framework developed by Cooper and Kaplanis (1986) to compute domestic and foreign biases. The domestic bias for a specific country j (DBIASj ) refers to the deviation of the proportion of country j’s investors in the local market from its world market capitalization weight. Therefore, DBIASj is defined as the log ratio of the

48

Table 3.1 Equity allocation for thirty countries, 2001 and 2002, percentages Panel A: First 15 countries Country

% WMP Argentina Austria Australia Belgium Brazil Canada Chile

Czech Denmark Finland France Germany Greece Hong Italy Kong Republic

Argentina

0.10

76.67

0.00

0.00

0.00

0.06

0.00

0.00

0.00

0.23

0.00

0.00

0.00

0.00

Australia

1.51

0.00

83.39

0.31

0.07

0.00

0.27

0.00

0.00

0.17

0.04

0.07

0.08

0.00

0.00 0.06 0.12 0.22

Austria

0.12

0.00

0.01

50.12

0.04

0.00

0.02

0.00

0.57

0.13

0.01

0.02

0.13

0.00

0.00 0.08

Belgium

0.74

0.00

0.03

0.80

74.12

0.00

0.04

0.00

1.18

0.26

0.06

0.97

0.21

0.01

0.00 0.14

Brazil

0.61

0.18

0.03

0.05

0.06

99.16

0.09

0.05

0.00

0.13

0.00

0.05

0.02

0.00

0.00 0.23

Canada

2.53

0.00

0.18

0.32

0.09

0.00

75.16

0.01

0.01

0.21

0.06

0.14

0.08

0.00

0.23 0.12

Chile

0.21

0.02

0.00

0.00

0.00

0.00

0.00

97.29

0.00

0.01

0.00

0.00

0.00

0.00

0.00 0.01

China P.R.

2.00

0.00

0.01

0.04

0.01

0.00

0.01

0.00

0.00

0.31

0.00

0.03

0.01

0.00

1.15 0.02

Czech Republic

0.05

0.00

0.00

0.17

0.03

0.00

0.00

0.00

94.53

0.03

0.00

0.00

0.01

0.00

0.00 0.01

Denmark

0.32

0.00

0.05

0.09

0.09

0.00

0.08

0.00

0.00

62.55

0.41

0.06

0.06

0.00

0.00 0.04

Egypt

0.10

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.01

0.00

0.00

0.00

0.00

0.00 0.00

Finland

0.65

0.09

0.08

0.92

0.63

0.00

0.20

0.00

0.06

1.01

79.46

0.66

1.24

0.02

0.01 0.48

France

3.00

0.05

0.65

3.40

7.17

0.03

1.03

0.01

0.39

2.49

1.65

73.87

4.37

0.11

0.09 2.65

Germany

3.44

0.07

0.42

11.58

1.76

0.02

0.59

0.04

0.75

1.54

1.00

3.01

73.56

0.09

0.06 2.19

Greece

0.30

0.00

0.01

0.08

0.03

0.00

0.01

0.00

0.00

0.03

0.01

0.01

0.02

98.71

0.00 0.03

Hong Kong

1.93

0.00

0.19

0.14

0.12

0.00

0.36

0.00

0.00

0.19

0.14

0.12

0.08

0.00

90.13

0.16

Hungary

1.02

0.00

0.00

0.81

0.01

0.00

0.00

0.00

0.11

0.06

0.00

0.01

0.01

0.01

0.00

0.02

India

0.47

0.00

0.02

0.01

0.01

0.00

0.02

0.00

0.00

0.06

0.00

0.01

0.02

0.00

0.00

0.04

Indonesia

0.11

0.00

0.00

0.02

0.01

0.00

0.01

0.07

0.00

0.01

0.00

0.01

0.00

0.00

0.00

0.03

Italy

1.99

0.15

0.20

0.86

0.96

0.01

0.35

0.00

0.16

1.04

0.43

1.67

1.36

0.02

0.01 77.53

Japan

8.61

0.00

1.06

1.81

0.68

0.00

1.68

0.00

0.00

1.93

0.72

1.08

0.76

0.01

0.44

1.63

Korea. Republic

0.82

0.00

0.05

0.15

0.05

0.00

0.25

0.00

0.00

0.49

0.04

0.09

0.11

0.01

0.31

0.21 0.03

Malaysia

0.27

0.00

0.01

0.03

0.01

0.00

0.01

0.00

0.00

0.07

0.00

0.02

0.01

0.00

0.14

Mexico

0.46

0.06

0.02

0.07

0.02

0.00

0.18

0.04

0.00

0.23

0.00

0.05

0.05

0.00

0.00

0.06

Netherlands

2.77

0.11

0.47

3.38

3.61

0.04

0.68

0.14

0.54

1.71

1.43

3.66

3.12

0.05

0.04

1.97

New Zealand

0.08

0.00

0.01

0.01

0.00

0.00

0.03

0.00

0.00

0.02

0.00

0.00

0.00

0.00

0.00

0.07

Norway

0.27

0.00

0.01

0.10

0.07

0.00

0.05

0.00

0.00

0.33

0.23

0.05

0.04

0.00

0.00

0.01

Pakistan

0.03

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.01

0.00

0.00

0.00

0.00

0.00

0.00

Philippines

0.08

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.01

0.00

0.00

0.00

0.00

0.01

0.00

Continued

49

50

Table 3.1 Continued Panel B: Second 15 countries Country

% WMP Argentina Austria Australia Belgium Brazil Canada Chile

Czech Denmark Finland France Germany Greece Hong Italy Kong Republic

Poland

0.11

0.00

0.00

0.20

0.02

0.00

0.00

0.00

0.05

0.01

0.00

0.00

0.01

0.00

0.00

0.02

Portugal

0.18

0.00

0.01

0.10

0.09

0.07

0.03

0.00

0.01

0.05

0.02

0.07

0.07

0.01

0.00

0.15

Russian Federation

0.41

0.00

0.01

0.24

0.03

0.00

0.02

0.00

0.04

0.10

0.09

0.02

0.08

0.01

0.00

0.05

Singapore

0.43

0.00

0.06

0.15

0.03

0.00

0.15

0.00

0.00

0.09

0.00

0.04

0.04

0.00

0.28

0.07

South Africa

0.66

0.00

0.05

0.06

0.05

0.00

0.06

0.00

0.00

0.06

0.00

0.04

0.04

0.00

0.00

0.04

Spain

1.85

1.08

0.14

0.84

1.11

0.20

0.34

0.00

0.04

0.73

0.29

1.93

1.23

0.02

0.01

0.74

Sweden

0.82

0.01

0.09

0.36

0.21

0.00

0.19

0.08

0.02

3.32

5.28

0.19

0.29

0.00

0.01

0.16

Switzerland

2.29

0.00

0.37

3.03

1.11

0.02

0.64

0.01

0.10

2.19

1.01

1.62

2.20

0.05

0.02

1.41

Taiwan

1.10

0.00

0.04

0.06

0.04

0.00

0.07

0.00

0.00

0.17

0.00

0.05

0.03

0.00

0.30

0.08

Thailand

0.16

0.00

0.01

0.06

0.06

0.00

0.02

0.00

0.00

0.07

0.00

0.02

0.03

0.00

0.15

0.03

Turkey

0.16

0.00

0.00

0.03

0.00

0.00

0.01

0.00

0.00

0.01

0.00

0.01

0.01

0.01

0.00

0.05

United Kingdom

7.79

0.17

1.57

5.61

2.96

0.06

2.74

0.04

0.30

5.98

3.20

3.95

4.25

0.37

4.73

2.80

49.44

21.32

10.74

13.99

4.61

0.30

14.58

2.20

1.11

11.96

4.43

6.39

6.35

0.46

1.74

6.36

0.02

0.00

0.00

0.00

0.00

0.01

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

United States Venezuela

Panel A: First 15 countries continued Country

% Japan Korea Malaysia Netherlands New Norway Portugal Singapore South Spain Sweden Switzerland UK WMP Zealand Africa

USA Venezuela

Argentina

0.10

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.04

0.00

0.02

0.02 0.00

0.00

Australia

1.51

0.16

0.00

0.05

0.34

5.28

0.84

0.00

0.77

0.06

0.00

0.25

0.13

0.76 0.28

0.00

Austria

0.12

0.00

0.00

0.00

0.06

0.00

0.09

0.09

0.00

0.00

0.00

0.06

0.13

0.03 0.01

0.00

Belgium

0.74

0.03

0.00

0.00

0.60

0.00

0.62

0.84

0.01

0.00

0.14

0.08

0.14

0.13 0.07

0.00

Brazil

0.61

0.01

0.00

0.00

0.06

0.00

0.05

0.26

0.00

0.00

0.10

0.03

0.04

0.15 0.14

0.00

Canada

2.53

0.14

0.01

0.01

0.27

0.40

0.43

0.02

0.15

0.02

0.01

0.22

0.37

0.21 0.59

0.00

Chile

0.21

0.00

0.00

0.00

0.05

0.00

0.00

0.00

0.00

0.00

0.06

0.00

0.00

0.02 0.01

0.00

China. P.R. 2.00

0.04

0.01

0.01

0.02

0.00

0.27

0.00

0.55

0.00

0.04

0.04

0.02

0.09 0.02

0.00

Czech Republic

0.05

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.01

0.01

0.01 0.00

0.00

Denmark

0.32

0.02

0.00

0.00

0.13

0.03

1.38

0.00

0.02

0.02

0.01

0.40

0.06

0.14 0.05

0.00 0.00

Egypt

0.10

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00 0.00

Finland

0.65

0.09

0.00

0.00

0.54

0.06

1.05

0.20

0.07

0.02

0.28

1.92

0.47

0.49 0.34

0.00

France

3.00

0.45

0.00

0.01

2.49

0.23

2.86

1.23

0.67

0.15

1.61

2.12

1.90

3.30 0.76

0.00

Germany

3.44

0.28

0.01

0.00

1.93

0.51

1.93

0.91

0.31

0.04

1.27

1.81

3.89

1.74 0.41

0.01 Continued

51

52

Table 3.1 Continued Panel B: Second 15 countries continued Country

% Japan Korea Malaysia Netherlands New Norway Portugal Singapore South Spain Sweden Switzerland UK USA Venezuela WMP Zealand Africa

Greece

0.30

0.01

0.00

0.00

0.04

0.00

0.08

0.00

0.00

0.01

0.01

0.02

0.03

0.05 0.02

0.00

Hong Kong

1.93

0.19

0.06

0.07

1.02

0.16

0.24

0.00

2.82

0.01

0.00

0.23

0.15

0.71 0.20

0.00

Hungary

1.02

0.00

0.00

0.00

0.01

0.00

0.01

0.00

0.00

0.00

0.00

0.03

0.01

0.03 0.01

0.00

India

0.47

0.00

0.00

0.01

0.12

0.00

0.00

0.03

0.25

0.00

0.05

0.01

0.01

0.10 0.06

0.00

Indonesia

0.11

0.00

0.01

0.02

0.01

0.00

0.00

0.00

0.86

0.00

0.00

0.00

0.01

0.04 0.02

0.00

Italy

1.99

0.16

0.00

0.00

1.00

0.08

0.85

0.76

0.06

0.03

0.83

0.57

0.57

0.91 0.24

0.00

Japan

8.61 90.51

0.10

0.01

1.43

2.23

4.05

0.14

1.56

0.14

0.50

1.97

1.14

2.87 1.28

0.00

Korea. Republic

0.82

0.02 99.31

0.01

0.16

0.09

0.36

0.00

1.07

0.00

0.00

0.17

0.14

0.46 0.24

0.00

Malaysia

0.27

0.02

0.05

99.23

0.05

0.00

0.01

0.00

4.40

0.00

0.00

0.04

0.02

0.09 0.02

0.00

Mexico

0.46

0.00

0.00

0.00

0.06

0.00

0.05

0.01

0.01

0.00

0.03

0.04

0.24

0.23 0.18

0.00

Netherlands 2.77

0.25

0.01

0.00

67.97

0.19

1.81

0.68

0.16

0.04

0.75

1.01

2.43

1.73 0.77

0.00

New Zealand

0.01

0.00

0.00

0.00

68.37

0.02

0.00

0.07

0.00

0.00

0.01

0.00

0.03 0.02

0.00

0.08

Norway

0.27

0.01

0.00

0.00

0.06

0.04

54.34

0.02

0.02

0.00

0.00

0.37

0.13

0.14 0.06

0.00

Pakistan

0.03

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00 0.00

0.00

Philippines

0.08

0.01

0.00

0.04

0.00

0.00

0.00

0.00

0.34

0.00

0.00

0.01

0.00

0.01 0.01

0.00

Poland

0.11

0.00

0.00

0.00

0.01

0.00

0.21

0.05

0.00

0.00

0.00

0.06

0.01

0.02 0.01

0.00

Portugal

0.18

0.01

0.00

0.00

0.06

0.00

0.11

87.49

0.01

0.00

0.26

0.03

0.03

0.13

0.02

0.00

Russian Federation

0.41

0.00

0.00

0.00

0.02

0.00

0.03

0.00

0.00

0.00

0.00

0.29

0.09

0.09

0.04

0.00

Singapore

0.43

0.04

0.00

0.38

0.11

0.07

0.20

0.00

74.08

0.01

0.00

0.09

0.04

0.29

0.17

0.00

South Africa

0.66

0.00

0.00

0.01

0.05

0.00

0.02

0.44

0.00

86.32

0.01

0.04

0.06

0.12

0.06

0.00

Spain

1.85

0.14

0.00

0.00

1.10

0.10

0.81

2.56

0.04

0.03

89.85

0.38

0.41

0.84

0.23

0.00

Sweden

0.82

0.07

0.00

0.00

0.34

0.13

2.95

0.06

0.06

0.04

0.05

64.65

0.36

0.66

0.13

0.01

Switzerland

2.29

0.32

0.00

0.00

1.60

0.19

1.81

0.27

0.26

0.10

0.34

1.83

75.75

1.55

0.58

0.00

Taiwan

1.10

0.02

0.00

0.01

0.09

0.00

0.28

0.00

1.03

0.00

0.00

0.12

0.08

0.35

0.14

0.00

Thailand

0.16

0.01

0.01

0.02

0.02

0.00

0.00

0.00

1.45

0.00

0.00

0.01

0.01

0.09

0.02

0.00

Turkey

0.16

0.00

0.00

0.00

0.01

0.00

0.01

0.00

0.00

0.00

0.02

0.00

0.04

0.11

0.01

0.00

United Kingdom

7.79

1.35

0.04

0.03

4.67

4.58

8.27

1.57

2.92

10.75

2.08

6.40

2.45

73.88

2.56

0.00

49.44

5.59

0.36

0.09

13.47

17.27

13.93

2.36

5.96

2.20

1.66

14.68

8.60

7.41

90.18

0.25

0.02

0.00

0.00

0.00

0.00

0.00

0.01

0.00

0.00

0.00

0.00

0.00

0.00

0.00

United States Venezuela

0.00 99.64

Notes: This table contains the distribution of 30 investing countries’ average investors’ allocations across 43 national markets for 2001 and 2002. The second column contains a country’s average stock market capitalization weight in the world market portfolio. Panel A – WMP: world market portfolio – Argentina, Australia, Austria, Belgium, Brazil, Canada, Chile, Czech Republic, Denmark, Finland, France, Germany, Greece, Hong Kong and Italy. Panel B – WMP: world market portfolio – Japan, Korea, Malaysia, Netherlands, New Zealand, Norway, Portugal Singapore, South Africa, Spain, Sweden, Switzerland, UK, USA, Venezuela.

53

54

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

Table 3.2 Average shares of domestic asset holdings by investors from different blocs American bloc

Asia/pacific bloc

African bloc

European bloc

89.68%

86.43%

86.32%

74.89%

Notes: American bloc: Argentina, Brazil, Canada, Chile, USA and Venezuela; Asia/Pacific: Australia, Korea, Hong Kong, Japan, Malaysia, New Zealand and Singapore; European bloc: Germany, Austria, Belgium, Denmark, Spain, Finland, France, Greece, Italy, Norway, Netherlands, Portugal, Czech Republic, UK, Sweden and Switzerland; African bloc: South Africa only.

Table 3.3 Shares of domestic asset holdings for developed and emerging countries Developed countries

Emerging countries

77.17%

93.24%

Notes: Developed countries: Germany, Australia, Austria, Belgium, Canada, Korea, Denmark, Spain, USA, Finland, France, Greece, Hong Kong, Italy, Japan, New Zealand, Norway, Netherlands, Portugal, UK, Singapore, Sweden, and Switzerland; Emerging countries: Argentina, Venezuela, Brazil, Chile, South Africa, Malaysia and Czech Republic.3

share of country j’s investors’ holdings in the domestic market (Wjj ) to the world market capitalization weight of country j (Wj∗ ): log

Wjj Wj∗

(3.11)

The foreign bias refers to the extent to which domestic investors underweight or overweight foreign markets in their asset holdings. The foreign bias (FBIASij ) is defined as log

Wij Wj∗

(3.12)

The average foreign bias of foreign investors in an investing country j is calculated by averaging FBIASj across all remaining countries. Table 3.4 exhibits the distribution of the domestic bias for domestic investors and the average foreign bias of foreign investors in an investing country, across thirty investing countries. Average values are computed for the sample periods 2001 and 2002. In general, we observe a significant cross-section variation in domestic bias and average foreign bias measures. The domestic bias fluctuates between 0.6 (USA) and 8.5 (Venezuela). The average foreign bias varies

FATHI ABID AND SLAH BAHLOUL

55

Table 3.4 Domestic bias of domestic investors and average foreign bias of foreign investors Country

Domestic bias

Average foreign bias

Argentina

6.71

−4.79

Australia

4.02

−3.82

Austria

6.07

−3.40

Belgium

4.61

−3.62

Brazil

5.09

−4.07

Canada

3.39

−4.05

Chile

6.15

−5.28

Czech Republic

7.59

−3.89

Denmark

5.27

−3.46

Finland

4.81

−2.24

France

3.21

−2.09

Germany

3.07

−2.44

Greece

5.80

−4.43

Hong Kong

3.85

−4.56

Italy

3.66

−3.05

Japan

2.35

−3.64

Korea

4.80

−3.99

Malaysia

6.14

−3.79

New Zealand

6.77

−4.19

Netherlands

3.20

−2.56

Norway

5.32

−3.58

Portugal

6.16

−2.95

Singapore

5.16

−3.82

South Africa

4.89

−4.71

Spain

3.88

−2.79

Sweden

4.37

−2.64

Switzerland

3.51

−2.69

UK

2.25

−1.97

USA

0.60

−2.64

Venezuela

8.50

−4.22

from −1.97 (UK) to −5.28 (Chile). The values of these measures are important relatively to those of Chan et al. (2005). We use aggregate cross-section data for institutional and individual investors. Chan et al. (2005) have used the data for mutual funds that are likely to invest more in foreign markets, so it seems acceptable to reach the end with different results.

56

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

Table 3.5 Domestic bias and average foreign bias Domestic bias

Average foreign bias

American bloc

5.07

−4.18

Asia/Pacific bloc

4.73

−3.97

European bloc

4.55

−2.99

African bloc

4.89

−4.71

Developed countries

4.21

−3.25

Developing countries

6.44

−4.39

The domestic and average foreign biases are calculated for investors from developed and developing countries as well as for investors from the American, European, Asia/Pacific and African blocs separately. Results are summarized in Table 3.5. The table shows that domestic bias and average foreign bias are less important for investors from the European bloc than for those from other regional blocs. Results corroborate once again the fact that European investors invest less in domestic assets. Developed countries also have a less important domestic bias and an average foreign bias compared to developing countries.

3.4 THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES There may be a large number of causes to explain home bias. We select several explanatory variables and regroup them into economic development, capital control, stock market development, information cost, familiarity, investor behavior, investor protection, and others. We calculate descriptive statistics of these variables for the years 2001 and 2002 separately. We find that an important number of these variables remain stable during this period. Consequently, we shall consider the descriptive statistics only for the year 2001, as shown in Table 3.6.

3.4.1 Economic development Chan et al. (2005) have found that the level of economic development of a country has a significant effect on the investment decisions of foreign investors. To study the impact of economic development on the home bias, we set a number of measures of economic development. We distinguish between gross domestic product (GDP) per capita in US dollars (GDPC); the real growth rate of gross domestic product (RGDP); the average of exports

Table 3.6 Summary statistics for the explanatory variables, 2001 Panel A: The first set of variables Country

Economic development GDP per capita US($)

Real GDP growth (%)

Trade volume (% of GDP)

Financial market development Foreign direct investment (% of GDP)

Transaction costs (basis points)

Stock market capitalization (% of GDP)

Capital control

Emerging market dummy

Capital flow Restrictions

Argentina

7,430

−4

17

0.01

70.8

0.12

1

5.8

Australia

18,995

4

35

0.01

51.0

1.02

0

6.1

Austria

23,603

1

77

0.03

45.6

0.13

0

8.1

Belgium

22,120

1

172

0.3

27.5

0.81

0

9.2

2,949

1

23

0.04

58.6

0.37

1

4.2

22,343

1

70

0.04

36.4

1.01

0

8.6

4,314

3

55

0.07

114.3

0.85

1

7.0

924

8

43

0.04

n.a.

0.45

1

2.7

5,593

3

122

0.10

72.9

0.16

1

7.0

29,713

1

61

0.06

41.3

0.53

0

9.0

1,511

4

17

0.01

n.a.

0.25

1

7.3

Finland

23,422

1

62

0.03

42.3

1.57

0

8.1

France

22,308

2

49

0.04

28.2

0.69

0

7.6

Germany

22,511

1

57

0.01

27.3

0.58

0

9.5

Greece

11,062

4

33

0.01

74.4

0.72

0

8.3

Hong Kong

24,213

0

241

0.15

53.2

3.11

0

9.6

5,088

4

124

0.05

103.8

0.20

1

8.8

India

463

5

19

0.01

44.4

0.23

1

2.0

Indonesia

676

3

62

−0.02

83.7

0.16

1

4.8

Brazil Canada Chile China Czech Republic Denmark Egypt

Hungary

57 Continued

Table 3.6 Continued Country

Italy Japan Korea, Republic of Malaysia Mexico Netherlands New Zealand Norway Pakistan Philippines Poland Portugal Russian Fed Singapore South Africa Spain Sweden Switzerland Taiwan Thailand Turkey UK USA Venezuela

58

Panel A: The first set of variables continued Economic development

Financial market development

Capital control

GDP per capita US($)

Real GDP growth (%)

Trade volume (% of GDP)

Foreign direct investment (% of GDP)

Transaction costs (basis points)

Stock market capitalization (% of GDP)

Emerging market dummy

Capital flow Restrictions

18,921 32,869 10,180 3,696 6,262 23,944 13,241 37,620 415 920 4,808 10,835 2,118 20,545 2,549 14,315 24,673 33,998 n.a. 1,888 2,119 24,211 35,118 5,123

2 0 4 0 0 1 3 2 3 3 4 2 5 −2 3 3 1 1 n.a. 2 −7 2 0 3

43 18 68 184 53 114 53 54 33 89 47 58 51 280 50 47 63 68 n.a. 110 50 42 19 36

0.01 0.00 0.01 0.01 0.04 0.13 0.04 0.01 0.01 0.01 0.03 0.05 0.01 0.13 0.06 0.05 0.06 0.04 n.a. 0.03 0.02 0.04 0.02 0.03

40.5 24.4 73.4 90.9 65.9 27.7 39.0 30.0 n.a. 113.2 n.a. 44.5 n.a. 74.0 88.5 39.4 29.3 41.5 59.7 87.4 45.3 46.6 28.5 102.7

0.48 0.54 0.40 1.35 0.20 1.81 0.35 0.41 0.08 0.29 0.14 0.42 0.25 1.36 1.29 0.80 1.08 2.15 n.a. 0.31 0.32 1.50 1.40 0.05

0 0 0 1 1 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 1 0 0 1

8.7 8.4 n.a. 3.7 5.1 9.5 8.9 7.8 0.8 4.6 3.8 7.6 3.2 7.6 4.1 6.9 7.2 9.1 7.6 4.3 5.8 9.1 8.2 7.8

Panel B: The second set of variables Country

Capital control

Other variables

Investor protection

Intensity of capital control

Rule of law

Accounting

Minority

Expropriation

Argentina

0.05

5.35

45

4

5.91

6

0

Australia

0.00

10

75

4

9.27

10

Austria

0.00

10

54

2

9.69

9.5

Belgium

0.00

10

61

0

9.63

9.5

Brazil

0.05

Canada

0.00

6.32 10

Efficiency

Legal system dummy

Lag 2-year return

Return correlation (average)

2.5

0.017

1

1.1

0.136

0

−12.0

0.004

0

−16.6

−0.028

54

3

7.62

5.75

0

22.1

0.126

74

4

9.67

9.25

1

19.7

0.202

Chile

0.11

7.02

52

3

7.5

7.25

0

13.4

0.056

China

0.59

n.a.

n.a.

n.a.

n.a.

n.a.

n.a.

29.4

−0.037

Czech Republic

0.02

n.a.

n.a.

n.a.

n.a.

n.a.

n.a.

1.3

0.162

Denmark

0.00

10

62

3

9.67

0

12.1

0.090 −0.029

Egypt

0.29

Finland

0.00

France

0.00

Germany Greece

4.17

10

24

2

6.3

0

23.1

77

2

9.67

10

0

n.a.

8.98

69

2

9.65

8

0

0.00

9.23

62

1

9.9

9

0

0.00

6.18

55

1

7.12

7

0

n.a.

n.a.

Hong Kong

0.00

8.22

69

4

8.29

10

1

20.8

0.169

Hungary

0.03

n.a.

n.a.

n.a.

n.a.

n.a.

n.a.

n.a.

India

0.59

4.17

57

2

7.75

8

1

20.0

0.144

Indonesia

0.14

3.98

65

2

7.16

2.5

0

14.0

0.056

10

6.5

n.a.

n.a.

5.9

0.051

1.5

0.145

59

Continued

Table 3.6 Continued Country

Italy Japan Korea, Republic of Malaysia Mexico Netherlands New Zealand Norway Pakistan Philippines Poland Portugal Russian Federation Singapore South Africa Spain Sweden Switzerland Taiwan Thailand Turkey UK USA Venezuela

60

Panel B: The second set of variables continued Capital control

Other variables

Investor protection

Intensity of capital control

Rule of law

Accounting

Minority

Expropriation

Efficiency

Legal system dummy

0.00 0.00 0.03 0.06 0.01 0.00 0.00 0.00 1.00 0.54 0.03 0.00 0.26 0.00 0.00 0.00 0.00 0.00 0.44 0.34 0.01 0.00 0.00 1.00

8.33 8.98 5.35 6.78 5.35 10 10 10 3.03 2.73 n.a. 7.8 n.a. 8.57 4.42 7.8 10 10 8.52 6.25 5.18 8.57 10 6.37

62 65 62 76 60 64 70 74 61 65 n.a. 36 n.a. 78 70 64 83 68 65 64 51 78 71 40

0 3 2 3 0 2 4 3 4 4 n.a. 2 n.a. 3 4 2 2 1 3 3 2 4 5 1

9.35 9.67 8.31 7.95 6.07 9.98 9.69 9.88 5.62 5.22 n.a. 8.9 n.a. 9.3 6.88 9.52 9.4 9.98 9.12 7.42 7 9.71 9.98 6.89

6.75 10 6 9 6 10 10 10 5 4.75 n.a. 5.5 n.a. 10 6 6.25 10 10 6.75 3.25 4 10 10 6.5

0 0 0 1 0 0 1 0 1 0 n.a. 0 n.a. 1 1 0 0 0 0 1 0 1 1 0

Lag 2-year return

−0.2 4.3 19.4 29.2 28.4 −2.9 n.a. −3.1 26.3 −23.9 n.a. −12.7 129.6 18.7 0.0 −13.3 18.5 −0.1 −9.4 −9.8 56.9 −4.2 4.8 43.8

Return correlation (average) 0.140 0.155 0.098 0.061 0.110 0.022 n.a. −0.062 0.084 0.066 n.a. 0.095 −0.003 0.153 n.a. 0.134 0.218 0.071 0.140 0.060 0.118 0.124 0.142 −0.002

Panel C: The third set of variables Country

Information costs

Investor behavior

Phone costs (By minute in $)

Investor behavior toward foreign market returns (basis points)

Distance (kilometers) (average)

Common language dummy (average)

Argentina

0.5743

−12

12,274

0.12

Australia

0.3881

−8

12,726

0.31

Austria

0.3323

−13

5,843

0.05

Belgium

0.3696

−28

5,970

0.12

Brazil

0.5861

41

11,417

0.02

Canada

0.2559

12

8,745

0.38

Chile

0.7472

−12

12,595

0.12

Czech Republic

0.6327

26

n.a.

0.00

Denmark

0.2706

−7

5,879

0.00

Finland

0.4442

N.A.

5,967

0.00

France

0.2997

−19

6,049

0.07

Germany

0.2417

−5

5 916

0.05

Greece

0.4996

N.A.

6 109

0.00

Hong Kong

0.2935

24

8 403

0.36

Italy

0.2991

Japan

0.5163

Korea, Republic of

0.3471

−2

Familiarity

6,044

0.00

12

8,809

0.05

20

8,324

0.05 Continued

61

62

Table 3.6 Continued Panel C: The third set of variables continued Country

Information costs

Investor behavior

Phone costs (by minute in $)

Investor behavior toward foreign market returns (basis points)

Familiarity Distance (kilometers) (average)

Common language dummy (average) 0.31

Malaysia

0.8926

5

8 846

Netherlands

0.3287

−19

5 956

0.05

New Zealand

0.2928

n.a.

13 996

0.31

Norway

0.3576

−25

5 997

0.00

Portugal

0.4850

−11

6 776

0.02

Singapore

0.3567

17

9 013

0.38

South Africa

0.3448

n.a.

9 536

0.31

Spain

0.3565

6 513

0.12

Sweden

0.2849

12

5 931

0.00

Switzerland

0.2556

−13

6 016

0.12

UK

0.3002

−9

6 093

0.31

USA

0.1560

−3

9 217

0.31

Venezuela

0.8008

−24

9 501

0.10

0.2

This table presents, summary statistics for each country, for eight groups of explanatory variables: (i) Economic development variables: GDP per capita, real GDP growth, trade volume (% of GDP) and foreign direct investment; (ii) Financial market development variables: transaction costs, stock market capitalization (% of GDP) and emerging market dummy variable; (iii) Capital control variables: capital flow restrictions and stock-holding restrictions; (iv) Investor protection variables: rule of law index, accounting standard index, minority investor protection index, risk of expropriation index and efficiency of judicial system index; (v) Other variables: past 2-year return and average return correlation; (vi) Information costs: average phone costs by minute; (vii) Investor behavior: average degree of pessimism toward foreign market return; (viii) Familiarity: average distance in kilometers and average common language dummy variable.4

FATHI ABID AND SLAH BAHLOUL

63

and imports scaled by GDP (TRADE); and foreign direct stock investment inward scaled by GDP (DI). All these variables are obtained from the World Development Indicators (WDI). Table 3.6 shows significant cross-sectional variation in the four measures of economic development. The country that has the highest value of GDPC is a developed country (Norway: US$37,620), while the most important value of RGDP is held by an emerging country (China: 8 percent). Belgium has the highest foreign direct stock investment inward relative to its GDP (DI), while the country that has the largest trade volume, as a percentage of GDP (TRADE) is Singapore.

3.4.2 Capital control Ahearne et al. (2004) suggest that, while capital controls have been greatly reduced in many countries, they still affect cross-border investment. In this chapter we measure capital control according to two parameters. The first is the intensity of capital control (RESTRICT) developed in Edison and Warnock (2003) and used by Ahearne et al. (2004). This measure is constructed by using International Finance Corporation (IFC) indexes. It equals one minus the ratio of the market capitalization of a country’s Investable (IFCI) and Global (IFCG) indexes.5 Restrictions vary greatly across developing countries. For industrial countries, the IFC does not publish investable indexes. We assume that, for these countries, Investable and Global indexes are identical. Table 3.6 shows that RESTRICT ranges between zero (in developed countries) to one (in Pakistan). The second variable (RFLOW) measures the restrictions of countries on capital flows. It is constructed by the Economic Freedom Network by assigning lower ratings to countries with more restrictions on foreign capital transactions and was used by Chan et al. (2005). Table 3.6 shows that RFLOW varies from 0.8 in Pakistan to 9.6 in Hong Kong.6 When a country imposes capital controls, this will stop, or at least discourage, foreign investors from holding stocks of companies in that country. When a country imposes higher capital control measures, the degree of foreign bias becomes higher (more negative FBIAS). Also, when a score on RFLOW is low (and the score on RESTRICT is important), domestic investors find it difficult to invest overseas, as it requires government approval. Then they will invest a large amount of their wealth in the domestic market and the domestic bias will consequently be important (more positive DBIAS).

3.4.3 Stock market development Chan et al. (2005) suggest that investors tend in general to invest more in developed stock markets. In fact, these markets present high liquidity and lower transaction costs. We measure the stock market development according to three distinct variables.

64

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

The first variable is the relative size of the stock market of each country, measured by the stock market capitalization as a percentage of the country’s GDP (SIZE). The value of SIZE varies from 0.05 percent in Venezuela to 3.11 percent in Hong Kong. The data on GDP are taken from the World Development Indicators (WDI) and data on market capitalization are from the International Federation of Stock Exchanges (FIBV). The second variable is the transaction costs associated with trading foreign securities (COST). The role of transaction costs in the explanation of the home bias has been neglected because of the existence of an important turnover rate of foreign assets compared to domestic assets (Tesar and Werner, 1995). However, Carmichael and Coen (2003) reveal, by using a simple OLG model of the world economy with transaction costs, that the introduction of very small transaction costs is sufficient to reproduce the large home bias observed in portfolios. We use the Elkins–McSherry Co. measure of transaction costs. The latter consists of three components: commissions, fees and market impact costs. This measure has been used by Ahearne et al. (2004) and Chan et al. (2005). Although the other explanatory variables are for the year 2001, we use the data for the year 1999.7 We assume that these transaction cost estimates do not change substantially, and we can utilize it in our analysis. We have transaction cost data for thirty-eight of the forty-three markets, ranging from 24.4 basis points for Japan to 114.3 for Chile. We also consider a dummy variable (DUMEMERG) that equals one for an emerging market, and zero otherwise. We expect that foreign investors will opt for investing in local markets which are large, developed and require low transaction costs. Foreign bias will be less important in these markets. The proportion of local asset holdings by domestic investors will be smaller and the domestic bias less important.

3.4.4 Information asymmetries Gehrig (1993) argues that one of the explanations for the home bias is that local investors spend too much on information about foreign markets. A high information cost discourages investors from investing abroad. Zhou (1998) shows that, with differential information, agents on average tilt their portfolio towards stocks about which they have better information. Portes and Rey (2005) use the volume of telephone call traffic as a proxy for information costs. To measure information costs, we shall consider the cost of international phone calls per minute from a country i to a country j.8 Table 3.6 shows that average values of information costs vary from US$0.1560 (USA) to US$0.8008 (Venezuela). High information costs between country i and country j make country i’s investors hold fewer assets in country j, so we expect an important FBIASij (more negative FBIAS).

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3.4.5 Investors’ behavior One explanation of the home bias is that investors in each country expect returns from their domestic equity market to be several hundred basis points higher than returns from foreign markets. French and Poterba (1991) show that investors may be relatively more optimistic about their home markets than are foreign investors. Strong and Xu (2003) find that fund mangers from the USA, UK, continental Europe and Japan show a significant relative optimism towards their equity markets. To measure an investor’s degree of optimism or pessimism towards a market, we use BEHAV, a variable that equals the difference between the market return implied by the actual portfolio holdings and the returns implied by an international value-weighted portfolio for each country. To determine this vector of return, we have used French and Poterba’s (1991) model: µ = λw∗

(3.13)

where µ is the vector of expected return; w∗ is optimal portfolio weights; is the covariance matrix; and λ is relative risk aversion. Average values of BEHAV vary from –28 basis points for Belgian investors to 41 basis points for Brazilian investors. In general, investors are pessimistic about foreign markets, which can help to explain why investors underweight foreign markets and overweight domestic markets.

3.4.6 Familiarity One explanation for the home bias is that investors may not be familiar with foreign markets. Huberman (2001) finds that “Shareholders of a Regional Bell Operating Company (RBOC) tend to live in the area which it serves, and an RBOC’s customers tend to hold its shares rather than other RBOCs’ equity”. Like Chan et al. (2005) and Sarkissan and Schill (2004), we use three proxies of familiarity9 variables for each pair of countries i and j. The first familiarity variable is common language. Data are obtained from the World Factbook 2001. For each pair of countries i (investing) and j, we construct a language dummy variable (DUMLANG) which equals one if i and j share the same language and zero otherwise. The second variable is geographical proximity (DISTANCE). Data are obtained from http://www.ksg.harvard.edu/people/sjwei. Average values of distance vary from 5,843 kilometers for Austria to 13,996 for New Zealand. The last variable is the amount of bilateral trades (TRADEB), with values ranging from 0 to 1. A value of 0.17 for TRADEB between the UK (investing) and the USA means that 17 percent of the total UK trade (imports

66

THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

and exports) is with the USA. (Data are obtained from the United Nations Statistics Division Databases.) For the familiarity variables, we expect that investors from country i, who are more familiar with country j, through sharing a common language, being close to each other, or having larger bilateral trade volumes, will display less foreign bias (FBIASij ). Familiarity variables do affect domestic bias. Investors from a country that is isolated from the rest of the world will also hold a large proportion of domestic assets.

3.4.7 Investor protection La Porta et al. (1997) argue that capital markets are narrow in countries with poorer investor protection. Giannetti and Koskinen (2004) show that foreign investors are reluctant to invest in a country where expropriating minority shareholders is easy, while wealthy investors have an incentive to become controlling shareholders by investing a large proportion of their wealth in the stock market in a country with poor investor protection. Similarly to Chan et al. (2005), we use six measures of investor protection based on La Porta et al. (1997, 1998, 2000).10 The first measure is the rule of law index (LAW), elaborated by the International Country Risk Agency. It arranges countries on a scale ranging from zero to 10, with lower scores for countries with less respect for law and order. The index varies from 2.73 in the Philippines to 10 in twelve countries across the world. The second measure is the accounting standard index (ACC). This defines the amount and transparency of information available to investors. Table 3.6 shows that Sweden has the highest score (83) while the lowest is for Egypt (24). The third measure is the anti-director rights (MINORITY). It indicates the degree of protection for minority investors. Values vary from zero for Italy, Belgium and Mexico, to 5 for the USA. The fourth measure is the risk of expropriation index (EXPROP). This index is constructed by the International Country Risk Agency. It has a scale ranging from zero to 10, with lower scores for higher risk. This index varies from 5.22 for the Philippines to 9.98 for the USA, Switzerland and the Netherlands. The fifth measure is the efficiency of the judicial system (EFFICIENCY). This index is constructed by the Business International Corporation. Values range from 2.5 for Indonesia to 10 for fourteen developed countries. The sixth measure is a dummy variable that considers the type of legal system (DUMLEGAL). It equals 1 for common-law countries and zero otherwise. In fact, La Porta et al. (1997) have found that the French civil law countries have the weakest investor protection, particularly when compared to common-law countries.

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3.4.8 Other variables In addition to the above variables, we include several others that account for the home bias. The first variable is the two-year return (RET2). Bohn and Tesar (1996) find that US investors exhibit returns-chasing behavior, with a tendency to underweight countries whose stock markets have performed poorly. The second variable is the correlation between the returns of two countries (CORR). For each pair of countries, i and j, we compute the correlation coefficient using country returns in US dollars from 1999 to 2001. Data for indexes are from Yahoo Finance, and data for exchange rates are from the Oanda website. The Correlation coefficient is used as a proxy for the diversification potential between two countries.

3.5 THE EMPIRICAL ANALYSIS Following Chan et al.’s (2005) methodology, this section studies the causes of domestic and foreign biases. In all tests, we stack up all the observations on domestic-bias measure (DBIASj ) and regress them against each set of the explanatory variables; then we do the same for the foreign bias measures (FBIASij ). The dependent variables (DBIAS and FBIAS) are average values for the 2001–02 period, while, explanatory variables are for the year 2001.

3.5.1 Results concerning domestic bias The second column of Table 3.7 shows the predicted signs of the coefficients. The other columns contain estimates of explanatory variables for the eight categories separately and for all variables. Table 3.7 shows that stock market development variables have the most important explanatory power, with adjusted R2 of 58 percent, while familiarity variables exhibit a low adjusted R2 of 17 percent. For the market development variables, we find that DBIAS is negatively related to the size of the market (SIZE) and positively linked to transaction costs (COST). Large markets have a higher visibility and attract foreign investors, so domestic investors will invest less in large domestic markets. If transaction costs are very high, foreign investors will tend to invest less in local markets, and investments in domestic market by local investors will then be important. Information costs have a significant positive effect on DBIAS; the coefficient is 6.35 with a t-ratio of 4.67. If information costs in foreign market are very important, local investors opt for local markets, and then the domestic bias will be very important.

68

Table 3.7 Regression analysis of domestic bias Predicted sign

Economic development variables Coeff.

Constant

4.00

GDPC RGDP TRADE DI

− − − −

RFLOW RESTRICT

− +

SIZE COST DUMEMERG

− + +

INFOR

+

BEHAV

+

DUMLANG DIST

− +

LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL

− − − − − −

RET2 CORR

− +

Adjusted R2

−0.001 0.03 0.00 −0.97

Capital control

Stock market development

Information costs

Investor behavior

Familiarity

Investor protection

Other variables

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

8.43

7.10

5.16

−5.34

−1.82

2.10

3.47

3.82

9.03

−21.03

−2.10

15.87

6.30

5.36

10.51

28.71

1.83

0.00 −0.01 0.01 −4.63

−0.05 −0.06 1.12 −0.94

0.01 1.14

0.19 0.51

−0.77 2.40 −2.00

−1.97 2.50 −0.87

4.79

1.61

2.72

1.35

2.79 −3.27

0.52 −2.37

1.55 0.01 0.29 −1.94 −0.35 −1.62

2.37 0.23 1.09 −3.29 −0.80 −1.16

−0.02 1.14

−0.56 0.21

−4.15 0.16 0.71 −0.19 −0.34 4.17

−1.97 2.77 −1.20 2.94 −0.70

−3.62 3.55 −0.85 6.35

4.67 4.19

2.61 −5.14 2.94

−2.17 2.60 0.57 −0.03 −0.32 −1.48 −0.01 0.36

1.21 −0.89 −1.19 −2.74 −0.03 0.42 0.05 −11.55

0.35

All variables

0.26

0.58

0.42

0.19

0.17

0.38

0.23

2.17 −2.65

t-stat

0.86

Notes: DBIAS j : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flow restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: average common language dummy variables; DIST : average of log geographical distances; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET2: past 2-year return; CORR: average of return correlations. * Bold numbers, indicate t-stat is significant.

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69

Concerning the investor-protection variables, only the index of risk expropriation is statistically significant at the 5 percent level with a coefficient reaching –1.48. Expropriation risk will have less of an impact on the decisions of local investors to invest in local markets than on foreign investors. As a result, when the expropriation risk is small (lager EXPROP), this will attract relatively more foreign investments to local markets. In this case, the domestic bias will be lower. Furthermore, the result of economic development variables shows that only GDP per capita (GDPC) has a significant negative coefficient. GDP per capita will reflect the development of a country and its financial markets. If GDPC is positive, a country will attract more foreign investors and domestic bias will consequently be less important. The two control variables (RFLOW) and (RESTRICT) are significant at the 5 percent level. The coefficient of RFLOW is positive. In fact, the lower the value of RFLOW, the more important are the restrictions facing the domestic investors in foreign markets and foreign investors in local markets. Then domestic investors will invest more in domestic markets and DBIAS will be important. The relation between DBIAS and RESTRICT is positive and this implies that if the restrictions on foreign investment are important, local investors will invest more in the domestic market. Investors’ behavior also has an impact on domestic bias. Its coefficient is 4.19 and the t-ratio 2.61. The more optimistic investors are about the local market, the more they invest in it; hence the importance of DBIAS. Common language (DUMLAN) and geographical proximity (DIST), which are proxies for familiarity, both have a significant impact on domestic bias. Countries that have the same language as a large number of countries in the world tend to have a smaller domestic bias. Countries that are farther away from the rest of the world have an important domestic bias. When, all explanatory variables are estimated jointly, results show significant coefficients generally for financial market development and for two variables of investor protection (LAW and EXPROP). Other variables, such as a common language, are not significant. Our findings corroborate those of Chan et al. (2005) especially regarding the importance of stock market development in the explanation of the domestic bias.

3.5.2 Results concerning foreign bias Table 3.8 shows regression results for the foreign bias measure, FBIASij . We introduce an additional independent variable, DBIASi , which controls the impact of domestic bias on foreign bias. If investors invest more in a domestic market, the proportion they could invest in other markets will be considerably lower. Then, if the domestic bias is important (DBIASi

Predicted sign

Economic development variables

Capital control

70

Table 3.8 Regression analysis of foreign bias Stock market development

Information costs

Investor behavior

Familiarity

Investor protection

Other variables

All variables

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff. t-stat

Coeff.

t-stat

−2.31 −0.51

−6.72 −12.2

1.01 −0.57

1.03 −13.0

−1.00 −0.18

−5.03 −4.06

−1.27 −0.52

−5.47 −10.9

8.46 −0.43

13.86 −10.6

−5.89 −0.55

−8.58 −12.8

−1.21 −5.05 −0.50 −10.7

15.32 −0.08

6.09 −1.48

0.00 0.05 0.00 −4.42

1.02 0.91 1.58 −2.12

−0.25 1.21

−2.57 2.21

0.30 −0.80 0.43

1.71 −1.87 0.77

Constant DBIAS

−

−2.15 −0.52

−8.36 −12.5

GDPC RGDP TRADE DI

+ + + +

0.001 −0.02 0.00 −0.99

9.12 −0.77 0.16 −0.62

RFLOW RESTRICT

+ −

SIZE COST DUMEMERG

+ − −

INFOR

−

BEHAV

+

DUMLANG TRADEB DIST

+ + −

LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL

+ + + + + +

RET2 CORR Adjusted R2

+ −

0.16 −1.31

4.23 −4.04 0.06 −0.42 −0.88

0.44 −1.55 −3.24 −4.86

−14.84 −2.83

−1.88 0.54 3.71 −1.21

2.89 2.80 −17.50

0.15

0.18

0.24

0.11

0.35

−9.96

−9.09

−4.78

0.86 4.26 −0.75 −0.54 −1.22 −15.98 0.04 0.03 −0.09 0.44 −0.11 −0.38

0.17

−4.92

0.20

0.40 4.03 −1.21 3.24 −1.91 −1.75 −0.02 −5.11 0.85 2.42 0.14

0.00 −0.01 −0.12 0.01 −0.21 0.59

−0.02 −0.80 −1.48 0.06 −3.13 2.21

−0.01 1.19 0.53

−1.49 2.70

Notes: FBIAS ij : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: common language dummy variables between two countries; DIST : log geographical distances between two countries; TRADE: bilateral trade volume between two countries; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET 2: past 2-year return; CORR: return correlations between two countries; DBIAS i : control variable.

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important), foreign bias will be important (more negative FBIASij ). The coefficients of DBIASi are negative and significant for almost all specification models. Among the eight different categories of explanatory variables, familiarity has the greatest influence on foreign bias. All familiarity variables are statistically significant at the 5 percent level. Foreign investors tend to invest in a country that is geographically close to them, that enjoys a large bilateral trade volume, and with which they share a language. The more familiar investors are with a foreign market, the more they invest abroad and the less the foreign bias (FBIAS important). Moreover, investors are less willing to invest in a foreign market that require a high information cost. The coefficient of INFOR is –4.86 and the t-ratio equals –14.84. Results concerning the investor protection variables show that the two variables ACC and EXPROP have a significant effect on foreign asset holdings. Foreign investors prefer markets with more transparency of information (ACC important) and a low expropriation risk. The more investors’ rights are maintained, the smaller the foreign bias. Economic market development (DUMEMERG) and financial market development (GDPC) have a significant impact on foreign bias. Foreign investors tend to invest more in a developed market (DUMEMERG = 0) and in a country with an important GDP per capita. In other words, foreign investors prefer large and developed markets with high levels of transparency. Capital control variables have a strong impact on foreign bias. The coefficient on RFLOW and RESTRICT are, respectively, 0.16 with a t-ratio of 4.23 and –1.31 with a t-ratio of –4.04. This result suggests that a country with fewer restrictions on capital flows (RFLOW important) or/and on foreign asset holding (RESTRICT low) attract more foreign investors, then foreign bias in this country will be less important (FBIAS important). When we regress the foreign bias on all variables, some of the coefficients are no longer statistically significant at the conventional level. Information costs and familiarity variables (except TRADEB) remain statistically significant; yet investor protection variables are insignificant. These results corroborate those found by Chan et al. (2005), in particular regarding the importance effect of familiarity on the foreign bias.

3.6 ADDITIONAL TESTS 3.6.1 Results concerning home bias In addition to domestic and foreign bias measures, we have used another measure of home bias developed by Ahearne et al. (2004) to confirm the preceding results.

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THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

The home bias of country i’s investors against country j (HBIASij ) is share of country j in the portfolio of country i’s investors HBIASij = 1 − (3.14) share of country j in the world market portfolio Regression results of the home bias on eight categories of variables are presented in Table 3.9. Table 3.9 shows that familiarity variables are the most important determinants of home bias. In fact, they present the most important adjusted R2 (0.13). Coefficients are −0.21 for common language (DUMLANG), −1.63 for bilateral trade volume (TRADEB) and 0.31 for geographic proximity (DIST), with t-ratios, respectively, of −2.64, −2.84 and 10.52. Familiarity variables seem to contribute similarly in explaining the foreign bias. Among the remaining factors, only four variables are statistically significant at the conventional level. The first variable is information costs, with a coefficient of 0.52 and a t-ratio of 4.49. Then, if foreign investors are less informed about the local market, they will invest less and the home bias will be very important. The second variable is related to investor protection (ACC). Foreign investors prefer markets with more information transparency (ACC important). The third variable is the restriction on asset holdings by foreign investors (RESTRICT). If foreign investors have this constraint, they will invest less abroad, and home bias will be very important. The last variable is trade volume scaled by GDP (TRADE). Its coefficient is −0.02, with a t-ratio of −2.84. Foreign investors tend to hold assets in a country with very important trade volume scaled by GDP. When all the explanatory variables are estimated jointly, apart from correlation coefficient (CORR), only familiarity variables present a statistically significant coefficient that confirms the hypothesis. Results corroborate those of Ahearne et al. (2004), especially for the impact of information costs on home bias.

3.6.2 Domestic, foreign and home biases according to a world float portfolio Dahlquist et al. (2003) show that the prevalence of closely held firms in most countries helps to explain why these countries exhibit a home bias in equity holdings. Based on their estimates of the percentage of closely held market capitalization, we construct a world float portfolio with country weights based on the free-floating shares available to investors. We calculate the float adjusted domestic bias (DBIAS_FLOAT), foreign bias (FBIAS_FLOAT) and home bias (HBIAS_FLOAT).

Table 3.9 Regression analysis of home bias Predicted sign

Economic development variables Coeff.

Constant GDPC RGDP TRADE DI

− − − −

RFLOW RESTRICT

− +

SIZE COST DUMEMERG

− + +

INFOR

+

BEHAV

−

DUMLANG TRADEB DIST

− − +

LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL

− − − − − −

RET2 CORR

− +

Adjusted R2

Capital control

Stock market development

t-stat

Coeff.

t-stat

Coeff.

t-stat

0.90

13.28

0.71

6.39

0.95

2.28

−0.00 −0.00 −0.02 1.18

−1.75 −0.04 −2.84 1.83 0.00 0.26

Information costs

Investor behavior

Familiarity

Investor protection

Other variables

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

0.55

9.92

0.75

22.50

−1.86

−7.11

1.24

4.65

0.72

18.35

−2.03

−1.38

0.00 −0.01 0.01 1.91

0.93 −0.32 −1.41 1.57

0.09 0.10

1.61 0.31

−0.08 −0.07 −0.05

−0.82 −0.30 −0.17

0.09 2.04 −0.03 −0.06 0.10

−0.55 −0.51 0.81 0.52

4.49 −0.65

−1.06 −0.21 −1.63 0.31

−2.64 −2.84 10.52 −0.02 −0.01 0.04 0.00 0.01 0.03

−0.40 −2.70 1.24 0.04 0.22 0.38 0.00 −1.01

0.01

0.01

0.01

All variables

0.04

0.01

0.13

0.01

0.00

1.46 −0.10

0.25

1.15

−1.92

−1.73

−0.30 −2.64 0.34

−2.52 −3.24 7.65

−0.11 0.00 0.00 0.04 0.00 0.11

−1.58 −0.40 0.08 0.38 −0.03 0.71

0.00 0.61

−0.29 2.39

0.15

73

Notes: HBIAS ij : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: common language dummy variables between two countries; DIST : log geographical distances between two countries; TRADE: bilateral trade volume between two countries; LAW : rule of law index: ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET 2: past 2-year return; CORR: return correlations between two countries.

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THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

Results concerning the float-adjusted domestic bias

Table 3.10 presents the result of the DBIAS_FLOAT estimation. The results of the float-adjusted domestic bias are almost the same as those of the unadjusted domestic bias. Variables that have a significant impact on domestic bias also have a significant effect on DBIAS_FLOAT. Moreover, the predictive power is generally more important for DBIAS_FLOAT than for DBIAS. The results confirm those of Dahlquist et al. (2003) for the impact of corporate governance structure on the home asset bias.

Results concerning the float-adjusted foreign bias

Table 3.11 shows the result of the FBIAS_FLOAT estimation for the eight categories of explanatory variables. The foreign bias calculated on the basis of the world float portfolio seems to be influenced by the same characteristics as the foreign bias. Except bilateral trade volume (TRADEB), all significant variables for the foreign bias are statistically significant at the 5 percent level for the foreign bias float.

Results for the float-adjusted home bias

Results found for the float-adjusted home bias are qualitatively the same as those of the home bias. Information costs and familiarity variables remain statistically significant for the home bias float. See Table 3.12.

3.7 CONCLUSION This chapter presented an analytical study of the bilateral asset holdings of investors from thirty investing countries in forty-three receiving countries. Similarly to Chan et al. (2005), we distinguish between domestic bias (domestic investors overweighting the local markets) and foreign bias (foreign investors under-or overweighting the overseas markets). We find that home bias is a large phenomenon for both developed and developing nations. The results show that stock market development and information costs have an important impact on domestic bias, while information costs and familiarity variables have an important effect on foreign bias. Economic development, capital control and investor protection variables have only a small effect on theses biases. However, investor behavior has a significant impact only on the domestic bias. Additional tests show that, only information costs and familiarity variables have an important effect on home bias. Investment behavior appears to be determined by multiple-factor models.

Table 3.10 Regression analysis of domestic bias based on the world float portfolio Economic development variables Coeff. Constant GDPC RGDP TRADE DI

Capital control

Stock market development

Information costs

Investor behavior

Familiarity

Other variables

All variables

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

7.09

8.34

7.91

4.84

−5.68

−1.68

2.06

2.91

3.99

8.47

−26.95

−2.38

18.15

6.35

5.73

9.72

25.75

2.08

0.01 0.05 0.01 −0.86

−4.49 0.28 0.76 −0.15

0.00 0.04 0.02 −7.14

−0.15 0.23 2.16 −1.18

0.15 1.00

0.60 0.57

−1.04 2.29 −1.33

−3.32 3.03 −0.73

−0.39 4.56

RFLOW RESTRICT

−1.98 2.54 −1.37 3.15 −0.46

SIZE COST DUMEMERG

−3.59 3.30 −0.49 7.41

INFOR

4.67 5.12

BEHAV

2.86 −5.76 3.65

DUMLANG DIST

−2.15 2.85 0.62 −0.04 −0.39 −1.68 −0.01 0.60

LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL RET2 CORR Adjusted R2

Investor protection

0.35

0.24

0.59

0.42

0.22

0.20

0.40

1.16 −0.95 −1.28 −2.74 −0.03 0.63 0.05 −12.34 0.22

2.15 −2.45

t-stat

4.95

2.11

3.80

2.39

5.03 −3.13

1.20 −2.29

1.71 0.02 0.34 −2.02 −0.40 −2.40

3.34 0.46 1.64 −4.36 −1.17 −2.18

−0.03 2.09 0.93

−0.99 0.49

75

Notes: DBIAS_FLOAT j : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: average common language dummy variables; DIST : average of log geographical distances; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET2: past 2-year return; CORR: average of return correlations.

Table 3.11 Regression analysis of foreign bias based on the world float portfolio 76

Economic development variables

Capital control

Stock market development

Information costs

Investor behavior

Familiarity

Investor protection

Other variables

All variables

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Constant DBIASFLOAT

−1.76 −0.43

−7.72 −13.6

−1.78 −0.42

−5.61 −13.5

0.46 −0.46

0.48 −14.0

−0.59 −0.22

−3.49 −6.6

−1.19 −0.43

−6.23 −11.8

8.12 −0.35

13.74 −11.6

−4.61 −0.44

−6.90 −13.8

−1.13 −0.42

−5.68 −11.7

13.34 −0.09

5.33 −2.1

GDPC RGDP TRADE DI

0.00 −0.03 0.00 −0.74

6.02 −0.89 0.47 −0.47

0.00 0.04 0.01 −4.45

0.54 0.76 2.66 −2.15

−0.20 1.29

−2.11 2.37

0.17 −0.76 0.93

0.95 −1.79 1.68

−4.78

−10.23

−9.16

−4.84

0.88 −0.95 −1.21

4.37 −0.68 −15.90

−0.01 −0.01 −0.02 0.23 −0.26 0.30

−0.04 −0.86 −0.25 1.23 −3.88 1.13

−0.01 1.19

−0.92 2.72

RFLOW RESTRICT

0.10 −1.16

2.66 −3.64 −0.08 −0.26 −0.69

SIZE COST DUMEMERG

−0.59 −0.98 −2.56 −4.16

INFOR

−13.13 −2.21

BEHAV

−1.50 0.51 2.41 −1.14

DUMLANG TRADEB DIST

2.82 1.85 −16.88 0.09 0.03 −0.10 0.31 −0.17 −0.19

LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL

−0.01 0.68

RET2 CORR Adjusted R2

0.96 3.75 −1.38 2.37 −2.99 −0.91

0.15

0.15

0.17

0.23

0.13

0.35

0.19

0.15

−3.77 1.99

0.53

Notes: FBIAS_FLOAT ij : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: common language dummy variables between two countries; DIST : log geographical distances between two countries; TRADE: bilateral trade volume between two countries; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET2: past 2-year return; CORR: return correlations between two countries; FLOAT _DBIAS j : control variable.

Table 3.12 Regression analysis of home bias based on the world float portfolio Economic development variables Coeff.

t-stat

Constant

0.78

7.02

GDPC RGDP TRADE DI

0.00 0.00 0.00 2.07

0.01 −0.04 −3.25 1.95

RFLOW RESTRICT

Capital control

Stock market development

Coeff.

t-stat

Coeff.

t-stat

0.42

2.26

1.21

1.76

0.02 0.42

Investor behavior

Familiarity

Investor protection

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

Coeff.

t-stat

0.37

4.01

0.62

10.92

−3.05

−7.06

1.04

2.39

0.66

INFOR

t-stat

Coeff.

t-stat

0.58

8.64

−3.14

−1.27

0.00 −0.03 −0.01 3.34

0.65 −0.55 −1.96 1.63

0.12 0.29

1.22 0.53

−0.04 −0.06 −0.40

−0.22 −0.14 −0.73

3.44 −0.93

−0.90 −0.35 −1.84 0.44

DUMLANG TRADEB DIST

−2.64 −1.97 8.98 −0.03 −0.01 0.06 0.01 0.04 −0.05

LAW ACC MINORITY EXPROP EFFICIENCY DUMLEGAL

−0.51 −2.09 1.20 0.11 1.06 −0.37 0.00 0.11

RET2 CORR 0.01

0.01

0.01

0.02

0.01

0.10

0.01

All variables

Coeff.

0.20 −0.86 0.43

BEHAV

Adjusted R2

Other variables

0.99 1.96 0.02 −0.17 0.09

SIZE COST DUMEMERG

Information costs

0.00

0.98 0.46

0.28

0.78

−2.73

−1.45

−0.49 −3.70 0.51

−2.46 −2.69 6.78

−0.16 0.00 −0.02 0.00 0.03 0.15

−1.38 −0.21 −0.24 0.03 0.52 0.56

0.00 0.91

−0.37 2.12

0.12

77

Notes: HBIAS_FLOAT ij : dependent variable; GDPC: log GDP per capita; RGDP: real GDP growth; TRADE: trade volume scaled by GDP; DI: foreign direct investment scaled by GDP; RFLOW : capital flows restrictions; RESTRICT : foreign asset holdings restrictions; SIZE: stock market capitalization scaled by GDP; COST : log transaction costs; DUMEMERG: emerging market dummy variable; INFOR: information cost; BEHAV : investor behavior; DUMLANG: common language dummy variables between two countries; DIST : log geographical distances between two countries; TRADE: bilateral trade volume between two countries; LAW : rule of law index; ACC: accounting standard index; MINORITY : minority investor protection index; EXPROP: risk of expropriation index; EFFICIENCY : efficiency of judicial system index; LEGAL: legal system dummy variable; RET2: past 2-year return; CORR: return correlations between two countries.

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THE DETERMINANTS OF DOMESTIC AND FOREIGN BIASES

Our results corroborate those of Chan (2005), namely about the effect of familiarity variables on domestic and foreign biases as well as the conclusion reached by Ahearne et al. (2004) as to the impact of information costs on the home bias.

NOTES 1. See Lewis (1999) and Karolyi and Stulz (2003) for a survey of literature on the home bias. 2. http://www.imf.org/external/np/sta/pi/datarsl.htm. 3. http://en.wikipedia.org/wki/developed_country. 4. For variable groups (vi), (vii) and (viii), average values are calculated for each investing country face other receiving countries. Average value of TRADEB (see page 000) is not presented with other familiarity variables because it is equal to 1/43 for each observation. 5. We thank Jack Glen for providing us with data. 6. Source: http://www.freetheword.com. 7. The data for the year 1999 are used by Bartram and Dufey (2001). 8. http://www.phone-rate-calculator.com. 9. In this chapter we use the term “familiarity” broadly to capture the effects of both asymmetric information and psychological factors. 10. Data are used by Chan et al. (2005).

REFERENCES Ahearne, A., Griever W. and Warnock F. (2004) “Information Costs and Home Bias: An Analysis of US Holding of Foreign Equities”, Journal of International Economics, 62(2): 313–36. Bartram, S. M. and Dufey, G. (2001) “International Portfolio Investment: Theory, Evidence and Institutional Framework”, Financial Market Institutions & Instruments, 10(3): 85–155. Black, F. (1974) “International Capital Market Equilibrium with Investment Barriers”, Journal of Financial Economics, 1(4): 337–52. Bohn, H. and Tesar, L. (1996) “US Equity Investment in Foreign Markets: Portfolio Rebalancing or Return Chasing”, American Economic Review, 86(2): 77–81. Carmichael, B. and Coen, A. (2003) “International Portfolio Choice in an Overlapping Generations Model with Transaction Costs”, Economic Letters, 80(2): 269–75. Central Intelligence agency (2001) The world factbook (Potomac Books). Chan, K., Covrig, V. and Ng, K. (2005) “What Determines the Domestic Bias and Foreign Bias? Evidence from Mutual Fund Equity Allocations Worldwide”, Journal of Finance, 60(3): 1495–534. Cooper, I. and Kaplanis, E. (1986) “Costs to Cross Border Investment and International Equity Market Equilibrium”, in J. Edwards, J. Franks, C. Mayer and S. Schaefer (eds) Recent Developments in Corporate Finance (Cambridge University Press). Dahlquist, M., Pinkowitz, L., Stulz, M. and Williamson, R. (2003) “Corporate Governance and the Home Bias”, Journal of Financial and Quantitative Analysis, 38(1): 87–110. Edison, H. and Warnock, F. (2003) “A Simple Measure of the Intensity of Capital Controls”, Journal of Empirical Finance, 10(1–2): 81–103. Faruqee, H., Li Sh. and Yan I. (2004) “The Determinants of International Portfolio Holdings and Home Bias”, Working Paper, WPL04/34.

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French, K. and Poterba, J. (1991) “Investor Diversification and International Equity Markets”, American Economic Review, 81(2): 222–26. Gehrig, T. (1993) “An Information Based Explanation of the Domestic Bias in International Equity Investment”, Scandinavian Journal of Economics, 95(1): 97–109. Giannetti, M. and Koskinen, Y. (2004) “Investor Protection and the Demand for Equity”, SSE/EFI Working Paper, Series in Economics and Finance, n.526. Glassman, D. A. and Riddick, L. A. (1996) “Why Empirical International Portfolio Models Fail: Evidence that Model Misspecification Creates Home Bias”, Journal of International Money and Finance, 15(2): 275–312. Huberman, G. (2001) “Familiarity Breeds Investment”, Review of Financial Studies, 14(3): 659–80. Karolyi, A. and Stulz, R. (2003) “Are Financial Assets Priced Locally or Globally?”, in G. Constantinides, M. Harris, and R. M. Stulz (eds), Handbook of the Economics of Finance, (Amsterdam: North-Holland). Kilka, M. and Weber, M. (2000) “Home Bias in International Stock Return Expectations”, Journal of Psychology and Financial Markets, 1(3–4): 176–92. La Porta, R., Lopez-De-Silanes, F., Shleifer, A. and Vishny, R. (1997) “Legal Determinants of External Finance”, Journal of Finance, 52(3): 1131–51. La Porta, R., Lopez-De-Silanes, F., Shleifer, A. and Vishny, R. (1998) “Law and Finance”, Journal of Political Economy, 106(6): 1113–55. La Porta, R., Lopez-De-Silanes, F., Shleifer, A. and Vishny, R. (2000) “Investor Protection and Corporate Governance”, Journal of Financial Economic, 58(1–2): 3–27. Lewis, K. (1999) “Trying to Explain Home Bias in Equities and Consumption”, Journal of Economic Literature, 37(2): 571–608. Lintner, J. (1965) “The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolio and Capital Budgets”, Review of Economics and Statistics, 47(1): 13–37. Portes, R. and Rey, H. (2005) “The Determinants of Cross-Border Equity Flows”, Journal of International Economics, 65(2): 269–96. Sarkissan, S. and Schill, M. (2004) “The Overseas Listing Decision: New Evidence of Proximity Preference”, Review of Financial Studies, 17(3): 769–809. Sharpe, W. F. (1964) “Capital Asset Prices: A Theory of Market Equilibrium under the Condition of Risk”, Journal of Finance, 19(3): 425–42. Strong, N. and Xu, X. (2003) “Understanding the Equity Home Bias: Evidence from Survey Data”, Review of Economics and Statistics, 85(2): 307–12. Stulz, R. M. (1981) “On the Effect of Barriers to International Investment”, Journal of Finance, 36(4): 923–34. Tesar, L. and Werner, I. (1995) “Home Bias and High Turnover”, Journal of International Money and Finance, 14(4): 467–92. Zhou, C. (1998) “Dynamic Portfolio Choice and Asset Pricing with Differential Information”, Journal of Economic Dynamic and Control, 22(7): 1027–51.

CHAPTER 4

The Critical Line Algorithm for UPM–LPM Parametric General Asset Allocation Problem with Allocation Boundaries and Linear Constraints Denisa Cumova, David Moreno and David Nawrocki

4.1 INTRODUCTION Assume that there are information costs and asymmetric information in the marketplace. These conditions have been associated with segmented markets, with investors having a preferred habitat. Therefore, investors will be searching for local minima and maxima in their preferred habitat rather than a global market optimization based on equilibrium market asset pricing. Investors will have unique utility functions depending on their preferred habitat, and attempt to maximize their utility through a localized solution. This idea is not new as it traces back to Simon (1955), who proposed satisficing investors who restrict their searches to localized searches with rationality bounded by the area of the search. Cyert and March (1963) followed with their behavioral theory which suggests that decision-makers 80

DENISA CUMOVA, DAVID MORENO AND DAVID NAWROCKI

81

break complex problems into a sequence of simpler problems, which they solve sequentially using local rationality. This is a key component of the Kahneman and Tversky (1979) prospect theory, where investors set up multiple mental accounts and optimize each account. Under these conditions, the traditional Markowitz (1959) portfolio theory may be used to optimize within a preferred habitat or mental account. This leads to the question of whether the quadratic utility function of traditional mean-variance analysis is the appropriate utility model for portfolio analysis. There is much evidence indicating that investors are more sensitive to losses than gains.1 This introduces a discontinuous change in the shape of the investor’s utility function at some target return and plays a role in prospect theory, developed by Kahneman and Tversky (1979) and Tversky and Kahneman (1991). However, there is evidence that investors are not risk-averse throughout the range of returns, and will exhibit risk-seeking behavior in special situations. Friedman and Savage (1948) and Markowitz (1952) argue that willingness to purchase both insurance and lottery tickets implies reverse S-shaped (both concave and convex) utility functions. A reverse S-shaped utility function provides an explanation for investors engaging in risk-averse behavior for losses and risk-seeking behavior for gains. Fishburn (1977) proposed the lower partial moment (LPM) a, τ model to explain risk-seeking and risk-averse behavior below a target return (τ). Investor behavior is explained through a coefficient (a) as a < 1 is risk-seeking behavior and a > 1 is risk-averse behavior, thus the LPM (a, τ) model. The LPM (a, τ) model proved to be a very useful risk measure because of its flexibility in capturing investor behavior (Nawrocki, 1999). However, it was not immune to criticism. Kaplan and Siegel (1994a, 1994b) zeroed in on its characteristic of a linear utility function above the target return, which assumes that the investor is risk-neutral to all above-target returns. A recent paper by Post and van Vliet (2002) found evidence that, while investors are risk-averse to below-target returns, they are risk-seeking above the target return. In order to apply more realistic behavior to above-target returns, Sortino et al. (1999) proposed a performance measure, the upper partial moment/lower partial moment (UPM/LPM) ratio. Given the potential usefulness of the UPM/LPM model, we develop a critical line UPM/LPM portfolio optimization algorithm (CLA) with bounded investment constraints for investors to generate optimal solutions within separate mental accounts or preferred habitats. This algorithm is important as it allows us to study the behavior of UPM/LPM portfolios in greater detail. Section 4.2 of the chapter derives the UPM/LPM CLA and provides a proof that it is consistent with Kuhn–Tucker conditions. Section 4.3 presents a short empirical test to demonstrate that the resulting algorithm does work relative to the traditional mean-variance (EV) algorithm, and section 4.4 offers a summary and conclusions.

82

CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES

4.2 THE UPSIDE POTENTIAL–DOWNSIDE RISK PORTFOLIO MODEL The UPM is also known as the upside potential, and the LPM is a family of downside risk measures. Therefore, the upside potential–downside risk (UPM/LPM) model may be formulated as follows: Maximize E(UPMp ) =

K

pt [max{0, E(Rpt ) − τ}]c

t=1

Minimize E(LPMp ) =

K

(4.1)

pt [max{0, τ − E(Rpt )}]a

t=1

subject to n

wi = 1

(4.2)

i=1

Although Markowitz (1959) developed the CLA for mean-variance optimization, this algorithm is not exclusive to the mean/variance problem; it can also be constructed for other risk–return portfolio models. In the next section, we derive the CLA for general asset allocation problem with bounds and linear equality constraints for the UPM–LPM portfolio model. Let r = (r1 , r2 , . . . rn )T be a vector of asset returns and x = (x1 , x2 , . . . xn ) a related vector of investment. It is assumed that x varies in the compact and convex set S. General asset allocation problems in the UPM–LPM framework can be formulated as looking for a legitimate investment vector x = (x1 , x2 , . . . xn ) with minimal downside risk for a given portfolio upper partial moment (b0 ). The investment vector x is legitimate whenever it fulfills the constraints. Therefore, the general optimization problem may be stated as Select x = (x1 , x2 , . . . . xn ) for which min E(LPMp ) =

n n

xi xj CLPMij

i=1 j=1

=

n i=1

xi2 LPMi +

n i =j

xi xj CLPMij

(4.3)

DENISA CUMOVA, DAVID MORENO AND DAVID NAWROCKI

83

subject to n n

E(UPMp ) =

xi xj CUPMij =

i=1 j=1

n i=1

xi2 UPMi +

a11 · x1 + a12 · x2 + · · · + a1n · xn = b1 a21 · x1 + a22 · x2 + · · · + a2n · xn .. .

n i=j

xi xj CUPMij

= b2 .. .

am1 · x1 + am2 · x2 + · · · + amn · xn = bm 0 ≤ x ≤ 1 ∀i = 1 . . . n where LPMi =

T

pt · [Max{0; (τ − rit )}]a

t=1

CLPMij =

T

pt · [Max{0; (τ − rit )}]a−1 (τ−rjt )

t=1

UPMi =

T

pt · [Max{0; (rit − τ)}]c

(4.4)

t=1

CUPMij =

T

pt · [Max{0, (rit − τ)}]c−1 (rjt − τ)

t=1

LPMi = CLPMij

for

∀i = j

UPMi = CUPMij

for

∀i = j

In matrix form, this optimization problem can be formulated as (a) min x

LPMp = xT · L · x

(b) UPMp = xT · U · x (c) A · x = b (d) 0 ≤ x with x = (x1 , x2 , . . . xn )T ⎞ ⎛ CLPM11 · · · CLPM1n ⎟ ⎜ .. .. .. L=⎝ ⎠ . . . CLPMn1 · · · CLPMnn

(4.5)

84

CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES

⎞ CUPM11 · · · CUPM1n ⎟ ⎜ .. .. .. U=⎝ ⎠ . . . CUPMn1 · · · CUPMnn ⎛

⎛

⎞ a11 · · · a1n ⎜ ⎟ A = ⎝ ... . . . ... ⎠ am1 · · · amn ⎛

⎞ b1 ⎜ ⎟ b = ⎝ ... ⎠ bm First, we have to prove whether this formulation is convex, as the Kuhn– Tucker conditions for finding a global optimum can only be applied in this case. The objective function E(LPMp ) is convex for all a ≥ 1 in bounded x ∈ S if and only if it is positive semi-definite for all bounded x ∈ S. The E(LPMp ) is a positive variable for all bounded x by a ≥ 1, which implies that E(LPMp ) (and the associated matrix L) is positive semi-definite. E(LPMp ) ≥ 02 for all bounded x by a ≥ 1 assures construction of LPM3 taking positive value or zero. Thus, we can state that the objective function E(LPMp ) is convex for all a ≥ 1 in bounded x ∈ S. With the exception of E(UPMp ) all constraints are linear, and hence convex. As the quadratic function E(UPMp ) ≥ 04 is bounded for all x by c ≥ 1, convexity is assured by the formulation of UPM taking positive values or zero. Therefore, E(UPMp ) (and the associated matrix U) is positive semi-definite, and therefore convex. Kuhn–Tucker conditions are based on Lagrangian multipliers. In this case, the Lagrangian function is5

1 1 T x · U · x − UPMp L = xT · L · x − λ · (A · X − b) − λu · 2 2 where λ = {λ1 , λ2 , . . . , λm } ∈ m and λu denote Lagrangian multipliers for constraints (Equations 4.5b and 4.5c). Using the matrix form, the Kuhn– Tucker conditions are now constructed.

4.2.1 Kuhn–Tucker conditions Equation (4.5) represents a convex quadratic minimization problem with convex constraints.6 The necessary and sufficient conditions for x to be a global optimum is that x fulfills the Kuhn–Tucker conditions.7

DENISA CUMOVA, DAVID MORENO AND DAVID NAWROCKI

85

Let η represent the vector of partial derivatives of the Lagrangian L with respect to the n decision variables xi (0 ≤ i ≤ 1):

∂L ∂L ∂L , ,..., η = (η1 , η2 , . . . , ηn ) = ∂x1 ∂x2 ∂xn The Kuhn–Tucker conditions for (4.5) are formulated as (a) η = L · x − λ · A − λu · (U · x) ≥ 0 (b) x ≥ 0, λ ≥ 0 (c)

∀ η · x = 0 ⇔ (η > 0 ∧ x = 0)

1≤i≤n

and

(η = 0 ∧ x > 0)

(4.6)

(d) A · x = b (e) xT · U · x = UPMp Similar to Markowitz’s CLA condition (4.6c) implies that the partial derivative ηi of L with respect to xi equals zero if and only if xi is greater than zero – that is, if asset i is included in the base solution. This is the optimality condition. When xi equals zero, then the respective partial derivative ηi is positive. Equations (4.6a) and (4.6d) can be summarized, then Equation (4.6) can be rearranged in Equation (4.7), where it is assumed that all partial derivatives are equal to zero:

X U 0 X 0 L A · − λu · = (a) A 0 −λ 0 0 −λ b (b) x ≥ 0, λ ≥ 0 (c)

∀

1≤i≤n

η > 0 ⇔ x = 0 and η = 0 ⇔ x > 0

(4.7)

(d) η = L · x − λ · A − λu · (U · x) ≥ 0 (e) xT · U · x = UPMp Equation (4.7d) is added to ascertain that (4.6a) remains satisfied. Each xi > 0 represents a base or “IN” variable. Each xi = 0 is a non-base or “OUT” variable. The first-order condition (Lagrangean function equals zero) remains appropriate for the variables that are in the solution, as their bounds are not binding and hence could have been omitted entirely (at least for the risk tolerance being examined). Assets xi = 0 will not have any impact on portfolio expected LPM, and therefore, a particular IN-set in the L matrix can replace its ith row by identity vector ei , for every i, which is not in the IN-set. Vector ei has a 1 in its ith position and zero in other positions. Let this matrix be L. Let U be the U matrix with its ith row replaced by 0 vector, for every i,

86

CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES

which is not in the IN-set. Next, the A matrix is defined as A matrix with the ith row replaced by a 0 vector, for every i not in the IN-set. In vector 0 , the ith zero position should be replaced with a downside boundary b for x in the OUT-position. But this downside boundary is zero (xi = 0), so

0 the vector does not change. Thus, it holds true that b

0 x x U 0 L A − λ = · · u b −λ −λ 0 0 A 0

L A

A 0

− λu

L − λu · U A

A 0

U 0

0 0

0 x = · b −λ

(4.8)

0 x = · b −λ

For example, if we have three assets and the second is IN, the first and the third are OUT, and the A · X = b constraint is simply x1 + x2 + x3 = 1, we shall have ⎞ ⎡⎛ 1 0 0 0 ⎢⎜ CLPM21 CLPM22 CLPM23 1 ⎟ ⎟ ⎢⎜ ⎣⎝ 0 0 1 0⎠ 1 1 1 0 ⎛ ⎞⎤ ⎛ ⎞ ⎛ ⎞ 0 0 0 0 x1 0 ⎜ CUPM21 CUPM22 CUPM23 0 ⎟⎥ ⎜ x2 ⎟ ⎜ 0 ⎟ ⎟⎥ · ⎜ ⎟=⎜ ⎟ − λu ⎜ ⎝ 0 0 0 0 ⎠⎦ ⎝ x3 ⎠ ⎝ 0 ⎠ 0 0 0 0 −λ 1 The matrix with the portfolio fractions emerges from

x −λ

=

L − λu · U A

A 0

−1

0 · b

Rearranging the inverse matrix gives

−1 0 0 A x = · −1 −1 −1 b −λ (A ) −(A ) · (L − λu · U) · (A ) After multiplying, we obtain

−1 A ·b x = −1 −1 −λ −(A ) · (L − λu · U) · (A ) · b

(4.9)

DENISA CUMOVA, DAVID MORENO AND DAVID NAWROCKI

87

It would be useful to divide this equation with and without λu :

−1 0 x A ·b = + λu −1 −1 −1 −1 −λ (A ) · U · (A ) · b −(A ) · L · (A ) · b Then, we can simply write

x = + λu · θ −λ

(4.10)

In addition to Equation (4.10), Equation (4.7d) has to be satisfied, thus

x η = ((L − λu · U) A ) · ≥0 −λ

x Substituting matrix gives −λ η = ((L − λu · U) A ) ·

−(A

−1 )

· L · (A

−1

A

−1

·b

) · b + λu · (A

−1 )

· U · (A

−1

)·b

≥0

Again, it would be useful to divide this equation with and without λu , and then simplify the denotation η=L·A

−1

· b − λu · U · A

+ λu · A · (A

−1

−1

) · U · (A

· b − A · (A

−1

)≥0

−1

) · L · (A

−1

)·b (4.11)

η = ϑ + λu · ϑ ≥ 0 where Equation (4.8) forces η = 0 + λu · 0

for

i ∈ IN-set.

Equations (4.10) and (4.11) provide the Kuhn–Tucker conditions (6a, 6b, 6d) expressed as linear functions of λu in the same way as the CLA for M–V. These linear functions are known as “critical lines”. Thus it is easy to compute efficient segments and corner portfolios in similar way as with the CLA for M–V. Portfolios with the same structure are defined as portfolios with the same i-assets in the “IN”-set. As we change the value of the λu -parameter expressing risk tolerance, the portfolio structure does not change with the value of λu , but remains the same for a certain interval. Portfolios, where the portfolio structure changes, are called “corner” portfolios by Markowitz (1959). These corner portfolios divide the λu (efficient frontier) into piecewise intervals, for which piecewise critical lines will be calculated. Hence, the efficient frontier is said to be segmented.

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CRITICAL LINE ALGORITHM WITH ALLOCATION BOUNDARIES

4.2.2 Efficient segments on the efficient frontier Each efficient segment is determined by the interval λlow ≤ λE ≤ λhi , with which Equation (4.10) and (4.11) are satisfied. For any critical line, the interval boundaries are defined as λhi = min(λc , λd ) λlow = max(λa , λb , 0) where

λa =

βi >0

−∞

λb =

δi >0

λc =

for

i = 1, · · · , n

if υi ≤ 0

for

i = 1, · · · , n

for

i = 1, · · · , n

for

i = 1, · · · , n

max −i /θi βi 9%

9% to 3%

3% to −3%

−3% to −9%

0 r>1

0.133768 0.018052

56.15137∗∗ 6.321137

AOI Day Close, SPI Night Open

r=0 r≤1

r>0 r>1

0.15986 0.017473

66.55955∗∗ 6.116847

SPI Day Open, AOI Day Open

r=0 r≤1

r>0 r>1

0.149069 0.017637

62.18897∗∗ 6.174793

AOI Day Close, SPI Day Close

r=0 r≤1

r>0 r>1

0.162063 0.017576

67.50692∗∗ 6.153183

AOI Day Close, SPI Day Open

r=0 r≤1

r>0 r>1

0.132571 0.01729

55.40282∗∗ 6.051996

Note: ∗∗ denotes rejection at 1% significant level.

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Table 9.9 Johansen cointegration test log results for the post-automation period Post-automation variables

H0

H1

Eigenvalues

Likelihood ratio

SPI Night Close, AOI Day Open

r=0 r≤1

r>0 r>1

0.171061 0.008075

66.54364** 2.756496

AOI Day Close, SPI Night Open

r=0 r≤1

r>0 r>1

0.102473 0.010521

40.47313** 3.60673

SPI Day Open, AOI Day Open

r=0 r≤1

r>0 r>1

0.177129 0.09194

69.62966** 3.149613

AOI Day Close, SPI Day Close

r=0 r≤1

r>0 r>1

0.076209 0.007512

29.6022** 2.571375

AOI Day Close, SPI Day Open

r=0 r≤1

r>0 r>1

0.087967 0.00663

33.56882** 2.261839

Note: ** denotes rejection at 1% significant level.

1999) that the behavior of index prices is consistent with the semi-strong form of the efficient market hypothesis. The results shown in Tables 9.8 and 9.9 provide evidence that the spot and futures prices are in a cointegrated equilibrium relationship. However, no previous empirical evidence has been provided to support the findings in this study for night open and close prices for a cointegrated relationship. Overall, results using a cointegration approach provide evidence supportive of a semi-strong form of the efficient market hypothesis.

Granger causality tests

As discussed earlier, the price discovery relationship is centered around the lead–lag behavior of the ASX All Ordinaries Index and the SFE SPI futures contracts. The causal relationship was examined using the Granger causality test to detect the direction of information flow as reflected in price change. The results for the Granger causality performed on the day-traded indexes are seen in Panel A in Table 9.10, and the night-traded indexes (SPI) are shown in Panel B. The causal relationships are tested on the open and close prices for the SPI futures and the All Ordinaries Index. As the unit root test results shown in Table 9.3 show that the variables are integrated of order one, I (1), the first differenced variables are used in the causality tests. It is observed from Panel A in Table 9.10 that the day traded All Ordinaries open value (day AOI open value) does cause the share price index (day open value), while the share price index (day close value) does cause the All Ordinaries Index (day close value). Causality seems to run both ways

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Table 9.10 Granger causality test results Null Hypothesis

Panel A: Day traded series SPI Day Open, SPI day open does not Granger AOI Open cause AOI open AOI open does not Granger cause SPI day open AOI Close, SPI Day Close

AOI close does not Granger cause SPI day close SPI day close does not Granger cause index close

Panel B: Night traded series AOI Close, AOI close does not Granger SPI Night Open cause SPI night open SPI night open does not Granger cause AOI close SPI Night Close, SPI night close does not Granger AOI Open cause AOI open AOI open does not Granger cause SPI night close AOI Close, AOI close does not Granger SPI Night Close cause SPI night close SPI night close does not Granger cause AOI close

Pre-automation Post-automation F-stat F-stat 44.6516∗∗∗ 0.13576

4.30455∗∗∗ 0.36373

1.94637 0.37811

2.48783∗ 1.58893

1402.57∗∗∗ 0.72121

0.59129 59.8435∗∗∗

33.2951∗∗∗ 0.82011

123.388*** 1.09953

5.46812∗∗∗ 0.38735

22.0058∗∗∗ 1.99863

Note: ∗∗∗ (∗ ) denotes significance at the 1% (10%) level.

(bi-directional causality) from/to the All Ordinaries Index (AOI) to/from share price index futures in the day trades. The night trades of series are shown in Panel B of Table 9.10. The share price index (SPI) night close causes the All Ordinaries Index day close value in the post-automation period, whereas the All Ordinaries Index (AOI) day close causes the share price index (SPI) night close in the pre-automation period. Thus causality seems to run both ways (bi-directional causality) from/to the All Ordinaries Index (AOI) to/from night traded share price index futures in the day trades in the night traded period. Comparing the pre-automation and the post-automation periods, the results suggest that, since automation, the structural alignment between the night traded futures and the day traded futures seems to have created a synergy from the 24-hour trading of the SPI futures contract. This can be seen in Table 9.10 (Panel B), with the shift to the SPI night open causing the AOI close in the “post-” period. The results broadly show bi-directional results over the 24-hour period before and after the introduction of electronic trading at the SFE. Thus Research Proposition 4 is supported in that

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the lead–lag relationship remains unchanged (still bi-directional) after the start of electronic trading at the SFE.

9.6 CONCLUSION This study examined whether the Sydney Futures Exchange has benefited from the introduction of electronic trading. Under the first research proposition, empirical tests were carried out on the liquidity of the Sydney Futures Exchange with the analysis of the “at the money” (ATM) SPI call options, and SPI futures contracts. This provided an insight to the change of liquidity between the two derivative markets. By classifying the option contracts by date to expiry and the closeness of the strike price to the spot price, the ATM volume was analyzed along with the SPI futures against the All Ordinaries Index (AOI). The tests were run by classifying the volumes into size groups, following the assumptions of the mixture of distributions hypotheses so as to provide relative levels of market volatility against which to compare the liquidity ratios. The results show that the “at the money” SPI options were more liquid in times of high volatility after the SFE became automated. The SPI futures are less liquid in times of medium to low market volatility. This overall result supports Research Proposition 1, that the SFE’s liquidity has changed with the introduction of electronic trading. Therefore it can be concluded that the liquidity of the Sydney Futures Exchange seems to have increased the operational efficiency within the SPI call options market, while there seems to have been a decline in the operational efficiency of SPI futures market. The importance of the analysis of liquidity in this study is that it was able to account for times of high volatility, such as the technology crash at the beginning of 2000. This was shown clearly by segmenting the option and futures market responses to the differing levels of market volatilities. The examination of the price discovery process was incorporated into the last three research propositions. Research Propositions 2 and 3 were used to test semi-strong form market efficiency. Under this assumption, the trading prices in the Australian Stock Exchange (ASX) and Sydney Futures Exchange (SFE) should have a long-run cointegrating relationship. The results confirm the belief that ASX and the SFE are semi-strong efficient. The existence of cointegration between the two markets before and after the introduction of electronic trading supported the semi-strong market efficiency. The presence of a bi-directional lead–lag relationship between the SPI futures price and the All Ordinaries Index price before and after the introduction of electronic trading supported the fourth research proposition. The automation of the SFE did alter the price discovery process. First, it appeared to synergize the night traded market. This is most probably a

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result from the SFE day and night structural alignment, of both instruments being traded on the same system. However, market leads and lags were bi-directional both before and after automation. This suggests that the electronic trading structure does not greatly enhance the price discovery price of the SFE. If it did, this would be observed in the SPI futures leading the AOI. Therefore, it can be concluded that a change in the liquidity is evident in the SPI futures and SPI call option contracts, but the price discovery process does not appear to have been enhanced by the automation of the Sydney Futures Exchange in the early stages up to August 2000.

APPENDIX Ratio variables Ratio_A =

Day_ATM_Volume Day_Index_Volume

Ratio_C =

Day_Futures_Volume Day_Index_Volume

Ratio_D =

Night_Futures_Volume Day_Index_Volume

Ratio_E =

Day_ATM_Volume Day_Futures_Volume

Day

= Traded during the day trading session

Night = Traded during the night trading session Futures = Share price index futures contract ATM

= “At the money” SPI call option contracts

Index = All Ordinaries Index

NOTES 1. Massimb and Phelps (1994) defined operational efficiency as the market’s ability to lower the costs of trading, and its execution speed of orders between buyer and seller. See Ates and Wang (2005) for the latest evidence on operational efficiency in the US futures market. 2. Informational efficiency means that all traders have equal access to all public information, and that the information is quickly reflected in trading prices (Tsang, 1999). 3. Overstatement of the lead–lag relationship will be discussed later in the chapter.

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4. Fama (1970) has identified three levels of market efficiency. The semi-strong form of the “efficient market hypothesis” states that prices reflect all publicly available information. 5. Market efficiency is discussed in detail in the following section. 6. Cost-of-carry is the cost involved in storing an asset and the interest lost on funds tied up therein. 7. A moneyness portfolio denotes the categorization of strike prices relative to the spot price, so as to create portfolios reflecting the moneyness at that point in time (Rubinstein, 1994). 8. A brief discussion on the mixture of distribution hypothesis was given earlier. 9. Variables definitions are given in the footnotes to Table 9.4.

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Epps, T. W. and Epps, M. L. (1976) “The Stochastic Dependence of Security Price Change and Transaction Volumes: Implications for the Mixture-of-Distributions Hypothesis”, Econometrica, 44(2): 305–21. Fama, E. F. (1970) “Efficient Capital Markets: A Review of Theory and Empirical Work”, Journal of Finance, 25(2): 383–417. Fama, E. F. (1991) “Efficient Capital Markets”, Journal of Finance, 46(5): 1575–617. Franke, G. and Hess, D. (1995) “Anonymous Electronic Trading Versus Floor Trading”, Working Paper, Series II, no. 285, Universitat Konstanz. Freund, W. C., Larrain, M. and Pagano, M. S. (1997) “Market Efficiency Before and After the Introduction of Electronic Trading at the Toronto Stock Exchange”, Review of Financial Economics, 6(1): 29–56. Frino, A., Bortoli, L., and Jarnecic, E. (2004), “Differences in the Cost of Trade Execution Serouson Floor-based and Electronic Future, Markets”, Journal of Financial Services Research, 26(1): 73–87. Frino, A. I. and Jarnecic, E. (2000) “An Empirical Analysis of the Supply of Liquidity by Locals in Futures Markets: Evidence from the Sydney Futures Exchange”, Pacific Basin Finance Journal, 8(3–4): 443–56. Frino, A., McInish, T. and Toner, M. (1998) “The Liquidity of Automated Exchanges: New Evidence from the German Bund Futures”, Journal of International Financial Markets, Institutions and Money, 8(3–4): 225–41. Granger, C. W. J. (1969) “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods”, Econometrica, 37(3): 424–38. Granger, C. W. J. (1981) “Some Properties of Time Series Data and Their Use in Econometric Model Specification”, Journal of Econometrics, 16(1): 121–30. Groenewold, N. (1997) “Share Market Efficiency: Tests Using Daily Data for Australia and New Zealand”, Applied Financial Economics, 7(6): 645–57. Grünbichler, A., Longstaff, F. A. and Schwartz, E. S. (1994) “Electronic Screen Trading and the Transmission of Information: An Empirical Examination”, Journal of Financial Intermediation, 3(2): 166–87. Harris, L. (1982) “Transaction Data Tests of the Mixture of Distributions Hypothesis”, Journal of Financial and Quantitative Analysis, 22(2): 127–41. Hemler, M. L. and Longstaff. F. (1991) “General Equilibrium Stock Index Futures Prices: Theory and Empirical Evidence”, Journal of Financial and Quantitative Analysis, 26(3): 287–308. Hinich, M. (1996) “Testing for Dependence in the Input to a Linear Time Series Model”, Journal of Nonparametric Statistics, 6(3): 205–21. Hurst, H. E. (1951) “The Long-term Storage Capacity of Reservoirs”, Transactions of the American Society of Civil Engineers, 116(3): 143–52. Jarnecic, E. (1999) “Trading Volume Relations between the ASX and ASX Options Market: Implication of Microstructure”, Australian Journal of Management, 24(1): 77–91. Johansen, S. (1991) “Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Autoregressive Models”, Econometrica, 59(6): 1551–80. Johansen, S. (1995) “Likelihood-based Inference in Cointegrated Vectors Autoregressive Models”, (Oxford University Press). Johansen, S. and Juselius, K. (1990) “Maximum Likelihood Estimation and Inference on Cointegration – with Application to the Demand for Money”, Oxford Bulletin of Economics and Statistics, 52(2): 169–210. Karpoff, J. M. (1987) “The Relation between Price Change and Trading Volume: A Survey”, Journal of Financial and Quantitative Analysis, 22(1): 109–26. Kempf, A. and Korn, O. (1998) “Trading System and Market Integration”, Journal of Financial Intermediation, 7(3): 220–39.

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Kofman, P. and Moser, J. (1997) “Spread Information Flows and Transparency Across Trading Systems”, Applied Financial Economics, 7(3): 281–94. MacKinnon, J. G. (1991) “Critical Values for Cointegration Tests”, in R. F. Engle, and C. W. J. Granger (eds), Modelling Long-run Economic Relationships (London: Oxford Publishing Company). Mananyi, A. and Struthers, J. J. (1997) “Cocoa Market Efficiency: A Cointegration Approach”, Journal of Economic Studies, 24(3): 141–51. Manaster, S. and Mann, S. C. (1998) “Life in the Pits: Competitive Market Making and Inventory Control”, Review of Financial Studies, 36(1): 51–64. Martell, T. F. and Wolf, A. S. (1985) “Determinants of Trading Volume in Futures Markets”, Working paper, Centre for the Study of Futures Markets, Columbia Business School, USA. Massimb, M. and Phelps, B. (1994) “Electronic Trading, Market Structure and Liquidity”, Financial Analysts Journal, 50(1): 39–50. O’Connor, S. M. (1993) “The Development of Financial Derivatives Markets: The Canadian Experience”, Technical Report, No. 62 (Ottawa: Bank of Canada). O’Hara, M. (1995) Market Microstructure Theory (Cambridge, Basil Blackwell). Perron, P. (1990) “Testing for a Unit Root in a Time Series with a Changing Mean”, Journal of Business and Economic Statistics, 8(1): 153–62. Peters, E. (1992) “R/S Analysis Using Logarithmic Returns: A Technical Note”, Financial Analysts Journal, 48(6): 81–2. Phillips, C. C. and Perron, P. (1988) “Testing for a Unit Root in Time Series Regression”, Biometrika, 8(2): 153–62. Pirrong, C. (1996) “Market Liquidity and Depth on Computerized and Open Outcry Trading Systems: A Comparison of DTB and LIFFE Bund Contracts”, Journal of Futures Markets, 16(5): 519–43. Ragunathan, V. and Peker, A. (1997) “Price Variability, Trading Volume and Market Depth: Evidence from the Australian Futures Market”, Applied Financial Economics, 7(5): 447–54. Richardson, M. and Smith, T. (1994) “A Direct Test of the Mixture of Distributions Hypothesis: Measuring the Daily Flow of Information”, Journal of Financial and Quantitative Analysis, 29(1): 101–16. Rubinstein, M. (1994) “Implied Binomial Trees”, Journal of Finance, 49(3): 771–818. Shyy, G. and Lee, J. (1996) “Price Transmission and Information Asymmetry in Bund Futures Markets: LIFFE vs. DTB”, Journal of Futures Markets, 15(1): 437–55. Sydney Futures Exchange (SFE) (1999) Full Electronic Trading: A Guide for Customers (Sydney, Australia: SFE). Tauchen, G. and Pitts, M. (1983) “The Price Variability–Volume Relationship on Speculative Markets”, Econometrica, 51(2): 485–505. Taylor, M. P. and Sarno, L. (1997) “Capital Flows to Developing Countries: Log and Short-term Determinants”, World Bank Economic Review, 11(3): 183–212. Tsang, R. (1999) “Open Outcry and Electronic Trading in Futures Exchanges”, Bank of Canada Review, Spring, 21–39. Turkington, J. and Walsh, D. (1999) “Price Discovery and Causality in the Australian Share Price Index Futures Market”, Australian Journal of Management, 24(2): 97–113.

C H A P T E R 10

How Does Systematic Risk Impact Stocks? A Study of the French Financial Market Hayette Gatfaoui

10.1 INTRODUCTION Systematic risk is known to affect the market prices of traded financial assets (Stulz, 1999a, 1999b, 1999c). Indeed, the capital asset pricing model (CAPM) theory argues that each financial asset bears an undiversifiable risk known as systematic or market risk, as introduced by Sharpe (1963, 1964, 1970) and Treynor (1961) among others.1 Such a risk can be estimated through a well-diversified portfolio so far as this portfolio presents as low as possible an idiosyncratic risk (French and Poterba, 1991). Recent literature focuses mainly on a sound assessment of the influence of systematic risk on financial assets, along with the beta coefficient in a CAPM framework. Koutmos and Knif (2002) estimate the influence of systematic risk while employing timevarying distributions (for example, conditional distributions depending on past innovations). Using market stock indices of the financial markets under consideration, they find that financial assets’ betas are stationary meanreverting processes with an average degree of persistence equal to four days. Gençay, Selçuk and Whitcher (2003) use wavelet techniques to assess the influence of systematic risk on any asset, or equivalently to compute its beta 183

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in a CAPM model. These authors use the S&P 500 index as a systematic risk benchmark. Therefore, common practice resorts to available stock indices as proxies for a well-diversified market portfolio, and pays little attention to the sound assessment of systematic risk itself.2 However, a study by Campbell et al. (2001) shows that the number of stocks in such an index has to be high enough to offset idiosyncratic risk. They find that the number of assets required to create a well-diversified portfolio has grown over time. Therefore, using market indices with an insufficient number of stocks (for example, small stock indices) may be an inaccurate and even wrong benchmark for systematic risk. Indeed, a market stock index represents a sub-set of the whole range of financial assets that should enter the composition of an actual market portfolio, according to the critique of Roll (1977). This author underlines that the actual market portfolio is non-identifiable, since a market portfolio should be composed of stocks, bonds, real estate and human capital assets, among others. However, Campbell et al. (2001) show that market volatility (which is that part of the global volatility related to market factor, and specifically their market factor proxy) tends to drive global volatility. Therefore, in this chapter we address the question of how to find a proxy for the market factor, such as the systematic risk factor, in markets where only small stock indices are available, and where options on such indices are traded. It is a hard task, since the undiversifiable risk is not directly observable and can only be estimated. Hence, lacking a portfolio diversified enough to represent the market factor accurately, we attempt to infer the fair level of market risk factor from only observed available stock indices and related European call prices. The chapter is organized as follows. Section 10.2 introduces the assumptions and theoretical framework aimed at finding a proxy for the systematic risk factor. Section 10.3 employs an empirical application of such a framework, focusing on the French financial market and its CAC40 stock index. Section 10.4 studies the impact of the implied market factor on a pool of French stocks. The impact of systematic risk is analyzed through a twostep methodology, namely a correlation study and a Granger causality test. For further investigation, section 10.5 attempts to test for a non-linear relationship between both prices and returns of the implied market factor, and French financial assets. This study is realized in two stages: a linear regression analysis and a volatility analysis. The linear regression analysis considers first simple regressions of returns, and then Jensen-type (1968, 1969) regressions. The volatility study considers weekly rolling volatilities of asset returns. Section 10.6 attempts to draw some conclusions while comparing our implied market factor with other available market stock indices. A two-step study is undertaken, considering first the explanatory power corresponding to each stock index. The empirical weekly forecasting performance underlying each available market proxy is then assessed, employing the average absolute relative error as a performance measure. Finally, the

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study ends with concluding remarks and suggestions for future research in section 10.7.

10.2 THEORETICAL FRAMEWORK In this section, we introduce our assumptions and the related theoretical framework allowing the induction of the market factor.

10.2.1 Valuation setting We assume that any small stock index is a non-perfect proxy of the systematic risk factor. Specifically, we suppose that a small stock index is a disturbed observation of the market factor.

Assumptions. Any small stock index It , at current time t, depends on market factor Xt such that It = t Xt

(10.1)

where t represents a (strictly) positive determinist scale factor that is time∗ . Moreover, is a continuous and derivable varying and bounded on R+ t function of time. We assume implicity that any small stock index is diversified so as to exhibit a sufficiently low level of idiosyncratic risk. Therefore, the scale factor encompasses this. This parameter is not purely, or mainly, driven by an idiosyncratic component. Hence the scale parameter can encompass many effects/factors such as liquidity phenomena, and short-term shocks resulting from some announcement effects or specific events occurring in the financial market. Further, all the assumptions of the Black and Scholes (1973) option valuation framework are supposed to hold. To sum up, trading is continuous; there are no dividend payments, no transaction costs and no taxes. Moreover, there is no arbitrage opportunity and a constant spot risk-free interest rate r prevails in the complete market.3 We also assume that the market fact tor follows a geometric Brownian motion such as dX Xt = µ dt + σ dWt where t is the current date; µ and σ are constant drift and volatility parameters of the systematic factor’s instantaneous rate of return;4 Wt is a standard Brownian motion under the historical probability.

Dynamic of the stock index. Applying Ito’s lemma in the risk-neutral universe and on-time sub-set [t, T], the stock index dynamic writes under risk neutral probability σ2 1 ∂t d ln (It ) = + r− dt + σ dWt∗ (10.2) t ∂t 2

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σ2 ∗ ∗ T (T − t) + σ W which rewrites IT = It exp r − T−t , where (Wt ) is a 2 t standard Brownian motion. Of course, we could estimate t and T while building a well-diversified portfolio. Such a portfolio should be a good proxy of market factor so that the market is complete. However, along with Roll’s critique and Campbell et al. (2001), we address the question of how to proxy the market factor from a small-stock index, which is an imperfect proxy of market factor. Hence, we consider the prices of options on a small index. Indeed, observed index prices and call market prices will give information about both scale and market factors.

10.2.2 Option pricing We introduce a call pricing formula for European calls on the small-stock index I.

Call’s dynamic in a no-dividend framework. We consider a European call on stock index I whose strike price and expiring date are, respectively, K and T. At maturity, such a call is valued C(T, IT ) = max(0, IT − K) = (IT − K)+ . Like Black and Scholes (1973), we apply the no-opportunity arbitrage valuation principle, which states that the current value of any contingent claim is equal to the discount expected value of its future cash flows under risk neutral probability. Then, our European call Q Q price writes C(t, It ) = Et e−r(T−t) (IT − K)+ , where Et [.] is the expectation operator under risk neutral probability Q, conditional on the information set Ft = σ{Ws ,0 ≤ s ≤ t} available at current date t. Therefore, from Equations (10.1) and (10.2) of the stock index, the pricing formula for a European call on stock index I at current date t reads: C(t, It ) ≡ C(T − t, K, It , r, t , T , σ) =

where N(.)

T It N(d1 ) − K e−r(T−t) N(d2 ) t (10.3)

is the cumulative distribution function of the stan

I σ2 ln T + ln Kt + r + 2 (T − t) √ t √ ; d = d − σ T −t= dard normal law; d1 = 2 1 σ T −t 2 I σ ln T + ln Kt + r − 2 (T − t) t √ . If we assume that the small-stock index is a σ T −t

perfect proxy of market factor, we get the classical Black and Scholes (1973) option pricing formula, since we have t = T = 1 for each date t < T. Therefore, introducing a disturbance in our modifies the classical Black setting

T and Scholes formula through ratio t . However, assuming a Black and Scholes (1973) setting to value a call on a stock index is inappropriate in so

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187

far as the no-dividend assumption is unrealistic. That is why we adapt the previous formula to account for a stock index comprising dividend-paying equities.

Call’s dynamic in a dividend framework. Since most of the stocks that constitute financial indices pay dividends, we assume that index I pays a dividend at a continuous annualized rate q (see Merton, 1973; Black, 1975). Therefore, under the Black and Scholes’ world and dividend-paying assumptions, the current price of underlying It has to be replaced with It e−q(T−t) . Then, adjusting the European call pricing formula in Equation (10.3) to become a dividend-paying framework, a European call on a dividend-paying stock index I is valued as C(T − t, K, It , r, t , T , σ) =

T It e−q(T−t) N(d1 ) − K e−r(T−t) N(d2 ) t (10.4)

where N (.) is the cumulative distribution function of the standard

I e−q(T − t) σ2 ln ( T ) + ln t K + r + 2 (T − t) √ t √ ; d2 = d1 − σ T − t = normal law; d1 = σ T − t −q(T−t) I e σ2 ln T + ln t K + r − 2 (T − t) t √ . In European call formula Equations σ T −t

(10.3) or (10.4), all parameters are known except the scale parameter at instants t and T (for example, t and T ), and volatility parameter σ. Therefore we shall use our knowledge about observed index prices and market prices of European index calls to extract information about the scale parameter and volatility parameter σ. Such a process will give information about the market factor itself.

10.3 EMPIRICAL STUDY We apply our European call pricing here to the French stock market and its CAC40 stock index.

10.3.1 Data We use Bloomberg daily closing data from January 2, 2002 to March 19, 2002, a total of 55 observations by series. We observe one-month r1M , twomonth r2M and three-month r3M risk-free interest rates, and consider market prices of the CAC40 French stock index. This index is composed of the forty most liquid and representative stocks listed on the French financial market, and pays a continuous annualized dividend rate q. The CAC40 INDEX is a weighted stock index whose weights are proportional to each of its forty

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Table 10.1 Index information Index

q(%)

nb K

Spread

CAC40

2.2650

3

4238.99–4682.79

Table 10.2 Features of CAC40 INDEX calls Call name

Strike price (€ )

CAC 3/02 C4000

4,000

CAC 3/02 C4500

4,500

CAC 3/02 C5000

5,000

stocks’ capitalization. We also obtain closing prices of three European calls on CAC40 while considering option contracts of the continuous listing class. These calls are traded on the French options market called MONEP (Marché des Options Négociables de Paris). Let q, nbK and spread be, respectively, the dividend rate, the number of different strike prices of CAC40 INDEX calls, and the variation bounds of the index value (that is, lowest–highest in euros) over the studied time period (see Table 10.1). European calls on the CAC40 INDEX, maturing on March 27, 2002, exhibit the features shown in Table 10.2. Over our time horizon, time to maturity of calls falls from 84 calendar days to 8 calendar days (6 working days). Part of these data will help us to compute the risk-free interest rate, which must be defined. Given our European call pricing formula, we compute the risk-free rate as a function of time to maturity. We choose a quadratic interpolation method to infer our short-term risk-free rate from the one-, two- and three-month term risk free rates. Let r(t, T) be the risk free rate at current time t for time horizon T. This rate is then described by relation r(t, T) = a(T − t)2 + b(T − t) + c with a = 72[r1M (t) − 2r2M (t) + r3M (t)], b = 12[r2M (t) − r1M (t) − (a/48)] = −30r1M (t) + 48r2M (t) − 18r3M (t) and c = r1M (t) − (a/144) − (b/12) = 3r1M (t) − 3r2M (t) + r3M (t). This method gives a good risk-free rate proxy, given that European calls’ time to maturity (for example, (T − t)) is, at most, three calendar months.

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This proxy is employed to infer the fair value of the market risk factor from the CAC40 INDEX and related European call prices. Incidentally, the lack of control of the CAC40 INDEX weights introduces size effects into this index, among others. Therefore, an important role is played by some specific effects that are peculiar to any given highly-capitalized firm belonging to the CAC40 INDEX. In this light, the market benchmark role of CAC40 is strongly compromised. Dow Jones STOXX market indices bypass such a bias by applying some weight constraint when a given stock’s weight exceeds some specific threshold among the indices under consideration (high free-floating market capitalization).

10.3.2 Induction of systematic risk We explain how to estimate the level of market factor from market prices of a small stock index and the closing prices of European calls on such an index. From Equation (10.4), the estimation of the market factor’s level requires the estimation of the scale parameter at instants t and T (that is, t and T ), and volatility parameter σ (the volatility of the market factor’s instantaneous rate of return). As we observe market prices of the CAC40 INDEX (the small-stock index) and closing prices of related European calls, one solution consists of inverting Equation (10.4) relative to the scale parameter at times t and T, and the volatility parameter. We estimate these implied parameters while minimizing the sum of squared valuation errors at each fixed date t as ⎧ ⎫ nbK ⎨

2 ⎬ Min CObs T − t, Kj , It − C T − t, Kj , It , r, t , T , σ ⎭ t , T , σ ⎩ j=1

where Kj ∈ {4000, 4500, 5000}, and CObs (T − t, Kj , It ) are the European call’s market price. We solve this non-linear minimization problem numerically with a quasi-Newton method, and a Davidon–Fletcher–Powell type of algorithm. First, we get T = 2.3050 and XT = 2033.8482. Second, results allow plotting the implied values of t and σ against time to maturity. Whaley, (1982) argues that valuation errors do not necessarily depend on options’ moneyness, hence we draw plots according to time rather than moneyness. The implied volatility parameter σ is time-varying with a quadratic trend (a “smirk” type trend). Moreover, implied time series t and σ exhibit the statistical profiles shown in Table 10.3. We then observe a non-normal behavior for t and σ, namely leptokurtic distributions. Specifically, the volatility of the systematic risk factor should be modeled by a non-normal stochastic process or time-varying series. This stylized fact is known as the Black and Scholes volatility bias characterizing non-normal observed market asset returns. Knowing the market trend, we can now characterize the impact of systematic risk on the French financial market.

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A STUDY OF THE FRENCH FINANCIAL MARKET

0.2350

2.305 2.3

0.2200

2.295

0.2150

2.29

0.2100

2.285

0.2050

2.28

0.2000

2.275

0.1950

2.27

0.1900

2.265

0.1850

2.26

84 80 78 76 72 70 66 64 62 58 56 52 50 48 44 42 38 36 34 30 28 24 22 20 16 14 10 8

Volatility level

0.2250

2.31 Implied volatility Implied lambda

Scale factor’s level

0.2300

Time to maturity (days)

Figure 10.1 Daily implied scale factor and market factor volatility

Table 10.3 Descriptive statistics t Mean

2.2881

σ 0.2069

Xt 1952.2699

Standard deviation

0.0086

0.0069

52.0441

Skewness

0.2144

1.1208

−0.1669

−1.0728

2.9231

−0.8446

3.0589

31.0976

1.8901

Excess Kurtosis Jarque-Bera Statistic

10.4 THE IMPACT OF SYSTEMATIC RISK Given our market factor’s estimation, we try to quantify its impact on prices of French stocks. Our primary econometric study is composed of a correlation study and a Granger causality test.

10.4.1 Correlation We study correlations between implied market factor and, on the one hand, French stock indices (CAC40, SBF120 and SBF250), and on the other ten French stocks: Air Liquide, Danone, L’Oréal, Renault, Schneider, Société Générale, Thomson, Totalfina Elf, Valéo and Vivendi (see Table 10.4). Most of commonly used descriptive statistics are valid only under the strong assumption of an elliptical distribution. When this is not the case, statistics

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Table 10.4 Correlation of assets with the implied market factor Asset return

Correlation coefficient

Asset return

Correlation coefficient

SBF120

0.9959

Valéo

0.5288

SBF250

0.9952

Société Générale

0.7078

CAC40

0.9967

L’Oréal

0.6775

Air Liquide

0.4667

Renault

0.5365

Danone

0.2002

Schneider

0.4736

Vivendi

0.7982

Thomson

0.5329

Totalfina Elf

0.6569

are false. Indeed, this point fits some of the current questions considered by the Basel Committee. Szego (2002) and Artzner et al. (1999, 2000), highlight the coherency problem of risk measures such as linear correlation or covariance. Such risk measures are valid only for, at least, stationary distributions when not elliptical. Specifically, leptokurtic distributions violate one main property ensuring risk measures’ coherency, namely the sub-additivity principle. Following this concern, we compute correlations between the return of the implied market factor and returns of French stocks. Returns of both series are stationary over the time period studied. We then study the link between evolutions of both the systematic risk factor’s return and French asset returns. The average correlation of our three stock indices is 0.9959. The implied market factor is highly correlated with stocks, whose correlation coefficients range from 0.2002 for Danone to 0.7982 for Vivendi. In the rest of the chapter, we study the dependency between systematic risk and French stocks.

10.4.2 Causality Any causality study needs a vector autoregressive (VAR) specification as a starting point. We first introduce our VAR specification and then apply a Granger causality test. VAR specification

We look for a link between the implied market risk’s return RX and French stock or index returns RS . Hence, we consider VAR representations linking RX to RS with S ∈ {SBF120,5 SBF250, CAC40, Air Liquide, Danone, L’Oréal, Renault, Schneider, Société Générale, Thomson, Totalfina Elf, Valéo and Vivendi}. A VAR model allows us to test for a statistical relation between variables. Moreover, any VAR process parameters have to be estimated for

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stationary time series such as our asset returns. The related bidimensional VAR with p lags, called VAR(p), writes Yt = A0 + A1 Yt−1 + A2 Yt−2 + · · · + Ap Yt−p + εt where Yt = [RXt RSt ] is the vector of variables; A0 = [a01 a02 ] is a vector of j

constant parameters; Ap = [aip ] 1 ≤ i,j ≤ 2 is the coefficient matrix for lag p; and εt = [ε1t ε2t ] is the vector of innovations that is assumed to follow a normal law. In practice, disturbances may be correlated contemporaneously with each other, without being correlated with, on the one hand, their own lagged values, and on the other, all the lagged values of the variables. When disturbances (εt ) are correlated, the variation of one error component has an impact on the other components – variables have a synchronous influence on each other. A causality analysis allows then to study the kind of influence variables have on each other. Moreover, the optimal lag is determined while minimizing Akaike and Schwarz information criteria. We investigate optimal lags of one to five days while looking for a weekly influence at most, as compared to the four days of persistence documented by Koutmos and Knif (2002) for beta estimates. The maximum likelihood method then gives an optimal lag p of one. Such a first-order relationship between asset returns may result from asynchronous trading in the financial markets or asset prices’ speed of adjustment to new private/public information. Indeed, large firms’ asset prices integrate information more easily and quickly than small ones’ asset prices, since large firms’ assets are usually more liquid: as large firms’ assets are usually traded more frequently than small firms’ assets, their prices adjust more quickly to the arrival of new information. Moreover, McKenzie and Faff (2003) show that trading volumes and market returns determine time-varying autocorrelations of asset returns. This setting leads to the results shown in Table 10.5. a01 a111 a211 In each column, the coefficients of returns are displayed as 0 1 a2 a21 a221 with their related Student statistics between brackets under each coefficient. Moreover, the R2 statistic related to the estimation of each univariate relation is displayed in percent as: [R2 (RXt ) R2 (RSt )] . Recall that we have the next VAR(1) bivariate specification:

RXt RS t

=

a01 a02

+

a111 a211 a121 a221

RXt−1 RSt−1

+

ε1t ε2t

(10.5)

Our VAR(1) specification does not exhibit any influence between the implied market factor’s return and returns of French indices. As a rough guide, we also compute the statistics and coefficients related to our ten stocks’ VAR(1) specification (see Table 10.6).

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Table 10.5 VAR results for stock indices Index return CAC40

SBF120

SBF250

R 2 (%)

Coefficients −0.0546

−1.1382

1.0206

(−0.3488)

(−0.6794)

(0.6428)

1.2644

−0.0219

−0.6581

0.5539

0.7644

(−0.1327)

(−0.3721)

(0.3305)

−0.0487

−0.4722

0.4102

(−0.2888)

(−0.3079)

(0.2666)

0.5899 0.4677

−0.0053

−0.4560

0.4037

(−0.0317)

(−0.2995)

(0.2643)

−0.0609

−0.5750

0.5389

(−0.3473)

(−0.4061)

(0.3619)

0.7086

−0.0030

−0.5750

0.4852

0.4455

(−0.0178)

(−0.3738)

(0.3441)

At the 5 percent level of Student test, Air Liquide and Renault stocks impact implied a systematic risk factor, while Société Générale stock influences implied a systematic risk factor at the 10 percent level. We further investigate these results through a causality test. Granger causality test

A natural application of VAR modeling is a causality test. Granger (1969) defines causality as follows: RXt is said to be the cause of RSt when taking into account the information set associated to RXt helps to improve predictions of RSt . Analyzing causality of RXt towards RSt is equivalent to realizing a test with constraints on the coefficients of RXt in its VAR representation (Equation (10.5)) (a restricted VAR specification for RXt , also known as RVAR). Specifically, consider assumption H0 : a121 = a221 = 0. If we accept H0 , then RXt does not cause RSt . To test assumption H0 , we compare the unrestricted VAR (for example, UVAR, in Equation 10.5) with the VAR specification restricted to H0 (RVAR). The related test statistic is the likelihood ratio L = (n − c) ln{|RVAR |/|UVAR |} where n is the number of observations; c is the numberof estimated coefficients in each univariate relation of the UVAR model; RVAR , UVAR are the covariance matrices of restricted and unrestricted VAR models, respectively; |A| represents the determinant of matrix A. In this case, L is assumed to follow a chi-square law with two degrees of freedom (for example, χ2 (2)). Therefore, we reject H0 assumption for a given test level α if L is greater than the critical value of the χ2 (2) law 2 for level α (for example, L > χcritical (2); see Hamilton, 1994).

194

Table 10.6 VAR results for stocks Stock return Air Liquide

Danone

Vivendi

Totalfina Elf

Valéo

R 2 (%)

Coefficients −0.0588 (−0.4002) 0.1104 (0.5906)

−0.2090 (−1.4016) −0.2699 (−1.4223)

0.2525 (2.0454) 0.0547 (0.3481)

8.1354 4.1368

−0.0352 (−0.2366) 0.0329 (0.2584)

−1.1071 (−0.7840) 0.1050 (0.8981)

0.2532 (1.5238) −0.1050 (−0.7379)

4.8662 2.2082

−0.0078 (−0.0477) −0.5975 (−1.5866)

−0.1317 (−0.5784) 0.0990 (0.1881)

0.0367 (0.3665) 0.0317 (0.1371)

0.7152 0.5276

−0.0326 (−0.2106) 0.1860 (1.1717)

−0.0787 (−0.4343) −0.0572 (−0.3077)

0.0208 (0.1160) −0.1560 (−0.8388)

0.4753 3.9099

−0.0281 (−0.1817) 0.2474 (1.0045)

−0.0611 (−0.3786) 0.0482 (0.1875)

−0.0047 (−0.0455) 0.0080 (0.0485)

0.4527 0.1308

Stock return Société Générale

L’Oréal

Renault

Schneider

Thomson

R 2 (%)

Coefficients −0.0808 (−0.5361) 0.1979 (0.7984)

−0.3013 (−1.5920) 0.1241 (0.3985)

0.2120 (1.7529) −0.0068 (−0.0341)

6.2121 0.5678

−0.0032 (−0.0209) 0.1839 (1.0659)

0.0924 (0.5057) 0.0766 (0.3671)

−0.1809 (−1.2728) −0.5131 (−3.1637)

3.5726 24.0139

−0.1423 (−0.9186) 0.4518 (1.6360)

−0.2426 (−1.5658) 0.0960 (0.3477)

0.1884 (2.1443) 0.1453 (0.9278)

8.8326 3.5828

−0.0233 (−0.1521) 0.1484 (0.5015)

−0.0384 (−0.2468) 0.4611 (1.5319)

−0.0278 (−0.3571) −0.1657 (−1.1000)

0.7017 4.8022

−0.0295 (−0.1934) 0.0393 (0.1082)

−0.0671 (−0.4149) 0.0559 (0.1451)

0.0017 (0.0250) −0.0438 (−0.2698)

0.4498 0.1454

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Table 10.7 Granger statistics for indices and stocks Asset return

L

Probability

CAC40

0.1385 0.4132

0.7114 0.5233

SBF120

0.0897 0.0711

SBF250

Stock return

L

Probability

Valéo

0.0352 0.0021

0.8520 0.9639

0.7658 0.7909

Société Générale

0.1588 3.0727

0.6920 0.0858

0.1397 0.1310

0.7101 0.7190

L’Oréal

0.1348 1.6199

0.7151 0.2090

Air Liquide

2.0229 4.1838

0.1612 0.0461

Renault

0.1209 4.5982

0.7296 0.0369

Danone

0.8066 2.3218

0.3734 0.1339

Schneider

2.3468 0.1275

0.1318 0.7226

Vivendi

0.0354 0.1343

0.8515 0.7156

Thomson

0.0211 0.0006

0.8852 0.9801

Totalfina Elf

0.0947 0.0135

0.7596 0.9081

Note: bold indicates a chi-squares results.

Studying relationships between an implied systematic risk factor’s return and French stock returns, we tested two assumptions, namely: “H0 : RXt does not Granger cause RSt ” and “H0∗ : RSt does not Granger cause RXt ”, and obtained the results shown in Table 10.7. For each asset, the first and second lines correspond to the results of H0 and H0∗ assumptions, respectively. At the 15 percent level, Air Liquide and Renault returns cause the implied market factor’s return (RXt ). Enlarging our test level to 40 percent, Société Générale also causes the implied market factor’s return (RXt ). Our study therefore shows a smaller impact of the implied market factor on French assets than was expected. Our results’ weakness may come from the small sample size used. For further investigation, we look for contemporaneous links between variables without lag consideration. Specifically, we test for a non-linear influence of the implied market factor’s price on the prices of French stocks and indices.

10.5 FURTHER INVESTIGATION We attempt to exhibit non-linear dependence and “quadratic” causality between implied market factor and French stocks. Non-linearity is captured through the study of returns. We proceed in two steps: a regression analysis of asset returns and a volatility analysis of these daily returns.

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10.5.1 Simple regression Focusing on a non-linear link between the price of the implied market factor and the price of an asset is equivalent to regressing this asset return on the return of the implied market factor. Specifically, we look for the following kind of relationship β

S t = αS X t S

(10.6)

where αS and βS are constant terms, and St ∈ {SBF120, SBF250, CAC40, Air Liquide, Danone, L’Oréal, Renault, Schneider, Société Générale, Thomson, Totalfina Elf, Valéo Moreover, we make the approximation that Vivendi}.

t−1 t RSt = StS−S for each time t ranging from 2 to 55, and rewrite ≈ ln SSt−1 t−1 Equation (10.6) as a logarithm variation between times (t − 1) and t

RSt = βS RXt

(10.7)

for t ∈ {2, … , 55}. Consequently, the non-linear link between Xt and St is equivalent to a linear regression of St return (RSt ) on Xt return (RXt ). Such a study is practical, given that returns are stationary variables here. Moreover, our methodology consists of applying a single-index model that translates into a one-factor model that is close to CAPM (see Tables 10.8 and 10.9)6 . Regressions of French asset returns on the return of the implied market factor are all significant at the 1 percent level, apart from Danone stock’s

Table 10.8 Regression results for stock indices β

Student t

R 2 (%)

CAC40

1.0514

86.3097

99.2930

SBF120

0.9925

73.1893

99.0179

SBF250

0.9460

64.8178

98.7496

Index return

Table 10.9 Regression results for stocks β

Student t

R 2 (%)

Air Liquide

0.5650

3.8326

21.2243

Danone

0.1645

1.4883

Vivendi

1.8302

9.0541

Totalfina Elf

0.6694

6.2260

Valéo

0.8384

4.4587

Stock return

β

Student t

R 2 (%)

Société Générale

1.1088

7.1866

48.4465

3.9693

L’Oréal

0.8737

6.6437

44.7481

58.9820

Renault

0.9527

4.3560

18.8525

40.7725

Schneider

0.9480

3.9014

21.6706

25.0821

Thomson

1.2628

4.5825

28.2329

Stock return

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regression. Among available French stock indices, the highest explanatory power is reached for CAC40 (for example, R2 (CAC40) = 99.2930%), whereas the highest explanatory power among French stocks is reached for Vivendi stock (for example, R2 (ex) = 58.8820%). Therefore, the implied market factor has an important influence, in terms of explaining daily returns, on all our financial assets apart from Danone. Such a pattern indicates that the residual risk factor (idiosyncratic risk factor) additional to the systematic risk factor explains the main part of Danone stock’s evolution. We also tested for the assumption “H0 : βS = 1” in Equation (10.7) for stocks. We found that βS has a significant unit value only for Valéo, Société Générale, L’Oréal, Renault Schneider and Thomson stocks. Therefore, these six assets are driven purely by market trends as represented by the implied market factor. Moreover, Air Liquide’s, Danone’s, and Totalfina Elf’s, stock returns absorb the influence of the implied market factor’s return, whereas Vivendi’s stock return amplifies such an impact. Finally, our ten stocks are globally market-driven, since their returns exhibit a positive link with that of the implied market factor. Such a finding is coherent with the work of Campbell et al. (2001). Brailsford and Faff (1997) found poor support for CAPM when studying Australian daily stock returns. Moreover, in a daily stock return setting, Koutmos and Knif (2002) show that the simple regression model works well for systematic risk measurement purposes (estimating the beta coefficient). However, a dynamic model with time-varying parameters is better for forecast purpose (forecasting efficient conditional beta estimates). Our main goal is to assess the impact of systematic risk on French stocks rather than value the validity and performance of CAPM (that is, to assess the mean-variance efficiency of our market proxy). Explanations about validity and performance of CAPM are proposed by Roll (1977) and Campbell et al. (1997), among others. Jagannathan and Wang (1996) also propose a good performance study. However, given the closeness of our single-index model to CAPM, we further investigate some linear dependency between returns of our implied market factor and French stocks. For this purpose, we employ the one-factor model of Jensen (1968, 1969) for each time t ranging from 2 to 55, namely RSt − r1M (t) = αS + βS (RXt − r1M (t)) + εt , where RXt is the return of the implied market factor X at time t; RSt is the return of stock S at time t; r1M (t) is the one-month French risk-free rate; βS is the sensitivity of stock S to the implied market factor X; αS is a constant term of regression; εt is a random normal error with zero expectation and constant variance; RSt − r1M (t) and RXt − r1M (t) are, respectively, stock S and implied market factor X market-risk premia. Jensen’s methodology allows the assessment of a risk-adjusted performance, the relevant risk measure being the beta of Sharpe (1963). The alpha coefficient of the previous regression is known as Jensen’s alpha and represents the abnormal return or excess return of a given stock relative to its CAPM return if this model were valid. In fact, alpha is the

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Table 10.10 Jensen’s regression results for stocks Stock return

α

Student t α

β

Student t β

R 2 (%)

CAC40

0.1945

4.6472

1.0511

88.2375

99.3366

SBF120

0.0191

0.4367

0.9922

79.8204

99.1905

SBF250

−0.1257

−2.7697

0.9457

73.1643

99.0379

Air Liquide

−1.3580

−2.6126

0.5640

3.8116

21.8380

Danone

−2.7691

−7.0750

0.1663

1.4924

4.1072

Vivendi

−0.2698

−0.5857

0.8737

6.6611

46.0417

Totalfina Elf

0.4514

0.6223

0.9511

4.6061

28.9774

Valéo

0.0167

0.0195

0.9475

3.8880

22.5227

Société Générale

0.5767

1.0732

1.1062

7.2304

50.1336

L’Oréal

0.9755

1.0011

1.2599

4.5414

28.3987

Renault

−0.9362

−2.5121

0.6680

6.2962

43.2574

Schneider

−0.2530

−0.3877

0.8357

4.4978

28.0081

Thomson

2.2339

3.3201

1.8333

9.5707

63.7881

non-equilibrium return that the stock brings in over the studied time horizon (see Table 10.10). Jensen’s alpha is significant for SBF250, CAC40, Air Liquide, Danone, Renault and Thomson stock returns. And the alpha is negative (that is, an abnormal return leading to a loss in value for an investment in the considered stock) for SBF250, Air Liquide, Danone, Vivendi, Renault and Schneider returns. Moreover, the beta coefficient is significant for all indices and stocks apart from Danone’s stock return, whose beta is close to zero. Hence, Danone’s stock evolution is uncorrelated or extremely low-correlated with the market, which means that this stock is low or not sensitive to market evolution. Put differently, Danone and Thomson stocks exhibit the lowest and highest beta coefficients (systematic risk), respectively, whereas beta estimates of CAC40, Société Générale, L’Oréal and Thomson returns lie above unity (for example, amplify the market effect). Finally, the explanatory power of our regressions is globally good in so far as CAC40 and Thomson stocks exhibit the highest explanatory power among indices and stocks, respectively (R2 (CAC40) = 99.3366% and R2 (Thomson) = 63.7881%). In contrast, Danone stock exhibits the lowest explanatory power (for example, R2 (Danone) = 4.1072%). Consequently, the implied market factor generally has a strong impact and influence in explaining stock return evolutions. Such an influence is nevertheless insufficient to explain the whole evolution of assets given both the limited explanatory power of regressions and the significance of Jensen’s alpha. Such a pattern can be explained by firm-specific features (for example, size effect) that are left aside while considering only systematic risk’s impact on French stocks (see Fama and French, 1992, 1993; Berk, 1995). This point is emphasized with

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Danone stock (Danone) whose evolution is not explained by the overall latent systematic risk factor prevailing in the French market. However, stronger evidence concerning the influence of the implied market factor on French stocks can be found while further investigating non-linear dependency between asset returns.

10.5.2 Volatility impact Investigating non-linear relationships between returns, we study the influence of the implied market factor on the volatility of our French assets. Indeed, linear causality analysis is unable to account for non-linear dependency between financial assets (for example, implied market factor and French stocks). Non-linear phenomena describing both financial markets and the underlying dynamics of the various assets composing such markets have been widely documented in the financial literature. Mele (1998) explains different kinds of non-linear dynamics, volatility and equilibrium that may describe a financial market. The simple existence of conditional heteroskedasticity in asset prices already describes some non-linear patterns in financial markets (Gourieroux and Jasiak, 2001). Put differently, exhibiting links between asset volatilities is a means of accounting for the non-linear features and effects that prevail between assets in markets. To this end, testing for a quadratic dependency between returns, we consider the weekly rolling volatilities of assets. As one calendar week represents five working days (a financial week), the weekly rolling volatility of return RSt at date t is written as σ(RSt ) = t t 1 2 with R = 1 (R − R ) (RSi ) for t ∈ {6, . . . , 55}. We analyze the S S S i t t 5 5 i=t−4

i=t−4

impact of the volatility of the implied market factor while considering the following first differences regressions: (10.8) σ RSt = aS σ RXt + ηt where ∀t ∈ {7, . . . , 55}, ∀ Xt , σ(RXt ) = σ(RXt ) − σ(RXt−1 ); aS is a constant coefficient; ηt is a “normal” disturbance; St ∈ {SBF120, SBF250, CAC40, Air Liquide, Danone, L’Oréal, Renault, Schneider, Société Générale, Thomson, Totalfina Elf, Valéo and Vivendi}. Results for first difference regressions (Equation (10.8)) of the weekly rolling volatilities of French assets on the weekly rolling volatility of the implied market factor are listed in Tables 10.11 and 10.12. Volatility regressions in Equation (10.8) are significant at a 1 percent level for CAC40, Vivendi, Totalfina Elf, Valéo, Société Générale, L’Oréal, Renault, SBF120 and SBF250. Among French indices, SBF250 presents the highest explanatory power (R2 (SBF250) = 97.1677%) whereas Vivendi exhibits the highest explanatory power (R2 (Vivendi) = 52.1625%) among stocks. Results

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Table 10.11 Volatility regression results for stock indices a

Student t

R 2 (%)

CAC40

1.0329

40.2860

97.1143

SBF120

0.9614

39.3774

96.9823

SBF250

0.9349

40.6838

97.1677

Index return

Table 10.12 Volatility regression results for stocks Stock return

a

Student t R 2 (%)

Stock return

a

Student t R 2 (%)

2.2470 Société Générale

0.8547

2.8667 14.4792

0.2888

0.2015 L’Oréal

0.4608

4.0223 23.6650

7.2466

52.1625 Renault

0.8075

2.7771 13.8271

0.3720

2.8797

14.7152 Schneider

−0.0938 −0.3519

0.3325

0.6732

2.7991

14.0162 Thomson

−0.5603 −1.3889

3.7211

Air Liquide

0.1634

1.1522

Danone

0.0402

Vivendi

1.7898

Totalfina Elf Valéo

suggest that the implied market factor has a strong influence on the weekly rolling volatilities of CAC40, L’Oréal, Renault, SBF120, SBF250, Société Générale, Totalfina Elf, Valéo, and finally Vivendi assets. However, from the explanatory power of regressions, the implied market factor fails to explain the whole evolution of assets. As shown by Campbell et al. (2001) and Goyal and Santa-Clara (2003), idiosyncratic risk should be the additional factor explaining the part of stock returns that is unexplained by systematic risk factor. Results require a global remark drawing a comparison between the global market information embedded in both the CAC40 stock index and its filtered counterpart, as represented by implied market factor X. The results we get while using the CAC40 stock index as a market proxy instead of an implied market factor give preliminary insights. Indeed, as the correlation between returns of the CAC40 stock index and implied market factor is 99.6666 percent, why should the CAC40 stock index not be the market? Given our results, the average (that is, arithmetic mean over time horizon) price of the CAC40 Index is 2.2881 times the average price of the implied market factor, with an average scale factor of 2.2883. Stated differently, the CAC40 INDEX average return is 3.8164 times the implied market factor’s average return. However, we also notice that the average ratio of the CAC40 INDEX return to the implied market factor’s return is 0.9636. These preliminary statistics suggest that CAC40 is a good market proxy if we assume that the implied market factor is the actual market portfolio. However, we cannot draw such

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a conclusion when considering the CAC40 INDEX as a market proxy. Moreover, being interested only in a view of the level of market return, CAC40, as well as the SBF120 and SBF250 indices, seem to be a convenient approximation of market return (as represented by implied market factor). Recall that both SBF120 and SBF250 indices are correlated almost as highly with the implied market factor as the CAC40 stock index. Nevertheless, for forecast purpose, this viewpoint changes greatly. Such results are introduced and summarized in the next section, which makes a comparison between our four different benchmarks.

10.6 MARKET BENCHMARK COMPARISON To answer the question about choosing between implied market factor and the CAC40 INDEX, we compare the results we get when employing successively Xt , CAC40, SBF250 and SBF120 indices as market benchmarks. First, we summarize the results obtained for our four different benchmarks, then we make a study of forecasting performance.

10.6.1 Basic empirical study We have estimated the three previous types of regressions that we called simple, Jensen and volatility regressions. We summarize the results in this section, displaying only relevant results to save space. We use two criteria to discriminate between market benchmarks. First, we consider the explanatory power of related regressions, and second, how close beta estimates of such regressions lie relative to the beta estimates we get for implied market factor X. In this way, we observe the impact of the bias, which comes from the fact that studied stocks are part of our three French stock indices. Recall that the Granger causality test allows for the classifying of benchmark-based relations with decreasing value of significance as Xt , SBF250, SBF120 and the CAC40 INDEX. We display the results for the explanatory power of our three types of regressions in Table 10.13. The first, second and third lines of each asset correspond to the simple, Jensen and volatility regressions, respectively. As regards simple regressions, SBF250-based regressions exhibit the highest explanatory power for 50 percent of stocks (Valéo, Société Générale, Schneider, Renault and Thomson returns) whereas CAC40-based regressions exhibit the highest explanatory power for 30 percent of stocks (Air Liquide, Totalfina Elf and L’Oréal returns). In a less powerful way, SBF120-based regressions exhibit the highest explanatory power for 10 percent of stocks (Danone returns) analogously to Xt -based regressions (Vivendi returns). As regards Jensen-type regressions, considering the proportion of stocks

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Table 10.13 Explanatory power of regressions (percentages) Stock

Xt

CAC40

SBF120

SBF250

Stock

21.2243 21.8380 2.2470

21.4424 21.9272 4.0148

20.1345 20.5040 3.1089

20.0694 20.3780 2.7512

Danone

3.9693 4.1072 0.2015

4.1214 4.2446 0.0354

4.1579 4.2681 0.1733

Vivendi

58.9820 46.0416 52.1625

57.6816 46.8405 47.8086

Totalfina Elf

40.7725 28.9774 14.7152

Valéo

25.0821 22.5227 14.0162

Air Liquide

Xt

CAC40

SBF120

SBF250

Société Générale

48.4465 50.1336 14.4792

50.4244 51.7562 17.1930

50.2269 51.2400 15.8954

50.4256 51.2706 16.1946

4.0683 4.1740 0.0841

L’Oréal

44.7481 28.3987 23.6650

45.8329 27.9946 26.9866

44.8268 29.3573 25.4735

44.0374 29.5735 23.1302

56.7106 45.6010 47.8346

56.4111 44.7000 49.3285

Renault

18.8525 43.2574 13.8271

18.8192 44.9391 14.9337

18.9169 42.8588 14.1422

19.4062 42.1550 13.6872

42.8574 28.2652 17.3273

41.1395 27.7130 15.9644

40.6282 27.8279 14.6540

Schneider

21.6706 28.0081 0.3325

22.2030 28.9954 0.3286

23.7321 30.7258 0.1534

24.1688 31.3020 0.0724

26.4305 22.8926 14.3045

28.5368 24.2689 14.3560

29.3201 24.6283 15.0222

Thomson

28.2329 63.7881 3.7211

27.9002 63.2092 3.4569

29.3254 63.0331 2.6751

29.5660 63.2194 3.1387

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that are best explained (highest explanatory power) through our four market benchmarks leads to classifying benchmarks with a decreasing proportion’s of value as CAC40 (40 percent of stocks, namely Air Liquide, Vivendi, Société Générale and Renault returns); SBF250 (30 percent of stocks, namely Valéo, L’Oréal and Schneider returns); Xt (20 percent of stocks, namely Totalfina Elf, and Thomson returns); and SBF120 (10 percent of stocks, namely Danone returns). As regards the explanatory power of volatility-based regressions, comparing the proportion of best-explained stocks through our different market proxies allows us to order results with decreasing proportion’s of value as CAC40 (50 percent of stocks, namely Air Liquide, Totalfina Elf, Société Générale, L’Oréal and Renault returns); Xt (40 percent of stocks, namely Danone, Vivendi and Thomson returns); SBF250 (for example, 10 percent of stocks, namely Valéo’s return); and SBF120. However, explanatory power-based results are probably upward-biased because of the weights of studied stocks that belong to available French stock indices (for example, important free-floating market capitalization weights in indices). To bypass such a bias, we consider how close the beta estimates of our different benchmark-based regressions are to the beta estimates of Xt -based regressions. For an overview, we display related beta estimates in Table 10.14. For each stock return, the first, second and third levels refer to simple, Jensen-type and volatility regressions, respectively. As regards simple regressions, SBF120-based regressions exhibit the closest estimates for Danone, Vivendi, Totalfina Elf, Société Générale, L’Oréal, Renault and Thomson stocks (70 percent of stocks) while CAC40- and SBF250-based regressions exhibit the closest estimates for Valéo and Schneider stocks (20 percent of stocks), and Air Liquide stock (10 percent of stocks), respectively. As regards Jensen-type regressions, SBF120-based regressions exhibit the closest beta estimates for Danone, Vivendi, Valéo, Totalfina Elf, Société Générale, L’Oréal, Renault and Thomson stocks (80 percent of stocks) while SBF250- and CAC40-based regressions exhibit the closest estimates for Air Liquide (10 percent of stocks) and Schneider stocks (10 percent of stocks), respectively. As regards volatility regressions, CAC40-based regressions exhibit the closest beta estimates for Danone, Totalfina Elf, Valéo, Société Générale, L’Oréal, Renault and Schneider stocks (70 percent of stocks) while SBF250- and SBF120-based regressions exhibit the closest estimates for Air Liquide and Thomson stocks (20 percent of stocks), and Vivendi-stock (10 percent of stock), respectively. Hence, given the closeness to the beta estimates of Xt -based regressions, the SBF120 index seems to be the best proxy for implied market factor for both simple and Jensen-type regressions (best for 80 percent, on average). Therefore, assuming that Xt is the actual market portfolio and given that the correlation between returns of Xt and SBF120 is 0.9959, employing SBF120 as a market proxy follows the findings of both Kandel and Staumbaugh (1987) and Shanken (1987). But differently, the CAC40 INDEX seems to be the best proxy of the implied market factor for

204

Table 10.14 Beta estimates for three types of regressions (1) Stock

Xt

CAC40

SBF120

SBF250

Stock

Air Liquide

0.5650 0.5640 0.1634*

0.5381 0.5359 0.2005*

0.5521 0.5486 0.1922*

0.5756 0.5734 0.1876*

Société Générale

Danone

0.1645*

0.1588*

0.1688*

0.1749*

L’Oréal

0.1663* 0.0402*

0.1603* 0.0448*

0.1702* 0.0605*

0.1764* 0.0543*

Vivendi

1.8302 0.8737 1.7898

1.7167 0.8356 1.6351

1.8017 0.8728 1.7560

1.8831 0.9059 1.8353

Renault

Valéo

0.6694 0.9511 0.3720

0.6495 0.8907 0.3851

0.6739 0.9337 0.3969

0.7020 0.9808 0.3914

Totalfina Elf

0.8384 0.9475 0.6732

0.8135 0.9058 0.6489

0.8907 0.9873 0.6979

0.9448 1.0427 0.7348

Note: *Non-significant estimates at a 5% test level.

Xt

CAC40

SBF120

SBF250

1.1088 1.1062 0.8547

1.0714 1.0658 0.8879

1.1312 1.1227 0.9170

1.1875 1.1773 0.9526

0.8737

0.8378

0.8767

0.9107

1.2599 0.4608

1.1861 0.4672

1.2858 0.4883

1.3530 0.4808

0.9527 0.6680 0.8075

0.9024 0.6457 0.8006

0.9562 0.6675 0.8365

1.0102 0.6940 0.8471

Schneider

0.9480 0.8357 −0.0938*

0.9091 0.8063 −0.0892*

0.9930 0.8786 −0.0610*

1.0497 0.9297 −0.0391*

Thomson

1.2628 1.8333 −0.5603*

1.1898 1.7305 −0.5161*

1.2901 1.8294 −0.4903*

1.3572 1.9207 −0.5445*

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volatility regressions. We investigate such preliminary results through a performance study.

10.6.2 Forecasting performance We attempt to discriminate between our four market proxies while considering their weekly forecasting performance in our three types of regressions. For this purpose, we first estimate regressions on the initial time horizon less one week of data (five observations). Then, we forecast corresponding returns or volatilities on the remaining week. Finally, we assess the related performance while computing the related average absolute relative error. Hence, we can assess the realized forecasting error relative to the actual level of return or volatility. Simple, Jensen-type and volatility regressions are successively estimated on t ∈ {2, … , 50}, {2, … , 50} and {7, … , 50} time horizons, respectively. We drop the last week of data for t ∈ {51, … , 55}. We display in Table 10.15 the beta estimates for our three types of regressions while employing successively our four market proxies. For each stock return, the first, second and third lines refer to simple, Jensen-type and volatility regressions, respectively. With regard to the closeness of our beta estimates to Xt -based beta estimates, a similar conclusion to that in the previous sub-section applies. On average (for 75 percent of stocks), the SBF120 index represents the best proxy of implied market factor for both simple and Jensen-type regressions whereas the CAC40 INDEX is the best proxy of Xt for volatility regressions. For the second part of our study, we first use previous regression estimates to forecast related returns and volatilities in the last week of our initial time horizon. Then, to assess the weekly forecasting performance of our benchmark-based regressions, we compute the corresponding average relative absolute error (average normalized absolute error). Such a performance measure allows us to highlight the percentage of forecasting errors relative to the actual level of both returns and volatilities under consideration. For this purpose, we compute respective forecasting errors eS of stock S during the last week of data as 55 ∗ ˆ St R − R 1 S eS = t ∗ RS 5 t t=51 ˆ St − rˆ1M (t) 55 R∗ − r1M (t) − R S 1 t eS = ∗ 5 RSt − r1M (t) t=51 and

55 ∗ R σ − σ ˆ RSt 1 S t eS = ∗ 5 σ RSt t=51

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Table 10.15 Beta estimates for three types of regressions (2) Xt

CAC40

SBF120

SBF250

Stock

Air Liquide

0.5789 0.5808

0.5520 0.5522

0.5664 0.5654

0.5911 0.5895

Société Générale

0.1397*

0.1748*

0.1740*

0.1714*

Danone

0.1644

0.1582

0.1630*

0.1672*

Stock

Vivendi

Valéo

Totalfina Elf

L’Oréal

Xt

CAC40

SBF120

SBF250

1.1122 1.1170

1.0748 1.0750

1.1314 1.1282

1.1878 1.1827

0.8820

0.9159

0.9537

0.9977

0.8533

0.8187

0.8569

0.8910

0.1654

0.1596

0.1650*

0.1696*

1.2302

1.1586

1.2588

1.3232

0.0200

0.0288

0.0465*

0.0363*

0.4747

0.4778

0.5062

0.5036

1.8708

1.7585

1.8422

1.9279

0.9135

0.8667

0.9235

0.9751

0.8577

0.8201

0.8565

0.8898

0.6808

0.6582

0.6842

0.7133

1.8490

1.6884

1.8192

1.9103

0.7195

0.7232

0.7699

0.7714

Renault

0.6756

0.6574

0.6868

0.7178

0.9949

0.9546

1.0407

1.0998

0.9291

0.8714

0.9200

0.9667

0.8648

0.8373

0.9086

0.9622

0.3930

0.4014

0.4228

0.4228

−0.1106*

−0.1105*

−0.1098*

−0.0954*

0.8592

0.8370

0.9125

0.9686

1.2321

1.1606

1.2607

1.3247

1.0017

0.9567

1.0395

1.0969

1.8570

1.7554

1.8495

1.9424

0.7090

0.6761

0.7209

0.7621

−0.6688*

−0.6184*

−0.6215*

−0.6960*

Note: *Non-significant estimates at a 5% test level.

Schneider

Thomson

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for simple, Jensen-type and volatility regressions. RS∗t , RS∗t − r1M (t) and σ ∗ (RSt ) are, respectively, observed market values of stock return, market risk premium and the first difference of weekly rolling volatility of stock St , while Rˆ St , (Rˆ St − rˆ1M (t)) and σ(R ˆ St ) are corresponding respective regression estimates of asset return, market risk premium and first-order weekly rolling volatility of asset St . We display our results in Table 10.16, where the first, second and third lines of each stock refer to simple, Jensen-type and volatility regressions, respectively. With regard to simple regressions, we classify benchmark-based regressions as an increasing function of the average relative absolute error, and get Xt implied factor, SBF250, SBF120 and CAC40 indexes. Indeed, Xt -based regressions exhibit the lowest absolute errors for Air Liquide, Totalfina Elf, Renault, and Schneider returns (40 percent of the stocks) while SBF250 and SBF120-based regressions exhibit the lowest ones for Danone, Vivendi and Thomson returns (30 percent of stocks), and Valéo, Société Générale, and L’Oréal returns (30 percent of stocks), respectively. Moreover, the mean of average absolute relative errors over all stocks for SBF250-based regressions is 130.8780 percent, and lies below the mean of SBF120-based regressions at is 131.2630 percent. On average, for 50 percent of cases, Xt average relative absolute errors are below the observed absolute errors of other stock indexbased simple regressions. With regard to Jensen-type regressions, ordering benchmark-based regressions with increasing value of average relative absolute errors results in Xt implied factor, SBF250, SBF120 and CAC40 indexes. In particular, Xt -based regressions exhibit the lowest average relative absolute errors for Totalfina Elf, Valéo, L’Oréal, Renault and Schneider returns (50 percent of the stocks), while SBF250 and SBF120-based regressions exhibit the lowest errors for Air Liquide, Danone, Société Générale and Thomson returns (40 percent of stocks), and Vivendi returns (10 percent of stocks), respectively. On average for 63.3334 percent of cases, Xt -based regressions exhibit average relative absolute errors that lie below the ones observed for the other stock index-based Jensen regressions. With regard to volatility regressions, ordering benchmark-based regressions with increasing value of average relative absolute errors results in Xt implied factor, SBF250, SBF120 and CAC40 indexes. Specifically, Xt -based regressions exhibit the lowest average relative absolute errors for Vivendi, Totalfina Elf, Société Générale and Renault returns (40 percent of stocks) while SBF250-based regressions exhibit the lowest for Air Liquide, Valéo and Schneider returns (30 percent of stocks). CAC40-based regressions exhibit lowest average relative absolute errors for L’Oréal and Thomson returns (20 percent of stocks), whereas SBF120-based regressions exhibit the lowest ones for Danone returns (10 percent of stocks). On average, for 50 percent of cases, Xt -based average relative absolute errors are lower than other benchmark-based average relative absolute errors.

208

Table 10.16 Average absolute relative errors for three types of regressions (percentages) Stock

Xt

CAC40

SBF120

SBF250

Air Liquide

190.2920 34.1998 93.6811

198.2960 34.4080 91.3792

199.7410 34.5058 89.5556

202.1020 34.1037 89.0935

Danone

108.7540 40.9120 100.5850

108.2980 40.8938 100.3970

106.8060 40.5194 97.5889

Vivendi

198.2090 21.3727 170.7980

203.5010 21.4338 181.0560

98.5802 75.7842 98.5722

Valéo

Mean error

Totalfina Elf

Stock

Xt

CAC40

SBF120

SBF250

Société Générale

156.1130 35.9794 156.3100

156.9040 34.9911 163.5170

145.9650 32.6395 165.2350

147.3510 32.2085 174.7310

105.9940 40.3935 97.6414

L’Oréal

58.8281 206.2060 228.1930

58.7250 208.0390 206.3310

56.9955 209.0130 242.5520

58.6216 206.6140 268.5740

179.0770 20.8868 285.0580

167.8610 21.3040 324.9240

Renault

118.5050 24.4820 76.1971

119.2220 24.7314 77.8722

123.4980 26.0944 101.8760

123.9320 26.5305 104.7220

99.4735 77.2388 103.1750

105.3530 78.5040 154.4290

107.1970 78.5563 187.6690

Schneider

178.6000 117.4490 134.1100

180.7920 119.7430 132.5430

200.0930 118.7390 127.7270

204.9850 119.3130 126.0830

119.8940 33.9410 124.9710

121.7060 34.1834 138.8180

118.5630 34.1897 122.2820

118.5890 33.9615 121.3980

Thomson

87.5991 93.7890 235.0740

84.4210 96.5114 91.5840

76.5346 90.8615 113.5140

72.1501 90.3767 164.9720

131.5370 68.4115 141.8490

133.1340 69.2174 128.6670

131.2630 68.5953 149.9820

130.8780 68.3362 165.9810

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The extremely good performance exhibited by the explanatory powers of some index-based regressions probably results from an upward bias. Such a bias comes from the non-negligible weights assigned to studied stocks from which French market indices are composed. Also, beta estimate-based analysis indicates globally that, among observed market proxies, SBF120based regressions exhibit the closest beta estimates relative to the ones we get for Xt -based regressions. On the other hand, CAC40-based regressions are far behind SBF120-based ones when considering linear relations between asset returns. Hence, in the prospect of an assessment of the impact of systematic risk on French stocks, the CAC40 INDEX represents a much less accurate approximation of market return than the implied market factor (fewer significant relations between returns). Finally, our forecasting performance study based on average relative absolute errors suggests that the Xt benchmark (implied market factor) is, on average, a more powerful proxy for systematic risk factor than our three French stock indices. Specifically, our implied market factor has a far more powerful forecasting performance than the CAC40 stock index. Such a viewpoint is sustained by the results of Jensen-type regressions when we employ the CAC40 INDEX as a market proxy (rather than implied market factor). Regarding the negative estimated abnormal returns (negative alpha coefficients): these have a higher absolute value for the CAC40 benchmark than for the implied market factor. Hence, the CAC40 benchmark leads to higher abnormal losses. On the other hand, regarding positive estimated alpha coefficients, abnormal returns brought in by French stocks are lower for the CAC40 benchmark than for the implied market factor. Moreover, there is a difference of sign for Jensen’s alpha of the stock Valéo. In conclusion, assuming that implied market factor X is an accurate proxy of the actual market portfolio, the CAC40 INDEX leads to an underestimation of positive abnormal returns and an overestimation of losses or negative abnormal returns brought in by French stocks. Consequently, CAC40 stock index does not represent the market, though it has been employed to infer the level of implied market risk factor (through some non-linear filtering methodology).

10.7 CONCLUSION Considering the wide literature about systematic risk initiated by Sharpe (1963) and the debate initiated by the famous critique by Roll (1977), we addressed the problem of finding a good market risk proxy when considering a small stock index with traded options on such an index. We proceeded in five steps: a theoretical framework; an empirical application of this setting; two empirical studies assessing impact of the implied systematic risk

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A STUDY OF THE FRENCH FINANCIAL MARKET

on French financial assets; and a critical study (forecasting performance analysis of available market benchmarks). First, our theoretical setting assumed that the small stock index was a disturbed observation of the actual market factor. This stock index depends on the market factor through a scale parameter, which is a continuous function of time. We further assumed that the market factor follows a geometric Brownian motion. Then, we induced an analytical formula pricing European calls on the stock index. All the parameters of our closed form formula are known apart from the scale parameter at times t and T, and the volatility of the market factor. Second, inverting our European call pricing formula given observed market prices of European index calls, we calculated the values of the scale factor at dates t and T, and the volatility parameter. These estimates allowed the computation of the implied market factor’s level from stock index prices. We applied this empirical study to the French financial market, and considered its CAC40 stock index. Results showed that the implied volatility parameter is time-varying, and the distributions of both volatility and market factor are leptokurtic. Third, we attempted to assess the implied market factor’s impact on a basket of French stocks and indices. We studied correlations between the implied market factor’s return and French asset returns. Results are poor in so far as our VAR study and the Granger causality test only show the strong influence of Air Liquide and Renault daily stock returns on the implied market factor’s return. Fourth, we investigated a non-linear relationship between French asset prices and the level of implied market factor. This led to the study of linear regressions of French asset returns on the implied market factor’s return. Our linear framework assumes that residual risk, which we assimilated to idiosyncratic risk, is normally distributed, with zero mean and constant variance. Results obtained are fruitful in that the implied market factor’s return appears to have a strong influence on French asset returns, apart from on Danone stock. Indeed, regressions exhibit high explanatory power. Further, we also estimated first differences regressions of French assets’ weekly rolling volatilities on the weekly rolling volatility of implied market factor. The implied market factor exhibits a strong link with CAC40, L’Oréal, Renault, SBF120, SBF250, Société Générale, Totalfina Elf, Valéo and Vivendi assets. However, it fails in explaining the whole evolution of assets, probably because idiosyncratic risk plays an important role. Indeed, such a risk factor can explain that part of assets’ evolution which remains unexplained by systematic risk factors, as suggested by Campbell et al. (2001) and Goyal and Santa-Clara (2003). Finally, we attempted to discriminate between implied market factor and French stock indices as a market proxy. Specifically, our forecasting performance study examines the highest correlation observed between both the CAC40 INDEX and implied market factor returns. Namely, the CAC40 INDEX can be a useful proxy to estimate current level of market return, whereas implied market factor is a more convincing market benchmark for

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forecast purpose and performance measure (systematic risk level), along with the CAPM setting. Suggested improvements are, first, the lengthening of the time period. A larger sample could give stronger and more significant results. Second, building a diversified portfolio (which replicates market factors accurately) would give a systematic risk benchmark to be compared with implied market factor. Prior to this, we should address what the optimal number of stocks and optimal composition of a well-diversified portfolio should be to achieve a sound and standardized assessment of systematic risk. Third, as in CAPM theory, firm-specific risk or unsystematic risk was not considered to explain realized stock returns. Our single factor framework ignores the part of any return’s global variance that is a result of firm-specific patterns. However, firm-specific factors are important explanatory variables for asset returns, as shown by Fama and French (1992, 1993) and Berk (1995). Hence, given Roll’s critique and advice in favor of multi-index models, future research should apply at least a two-factor model accounting for both systematic and idiosyncratic risk factors.

NOTES 1. Improved versions of CAPM are also given by Mossin (1966), Lintner (1965, 1969) and Black (1972). Dynamic versions of CAPM are also proposed, along with intertemporal models like that in Merton (1974). 2. Milevsky (2002) studies two dimensions of diversification, namely the number of stocks in a portfolio, and the time horizon for investment. The author discusses the benefits of the number of stocks diversification versus time horizon diversification. 3. Under completeness, financial asset prices can be reached (for example, each market variable is observable or has a proxy). 4. Drift and volatility parameters must satisfy the Lipschitz conditions, which ensure the existence and uniqueness of the solution to the stochastic differential equation satisfied by the market factor’s dynamic (given a starting value). 5. The SBF120 index is a weighted stock index composed of the forty values of CAC40 INDEX and another eighty most liquid French stocks. The SBF250 index is a weighted index composed of the 120 stocks of the SBF120, and 130 stocks selected for their importance and sector representativity. There is no traded option on such indices in the MONEP. 6. We also performed regressions with a constant term. Unfortunately, the constant coefficient does not generally appear to be significant, and its estimated value is very different from the levels of the one-, two- or three-month French risk-free rates observed in the market. We notice that constant term of regression =(1 – β) rf 55 1 for rf ∈ {¯r1M , r¯2M , r¯3M } with r¯iM = 54 riM (t) whatever i = 1, 2, 3. t=2

ACKNOWLEDGMENTS I would like to thank participants and referees at the MODSIM conference (Townsville, Australia, July 2003) and Professor Peter Verhoeven (Curtin University of Technology)

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for their helpful comments. I am also grateful to Professor Pascal St-Amour (HEC Montreal) and attendees of the EFMA annual meeting (Basle, Switzerland, July 2004) for their interesting remarks and suggestions. Finally, I thank participants at the 17th IAE National Days (Lyon, France, September 2004) and Deloitte Risk Management Conference (Antwerp, Belgium, May 2005) for their interesting comments. The usual disclaimer applies.

REFERENCES Artzner, P., Delbaen, F., Eber, J. M. and Heath, D. (1999) “Coherent Measures of Risk”, Mathematical Finance, 9(3): 203–28. Artzner, P., Delbaen, F., Eber, J. M. and Heath, D. (2000) “Risk Management and Capital Allocation with Coherent Measures of Risk”, Working paper, ETH Zentrum. Berk, J. B. (1995) “A Critique of Size-Related Anomalies”, Review of Financial Studies, 8(2): 275–86. Black, F. (1972) “Capital Market Equilibrium with Restricted Borrowing”, Journal of Business, 45(3): 444–55. Black, F. (1975) “Fact and Fantasy in the Use of Options”, Financial Analysts Journal, 31(July/August): 36–72. Black, F. and Scholes, M. (1973) “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, 81(1): 637–54. Brailsford, T. J. and Faff, R. W. (1997) “Testing the Conditional CAPM and the Effect of Intervaling: A Note”, Pacific-Basin Finance Journal, 5(5): 527–37. Campbell, J. Y., Lo, A. W. and MacKinlay, A. C. (1997) The Econometrics of Financial Markets (Princeton, NJ: Princeton University Press). Campbell, J. Y., Lettau, M., Malkiel, B. G. and Xu, Y. (2001) “Have Individual Stocks become More Volatile? An Empirical Exploration of Idiosyncratic Risk”, Journal of Finance, 56(1): 1–43. Fama, E. and French, K. (1992) “The Cross-Section of Expected Stock Returns”, Journal of Finance, 47(2): 427–65. Fama, E. and French, K. (1993) “Common Risk Factors in the Returns on Stocks and Bonds”, Journal of Financial Economics, 33: 3–56. French, K. R. and Poterba, J. M. (1991) “International Diversification and International Equity Markets, American Economic Review, 81(2): 222–6. Gençay, R., Selçuk, F. and Whitcher, B. (2003) “Systematic Risk and Timescales”, Quantitative Finance, 3, April: 108–16. Gourieroux, C. and Jasiak, J. (2001) Financial Econometrics: Problems, Models, and Methods (Princeton, NJ: Princeton University Press). Goyal, A. and Santa-Clara, P. P. (2003) “Idiosyncratic Risk Matters!”, Journal of Finance, 58(3): 975–1007. Granger, C. W. J. (1969) “Investigating Causal Relations by Econometric Models and Cross Spectral Methods”, Econometrica, 37(3): 424–38. Hamilton, J. D. (1994) Time Series Analysis (Princeton, NJ: Princeton University Press). Jagannathan, R. and Wang, Z. (1996) “The Conditional CAPM and the Cross-Section of Expected Returns”, Federal Reserve Bank of Minneapolis, Research Department Staff Report 208. Jensen, C. M. (1968) “The Performance of Mutual Funds in the Period 1945−1964”, Journal of Finance, 23(2): 389–415. Jensen, C. M. (1969) “Risk, the Pricing of Capital Assets, and the Evaluation of Investment Portfolios”, Journal of Business, 42(2): 167–247.

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Kandel, S. and Staumbaugh, R. (1987) “On Correlations and Inferences about MeanVariance Efficiency”, Journal of Financial Economics, 18(1): 61–90. Koutmos, G. and Knif, J. (2002) “Estimating Systematic Risk Using Time-Varying Distributions”, European Financial Management, 8(1): 59–73. Lintner, J. (1965) “The Valuation of Risky Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets”, Review of Economics and Statistics, 47(1): 13–37. Lintner, J. (1969) “The Aggregation of Investor’s Diverse Judgments and Preferences in Purely Competitive Security Markets”, Journal of Financial and Quantitative Analysis, 4(4): 347–400. McKenzie, M. D. and Faff, R. W. (2003) “The Determinants of Conditional Autocorrelations in Stock Returns”, Journal of Financial Research, 26(2): 259–74. Mele, A. (1998) Dynamiques Non Linéaires, Volatilité et Equilibre, Collection Approfondissement de la Connaissance Economique, Economica. Merton, R. C. (1973) “The Theory of Rational Option Pricing”, Bell Journal of Economics & Management Science, 4(1): 141–83. Merton, R. C. (1974) “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates”, Journal of Finance, 29(2): 449–70. Milevsky, M. A. (2002/03) “Space-Time Diversification: Which Dimension is Better?”, Journal of Risk, 5(2) Winter: 45-71. Mossin, J. (1966) “Equilibrium in a Capital Asset Market”, Econometrica, 34: 768–83. Rogers, L. C. G. and Williams, D. (1994a) Diffusions, Markov Processes and Martingales: Foundations, vol. 1 (Cambridge University Press). Rogers, L. C. G. and Williams, D. (1994b) Diffusions, Markov Processes and Martingales: It¯o Calculus, vol. 2 (Cambridge University Press). Roll, R. (1977) “A Critique of the Asset Pricing Theory’s Tests”, Journal of Financial Economics, 4(2): 129–76. Shanken, J. (1987) “Multivariate Proxies and Asset Pricing Relations: Living with the Roll Critique”, Journal of Financial Economics, 18(1): 91–110. Sharpe, W. F. (1963) “A Simplified Model for Portfolio Analysis”, Management Science, 9: 499–510. Sharpe, W. F. (1964) “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk”, Journal of Finance, 19(3): 425–42. Sharpe, W. F. (1970) Portfolio Theory and Capital Markets (New York: McGraw-Hill). Stulz, R. M. (1999a) “International Portfolio Flows and Security Markets”, In M. Feldstein, (ed.), International Capital Flows (Chicago: University of Chicago Press). Stulz, R. M. (1999b) “Globalization, Corporate Finance, and the Cost of Capital”, Journal of Applied Corporate Finance, 12: 8–25. Stulz, R. M. (1999c) “Globalization of Equity Markets and the Cost of Capital”, Working paper, Dice Center, Ohio State University. Szego, G. (2002) “Measures of Risk”, Journal of Banking and Finance, 26(7): 1253–72. Treynor, J. (1961) “Toward a Theory of the Market Value of Risky Assets”, unpublished manuscript. Whaley, R. (1982) “Valuation of American Call Options on Dividend-paying Stocks”, Journal of Financial Economics, 10(1): 29–58.

C H A P T E R 11

Matrix Elliptical Contoured Distributions versus a Stable Model: Application to Daily Stock Returns of Eight Stock Markets Taras Bodnar and Wolfgang Schmid

11.1 INTRODUCTION The assumptions of independency and normality are not appropriate in many situations of practical interest, especially in modeling financial data from emerging markets. It was pointed out in numerous studies that daily financial data is heavily tail distributed (Blattberg and Gonedes, 1974; Fama, 1976; Engle, 1982; Bollerslev, 1986; Nelson, 1991; Rachev and Mittnik, 2000). These studies proposed to pick up the assumptions of t-distribution, symmetric stable distribution, or the autoregressive conditional heteroskedasticity (ARCH) process instead of normality. In this chapter, the much weaker assumption of matrix ellipticalcontoured distribution is imposed on the asset returns. This family covers a wide range of distributions – for example, the matrix normal distribution, the matrix mixture of normal distribution, the matrix t-distribution and 214

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the matrix symmetric stable distribution. Elliptical distributions, whose contours of equal densities have the same elliptical shape as the normal distribution, provide an attractive alternative to the multivariate stable law. These distributions have been already discussed in financial literature (Chamberlain, 1983; Owen and Rabinovitch, 1983; Zhou, 1993; Berk, 1997; Bodnar and Schmid, 2003, 2004). For instance, Owen and Rabinovitch (1983) showed that Tobin’s separation theorem, Bawa’s rules of ordering certain prospects can be extended to elliptically contoured distributions. While Chamberlain (1983) showed that elliptical distributions imply mean– variance utility functions, Berk (1997) argued that one of the necessary conditions for the capital asset pricing model (CAPM) is an elliptical distribution for the asset returns. Furthermore, Zhou (1993) extended findings of Gibbons et al. (1989) by applying their test for the validity of the CAPM to elliptically distributed returns. The first paper dealing with the application of matrix elliptically contoured distributions in finance, however, seems to be Bodnar and Schmid (2003). They introduced a test for the global minimum variance. It is analyzed whether the lowest risk is larger than a given benchmark value or not. The aim of the present study is to derive the statistical procedures for testing the elliptical symmetry of multivariate sample, for example, matrix ellipticity. While there are several procedures for testing the multivariate elliptical and spherical symmetry under independency assumptions (Beran, 1979; Baringhaus, 1991; Fang et al., 1993; Heathcote, et al., 1995; Koltchinskii and Li, 1998; Manzotti et al., 2002; Zhu and Neuhaus, 2003), the matrix elliptical symmetry was not treated in literature up to now. Furthermore, only the limiting distributions of above-mentioned statistics were derived in the studies. Conversely, our approaches are based on the small sample tests. Empirically we show that daily returns of seven developed countries follow a matrix elliptical distribution. This result is in line with the findings of Andersen et al. (2001), who, using the distributional properties of realized volatility, argue that daily returns can be well approximated by the mixture of normal distributions (the partial case of elliptical family). The remainder of the chapter is organized as follows. The main results are presented in section 11.2. Under the null hypothesis of matrix elliptical symmetry, the finite sample distributions of the proposed statistics are derived in all cases when the type of elliptical symmetry, location vector and scale matrix are known or unknown. The test’s powers are considered in section 11.3. In section 11.4 we implement our findings empirically by considering daily returns of seven developed stock markets. We show that the null hypothesis of the matrix ellipticity cannot be rejected for this data. Furthermore, as the test power for the symmetric stable distribution is always very high, a practitioner should be very careful with modeling daily financial data by using the stable law. Final remarks and conclusions are presented in Section 11.5. All proofs are given in the Appendix.

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11.2 SMALL SAMPLE TESTS Before presenting the main results of the study, we introduce the family of the matrix elliptical contoured distributions and briefly discuss their main properties. Following Gupta and Varga (1993) a random matrix X of dimensions k × n is said to have a matrix variate elliptically contoured distribution if its characteristic function has the form

(T) = exp(tr(iT M)) (tr(T T)) where T and M are k × n matrices, is a k × k positive semidefinite matrix, and : [0, ∞) → R. The symbol tr stands for the trace of a matrix. This family of distributions we denote by Ek,n (M, , ). If X = (X1 , … ,Xn ) ∼ Ek,n (M, , ) and if its second moments exist then it holds with M = (µ1 , … ,µn ) that the random vectors X1 , … , Xn are uncorrelated and Xi ∼ Ek,1 (µi , , ) (see Gupta and Varga, 1993, theorem 2.4.1, corollaries 2.4.1.1 and 2.4.1.2, theorem 2.3.2). Thus the columns of X follow a vector elliptically contoured distribution. It holds that E(X) = M and = Cov(Xi ) = −2 (0) (see Fang and Zhang, 1990, theorem 2.6.5). Note that the columns are independent if X follows a matrix variate normal distribution, for example, if is taken as being equal to exp(−x/2) (Gupta and Varga, 1993, theorem 2.1.5). Assuming X to be absolutely continuous, it follows that X ∼ Ek,n (M, , ) if and only if the density of X has the form f (X) = det()−n/2 h(tr((X − M) −1 (X − M)), where h and determine each other for specified k and n (see Gupta and Varga, 1993, theorem 2.2.1). The matrix elliptically contoured distributions possess several desirable properties that have been observed for financial assets. It presents an extension of the assumption of an independent normal sample. First, the returns must not be independent, and, second, they may have heavy tails. The aim of our study is to compare the ability of the matrix elliptical and symmetric stable distributions to explain the stochastic behavior of daily asset returns. For these purposes, we derive statistical procedures for testing the matrix elliptical symmetry of a multivariate sample. The cases of known and unknown nature of elliptical symmetry, scale parameters and location vector are considered. Let us denote the set of distribution functions with the known location vector, the known scale matrix, and the known characteristic function by µ,, = {F : F ∈ Ek,n (µ × 1, , )}

(11.1)

If some parameters are not precisely known we put points on the corresponding places in Equation (11.1). For example, when the location vector µ is unknown, we put .,, = {F : F ∈ Ek,n (µ × 1, , ), µ ∈ Rk }

(11.2)

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In a similar way, the testing hypotheses are denoted. When the location vector is unknown we obtain H0,.,, : X ∼ F ∈ .,,

against H1,.,, : X ∼ F ∈ .,,

(11.3)

11.2.1 Known type of elliptical symmetry (known characteristic function) In this section we derive several procedures to test the ellipticity of the sample with the precise definition of the type of elliptical symmetry, for example, to test if X ∼ Ek,n (µ × 1 , , ) with the known characteristic function . First, we consider the case of the known scale matrix and unknown location vector µ. The test hypothesis is H0,.,, : X ∼ F ∈ .,,

against H1,.,, : X ∼ F ∈ .,, .

(11.4)

Let us denote the following random variable ˆ τ τ Q1 = τ τ

(11.5)

where τ is a nonzero vector of constants and is estimated by ˆ =

n 1 1 1 (Xt − X)(Xt − X) = X(I − 11 )X n−1 n−1 n

(11.6)

t=1

The stochastic representation of the random matrix X is essential for deriving the distribution of Q1 . Let be positive definite. It holds that X ∼ Ek,n (M, , ) if and only if X has the same distribution as M + R 1/2 U, where U is a k × n random matrix and vec (U ) is uniformly distributed on the unit sphere in Rkn , R is a non-negative random variable, and R and U are independent (see Gupta and Varga, 1993, theorem 2.5.2). The expression M + R 1/2 U is a stochastic representation of X, that is, it holds that X ≈ M + R 1/2 U. The symbol A ≈ B says that the two random variables A and B have the same distribution. The variable R is called the generating variable of X. The distribution of R2 is equal to the distribution of ni=1 (Xi − µi ) −1 (Xi − µi ). If X is absolutely continuous, then R is also absolutely continuous and its density is fR (r) =

2πnk/2 nk−1 r h(r2 ) nk 2

(11.7)

for r ≥ 0 (see Gupta and Varga, 1993, theorem 2.5.5). Note that for the matrix 2 ∼ χ2 . The index N refers to the normal variate normal distribution RN nk distribution.

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Lemma 1. Let X = (X1 . . . Xn ) ∼ Ek,n (M,,) and n > k. Let be positive definite and suppose that X is absolutely continuous. Then it holds that (n − 1) Q1 has a stochastic representation R2 b, for example, (n − 1)Q1 ≈ R2 b with R being the generating variable of X, nk−n+1 b ∼ B( n−1 ) (Beta distribution), and the random variables R and b are independent. 2 2 ,

The proof of the lemma is given in the Appendix. From the result of Lemma 1, the moment sequence of random variable Q is calculated mi = E(Qi ) =

k E(R∗2i ) (i + n−1 2 ) ( 2 ) k (n − 1)i ( n−1 2 ) (i + 2 )

(11.8)

where R∗ is the generating variable of X1 . Finally, from Lemma 1 we obtain Theorem 1 Let X = (X1 , . . . , Xn ) ∼ Ek,n (µ × 1 , , ) and n > k. Then it follows that under the null hypothesis H0,.,, the test statistic T1 = (n−1)Q1 has the same distribution as R˜ 1 , where R˜ 1 is the generating variable of En−1,1 (.,., ).

In case of the known scale matrix and the known location vector µ we consider the following random variable Q2 =

ˆ ˜ τ τ τ τ

with

n ˆ˜ = 1 (X − µ)(X − µ) t t n

(11.9)

t=1

The test hypothesis is given by H0,µ,, : X ∼ F ∈ µ,,

against

H1,µ,, : X ∼ F ∈ µ,,

ˆ˜ has a Wishart distriFrom the result of Lemma 1 and the fact that n bution with n degrees of freedom and the parameter matrix , namely, ˆ˜ ∼ W (n, ), it follows n k Theorem 2 Let X = (X, . . . Xn ) ∼ Ek,n (µ × 1 , , ) and n > k. Then it follows that under the null hypothesis H0,µ,, the test statistic T2 = nQ2 has the same distribution as R˜ 2 , where R˜ 2 is the generating variable of En,1 (.,.,).

11.2.2 Unknown type of elliptical symmetry (unknown characteristic function) In contrast to the distributional properties of the proposed statistics in section 11.2.1, the test statistics presented in this section are distributional free within the class of matrix elliptical contoured distributions. Their distributions, in general, are presented by central F-distributions with some degrees of freedom. These results make them available to be applied without specifying the concrete type of elliptical symmetry. Again, the cases with known

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and unknown location vectors are treated separately. We omit the moments requirements imposed on the asset returns. The proposed approaches can be applied to rather heavy-tailed distributions, even which do not possess the first and higher moments. Let X = (X(1) , X(2) ), where X(1) = (X1 , . . . , Xn1 ) and X(2) = (X1+n1 , . . . , Xn ) with n2 = n − n1 . First, we treat the case of unknown location vector µ. When the scale matrix is unknown one has to estimate it by previous observations. Then based on the first n1 observations we estimate ˆ 1) = (n

n1 1 1 1 (Xt − X¯ (1) )(Xt − X¯ (1) ) = X(1) (I − 11 )X(1) n1 − 1 n1 − 1 n1 t=1

Using the rest of the n2 observations we obtain ˆ 2) = (n

1 n2 − 1

n

(Xt − X¯ (2) )(Xt − X¯ (2) ) =

t=n1 +1

1 1 X(2) (I − 11 )X(2) n2 − 1 n2

Finally, to test the null hypothesis H0,.,.,. : X ∼ F ∈ .,.,.

against

H1,.,.,. : X ∼ F ∈ .,.,.

(11.10)

we use the result of the following theorem Theorem 3 Let X = (X1 . . . Xn ) ∼ Ek,n (µ × 1 , , ) and n > k. Then it follows that ˆ 2 )τ/τ (n ˆ 1 )τ has a central under the null hypothesis H0 ,.,.,. the test statistic T3 = τ (n F-distribution with n2 − 1 and n1 − 1 degrees of freedom.

The proof of the theorem is given in the Appendix. If the location vector µ is known, a similar statistic as above is considered. However, the estimators of the scale parameters are given by ˆ˜ (n 1) =

n1 1 (Xt − µ)(Xt − µ) n1 − 1

ˆ˜ and (n 2)

t=1

n 1 = (Xt − µ)(Xt − µ) n2 − 1 t=n1 +1

correspondingly. From Muirhead (1982, theorem 3.2.8) and Fang and Zhang (1990, theorem 5.1.1) we obtain Let X = (X1 , . . . , Xn ) ∼ Ek,n (µ × 1 , , ) and n > k. Then it follows that ˆ ˆ˜ ˜ 2 )τ/τ (n under the null hypothesis H0,µ,.,. the test statistic T4 = τ (n 1 )τ has a central F-distribution with n2 and n1 degrees of freedom. Theorem 4

Note that the distributions of T1 , T2 , T3 and T4 statistics do not depend on τ.

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MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS

11.2.3 Further statistics If the type of ellipticity is unknown, additionally, the following four statistics are considered: −1 ˆ τ −1 τ n − k ˆ −1 ˆ )−1 − −1 T5 = (LR()L (τ τ) −1 p τ τ ˆ τ τ −1 ˆ τ −1 τ × − −1 τ τ ˆ −1 τ τ ˆ˜ −1 τ −1 τ n − k ˆ −1 ˆ˜ )−1 ˜ τ) − −1 (τ (LR()L T6 = ˆ˜ −1 τ p τ τ τ ˆ˜ −1 τ −1 τ × − −1 ˆ −1 τ τ ˜ τ τ −1 ˆ (n2 )τ n − k ˆ −1 T7 = (τ (n2 )τ) (n2 ) − p ˆ −1 τ τ −1 ˆ (n2 )τ −1 ˆ 2 ))L ) × (LR((n − ˆ −1 (n2 )τ τ

ˆ −1 (n1 )τ

ˆ −1 (n1 )τ τ

ˆ −1 (n1 )τ

ˆ −1 (n1 )τ τ

ˆ˜ −1 (n )τ ˆ˜ −1 (n )τ n − k ˆ −1 2 1 ˜ (n2 )τ) T8 = − (τ ˆ −1 ˆ −1 p ˜ ˜ τ (n2 )τ τ (n1 )τ ˆ˜ −1 × (LR((n 2 ))L )

ˆ˜ −1 (n )τ ˆ˜ −1 (n )τ 2 1 − ˆ −1 ˆ −1 ˜ (n2 )τ ˜ (n1 )τ τ τ

where R(A) = A−1 − A−1 ττ A−1 /τ A−1 τ. The statistic T5 is used to test the null hypothesis H0,.,,. : X ∼ .,,. , while the statistic T6 corresponds to the hypothesis H0,µ,,. : X ∼ µ,,. , the statistic T7 to H0,.,.,. : X ∼ .,.,. , and the statistic T8 to H0,µ,.,. : X ∼ µ,.,. . L is a p × k matrix of constants, p ≤ k − 1, such that (L,τ) is of full rank p + 1. The distributions of these statistics have already been derived in Bodnar (2004) and Bodnar and Schmid (2004). They do not depend on the type of elliptical symmetry within the class of matrix elliptical contoured. The critical values of the T7 - and T8 statistics can be obtained by numerical integration from the mathematical software package Mathematica. They are twice as large as the corresponding ones of central F-distributions (T5 - and T6 statistics).

Power

TARAS BODNAR AND WOLFGANG SCHMID

0.40 0.39 0.38 0.37 0.36 0.35 0.34 0.33 0.32 0.31 0.30 0.29 0.28 0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.20 0.19 0.18 0.17 0.16 0.15 0.14 0.13

221

T4 T8

0

1

2

3

4

5

6

7

8

9

10

Degrees of freedom

Figure 11.1 Power functions of the T4 and T8 tests for testing sample elliptical symmetry against independent multivariate t-distribution with different degrees of freedom

11.3 ANALYSIS OF THE POWER FUNCTIONS In this section we deal with the power functions of the proposed tests. Note that the rejection of the null hypothesis may be caused by a change in the covariance matrix or misspecification of the underlying distribution. As the first case is not of interest to us, we fix the covariance matrix of the process. Then the rejection of the null hypothesis is the result of an incorrect specification. Since a huge number of alternative hypotheses can be modeled, we do not consider all of them. Furthermore, because of the analytical difficulties of deriving the distributional function under the alternative hypothesis H1 to calculate the power of the test, we apply a Monte Carlo study. Several situations are modeled by drawing a sample of an independent multivariate t-distribution in the first case and an independent symmetric multivariate stable distribution in the second one. The location vector is chosen to be 0, and the scale matrix is identical. This choice is not restrictive, as neither of the T4 and T8 statistics depend on τ, and T8 is independent of L. In all cases, 104 seven-dimensional vectors of the corresponding distributions are drawn. The procedures for generating a multivariate symmetric stable distribution and a multivariate t-distribution are discussed in the Appendix. The powers of the T4 and T8 tests are shown in Figure 11.1 for multivariate t-distributions with different degrees of freedom. The powers of both tests decrease as the degrees of freedom increase. It is not surprising that the t-distribution converges to the normal when the degree of freedom tends

Power

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MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS

0.66 0.64 0.62 0.60 0.58 0.56 0.54 0.52 0.50 0.48 0.46 0.44 0.42 0.40 0.38 0.36 0.34 0.32 0.30 0.28

T8

T4

0.6

0.8

1.0

1.2 1.4 Stability index

1.6

1.8

2.0

Figure 11.2 Power function of the T4 and T8 tests for testing sample elliptical symmetry against independent symmetrical multivariate stable distribution with different stability indices to infinity. Finally, the power functions of the T4 and T8 statistics are almost the same for the different degrees of freedom. Figure 11.2 contains the power functions of the T4 and T8 tests for independent symmetric stable distributions with different stability indices. The power of a T8 test is always higher than the power of T4 . When the stability index is around one, the power is over 0.6. For larger values of stability indices this probability decreases. It is around 35 percent when the stability index values are around 1.65, the recommended value for describing daily data (Blattberg and Gonedes, 1974). We make use of these results in the next section, when an empirical example of the daily returns of seven developed stock markets is discussed.

11.4 EMPIRICAL STUDY In this section, the results of the empirical study are presented. Because the location parameter, the scale matrix and the type of elliptical symmetry are usually unknown in a practical situation, we make use of the T8 statistic for testing the null hypothesis that daily returns follow a matrix ellipticalcontoured distribution. It is seen how the finite sample properties of these statistics can be used. We consider the daily price data from Morgan Stanley Capital International for the equity markets of seven developed countries (France, Germany, Italy, Japan, Spain, the UK and the USA) for the period January 1, 1994 to December 31, 2000. We group our data set by half-year

TARAS BODNAR AND WOLFGANG SCHMID

223

Table 11.1 The 5% and 10% critical values for the T8 statistic depending on the sample sizes n1 , n2 (k = 1) 0.05

α n1 \n2

63

0.1

0.05

63

64

0.1 64

0.05

0.1

65

65

63

8.195

5.695

7.695

5.495

7.265

5.287

64

8.815

5.927

8.190

5.691

7.689

5.498

65

9.581

6.192

8.792

5.927

8.176

5.684

Table 11.2 Value of the T8 statistic for different linear restrictions Year\Test

T 8, l 1

T 8, l 2

T 8, l 3

T 8, l 5

T 8, l 6

1994, I

2.608

1.283

5.175

1994, II

0.106

3.781

0.271

0.840

6.629

0.418

5.597

0.008

1995, I

2.616

1.702

0.168

1995, II

3.869

0.128

2.482

4.285

0.123

0.650

0.014

2.981

5.399

0.0

5.077

1996, I

3.552

3.827

1996, II

2.883

5.263

4.098

2.190

0.137

0.667

0.305

8.092

1997, I

0.001

3.569

4.247

0.025

0.170

0.761

1.146

1997, II

0.084

1.034

1998, I

0.270

1.808

0.255

1998, II

0.075

0.899

1999, I

0.108

1999, II

2.756

2000, I 2000, II

0.599

T 8, l 4 11.50

13.79 3.209

0.809 0.005

0.017

3.436

2.668

0.159

1.726

4.466

2.224

5.387

0.786

3.592

0.152

0.021

3.736

0.027

0.206

2.605

1.968

1.149

1.629

0.136

1.743

0.734

0.518

0.019

0.637

0.142

2.176

0.675

2.753

8.155

2.568

3.298

0.314

25.63

20.73

4.815 11.93

T 8, l 7

11.57

0.673 4.682

11.18

data. Furthermore, each group is additionally partitioned into two subsamples of three-month data sets. The first sub-sample is used to calculate ˆ 1 ) and the second for (n ˆ 2 ). (n In Table 11.1, the 5 percent and 10 percent critical values of the T8 statistics for different sample sizes, n1 , n2 ∈ 63,64,65 are presented. These change significantly with a change of sample size. However, the critical values are almost the same on the diagonals that are parallel to the main diagonal of the table. The values of the T8 statistic are presented in Table 11.2. They are calculated for different linear restrictions, for example, l1 = (1, 0, 0, 0, 0, 0, 0), . . . , l7 = (0, 0, 0, 0, 0, 0, 1). The values of the statistics that are greater than the 5 percent critical value are indicated in bold type. The null hypothesis of matrix elliptical symmetry is rejected in null cases out of fourteen for the T8

224

MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS

statistic with the l1 and l2 linear restrictions. In other cases there are only two rejections per a column. In general we observe seven rejections out of ninety-eight for the 5 percent level of significance, and ten rejections out of ninety-eight for the 10 percent level. Keeping in mind that the rejection of the null hypothesis can also be effected by changes in the covariance structure of the stock returns process, we are unable to reject the matrix ellipticity of the considered returns. From the other side, using the results of section 11.3 (the test power is very high for the multivariate symmetric stable distribution), one should be very careful with the assumption that daily stock returns follow a multivariate stable law.

11.5 CONCLUSION In this chapter, several statistics for testing the null hypothesis of a matrix elliptical-contoured distribution are proposed. The finite sample properties are derived in all cases of known and unknown types of elliptical symmetry, scale matrix and location vector. The T1 - and T2 statistics do not possess the invariance property with respect to matrix elliptical-contoured distributions and their null distributions are specified by the corresponding generating variables. From the other side, the statistics from T3 to T8 are distributionally free. Their stochastic properties, apart from T7 and T8 , are based on the central F-distribution with some degrees of freedom. The control limits of the T7 - and T8 statistics can be obtained by numerical calculations in the software package Mathematica. We applied the T8 statistic in a situation of practical interest by considering the daily stock returns of seven developed stock markets. The null hypothesis of the matrix elliptical symmetry is rejected ten times out of ninety-eight for the 10 percent level of significance. In addition, the rejection of the null hypothesis may be caused by changes in the covariance structure of the underlying distribution. Keeping everything together, we conclude that the results of the empirical study provide support to model the daily data by matrix elliptical-countered distributions. They are in line with the suggestions of Andersen et al. (2001) and Andersen et al. (2004), who argued that daily returns normalized by the realized volatility can be well approximated by normal distribution. Furthermore, researchers should be very careful with the application of the multivariate symmetric stable law in modeling daily data. Instead, the assumption of a matrix elliptical-contoured distribution should be maintained.

APPENDIX In this section, the proofs of Lemma 1 and Theorem 3 are given. Furthermore, we deal with the problem of generating independent multivariate t- and symmetrical stable distributed random vectors.

TARAS BODNAR AND WOLFGANG SCHMID

225

We say that a characteristic function belongs to the class (U) if it can be equal to another characteristic function in the neighborhood of zero without being identical to it. Correspondingly, a characteristic function belongs to the class (U) if it does not belong to (U). We denote a set of orthogonal matrices of the order k by O(k).

Proof of Lemma 1 We have it that Q ≈ R2 τ

(1/2 U(I − 11 /n)U 1/2 )τ τ τ

= R2 Q∗ . A similar presentation is obtained

2 Q . The index N is used to when X is matrix normally distributed; for example, QN ≈ RN ∗ 2 ∼ χ2 , R2 Q ∼ χ2 , it follows from Fang and Zhang indicate the normal case. Because RN N ∗ n−1 nk 1 nk − n + 1 2 Q ≈ R2 b . Furthermore, we ) exists such that RN (1990, p. 59) that b∗ ∼ B( n − ∗ N ∗ 2 , 2 2 > 0) = 1 and P(b > 0) = 1. have it that P(RN ∗ ˆ is positive definite with From the assumption of the lemma, it follows that probability 1 (see Muirhead, 1982, theorem 3.1.4). Hence, Q is greater than 0 with probability 1, and therefore, P(Q∗ > 0) = 1. From the above consideration, it follows that the density of ln b∗ is

fln b∗ (t) =

n−1

et 2 (1 − et ) 0,

nk−n+1 −1 2

,

if if

t≤0 t>0

∞ Hence, for any positive r it holds 0 ert fln b∗ (t)dt = 0 < ∞. Using the results of Fang and Zhang (1990), it follows that characteristic function φln b∗ ∈ (U). Thus, from the property of the operation ≈ (see Fang and Zhang, 1990, p. 38) it follows that Q∗ ≈ b∗ . As a 1 nk − n + 1 result we obtain Q∗ (R2 ) ≈ R2 Q∗ ≈ R2 b∗ , where b∗ ∼ B( n − ) and R2 and b∗ are 2 , 2 independently distributed.

Proof of Theorem 3 Let us consider T3 =

ˆ 2 )τ ˆ 2 )τ τ τ τ (n τ (n = ˆ 1 )τ ˆ 1 )τ τ τ τ (n τ (n

The rest of the proof follows from Muirhead (1982, theorem 3.2.8) and Fang and Zhang (1990, theorem 5.1.1).

Drawing samples from multivariate t- and symmetric stable distributions The way of generating samples of independent multivariate t-distributions and symmetric stable distributions follow immediately from their stochastic representations. From Fang et al. (1990, p. 85) we obtain that the k-dimensional t-distributed random vector Y is equal to Y = Z/

χ n

(11.11)

226

MATRIX ELLIPTICAL CONTOURED DISTRIBUTIONS

where Z has a k variate normal distribution, χ has a chi-squared distribution with n degrees of freedom, and Z and χ are independently distributed. The stochastic representation of the k-dimensional stable random vector with the index of stability equal to α, 0 < α < 2 is given by S=

√ AZ

(11.12)

where Z has a k variate normal distribution, A has an univariate α/2-stable distribution with skewness parameter equal to 1, the location parameter 0, and the scale parameter (cos(πα/4))2/α , and Z and A are independently distributed (see Samorodnitsky and Taqqu, 1994, p. 77). Following Kantner (1975), the stochastic representation of A is

A = cos (πα/4)

1/α

sin ((1 − α/2)θ) sin (αθ/2)α/(2−α) sin (θ)2/(2−α) W

(2−α)/α (11.13)

where θ is uniform on (0,π),W has a standard exponential distribution, and θ and W are independently distributed.

REFERENCES Andersen, T. G., Bollerslev, T. and Diebold, F. X. (2004) “Parametric and Nonparametric Measurements of Volatility”, in Y. Aït-Sahalia and L. P. Hansen (eds), Handbook of Financial Econometrics, (Amsterdam: North-Holland). Andersen, T. G., Bollerslev, T., Diebold, F. X. and Ebens, H. (2001) “The Distribution of Realized Exchange Rate Volatility”, Journal of the American Statistical Association, 96(453): 42–55. Baringhaus, L. (1991) “Testing for Spherical Symmetry of a Multivariate Distribution”, The Annals of Statistics, 19(2): 899–917. Beran, R. (1979) “Testing Ellipsoidal Symmetry of a Multivariate Density”, The Annals of Statistics, 7(1): 150–62. Berk, J. B. (1997) “Necessary Conditions for the CAPM”, Journal of Economic Theory, 73(1): 245–57. Blattberg, R. C. and Gonedes, N. J. (1974) “A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices”, The Journal of Business, 47(2): 244–80. Bodnar, T. (2004) “Optimal Portfolios in an Elliptical Model – Statistical Analysis and Tests for Efficiency”, Ph.D. thesis, Europa University Viadrina, Frankfurt (Oder), Germany. Bodnar, T. and Schmid, W. (2003) “The Distribution of the Global Minimum Variance Estimator in Elliptical Models”, EUV Working paper 22. Bodnar, T. and Schmid, W. (2004) “A Test for the Weights of the Global Minimum Variance Portfolio in an Elliptical Model”, EUV Working paper 2. Bollerslev, T. (1986) “Generalized Autoregressive Conditional Heteroskedasticity”, Journal of Econometrics, 31(3): 307–27. Chamberlain, G. A. (1983) “A Characterization of the Distributions that Imply MeanVariance Utility Functions”, Journal of Economic Theory, 29(1): 185–201. Engle, R. F. (1982) “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation”, Econometrica, 50(4): 987–1008. Fang, K. T. and Zhang, Y. T. (1990) Generalized Multivariate Analysis (Berlin: Springer-Verlag Beijing: Science Press). Fang, K. T., Kotz, S. and Ng, K. W. (1990) Symmetric Multivariate and Related Distributions (London: Chapman & Hall).

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Fang, K. T., Zhu, L. X. and Bentler, P. M. (1993) “A Necessary Test of Goodness of Fit for Sphericity”, Journal of Multivariate Analysis, 45(1): 34–55. Fama, E. F. (1965) “The Behavior of Stock Market Prices”, Journal of Business, 38(1): 34–105. Fama, E. F. (1976) Foundations of Finance (New York: Basic Books). Gibbons, M. R., Ross, S. A. and Shanken, J. (1989) “A Test of the Efficiency of a Given Portfolio”, Econometrica, 57(5): 1121–52. Gupta, A. K. and Varga, T. (1993) Elliptically Contoured Models in Statistics (Dondrecht: Kluwer Academic). de Haan, L. and Rachev, S. T. (1989) “Estimates of the Rate of Convergence for Max-Stable Processes”, The Annals of Probability, 17(2): 651–77. Heathcote, C. R., Cheng, B. and Rachev, S. T. (1995) “Testing Multivariate Symmetry”, Journal of Multivariate Analysis, 54(2): 91–112. Kantner, M. (1975) “Stable Densities under Change of Scale and Total Variation Inequalities”, The Annals of Probability, 3(4): 697–707. Koltchinskii, V. I. and Li, L. (1998) “Testing for Spherical Symmetry of a Multivariate Distribution”, Journal of Multivariate Analysis, 65(2): 228–44. Manzotti, A., Perez, F. J. and Quiroz, A. J. (2002) “A Statistic for Testing the Null Hypothesis of Elliptical Symmetry”, Journal of Multivariate Analysis, 81(2): 274–85. Muirhead, R. J. (1982) Aspects of Multivariate Statistical Theory (New York: John Wiley). Nelson, D. (1991) “Conditional Heteroscedasticity in Stock Returns: A New Approach”, Econometrica, 59(2): 347–70. Owen, J. and Rabinovitch, R. (1983) “On the Class of Elliptical Distributions and Their Applications to the Theory of Portfolio Choice”, The Journal of Finance, 38(3): 745–52. Rachev, S. T. and Mittnik, S. (2000) Stable Paretian Models in Finance. (New York: John Wiley). Samorodnitsky, G. and Taqqu, M. S. (1994) Stable Non-Gaussian Random Processes, Stochastic Models with Infinite Variance (New York/London: Chapman & Hall). Tu, J. and Zhou, G. (2004) “Data-generating Process Uncertainty: What Difference Does It Make in Portfolio Decisions?”, Journal of Financial Economics, 72(2): 385–421. Zhou, G. (1993) “Asset-pricing Tests under Alternative Distributions”, The Journal of Finance, 48(5): 1927–42. Zhu, L. X. and Neuhaus, G. (2003) “Conditional Tests for Elliptical Symmetry”, Journal of Multivariate Analysis, 84(2): 284–98.

C H A P T E R 12

The Modified Sharpe Ratio Applied to Canadian Hedge Funds Greg N. Gregoriou

12.1 INTRODUCTION The assessment of portfolio performance is fundamental for both investors and fund managers, and this applies also to Canadian hedge funds. Traditional portfolio measures present some limitations when applied to hedge funds. For example, the Sharpe ratio uses the excess reward per unit of risk as a measure of performance, with risk represented by the standard deviation. The mean-variance approach to the portfolio selection problem developed by Markowitz (1952) has frequently been the subject of undue criticism because of its utilization of variance as a measure of risk exposure when examining the non-normal returns of hedge funds. The value-at-risk (VaR) measure for financial risk has recently grown to be accepted as a traditional measure in investment firms, large banks and pension funds. As a result of the recurring frequency of down-markets since the collapse of Long-Term Capital Management (LTCM) in August 1998, VaR has played a paramount role as a risk management tool, and is considered to be a mainstream technique for estimating and conveying the exposure a hedge fund has to market risk. With the wide acceptance of VaR, and specifically, of modified VaR as a relevant risk-management tool, a more suitable portfolio performance measure for hedge funds can be formulated in terms of a modified Sharpe ratio.1 228

GREG N. GREGORIOU

229

Using the traditional Sharpe ratio to rank hedge funds will underestimate the tail risk, and then overestimate performance. Therefore, the further the distribution is from the normal, the greater the risk of underestimation. In this chapter, we rank nine funds according to the Sharpe and modified Sharpe ratios. Our results indicate that the modified Sharpe is lower and more accurate when examining non-normal returns.

12.2 LITERATURE REVIEW Many hedge funds produce statistical reports for clients using the traditional Sharpe ratio, which can be misleading because funds will have a tendency to look better in terms of risk-adjusted returns. The drawback of using a traditional Sharpe ratio is that no distinction is made between upside and downside risk, but rather a fund is penalized for upside risk as much as downside risk and does not differentiate irregular losses compared to repeated losses. VaR has emerged progressively in the finance literature as a prevailing measure of risk. However, its simple version also presents some limitations because of the skewed returns hedge funds possess. Methods of measuring VaR, such as the delta-normal method developed by Jorion (2000), are straightforward and simple to apply. However, the formula has its drawbacks, since the assumptions of normality of the distributions are violated largely because of the use of short-selling and derivatives strategies, such as futures and options, frequently used by hedge funds. Current methods have been proposed to properly assess the VaR for nonnormal returns as developed by Rockafellar and Uryasev (2001) using a conditional VaR for general loss distributions, while Agarwal and Naik (2004) construct a mean conditional VaR demonstrating that mean-variance analysis also underestimates tail risk. Furthermore, Favre and Galeano (2002) also developed a technique to properly assess funds with non-normal distributions. The authors demonstrate that the modified VaR (MvaR) can significantly improve the accuracy of the traditional VaR. The difference between the modified VaR and the traditional VaR is that the latter considers only mean and standard deviation, while the former takes into account higher moments such as skewness and kurtosis. In addition, it is possible to reduce the probability of large negative returns by at least 15 percent (Favre and Singer, 2002). The modified VaR allows us to define a modified Sharpe ratio, which is more suitable for hedge funds. For example, when two portfolios have the same mean and standard deviation, in essence they may be different because of extreme losses. Therefore an advantage exists when using the modified VaR measure and modified Sharpe ratio.

230

THE MODIFIED SHARPE RATIO AND CANADIAN HEDGE FUNDS

12.3 DATA AND METHODOLOGY The dataset we use contains hedge fund returns for fifty funds in Canada. However, the majority of the funds commenced operations in 2001 and have been discarded because of the small number of data points available at the time of writing. Only nine live Canadian hedge funds reporting monthly performance figures spanning the period January 1998–December 2002 have been investigated. We obtain data from Beck and Nagy (2003). This period contains the extreme market event of August 1998 as well as the September 11, 2001 terrorist attacks. We use the Extreme Metrics software and assume a risk-free rate of 0 percent to compute the results, using a 95 percent VaR probability. This means that the investor is able to borrow and reinvest in the market portfolio at zero cost in order to move along the capital market line. This assumption simplifies the ranking of assets, especially when some of them have an average return below the risk-free rate, which yields a negative Sharpe or modified Sharpe ratio. The difference between the traditional and modified Sharpe ratios is that, in the latter, the standard deviation is replaced by the modified VaR (at 95 percent) in the denominator. The traditional Sharpe ratio is generally defined as the excess return per unit of standard deviation, as represented by the following equation: Sharpe ratio =

R p − RF σ

(12.1)

where RP = return of the portfolio; RF = risk-free rate; and σ = standard deviation. Since Equation (12.1) presents several limitations for non-normal distribution, a modified Sharpe ratio can be defined in term of modified VaR, as follows: Modified Sharpe ratio =

Rp − RF , MVaR

(12.2)

with 1 1 MVaR = W[µ − {zc + (zc2 − 1)S + (zc3 − 3zc )K 6 24 1 − (2zc3 − 5zc )S2 }σ] 36

(12.3)

where RP = return of the portfolio; RF = risk-free rate; σ = standard deviation; Zc = is the critical value for probability (1 − α) − 1.96 for a 95 percent probability; S = skewness; and K = excess kurtosis. The detailed derivation of the formula for modified VaR is beyond the scope of this chapter. Readers are guided to Favre and Galeano (2002) for a more detailed explanation.

GREG N. GREGORIOU

231

12.4 EMPIRICAL RESULTS 12.4.1 Descriptive statistics Table 12.1 displays monthly statistics on mean return, standard deviation, skewness, excess kurtosis, normal and modified VaR, Jarque–Bera statistic and compounded returns of the hedge funds during the examination period. The average of the compounded returns and mean monthly returns are greatest in the highest group and least in the lowest group – an expected finding. In addition, we find that positive skewness is more pronounced in the lowest group, yielding more positive monthly returns, whereas the top group has the least average positive skewness. A likely explanation is that smaller hedge funds can better control skewness in negative extreme market events, and on average will have less negative monthly returns. The lowest group (see Table 12.1, Panel C) has the highest volatility and lowest returns, which could be attributed to hedge funds taking on more risk to achieve greater returns while increasing assets under management.

12.4.2 Performance discussion Market risk and performance results are also presented in Table 12.1. First we observe that the middle group has, in absolute value, the lowest normal and modified VaR, so is less exposed to extreme market losses. Furthermore, we find that the non-normality when skewness and kurtosis are considered simultaneously using the Jarque–Bera tend to be the largest for small hedge funds. With regard to performance, we notice that the lowest group has the lowest traditional Sharpe and modified Sharpe ratios. It appears that medium-sized hedge funds can do a better job in controlling riskadjusted performance than either small or large funds. Since medium-sized hedge funds receive a greater inflow of money than small funds, they can alter their allocation more frequently. However, there exists a huge difference of assets under management between large and medium funds. When receiving a vast inflow of capital, large hedge funds could be overwhelmed and might experience trouble in producing superior risk-adjusted returns than medium-sized hedge funds. Capacity constraints may exist, since the Toronto Stock Exchange is relatively small compared to the US markets, and trading securities may further restrict large Canadian hedge funds, thus making trading sporadic, especially when leverage and short-selling is involved. Smaller hedge funds with fewer assets might have no choice but to hold their portfolio for a long period of time, irrespective of changing economic conditions.

Fund name

232

Table 12.1 Descriptive statistics of Canadian hedge funds, 1998–2002 Assets Mean Std. dev. Skewness Excess Modified Normal Traditional Modified Jarque–Bera Compound (millions (%) (%) kurtosis VaR VaR Sharpe sharpe statistic return $) 99% 99% ratio ratio (%)

Panel A: Sub-sample 1: Top 3 funds Arrow Clocktower

325

1.4

3.5

0.2

0.00

−6.3

−6.8

0.18

0.16

Goodwood Fund

200

1.6

4.6

0.4

0.5

BPI Global Opportunities

195

1.5

5.9

0.9

1.1

−8.2

−9.1

0.15

−8.3

−12.3

0.14

Average

240

1.5

4.67

0.5

0.53

−7.60

−9.4

−3.9

0.26

127.73

0.13

1.92

137.07

0.09

10.88

119.97

0.16

0.13

4.35

128.26

−4.6

0.14

0.12

0.97

65.10

Panel B: Sub-sample 2: Middle 3 funds −0.4

Horizons Mondiale

125

0.9

2.3

0.2

Horizons Univest 2

107

0.9

0.7

0.2

1.2

−0.8

−0.9

0.75

0.66

3.74

74.44

82

1.8

5.7

0.00

3.4

−11.5

−16.2

0.13

0.09

28.57

162.52

104.67

1.2

2.9

0.13

1.4

5.4

7.2

0.34

0.29

11.09

100.69

0.10

0.07

24.79

112.03

Vertex Average

Panel C: Sub-sample 3: Bottom 3 funds Friedberg TT Equity Hedge

6

1.6

8.0

1.1

2.3

−11.9

−17.1

Horizons Strategic

3

1.8

7.0

3.2

18.4

−0.1

−14.5

0.10

0.09

948.02

77.22

Hillsdale Market Neutral ($US)

2

0.2

4.2

0.4

1.8

−9.6

−9.7

−0.02

−0.02

10.18

6.18

Average

3.67

1.2

6.4

1.57

7.5

−7.2

−13.76

0.06

0.05

327.66

65.14

GREG N. GREGORIOU

233

When we compare the results between the traditional and the modified Sharpe ratios, we find that the traditional Sharpe ratio is higher, confirming that tail risk is underestimated.

12.5 CONCLUSION It is of critical importance to understand that complications will arise when a traditional measure of risk-adjusted performance, such as the traditional Sharpe ratio, is used to investigate fat tails and non-normal returns of hedge funds. Institutional investors must use the modified Sharpe ratio to measure the risk-adjusted returns correctly; and the modified VaR is recommended to measure extreme negative returns because the normal VaR only considers the first two moments of a distribution, namely mean and standard deviation. The modified VaR, however, takes into consideration the third and fourth moments of a distribution – skewness and kurtosis. Using both the modified Sharpe and modified VaR will enable investors to obtain a more accurate picture without any bias. Furthermore, the modified VaR is lower than the normal VaR because of negative skewness in hedge fund returns and the small excess positive kurtosis. The statistics we have presented can be applied to all hedge fund and commodity trading adviser (CTA) classifications to evaluate non-normal returns. We believe many institutional investors wanting to add hedge funds and funds of hedge funds to traditional stock and bond portfolios must request additional and more appropriate statistics such as the modified Sharpe ratio in analyzing the returns of hedge funds.

NOTE 1. The standard VaR, which assumes normality and uses the traditional standard deviation measure, looks only at the tail of the distribution of extreme events. This is common when examining mutual funds, but when applying this technique to funds of hedge funds, difficulties arise because of the non-normality of returns (Favre and Galeano, 2002). The modified VaR takes into consideration the mean, standard deviation, skewness and kurtosis to evaluate correctly the risk-adjusted returns of funds of hedge funds. Computing the risk of a traditional investment portfolio consisting of 50% stocks and 50% bonds with the traditional standard deviation measure could underestimate the risk by as much as 35% (Favre and Singer, 2002).

REFERENCES Agarwal, V. and Naik, N. (2004) “Risks and Portfolio Decisions Involving Hedge Funds”, Review of Financial Studies. 17(1): 63–98. Beck, P. and Nagy, M. (2003) Hedge Funds for Canadians (Toronto: John Wiley).

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THE MODIFIED SHARPE RATIO AND CANADIAN HEDGE FUNDS

Favre, L. and Galeano, J. A. (2002) “Mean-Modified Value-at-Risk with Hedge Funds”, Journal of Alternative Investments, 5(2): 21–5. Favre, L. and Singer, A. (2002) “The Difficulties in Measuring the Benefits of Hedge Funds”, Journal of Alternative Investments, 5(1): 31–42. Jorion, P. (2000) Value at Risk (New York: McGraw-Hill). Markowitz, H. (1952) “Portfolio Selection”, Journal of Finance, 77(1): 77–91. Rockafellar, R. T. and Uryasev, S. (2001) “Conditional Value-at-Risk for General Loss Distributions”, Research Report, ISE Dept, University of Florida.

Index

ABN Amro 1 accounting standard index 59–60, 66, 67–74, 75–7 Ackert, L.F. 157, 164, 174, 175 African bloc 47, 54, 56 Agarwal, V. 229 Ahearne, A. 42, 43, 63, 64, 71–2, 78 Air Liquide 190–209, 210 alpha art market 9–10 Jensen’s alpha 197–8 American bloc 47, 54, 56 Amin, G. 140 analysis of variance (ANOVA) 161, 162, 167–73 Andersen, T.G. 215, 224 Ang, A. 17, 33, 37 Anthony, J.H. 155 anti-director rights 59–60, 66, 67–74, 75–7 arbitrage pricing theory (APT) 114–15, 116–17 Argentina 104 art 1–15 art market 3–4; defining a bubble in 5–6 data 6–8 empirical studies 4–5 methodology 9–10 results 10–11 Art Market Research (AMR) database 6 Artzner, P. 191 Asia/Pacific bloc 47, 54, 56 Asian currency crisis 96–102 at the money (ATM) option contracts 160, 167, 168–9, 172 Ates, A. 153

auction houses 3 Augmented Dickey–Fuller (ADF) test 163, 174 Australian Statex Actuaries Price Index 157 Australian Stock Exchange (ASX) 151–2 All Ordinaries Index 154, 157, 158, 159, 161, 165–7, 168, 169, 175, 176–8 lead–lag relationship with SFE 155, 158–9, 164–5, 176–8 multifactor model 114–36; cross-sectional regressions 124–7; data 119–27; data analysis and results 127–32; parsimonious model 119, 130–2; portfolio characteristics 124; returns to be explained 123–4 Austria 47 automation 151–82 analysis of results 165–78; ANOVA results 167–73; descriptive statistics 165–7; price discovery analysis 174–8 sample design 159–65; cointegration 163–5; data sources 159–60; methodology 160; model and statistical procedures 160–3 Baig, T. 97, 104 Barbone, L. 96, 104 Bauer, R. 5 Baumol, W. 4 bear state 23–6, 27 Beck, P. 230 behavioral anomalies

12–13 235

236

INDEX

behavioral theory 80–1 Bekaert, G. 17, 33, 37 Belgium 47, 63 benchmarks benchmark assets in efficiency gain/loss methodology 140–1; return distribution 144 market benchmarks for French stocks 201–9; forecasting performance 205–9 Benjamin, W. 12 bequests 13 Berk, J.B. 211, 215 Berkowitz, J. 22 beta 197–8, 203–5, 206 bilateral trades 65–6, 67–74, 75–7 Black, F. 17, 185, 186 Black and Scholes volatility bias 189 Blattberg, R.C. 222 blocs 47, 54, 56 Bodnar, T. 215, 220 Bohn, H. 67 bond market–stock market linkages 103–15 book-to-market value 118–19, 121–2, 123–4, 125, 126–32, 133 Bortoli, L. 153 bounded rationality 80 Brady bonds 103–13 stripped-yield spreads 104, 105, 107, 108, 109–11 Brailsford, T. 154, 197 Brandt, M.W. 21 Brazilian bond market–stock market linkages 103–13 Brooks, C. 158, 159 Brownian motion 185–6 bubbles 11–14 art market 1–2, 5, 8, 10–11; defining a bubble in the art market 5–6 Internet 32–3 real estate 6 “bubbliness”, degree of 13–14 CAC40 index 187–209, 210–11 call pricing formula 186–7 Campbell, J.Y. 184, 186, 197, 200, 210 Campbell, R.A. 4, 8 Canadian hedge funds 228–34 capital asset pricing model (CAPM) 114, 115–16, 183, 197, 215 international (ICAPM) 42 capital controls 43, 44, 57–8, 59–60, 63, 67–74, 75–7

capital flow restrictions 57–8, 63, 67–74, 75–7 Carmichael, B. 64 Case, K.E. 6 causality analysis impact of systematic risk on French stocks 191–5; Granger causality test 193–5; VAR specification 191–3 lead–lag relationship 164–5, 176–8 Central and Eastern European investment funds 143–50 certainty-equivalent compensation 28–30 Chamberlain, G.A. 215 Chan, K. 43, 44, 45, 46, 47, 55, 56, 63, 64, 65, 67, 69, 74, 78, 127 Chanel, O. 4 Chen, J. 17 Chen, N.F. 116, 117, 118, 119, 125, 126, 127, 130, 132 Cheng, C.S. 119 China 63 Chordia, T. 105 Clare, A.D. 127, 130 Clark, P.K. 154 cocoa prices 156–7 Coen, A. 64 cointegration 156–8, 163–5, 175–8 Johansen test 163–4, 175–6 lead–lag relationship 155, 158–9, 164–5, 176–8 unit root tests 163, 174 collateral, art as 8 compensation, certainty-equivalent 28–31 Connor, G. 117 contagion 104 currency crises and portfolio selection 96–102 Cooper, I. 44–5, 47 Coordinated Portfolio Investment Survey (CPIS) dataset 43 Copeland, L. 154, 156 corner portfolios 87, 91 correlation studies correlation coefficient and home bias 67–74, 75–7 impact of systematic risk on French stocks 190–1 peer group analysis 148–9 stock market returns 97, 99–100, 101 cost-of-carry 158, 180 creditworthiness 103–4

INDEX

critical line UPM/LPM portfolio optimization algorithm (CLA) 80–95 derivation 82–4 efficient segments on the efficient frontier 88; adjacent efficient segments 89–92 empirical example 92–4 Kuhn–Tucker conditions 84–7 cross-sectional regressions 124–32 CRR (Chen, Roll and Ross) macroeconomic variables 122–3, 126, 127–32, 133 cumulative wealth 32–4 currency crises 96–102, 104 portfolio performance 100–1 stock market average rates of return and average volatility 97–9, 101 stock market correlations 97, 99–100, 101 currency hedging 16–41 economic importance of regimes 28–31 estimation results for regime-switching models 21–6, 27 optimal foreign investment 38, 39 out-of-sample test for regime-switching strategies 31–7, 39 Cyert, R.M. 80 Dahlquist, M. 43, 72 daily financial data 214 matrix elliptical contoured distributions 222–4 Dales, A. 17 Danone 190–209, 210 DAX 157 DeBondt, W.F.M. 117 default probabilities 104 depreciation of the dollar 33, 36 descriptive statistics automation of SFE 161, 165–7 Canadian hedge funds 231, 232 developed countries 47, 54, 56 Diacogiannis, G.P. 116 Diamandis, P. 116 disposition effect 13 distributional price 139 dividend-paying framework 187 dollar, depreciation of the 33, 36 domestic bias 42–79 data sources 46

237

determinants of 56–67; capital control 57–8, 59–60, 63; economic development 56–63; familiarity 61–2, 65–6; information costs 61–2, 64; investor protection 59–60, 66; investors’ behavior 61–2, 65; other variables 67; stock market development 57–8, 63–4 results of empirical analysis 67–9 statistics on 47–56 statistics on investor holdings 46–53, 54 theoretical framework 44–6 world float portfolio 72–4, 75 double-lognormal (DLN) framework 138, 142–3, 144, 150 Dow Jones STOXX market indices 189 downside risk 229 time-varying 1–15 Durbin–Watson (DW) test 130, 131 Dybvig, P.H. 139 East Asian stock markets 96–102 economic development 43, 44, 56–63, 67–77 economic importance of regimes 28–31 Ederington, L.H. 143 Edison, H. 63 efficiency gain/loss 138, 140–3, 149–50 benchmark 141–2 definition 140–1 higher moment performance characteristics 145–7 underlying 142–3 efficient frontiers 87–94 efficient segments on 88; adjacent efficient segments 89–92 mean-variance and UPM/LPM models 92–4 efficient market hypothesis (EMH) 114 semi-strong form 156, 175–6, 180 Eichenberger, R. 12, 13 Eichengreen, B. 96 electronic trading see automation elliptical distributions see matrix elliptical contoured distributions emerging markets Brady bonds see Brady bonds domestic and foreign biases 47, 54, 56, 64, 67–77 performance evaluation 137–50

238

INDEX

endowment effect 12 Engle, R.F. 163 Epps, M.L. 154 Epps, T.W. 154 Erb, C.B. 96–7 European bloc 47, 54, 56 expected inflation, change in 122–3, 126, 127–32, 133 expropriation, risk of 59–60, 66, 67–77 extreme value theory (EVT) 9 Faff, R.W. 118, 119, 192, 197 Fama, E.F. 115, 117–19, 121, 123, 125, 126, 127, 130, 132, 156, 158, 180, 211 familiarity 43, 44, 61–2, 65–78 Fang, K.T. 219, 225 far in the money (FITM) option contracts 160, 168–9 Faruquee, H. 43 Favre, L. 229, 230 firm-attribute factors 211 multifactor models 117, 118–19; ASX 121–32, 133 Fishburn, P.C. 81 Fleming, J. 97, 105 foreign bias 42–79 data sources 46 determinants of 56–67; capital control 57–60, 63; economic development 56–63; familiarity 61–2, 65–6; information costs 61–2, 64; investor protection 59–60, 66; investors’ behavior 61–2, 65; other variables 67; stock market development 57–8, 63–4 results of empirical analysis 69–71 statistics on 47–56 statistics on investor holdings 46–53, 54 theoretical framework 44–6 world float portfolio 72–4, 76 foreign direct investment (FDI) 56–63, 67–77 Forni, L. 96, 104 France 47 impact of systematic risk on stocks in French financial market 183–213 Fraser, P. 116, 118, 119 French, K.R. 43, 65, 115, 117–19, 121, 123, 125, 126, 127, 130, 132, 183, 211 Freund, W.C. 156, 161, 163 Frey, B.S. 12, 13 Friedman, M. 81

Friend, I. 116 Frino, A. 154 Froot, K. 16 Fund of Art Funds 1 futures see Sydney Futures Exchange (SFE) gamma estimates 9, 10–11, 13 Galeano, J.A. 229, 230 Garcia, R. 17 GDP per capita 56–63, 67–77 Gehrig, T. 64 Gençay, R. 183–4 geographical proximity 47, 61–2, 65, 67–74, 75–7 Germany 47 currency hedging and regime switching 21–6, 27, 33–6; optimal hedge ratio 36–7 DAX 157 Giannetti, M. 66 Gibbons, M.R. 215 Glassman, D.A. 42 Glen, J. 17 Glosten, L.R. 140 Goetzmann, W.N. 4 Goldfajn, I. 97, 104 Gonedes, N.J. 222 Gourieroux, C. 199 Goyal, A. 200, 210 Granger, C.W.J. 163 Granger causality test automation of SFE and lead–lag relationship 164–5, 176–8 systematic risk and French stocks 193–5 Gray, S. 17 Groenewold, N. 116, 118, 119, 157, 174, 175 Grünbichler, A. 158 Guidolin, M. 17, 28 Gupta, A.K. 216, 217 Halliwell, J. 118, 119, 123 Hamilton, J.D. 20 Hartmann, P. 96–7 Harvey, C.R. 96–7 He, J. 119, 131–2 hedge funds, Canadian 228–34 hedging, currency see currency hedging; optimal currency hedging hidden regime switches 20–1 high correlation state 25–6, 27

INDEX

higher moment performance analysis 138–40, 145–7 portfolio replication 139–40 rationale 139 role of higher moments 138–9 Hill, B. 9 home bias 42–79 Ahearne measure 71–2, 73 causes 56–67 data and preliminary statistics 44–56 empirical analysis 67–71 theoretical framework of domestic and foreign biases 44–6 world float portfolio 72–4, 77 Hong Kong 97–100 horse race (out-of-sample test) 31–7, 39 house prices 6 Huberman, G. 43, 65 Huisman, R. 9 Hungarian investment funds 143–50 in the money (ITM) option contracts 160, 168–9 Indonesia 97–100 industrial production growth rate, unexpected change in 122–3, 126, 127–32, 133 inflation change in expected 122–3, 126, 127–32, 133 unexpected inflation rate 122–3, 126, 127–32, 133 information costs 43, 44, 61–2, 64, 67–78 information flow 154, 155, 164–5 informational efficiency 152, 179 intensity of capital control 63, 67–74, 75–7 interest rates risk-free 188–9 unexpected change in term structure 122–3, 126, 127–32, 133 international capital asset pricing model (ICAPM) 42 International Finance Corporation (IFC) 63 Internet bubble 32–3 investor behavior domestic and foreign biases 43, 44, 61–2, 65, 67–77 UPM/LPM critical line algorithm 80–95

investor protection 67–77 Izvorski, I. 104

239

43, 44, 59–60, 66,

Jagannathan, R. 140, 197 Japan 5, 8, 97–100 Jarnecic, E. 155 Jarque–Bera test statistic 22, 24, 145, 231, 232 Jasiak, J. 199 Jegadeesh, N. 117 Jensen, C.M. 184, 197 Jensen-type regressions 197–8, 201–3, 204, 207, 208 Johansen, S. 164, 175 Johansen cointegration test 163–4, 175–6 Jorion, P. 17, 229 judicial system efficiency 59–60, 66, 67–77 Juselius, K. 164, 175 Kahneman, D. 12, 13, 81 Kantner, M. 226 Kaplan, P.D. 81 Kaplanis, E. 44–5, 47 Karolyi, G.A. 96 Karpoff, J.M. 154 Kat, H.M. 139, 140 Kelly, J.M. 105 Kempf, A. 157 Kilka, M. 43 Knif, J. 183, 192, 197 known characteristic function 216, 217–18 known location vector 216, 218, 219 Kofman, P. 152, 158 Korea 97–100 Korn, O. 157 Koskinen, Y. 66 Koutmos, G. 183, 192, 197 Kuhn–Tucker conditions 84–7 kurtosis 229, 231, 232, 233 La Porta, R. 66 Lagrange multiplier tests 130, 131 language, common 61–2, 65, 67–77 law, rule of 59–60, 66, 67–77 lead–lag relationship 155, 158–9, 164–5, 176–8 Lee, J. 153 legal system, type of 66, 67–77 leptokurtic distributions 189, 191

240

INDEX

likelihood ratio tests 22, 24 Lintner, J. 42, 115 liquidity 151–82 options data volume as a proxy for 154–9; price discovery and operational efficiency of a market structure 156–9 liquidity ratios 162, 165–7, 168, 169 and market volatility 169–71 London Futures and Options Exchange 156–7 Long-Term Capital Management (LTCM) 228 Longstaff, F.A. 21 L’Oréal 190–209 loss aversion 12 low correlation state 25–6, 27 lower partial moment (LPM) model 81 see also upside potential–downside risk portfolio model MacBeth, J.D. 119, 125 MacKinlay, C. 115 MacKinnon, J.G. 174 macroeconomic variables 117, 118–19 multifactor model for ASX 122–3, 126, 127–32, 133 Malaysia 97–100 Mananyi, A. 156–7 March, J.G. 80 market efficiency 156 market factor 185–7, 189–90 impact of systematic risk on French stocks 190–211 market return index 126, 127–32, 133 market structure 151–82 dynamics of a changing market structure 152–3 price discovery and operational efficiency of 156–9 Markowitz, H. 81, 82, 87, 88, 228 Massimb, M. 152–3 Masson, P. 96 matrix elliptical contoured distributions 214–27 analysis of the power functions 221–2 empirical study 222–4 small sample tests 216–20; further statistics 220; known type of elliptical symmetry 217–18; unknown type of elliptical symmetry 218–19

McKenzie, M.D. 192 mean–variance analysis 81, 228 critical line UPM/LMP model and 92–4 Meese, R. 17 Mei, J. 4 Mele, A. 199 Merton, R.C. 130 Mexico bond market–stock market linkages 103–13 currency crisis 96, 104 Min, H. 103, 112 mixture of distributions (MDH) hypothesis 154, 161, 170 modified Sharpe ratio 228–34 modified VaR (MvaR) 228, 229, 230, 231, 232, 233 Mody, A. 96 MONEP (Marché des Options Négociables de Paris) 188 moneyness portfolios 154–5, 159–60, 161, 167, 168–9 monsoonal effect 96 Moser, J. 152, 158 Moses, M. 4 Mossin, J. 115 Muirhead, R.J. 219, 225 multifactor arbitrage pricing theory (APT) 114–17 multifactor models (MFM) 114–36 data 119–27; cross–sectional regressions 124–7; CRR macroeconomic variables 122–3; explanatory returns 119–22; portfolio characteristics 124; returns to be explained 123–4 data analysis and results 127–32 existing evidence 115–18 parsimonious model for ASX 119, 130–2 multivariate t-distributions 221–2, 225–6 Nagy, M. 230 Naik, N. 229 Nawrocki, D. 81 New Zealand Gross Index 157 Ng, L.K. 119, 131–2 no-opportunity arbitrage valuation principle 186 non-linearity 195–201 Norway 63 NZSE-40 Index 157

INDEX

omission bias 13 open outcry 152, 153 operational efficiency 152, 153, 179 of a market structure 156–9 opportunity cost effect 12 optimal currency hedging 16–41 certainty equivalent compensation 28–31 estimation results for regime-switching models 21–6, 27 horse race for regime-switching strategies 31–7, 39 optimal foreign investment 38, 39 optimal hedge ratios 36–7 optimal weights 35–6 options benchmark assets 141–2; return distribution 144 option moneyness portfolios 154–5, 159–60, 161, 167, 168–9 options data volume as a proxy for liquidity 154–9 portfolio replication 139–40 pricing 186–7; dividend framework 187; no-dividend framework 186–7 out-of-sample test (horse race) 31–7, 39 out of the money (OTM) option contracts 160, 168–9 Owen, J. 215 ownership effect 12 Pacific/Asia bloc 47, 54, 56 Palaro, H.P. 139 parsimonious multifactor model 119, 130–2 payoff distribution pricing model 139 peer group analysis 148–9 Perez-Quiros, G. 17 perfect knowledge 18–20 performance Canadian hedge funds 231–3 forecasting 205–9 performance evaluation 137–50 efficiency gain/loss methodology 140–3 higher moment performance analysis 138–40 testing results 143–9; basic performance characteristics 145; data for the analysis 143–4; higher moment performance

241

characteristics 145–7; peer group analysis 148–9; return distribution of the benchmark asset 144 Perold, A. 16 Perron, P. 17 Phelps, B. 152–3 Philippines 97–100, 104 Phillip–Perron (PP) test 163, 174 phone call costs 61–2, 64, 67–77 Pirrong, C. 153 Pitts, M. 154 Portes, R. 64 portfolio replication 139–40, 145–7, 150 Post, T. 81 Poterba, J. 43, 65, 183 power functions 221–2 price art price indices 6–8 and trading volume 154 price discovery analysis 156–9, 174–9 prospect theory 81 Rabinovitch, R. 215 Racine, M.D. 157, 164, 174, 175 ratio analysis 162, 172–3, 179 real-estate bubble 6 real GDP growth rate 56–63, 67–77 regime switching 16–41 economic importance of regimes 28–31 estimation results 21–6, 27; data 21–2, 23; parameter estimates 22–6; specification test 22, 24 implications on asset allocation 26–38 model 18–21; portfolio selection under hidden regime switches 20–1; portfolio selection with perfect knowledge of the active state 18–20 optimal foreign investment 38, 39 strategies in competition 31–7, 39; cumulative wealth and Sharpe ratio 32–4; optimal hedge ratio 36–7; optimal weights 35–6 regression analysis cross-sectional regressions 124–32 impact of systematic risk on French stocks 196–209 liquidity and automation of SFE 162–3, 172–3 Reinganum, M.R. 115, 116, 119

242

INDEX

Renault 190–209, 210 replication, portfolio 139–40, 145–7, 150 return correlations 96 East Asian economies 97, 99–100, 101 returns Australian stock market 123–4, 127–32 bond and stock market linkages 105–12 East Asian stock markets average rates of return 97, 97–9, 101 return distribution of benchmark asset for Hungarian investment funds 144 reverse S-shaped utility functions 81 Rey, H. 64 Richardson, M. 161 Riddick, L.A. 42 risk downside see downside risk upside 229 upside potential–downside risk portfolio model 80–95 risk aversion 81, 92, 93, 94 risk-free interest rate 188–9 risk premiums, unexpected change in 122–3, 126, 127–32, 133 risk-seeking behavior 81 Rockafellar, R.T. 229 Roll, R. 114, 116, 184, 197, 209 Ross, S.A. 114, 116 Rubinstein, M. 154, 160, 161 rule of law 59–60, 66, 67–74, 75–7 S&P 500 index 157 Samorodnitsky, G. 226 Samuelson, W. 12 Santa-Clara, P.P. 200, 210 Sarkisson, S. 65 Sarno, L. 164 Savage, L.J. 81 SBF120 index 190–209, 210 SBF250 index 190–209, 210 scale factor 185–7, 189–90 Schill, M. 65 Schmid, W. 215, 220 Schneider 190–209 Scholes, M. 185, 186 Schulman, E. 16 Schwartz, E. 21 Selçuk, F. 183–4 self-deception theory 13

Sharpe, W.F. 42, 115, 183, 197, 209 Sharpe ratio 138, 228, 229, 230, 231, 232, 233 CEE investment funds 147, 150 modified 228–34 regime switching and optimal currency hedging 32–4 Shefrin, H. 13 short selling 145–7 Shyy, G. 153 Siegel, J.J. 5 Siegel, L.B. 81 Simon, H.A. 80 simple regression analysis 196–9, 201, 202, 203, 204, 206, 207, 208 Singapore 63 Singer, A. 229 size 118–19, 121–33 skewness 229, 231, 232, 233 Smith, T. 161 Société Générale 190–209, 210 Solnik, B. 117 Sortino, F. 81 spillover effect 96 SPI futures/All Ordinaries Index ratio 162–3, 172–3 Statex Actuaries Accumulation Index 157 statistical multifactor models 117 Statman, M. 13 status quo bias 12 Stiglitz, J.E. 5 stock index dynamic 185–6 stock market capitalization 57–8, 64, 67–77 stock market development 43, 44, 57–8, 63–4, 67–77 stock markets linkages to bond markets 103–13 currency crises, contagion and portfolio selection 96–102 Strong, N. 65 Struthers, J.J. 156–7 Stulz, R.M. 96, 183 sunk cost effect 12–13 Sydney Futures Exchange (SFE) 151–82 lead–lag relationship with ASX 155, 158–9, 164–5, 176–8 Share Price Index (SPI) 157, 158, 159, 161, 165–7, 175, 176–8 SPI futures/All Ordinaries Index ratio 162–3, 172–3

INDEX

symmetric stable distributions 221, 222, 225–6 systematic risk 183–213 empirical study 187–90; data 187–9; induction of systematic risk 189–90 impact 190–5; causality 191–3; correlation 190–1; Granger causality test 193–5 market benchmark comparison 201–9; basic empirical study 201–5; forecasting performance 205–9 non-linearity 195–201; simple regression analysis of asset returns 196–9; volatility impact 199–201 theoretical framework 185–7; option pricing 186–7; valuation setting 185–6 Szego, G. 191 tail index estimator 8, 9–12 Taiwan 97–100 Taqqu, M.S. 226 Tauchen, G. 154 Taylor, M.P. 164 term structure, unexpected change in 122–3, 126, 127–32, 133 Tesar, L. 64, 67 Thaler, R.H. 12, 13, 117 Thomas, S.H. 127, 130 Thomson 190–209 time series properties 174–5 time-varying downside risk 1–15 Timmerman, A. 17, 28 Titman, S. 117 Toronto Stock Exchange (TSE) 156, 231 Totalfina Elf 190–209, 210 trade, scaled by GDP 56–63, 67–77 trading volume 154, 155, 161, 165–7 market volatility and 169–71 transaction costs 64, 67–77 Treynor, J. 183 Turkington, J. 157, 158 Turner, C. 17 Tversky, A. 12, 81 two-year return 67–77 underlying distribution 142–3 unexpected inflation rate 122–3, 126, 127–32, 133 uniqueness of art works 3

243

unit root tests 163, 174 United Kingdom art market 6–8, 10 currency hedging and regime switching 21–6, 27, 33–6; optimal hedge ratio 36–7 United States art market 6–8, 10 regime switching and currency hedging 21–7, 33–6 stock market levels and returns 107 unknown characteristic function 218–19 unknown location vector 216–19 upper partial moment/lower partial moment (UPM/LPM) ratio 81 see also upside potential–downside risk portfolio model upside potential–downside risk portfolio model 80–95 efficient segments on the efficient frontier 88; adjacent efficient segments 89–92 empirical example 92–4 Kuhn–Tucker conditions 84–7 upside risk 229 upward bias 209 Uryasev, S. 229 Valéo 190–209, 210 valuation 185–6 value, art and 3–4 value-at-risk (VaR) 228, 229, 231, 232, 233 modified (MvaR) 228, 229, 230, 231, 232, 233 Van Vliet, P. 81 Varga, T. 216, 217 variance decompositions 109, 110, 111 vector autoregressive (VAR) models 164 Brady bonds 105–12 systematic risk and French stocks 191–3, 194 Venezuela 43, 47, 104 Vivendi 190–209, 210 volatility Asian currency crisis 97–9, 101 automation of SFE 161, 169–71 Black and Scholes volatility bias 189

244

INDEX

volatility continued impact of systematic risk on French stocks 199–205, 207, 208 volatility parameter 185, 189–90

Whaley, R. 189 Whitcher, B. 183-4 White, H. 127 world float portfolio

Walsh, D. 157, 158 Wang, G.H.K. 153 Wang, Z. 197 Warnock, F. 63 Weber, M. 43 Werner, I. 64

Xu, X.

65

Zeckhauser, R. 12 Zhang, Y.T. 219, 225 Zhou, C. 64 Zhou, G. 215

72–7

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