Cost of Capital: Applications and Examples

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COST OF CAPITAL Applications and Examples Third Edition SHANNON P. PRATT ROGER J. GRABOWSKI

John Wiley & Sons, Inc.

Cost of Capital Applications and Examples

COST OF CAPITAL Applications and Examples Third Edition SHANNON P. PRATT ROGER J. GRABOWSKI

John Wiley & Sons, Inc.

1 This book is printed on acid-free paper. Copyright # 2008 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201-748-6011, fax 201-748-6008, or online at http://www.wiley.com/go/permissions. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services, or technical support, please contact our Customer Care Department within the United States at 800-762-2974, outside the United States at 317-572-3993 or fax 317-5724002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our Web site at http://www.wiley.com. Library of Congress Cataloging-in-Publication Data: Pratt, Shannon P. Cost of capital: applications and examples/Shannon P. Pratt, Roger J. Grabowski.—3rd ed. p. cm. ‘‘Published simultaneously in Canada.’’ Previous editions had subtitle: Estimation and applications. Includes bibliographical references and index. ISBN 978-0-470-17115-8 (cloth: alk. paper) 1. Capital investments. 2. Business enterprises—Valuation. 3. Capital investments—United States. 4. Business enterprises—Valuation—United States. I. Grabowski, Roger J. II. Title. HG4028.C4P72 2008 658.15’2—dc22 2007020132 Printed in the United States of America 10 9

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To our families for their support and encouragement, without which our careers and this book would not have been possible Millie Son Mike Pratt Daughter-in-law Barbara Brooks Randall Kenny Portland, OR

Daughter Susie Wilder Son-in-law Tim Wilder John Calvin Meg Springfield, VA Daughter Georgia Senor Son-in-law Tom Senor Elisa Katie

Son Steve Pratt Daughter-in-law Jenny Pratt Addy Zeph Portland, OR

Graham Fayetteville, AR Mary Ann Son Roger Grabowski Jr. Daughter-in-law Misako Takahashi Rob Tokyo, Japan

Daughter Sarah Harte Son-in-law Mike Harte Kevin Evanston, IL

Daughter Julia Grabowski, MD San Diego, CA

Son Paul Grabowski New York, NY

Contents About the Authors Foreword Preface

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Acknowledgments Introduction

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Notation System and Abbreviations Used in This Book Part 1. Cost of Capital Basics

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1. Defining Cost of Capital 3 2. Introduction to Cost of Capital Applications: Valuation and Project Selection 3. Net Cash Flow: Preferred Measure of Economic Income 15 4. Discounting versus Capitalizing 23 5. Relationship between Risk and the Cost of Capital 39 Appendix 5A. FASB’s Concepts Statement No. 7: Cash Flows and Present Value Discount Rates 48 6. Cost Components of a Company’s Capital Structure 51 Part 2. Estimating the Cost of Equity Capital and the Overall Cost of Capital

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7. Build-up Method 69 8. Capital Asset Pricing Model 79 9. Equity Risk Premium 89 Appendix 9A. Bias Issues in Compounding and Discounting 114 10. Beta: Differing Definitions and Estimates 117 Appendix 10A. Formulas and Examples for Unlevering and Levering Equity Betas 143 Appendix 10B. Examples of Computing OLS Beta, Sum Beta, and Full-Information Beta Estimates 151 11. Criticism of CAPM and Beta versus Other Risk Measures 161 Appendix 11A. Example of Computing Downside Beta Estimates 176 12. Size Effect 179 13. Criticisms of the Size Effect 209 Appendix 13A. Other Data Issues Regarding the Size Effect 220 14. Company-Specific Risk 225 15. Alternative Cost of Equity Capital Models 243 16. Implied Cost of Equity Capital 255 17. Weighted Average Cost of Capital 265 Appendix 17A. Iterative Process Using CAPM to Calculate the Cost of Equity vii

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Component of the Weighted Average Cost of Capital When Capital Structure Is Constant 283 Appendix 17B. Iterative Process Using CAPM to Calculate the Cost of Equity Component of the Weighted Average Cost of Capital When Capital Structure Is Changing 297 18. Global Cost of Capital Models 309 19. Using Morningstar Cost of Capital Data 331 Part 3. Corporate Finance Officers: Using Cost of Capital Data 20. 21. 22. 23. 24.

Capital Budgeting and Feasibility Studies 363 Cost of Capital for Divisions and Reporting Units 369 Cost of Capital in Evaluating Acquisitions and Mergers 385 Cost of Capital in Transfer Pricing 393 Central Role of Cost of Capital in Economic Value Added 409

Part 4. Cost of Capital for Closely Held Entities 25. 26. 27. 28. 29.

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Handling the Discount for Lack of Marketability for Operating Businesses 419 Private Company Discount 429 Cost of Capital of Interests in Pass-Through Entities 437 Cost of Capital in Private Investment Companies 449 Relationship between Risk and Returns in Venture Capital Investments 469

Part 5. Other Topics

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30. Minority versus Control Implications of Cost of Capital Data 481 31. How Cost of Capital Relates to the Excess Earnings Method of Valuation 32. Adjusting the Discount Rate to Alternative Economic Measures 501 33. Estimating Net Cash Flows 505 Appendix 33A. Estimating the Value of a Firm in Financial Distress 518 34. Common Errors in Estimation and Use of Cost of Capital 529 35. Cost of Capital in the Courts 539 Part 6. Real Estate and Ad Valorem

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36. Cost of Capital of Real Property–Individual Assets 561 Appendix 36A. Valuing Real Property 580 37. Cost of Capital of Real Estate Entities 587 Appendix 37A Valuing Real Estate Entities 615 38. Cost of Capital in Ad Valorem Taxation 623 Part 7. Advice to Practitioners

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39. Dealing with Cost of Capital Issues 641 40. Questions to Ask Business Valuation Experts 647 Appendices

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I. Bibliography 655 II. Data Resources 683

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III. International Glossary of Business Valuation Terms 693 IV. Sample Report Submitted to U.S. Tax Court by Roger J. Grabowski V. ValuSource Valuation Software 725 VI. Review of Statistical Analysis 733 Index

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About the Authors Dr. Shannon P. Pratt, CFA, FASA, MCBA, CM&AA is the chairman and CEO of Shannon Pratt Valuations, Inc., a nationally recognized business valuation firm headquartered in Portland, Oregon. He is also the founder and editor emeritus of Business Valuation Resources, LLC, and one of the founders of Willamette Management Associates, for which he was a managing director for almost 35 years. He has performed valuation assignments for these purposes: transaction (acquisition, divestiture, reorganization, public offerings, public companies going private), taxation (federal income, gift, and estate and local ad valorem), financing (securitization, recapitalization, restructuring), litigation support and dispute resolution (including dissenting stockholder suits, damage cases, and corporate and marital dissolution cases), and management information and planning. He has also managed a variety of fairness opinion and solvency opinion engagements. Dr. Pratt has testified on hundreds of occasions in such litigated matters as dissenting stockholder suits, various types of damage cases (including breach of contract, antitrust, and breach of fiduciary duty), divorces, and estate and gift tax cases. Among the cases in which he has testified are Estate of Mark S. Gallo V. Commissioner, Charles S. Foltz, et al. V. U.S. News & World Report et al., Estate of Martha Watts V. Commissioner, and Okerlund V. United States. He has also served as appointed arbitrator in numerous cases. PREVIOUS EXPERIENCE Before founding Willamette Management Associates in 1969, Dr. Pratt was a professor of business administration at Portland State University. During this time, he directed a research center known as the Investment Analysis Center, which worked closely with the University of Chicago’s Center for Research in Security Prices. EDUCATION Doctor of Business Administration, Finance, Indiana University Bachelor of Arts, Business Administration, University of Washington PROFESSIONAL AFFILIATIONS Dr. Pratt is an Accredited Senior Appraiser and Fellow (FASA), Certified in Business Valuation, of the American Society of Appraisers (their highest designation). He is also a Chartered Financial Analyst (CFA) of the Institute of Chartered Financial Analysts, a Master Certified Business Appraiser (MCBA) of the Institute of Business Appraisers, a Master Certified Business Counselor (MCBC), and is Certified in Mergers and Acquisitions (CM&AA) with The Alliance of Merger and Acquisition Advisors. Dr. Pratt is a life member of the American Society of Appraisers, a life member of the Business Valuation Committee of that organization and teaches courses for the organization. He is also a lifetime member emeritus of the Advisory Committee on Valuations of The ESOP Association. He is a recipient of the magna cum laude award of the National Association of Certified Valuation Analysts for service to

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the business valuation profession. He is also the first life member of the Institute of Business Appraisers. He is a member and a past president of the Portland Society of Financial Analysts, the recipient of the 2002 Distinguished Achievement Award, and a member of the Association for Corporate Growth. Dr. Pratt is a past trustee of The Appraisal Foundation and is currently an outside director and chair of the audit committee of Paulson Capital Corp., a NASDAQ-listed investment banking firm specializing in small initial public offerings (usually under $50 million). PUBLICATIONS Dr. Pratt is the author of Valuing a Business: The Analysis and Appraisal of Closely Held Companies, 5th edition (New York: McGraw-Hill, 2007); coauthor, Valuing Small Businesses and Professional Practices, 3rd edition with Robert Schweihs and Robert Reilly (New York: McGraw-Hill, 1998); coauthor, Guide to Business Valuations, 18th edition with Jay Fishman, Cliff Griffith and Jim Hitchner (Fort Worth, TX: Practitioners Publishing Company, 2008); coauthor, Standards of Value, with William Morrison and Jay Fishman (New York: John Wiley & Sons, 2007); coauthor, Business Valuation and Taxes: Procedure, Law, and Perspective, with Judge David Laro (New York: John Wiley & Sons, 2005); author, Business Valuation Discounts and Premiums (New York: John Wiley & Sons, 2001); Business Valuation Body of Knowledge: Exam Review and Professional Reference, 2nd edition (New York: John Wiley & Sons, 2003); The Market Approach to Valuing Businesses, 2nd edition (New York: John Wiley & Sons, 2005); and The Lawyer’s Business Valuation Handbook (Chicago: American Bar Association, 2000). He has also published nearly 200 articles on business valuation topics. Roger Grabowski, ASA, is a managing director of Duff & Phelps, LLC. Mr. Grabowski has directed valuations of businesses, partial interests in businesses, intellectual property, intangible assets, real property, and machinery and equipment for various purposes including tax (income and ad valorem) and financial reporting; mergers, acquisitions, formation of joint ventures, divestitures, and financing. He developed methodologies and statistical programs for analyzing useful lives of tangible and intangible assets, such as customers and subscribers. His experience includes work in a wide range of industries including sports, movies, recording, broadcast and other entertainment businesses; newspapers, magazines, music, and other publishing businesses; retail; banking, insurance, consumer credit, and other financial services businesses; railroads and other transportation companies; mining ventures; software and electronic component businesses; and a variety of manufacturing businesses. Mr. Grabowski has testified in court as an expert witness on the value of closely held businesses and business interests, matters of solvency, valuation, and amortization of intangible assets, and other valuation issues. His testimony in U.S. District Court was referenced in the U.S. Supreme Court opinion decided in his client’s favor in the landmark Newark Morning Ledger income tax case. Among other cases in which he has testified was Herbert V. Kohler Jr., et al., v. Comm. (value of stock of The Kohler Company); The Northern Trust Company, et al., v. Comm. (the first U.S. Tax Court case that recognized the use of the discounted cash flow method for valuing a closely held business); Oakland Raiders v. Oakland-Alameda County Coliseum Inc. et al. (valuation of the Oakland Raiders); ABCNACO, Inc. et al., Debtors, and The Official Committee of Unsecured Creditors of ABC-NACO v. Bank of America, N.A. (valuation of collateral); Wisniewski and Walsh v. Walsh (oppressed shareholder action); and TMR Energy Limited v. The State Property Fund of Ukraine (arbitration on behalf of world’s largest private company in Stockholm, Sweden, on cost of capital for oil refinery in Ukraine in a contract dispute).

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PREVIOUS EXPERIENCE He was formerly managing director of the Standard & Poor’s Corporate Value Consulting practice, a partner of PricewaterhouseCoopers, LLP and one of its predecessor firms, Price Waterhouse (where he founded its U.S. Valuation Services practice and managed the real estate appraisal practice). Prior to Price Waterhouse, he was a finance instructor at Loyola University of Chicago, a cofounder of Valtec Associates, and a vice president of American Valuation Consultants. EDUCATION Mr. Grabowski received his B.B.A.—Finance from Loyola University of Chicago and completed all coursework in the doctoral program, Finance, at Northwestern University, Chicago. PROFESSIONAL AFFILIATIONS He serves on the Loyola University School of Business Administration Dean’s Board of Advisors. Mr. Grabowski is an Accredited Senior Appraiser of the American Society of Appraisers (ASA) certified in business valuation. PUBLICATIONS Mr. Grabowski coauthors the annual Duff & Phelps’ Risk Premium Report. He lectures and publishes regularly. He is the coauthor of three chapters (on equity risk premium, valuing pass-through entities, and valuing sports teams) in Robert Reilly and Robert P. Schweihs, The Handbook of Business Valuation and Intellectual Property Analysis (New York: McGraw-Hill, 2004). He teaches courses for the American Society of Appraisers including Cost of Capital, a course he developed. He is the editor of the Business Valuation Review, the quarterly Journal of the Business Valuation Committee. David Fein is the CEO and president of ValuSource, which for over 20 years has been the leading provider of business valuation software, data, and report writers for CPAs, M&A professionals, and business owners. Mr. Fein’s mission is to create state-of-the-art technology to automate and standardize complex financial analysis and reporting tasks. He has a bachelor’s degree in computer science and an MBA. You may reach him at 1.800.825.8763 100, or at [email protected]. William H. Frazier is a principal and founder of the firm of Howard Frazier Barker Elliott, Inc, and manages its Dallas office. He has 30 years of experience in business valuation and corporate finance. Mr. Frazier has been an Accredited Senior Appraiser of the American Society of Appraisers (ASA) since 1987 and serves on the ASA’s Business Valuation Committee as secretary. He has participated as an appraiser and/or expert witness in numerous U.S. Tax Court cases, including testimony in Jelke, McCord, Dunn and Gladys Cook. Mr. Frazier has written numerous articles on the subject of business valuation for tax purposes, appearing in such publications as the Business Valuation Review, Valuation Strategies, BV E-Letter, Shannon Pratt’s Business Valuation Update, and Estate Planning. He is the coauthor of the chapter on valuing family limited partnerships in Robert Reilly and Robert P. Schweihs, eds., The Handbook of Business Valuation and Intellectual Property Analysis (New York: McGraw-Hill, 2004). Mr. Frazier serves on the Valuation Advisory Board of Trusts & Estates Journal. James Harrington, MBA, is a product manager in Morningstar’s Individual Investor Business segment. He leads the group that produces the widely used and cited SBBI Classic and Valuation Yearbooks, the Cost of Capital Yearbook, the Beta Book, and various international and domestic reports.

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At Morningstar since March of 2006, Mr. Harrington has expanded the product offerings and increased sales for Morningstar publications and reports and is an accomplished financial writer and analyst. Immediately prior to his tenure at Morningstar, he was a product manager in the financial communications group at Ibbotson Associates. Before that, he was a bond and bond portfolio analyst, worked at the Chicago Board of Trade in the bond options pit for a filling group, managed inbound and outbound dock workers at a large trucking firm, and was even a Teamster for a year. Mr. Harrington holds a bachelor’s degree in marketing from Ohio State University and an MBA in both finance and economics from the University of Illinois at Chicago, where he graduated at the top in his class. Carl Hoemke started in valuation when he accepted a position as a tax appraiser for the local tax assessor’s office. His education is in architecture and business. After working for the tax assessor’s office for a period of about eight years, he joined a property tax consulting firm. He later became a partner at one of the big four accounting firms, first as a tax partner then ultimately as a corporate finance partner responsible for valuations primarily in energy and telecommunications. He is currently a managing director with Duff & Phelps, where he leads the firm’s State & Local Tax practice while continuing to provide valuation for other purposes including financial reporting and litigation support. Jim MacCrate, MAI, CRE, ASA owns his own boutique real estate valuation and consulting company, MacCrate Associates, LLC, located in the New York City Metropolitan area, concentrating on complex real estate valuation issues. Formerly, he was the Northeast regional practice leader and director of the Real Estate Valuation/Advisory Services Group at Price Waterhouse LLP and PricewaterhouseCoopers LLP. He received a B.S. Degree from Cornell University and M.B.A. from Long Island University, C. W. Post Center. He has written numerous articles for Price Waterhouse LLP, ‘‘The Counselors of Real Estate,’’ and has contributed to the Appraisal Journal. He initiated the Land Investment Survey that has been incorporated into The PricewaterhouseCoopers Korpacz Real Estate Investor Survey. He is on the national faculty for the Appraisal Institute and adjunct professor at New York University. Harold G. Martin, Jr., MBA, CPA, ABV, ASA, CFE, is the principal-in-charge of the Business Valuation, Forensic, and Litigation Services Group for Keiter, Stephens, Hurst, Gary & Shreaves, P.C. He has over 25 years of experience in financial consulting, public accounting, and financial services. He has appeared as an expert witness in federal and state courts, served as a court-appointed neutral business appraiser, and also served as a federal court-appointed accountant for a receivership. Mr. Martin is an adjunct faculty member of The College of William & Mary Mason Graduate School of Business and teaches valuation and forensic accounting in the Master of Accounting program. Prior to joining Keiter Stephens, he was affiliated for 13 years with Price Waterhouse and Coopers & Lybrand. He is a former member of the American Institute of Certified Public Accountants Business Valuation Committee and is a two-time recipient of the AICPA Business Valuation Volunteer of the Year Award. He is an editorial advisor for the AICPA CPA Expert, a national instructor for the AICPA’s valuation education program, an AICPA faculty member for the National Judicial College, a former member of the Appraisal Standards Board USPAP BV Task Force, and former editor of the AICPA ABV e-Alert. He is a frequent speaker and author on valuation topics and is a coauthor of Financial Valuation: Applications and Models, 2nd edition, published by John Wiley & Sons. Mr. Martin received his A.B. degree in English from The College of William and Mary and his M.B.A. degree from Virginia Commonwealth University. James Morris is a professor of finance at the University of Colorado at Denver, where he teaches courses in business valuation and financial modeling. He is an AM and holds a Ph.D. in Finance

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from University of California, Berkeley. In addition to teaching, he provides valuation services to the business community. Mark Shirley is a licensed certified public accountant and has earned advanced accreditations: Accreditation in Business Valuation (BV), Certified Valuation Analyst (CVA), Certified Forensic Financial Analyst (CFFA), and Certified Fraud Examiner (CFE). After leaving the Internal Revenue Service in 1984, Mr. Shirley’s consulting practice has concentrated on the disciplines of business valuation, forensic/investigative accounting, and financial analysis/modeling. Professional engagements have included business valuation, valuation of options/warrants, projections and forecasts, statistical sampling, commercial damage modeling, personal injury loss assessment, and the evaluation of proffered expert testimony under Daubert and the Federal Rules of Evidence. Since 1988, technical contributions have been published by Wiley Law Publications, Aspen Legal Press, and in professional periodicals, including The Valuation Examiner, BewertungsPraktiker Nr. (a German-language business valuation journal), The Practical Accountant, CPA Litigation Services Counselor, The Gatekeeper Quarterly, The Journal of Forensic Accounting, and local legal society publications. Since 1997 Mr. Shirley has authored courses for NACVA’s Fundamentals, Techniques & Theory; Forensic Institute, and Consultant’s Training Institute. He also has developed several advanced courses for the NACVA in applied statistics and financial modeling. A charter member of the LA Society of CPA’s Litigation Services Committee, Mr. Shirley has remained active since the committee’s formation. He is an adjunct faculty member at the National Judicial College, University of Nevada, Reno, since 1998. Mr. Shirley also serves on the Advisory Panel for Mdex Online; The Daubert Tracker, an online Daubert research database; and the Ethics Oversight Board for the NACVA. Since 1985, Mr. Shirley has provided expert witness testimony before the U.S. Tax Court, Federal District Court, Louisiana district courts, Tunica-Biloxi Indian Tribal Court, and local specialty courts. Court appointments have been received in various matters adjudicated before the Louisiana Nineteenth Judicial District Court. The NACVA has recognized Mr. Shirley’s contributions to professional education by awarding him the Circle of Light in 2002, Instructor of the Year in 2000/2001, and multiple recognitions as Outstanding Member and Award of Excellence. David M. Ptashne, CFA, is a Senior Associate with the Chicago office of Duff & Phelps and has worked directly with Roger Grabowski since 2003. Mr. Ptashne has performed numerous valuation studies of businesses and interests in businesses and intangible assets across various industries including advertising and communications, consumer products, technology, financial services, integrated oil and gas, retail, and healthcare. Mr. Ptashne enjoys researching international cost of capital issues and currently manages Duff & Phelps’ international cost of capital model. Mr. Ptashne is a member of the CFA Institute and CFA Society of Chicago. He received a Bachelor of Science degree in Finance with High Honors from the University of Illinois at Urbana-Champaign.

Foreword Given the central role of the cost of capital in discounted cash flow valuation, it is surprising how little discussion there is on the central inputs that go into its estimation. Shannon Pratt and Roger Grabowski have not only brought together all of the issues in the cost of capital computation but have done so in a way that melds timely advice for practitioners with serious debate about the best practices in the area. Both authors have decades of experience, and their expertise and knowledge show not only in how they present the material but also in the examples they use. The book has been significantly updated from the second edition, both in the numbers and also in the coverage of issues. In general, books on valuation-related topics take one of two paths. One is to follow the cookbook style and provide the reader with the ‘‘right answers’’ to questions, even though there may be debate about what is ‘‘right.’’ The other is to present the alternatives, explain the pros and cons, and trust the reader to make the right choices at the end. This book adopts the latter approach and provides comprehensive discussions not only of the standard inputs—risk-free rates, risk premiums, and debt ratios—but also of issues that come up infrequently in valuations but often enough that practitioners look for guidance. While this book is grounded in solid theory, it is a book for practitioners, and consequently it provides practical solutions to estimation problems. In the process, though, it respects their intelligence and capacity to handle ambiguity by providing a balanced discussion of the choices. It belongs on the bookshelves of serious valuation practitioners, and I would expect it to be widely referenced in the coming years. ASWATH DAMODARAN

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Why did we undertake writing this book? In a recent speech, Geoff Colvin, Senior Editor at Large with Fortune, stated that one of the four traits of great executives is their understanding of the fundamental reality of wealth creation—successful organizations invest to earn a rate of return in excess of their cost of capital.1 These executives ingrain that understanding in the organization, and the managers at all levels come to understand their cost of capital. Our experience tells us that practitioners need assistance in better understanding and estimating the cost of capital and in communicating their results, not from the view of portfolio management but from the view of business owners and managers. No other valuation text designed for the practitioner treats the cost of capital in the breadth and depth that this one does. In terms of breadth, this text treats cost of capital for uses in business valuation, project assessment and capital budgeting, divisional cost of capital, reporting unit valuation and goodwill impairment testing, transfer pricing, utility and other regulated industry rate setting, and ad valorem (property) taxation. Emphasis is on the cost of equity capital. In addition to detailed exposition of the build-up and Capital Asset Pricing Models for estimating the cost of capital, we present in-depth analysis of the components, including the equity risk premium, beta, and the size effect. We also analyze criticism of major models for developing estimates of the cost of capital in use today and present procedures for a number of alternative models. We discuss the Duff & Phelps Risk Premium studies, which are becoming more widely used tools in estimating the cost of equity capital.

WHAT’S NEW IN THIS EDITION Throughout the book, we summarize the results and practical implications of the latest research, much of which has been gleaned from unpublished academic working papers. EQUITY RISK PREMIUM 4 TO 6 PERCENT Importantly, based on empirical research on the magnitude of the equity risk premium, we conclude that it is in the range of 4% to 6% rather than above the 7% that many analysts have used in recent years. We believe that readers will find this research convincing, as we did. PRIVATE COMPANY DISCOUNTS OVER 25 PERCENT We were surprised to find academic research that concludes that the typical private company sells for about a 25% discount compared with otherwise comparable public companies. This is after controlling for variables such as size and industry. The authors of the studies surmise that the reasons for this phenomenon could relate to lack of exposure to the market and lower quality of information (e.g., lack of a track record of audited financial statements). 1

Summarized from a series of Fortune articles on ‘‘Lessons in Leadership.’’

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MUCH NEW DATA AND LITERATURE We not only describe the practical procedures that can be used to apply the various theories but also describe in detail the databases available to derive the numbers to put into the models. We summarize the most important and convincing of the proliferation of literature, both published and unpublished, in recent and prior years. The footnotes and bibliography tell the reader who wishes to get the original studies where to find them. We also added a chapter on global cost of capital models. Much of the new research cited in this book is from working papers, to locate a working paper, search online using any major search engine (i.e., Google or Yahoo) by author and title. An example: the first link provided using the Google search engine when a search for ‘‘Adrian, Tobias, and Francesco Franzoni. Learning About Beta: an Explanation of the Value Premium’’ will be ‘‘SSRN–Learning About Beta: An Explation of the Value Premium by. . .’’ This link will direct you to the Social Science Research Network, where the article is downloadable. CORPORATE FINANCE AND ADVICE TO PRACTITIONERS We have added a five-chapter section specifically for corporate finance officers. This includes, for example, capital budgeting and feasibility studies, cost of capital for divisions and reporting units, and valuation in mergers and acquisitions. There is also a chapter on advice from the authors about dealing with controversial cost of capital issues, a major chapter on cost of capital in the courts, and a chapter on cross examining experts on cost of capital.

AUDIENCES FOR THE BOOK In addition to the traditional professional valuation practitioner, this book is designed to serve the needs of:

Attorneys and judges who deal with valuation issues in mergers and acquisitions, shareholder and partner disputes, damage cases, solvency cases, bankruptcy reorganizations, property taxes, rate setting, transfer pricing, and financial reporting Investment bankers for pricing, public offerings, mergers and acquisitions, and private equity financing

Corporate finance officers for pricing or evaluating mergers and acquisitions, raising private or public equity, property taxation, and stakeholder disputes Academicians and students who wish to learn anywhere from the basic theory to the latest research

CPAs who deal with either valuation for financial reporting or client valuations issues

PRACTICAL APPLICATIONS The book is designed to enhance the insights of users of cost of capital applications as well as originators of such applications. Most formulas are accompanied by examples. Several chapter appendices present detailed expositions of the more complex procedures.

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Finally, the book is comprehensively indexed to serve as a reference for specific concepts and procedures within the general topic of cost of capital. Please contact the authors with any questions, comments, or suggestions for the next edition. Shannon P. Pratt, CFA, FASA, MCBA, CM&AA Shannon Pratt Valuations, Inc. 6443 S.W. Beaverton Hillsdale Highway, Suite 432 Portland, Oregon 97221 (503) 459-4700 E-mail: [email protected] www.shannonpratt.com Roger J. Grabowski, ASA Duff & Phelps, LLC 311 S. Wacker Drive, Suite 4200 Chicago, Illinois 60606 (312) 233-6820 E-mail: [email protected] www.duffandphelps.com

Acknowledgments This book has benefited immensely from review by many people with a high level of knowledge and experience in cost of capital and valuation. These people reviewed the manuscript, and the book reflects their invaluable efforts and legions of constructive suggestions: Thomas Blake CRA International Inc Boston, MA

Mark Lee Eisner LLP New York, NY

Stephen J. Bravo The Financial Valuation Group Framingham, MA

Gilbert E. Matthews Sutter Securities, Inc. San Francisco, CA

James Budyak Valuation Research Corp. & Company Milwaukee, WI

Dan McConaughy Grobstein, Horwath LLP Sherman Oaks, CA

Donald A. Erickson Erickson Partners, LLC Dallas, TX

Professor James Morris University of Colorado Denver, CO

Professor Thomas J. Frecka University of Notre Dame Notre Dame, IN

George Pushner Duff & Phelps LLC New York, NY

Michael Hamilton FTI Consulting New York, NY

Jeffrey Tarbell Houlihan Lokey Howard & Zukin San Francisco, CA

Jim Hitchner The Financial Valuation Group Atlanta, GA

Richard M. Wise Wise, Blackman, LLP Montreal (Quebec), Canada

In addition, we thank:

Kimberly Short of Shannon Pratt Valuations, Inc., for assistance with editing and research, including updating of the bibliography; updating and shepherding the manuscript among reviewers, contributors, authors, and publisher; typing; obtaining permissions; and other invaluable help. David Turney of Duff & Phelps, LLC, for preparing numerous tables and calculations that appear throughout the book. Harold Martin of Keiter, Stephens, Hurst, Gary & Shreaves, P.C., for contributing Appendix 17.1 on using the weighted average cost of capital with a constant capital structure. Professor James Morris, University of Colorado at Boulder, for contributing Appendix 17.2 on using the weighted average cost of capital with a changing capital structure. David Ptashne, CFA, of Duff & Phelps, LLC, for contributing Appendices 10.1, 10.2, and 11.1 on beta calculations. James Harrington with Morningstar for contributing Chapter 19 on using Morningstar Inc. cost of capital data as well as assisting in obtaining permissions for Morningstar exhibits used herein. xxiii

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Acknowledgments

Joel M. Stern, G. Bennett Stewart III, and Donald H. Chew Jr. for contributing Chapter 24 on economic value added. William Frazier of Howard Frazier Barker Elliott, Inc., for contributing Chapter 28 on the cost of capital in private investment companies. Jim MacCrate, MAI, of MacCrate Associates, LLC, for his contribution of Chapters 37, Appendix 37.1, and Chapter 36 and Appendix 36.1 on the cost of capital in real estate. Carl Hoemke of Duff & Phelps, LLC, for contributing Chapter 38 on the cost of capital in ad valorem taxation. David Fein of ValuSource for contributing the revised and updated Appendix E on ValuSource Pro. Mark Shirley of V & L Consultants, LLP, for contributing Appendix F on statistical analysis. Noah Gordon of Shannon Pratt Valuations, Inc., for assistance with researching and updating legal cases, as well as general editorial assistance.

For the granting of permissions, we would like to thank:

Professor Edwin Burmeister, Duke University

Business Valuation Resources, LLC Center for Research in Security Prices, University of Chicago

Professors Elroy Dimson, Paul Marsh, and Mike Staunton

Duff & Phelps, LLC John D. Emory Sr. and John D. Emory Jr., Emory & Co.

FactSet Mergerstat, LLC FMV Opinions, Inc.

Professor Arthur Korteweg, University of Chicago

Jim MacCrate, MacCrate Associates, LLC The McGraw-Hill Companies, Inc.

Morningstar, Inc. Glen Mueller, Dividend Capital Group

National Association of Real Estate Investment Trusts Pluris Valuation Advisors, LLC

PricewaterhouseCoopers, LLP

Standard & Poor’s (a division of McGraw-Hill) Thomson Corporation

Valuation Advisors, LLC

Thank you to those whose ideas contributed to several of the analyses incorporated herein:

David King, Mesirow Financial Consulting LLC

Professor Timothy Leuhrman, Harvard University

We thank all of the people singled out above for their assistance of course, any errors herein are our responsibility. Shannon Pratt Roger Grabowski

Introduction

PURPOSE AND OBJECTIVE OF THIS BOOK The purpose of this book is to present both the theoretical development of cost of capital estimation and its practical application to valuation, capital budgeting, forecasting of expected investment returns, and rate-setting problems encountered in current practice. It is intended both as a learning text for those who want to study the subject and as a handy reference for those who are interested in background or seek direction in some specific aspect of cost of capital. The objective is to serve two primary categories of users: 1. The practitioner who seeks a greater understanding of the latest theory and practice in cost of capital estimation 2. The reviewer who needs to make an informed evaluation of another party’s methodology and data used to produce a cost of capital estimate

OVERVIEW In this text, the reader can expect to learn about:

The theory of what drives the cost of capital

The models currently in use to estimate cost of capital The data available as inputs to the models to estimate cost of capital

How to use the cost of capital estimate in: Valuation

Feasibility studies Corporate finance decisions

Forecasting expected investment returns

How to reflect minority/control and marketability considerations Explanation of terminology, with its unfortunately varied and sometimes ambiguous usage in current-day financial analysis

IMPORTANCE OF THE COST OF CAPITAL The cost of capital estimate is the essential link that enables us to convert a stream of expected income into an estimate of present value. Doing this allows us to make informed pricing decisions for purchases and sales and to compare one investment opportunity against another.

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Introduction

COST OF CAPITAL ESSENTIAL IN THE MARKETPLACE In valuation and financial decision making, the cost of capital estimate is just as important as the estimate of the expected amounts of income to be discounted or capitalized. Yet we continually see income estimates laboriously developed and then converted to estimated value by a cost of capital that is practically pulled out of thin air. In the marketplace, better-informed cost of capital estimation will improve literally billions of dollars’ worth of financial decisions every day. For example, small differences in discount rates, and especially small differences in capitalization rates, can make very large differences in concluded values.

SOUND SUPPORT ESSENTIAL IN THE COURTROOM In the courts, billions of dollars turn on experts’ disputed cost of capital estimates in many contexts:

Gift, estate, and income tax disputes

Dissenting stockholder suits Corporate and partnership dissolutions

Marital property settlements Employee stock ownership plans (ESOPs)

Ad valorem (property) taxes

Utility rate-setting Damages calculations

Fortunately, courts are becoming unwilling to accept the statement ‘‘Trust me, I’m a great expert’’ in these disputes and instead are carefully weighing the quality of supporting evidence presented by opposing sides. Because cost of capital is critical to the valuation of any ongoing business, the thorough understanding, analysis, and presentation of cost of capital issues will go a long way toward carrying the day in a battle of experts in a legal setting.

ORGANIZATION OF THIS BOOK PART I. COST OF CAPITAL BASICS Chapter 1 defines the concept of cost of capital. Chapter 2 describes, in a general sense, how it is used in business valuation and capital budgeting. Chapter 3 defines net cash flow, explains why it is the preferred economic income variable for valuation and capital budgeting, and discusses issues relative to measuring expected net cash flow. Chapter 4 explains the difference between discounting and capitalizing. Chapter 5 addresses the concept of risk and the impact of risk on the cost of capital. Chapter 6 discusses the various components of a company’s capital structure. PART II. ESTIMATING THE COST OF EQUITY CAPITAL The second part explores cost of capital estimation. We begin with the build-up model (Chapter 7) and the Capital Asset Pricing Model (CAPM) (Chapter 8), two of the most widely used models for estimating cost of equity capital. With the backgrounds of those chapters, we then discuss a major input into these

Introduction

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models (and in all other cost of capital models), the equity risk premium, in Chapter 9. We then discuss other inputs into these models: making beta estimates for CAPM (Chapter 10) and one correction to the CAPM called the size effect (Chapter 12). In Chapter 11 we discuss the criticism of CAPM and beta as sole risk measures, and present alternative risk measures, while in Chapter 13 we discuss the criticisms of the size effect. In Chapter 14 we discuss company-specific risk. In Chapter 15 we then present alternative cost of equity capital models, such as the Fama-French 3-factor model. In Chapter 16 we present methods for deriving implied cost of capital estimates using discounted cash flow (DCF) and residual income models. In Chapter 17 we discuss the overall cost of capital based on the concept of a weighted average of the cost of each component of the capital structure (commonly referred to as the weighted average cost of capital) and how changes in the capital structure affect the cost of equity capital. We include appendices with detailed explanations of the iterative process for cost of equity capital estimation for nonpublic businesses (divisions, reporting units, closely held firms) in the context of the weighted average cost of capital. The models presented thus far are based on estimating the cost of equity for companies and investments in developed economies. In Chapter 18 we discuss alternative models for estimating the cost of equity capital for companies and investments in developing economies. Chapter 19 contains a discussion of the various data sources available from Ibbotson Associates, now part of Morningstar, for use in estimating the cost of capital. PART III. CORPORATE FINANCIAL OFFICERS—USING COST OF CAPITAL DATA The third part explores use of the cost of capital data in capital budgeting and feasibility studies in Chapter 20, determining cost of capital for divisions and reporting units in Chapter 21, using the data in evaluating mergers and acquisitions in Chapter 22, cost of capital in transfer pricing in Chapter 23, and using the data within the framework of an Economic Value Added (EVA1) financial management system in Chapter 24. PART IV. COST OF CAPITAL FOR CLOSELY HELD ENTITIES Part IVaddresses commonly encountered variations in cost of capital application particularly within the context of closely held entities. Chapter 25 covers handling discounts for lack of marketability. Chapters 26 to 28 address cost of capital for pass-through entities: partnerships, limited liability corporations, S corporations, and private investment companies. Chapter 29 focuses on venture capital investments. PART V. OTHER TOPICS Chapter 30 discusses minority versus controlling interest valuations. In Chapter 31 we discuss how the cost of capital relates to the excess earnings valuation method. Chapter 32 discusses adjusting the discount rate when the measure of economic income is some measure other than net cash flow. While this book is not a treatise on forecasting, we discuss issues in estimating net cash flows in Chapter 33. Chapter 34 covers common errors in estimating discount rates and future economic income, and Chapter 35 presents court case examples of cost of capital issues. PART VI. REAL ESTATE AND AD VALOREM Given the specialized nature of real estate, we present information on developing cost of capital estimates for real property investments, real estate investment trusts, and are within the context of ad valorem taxation in Chapters 36 through 38.

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Introduction

PART VII. ADVICE TO PRACTITIONERS Chapter 39 provides issue-by-issue advice on handing real-world cost of capital estimation issues. Chapter 40 provides questions for attorneys to use in cross-examining experts on cost of capital. APPENDICES The appendices provide sources for follow-up to this book, including a detailed bibliography in Appendix I, sources for the current data needed to implement cost of capital estimation in Appendix II, a glossary of business valuation terms in Appendix III, a sample report in Appendix IV, information on how to use ValuSource valuation software in Appendix V, and a review of statistical analysis in Appendix VI.

SUMMARY The book is designed to serve as both a primer and a reference source. Part I covers cost of capital basics. Part II covers the methods generally used to estimate cost of equity capital. Part III covers a variety of topics commonly encountered by the corporate financial officer. Part IV covers issues peculiar to closely held entities. Part V covers a variety of topics integral to users of cost of capital data. Part VI covers real property cost of capital issues. Part VII covers real-world cost estimation issues. The appendices provide a directory for further study and data sources.

Notation System and Abbreviations Used in This Book A source of confusion for those trying to understand financial theory and methods is that financial writers have not adopted a standard system of notation. The notation system used in this volume is adapted from the fifth edition of Valuing a Business: The Analysis and Appraisal of Closely Held Companies, by Shannon P. Pratt (New York: McGraw-Hill, 2007).

VALUE AT A POINT IN TIME PV ¼ Present value (seen as Pi in Chapter 19) PVb ¼ Present value of net cash flows due to business operations before cost of financing PVeu ¼ Present value of unlevered equity PVTSD ¼ Present value of tax savings on debt PVe ¼ Present value of equity FV ¼ Future value MVIC ¼ Market value of invested capital ¼ Enterprise value ¼ Me þ Md þ M p BV ¼ Book value of net assets FVRU ¼ Fair value of reporting unit FVNWCRU ¼ Fair value of net working capital of the reporting unit FVICRU ¼ Fair value of invested capital of the reporting unit FVFARU ¼ Fair value of fixed assets of the reporting unit FVIARU ¼ Fair value on intangible assets, identified and individually valued, of the reporting unit FVUIVRU ¼ Fair value of unidentified intangibles value (i.e., ‘‘goodwill’’) of the reporting unit FVdRU ¼ Fair value of debt capital of the reporting unit FVeRU ¼ Fair value of equity capital of the reporting unit FMVBE ¼ Fair market value of the business enterprise FMVNWC ¼ Fair market value of net working capital FMVFA ¼ Fair market value of fixed assets FMVIA ¼ Fair market value on intangible assets FMVUIV ¼ Fair market value of unidentified intangibles value (i.e., ‘‘goodwill’’) FMVe ¼ Fair market value of equity capital

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Notation System and Abbreviations Used in This Book

COST OF CAPITAL AND RATE OF RETURN VARIABLES k ¼ Discount rate (generalized) kc ¼ Country cost of equity ke ¼ Discount rate for common equity capital (cost of common equity capital) (seen as ki in Chapter 19). Unless otherwise stated, it generally is assumed that this discount rate is applicable to net cash flow available to common equity. ke500 ¼ Cost of equity for the S&P 500 keu ¼ Cost of equity capital, unlevered (cost of equity capital assuming firm financed with all equity) kni ¼ Discount rate for equity capital when net income rather than net cash flow is the measure of economic income being discounted kð ptÞ ¼ Discount rate applicable to pretax cash flows keð ptÞ ¼ Cost of equity prior to tax affect k p ¼ Discount rate for preferred equity capital kd ¼ Discount rate for debt (net of tax affect, if any) (Note: For complex capital structures, there could be more than one class of capital in any of the preceding categories, requiring expanded subscripts.) ¼ kdð ptÞ (1 – tax rate) kdð ptÞ ¼ Cost of debt prior to tax effect kTS ¼ Rate of return used to present value tax savings due to deducting interest expense on debt capital financing kNWCð ptÞ ¼ Rate of return for net working capital financed with debt capital (measured pretax) and equity capital kFAð ptÞ ¼ Rate of return for fixed assets financed with debt capital (measured pretax) and equity capital kdRU ¼ After-tax rate of return on debt capital of the reporting unit ¼ kdð ptÞRU (1 – tax rate) kdð ptÞRU ¼ Rate of return on debt capital of the reporting unit without taking into account the tax deduction on interest expense (pretax cost of debt capital) keRU ¼ After-tax rate of return on equity capital of the reporting unit kNWC ¼ Rate of return for net working capital kNWCRU ¼ Rate of return for net working capital of the reporting unit financed with debt capital (return measured after-tax) and equity capital kNWCð ptÞRU ¼ Rate of return for net working capital of the reporting unit financed with debt capital (measured pretax) and equity capital kFA ¼ Rate of return for fixed assets kFARU ¼ Rate of return for fixed assets financed with debt capital (return measured after tax) and equity capital kFAð ptÞRU ¼ Rate of return for fixed assets of the reporting unit financed with debt capital (measured pretax) and equity capital


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kIA ¼ Rate of return for intangible assets kIARU ¼ Rate of return for identified and individually valued intangible assets financed with debt capital (return measured after tax) and equity capital kUIV ¼ Rate of return for unidentified intangibles value kUIVRU ¼ Rate of return for unidentified intangibles value of the reporting unit financed with debt capital (return measured after tax) and equity capital kIAþUIV ¼ After-tax rate of return on all intangible assets, identified and individually valued, plus the unidentified intangible value kIAþUIVð ptÞ ¼ Pretax rate of return on all intangible assets, identified and individually valued, plus the unidentified intangible value financed with debt capital (measured pretax) and equity capital kIAþUIVRU ¼ After-tax rate of return on all intangible assets, identified and individually valued, of the reporting unit plus the unidentified intangible value of the reporting unit kIAþUIVð ptÞRU ¼ Pretax rate of return on all intangible assets, identified and individually valued, plus the unidentified intangible value of the reporting unit financed with debt capital (measured pretax) and equity capital kTSRU ¼ Rate of return used to present value tax savings due to deducting interest expense on debt capital financing of the reporting unit c ¼ Capitalization rate ce ¼ Capitalization rate for common equity capital. Unless otherwise stated, it generally is assumed that this capitalization rate is applicable to net cash flow available to common equity. Cni ¼ Capitalization rate for net income cð ptÞ ¼ Capitalization rate on pretax cash flows c p ¼ Capitalization rate for preferred equity capital cd ¼ Capitalization rate for debt (Note: For complex capital structures, there could be more than one class of capital in any of the preceding categories, requiring expanded subscripts.) D=P ¼ Dividend yield on stock DR j ¼ Downside risk in the local market (U.S. dollars) DRw ¼ Downside risk in global (‘‘world’’) market (U.S. dollars) Pn ¼ Stock price in period n P0 ¼ Stock price at valuation period R ¼ Rate of return R f ¼ Rate of return on a risk-free security R f ;n ¼ Risk-free rate in current month R f local ¼ Return on the local country government’s (default-risk-free) paper R f u:s: ¼ U.S. risk-free rate Rlocaleuro$issue ¼ Current market interest rate on debt issued by the local country government denominated in U.S. dollars (‘‘euro-dollar’’ debt), same maturity as debt issued by the local country government denominated in U.S. dollars

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Rn ¼ Return on individual security subject stock in current month Rm ¼ Historical rate of return on the ‘‘market’’ RP ¼ Risk premium RPm ¼ Risk premium for the ‘‘market’’ (usually used in the context of a market for equity securities, such as the NYSE or S&P 500) RPs ¼ Risk premium for ‘‘small’’ stocks (usually average size of lowest quintile or decile of NYSE as measured by market value of common equity) over and above RPm RPmþs ¼ Risk premium for the market plus risk premium for size (Duff & Phelps Risk premium report data for use in build-up method) RPu ¼ Risk premium for company specific or unsystematic risk attributable to the specific company RPw ¼ The equity risk premium on a ‘‘world’’ diversified portfolio RPi ¼ Risk premium for the ith security (seen in Chapter 19 as IRPi) RPlocal ¼ Equity risk premium in local country’s stock market RIi ¼ Risk index (full-information beta) for industry i RIiL ¼ Full-information levered beta estimate of the subject company EðRÞ ¼ Expected rate of return EðRm Þ ¼ Expected rate of return on the ‘‘market’’ (usually used in the context of a market for equity securities, such as the New York Stock Exchange [NYSE] or Standard & Poor’s [S&P] 500) EðRi Þ ¼ Expected rate of return on security i EðRdiv Þ ¼ Expected rate of return on dividend EðRcapgains Þ ¼ Expected rate of return on capital gains B ¼ Beta (a coefficient, usually used to modify a rate of return variable) BL ¼ Levered beta for (equity) capital BU ¼ Unlevered beta for (equity) capital BLS ¼ Levered segment beta Bd ¼ Beta for debt capital BUi ¼ Beta unlevered for industry (or guideline companies) equity capital BLi ¼ Beta levered for industry (or guideline companies) equity capital Be ¼ Beta (equity) expanded Bop ¼ Operating beta (beta with effects of fixed operating expense removed) Bi ¼ Beta of company i (F-F Beta) Bn ¼ Estimated market coefficient based on sensitivity to excess returns on market portfolio in current month Bn1 ¼ Estimated lagged market coefficient based on sensitivity to excess returns on market portfolio last month Blocal ¼ Market risk of the subject company measured with respect to the local securities market Bw ¼ Market or systematic risk measured with respect to a ‘‘world’’ portfolio of stocks


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bcw ¼ Country covariance with world bcr ¼ Country covariance with region K1 :::Kn ¼ Risk premium associated with risk factor 1 through n for the average asset in the market (used in conjunction with arbitrage pricing theory) si ¼ Small-minus-big coefficient in the Fama-French regression SMBP ¼ Expected small-minus-big risk premium, estimated as the difference between the historical average annual returns on the small-cap and large-cap portfolios hi ¼ High-minus-low coefficient in the Fama-French regression HMLP ¼ Expected high-minus-low risk premium, estimated as the difference between the historical average annual returns on the high book-to-market and low book-to-market portfolios Fd ¼ Face value of outstanding debt WACCð ptÞ ¼ Weighted average cost of capital (pretax) WACCBE ¼ Overall rate of return for the business enterprise WACCð ptÞBE ¼ Pretax WACC of the business enterprise WACCRU ¼ Overall rate of return for the reporting unit WACCð ptÞRU

¼ Weighted average cost of capital for the reporting unit ¼ Pretax WACC of the reporting unit

Me ¼ Market value of equity capital (stock) Md ¼ Market value of debt capital M p ¼ Market value of preferred equity s 2 ¼ Variance of returns for subject company stock s 2M ¼ Variance of the returns on the market portfolio (e.g., S&P 500) s 2e ¼ Variance of error terms s ¼ Standard deviation s B ¼ Standard deviation of operating cash flows of business before cost of financing s rev ¼ Standard deviation of revenues of output s local ¼ Volatility of subject (local) stock market s u:s: ¼ Volatility of U.S. stock market s stock ¼ Volatility of local country’s stock market s bond ¼ Volatility of local country’s bond market dr ¼ Regional risk not included in RPw CCRc ¼ Country credit rating of country l ¼ Company’s exposure to the local country risk t ¼ Tax rate (expressed as a percentage of pretax income) ti ¼ Federal and state income tax rate for industry (or guideline companies) r ¼ Property tax rate (expressed as a percentage of total fair market value) C ¼ Proportion of the entity that is assessed property tax h ¼ Holding period

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INCOME VARIABLES E ¼ Expected economic income (in a generalized sense; i.e., could be dividends, any of several possible definitions of cash flows, net income, etc.) F ¼ Fixed operating assets (without regard to costs of financing) Fc ¼ Fixed operating costs of the business NI ¼ Net income (after entity-level taxes) CF ¼ Cash flow for a specific period NCFe ¼ Net cash flow (free cash flow) to equity NCF f ¼ Net cash flow (free cash flow) to the firm (to overall invested capital, or entire capital structure, including all equity and long-term debt) NCFue ¼ Net cash flow to unlevered equity PMT ¼ Payment (interest and principal payment on debt security) D ¼ Dividends T ¼ Tax (in U.S. dollars) TS ¼ Present value of tax savings due to deducting interest expense on debt capital financing GCF ¼ Gross cash flow (usually net income plus noncash charges) EBT ¼ Earnings before taxes EBIT ¼ Earnings before interest and taxes EBITD ¼ Earnings before depreciation, interest, and taxes (‘‘Depreciation’’ in this context usually includes amortization. Some writers use EBITDA specifically to indicate that amortization is included.) EBITDA ¼ Earnings before interest, taxes, depreciation, and amortization V ¼ Variable operating assets

PERIODS OR VARIABLES IN A SERIES i ¼ ith period or ith variable in a series (may be extended to the jth variable, the kth variable, etc.) n ¼ Number of periods or variables in a series, or the last number in a series 0 ¼ Period 0, the base period, usually the latest year immediately preceding the valuation date py ¼ Partial year of first year following the valuation date

WEIGHTINGS W ¼ Weight We ¼ Weight of common equity in capital structure WeRU

¼ Me =Me þ Md ¼ Weight of equity capital in structure of reporting unit


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¼ Fair value of equity capital/FVRU W p ¼ Weight of preferred equity in capital structure ¼ M p =Me þ Md þ M p Wd ¼ Weight of debt in capital structure ¼ Md =Me þ Md (Note: For purposes of computing a weighted average cost of capital [WACC], it is assumed that preceding weightings are at market value.) Wdi ¼ Weight of interest-bearing debt in capital structure at market for industry (or guideline companies) WdRU ¼ Weight of debt capital in capital structure of reporting unit ¼ Fair value of debt capital/FVRU Wei ¼ Weight of common equity in capital structure at market for industry (or guideline companies) Ws ¼ Weight of segment data to total business (e.g., sales, operating income) WNWC ¼ Weight of net working capital in FMVBE ¼ FMVNWC =FMVBE WNWCRU ¼ Weight of net working capital in FVRU ¼ FVNWCRU =FVRU WFA ¼ Weight of fixed assets in FMVBE WFARU WIA

¼ FMVFA =FMVBE ¼ Weight of fixed assets in FVRU ¼ FVFARU =FVRU ¼ Weight of intangible assets in FMVBE ¼ FMVIA =FMVBE

WIARU ¼ Weight of intangible assets in FVRU ¼ FVIARU =FVRU WUIV ¼ Weight of unidentified intangibles value FMVBE ¼ FMVUIV (i.e., ‘‘goodwill’’)=FMVBE WUIVRU ¼ Weight of unidentified intangibles value FVRU WIAþUIV

WIAþUIVRU

¼ FVUIVRU (i.e., ‘‘goodwill’’)=FVRU ¼ Weight of intangible assets in FMVBE plus the weight of unidentified intangible value in FMVBE ¼ ðFMVIA þ FMVUIV Þ=FMVBE ¼ Weight of intangible assets in FVRU plus the weight of unidentified intangible value in FVRU

¼ ðFVIARU þ FVUIVRU Þ=FVRU WTS ¼ Weight of TS in FMVBE WTSRU

¼ TS=FMVBE ¼ Weight of TS in FVRU ¼ TS=FVRU

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GROWTH g ¼ Rate of growth in a variable (e.g., net cash flow) gni ¼ Rate of growth in net income

MATHEMATICAL FUNCTIONS S ¼ Sum of (add all the variables that follow) \ ¼ Product of (multiply together all the variables that follow) X ¼ Mean average (the sum of the values of the variables divided by the number of variables) G ¼ Geometric mean (product of the values of the variables taken to the root of the number of variables) a ¼ Regression constant e ¼ Regression error term 1 ¼ Infinity

NOTATION FOR REAL PROPERTY VALUATION (CHAPTER 36) DSCR ¼ Debt service coverage ratio EGIM ¼ Effective gross income multiplier NOI; Ip ¼ Net operating income OER ¼ Operating expense rates ke ¼ Equity discount or yield rate (dividend plus appreciation) km ¼ Mortgage interest rate kp ¼ Overall property discount rate cp ¼ Overall property capitalization rate ce ¼ Dividend to equity capitalization rate cm ¼ Mortgage capitalization rate or constant cn ¼ Terminal or residual or going-out capitalization rate cB ¼ Building capitalization rate cL ¼ Land capitalization rate cLF ¼ Leased fee capitalization rate cLH ¼ Leasehold capitalization rate A ¼ Change in income and value (adjustment factor) P ¼ Principal paid off over the holding period 1=Sn ¼ Sinking fund factor at the equity discount or yield rate (ke ) D0 ¼ Change in value over the holding period SC% ¼ Cost of sale PGI ¼ Potential gross income PGIM ¼ Potential gross income multiplier


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EGI ¼ Effective gross income Fd =PVp ¼ Face value of debt (loan amount outstanding) to value ratio ½1 ðFd =PVp Þ ¼ Equity to value ratio MB ¼ Building value Mm ¼ Mortgage value ML ¼ Land value MLF ¼ Leased fee value MLH ¼ Leasehold value Ip ¼ Overall income to the property IL ¼ Residual income to the land IB ¼ Residual income to the building Ie ¼ Equity income Im ¼ Mortgage income ILF ¼ Income to the leased fee ILH ¼ Income to the leasehold

ABBREVIATIONS ERP ¼ Equity risk premium (usually the general equity risk premium for which the benchmark for equities is either the S&P 500 stocks or the NYSE stocks) WACC ¼ Weighted average cost of capital WARA ¼ Weighted average return on assets T-Bill ¼ U.S. government bill (usually 30-day, but can be up to one year) STRIPS ¼ Separate trading of registered interest and principal of securities CRSP ¼ Center for Research in Security Prices, at the University of Chicago PIPE ¼ Private Investment in Public Equity SBBI ¼ Stocks, Bonds, Bills, and Inflation, published annually by Ibbotson Associates (now Morningstar) in both a ‘‘Classic edition’’ and a ‘‘Valuation edition’’ CAPM ¼ Capital Asset Pricing Model DCF ¼ Discounted cash flow DDM ¼ Discounted dividend model TIPS ¼ Treasury Inflation-Protected Security NCF ¼ Net cash flow (also sometimes interchangeably referred to as FCF, free cash flow) BE ¼ Business enterprise or reporting unit NWC ¼ Net working capital FA ¼ Fixed assets IA ¼ Intangible assets UIV ¼ Unidentified intangible value (i.e., ‘‘goodwill’’) NOPAT ¼ Net operating profit after taxes

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PAT ¼ Profit after tax ¼ Net Income RI ¼ Residual income EVA ¼ Economic value added DY ¼ Dividend yield RE ¼ Residual earnings AEG ¼ Abnormal earnings growth ROCE ¼ Return on common equity RNOA ¼ Return on net operating assets FLEV ¼ Net financial obligations/(Net operating assets net financial obligations) (i.e., financial leverage) SPREAD ¼ RNOA Net borrowing costs [(financial expense financial income, after tax)/(financial obligations financial assets)] io ¼ Implicit interest charges on operating liabilities (other than deferred taxes) OI ¼ Operating income OA ¼ Operating assets OL ¼ Operating liabilities OI ¼ Operating income NICE ¼ Nonmarketable investment company evaluation REIT ¼ Real estate investment trusts

Part 1

Cost of Capital Basics

Chapter 1

Defining Cost of Capital Introduction Components of a Company’s Capital Structure Cost of Capital is a Function of the Investment Cost of Capital is Forward Looking Cost of Capital is Based on Market Value, Not Book Value Cost of Capital is Usually Stated in Nominal Terms Cost of Capital Equals the Discount Rate Discount Rate is Not the Same as Capitalization Rate Summary

INTRODUCTION Cost of capital is the expected rate of return that the market participants require in order to attract funds to a particular investment. In economic terms, the cost of capital for a particular investment is an opportunity cost—the cost of forgoing the next best alternative investment. In this sense, it relates to the economic principle of substitution—that is, an investor will not invest in a particular asset if there is a more attractive substitute. The term market refers to the universe of investors who are reasonable candidates to provide funds for a particular investment. Capital or funds are usually provided in the form of cash, although in some instances capital may be provided in the form of other assets. The cost of capital usually is expressed in percentage terms, that is, the annual amount of dollars that the investor requires or expects to realize, expressed as a percentage of the dollar amount invested. Put another way: Since the cost of anything can be defined as the price one must pay to get it, the cost of capital is the return a company must promise in order to get capital from the market, either debt or equity. A company does not set its own cost of capital; it must go into the market to discover it. Yet meeting this cost is the financial market’s one basic yardstick for determining whether a company’s performance is adequate.1

As the quote suggests, most of the information for estimating the cost of capital for any company, security, or project comes from the investment markets. The cost of capital is always an expected (or forward-looking) return. Thus, analysts and would-be investors never actually observe it. We analyze many types of market data to estimate the cost of capital for a company, security, or project in which we are interested.

1

Mike Kaufman, ‘‘Profitability and the Cost of Capital,’’ in Chapter 8 of Handbook of Budgeting, 4th ed., ed. Robert Rachlin (New York: John Wiley & Sons, 1999), 8–3.

3

4

Cost of Capital

As Roger Ibbotson put it, ‘‘The Opportunity Cost of Capital is equal to the return that could have been earned on alternative investments at a specific level of risk.’’2 In other words, it is the competitive return available in the market on a comparable investment, with risk being the most important component of comparability.

COMPONENTS OF A COMPANY’S CAPITAL STRUCTURE The term capital in this context means the components of an entity’s capital structure. The primary components of a capital structure include:

Debt capital Preferred equity (stock or partnership interests with preference features, such as seniority in receipt of dividends or liquidation proceeds) Common equity (stock or partnership interests at the lowest or residual level of the capital structure)

There may be more than one subcategory in any or all of the listed categories of capital. Also, there may be related forms of capital, such as warrants or options. Each component of an entity’s capital structure has its unique cost, depending primarily on its respective risk. The next quote explains how the cost of capital can be viewed from three different perspectives: On the asset side of a firm’s balance sheet, it is the rate that should be used to discount to a present value the future expected cash flows. On the liability side, it is the economic cost to the firm of attracting and retaining capital in a competitive environment, in which investors (capital providers) carefully analyze and compare all return-generating opportunities. On the investor’s side, it is the return one expects and requires from an investment in a firm’s debt or equity. While each of these perspectives might view the cost of capital differently, they are all dealing with the same number.3

When we talk about the cost of ownership capital (e.g., the expected return to a stock or partnership investor), we usually use the phrase cost of equity capital. When we talk about the cost of capital to the firm overall (e.g., the average cost of capital for both equity ownership interests and debt), we commonly use the phrases weighted average cost of capital (WACC) or blended cost of capital overall cost of capital. Simply and cogently stated, ‘‘The cost of equity is the rate of return investors require on an equity investment in a firm.’’4 Recognizing that the cost of capital applies to both debt and equity investments, a well-known text states: Since free cash flow is the cash flow available to all financial investors (debt, equity, and hybrid securities), the company’s Weighted Average Cost of Capital (WACC) must include the required return for each investor.5 2 3 4

5

Ibbotson Associates, Cost of Capital Workshop (Chicago: Ibbotson Associates, 1999). Stocks, Bonds, Bills and Inflation Valuation Edition 2007 Yearbook (Chicago: Morningstar, 2007), 23. Aswath Damodaran, Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2002), 182. Tim Koller, Marc Goedhart, and David Wessels, Valuation: Measuring and Managing the Value of Companies, 4th ed. (Hoboken, NJ: John Wiley & Sons, 2005), 291.

Cost of Capital is Forward Looking

5

COST OF CAPITAL IS A FUNCTION OF THE INVESTMENT As Ibbotson puts it, ‘‘The cost of capital is a function of the investment, not the investor.’’6 The cost of capital comes from the marketplace. The marketplace is the universe of investors ‘‘pricing’’ the risk of a particular asset. Allen, Brealey, and Myers state the same concept: ‘‘The true cost of capital depends on the use to which the capital is put.’’7 They make the point that it would be an error to evaluate a potential investment on the basis of a company’s overall cost of capital if that investment were more or less risky than the company’s existing business. ‘‘Each project should in principle be evaluated at its own opportunity cost of capital.’’8 When a company uses a given cost of capital to evaluate a commitment of capital to an investment or project, it often refers to that cost of capital as the hurdle rate. The hurdle rate is the minimum expected rate of return that the company would be willing to accept to justify making the investment. As noted, the hurdle rate for any given prospective investment may be at, above, or below the company’s overall cost of capital, depending on the degree of risk of the prospective investment compared to the company’s overall risk. The most popular focus of contemporary corporate finance is that companies should be making investments, either capital investments or acquisitions, from which the returns will exceed the cost of capital for that investment. Doing so creates economic value added, economic profit, or shareholder value added.9

COST OF CAPITAL IS FORWARD LOOKING The cost of capital represents investors’ expectations. There are two elements to these expectations: 1. The risk-free rate, which includes: The ‘‘real’’ rate of return—the amount (excluding inflation) investors expect to obtain in exchange for letting someone else use their money on a risk-free basis. Expected inflation—the expected depreciation in purchasing power while the money is in use. 2. Risk—the uncertainty as to when and how much cash flow or other economic income will be received. It is the combination of the first two items that is sometimes referred to as the time value of money. While these expectations, including assessment of risk, may be different for different investors, the market tends to form a consensus with respect to a particular investment or category of investments. That consensus determines the cost of capital for investments of varying levels of risk. The cost of capital, derived from investors’ expectations and the market’s consensus of those expectations, is applied to expected economic income, usually measured in terms of cash flows, in order to estimate present values or to compare investment alternatives of similar or differing levels of risk. Present value, in this context, refers to the dollar amount that a rational and well-informed investor 6 7

8 9

Ibbotson Associates, Cost of Capital Workshop (Chicago: Ibbotson Associates, 1999). Richard A. Brealey, Stewart C. Myers, and Franklin Allen, Principles of Corporate Finance, 8th ed. (Boston: Irwin McGrawHill, 2006), 216. Ibid. See, for example, Tim Koller, Marc Goedhart, and David Wessels, Valuation: Measuring and Managing the Value of Companies, 4th ed. (Hoboken, NJ: John Wiley & Sons, 2005); also see Alfred Rappaport, Creating Shareholder Value, rev. ed. (New York: The Free Press, 1997).

6

Cost of Capital

would be willing to pay today for the stream of expected economic income. In mathematical terms, the cost of capital is the percentage rate of return that equates the stream of expected income with its present cash value (see Chapter 4).

COST OF CAPITAL IS BASED ON MARKET VALUE, NOT BOOK VALUE The cost of capital is the expected rate of return on some base value. That base value is measured as the market value of an asset, not its book value. For example, the yield to maturity shown in the bond quotations in the financial press is based on the closing market price of a bond, not on its face value. Similarly, the implied cost of equity for a company’s stock is based on the market price per share at which it trades, not on the company’s book value per share of stock. It was noted earlier that the cost of capital is estimated from market data. This data refers to expected returns relative to market prices. By applying the cost of capital derived from market expectations to the expected cash flows (or other measure of economic income) from the investment or project under consideration, the market value can be estimated.

COST OF CAPITAL IS USUALLY STATED IN NOMINAL TERMS Keep in mind that we have talked about expectations, including inflation. The return an investor requires includes compensation for reduced purchasing power of the dollar over the life of the investment. Therefore, when the analyst or investor applies the cost of capital to expected returns in order to estimate value, he or she must also include expected inflation in those expected returns. This obviously assumes that investors have reasonable consensus expectations regarding inflation. For countries subject to unpredictable hyperinflation, it is sometimes more practical to estimate cost of capital in real terms rather than in nominal terms.10

COST OF CAPITAL EQUALS THE DISCOUNT RATE The essence of the cost of capital is that it is the percentage return that equates expected economic income with present value. The expected rate of return in this context is called a discount rate. By discount rate, the financial community means an annually compounded rate at which each increment of expected economic income (e.g., net cash flow, net income, or some other measure of economic benefits) is discounted back to its present value. A discount rate reflects both time value of money and risk and therefore represents the cost of capital. The sum of the discounted present values of each future period’s incremental cash flow or other measure of return equals the present value of the investment, reflecting the expected amounts of return over the life of the investment. The terms discount rate, cost of capital, and required rate of return are often used interchangeably. The economic income referenced here represents total expected benefits. In other words, this economic income includes increments of cash flow realized by the investor while holding the investment as well as proceeds to the investor upon liquidation of the investment. The rate at which these expected future total returns are reduced to present value is the discount rate, which is the cost of capital (required rate of return) for a particular investment. 10

We discuss the problems with estimating cash flows and cost of capital in real terms in Chapter 18.

Summary

7

DISCOUNT RATE IS NOT THE SAME AS CAPITALIZATION RATE Because some practitioners confuse the terms, we point out here that discount rate and capitalization rate are two distinctly different concepts. As noted in the previous section, discount rate equates to cost of capital. It is a rate applied to all expected incremental returns to convert the expected return stream to a present value. A capitalization rate, however, is merely a divisor applied to one single element of return to estimate a present value. The only instance in which the discount rate is equal to the capitalization rate is when each future cash flow is equal (i.e., no growth), and the expected returns are in perpetuity. One of the few examples would be a preferred stock paying a fixed amount of dividend per share in perpetuity. The relationship between discount and capitalization rates is discussed in Chapter 4.

SUMMARY As stated in the introduction, ‘‘The cost of capital estimate is the essential link that enables us to convert a stream of expected income into an estimate of present value.’’ Cost of capital has several key characteristics:

It is market driven. It is the expected rate of return that the market requires to commit capital to an investment.

It is a function of the investment, not a particular investor; to make the discount rate a function of the particular investor would imply changing the standard of value to what is commonly termed investment value rather than fair market value. It is forward looking, based on expected returns. Past returns are, at best, to provide guidance as to what to expect in the future.

The base against which cost of capital is measured is market value, not book value. It is usually measured in nominal terms, that is, including expected inflation. It is the link, called a discount rate, that equates expected future returns for the life of the investment with the present value of the investment at a given date.

Chapter 2

Introduction to Cost of Capital Applications: Valuation and Project Selection Introduction Net Cash Flow is the Preferred Economic Income Measure Cost of Capital is the Proper Discount Rate Present Value Formula Example: Valuing a Bond Applications to Businesses, Business Interests, Projects, and Divisions Summary

INTRODUCTION Cost of capital has many applications, the two most common being valuation and capital investment project selection. These two applications are very closely related. This chapter discusses these two applications in very general terms so the reader can quickly understand how a proper estimation of the cost of capital underlies valuations and financial decisions worth billions of dollars every day. Later chapters discuss these applications in more detail.

NET CASH FLOW IS THE PREFERRED ECONOMIC INCOME MEASURE For the purpose of this chapter, we will assume that the measure of economic income to which cost of capital will be applied is net cash flow (sometimes called free cash flow). Net cash flow represents discretionary cash available to be paid out to stakeholders of a firm (providers of capital to the firm) (e.g., interest, debt payments, dividends, withdrawals, discretionary bonuses) without jeopardizing the projected ongoing operations of the business. We will provide a more exact definition of net cash flow in Chapter 3. Net cash flow to equity is that cash flow available to the equity holders, usually common equity. Net cash flow is the measure of economic income on which most financial analysts today prefer to focus for both valuation and capital investment project selection. Net cash flow represents money available to stakeholders, assuming the business is a going concern and the company is able to support the projected operations. Although the contemporary literature of corporate finance widely embraces a preference for net cash flow as the relevant economic income variable to which to apply cost

9

10

Cost of Capital

of capital for valuation and decision making, there is still a contingent of analysts who prefer to focus on reported or adjusted accounting income.1

COST OF CAPITAL IS THE PROPER DISCOUNT RATE At the end of Chapter 1, we said that the cost of capital is customarily used as a discount rate to convert expected future returns to a present value. This concept is summarized succinctly by Allen, Brealey, and Myers: ‘‘When you discount [a] project’s expected cash flow at its opportunity cost of capital, the resulting present value is the amount investors would be willing to pay for the project.’’2 In this context, let us keep in mind critical characteristics of a discount rate: Definition: A discount rate is a yield rate used to convert anticipated future payments or receipts into present value (i.e., a cash value as of a specified valuation date). A discount rate represents the total expected rate of return that the investor requires to realize on the amount invested. The use of the cost of capital to estimate present value thus requires two sets of estimates: 1. The numerator. The expected amount of return (e.g., the net cash flow) on the investment in each future period over the life of the investment. 2. The denominator. A function of the discount rate, which is the cost of capital, which, in turn, is the required rate of return. This function is usually written as ð1 þ kÞn . where: k ¼ Discount rate n ¼ Number of periods into the future when the returns are expected to be realized Usually analysts and investors make the simplifying assumption that the cost of capital is constant over the life of the investment and use the same cost of capital to apply to each increment of expected future return. There are, however, special cases in which analysts might choose to estimate a discrete cost of capital to apply to the expected return in each future period. (An example is when the analyst anticipates a changing weighted average cost of capital because of a changing capital structure.) Notwithstanding, well-known author, professor, and consultant Dr. Alfred Rappaport espouses a constant cost of capital in his book Creating Shareholder Value: The appropriate rate for discounting the company’s cash flow stream is the weighted average of the costs of debt and equity capital. . . . It is important to emphasize that the relative weights attached to debt and equity, respectively, are neither predicated on dollars the firm has raised in the past, nor do they constitute the relative proportions of dollars the firm plans to raise in the current year. Instead, the relevant weights should be based on the proportions of debt and equity that the firm targets for its capital structure over the long-term planning period.3

The latter view is most widely used in practice. 1

2

3

See, for example, Z. Christopher Mercer, Valuing Financial Institutions (Homewood, IL: Business One Irwin, 1992), Chapter 13; and his article ‘‘The Adjusted Capital Asset Pricing Model for Developing Capitalization Rates,’’ Business Valuation Review (December 1989): 147ff. Franklin Allen, Richard A. Brealey, and Stewart C. Myers, Principles of Corporate Finance, 8th ed. (Boston: Irwin McGrawHill, 2006), 20. Alfred Rappaport, Creating Shareholder Value: A Guide for Managers and Investors, rev. ed. (New York: The Free Press, 1997), 37.

Example: Valuing a Bond

11

PRESENT VALUE FORMULA Converting the concepts just discussed into a mathematical formula, we have the following formula, which is the essence of using cost of capital to estimate present value. (Formula 2.1) PV ¼

NCF1 NCF2 NCFn þ þ þ ð1 þ kÞ ð1 þ kÞ2 ð1 þ kÞn

where: PV ¼ Present value NCF1. . . NCFn ¼ Net cash flow (or other measure of economic income) expected in each of the periods 1 through n, n being the final cash flow in the life of the investment k ¼ Cost of capital applicable to the defined stream of net cash flow n ¼ Number of periods The critical job for the analyst is to match the cost of capital estimate to the definition of the economic income stream being discounted. This is largely a function of reflecting in the cost of capital estimate the degree of risk inherent in the expected cash flows being discounted. The relationship between risk and the cost of capital is the subject of Chapter 5.

EXAMPLE: VALUING A BOND A simple example of the use of Formula 2.1 is valuing a bond for which a risk rating has been estimated. Let us make five assumptions: 1. The bond has a face value of $1,000. 2. It pays 8% interest on its face value. 3. The bond pays interest once a year, at the end of the year. (This, of course, is a simplifying assumption. Some bonds and notes pay only annually, but most publicly traded bonds pay interest semiannually.) 4. The bond matures exactly three years from the valuation date. 5. As of the valuation date, the market yield to maturity (i.e., total rate of return, including interest payments and price appreciation) for bonds of the same risk grade as the subject bond is 10%. Note three important implications of this scenario: 1. The issuing company’s embedded cost of capital for this bond is only 8%, although the market cost of capital (yield to existing, sometimes referred to as nominal, maturity) at the valuation date is 10%. The discrepancy may be because the general level of interest rates was lower at the time of issuance of this particular bond, or because the market’s rating associated with this bond was lowered between the date of issuance and the valuation date. 2. If the issuing company wanted to issue new debt on comparable terms as of the valuation date, it presumably would have to offer investors a 10% yield, the current market-driven cost of capital for bonds of that risk grade, to induce investors to purchase the bonds.

12

Cost of Capital

3. For purposes of valuation and capital budgeting decisions, when we refer to cost of capital, we mean market cost of capital, not embedded cost of capital. (Embedded cost of capital is sometimes used in utility rate-making, but this chapter focuses only on valuation and capital budgeting applications of cost of capital.) Substituting numbers derived from the preceding assumptions into Formula 2.1 gives us: (Formula 2.2) PV ¼

$80 $80 $80 $1;000 þ þ þ 2 3 ð1 þ :10Þ ð1 þ :10Þ ð1 þ :10Þ ð1 þ :10Þ3

$80 $80 $80 $1;000 þ þ þ ð1:10Þ ð1:21Þ ð1:331Þ ð1:331Þ ¼ $72:73 þ $66:12 þ $60:11 þ $751:32 ¼ $950:28 ¼

In this example, the fair market value of the subject bond as of the valuation date is $950.28. That is the amount that a willing buyer would expect to pay and a willing seller would expect to receive (before considering any transaction costs).

APPLICATIONS TO BUSINESSES, BUSINESS INTERESTS, PROJECTS, AND DIVISIONS The same framework can be used to estimate the value of an equity interest in a company or a company’s entire invested capital. You project the cash flows available to the interest to be valued and discount those cash flows at a cost of capital (discount rate) that reflects the risk associated with achieving the particular cash flows. Details of this procedure for valuing entire companies or interests in companies are presented in later chapters. Similarly, the same construct can be applied to evaluating a capital budgeting decision, such as building a plant or buying equipment. In that case, the cash flows to be discounted are incremental cash flows (i.e., cash flows resulting specifically from the decision that would not occur absent the decision). The early portions of the cash flow stream may be negative while funds are being invested in the project. The primary relationship to remember is that cost of capital is a function of the investment, not of the investor. Therefore, the analyst must evaluate the risk of each project under consideration. If the risk of the project is greater or less than the company’s overall risk, then the cost of capital by which that project is evaluated should be commensurately higher or lower than the company’s overall cost of capital. Although some companies apply a single ‘‘hurdle rate’’ to all proposed projects or investments, the consensus in the literature of corporate finance is that the rate by which to evaluate any investment should be based on the risk of that investment, not on the company’s overall risk that drives its cost of capital. We agree with this consensus. If the company invests in something riskier than its normal operations, the company’s risk will increase marginally. When this increased risk is recognized and reflected in the market, it will raise the company’s cost of capital. If the returns on the riskier new investment are not great enough to achieve higher returns commensurate with this higher cost of capital, the result will be a decrease in the stock price and a loss of shareholder value.

Summary

13

SUMMARY The most common cost of capital applications are valuation of an investment or prospective investment and project selection decisions (the core component of capital budgeting). In both applications, returns expected from the capital outlay are discounted to a present value by a discount rate, which should be the cost of capital applicable to the specific investment or project. The measure of returns generally preferred today is net cash flow, as discussed in the next chapter.

Chapter 3

Net Cash Flow: Preferred Measure of Economic Income

Introduction Defining Net Cash Flow Net Cash Flow to Common Equity Net Cash Flow to Invested Capital Net Cash Flows Should Be Probability-Weighted Expected Values Why Net Cash Flow Is the Preferred Measure of Economic Income Residual Earnings Summary Additional Reading

INTRODUCTION Cost of capital is a meaningless concept until we define the measure of economic income to which it is to be applied. Based on the tools of modern finance, the variable of choice for most financial decision making is net cash flow. This, obviously, poses two critical questions: 1. How do we define net cash flow? 2. Why is net cash flow considered the best economic income variable to use in net present value analysis? There are two general frameworks for valuing a business: valuing net cash flow to common equity and valuing net cash flow to invested capital. When valuing net cash flow to common equity, the discount rate should be the cost of equity capital. When valuing net cash flow to invested capital, the discount rate should be the overall cost of capital (commonly referred to as the weighted average cost of capital, or WACC).

DEFINING NET CASH FLOW Net cash flow is generally defined as cash that a business or project does not have to retain and reinvest in itself in order to sustain the projected levels of cash flows in future years. In other words, it is cash available to be paid out in any year to the owners of capital without jeopardizing the company’s expected-cash-flow-generating capability in future years. It must be distributed or dividended to the investors or reinvested in some incremental project not reflected in the cash flows that have been discounted to become incremental value. (Net cash flow is sometimes called free cash flow. It is 15

16

Cost of Capital

also sometimes called net free cash flow, although this phrase seems redundant. Finance terminology being as ambiguous as it is, minor variations in the definitions of these frequently arise.)

NET CASH FLOW TO COMMON EQUITY In valuing equity by discounting or capitalizing expected cash flows (keeping in mind the important difference between discounting and capitalizing, as discussed elsewhere), net cash flow to equity (NCFe in our notation system) is defined as: (Formula 3.1) Net income to common equityðafter taxÞ þ Noncash chargesðe:g:; depreciation; amortization; deferred revenue; deferred taxesÞ Capital expenditures Additions to net working capital Dividends on preferred stock Changes in long-term debtðþcash from borrowing; repaymentsÞ ¼ Net cash flow to common equity

NET CASH FLOW TO INVESTED CAPITAL In valuing the entire invested capital of a company or project by discounting or capitalizing expected cash flows, net cash flow to invested capital (NCFf in our notation system) is defined as: (Formula 3.2) Net income to common equityðafter taxÞ þ Noncash chargesðe:g:; depreciation; amortization; deferred revenue; deferred taxesÞ Capital expenditures Additions to net working capital þ Dividends on preferred stock þ Interest expenseðnet of the tax deduction resulting from interest as a tax-deductible expenseÞ ¼ Net cash flow to invested capital In other words, NCFf includes interest (tax effected, because interest is a deductible expense for tax purposes), because invested capital includes the debt on which the interest is paid, whereas net cash flow to equity does not. Occasionally an analyst treats earnings before interest, taxes, depreciation, and amortization (EBITDA) as if it were free cash flow. This error is not a minor matter, since the analyst has added back the noncash charges without deducting the capital expenditure investments, not to mention additions to working capital necessary to keep the operation functioning as expected. When we discount net cash flow to equity, the appropriate discount rate is the cost of equity capital. When we discount net cash flow to all invested capital, the appropriate discount rate is the WACC.

Only

amounts necessary to support projected revenues:

Net Cash Flows Should be Probability-Weighted Expected Values

17

NET CASH FLOWS SHOULD BE PROBABILITY-WEIGHTED EXPECTED VALUES Net cash flows to be discounted or capitalized should be statistical expected values, that is, (mean) probability-weighted cash flows. In the real world, it is far more common for realized net cash flows to be below forecast than above. A valuation that does not take this factor into account will overvalue a business. If the distribution of possible cash flows in each period is symmetrical above and below the most likely cash flow in that period, then the most likely cash flow is equal to the probability-weighted cash flow (the mathematical expected value of the distribution). However, many distributions of possible cash flows are skewed. This is where probability weighting comes into play. Exhibit 3.1 tabulates the probability-weighted expected values of projected cash flows under a symmetrically distributed scenario and a skewed distribution scenario. Exhibit 3.2 portrays the information in Exhibit 3.1 graphically. In both scenario A and scenario B, the most likely cash flow is $1,000. In scenario A, the expected value (probability weighted) is also $1,000. But in scenario B, the expected value is only $901. In scenario B, $901 is the figure that should appear in the numerator of the discounted cash flow formula, not $1,000. Most analysts do not have the luxury of a probability distribution for each expected cash flow, and it is not a common practice. However, they should be aware of the concept when deciding on the amount of each expected cash flow to be discounted.

Exhibit 3.1

Cash Flow Expectation Tables Scenario A—Symmetrical Cash Flow Expectation

Expected Value of Cash Flow $1,600.00 1,500.00 1,300.00 1,000.00 700.00 500.00 400.00

Probability of Occurrence

Weighted Value

0.01 0.09 0.20 0.40 0.20 0.09 0.01

$16 135 260 400 140 45 4

100%

$1,000

Scenario B—Skewed Cash Flow Expectation Expected Value of Cash Flow $1,600.00 1,500.00 1,300.00 1,000.00 700.00 500.00 (100.00) (600.00)


Weighted Value

0.01 0.04 0.20 0.35 0.25 0.10 0.04 0.01 100%

$16 60 260 350 175 50 (4) (6) $901

18

Cost of Capital


Scenario A—Symmetrical Cash Flow Expectation 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 $ 400

Expected value (mean)

500

700

1,000

1,300

1,500

1,600

Expected Cash Flow


Scenario B—Skewed Cash Flow Expectation 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 $ –600

Expected value (mean)

–100

500

700

901 1,000

1,300

1,500

1,600

Expected Cash Flow

Exhibit 3.2

Cash Flow Expectation Graphs

As we pointed out in Chapter 2, in calculating the present value of economic benefits, the numerator is the expected economic benefits. We have suggested that net cash flow is the preferred measure of economic income. While this is not a course in forecasting, the analyst may need to facilitate the preparation of expected net cash flows and/or test the reasonableness of net cash flows projected. In Chapter 33 we discuss projections of expected future economic benefits, focusing on net cash flows.

WHY NET CASH FLOW IS THE PREFERRED MEASURE OF ECONOMIC INCOME There are two reasons why the financial community tends to focus on net cash flow as the preferred measure of economic income to be discounted by the opportunity cost of capital. They are: 1. Conceptual. Net cash flow is what an investor actually expects to receive in a discrete period of time. In a valuation context, it is important that the numerator gives the most accurate estimate of what the investor expects to get. 2. Empirical. It is the economic income measure that best matches the historical data available to estimate a discount rate by ex post (historical) methods.

Residual Earnings

19

Morningstar in the Stocks, Bonds, Bills and Inflation (SBBI) 2007 Valuation Edition Yearbook clearly state the case for preferring what they term ‘‘free cash flows’’ (i.e., net cash flows after tax) as the appropriate economic income measure to discount: Several things can be noted about free cash flow. First, it is an after-tax concept.. . . Secondly, pure accounting adjustments need to be added back into the analysis .. . . Finally, cash flows necessary to keep the company going forward must be subtracted from the equation . These cash flows represent necessary capital expenditures to maintain plant, property, and equipment or other capital expenditures that arise out of the ordinary course of business. Another common subtraction is reflected in changes in working capital. The assumption in most business valuation settings is that the entity in question will remain a long-term going concern that will grow over time. As companies grow, they accumulate additional accounts receivable and other working capital elements that require additional cash to support. Free cash flow is the relevant cash flow stream because it represents the broadest level of earnings that can be generated by the asset. With free cash flow as the starting point, the owners of a firm can decide how much of the cash flow stream should be diverted toward new ventures, capital expenditures, interest payments, and dividend payments. It is incorrect to focus on earnings as the cash flow stream to be valued because earnings contain a number of accounting adjustments and already include the impact of the capital structure.1

If the SBBI data or Duff & Phelps data are used to develop a common equity discount rate—using either the build-up model or the Capital Asset Pricing Model (CAPM)—the discount rate is applicable to net cash flow available to the common equity investor. This is because the SBBI and Duff & Phelps return data have two components: 1. Dividends to the common stock 2. Change in common stock prices The investor receives the dividends, so their utilization is entirely at the investor’s discretion. For actively traded stock, the investor’s realization of the change in stock price is equally discretionary because the stocks are highly liquid (i.e., they can be sold at their market price at any time, with the seller receiving the proceeds in cash within three business days).

RESIDUAL EARNINGS An alternative formulation of economic income that has arisen in the literature is residual earnings (RE)2 and abnormal earnings growth (AEG)3 models. The RE and AEG models always yield the same valuation and yield the same value as does the discounted net cash flow method when applied with the same valuation assumptions. Both the RE and AEG models begin with adjusted statements of accounting income. RE is the return on common equity (expressed in dollars) in excess of the cost of equity capital. AEG is earnings (assuming reinvestment of dividends) in excess of earnings growing at the cost of equity capital. (Formula 3.3) NIn ðke BVn1 Þ RE ¼ BVn1 1 2 3

SBBI Valuation Edition Yearbook (Chicago: Morningstar, 2007), 13. Stephen H. Penman, Financial Statement Analysis and Security Valuation, 3rd ed. (New York: McGraw-Hill, 2007), Chapter 5. J. Feltham and J. Ohlson, ‘‘Valuation and Clean Surplus Accounting for Operating and Financing Activities,’’ Contemporary Accounting Research, vol. 11 (1995): 689–731.

20

Cost of Capital

where: RE ¼ Residual earnings NI ¼ Net income BV ¼ Book value of net assets ke ¼ Cost of equity capital The RE is based on clean-surplus accounting statement: (Formula 3.4) BVn ¼ BVn1 þ NIn Dn where: D ¼ Dividends NIn ¼ not reported earnings, but rather comprehensive income, which includes income terms reported directly in the equity account rather than in the income statement (Formula 3.5) AEG ¼ REn REn1 where: AEG ¼ Abnormal earnings growth In Chapter 4 we demonstrate the conditions for equality between valuations using net cash flow and residual earnings. While theoreticians and practitioners alike accept the primacy of cash flows in valuation,4 it is the subject of recent studies. In one study, the authors question whether multiples developed from earnings per share or from operating cash flows per share (net income plus depreciation and amortization plus net working capital divided by the weighted average number of common shares outstanding for the year) more accurately reflect stock prices. Their results suggest that valuations based on earnings forecasts provide better valuations where consensus earnings forecasts of analysts are available.5 Another study examines whether accounting variables explain stock price movements by assisting users of accounting information to better forecast cash flows.6 They find that changes in four accounting variables explain about 20% of the differences in stock returns. Finally, another study finds that when investors are provided complete cash flow data, stock prices fully reflect that data.7

4

5

6

7

See, for example, Steven J. Kaplan and Richard S. Ruback, ‘‘The Valuation of Cash Flow Forecasts: An Empirical Analysis,’’ Journal of Finance 50, No. 4 (1995): 1059–1093. Jing Liu, Doron Nissim, and Jacob Thomas, ‘‘Is Cash Flow King in Valuations?’’ Financial Analysts Journal 63, No. 2 (2007): 56–68. Peter F. Chen and Guochang Zhang, ‘‘How Do Accounting Variables Explain Stock Price Movements? Theory and Evidence,’’ Journal of Accounting and Economics (July 2007): 219–244. Keren Bar-Hava, Roni Ofer, and Oded Sarig, ‘‘New Tests of Market Efficiency Using Fully Identifiable Equity Cash Flows,’’ Working paper, February 2007.

Additional Reading

21

SUMMARY Net cash flow is the measure of economic income that most financial analysts prefer to use today when using the cost of capital for valuation or project selection. If valuing cash flows to equity, the discount rate should be the cost of equity capital. If valuing cash flows to debt, the discount rate should be the cost of the debt capital. If valuing cash flows available for all invested capital, the discount rate should be the weighted average cost of capital. Net cash flows should be measured as the mathematical expected value of the probabilityweighted distribution of expected outcomes for each projected period of returns, not the most likely value. In Chapter 5 we define risk as uncertainty of possible outcomes, a definition intended to encompass the entire range of possible returns for each future period.

ADDITIONAL READING Brief, Richard P. ‘‘Accounting Valuation Models: A Short Primer.’’ Abacus, 43 no. 4 (2007): 429–437. Estridge, Juliet, and Babara Lougee. ‘‘Measuring Free Cash Flows for Equity Valuation: Pitfalls and Possible Solutions.’’Journal of Applied Corporate Finance (Spring 2007): 60–71.

Chapter 4

Discounting versus Capitalizing Introduction Capitalization Formula Example: Valuing a Preferred Stock Functional Relationship between Discount Rate and Capitalization Rate Major Difference between Discounting and Capitalizing Constant Growth or Gordon Growth Model Combining Discounting and Capitalizing (Two-Stage Model) Equivalency of Discounting and Capitalizing Models Midyear Convention Midyear Discounting Convention Midyear Capitalization Convention Midyear Convention in the Two-Stage Model Seasonal Businesses Matching Projection Periods to Financial Statements: Partial First Year Capitalized Residual Earnings Summary

INTRODUCTION The first two chapters explained that the cost of capital is used as a discount rate to discount a stream of future economic income to a present value. This valuation process is called discounting. In discounting, we project all expected economic income (cash flows or other measure of economic income) from the subject investment to the respective class or classes of capital over the life of the investment. Thus, the percentage return that we call the discount rate represents the total compound rate of return that an investor in that class of investment requires over the life of the investment. There is a related process for estimating present value, which we call capitalizing. In capitalizing, instead of projecting all future returns on the investment to the respective class (es) of capital, we focus on the return of just one single period, usually the return expected in the first year immediately following the valuation date. We then divide that single number by a divisor called the capitalization rate. As will be seen, the process of capitalizing is really just a shorthand form of discounting. The capitalization rate, as used in the income approach to valuation or project selection, is derived from the discount rate. (This differs from the market approach to valuation, where capitalization rates for various economic income measures are observed directly in the marketplace.) A common error is to use the discount rate as a capitalization rate. This is correct only if the expected cash flows are the same from the year following the valuation date into perpetuity, as in the case of a perpetual preferred stock. The balance of this chapter presents the differences between discounting and capitalizing and alternative discounting and capitalizing conventions.

23

24

Cost of Capital

CAPITALIZATION FORMULA Putting the capitalization concept into a formula, we have: (Formula 4.1) PV ¼

NCF1 c

where PV ¼ Present value NCF1 ¼ Net cash flow expected in the first period immediately following the valuation date c ¼ Capitalization rate

EXAMPLE: VALUING A PREFERRED STOCK A simple example of applying Formula 4.1 uses a preferred stock for which a risk rating has been estimated. Let us make five assumptions: 1. The preferred stock pays dividends of $5 per share per year. 2. The preferred stock is issued in perpetuity and is not callable. 3. It pays dividends once a year, at the end of the year. (This, of course, is a simplifying assumption. Some privately owned preferred stocks pay dividends only annually, but most publicly traded preferred stocks pay dividends quarterly.) 4. As of the valuation date, the market yield for preferred stocks of the same risk grade as the subject preferred stock is 10% per annum. (We also must assume comparable rights, such as voting, liquidation preference, redemption, conversion, participation, cumulative dividends, etc.) 5. There is no prospect of liquidation. Note that the par value of the preferred stock is irrelevant, since the stock is issued in perpetuity and there is no prospect of a liquidation. The entire cash flow an investor can expect to receive over the life of the investment (perpetuity in this case) is the $5 annual per-share dividend. Substituting numbers derived from the preceding assumptions into Formula 4.1 produces: (Formula 4.2) $5:00 0:10 ¼ $50:00

PV ¼

In this example, the estimated fair market value of the subject preferred stock is $50 per share. That is the amount a willing buyer would expect to pay and a willing seller would expect to receive (before considering any transaction costs).

FUNCTIONAL RELATIONSHIP BETWEEN DISCOUNT RATE AND CAPITALIZATION RATE The preceding example presented the simplest possible scenario in which to apply the cost of capital using the capitalization method: a fixed cash flow stream in perpetuity. This is the one unique situation in which the discount rate (cost of capital) equals the capitalization rate. The discount rate equals

Functional Relationship between Discount Rate and Capitalization Rate

25

the capitalization rate because no growth or decline in the investor’s cash flow is expected. Most realworld investments are not quite that simple. In the case of an investment in common stock, a partnership interest, or a capital budgeting project in an operating company, investors often are expecting some level of growth over time in the cash flows available to pay dividends or partnership withdrawals. Even if unit volume is expected to remain constant (i.e., no real growth), investors still might expect cash flows to grow at a rate approximating expected inflation. If the expected annually compounded rate of growth is stable and sustainable over a long period of time, then the discount rate (cost of capital) can be converted to a capitalization rate. As stated earlier, the capitalization rate is a function of the discount rate. This obviously raises the question: What is the functional relationship between the discount rate and the capitalization rate? Assuming stable long-term growth in net cash flows available to the investment being valued, the capitalization rate equals the discount rate minus the expected long-term growth rate. In a formula, this functional relationship can be stated as: (Formula 4.3) c¼kg where: c ¼ Capitalization rate k ¼ Discount rate (cost of capital) for the subject investment g ¼ Expected long-term sustainable growth rate in the cash flow available to the subject investment The critical assumption in this formula is that the growth in the net cash flow available to the capital is relatively constant over the long term (technically in perpetuity). Caveat: As explained in Chapter 3, in estimating the net cash flow to capitalize, we deduct investments such as capital expenditures and additional net working capital needed to realize the projected future revenues of the existing business investment. In this formulation, we are valuing the existing business, not currently unknown investments that may be made in future years from investing these net cash flows in new business ventures. Now we know two essential things about using the cost of capital to estimate present value using the capitalization method, assuming relatively stable long-term growth in the return available to the investor: 1. Present value equals the next period’s expected cash flow divided by the capitalization rate. 2. The capitalization rate is the discount rate (cost of capital) less the sustainable expected long-term rate of growth in the cash flow. (Technically, sustainable growth in this context means in perpetuity. However, after 15 or 20 years, the remaining rate of growth has minimal impact on the present value, due to very small present value factors.) We can combine these two relationships into a single formula as: (Formula 4.4) PV ¼

NCF1 kg

where: PV ¼ Present value NCF1 ¼ Net cash flow expected in period 1, the period immediately following the valuation date

26

Cost of Capital

k ¼ Discount rate (cost of capital) g ¼ Expected long-term sustainable growth rate in net cash flow to investor A simple example of substituting numbers into Formula 4.4 is an equity investment with a constant expected growth in net cash flow. Let us make three assumptions: 1. The net cash flow in period 1 is expected to be $100. 2. The cost of capital (i.e., the market-required total return or the discount rate) for this investment is estimated to be 13%. 3. The sustainable rate of growth in net cash flow from year 1 to perpetuity is expected to be 3%. Substituting numbers from the preceding assumptions into Formula 4.4 gives us: (Formula 4.5) $100 PV ¼ 0:13 0:03 $100 ¼ 0:10 ¼ $1;000 In this example, the estimated fair market value of the investment is $1,000. That is the amount a willing buyer would expect to pay and a willing seller would expect to receive (before considering any transaction costs).

MAJOR DIFFERENCE BETWEEN DISCOUNTING AND CAPITALIZING From the preceding discussion, we can now deduce a critical insight: The difference between discounting and capitalizing is in how we reflect changes over time in expected future cash flows. In discounting: Each future increment of return is estimated specifically and included in the numerator. In capitalizing: Estimates of changes in future returns are lumped into one annually compounded growth rate, which is then subtracted from the discount rate in the denominator. If we assume that there really is a constant compounded growth rate in net cash flow to the investor in perpetuity, then it is a mathematical truism that the discounting method and the capitalizing method will produce identical values. (See the section in this chapter titled ‘‘Equivalency of Discounting and Capitalizing Models’’ for an illustration of how this equality works.)

CONSTANT GROWTH OR GORDON GROWTH MODEL One frequently encountered minor modification to Formulas 4.4 and 4.5 is to use as the ‘‘base period’’ the period just completed prior to the valuation date, instead of next period’s estimate. The assumption is that cash flows will grow evenly in perpetuity from the period immediately preceding the valuation date. This scenario is stated in a formula commonly known as the Gordon Growth Model (named for Professor Myron Gordon who popularized this formulation): (Formula 4.6) PV ¼

NCF0 ð1 þ gÞ kg

Combining Discounting and Capitalizing (Two-Stage Model)

27

where: PV NCF0 k g

¼ ¼ ¼ ¼

Present value Net cash flow in period 0, the period immediately preceding the valuation date Discount rate (cost of capital) Expected long-term sustainable growth rate in net cash flow to investor

Note that for this model to make economic sense, NCF0 must represent a normalized amount of cash flow from the investment for the previous year, from which a steady rate of growth is expected to proceed. Therefore, NCF0 need not be the actual cash flow for period 0 but may be the result of certain normalization adjustments, such as elimination of the effect of one or more nonrecurring factors. In fact, if NCF0 is the actual net cash flow for period 0, the valuation analyst must take reasonable steps to be satisfied that NCF0 is indeed the most reasonable base from which to start the expected growth embedded in the growth rate. Furthermore, the valuation report should state the steps taken and the assumptions made in concluding that last year’s actual results are the most realistic base for expected growth. Mechanistic acceptance of recent results as representative of future expectations is one of the most common errors in implementing the capitalization method of valuation. For a simple example of using numbers in Formula 4.6, accept all assumptions in the previous example, with the exception that the $100 net cash flow expected in period 1 is instead the normalized base cash flow for period 0. (The $100 is for the period just ended, rather than the expectation for the period just starting.) Substituting the numbers with these assumptions into Formula 4.6 produces: (Formula 4.7) $100ð1 þ 0:03Þ 0:13 0:03 $103 ¼ 0:10

PV ¼

¼ $1;030 In this example, the estimated fair market value of the investment is $1,030. That is the amount a willing buyer would expect to pay and a willing seller would expect to receive (before considering any transaction costs). Note that the relationship between this and the previous example is simple and straightforward. We backed up the receipt of the $100 by one period, and the value of the investment was higher by 3%, the growth rate. In a constant growth model, assuming that all of the available cash flows are distributed, the value of the investment grows at the same rate as the rate of growth of the cash flows. The reason is that, in defining net cash flow (as we did in the previous chapter), we have already subtracted the amount of capital expenditures and additions to net working capital necessary to support the projected growth. The investor in this example thus earns a total rate of return of 13%, comprised of 10% current return (the capitalization rate) plus 3% annually compounded growth in the value of the investment.

COMBINING DISCOUNTING AND CAPITALIZING (TWO-STAGE MODEL) For many investments, even given an accurate estimate of the cost of capital, there are practical problems with either a pure discounting or a pure capitalizing method of valuation.

28

Cost of Capital

Problem with discounting. There are few equity investments for which returns for each specific incremental period can be projected with accuracy many years into the future. Problem with capitalizing. For most equity investments, it is not reasonable to expect a constant growth rate in perpetuity from either the year preceding or the year following the valuation date.

This dilemma typically is dealt with by combining the discounting method and the capitalizing method into a two-stage model. The idea is to project discrete cash flows for some number of periods into the future and then to project a steady growth model starting at the end of the discrete projection period. Each period’s discrete cash flow is discounted to a present value, and the capitalized value of the projected cash flows following the end of the discrete projection period is also discounted back to a present value. The sum of the present values is the total present value. The capitalized value of the projected cash flows following the discrete projection period is called the terminal value or residual value. The preceding narrative explanation of a two-stage model is summarized in seven steps: Step 1. Decide on a reasonable length of time for which discrete projections can be made. Step 2. Estimate specific amounts of expected cash flow for each of the discrete projection periods. Step 3. Estimate a long-term sustainable rate of growth in cash flows from the end of the discrete projection period forward. Step 4. Use the Gordon Growth Model (Formulas 4.6 and 4.7) to estimate value as of the end of the discrete projection period. Step 5. Discount each of the increments of cash flow back to a present value at the discount rate (cost of capital) for the number of periods until it is received. Step 6. Discount the terminal value (estimated in step 4) back to a present value for the number of periods in the discrete projection period (the same number of periods as the last increment of cash flow). Step 7. Sum the value derived from steps 5 and 6. These steps can be summarized in the next formula, which assumes that net cash flows are received at the end of each year: (Formula 4.8) NCFn ð1 þ gÞ NCF1 NCF2 NCFn kg þ þ þ PV ¼ nþ 2 ð1 þ kÞ ð1 þ kÞ ð1 þ kÞ ð1 þ kÞn where: NCF1. . . NCFn ¼ Net cash flow expected in each of the periods 1 through n, n being the last period of the discrete cash flow projections k ¼ Discount rate (cost of capital) g ¼ Expected long-term sustainable growth rate in net cash flow, starting with the last period of the discrete projections as the base year The discrete projection period in the two-stage model is commonly between 5 and 10 years. However, for simplicity in applying Formula 4.8, we will just use a three-year discrete projection period. Let us make three assumptions: 1. Expected net cash flows for years 1, 2, and 3 are $100, $120, and $140, respectively. 2. Beyond year 3, cash flow is expected to grow fairly evenly at a rate of about 5% in perpetuity. 3. The cost of capital for this investment is estimated to be 12%.

Equivalency of Discounting and Capitalizing Models

29

Substituting numbers derived from these assumptions into Formula 4.8 produces: (Formula 4.9) $140ð1 þ 0:05Þ $100 $120 $140 PV ¼ þ þ 0:12 0:053 þ ð1 þ 0:12Þ ð1 þ 0:12Þ2 ð1 þ 0:12Þ3 ð1 þ 0:12Þ $147 $100 $120 $140 þ þ þ 0:07 ¼ 1:12 1:2544 1:4049 1:4049 $2;100 ¼ $89:30 þ $95:66 þ $99:65 þ 1:4049 ¼ $89:30 þ $95:66 þ $99:65 þ $1;494:77 ¼ $1;779 Thus, the estimated fair market value of this investment is $1,779. This is the amount a willing buyer would expect to pay and a willing seller would expect to receive (before considering any transaction costs). A common error is to discount the terminal value for n þ 1 periods instead of n periods. The assumption we have made is that the nth period cash flow is received at the end of the nth period, and the terminal value is the amount for which we estimate we could sell the investment as of the end of the nth period. The end of one period and the beginning of the next period are the same moment in time, so they must be discounted for the same number of periods. Note that, in the preceding example, the terminal value represents 84% of the total present value ($1,495 $1,779 ¼ 0.84). The analyst should always keep in mind two relationships when using cost of capital in a two-stage model for valuation: 1. The shorter the projection period, the greater the impact of the terminal value on the total present value. The length of the projection period should be the number of periods until the company is expected to reach a steady state, that is, until the company is expected to reach a normalized level of cash flow that it can grow at a more or less constant rate over a long period of time. There is no ‘‘magic’’ in using 5 years or 10 years for the projection period. 2. The closer the estimated growth rate is to the cost of capital, the more sensitive the model is to changes in assumptions regarding the growth rate. (This is true for the straight capitalization model as well as the two-stage model.) Of course, if the assumed growth rate exceeds the cost of capital, the model implodes and is useless. In some cases, the terminal value may not be a perpetuity model. For example, you might assume liquidation at that point, and the terminal value could be a salvage value. For example, the license to operate the business may have a finite life at which the operating business ends.

EQUIVALENCY OF DISCOUNTING AND CAPITALIZING MODELS As stated earlier, if all assumptions are met, the discounting and capitalizing methods of using the cost of capital will produce identical estimates of present value. Let us test this on the example used in Formula 4.5. Recall that we assumed cash flow in period 1 of $100, growing in perpetuity at 3%.

30

Cost of Capital

The cost of capital (the discount rate) was 13%, so we subtracted the growth rate of 3% to get a capitalization rate of 10%. Capitalizing the $100 (period 1 expected cash flow) at 10% gave us an estimated present value of $1,000 ($100 0.10 ¼ $1,000). Let us take these same assumptions and put them into a discounting model. For simplicity, we will only use three periods for the discrete projection period, but it would not make any difference how many discrete projection periods we used. (Formula 4.10) $100ð1:03Þ3 $100 $100ð1:03Þ $100ð1:03Þ þ þ 0:13 0:03 þ PV ¼ ð1 þ 0:13Þ ð1 þ 0:13Þ2 ð1 þ 0:13Þ3 ð1:13Þ3 2

$109:27 $100 $103 $106:09 þ þ þ 0:10 ¼ 1:13 1:2769 1:4429 1:4429 $1;092:73 ¼ $88:50 þ $80:66 þ $75:53 þ 1:4429 ¼ $88:50 þ $80:66 þ $73:53 þ $757:31 ¼ $1; 000 This example, showing the equivalency of using cost of capital in either the discounting or the capitalizing model, when all assumptions are met, demonstrates the point that capitalizing is really just a shorthand form of discounting. Capitalization usually is used when we do not have enough information to implement a discounting model. Nevertheless, when using a capitalizing model, the analyst should consider whether the answer would work out the same if it were expanded to a full discounting model. If not, it may be propitious to review and possibly adjust certain assumptions. If the discounting and capitalization models produce different answers using the same cost of capital and the same inputs, there may be some kind of internal inconsistency.

MIDYEAR CONVENTION In all of our previous examples, we have assumed that cash flows are received at the end of each year. Even if a company realizes cash flows throughout the year, payouts to the investors may be made only at the end of the year when the managers have seen the results of the entire year and have an idea about next year’s projections. For some companies or investments, however, it may be more reasonable to assume that the cash flows are distributed more or less evenly throughout the year. To accommodate this latter assumption, we can modify our formulas for what we call the midyear convention. MIDYEAR DISCOUNTING CONVENTION We can make a simple modification to Formula 2.1 (discounting) to what we call the midyear discounting convention. We merely subtract a half year from the exponent in the denominator of the equation. Formula 2.1, the discounting equation, now becomes: (Formula 4.11) PV ¼

NCF1 ð1 þ kÞ0:5

þ

NCF2 ð1 þ kÞ1:5

þ þ

NCFn ð1 þ kÞn0:5

Midyear Convention

31

MIDYEAR CAPITALIZATION CONVENTION Similarly, we can make a modification to the capitalization formula to reflect the receipt of cash flows throughout the year. The modification to Formula 4.4, the capitalization equation, is handled by accelerating the returns by a half year in the numerator:1 (Formula 4.12) PV ¼

NCF1 ð1 þ kÞ0:5 kg

Formula 4.12 is a mathematical equivalent of Formula 4.13. (Formula 4.13) NCFn ð1 þ gÞ NCF1 NCF2 NCFn kg PV ¼ þ þ þ þ ð1 þ kÞ0:5 ð1 þ kÞ1:5 ð1 þ kÞn0:5 ð1 þ kÞn0:5

MIDYEAR CONVENTION IN THE TWO-STAGE MODEL Combining discrete period discounting and capitalized terminal value into a two-stage model as shown in Formula 4.8, the midyear convention two-stage equation becomes: (Formula 4.14) NCFn ð1 þ gÞð1 þ kÞ0:5 NCF1 NCF2 NCFn kg PV ¼ þ þ þ þ 0:5 1:5 n0:5 ð1 þ kÞn ð1 þ kÞ ð1 þ kÞ ð1 þ kÞ Using the same assumptions as in Formula 4.9 (where the value using the year-end convention was $1,779) produces: (Formula 4.15) $140ð1 þ 0:05Þð1 þ 0:12Þ0:5 $100 $120 $140 0:12 0:05 þ þ þ PV ¼ 0:5 1:5 ð1 þ 0:12Þ3 ð1 þ 0:12Þ ð1 þ 0:12Þ ð1 þ 0:12Þ2:5 $155:527 $100 $120 $140 ¼ þ þ þ 0:07 1:0583 1:1853 1:3275 1:4049 $2;221:81 ¼ $94:49 þ $101:24 þ $105:46 þ 1:4049 ¼ $94:49 þ $101:24 þ $105:46 þ $1;581:47 ¼ $1;883

1

Proof of the accuracy of this method was presented in Todd A. Kaltman, ‘‘Capitalization Using a Mid-Year Convention,’’ Business Valuation Review (December 1995): 178–182. Also see, Michael Dobner, ‘‘Mid-year Discounting and Seasonality Factors,’’ Business Valuation Review (March 2002): 16–18; Jay B. Abrams, and R.K. Hiatt, ‘‘The Bias in Annual (Versus Monthly) Discounting is Immaterial,’’ Business Valuation Review (September 2003): 127–135.

32

Cost of Capital

In this case, using the midyear convention increased the value by $104 ($1,883 $1,779 ¼ $104) or 5.8% ($104 $1,779 ¼ 0.058). An alternative version of the terminal value factor in the two-stage model actually is equivalent to that used in the preceding formula. Instead of using the modified capitalization equation in the numerator of the terminal value factor, the normal terminal value capitalization equation is used, and the terminal value is discounted by n – 0.5 years instead of n years. This equation reads: (Formula 4.16) NCFn ð1 þ gÞ NCF1 NCF2 NCFn kg þ þ þ þ PV ¼ 0:5 1:5 n0:5 ð1 þ kÞ ð1 þ kÞ ð1 þ kÞ ð1 þ kÞn0:5 Using the same numbers as in Formula 4.15, this works out to: (Formula 4.17) $140ð1 þ 0:05Þ þ þ þ 0:12 0:05 PV ¼ 0:5 1:5 2:5 ð1 þ 0:12Þ ð1 þ 0:12Þ ð1 þ 0:12Þ ð1 þ 0:12Þ2:5 $100

$120

$140

$147 $100 $120 $140 þ þ þ 0:07 ¼ 1:0583 1:1853 1:3275 1:3275 $2;100 ¼ $94:49 þ $101:24 þ $105:46 þ 1:3275 ¼ $94:49 þ $101:24 þ $105:46 þ $1;581:92 ¼ $1;883 (The difference is a matter of rounding.) Note that using the midyear convention will always produce a higher value when the annual projected cash flows are the same, because of the time value of money. The assumption underlying the midyear convention is that investors receive the cash flows earlier than is the case under the year-end convention. A quick way to handle the midyear convention is simply to multiply the value without midyear discounting by (1 + k)0.5. SEASONAL BUSINESSES The midyear convention formulas can be modified for seasonal businesses. For example, assume that you analyze monthly income and cash flows and determine that springtime is the period which is the weighted average of the monthly cash flows during the year. You can substitute n ¼ 0.3 for n ¼ 0.5 in the midyear convention formula. The important point is that you need to understand the timing of the cash flows through the year before adopting any convention, annual, midyear, or other.

MATCHING PROJECTION PERIODS TO FINANCIAL STATEMENTS: PARTIAL FIRST YEAR Often our valuation date is not at the beginning of an accounting year; rather the valuation date is in the middle of the accounting year. For presentation purposes, it is often helpful to match the projection periods to the financial statement fiscal years. For example, the company may assemble

Matching Projection Periods to Financial Statements: Partial First Year

0

Year 1

2

3

4

^

^

^

^

33

where: 0 ¼ Valuation Date ^ ¼ Point in year where cash flows assumed to be realized Exhibit 4.1

Timeline of Net Cash Flows Equivalent to Formula 4.8

long-range plans. Those projections typically match the periods included in future financial statement fiscal years. We can adapt the principles of midyear discounting to this special case. It is helpful to present the projection periods in terms of timelines. Exhibit 4.1 presents the timeline of net cash projections valued in Formula 4.8 (cash flows assumed to be realized at the end of each of the future years). Exhibit 4.2 presents the timeline of net cash flow projections valued in Formula 4.13 (cash flows assumed to be realized at the midpoint of each of the future years or uniformly during those future years). Now assume, for example, that the valuation date is at the beginning of the fifth month of the current financial reporting year and net cash flow projections are similarly assumed to be realized at the midpoint of each of the future periods. For the remaining ‘‘partial period’’ (matching the remainder of the current financial reporting year), the net cash flows are expected to be realized 3½ months after the valuation date and the net cash flows in the first full year following the partial period are expected to be realized 13 months (midpoint of 7 remaining months of the remaining financial statement fiscal year plus 6 months into the first full year thereafter) after the valuation date, the net cash flows in the second full year are expected to be realized 20 months after the valuation date, and each subsequent year’s net cash flows is expected to be realized 12 months thereafter. Exhibit 4.3 presents the timeline of net cash flows expected in this example. Formula 4.18, a variation of Formula 4.13, displays the calculation of present value of net cash flows where the first projection period is a partial year. (Formula 4.18) NCFn ð1 þ gÞ ðNCF1 pyÞ NCF2 NCFn ðk gÞ þ þ þ þ PV ¼ py pyþ0:5 pyþðn0:5Þ 2 ð1 þ kÞ ð1 þ kÞ ð1 þ kÞ pyþðn0:5Þ ð1 þ kÞ where: py ¼ months of partial first year expressed as a decimal Year 1 0

^

2 ^

3 ^

4 ^



34

Cost of Capital

Partial Full Year Year 1

0 ^

^

2

^

3

^

4

^



7 ¼ 0:5833 of the first year, and the partial year factor for In the example, the partial year represents 12 ð70:5Þ the present value of the net cash flows in the partial first year equals 12 ¼ 0:2917. That is, the firstperiod net cash flows are expected to be received 0.2917 of a year following the valuation date (3½ months following the valuation date). The exponent for the present value of the net cash flows expected during the first full year following the valuation date equals ð0:5833 þ 0:5Þ ¼ 1:0833. That is, the net cash flows are expected to be received 0.5 years after the end of the partial first year (7 months). Applying Formula 4.16 using the same assumptions as in Formula 4.15 except for the partial first year, we get: (Formula 4.19) 7 $140ð1 þ 0:05Þ $100 $120 $140 12 ð0:12 0:05Þ þ þ þ PV ¼ 7 7 7 0:2917 þ0:5 þð20:5Þ ð1 þ 0:12Þ ð1 þ 0:12Þ12 ð1 þ 0:12Þ12 ð1 þ 0:12Þ12þð20:5Þ

¼

$58:33 $120 $140 $2;100 þ þ þ 1:083 2:083 1:034 ð1 þ 0:12Þ ð1 þ 0:12Þ ð1 þ 0:12Þ2:083

¼ $56:41 þ

$120 $140 $2;100 þ þ 1:131 1:266 1:266

¼ $56:41 þ $106:14 þ $110:56 þ $1;658:43 ¼ $1;931:55

CAPITALIZING RESIDUAL EARNINGS As we discussed in Chapter 3, the literature includes an alternative formulation of the valuation of net cash flows based on residual earnings. The equivalent RE valuation to Formula 4.7 as applied to net cash flows to equity capital is: (Formula 4.20) RE1 PV ¼ BV0 þ ðke gÞ where: PV ¼ Present value BV0 ¼ Book value (net asset value) for period 0, the period immediately preceding the valuation date

Summary

35

RE1 ¼ Residual earnings for period 0 BV0 ¼ ke ke ¼ Cost of equity capital g ¼ Expected long-term sustainable growth rate in net cash flow to equity investors Exhibit 4.4 shows an example of valuation using residual earnings consistent with the example shown in Formula 4.7. The equivalent abnormal earnings growth (AEG)–based valuation to Formula 4.7 is applied to net cash flows to equity capital: (Formula 4.21) 1 AEG2 NI þ PV ¼ ðke gÞ ke where: AEG2 ¼ RE2 RE1. The other variables are as defined in Formula 4.20. Exhibit 4.5 continues the example in Exhibit 4.4 for abnormal earnings growth. We can also reconcile the two-stage model of valuation using net cash flows to equity and the capitalization and discounting of net cash flows to invested capital.2 Why would we use the residual earnings model? This formulation causes the analyst to focus on the amount of capital invested (net assets) and the return on that investment. It highlights whether the firm is earning returns in excess of its cost of equity capital. It also ties the valuation to the financial statements and treats investments ( use of cash) instead of simply reductions of net cash flow. Finally, it most often results in more of the value being attributed to the existing investments (net assets) than to the terminal or residual value. While some may criticize the approach because it appears to place too much relevance on the accuracy of the balance sheet and the net asset amount, the residual earnings will be reduced if the carrying components of net assets overstates their value, reducing the present value of residual earnings. The residual earnings model will always be equivalent to a dividend discount valuation and a DCF valuation if we somehow could forecast dividends and cash flow for very long (infinite) horizons, or if we somehow could get the correct (but different) growth rates for each model. However, in separating what we know from speculation, this model breaks down the components of the valuation differently. We now have a component (1), the book value, which we observe in the present. If mark-to-market accounting is applied, the book value gives the complete valuation, as in the case of an investment fund where one trades as net asset (book) value. More generally, book value is not sufficient so one adds forecasts of residual earnings for the near term, component (2), and speculation about the long term, component (3), to estimate the difference between value and book value.3

The cost of equity capital and the overall cost of capital are the same whether we are using the present value of net cash flow or the residual earnings formulation of valuation.

SUMMARY This chapter has shown the mechanics of discounting and capitalizing and has defined the difference between a discount rate and a capitalization rate. 2 3

Ronald S. Longhofer, ‘‘The Residual Income Method of Business Valuation,’’ Business Valuation Review (June 2005): 65–70. Stephen H. Penman, ‘‘Handling Valuation Models,’’ Journal of Applied Corporate Finance (Spring 2006): 51.

36

Cost of Capital

Exhibit 4.4

Example of Valuation Using Residual Earnings

For company A we have: Income Statement

Year 1

EBIT Interest Expense EBT Taxes NI

Growth Rate

$228 16 $212 85 $127

3% 3% 3% 3% 3%

where: EBIT ¼ Earnings before interest and taxes EBT ¼ Earnings before taxes NI ¼ Net income Balance Sheet

Year 0

Year 1

Growth Rate

Current Assets Fixed and Intangible Assets Total Assets Current Liabilities Long-term Debt Book Value of Equity (BV) Liabilities plus Equity

$300 900 $1,200 $200 200 800 $1,200

$309 927 $1,236 $206 206 824 $1,236

3% 3% 3% 3% 3% 3%

Applying Formula 4.20 we get:

$23 ð0:13 0:03Þ ¼ $800 þ $230 ¼ $1;030

PV ¼ $800 þ

where: RE1 ¼ NI1 (BV0 ke) ¼ $127 ($800 0.13) ¼ $127 $104 ¼ $23 This is the same result we obtained in Formula 4.7. Using the clean surplus accounting statement we get: NCF1 ¼ BV0 þ NI1 BV1 ¼ $800 þ $127 $824

¼ $103 where: NCF1 ¼ Net cash flow to equity in period 1 NI1 ¼ Net income in period 1 (comprehensive income) BVn ¼ Book value of equity This is the same NCF we capitalized in Formula 4.7.

Summary

Exhibit 4.5

37

Example of Valuation Using Abnormal Earnings Growth RE2 ¼ RE1 ð1:03Þ ¼ $23 ð1:03Þ ¼ $23:69

Applying Formula 4.21 we get: PV ¼

1 ð$23:69 $23Þ $127 þ 0:13 ð0:13 0:03Þ

1 ½$127 þ $6:9 0:13 1 ¼ ½$133:9 0:13 ¼

¼ $1;030 This is the same result as we obtained in Formula 4.7.

It has shown that capitalizing is merely a short-form version of discounting. The essential difference between the discounting method and the capitalizing method is how changes in expected cash flows over time are reflected in the respective formulas. All things being equal, the discounting method and the capitalizing method will yield identical results. However, the validity of the capitalizing method in the income approach to valuation depends on the assumption that the difference between the discount rate and the capitalization rate represents a long-term average rate of growth in the income variable being capitalized. Because many companies are likely to expect near-term changes in levels of their returns that are not expected to be representative of longer-term expectations, many analysts use a combination of discounting and capitalizing for valuation. To accomplish this, they implement five steps: Step 1. Project discrete amounts of return for some period of years until the company is expected to reach a stabilized level from which relatively constant growth may be expected to proceed. Step 2. Use the Gordon Growth Model (or some other method) to estimate a ‘‘terminal value’’ as of the end of the discrete projection period. Step 3. Discount each discrete projected cash flow to a present value at the cost of capital for the number of periods until it is expected to be received. Step 4. Discount the terminal value to a present value at the cost of capital for the number of periods in the discrete projection period (the beginning of the assumed stable growth period). Step 5. Add the values from steps 3 and 4. Most discounting and capitalization formulas reflect the implicit assumption that investors will realize their cash flows at the end of each year. If it is assumed that investors will receive cash flows more or less evenly throughout the year, the formulas can be modified by the midyear convention.

Chapter 5

Relationship between Risk and the Cost of Capital Introduction Defining Risk How Risk Impacts the Cost of Capital Valuation of Risky Net Cash Flows Estimating Risk of Company Operations and Assets Types of Risk Maturity Risk Market Risk Unique Risk Other Risk Cost of Equity Capital Cost of Invested Capital or Overall Cost of Capital Summary Appendix 5A

INTRODUCTION The cost of capital for any given investment is a combination of two basic factors1: 1. A risk-free rate. By ‘‘risk-free rate,’’ we mean a rate of return that is available in the market on an investment that is free of default risk, usually the yield to maturity on a U.S. government security, which is a ‘‘nominal’’ rate (i.e., it includes expected inflation). 2. A premium for risk. This is an expected amount of return over and above the risk-free rate to compensate the investor for accepting risk. The generalized cost of capital relationship is as: (Formula 5.1) EðRi Þ ¼ R f þ RPi where: E(Ri) ¼ Expected return of security i Rf ¼ Risk-free rate RP ¼ Risk premium for security i

1

A third factor is liquidity, but that is usually treated as a separate adjustment, as discussed in Chapter 25.

39

40

Cost of Capital

Quantifying the amount by which risk affects the cost of capital for any particular company or investment is arguably one of the most difficult analyses in the field of corporate finance, including valuation and capital budgeting. Estimating the cost of capital is first and foremost an exercise in pricing risk.

DEFINING RISK Probably the most widely accepted definition of risk in the context of business valuation is the degree of uncertainty (or lack thereof) of achieving future expectations at the times and in the amounts expected.2 This means uncertainty as to both the amounts and the timing of expected economic income. Note that the definition implies as the reference point expected economic income. By expected economic income, in a technical sense, we mean the expected value (mean average) of the probability distribution of possible economic income for each forecast period. This concept was explained in Chapter 3 in the discussion of net cash flow. The point to understand here is that the uncertainty encompasses the full distribution of possible economic income for each period both above and below the expected value. Inasmuch as uncertainty is within the mind of each individual investor, we cannot measure the risk directly. Consequently, participants in the financial markets have developed ways of measuring factors that investors normally would consider in their effort to incorporate risk into their required rate of return. Throughout this book we are equating risk with uncertainty, as does most of the literature. However, some analysts make a useful distinction between the two terms. That is, ‘‘risk’’ is present where the parameters of uncertainty are defined (i.e., when the generating function is known with certainty), as in a coin toss (e.g. if forecasters all agree that recession will occur next year then the subject company’s net cash flows will still vary, but within the forecast of recession). ‘‘Uncertainty beyond risk’’ is when analysts have the possibility of an infinite number of subjective inputs (e.g. wide divergence of opinions among forecasters as to whether there will be a recession next year or not).3 No matter how many probability distributions or Monte Carlo simulations are used to create the forecasts, all risk cannot be eliminated. Therefore, forecasts cannot be discounted at the risk-free rate.

HOW RISK IMPACTS THE COST OF CAPITAL As noted earlier, the cost of capital has two components:4 1. A risk-free rate, Rf 2. A risk premium, RP As the market’s perception of the degree of risk of an investment increases, the risk premium, RP, increases so that the rate of return that the market requires (the discount rate) increases for a given set 2

3

4

David Laro and Shannon P. Pratt, Business Valuation and Taxes: Procedure, Law, and Perspective (Hoboken, NJ: John Wiley & Sons, 2005), 160. Evan W. Anderson, Eric Ghysels, and Jennifer L. Juergens, ‘‘The Impact of Risk and Uncertainty on Expected Returns,’’ Working paper, April 11, 2007. As noted in note 1, a third element—lack of marketability or liquidity—may be embedded in the discount rate, but more often it is treated as a separate adjustment to value. This is covered in Chapter 25.

How Risk Impacts the Cost of Capital

41

of expected cash flows. The greater the market’s required rate of return, the lower the present value of the investment. Risk is the ultimate concern to investors. The risk-free rate compensates investors for renting out their money (i.e., for delaying consumption over some future time period and receiving back dollars with less purchasing power). This component of the cost of capital is readily observable in the marketplace and generally differs from one investment to another only to the extent of the time horizon (maturity) selected for measurement of the risk-free rate. The risk premium is due to the uncertainty of expected returns and varies widely from one prospective capital investment to another. We could say that the market abhors uncertainty and consequently demands a high price (in terms of required rate of return or cost of capital) to accept uncertainty. Since uncertainty as to timing and amounts of future receipts is greatest for equity investors, the high risk forces equity as a class to have the highest cost of capital. VALUATION OF RISKY NET CASH FLOWS In Chapter 3 we discussed measuring future net cash flows in terms of expected net cash flows, and in Chapter 4 we discussed the valuation processes of discounting and capitalization. Combining the concepts, we can better understand the valuation process under conditions of risk. For example, Exhibit 5.1 represents the valuation process for a series of expected cash flows over the life of a typical five-year business project. Each year the cash flows have the potential to vary (the distribution of net cash flows). When viewed in terms of the valuation date, these distributions generally can be expected to be increasingly risky (increasing variability of possible net cash flows). The goal of the valuation process is to estimate the ‘‘price the market would pay’’ for the distributions of estimated net cash flows. In terms of Exhibit 5.1, we are estimating how much the market will pay as of the valuation date for the distribution of net cash flows in periods n ¼ 1, n ¼ 2, and so on. Our task is to determine the marketplace’s pricing of risk as of the valuation date for the comparable distribution of expected net cash flows. We need to first measure the risk and then measure the market’s pricing of those risks (i.e., what is the cost of capital for the net cash flows with comparable risk characteristics). n= 0

n= 1

n= 2

n=3

n=4

n=5

PV1 PV2 PV3 PV4 PV5 PVTotal

Exhibit 5.1

Valuation of Increasingly Risky Net Cash Flows with Symmetric Distributions

42

Cost of Capital

n= 0

n= 1

n= 2

n=3

n=4

n=5

PV1 PV2 PV3 PV4 PV5 PVTotal

Exhibit 5.2

Valuation of Increasingly Risky Net Cash Flows with Skewed Distributions

Exhibit 5.2 represents the same process but for a series of expected skewed distributions of net cash flows. Often net cash flows of a business reach the upper limit because of capacity constraints, pricing limitation, and so on, making such skewed distributions more representative of possible outcomes. In either case, calculating a measure of central tendency (e.g., expected value) by probability weighting the expected cash flows does not eliminate the risk of the distributions. The appropriate discount rate is not a risk-free rate of return. Would the market only demand the risk-free rate of return for taking on the variability of the cash flows? The answer is no. The market will demand compensation (added return) for accepting the risk that the actual cash flows will differ from the expected cash flows in future periods and the added return will increase depending on the amount of expected dispersion that could occur. That is, one would expect that the greater the dispersion of expected cash flows the greater the discount rate. We discuss the Financial Accounting Standards Board’s Concepts Statement No. 7 (Con 7) in Appendix 5A; misreading that statement has added to this confusion. Con 7 is sometimes interpreted to advocate discounting at the risk-free rate when the expected value of a probability distribution is used in the numerator. This is not what it says, as explained in Appendix 5A. ESTIMATING RISK OF COMPANY OPERATIONS AND ASSETS Business risk is the risk of the company operations. Business risk can be thought of in terms of the various underlying business operations: sales risk (risk of decrease in unit sales or in unit sales growth), profit margin risk (pricing and expense risks), and operating leverage risk. It can also be expressed in terms of the risk of the underlying assets of the business. Operating leverage is the variability of net cash flow from business operations (i.e., without regard to the cost of financing the business) as output or revenues change. Net cash flow from operations can be broken down in terms of revenues, variable costs, and fixed costs. Variable costs are those that are dependent on the rate of output or revenues of the firm. Fixed costs occur regardless of the level of

How Risk Impacts the Cost of Capital

43

output or revenue of the firm. Business risk can be quantified in terms of variability of revenue in this way:5 (Formula 5.2) Fc s rev sB ¼ 1 þ PVb where: sB ¼ Standard deviation of operating cash flows of the business before cost of financing Fc ¼ Fixed operating costs of the business PVb ¼ Present value of net cash flows from business operations (before costs of financing) srev ¼ Standard deviation of revenues derived from output That is, as the level of fixed costs rise relative to total costs, the variability of operating cash flows increases. All else being equal, a firm with high operating leverage has high fixed costs and low variable costs; each dollar of revenue from each additional unit of output is offset by a relatively small increase in operating costs. Alternatively, any company can be thought of as a portfolio of assets. But we generally are unable to directly observe rates of return appropriate for the risk of the underlying assets of the business (particularly intangible assets, including the goodwill of the business). We can depict the risk hierarchy of the asset mix of a business generally as shown in Exhibit 5.3. Generally, the risks of investing in net working capital are the least risky of the business assets. Net working capital can be converted to cash over the shortest time frame and with the least expected variance from carrying values. Property, plant, and equipment typically can be utilized in a variety of businesses and in producing a variety of different products. Further, if need be, they can be sold to other businesses; but the proceeds from any such sale likely will vary more from their carrying values than will, for example, net working capital. Finally, the value of intangible assets are most often Risk to the Company Lower Rate of Return Lower Risk Net Working Capital

Property, Plant, and Equipment

Higher Rate of Return

Higher Risk Intangible Assets

Exhibit 5.3 5

Risk of Company’s Asset Mix

Hazem Daouk and David Ng, ‘‘Is Unlevered Firm Volatility Asymmetric?’’ AFA 2007 Chicago Meetings, January 11, 2007.

44

Cost of Capital

Risk to the Investor Lower Rate of Return

Lower Risk Senior Debt Mezzanine (Subordinate) Debt

Preferred Equity

Higher Rate of Return

Higher Risk Common Equity

Exhibit 5.4

Risks of the Components of the Company Capital Structure

dependent on the success of the specific operations of the subject business. They generally have little or no value outside of the existing going concern. How can we estimate an appropriate risk premium for a company as a whole and for its component assets? We can look at the capital structure of the company and the company’s overall cost of capital as a mirror of the business risk. Think of the mix of business assets as the left-hand side of the company balance sheet and the company capital structure as the right-hand side of the company balance sheet. By determining the overall company cost of capital, we can then impute the overall return required from the business operations in order to provide investors (the suppliers of capital to the company) with their expected returns. We can observe market returns investors have received in the past and impute implied returns expected by investors from investments in companies with similar business risks. We are imputing the risk of the investment (company business) from the risks of the securities used to supply the investment and the pricing of risk implied from the returns on those securities. The capital structure of the company adds another layer of risk, financial risk. Financial risk is the added volatility providers of equity capital will experience because returns to bond holders and other preferred investors generally are fixed and are senior to returns to common equity. The leverage of financing increases the volatility to returns on common equity. We can depict the risk hierarchy of the risk of the components of the company capital structure generally as shown in Exhibit 5.4. This book is about measuring and pricing the risks of the assets and components of the capital structure of a business. We discuss separating business risk and financial risks in Chapter 10.

TYPES OF RISK Although risk arises from many sources, this chapter addresses risk in the economic sense, as used in the conventional methods of estimating cost of capital. In this context, capital market theory divides risk into three components:

Types of Risk

45

1. Maturity risk 2. Market risk 3. Unique risk 6 MATURITY RISK Maturity risk (also called horizon risk or interest rate risk) is the risk that the value of the investment may increase or decrease because of changes in the general level of interest rates. The longer the term of an investment, the greater the maturity risk. For example, market prices of long-term bonds fluctuate much more in response to changes in levels of interest rates than do short-term bonds or notes. When we refer to the yields on U.S. government bonds as risk-free rates, we mean that we regard them as free from the prospect of default, but we recognize that they do incorporate maturity risk: The only part of the yield that is risk-free is the income return component. That is, the interest payments promised are risk-free. But the market price or value of the bonds move up or down as interest rates move, creating capital loss or gain. Thus there is a risk to capital embedded in these bonds. The longer the maturity, the greater the susceptibility to change in market price in response to changes in market rates of interest. With regard to interest rates, much of the uncertainty derives from the uncertainty of future inflation levels. MARKET RISK Market risk (also called systematic risk or undiversifiable risk) is the uncertainty of future returns due to the sensitivity of the return on a subject investment to variability in returns for the investment market as a whole. Although this is a broad conceptual definition, for U.S. companies, the investment market as a whole is generally limited to the U.S. equity markets and typically is measured by returns on either the New York Stock Exchange (NYSE) Composite Index or the Standard & Poor’s (S&P) 500 Index. Some theoreticians say that the only risk the capital markets reward with an expected premium rate of return is market risk, because unique or unsystematic risk can be eliminated by holding a well-diversified portfolio of investments. Recent research has shown that it may be difficult or nearly impossible to be fully diversified. The chapters on the various methods of estimating the cost of capital show that market risk is a factor specifically measured for a particular company or industry in some methods but not at all or not necessarily in others. For example, market or systematic risk is taken into consideration in the Capital Asset Pricing Model (CAPM), which is the subject of Chapter 8, and in other methods of estimating the cost of capital. The term that is commonly used for sensitivity to market risk is beta. While beta has come to have a specific meaning in the context of the CAPM, beta is used in the literature of finance as a more general term meaning the sensitivity of an investment to the market factor. Bonds have beta risks (e.g., to interest rates, to general economic conditions as reflected in the broad stock market, etc.). Individual stocks have beta risks (e.g., to general economic conditions as reflected in the broad stock market, to the relative risks of large company stocks to small company stocks, etc.). In the context of the CAPM, beta attempts to measure the sensitivity of the returns realized by a security in company or an industry to movements in returns of ‘‘the market,’’ usually defined as either the S&P 500 Index or the NYSE Composite Index.

6

See Richard A. Brealey, Stewart C. Myers, and Franklin Allen, Principles of Corporate Finance, 8th ed. (New York: McGrawHill, 2006), 162.

46

Cost of Capital

UNIQUE RISK Unique risk (also called unsystematic risk, residual risk, or company-specific risk) is the uncertainty of expected returns arising from factors other than those factors correlated with the investment market as a whole. These factors may include characteristics of the industry and the individual company. In international investing, they also can include characteristics of a particular country. Some of the unique risk of an investment may be captured in the size premium, which is an adjustment to the textbook CAPM and is the subject of Chapter 12. Fully capturing unique risk in the discount rate requires analysis of the company in comparison with other companies, which is discussed in Chapter 14. However, while the size premium captures many risk factors, the analyst must be careful to capture all the risk factors and at the same time avoid double-counting. OTHER RISK Capital market theory assumes efficient markets. That is, it assumes prices change concurrent with changes in the economic fundamentals (economy, industry, or company factors) such that the market prices of publicly traded stocks represent the consensus of investors as to the present value of cash flows and that changes in such fundamentals are ‘‘instantly’’ recognized in market prices. One recent study supports the rationality of stock prices where data on expected cash flows is available to investors.7 But market inefficiency can and does occur for publicly traded stocks, particularly for smaller company stocks that do not have the investor following such that their prices do not react to changes in fundamentals. We do recognize that market prices may not correctly or fully account for the fundamentals of a smaller, thinly traded public company at particular points in time. We point out problems with textbook theories that fall down in such circumstances. Capital market theory also assumes liquidity of investments. Many of the observations about risk and return are drawn from information for liquid investments. Investors desire liquidity and require greater returns for illiquidity. We specifically address issues pertaining to illiquidity risk in Chapter 25 for minority interests and Chapter 26 for entire businesses.

COST OF EQUITY CAPITAL When using the build-up method (Chapter 7), the Capital Asset Pricing Model (CAPM) (Chapter 8), or another model such as the Fama-French 3-factor model (FF) (Chapter 15), we estimate one or more components of a risk premium and add the total risk premium to the risk-free rate in order to estimate the cost of equity capital. When using publicly traded stock data to imply the cost of equity capital (e.g., the discounted cash flow (DCF) method discussed in Chapter 16) we get a total cost of equity capital without any explicit breakdown regarding how much of it is attributable to a risk-free rate and how much is attributable to the risk premium.

COST OF INVESTED CAPITAL OR OVERALL COST OF CAPITAL The cost of invested capital is a blending of the costs of each component, commonly referred to as the weighted average cost of capital (WACC). Chapter 6 discusses each component in the capital structure, and Chapter 17 addresses the weighted average cost of capital. 7

Keren Bar-Hava, Roni Ofer, and Oded Sarig, ‘‘New Tests of Market Efficiency Using Fully Identifiable Equity Cash Flows,’’ Working paper, February 2007.

Summary

47

SUMMARY The cost of capital is a function of the market’s risk-free rate plus a premium for the risk associated with the investment. Risk is the degree of uncertainty regarding the realization of the expected returns from the investment at the times and in the amounts expected. The authors observe a common error of discounting probability-weighted net cash flows using the risk-free rate number. The false assumption is that the probability weighting accounts for risk. It does not. In an economic sense, the market distinguishes between types of risks of a company or investment: market or systematic risk and unique or unsystematic risk. Market risk is the sensitivity of returns on the subject investment to returns on the overall market. Unique risk is the specific risk of the subject company or industry as opposed to the market as a whole (i.e., the risk that remains after taking into account the market risk). Risk impacts the cost of each of the components of capital: debt, senior equity, and common equity. Because risk has an impact on each capital component, it also has an impact on the weighted average cost of capital. As risk increases, the cost of capital increases, and value decreases. Because risk cannot be observed directly in the market, it must be estimated. The impact of risk on the cost of capital is at once one of the most essential and one of the most difficult analyses in corporate finance and investment analysis.

Appendix 5A

FASB’s Concepts Statement No. 7: Cash Flows and Present Value Discount Rates The Financial Accounting Standards Board’s Concepts Statement No. 7 (Con 7), Using Cash Flow and Present Value in Accounting Measures, addresses issues surrounding the use of cash flow projections and present value techniques in accounting measurement. Practitioners are reading the Statement especially for guidance on implementation of the Statement of Financial Accounting Standard (SFAS) No. 142. The guidance was clarified in SFAS No. 157 Fair Value Measurements, Appendix B of FASB No. 157. Two particular elements of the Con 7 seem to be generating confusion: 1. The comparisons of ‘‘traditional’’ and ‘‘expected cash flow’’ approaches to present value (and Con 7’s endorsement of the latter). [paragraphs 42–61]. 2. The use of the risk-free rate to discount expected cash flows [Appendix A and paragraphs 114– 116]. This second point is probably the more confusing. Con 7 observes that when values are uncertain, accountants are trained to use ‘‘most likely’’ values or ‘‘best estimates.’’ The Statement refers to this practice of using ‘‘most likely’’ values as the traditional method. Then it correctly points out that when probability distributions are asymmetric, the ‘‘most likely’’ cash flow is not the same as the ‘‘expected’’ cash flow (the probability-weighted mean of the distribution of all possible outcomes). Con 7 refers to the use of ‘‘expected’’ cash flows as the expected value method. Note, though, that the risk-free rate alone is generally not the correct discount rate for either method, though it works for other present value methods, as is discussed later. Further, all of the standard finance theory for estimating risk-adjusted discount rates that are most commonly applied in a present value analysis (weighted average cost of capital [WACC], Capital Asset Pricing Method [CAPM], betas, etc.) was developed for the so-called expected value method, not for the traditional method. In fact, most finance practitioners and academics would have presumed that ‘‘traditional method’’ referred to this body of work, not to what Con 7 is referring to. Applying standard finance tools to develop discount rates for ‘‘most likely’’ cash flows is flawed unless the probability distribution is symmetric. There are two alternative valid approaches to discounting uncertain future cash flows. Consistently applied, they give the same result. 1. The risk-adjusted discount rate approach adds a risk premium to the discount rate which is then applied to expected cash flows. (Formula 5A.1) Eðcash flowsÞ PV ¼ ð1 þ kÞ where: k ¼ Risk-adjusted discount rate. Where k > risk-free rate of return (Rf ). 48

FASB’s Concepts Statement No. 7: Cash Flows and Present Value Discount Rates

49

This is the approach most commonly presented in finance texts as the ‘‘standard’’ present value method. Risk premia are typically estimated using a model (e.g., the build-up method or CAPM). 2. The certainty-equivalent approach subtracts a cash risk premium from the expected cash flows and then discounts at the risk-free rate. This appears to be what Con 7 is advocating. (Formula 5A.2)

PV ¼

½Eðcash flowsÞ cash risk premium ð1 þ R f Þ

The approach, though rarely used in companies, also is a present value method. It is theoretically appealing. The numerator is called a certainty equivalent. Here also, CAPM or other models can be used to estimate the cash risk premium. Although Con 7 does not say so explicitly, this is the approach set forth in its Appendix A. What Con 7 calls traditional versus expected value is misleading from a finance perspective. The so-called traditional incorporates probabilities only to the extent of noting which outcome is most likely; all other information in the probability distribution is ignored. In contrast, ‘‘expected value’’ is a probability-weighted average of all possible values the random variable can reach at a given point in time. It uses all the information in the probability distribution. Performing the probability weighting to arrive at the expected value is not by itself a sufficient treatment of risk for discounted cash flow (DCF) purposes. It is necessary but not sufficient. Neither the ‘‘most likely’’ cash flow nor the ‘‘expected’’ cash flow may be discounted at the risk-free rate without further adjustment. Expected cash flows may be discounted at a risk-adjusted discount rate, or they may be charged a cash risk premium and then discounted at the risk-free rate. The ‘‘most likely’’ cash flow should not be incorporated in a present value analysis unless the probability distribution is plausibly symmetric or unless some other accommodation is made for the other possible outcomes. How is the cash risk premium determined? Either:

Conduct interviews with investors (e.g., ask ‘‘What lesser amount of risk-free cash would make you indifferent between the risky gamble and the risk-free cash?’’)

It can be computed formulaically using capital market data as shown in Formula 5A.3: (Formula 5A.3) Eðcash flowÞ ð1 þ kÞ ¼

½Eðcash flowÞ1 ðcash risk premiumÞ ð1 þ R f Þ ¼

Certainty Equivalent ð1 þ R f Þ

50

Cost of Capital

Therefore, to get from the expected cash flow to its certainty equivalent, just multiply the former by the ratio: ½ð1 þ Rf Þ=ð1 þ kÞ, where k is a risk-adjusted discount rate that can be computed in the usual way. (Formula 5A.4) ½Eðcash flowÞ1 ð1 þ R f Þ ¼ Certainty equivalent ð1 þ WACCÞ Con 7 does not explain this, but it is part of widely available and accepted corporate finance theory.1 It is not controversial. It works for all the examples shown here and for broad classes of distributions.

1

See, for example, Richard A. Brealey, Stewart C. Myers, and Franklin Allen, Principles of Corporate Finance, 8th ed. (Boston: Irwin McGraw-Hill, 2006), Chapter. 9.

Chapter 6

Cost Components of a Company’s Capital Structure

Introduction Debt Capital Estimating Current Market Yields on Debt Tax Effect Lowers Cost of Debt Leases Are Debt Personal Guarantees Postretirement Obligations Risky Debt Preferred Equity Convertible Debt and Preferred Equity Employee Stock Options Common Equity Summary

INTRODUCTION The capital structure of many companies includes two or more components, each of which has its own cost of capital. Such companies may be said to have a complex capital structure. The major components commonly comprising a company’s capital structure are:

Debt capital Preferred equity

Common equity

Similarly, a project being considered in a capital budgeting decision may be financed by multiple components of capital. In a complex capital structure, each of these general components may have subcomponents, and each subcomponent may have a different cost of capital. In addition, there may be hybrid or special securities, such as convertible debt or preferred stock, warrants, options, or leases. Ultimately, a company’s or project’s overall cost of capital is a result of the blending of the individual costs of each of these components. This chapter briefly discusses each of the capital structure components, and Chapter 17 shows the process of blending them into a company’s or project’s overall cost of capital, which is called the weighted average cost of capital (WACC). Estimation of the costs of conventional fixed-income components of the capital structure, that is, straight debt and preferred stock, is relatively straightforward, because costs of capital for securities of comparable risk usually are directly observable in the market and the company’s actual embedded 51

52

Cost of Capital

cost is often at or very close to current market rates. Although there can be many controversies surrounding costs of fixed-income capital, especially if unusual provisions exist, we discuss these components only briefly here. This book is not intended to be a comprehensive treatise of debt, preferred and hybrid capital instruments. The rest of this book deals primarily with the critically important but highly elusive and often controversial issue of the cost of equity.

DEBT CAPITAL Conceptually, only ‘‘long-term’’ liabilities are included in a capital structure. However, many closely held companies, especially smaller ones, use what is technically short-term interest-bearing debt as if it were long-term debt. In these cases, it becomes a matter of the analyst’s judgment whether to reclassify the short-term debt as long-term debt and include it in the capital structure for the purpose of estimating the company’s overall cost of capital (WACC). The ‘‘long-term’’ debt would include the current portion of long-term debt and short-term debt used as if it were long-term debt. ESTIMATING CURRENT MARKET YIELDS ON DEBT Usually the cost of debt is equivalent to the company’s interest expense (after adjusting for the tax deductibility of interest) and is readily ascertainable from the footnotes to the company’s financial statements (if the company has either audited or reviewed statements or compiled statements with footnote information). If the rate the company is paying is not a current market rate (e.g., long-term debt issued at a time when market rates were significantly different), then the analyst should estimate what a current market rate would be for that component of the company’s capital structure. The interest rate should be consistent with the financial condition of the subject company based on a comparative analysis of it’s average ratios. Standard & Poor’s publishes debt rating criteria along with the Standard & Poor’s Bond Guide. Standard & Poor’s Corporate Ratings Criteria indicates median ratios by rating. CreditScore is a Standard & Poor’s application that gives a hypothetical credit rating based on the financial metrics of a subject company. The analyst can see where the investment would fit within the bond rating system, then check the financial press to find the yields for the estimated rating. Exhibit 6.1 displays the statistics available on debt ratings for industrials and utilities from Corporate Ratings Criteria. Once you have a debt rating, you can estimate the cost of debt capital using yield curve analysis. If the subject company does not have rated debt, you must estimate the debt rating. Interest rates vary depending on the years to maturity. That relationship is called the yield curve. For example, if short-term U.S. government interest rates for bonds with one-year to maturity have a current yield-to-maturity less than the yield-to-maturity on U.S. government bonds with 10 years to maturity, the yield curve is upward sloping. This is the most common slope for the yield curve over the years. But the yield curve can be inverted or downward sloping at times. Exhibit 6.2 shows an example of determining the weighted average current yield to maturity for a company’s bonds using a yield curve analysis. Assume that the yield curve is represented in the top panel of the exhibit. As you see, the yield curve developed from the example market data is upward sloping. Assume that the subject company debt is rated in the lowest rating categories and that the company’s outstanding debt has maturities as shown in the first column of the bottom panel of Exhibit 6.2. You can estimate the weighted average current yield by applying the appropriate yield to maturity from the third line of the top panel of Exhibit 6.2 to the company’s debt, as shown in columns three and four of the bottom panel of the exhibit.

Debt Capital

Exhibit 6.1

53

Key Industrial and Financial Ratios, Long-term Debt

Table 1 Key Industrial Financial Ratios, Long-term Debt Three-year (2002–2004) medians AAA AA EBIT interest coverage (x) EBITDA interest coverage (x) FFO/total debt (%) Free operating cash flow/total debt (%) Total debt/EBITDA (x) Return on capital (%) Total debt/total debt þ equity (%)

23.8 25.5 203.3 127.6 0.4 27.6 12.4

Table 2 Key Utility Financial Ratios, Long-term Debt Three-year (2002–2004) medians AA EBIT interest coverage (x) FFO interest coverage (x) Net cash flow/capital expenditures (%) FFO/average total debt (%) Total debt/total debt þ equity (%) Common dividend payout (%) Return on common equity (%)

4.4 5.4 86.9 30.6 47.4 78.2 11.3

19.5 24.6 79.9 44.5 0.9 27.0 28.3

A

BBB

BB

B

CCC

8.0 10.2 48.0 25.0 1.6 17.5 37.5

4.7 6.5 35.9 17.3 2.2 13.4 42.5

2.5 3.5 22.4 8.3 3.5 11.3 53.7

1.2 1.9 11.5 2.8 5.3 8.7 75.9

0.4 0.9 5.0 2.1 7.9 3.2 113.5

A

BBB

BB

B

3.1 4.0 76.2 18.2 53.8 72.3 10.8

2.5 3.8 100.2 18.1 58.1 64.2 9.8

1.5 2.6 80.3 11.5 70.6 68.7 4.4

1.3 1.6 32.5 21.6 47.2 4.8 6.0

Table 3 Key Ratios Formulas EBIT interest coverage EBITDA interest coverage

Funds from operations (FFO)/ total debt Free operating cash flow/ total debt Total debt/total debt þ equity

Return on capital

Total debt/EBITDA

Earnings from continuing operations* before interest and taxes/gross interest incurred before subtracting capitalized interest and interest income Adjusted earnings from continuing operationsy before interest, taxes, depreciation, and amortization/gross interest incurred before subtracting capitalized interest and interest income Net income from continuing operations, depreciation and amortization, deferred income taxes, and other noncash items/long-term debtz þ current maturities þ commercial paper, and other short-term borrowings FFO- capital expenditures— (þ) increase (decrease) in working capital (excluding changes in cash, marketable securities, and short-term debt)/long-term debtz þ current maturities, commercial paper, and other short-term borrowings Long-term debtz þ current maturities, commercial paper, and other short-term borrowings/long-term debtz þcurrent maturities, commercial paper, and other short-term borrowings þ shareholders’ equity (including preferred stock) þ minority interest EBIT/Average of beginning of year and end of year capital, including short-term debt, current maturities, long-term debt,z non-current deferred taxes, minority interest, and equity (common and preferred stock) Long-term debtz þ current maturities, commercial paper, and other short-term borrowings/adjusted earnings from continuing operations before interest, taxes, and D&A

*Including interest income and equity earnings; excluding nonrecurring items. y Excludes interest income, equity earnings, and nonrecurring items; also excludes rental expense that exceeds the interest component of capitalized operating leases. z Including amounts for operating lease debt equivalent, and debt associated with accounts receivable sales/securitization programs. Source: Standard & Poor’s Corporate Ratings Criteria 2006 (New York: Standard & Poor’s, a division of McGraw-Hill Companies, Inc.) copyright # 2006: 20, 43. Used with permission. All rights reserved.

54 Exhibit 6.2

Cost of Capital Yield Curve Approach to Determining Current Cost of Debt Capital Year(s) Until Debt Matures

AAA, AA, A BBB BB, B, CCC, CC, C, D

One

Two

Three

Four

Five

Six+

6.49% 7.45% 9.36%

7.15% 8.10% 10.01%

7.29% 8.25% 10.16%

7.38% 8.33% 10.27%

7.44% 8.40% 10.31%

7.60% 8.56 10.47

Yield Curve Approach

Exhibit 6.2 Face Value 1 Year 2 Year 3 Year 4 Year 5 Year Over 5 Years

$180 $166 $45,978 $108 $48 $8,400 $54,880

Yield

Weighted Average

9.36% 10.01% 10.16% 10.27% 10.31% 10.47%

0.03% 0.03% 8.51% 0.02% 0.01% 1.60% 10.20%

The analyst should consider that smaller companies may have higher costs of debt than larger companies because, on average, larger companies have higher credit ratings than smaller companies. Also, smaller companies may not be able to borrow as great a proportion of their capital structure as larger companies. Some companies have more than one class of debt, each with its own cost of debt capital (e.g., senior, subordinate, etc.). Traditionally, the relevant market ‘‘yield’’ has been either the yield to maturity or the yield-to-call date. Either of these yields represents the total return the debt holder expects to receive over the life of the debt instrument, including current yield and any appreciation or depreciation from the market price, to the redemption of the debt at either its maturity or call date, if callable. If the stated interest rate is above current market rates, the bond would be expected to sell at a premium. The yield-to-call date likely would be the appropriate yield, because it probably would be in the issuer’s best interest to call it (redeem it) as soon as possible and refinance it at a lower interest cost. If the stated interest rate is below current market rates, then it usually would not be attractive to the issuer to call it, and the yield to maturity would be the most appropriate rate. Credit quality is not your only criteria when determining the appropriate yield for debt instruments. The period over which cash flows (principal and interest) are expected to be received is also important. If you are matching nontraded debt instruments to traded debt instruments to obtain market observations of yields, you need to estimate the credit quality and the length of time over which cash flows are expected to be received. Increasingly, the debt markets have introduced instruments with varying schedules for paying interest and repaying principal. For example, to compare zero-coupon bonds to bonds paying periodic interest payments, you need to measure the length of time over which you will receive cash flow (interest and principal). With the variety of debt instruments that have become common, you need a method to equate the various instruments. One measure of the length of time over which cash flows are expected is the duration of the cash flows:1

1

For an explanation of duration in the context of bond valuation, see, e.g., Richard A. Brealey, Stewart C., and Franklin Allen, Principles of Corporate Finance, 8th ed. (Boston: Irwin McGraw-Hill, 2006), 632–635; Aswath Damodaran, Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2002), 891–892.

Debt Capital

55

(Formula 6.1) n X n Eðcash flowÞ

Duration ¼

ð1 þ kÞn

1

n X Eðcash flowÞ 1

ð1 þ kÞn

n

n

where: n ¼ Periods of expected receipt of the cash flow from 1 through n E(cash flow) ¼ Period cash flow expected from the security, project, or company k ¼ Discount rate used to convert security, project, or company expected cash flows to present value Exhibit 6.3 is a simple example of calculating the duration of a bond. The $1,000 face value bond, issued several years earlier, has a coupon rate of 10% and will mature in 10 years. (We use the simplifying assumption that interest is paid annually, although interest is typically paid made frequently.) The expected cash flows are $100 per year for 9 years and $1,100 in year 10. Make these assumptions:

Assume that the current market rate of interest, given current interest rates and the risk of the issuing company, is now 15%.

The duration is the weighted present value of the cash flows with the weights being the year. The duration of the bond, as shown in Exhibit 6.3, is 6.24 years. This differs from the maturity date.

The duration is an average time over which you expect to receive the cash flow.

Duration can be used as a tool to measure the effective time over which expected cash flows from any investment will be received. TAX EFFECT LOWERS COST OF DEBT Because interest expense is a tax-deductible expense to a company, the net cost of debt to the company is the interest paid less the tax savings resulting from the deductible interest payment. This after-tax cost of debt can be expressed by Formula 6.2: Exhibit 6.3 (1) Year

Example of Calculating Duration of a Bond (2) Expected Cash Flow

(3) Present Value Factor

1 $100 0.8696 2 $100 0.7561 3 $100 0.6575 4 $100 0.5718 5 $100 0.4972 6 $100 0.4323 7 $100 0.3759 8 $100 0.3269 9 $100 0.2843 10 $1,100 0.2472 Total Duration ¼ Sum of (5)/sum of (4) ¼ 6.24 Years

(4) Present Value of Expected Cash Flow

(5) (5) ¼ (1) (4)

86.96 75.61 65.75 57.18 49.72 43.23 37.59 32.69 28.43 271.90 749.06

86.96 151.23 197.25 228.70 248.59 259.40 263.16 261.52 255.84 2,719.03 4,671.67

56

Cost of Capital

(Formula 6.2) kd ¼ kdð ptÞ ð1 tÞ where: kd ¼ Discount rate for debt (the company’s after-tax cost of debt capital) kd(pt) ¼ Rate of interest on debt (pretax) t ¼ Tax rate (expressed as a percentage of pretax income) For decision-making purposes, most corporate finance theoreticians recommend using the marginal tax rate (the rate of tax paid on the last incremental dollar of taxable income) if that differs from the company’s effective tax rate.2 That makes sense, since the marginal rate will be the cost incurred as a result of the investment. However, the focus should be on the marginal rate over the life of the investment, if that is different from the marginal cost incurred initially. Common practice assumes that the top statutory rate is the applicable rate because the typical assumption is that with the long-term horizon, companies will be profitable and will pay income taxes. But we know from historical records that many companies do not pay the top marginal rate. Simulations of expected income tax rates for public companies are available through Professor John Graham.3 The simulations take into account expected taxable income from current operations, carryover of net operating losses from prior periods, and interest expense from outstanding debt.4 LEASES ARE DEBT Capitalized leases are included in reported debt. But operating leases are a substitute for debt. You should generally include all debt (including off-balance sheet leases) in measuring the debt capital of the company.5 Financial Accounting Standards (FAS) Statement No. 13, Accounting for Leases (October 1975), requires footnote disclosure for noncancellable long-term operating leases.6 The disclosure includes:

Aggregate future minimum payments

Minimum payments for first five fiscal years Aggregate future minimum sublease rentals

Historical rental expense

For example, the Standard & Poor’s Ratings Services group routinely capitalizes operating leases for purposes of calculating comparative ratios.7 An excerpt from their Web site describing their methodology follows.

2 3 4

5 6 7

See, e.g., Allen, Brealey, and Myers, Principles of Corporate Finance, 8th ed., 461. [email protected] John R. Graham, ‘‘Debt and the Marginal Tax Rate,’’ Journal of Financial Economics, (May 1996): 41–73; John R. Graham and Mike Lemmon, ‘‘Measuring Corporate Tax Rates and Tax Incentives: A New Approach,’’ Journal of Applied Corporate Finance (Spring 1998): 54–65. Aswath Damodaran, Damodaran on Valuation, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2006), 71–72. FAS No. 13, Accounting for Leases, (October 1975), paragraphs 16, 122. See also ‘‘Off–Balance Sheet Leases: Capitalization and Ratings Implications,’’ Moody’s Investors Service (October 1999).

Debt Capital

57

Corporate Ratings Criteria 2006: To improve financial ratio analysis, Standard & Poor’s uses a financial model that capitalizes off-balancesheet operating lease commitments and allocates minimum lease payments to interest and depreciation expenses. Not only are debt-to-capital ratios affected, but so are interest coverage, funds from operations to debt, total debt to EBITDA, operating margins, and return on capital. . . . The operating lease model is intended to make companies’ financial ratios more accurate and comparable by taking into consideration all assets and liabilities, whether they are on or off the balance sheet. In other words, all rated firms are put on a level playing field, no matter how many assets are leased and how the leases are classified for financial reporting purposes. (We view the distinction between operating leases and capital leases as artificial. In both cases, the lessee contracts for the use of an asset, entering into a debt-like obligation to make periodic rental payments.) The model also helps improve analysis of how profitably a firm employs both its leased and owned assets. By adjusting the capital base for the present value of lease commitments, the return on capital better reflects actual asset profitability.

Exhibit 6.4 shows an example of the methodology. Exhibit 6.5 displays the lease disclosure for a public company and the analysis resulting from capitalizing operating leases.8 Capitalizing operating leases is essential to accurately determine the implied coverage, rating, and market interest rate on outstanding company debt. For some companies, no adjustment is needed because the amount of lease financing used is not significant. But for other companies (e.g., airlines), off-balance sheet lease financing is significant and you must make appropriate adjustments if you hope to calculate a reasonably accurate cost of capital.9 PERSONAL GUARANTEES When estimating the cost of debt for a closely held company, the analyst should ascertain whether the debt is secured by personal guarantees. If so, this is an additional cost of debt that is not reflected directly in the financial statements (or, in some cases, might not even be disclosed). Such guarantees would justify an upward adjustment in the company’s cost of debt to what it would be without the guarantees (assuming that the debt would be available without guarantees). That is, you are interested in the cost of company debt without the influence of guarantor’s pledge of personal assets. You can use the estimated ratings methodology in the earlier section on debt capital to estimate the current implicit credit rating on company debt, the appropriate interest rate, and the implicit market value of outstanding debt.10 For example, in the late 1990s, insurance companies offered guarantees on seller financing. That is, when a company was sold with some percentage of the price as a down payment and the buyer gave the seller a promissory note for the balance (a common procedure in the sale of small businesses and professional practices), the insurance company would guarantee the note to the seller. The required down payment was at least 30% of the purchase price, and the insurance premium was about 3% per annum of the face value of the note. Perhaps 3% can be used as a shortcut estimate for adding to the cost of debt to reflect personal guarantees. Without personal guarantees, many times no debt would be available, and all the company’s capital structure should be discounted at the cost of equity.

8 9

10

This example appeared in a report submitted by Roger Grabowski in a disputed matter. See, e.g., Kirsten M. Ely, ‘‘Operating Lease Accounting and the Market’s Assessment of Equity Risk,’’ Journal of Accounting Research (Autumn 1995): 397–415; and Eugene A. Imhoff, Jr., Robert C. Lipe, and David W. Wright, ‘‘Operating Leases: Income Effects of Constructive Capitalization,’’ Accounting Horizons (June 1997): 12–32. See, for example, Brealey, Myers, and Allen, Principles of Corporate Finance, 8th ed., 655–656.

58

Cost of Capital

Exhibit 6.4

Example of Operating Lease Capitalizations (2004)

Table 1 provides data that would typically appear in the financial statement disclosure. Table 1 Lease Model Calculation* Payment Period

Reporting Year 2004

2005

Year 1 Year 2 Year 3 Year 4 Year 5 Thereafter Total Payments

61.0 54.0 46.1 42.6 38.7 177.9 420.3

65.8 53.3 46.5 41.9 39.6 177.9 425

Source: Standard & Poor’s Corporate Ratings Criteria 2006 (New York: Standard & Poor’s, a division of McGraw-Hill Companies, Inc.) copyright # 2006: 93. Used with permission. All rights reserved. *Reported figures: Future minimum lease commitments (mil. $).

The debt equivalent of the leases is based on discounting future lease commitment data gathered from the notes to financial statements using (1) annual lease payments for the first five years are set forth in the notes; and (2) for the remaining lease years, the model assumes the lease payments approximate the minimum payment due in year five. The number of years remaining under the leases is simply the amount ‘‘thereafter’’ divided by the minimum fifth-year payment. The result is rounded to the nearest whole number. The present value of this payment stream is then determined. The interest rate used is generally the issuer’s average interest rate. Adjustments used in the Standard & Poor’s Ratings model for calculating financial ratios:

Selling, General, and Administrative Expenses (SG&A) adjustment

Average of first-year minimum lease payments in the current and previous years.

SG&A is then reduced by this amount.

Implicit interest *

*

Depreciation expense *

*

Multiply the average (current and previous years) PV of operating leases by the interest rate. In Table 2 we have ($336.5 + $318.7)/2 = $327.6. This figure is then added to the firm’s total interest expense.

Calculated by subtracting the implicit interest from the SG&A adjustment. The lease depreciation is then added to reported depreciation expense.

The interest and depreciation adjustments attempt to allocate the annual rental cost of the operating leases. There is ultimately no change to reported net income as a result of applying the Standard & Poor’s lease analytical methodology.

Table 2 demonstrates the adjustments of the Standard & Poor’s ratings lease model. Table 2 Calculation of Operating Lease Adjustments for 2004

Total debt (reported) Total interest (incl. capitalized interest) Implied interest rate

2004

2003

2002

659.4 36.2 5.5

664.9 40.2 5.6

766.8

Debt Capital

59

Future minimum lease commitments (mil. $) 2005 61 65.8 2006 54 53.3 2007 46.1 46.5 2008 42.6 41.9 2009 38.7 39.6 2010–2014 38.7 2009–2012 39.6 Net present value (NPV) 336.5 318.7 2004 implicit interest Avg. NPV ($327.6) interest rate (5.5%) ¼ $17.9 Lease depreciation expense Adjustment to SG&A* implicit interest ¼ $63.4 $17.9 ¼ $45.5 Adjustment to SG&A—rent Avg. first-year min. payments ($61.0 þ $65.8)/2 ¼ $63.4 Source: Standard & Poor’s Corporate Ratings Criteria 2006 (New York: Standard & Poor’s, a division of McGraw-Hill Companies, Inc.) copyright # 2006: 94. Used with permission. All rights reserved. *SG&A—Selling, general, and administrative expenses.

If you adjust the ‘‘debt’’ balance, you need to adjust the income statement. The imputed ‘‘rent’’ on these assets becomes imputed interest plus depreciation expense. This changes EBIT and EBITDA. (both go up). We can see the impact on the debt and ratios in Table 3. Table 3 Sample Calculation Results

Oper. income/sales (%) EBIT interest coverage (x) EBITDA interest coverage (x) Return on capital Funds from oper./total debt (%) Total debt/EBITDA (x) Total debt/capital (%)

Without Capitalization

With Capitalization

18.6 8.7 12.3 18.9 54.1 1.5 37.6

21.2 6.2 8.6 15.6 40.4 2.1 41.0

Source: Standard & Poor’s Corporate Ratings Criteria 2006 (New York: Standard & Poor’s, a division of McGraw-Hill Companies, Inc.) copyright # 2006: 95. Used with permission. All rights reserved.

POSTRETIREMENT OBLIGATIONS Unfunded liabilities relating to defined benefit pension plans and retiree medical plans are debtlike in nature.11 Employees become the equivalent of creditors of the company because they accepted a portion of their compensation as these deferred benefits. Defined benefit plans differ from defined contribution plans, which are funded on a current basis, because with the latter the sponsor company does not bear the risk of ongoing performance of the assets set aside to fund the obligations. Because of the assumptions necessary for their measurement, you must be cognizant of the relative uncertain nature of accounting for postretirement obligations. When assessing assumptions, you can focus on differences among companies’ disclosures. The analysis requires that you compare the current value of the company’s plan assets to the projected benefit obligation for pensions (PBO) and the accumulated postretirement benefit obligations for retiree medical obligations (APBO). The PBO may understate the true economic liability because it 11

This section is drawn from Standard & Poor’s Corporate Rating Critera 2006 (New York: Standard & Poor’s/McGraw-Hill, 2006), 96–111.

60

Cost of Capital

Exhibit 6.5

Sample Capitalization of Operating Leases

Leases The Company distributes petroleum products throughout its marketing areas through a combination of owned and leased terminals. Leases for product distribution terminals are generally for short periods of time and continue in effect until canceled by either party with contracted days of notice, generally 30 to 60 days. Most product distribution terminal leases are subject to escalations based on various factors. The Company subleases a portion of its leased product distribution terminals. During December 1997, the Company purchased the Riverhead Terminal pursuant to a purchase option in the lease. Additionally, the Company leases two of its refining processing units pursuant to long-term operating leases. The Company has long-term leases with special purpose entities for land and equipment at the Company’s BP California, Exxon Arizona, and certain 76 Products sites. These leases provide the Company the option to purchase, at agreed-upon contracted prices, (a) not less than all of the leased assets at annual anniversary dates, and (b) a portion of the leased assets for resale to unaffiliated parties at quarterly lease payment dates. The Company may cancel the leases provided that lessors receive minimum sales values for the assets. The contracted purchase option price and minimum guaranteed sales values decline over the term of the leases. Minimum annual rentals vary with a reference interest rate (LIBOR). The Company leases the majority of its stores and certain other property and equipment. The store leases generally have primary terms of up to 25 years with varying renewal provisions. Under certain of these leases, the Company is subject to additional rentals based on store sales as well as escalations in the minimum future lease amount. The leases for other property and equipment are for terms of up to 15 years. Most of the Company’s lease arrangements provide the Company an option to purchase the assets at the end of the lease term. The Company may also cancel certain of its leases provided that the lessor receives minimum sales values for the leased assets. Most of the leases require that the Company provide for the payment of real estate taxes, repairs and maintenance, and insurance. At December 31, 1997, future minimum obligations under non-cancelable operating leases and warehousing agreements are as follows: (Thousands of Dollars) 1998 $159,441 1999 $147,674 2000 $134,432 2001 (a) $107,610 2002 (a) $44,023 Thereafter $390,376 ———— Total Payments $983,556 Less future minimum sublease income $110,855 ———— Net Total Payments $872,701 ———— ———— (a) Excludes guaranteed residual payments, totaling $123,221,000 (2001) and $191,522,000 (2002) due at the end of the lease term, which will be reduced by the fair market value of the leased assets. Source: Tosco Corporation 10k, December 1997.

Using the data from the disclosure, we can calculate the discounted present value of lease commitments at 7% discount rate (current market rate for borrowing), net of sublease income and including guaranteed residual payments: Present Value of Lease Payments @ 7%

$702,703

Future Sublease Income as % of Total Lease Payments Estimated Value of Future Sublease Income

11.3% $79,405

Debt Capital

61

Present Value of Lease Payments @ 7% Less: Estimated Value of Future Sublease Income Plus: Present Value of Guaranteed Residual Payments @ 7% Present Value of Lease Commitments Net of Sublease Value Adjusting the balance sheet we get the following: Total Debt per Balance Sheet (12/97) Market Value of Balance Sheet Debt1 Value of Operating Leases Market Value of Debt Plus Operating Leases 1

$702,703 (79,405) 230,557 $853,855 $1,893,165 $2,075,200 853,855 $2,929,055

See Chapter 17 for calculation.

We can calculate the ratios of market value of invested capital (MVIC) to earnings before interest, taxes, depreciation, and amortization (EBITDA) and debt to MVIC as shown next: MVIC/EBITDA Debt/MVIC 1 2

Book1 9.4 24%

Adjusted2 9.0 33%

Using book value of debt and unadjusted EBITDA. Using market value of debt plus operating leases and EBITDA adjusted for lease rent expense.

does not take into account future benefit improvements, even if probable, unless provided for in the current labor agreement. The PBO may differ from the accumulated benefit obligation (ABO), which is a measure of the present value of all the benefits earned to date. It approximates the value of the benefits if the company were to terminate the plan (similar to a ‘‘shutdown’’ scenario). The PBO also accounts for the effect of salary and wage increases on benefit payouts that are linked to future compensation amounts by formula. The PBO measures the pension promise at the amount that will ultimately be settled as the company continues (a ‘‘going concern’’ scenario). Under FAS Statement No. 87, PBO is the basis for expense recognition but ABO serves as a basis for balance sheet recognition of the accumulated, but unfunded liability. PBO, though, is the better measure of the true economic liability. Standard & Poor’s Ratings Services considers that: Companies with the same funding ratios in their benefit plans do not, however, necessarily bear the same risks related to their plans. The size of the gross liability is also important because, where the gross liability is large relative to the company’s assets, any given percentage change in the liability or related plan assets will have a much more significant effect than if the gross liability had been less substantial.12

Any adjustment made for unfunded pension liabilities, health care obligations, and other forms of deferred compensation are similar to debt but differ from debt instruments because the full amount of the expense incurred in meeting the obligations will result in tax deductions when made. This is equivalent to being able to expense both interest and principal of a debt obligation. Thus you need to factor in such benefit liabilities on an after-tax basis. Exhibit 6.6 displays an example of the adjustment to the debt because of unfunded PBOs. In Exhibit 6.6, the debt is increased by the amount of the unfunded projected benefit obligations, with the effect of a reduction to equity. This causes the capitalization to change (increase in debt to book value of equity) and the company’s debt rating is likely to decline, raising the cost of debt capital.

12

Standard & Poor’s Corporate Rating Critera 2006 (New York: Standard & Poor’s/McGraw-Hill, 2006), 98.

62

Cost of Capital

Exhibit 6.6

Example of Adjustment to Debt Due to Unfunded PBO

Capitalization Adjustments XYZ Co.* Debt totals $1.0 billion and equity $600 million at Dec. 31, 200X. Tax rate: 33%-1/3%. Projected benefits obligation (PBO) exceeds fair value of plan assets by $1.1 billion at year-end 200X, up from $700 million at the previous year-end. Change in benefits obligation (Mil. $) PBO, beginning of year Current service cost Interest cost (7% 2,000) Actuarial adjustments Benefits paid

2,000.0 60.0 140.0 100.0 300.0

PBO, end of year

2,000.0

Change in plan assets Fair value of plan assets, beginning of year Actual return on plan assets Benefits paid Fair value of plan assets, end of year Unfunded PBO

1,300.0 100.0 300.0 900.0 1,100.0

Source: Standard & Poor’s Corporate Ratings Criteria 2006 (New York: Standard & Poor’s, a division of McGraw-Hill Companies, Inc.) copyright # 2006: 107. Used with permission. All rights reserved.

Assuming only $800 million of the $1.1 billion unfunded accumulated benefits obligation was recognized on the balance sheet at Dec. 31, 200X, adjusted debt leverage is computed as follows: Adjusted debt and debt-like liabilities¼ Adjusted equity¼

Adjusted debt and debt-like liabilities/total capitalization This compares with unadjusted total debt to capitalization of:

Total debt þ [(1 tax rate) (unfunded PBO)] Book equity [(1 tax rate) (unfunded PBO liability already recognized on balance sheet)]

$1.0 bil. þ (66 2/3% $1.1 bil.) ¼ $1.733 bil. $600 mil. [66 2/3% ($1.1 bil. $800 mil.)] ¼ $400 mil. $1.733 bil./($1.733 bil. þ $400 mil.) ¼ 81.2% $1.0 bil./($1.0 bil. þ $600 mil.) ¼ 62.5%

Source: Standard & Poor’s Corporate Ratings Criteria 2006 (New York: Standard & Poor’s, a division of McGraw-Hill Companies, Inc.) copyright # 2006: 107. Used with permission. All rights reserved. *XYZ Co. operates in a country where benefits plans are prefunded and plan contributions are tax-deductible. Any intangible pension asset account relating to previous service cost would be eliminated against equity. This would also be tax-affected.

RISKY DEBT While a common approach in estimating the cost of debt capital is to use the promised yield on newly issued debt of the company (or comparably rated debt of other companies) in theory, the expected return on debt should reflect the promised yield net of expected default loss. The expected default loss (net of expected recovery) should not be included in the cost of debt because it is not part of the expected return.

Convertible Debt and Preferred Equity

63

One commonly used approach to estimating net default loss is using studies of historical default rates and recovery rates. But market expectations may differ from historical rates.13 One can look upon the value of equity as a call option on the company’s assets and use volatility for public companies’ stock to infer the portion of the yield that equates to the expected default loss on debt.14 Alternatively, if one considers risky debt as a combination of a safe bond and a short position in a put option (i.e., the company has the option of defaulting when the value of the operations and assets decline to amounts below the face value of the debts), then one can use volatility of public debt to infer the portion of the yield that equates to expected default loss on debt (i.e., difference between face value and portion of market value representing return of principal).15

PREFERRED EQUITY If the capital structure includes preferred equity, the yield rate can be used as the cost of that component. If the dividend is at or close to the current market rate for preferred stocks with comparable features and risk, then the stated rate can be a proxy for market yield. If the rate is not close to a current market yield rate, then the analyst should estimate what a current market yield rate would be for that component of the company’s capital structure. Standard & Poor’s publishes preferred stock rating criteria along with the Standard & Poor’s Stock Guide. Using this publication, analysts can see where the company’s preferred stock would fit within the preferred stock rating system, then check the financial press to find the yields for preferred stocks with similar features and estimated rating. Analysts must adjust for any differences in features often found in privately issued preferred equity, such as special voting or liquidation rights. If the preferred stock is callable, the same analysis (of the market rate of dividend compared to the dividend relative to call price as discussed with respect to debt) applies to the preferred stock.16

CONVERTIBLE DEBT AND PREFERRED EQUITY Convertible debt and convertible preferred equity are hybrid instruments that are essentially two securities combined into one: a straight debt or preferred equity element plus a warrant. Typically the instrument is callable at the request of the issuer. This feature is for the benefit of the issuer. The call forces conversion of the bond or preferred instrument earlier than the investor might chose. The cost of capital for the convertible instrument is the sum of the costs of these two elements. A warrant is a long-term call option issued by a company on a specific class of its own common equity, usually at a fixed price. Convertibles are easiest to understand if they are analyzed first as debt or nonconvertible preferred equity and then the value is adjusted for the value of the warrants (long-term call options).17 We discuss the overall cost of capital for a firm with convertible debt or convertible preferred equity in Chapter 17. There are several theories why companies issue convertible instruments. One theory is that convertible instruments offer a cheaper source of financing than straight debt financing or preferred 13

14

15 16 17

Ian A. Cooper and Sergei A. Davydenko, ‘‘Estimating the Cost of Risky Debt,’’ Journal of Applied Corporate Finance (Summer 2007): 90–94. Jens Hilscher, ‘‘Is the Corporate Bond Market Forward Looking?’’ European Central Bank Working Paper Series No. 800 (August 2007). Ibid, 92–94. See, for example, Damodaran, Investment Valuation, 2nd ed., 212–213. Ibid, 806–914.

64

Cost of Capital

equity financing. The convertible feature offers the issuer (1) the probability of a hybrid price for common stock not just the common stock price at the time the convertible instrument is issued and (2) the possibility of issuing debt or preferred equity at a lower yield than would be the case were the instruments not convertible. But the issuer is giving up a valuable right: the right to buy stock in the future at a predetermined price (the conversion price, which may change over time). That right has value and the value given up must be balanced with the seeming benefits.18 New valuation models for these hybrid instruments are being studied employing advanced methodologies for measuring risk. For example, one study adapts simulation models to their valuation.19 Another study proposes use of advanced binomial warrant (option) pricing model.20 Another study compares various models to observed market prices.21

EMPLOYEE STOCK OPTIONS Employee stock options are equity. Outstanding employee stock options will generate capital for the company once they are exercised; they represent a part of the equity capital of the company. Issuing employee stock options is a company expense, and the income statement should reflect the cost of issuing options to employees. Employee stock options are part of the cost of attracting and retaining employees.22 The topic of employee stock option valuation has received considerable attention since the Financial Accounting Standards Board proposed and later adopted the requirement to expense employee stock options.23 Valuation models have been proposed: binomial lattice models, modified BlackScholes models, and so on, to incorporate the nuances of valuing employee stock options compared to traded options.24 Finally, integral to pricing options is the forecasting of volatilities of the common stock on which the option pricing models depend.25

COMMON EQUITY Part II of this book is devoted to estimating the cost of common equity capital. Unlike yield to maturity on debt or yield on preferred equity, the cost of common equity for specific companies or risk categories cannot be directly observed in the market. The cost of equity capital is the expected rate of return needed to induce investors to place funds in a particular equity investment. As with the returns on bonds or preferred stock, the returns on common equity have two components:

18

19

20

21

22 23 24

25

Igor Loncarski, Jenke ter Horst, and Chris Veld, ‘‘Why Do Companies Issue Convertible Bonds? A Review of Theory and Empirical Evidence,’’ in Advances in Corporate Finance and Asset Pricing, ed. L. D. R. Renneboog (Amsterdam: Elsevier, 2006), 311–339. Dmitri Lvov, Ali Bora Yigitbasioglu, and Naoufel El Bachir, ‘‘Pricing Convertible Bonds by Simulation,’’ Working paper, December 2004. Zhiguo Tan and Yiping Cai, ‘‘Risk Equilibrium Binomial Model for Convertible Bonds Pricing,’’ South West University of Finance and Economics, Working paper January 28, 2007. Yuriy Zabolotnyuk, Robert Jones, and Chris Veld, ‘‘An Empirical Comparison of Convertible Bond Valuation Models,’’ Working paper, June 18, 2007. See, e.g., Damodaran, note 6 above pages 72–73 and note 13 above pages 440–450. See, e.g., Mark H. Lang, ‘‘Employee Stock Options and Equity Valuation,’’ Research Foundation of CFA Monograph (2004). Jaksˇa Cvitanic´, Zvi Wiener, and Fernando Zapatero, ‘‘Analytic Pricing of Employee Stock Options,’’ Working paper, July 19, 2006. George J. Jiang and Yisong S. Tian, ‘‘Volatility Forecasting and the Expensing of Stock Options,’’ Working paper, November 21, 2005.

Summary

65

1. Dividends or distributions 2. Changes in market value (capital gains or losses) Because the cost of capital is a forward-looking concept, and because these expectations regarding amounts of return cannot be directly observed, they must be estimated from current and past market evidence. Analysts primarily use theoretical-based methods of estimating the cost of equity capital from market data, each with variations:

Build-up methods (Chapter 7)

Capital asset pricing model (Chapter 8)

Arbitrage pricing theory (Chapter 15) Fama-French 3-factor model (Chapter 15)

Market-derived capital pricing model (Chapter 15) Yield-spread model (Chapter 15)

Or they derive an implied cost of equity capital from the current market price of the common stock (for public companies) (Chapter 16).

SUMMARY The typical components of a company’s capital structure are summarized in Exhibit 6.7. In addition to the straight debt, preferred equity, and common equity shown, some companies have hybrid securities, such as convertible debt or preferred stock and options or warrants. Chapter 17 explains how to combine the costs of each of these components to derive a company’s overall cost of capital, the weighted average cost of capital. Whereas this chapter has addressed briefly the cost of each component, the rest of the book focuses primarily on the many ways to estimate the cost of equity capital.

Exhibit 6.7

Capital Structure Components

Short-term notes Long-term debt Capital leases Unfunded postretirement obligations Preferred equity Common equity Additional paid-in capital Retained earnings Off–balance sheet financing Warrants Operating leases

Not technically part of the capital structure, but may be included in many cases, especially if being used as if long term (e.g., officer loans) YES (including current portion) Normally YES Normally YES YES YES—all part of common equity YES—all part of common equity YES—all part of common equity Normally YES YES Normally YES

Part 2

Estimating the Cost of Equity Capital and the Overall Cost of Capital

Chapter 7

Build-up Method Introduction Formula for Estimating the Cost of Equity Capital by the Build-up Method Risk-free Rate Risk-free Rate Represented by U.S. Government Securities Components of the Risk-free Rate Why Only Three Specific Maturities? Selecting the Best Risk-free Maturity Equity Risk Premium Small-Company Premium Company-Specific Risk Premium Size Smaller than the Smallest Size Premium Group Incorporating an Industry Risk Factor into the Build-up Method Volatility of Returns Leverage Other Company-Specific Factors Example of the Build-up Method Using Morningstar Data Example of the Build-up Method Using Duff & Phelps Size Study Data Summary

INTRODUCTION Previous chapters discussed the cost of capital in terms of its two major components, a risk-free rate and a risk premium. This chapter examines these components in general, dividing the equity risk premium into three principal subcomponents. The typical ‘‘build-up model’’ for estimating the cost of common equity capital consists of two primary components, with three subcomponents: 1. A ‘‘risk-free’’ rate 2. A premium for risk, including any or all of these subcomponents:

A general equity risk premium A small company premium A company-specific risk premium

In international investing, there may also be a country-specific risk premium, reflecting uncertainties owing to economic and political instability in the particular country to the extent that the instability is greater than in the U.S. We discuss the cost of capital in developing economies in Chapter 8.

69

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Cost of Capital

FORMULA FOR ESTIMATING THE COST OF EQUITY CAPITAL BY THE BUILD-UP METHOD Stating the preceding concept in a formula, the equity cost of capital can be estimated by the build-up method as: (Formula 7.1) EðRi Þ ¼ R f þ RPm þ RPs þ RPu where: E(Ri) ¼ Expected (market required) rate of return on security i Rf ¼ Rate of return available on a risk-free security as of the valuation date RPm ¼ General equity risk premium (ERP) for the ‘‘market’’ RPs ¼ Risk premium for smaller size RPu ¼ Risk premium attributable to the specific company or to the industry (the u stands for unsystematic risk, as defined in Chapter 5) After discussing how to develop each of these four components, we will substitute some numbers into the formula to reach an estimated cost of equity capital for a sample company. An additional possible component, industry risk, is discussed in a later section in this chapter.

RISK-FREE RATE A ‘‘risk-free rate’’ is the return available as of the valuation date on a security that the market generally regards as free of the risk of default. RISK-FREE RATE REPRESENTED BY U.S. GOVERNMENT SECURITIES In the build-up method (as well as in other methods), analysts typically use the yield to maturity on U.S. government securities, as of the valuation date, as the risk-free rate. They generally choose U.S. government obligations of one of these maturities:

30 days 5 years

20 years

Sources for yields to maturity for maturities of any length as of any valuation date can be found in the daily financial press. (When it is not possible to find yields to match the exact length of maturity, choose the closest maturity available.) To obtain a yield on long-term government bonds—for example, a 20-year yield, which is commonly used as the default long-term government bond—most analysts go to the financial press (e.g., The Wall Street Journal or The New York Times) as of the valuation date and find the yield on a bond originally issued for 30 years with approximately 20 years left to maturity. The Federal Reserve Statistical Release tracks 20-year yields. The link to its Web site is http://federalreserve.gov/release/ h15. The St. Louis branch of the Federal Reserve Bank also tracks 20-year yields. The link to its Web site is: http://research.stlouisfed.org/fred2/series/GS20. Alternatively, you can use the returns on

Risk-free Rate

71

zero-coupon government STRIPS.1 Long-term government bonds make interim interest payments, which results in their duration being less than their stated maturity. COMPONENTS OF THE RISK-FREE RATE The so-called risk-free rate reflects three components: 1. Rental rate. A real return for lending the funds over the investment period, thus forgoing consumption for which the funds otherwise could be used. 2. Inflation. The expected rate of inflation over the term of the risk-free investment. 3. Maturity risk or investment rate risk. As discussed in Chapter 5, the risk that the principal’s market value will rise or fall during the period to maturity as a function of changes in the general level of interest rates. All three of these economic factors are embedded in the yield to maturity for any given maturity length. However, it is not possible to observe the market consensus about how much of the yield for any given maturity is attributable to each of these factors. Very importantly, note that this basic risk-free rate includes inflation. Therefore, when this rate is used to estimate a cost of capital to discount expected future cash flows, those future cash flows also should reflect the expected effect of inflation. In the economic sense of nominal versus real dollars, we are building a cost of capital in nominal terms, and it should be used to discount expected returns that also are expressed in nominal terms. WHY ONLY THREE SPECIFIC MATURITIES? The risk-free rate typically is chosen from one of only three specific maturities because the build-up model incorporates a general equity risk premium often based on historical data developed by Morningstar. Morningstar data provides short-term, intermediate-term, and long-term historical risk premium series, based on data corresponding to the aforementioned three maturities. Twenty years is the longest maturity because Morningstar’s data goes all the way back to 1926, and 20 years was the longest U.S. government obligation issued during the earlier years of that time period. Data in the Duff & Phelps Studies can be used as an alternative to using Morningstar data in the build-up method. The risk premiums for the build-up method in the Duff & Phelps Studies include a general equity risk premium and size premium in one number, measured in terms of a premium over long-term U.S. government bonds (20-year). SELECTING THE BEST RISK-FREE MATURITY In valuing ‘‘going-concern’’ businesses and long-term investments made by businesses, practitioners generally use long-term government bonds as the risk-free security and estimate the equity risk premium (ERP or notationally RPm) in relation to long-term government bonds. This convention represents a realistic, simplifying assumption. Most business investments have long durations and suffer from a reinvestment risk comparable to that of long-term government bonds. As such, the use of long-term government bonds and an ERP estimated relative to long-term bonds more closely matches the investment horizon and risks confronting business managers in capital decisions and valuators in valuation problems than reference to Treasury bills. 1

STRIPS stands for ‘‘Separate Trading of Registered Interest and Principal of Securities.’’ STRIPS allow investors to hold and trade the individual components of U.S. government bonds and notes as separate securities. See, e.g., Brian P. Sack, ‘‘Using Treasury STRIPS to Measure the Yield Curve,’’ FEDS Working Paper No. 2000-42 (October 2000).

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Cost of Capital Exhibit 7.1

Yields on 10-Year, 20-Year, and 30-Year U.S. Government Bonds Yields

Period 6/15/2007 4/6/2006 2005 2004 2003 2002 2001 2000 1999 1998 1997

10-Year

20-Year

30-Year

5.2 4.6 4.3 4.3 4.2 3.9 5.1 5.1 6.4 4.9 5.8

5.3 4.8 4.6 4.9 5.0 4.9 5.8 5.6 6.8 5.6 6.0

5.2 4.7 n/a n/a n/a n/a 5.5 5.4 6.4 5.2 5.9

The consensus of financial analysts today is to use the 20-year U.S. government bond yield to maturity as of the effective date of valuation because:

It most closely matches the often-assumed perpetual lifetime horizon of an equity investment. The longest-term yields to maturity fluctuate considerably less than short-term rates and thus are less likely to introduce unwarranted short-term distortions into the actual cost of capital.

People generally are willing to recognize and accept that the maturity risk is embedded in this base, or otherwise risk-free, rate.

It matches the longest-term bond over which the equity risk premium is measured in the Morningstar data series.

Many analysts use either a 10-year or a 30-year yield, but as a practical matter it usually does not differ greatly from the 20-year yield. Exhibit 7.1 summarizes the yields on 10-year, 20-year, and 30-year government bonds for the last decade. Sometimes analysts select a 5-year rate to match the perceived investment horizon for the subject equity investment. The 30-day rate is the purest risk-free base rate because it contains virtually no maturity risk. If inflation is high, it does reflect the inflation component, but it contains little compensation for inflation uncertainty.

EQUITY RISK PREMIUM On an equity investment, the return on investment that the investor will (or has the opportunity to) realize usually has two components: 1. Distributions during the holding period (e.g., dividends or withdrawals) 2. The capital gain or loss in the value of the investment (For an active public security, it is considered part of the return whether the investor chooses to realize it or not, because the investor has that choice at any time.) Obviously, these expected returns on equities are much less certain (or more risky) than the interest and maturity payments on U.S. government obligations. This difference in risk is well documented by

Company-Specific Risk Premium

73

much higher standard deviations (year-to-year volatility) in returns on the stock market compared with the standard deviation of year-to-year returns on U.S. government obligations. To accept this greater risk, investors demand higher expected returns for investing in equities than for investing in U.S. government obligations. As discussed earlier, this differential in expected return on the broad stock market over U.S. government obligations (sometimes referred to as the excess return, but not to be confused with the excess earnings method) is called the equity risk premium (ERP) (or interchangeably market risk premium). See Chapter 9 for a complete discussion on estimating the equity risk premium.

SMALL-COMPANY PREMIUM Studies have provided evidence that the degree of risk and corresponding cost of capital increase with the decreasing size of the company. The studies show that this addition to the realized market premium is over and above the amount that would be warranted solely for the companies’ systematic risk. Chapter 12 discusses the results of research on this phenomenon as well as the data sources. Many practitioners use the small-company premium in the build-up method (difference between the realized returns on small company stocks and large company stocks). Data in the Duff & Phelps studies can be used as an alternative to using Morningstar data in the build-up method. The risk premiums for the build-up method in the Duff & Phelps studies include a general equity risk premium and size premium in one number. The Duff & Phelps study is discussed in Chapter 12.

COMPANY-SPECIFIC RISK PREMIUM To the extent that the subject company’s risk characteristics are greater or less than the typical risk characteristics of the companies from which the equity risk premium and the size premium were drawn, a further adjustment may be necessary to estimate the cost of capital for the specific company. Such adjustment may be based on (but not necessarily limited to) analysis of five factors: 1. 2. 3. 4. 5.

Size smaller than the smallest size premium group Industry risk Volatility of returns Leverage Other company-specific factors

SIZE SMALLER THAN THE SMALLEST SIZE PREMIUM GROUP As will be seen in Exhibits 12.7 and 12.8 from the Duff & Phelps studies, the smallest size group for which we have specific size premium data averages $76 million in market value of equity, $54 million in sales, and so forth. If the subject company is somewhat smaller than this cutoff, most observers believe that a further size premium adjustment is warranted, but there have not yet been adequate empirical studies to quantify this amount. The Duff & Phelps studies do provide regressions of the observed relationships between size and returns to use for extrapolating to smaller firms. Alternatively, a conservative approach may be appropriate, perhaps adding a point or two to the discount rate for a significantly smaller company and leaving any greater adjustments to be attributed to other specifically identifiable risk factors.

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Cost of Capital

INCORPORATING AN INDUSTRY RISK FACTOR INTO THE BUILD-UP METHOD The SBBI Valuation Edition 2006 Yearbook presents an expanded alternative build-up model that includes a separate variable for the industry risk premium. This model is shown in Formula 7.2. (Formula 7.2) EðRi Þ ¼ R f þ RPm þ RPs þ= RPi þ RPu where: E(Ri) ¼ Expected rate of return Rf ¼ Risk-free rate of return RPm ¼ Equity risk premium (market risk) RPs ¼ Size premium RPi ¼ Industry risk premium RPu ¼ Company-specific risk premium (unsystematic risk) The industry in which the company operates may have more or less risk than the average of other companies in the same size category. This differential is very hard to quantify in the build-up model. However, if the company is obviously in a very low-risk industry (e.g., water distribution) or a very high-risk industry (e.g., airlines), a point or two adjustment, either downward or upward, for this factor may be warranted. In an attempt to make the build-up method more closely approximate the Capital Asset Pricing Method (CAPM), Morningstar since 2000 has published industry risk adjustment factors (see Chapter 14). These ‘‘industries’’ are based on Standard Industrial Classification (SIC) codes. The industry premia have been adjusted quarterly since 2005. Each company’s contribution to the adjustment shown is based on a full-information beta (see Chapter 10). Morningstar calculates each company’s contribution to the full-information beta based on the segment sales reported in the company’s 10-K for that SIC code. A listing of each company included in each industry is available for downloading for free from the Morningstar Web site, http://corporate.morningstar.com/ib/asp/detail .aspx?xmlfile¼1431.xml. These industry adjustments are valid only to the extent that the subject company’s risk characteristics are similar to the weighted average of the companies that make up the ‘‘industry’’ for the SIC code shown. Any analyst contemplating using the Morningstar industry adjustments in the build-up method should download the list of companies included in the industry and make a judgment as to whether the risk characteristics of the companies are substantially similar to the subject company to make the adjustment reliable. In order to aid this judgment, the analyst may wish to go to the 10-K filings for the companies included. The description of the companies included give the analyst a much better picture of the similarities of the companies included in the industry to the subject. Also, the segment information in the 10-K will show the proportionate contribution to earnings, which may be very different than the proportionate contribution to revenue. But stock returns are a function of profit, not revenue, and use of revenue may result in an overweight of a low-profit segment. VOLATILITY OF RETURNS High volatility of returns (usually measured by the standard deviation of historical returns over some period) is another risk factor. However, without comparable data for the average of the other companies in the size category and/or industry, it is not possible to make a quantified comparison. If the

Example of the Build-up Method Using Morningstar Data

75

analyst perceives that the subject company returns are either unusually stable or unusually volatile compared with others in the size category and/or industry, some adjustment for this factor may be warranted, which would be a factor to consider in the company-specific risk premium. LEVERAGE Leverage is clearly a factor that can be compared between the subject company and its size peers. Exhibit 12.7 gives the market value of equity for each size category. For example, the smallest size category averages $76 million in market value of equity with a capital structure of roughly 30% debt and 70% equity, at market value. Size breakdowns of other size measures show generally similar capital structures. If the subject company’s capital structure significantly departs from this average, some upward or downward adjustment to the cost of equity relative to the average company in the size category would seem warranted. For example, highly leveraged companies should have higher equity costs of capital compared with companies with lower debt levels, all else being equal. Of course, a decrease in the required equity return might be warranted if the subject’s capital structure has little or no debt. OTHER COMPANY-SPECIFIC FACTORS Other factors specific to a particular company that affect risk could include, for example:

Concentration of customer base Key person dependence

Key supplier dependence

Abnormal present or pending competition Pending regulatory changes

Pending lawsuits Volatility of returns

Strengths/weaknesses of company management

A wide variety of other possible specific factors

Because the size premium tends to reflect some factors of this type, the analyst should adjust further only for specific items that are truly unique to the subject company. Unfortunately, despite the widespread use by analysts and appraisers of a company-specific risk premium in a build-up (or CAPM) model, there is limited academic research on the topic, and the company-specific risk premium remains in the realm of the analyst’s judgment. We discuss the research in Chapter 14.

EXAMPLE OF THE BUILD-UP METHOD USING MORNINGSTAR DATA Now that we have discussed the factors in the build-up model, we can substitute some numbers into the method. We start with five assumptions about Shannon’s Bull Market (SBM), a closely held, regional steakhouse chain with excellent food and drink and noted for its friendly service. We first use the Morningstar data in Formula 7.2: EðRi Þ ¼ R f þ RPm þ RPs þ= RPi þ RPu

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Cost of Capital

1. Risk-free rate. We will use the 20-year U.S. government bond, for which the yield to maturity at the valuation date of December 31, 2005, was 4.6%. 2. Equity risk premium. We will use the results of the research on the expected equity risk premium discussed in Chapter 9 and use an RPm estimate of 5.0% for this example. 3. Size premium. The SBBI Valuation Edition 2006 Yearbook shows that the size premium for the tenth decile—smallest 10% of New York Stock Exchange (NYSE) stocks with American Stock Exchange (AMEX) and Nasdaq Stock Market (Nasdaq) stocks included—over and above the return estimated by CAPM is 6.36%.2 4. Industry adjustment factor. SBM is in the SIC code 58, Eating and Drinking Places. The Industry Risk premia for that industry, developed using the full-information beta with contributions to that beta from eighty companies, is 2.07%. 5. Company-specific risk premium. SBM is considerably smaller than the average of the smallest 10% of NYSE stocks, and our analyst perceives that the restaurant industry is riskier than the average for the companies included in the subject company industry adjustment factor. Although the assessment is somewhat subjective, our analyst recommends adding a company-specific risk factor of 3.0% because of risk factors identified as unique to this company. Substituting the preceding information in Formula 7.2 we have: (Formula 7.3) EðRi Þ ¼ 4:6% þ 5:0% þ 6:36% þ ð2:07%Þ þ 3:0% ¼ 16:89%; rounded to 17% The indicated cost of capital for SBM is approximately 17%. Some analysts prefer to present these calculations in tabular form, as shown.

Build-up Cost of Equity Capital for SBM Using Morningstar Data Risk-free rate Equity risk premium Size premium Industry risk premium Company-specific risk premium SBM indicated cost of equity capital

4.6% 5.0% 6.36% 2.07% 3.00% 17% (rounded)

EXAMPLE OF THE BUILD-UP METHOD USING DUFF & PHELPS SIZE STUDY DATA As an alternative to Formula 7.2 for the build-up method, EðRi Þ ¼ R f þ RPm þ RPs þ RPu , where a general risk premium is added for the ‘‘market’’ (equity risk premium) and a risk premium for small size to the risk-free rate, you can use the Size Study to develop a risk premium for the subject company that measures risk in terms of the total effect of market risk and size.

2

Morningstar recommends using the size premium (return in excess of CAPM) analysis for both the build-up and CAPM cost of equity estimates. These data can be seen in the SBBI Valuation Edition 2006 Yearbook, at 38–39. See Chapter 12 in this volume for more information.

Example of the Build-up Method Using Duff & Phelps Size Study Data

77

The formula then is modified to be: (Formula 7.4) EðRi Þ ¼ R f þ RPmþs þ RPu where: E(Ri) ¼ Expected (market required) rate of return on security i Rf ¼ Rate of return available on a risk-free security as of the valuation date RPm+s ¼ risk premium for the ‘‘market’’ plus risk premium for size RPu ¼ Risk premium attributable to the specific company or to the industry The Size Study sorts companies by eight size measures (see Chapter 12 for a list of the measures), breaking the NYSE universe of companies into 25 size-ranked categories or portfolios and adding AMEX- and Nasdaq-listed companies to each category based on their respective size measures. We use four assumptions: 1. Risk-free rate. We will use the 20-year U.S. government bond, for which the yield to maturity at the valuation date of December 31, 2005, was 4.6%. 2. Risk premium. The Size Study incorporates both the equity risk premium and the size premium into a single number. The Duff & Phelps Risk Premium Report Size Study indicates that the risk premium for the smallest companies are as shown in the table. We use only six of the eight size measures listed in the report because SBM is closely held. Size as Measured by Book Value of Common Equity 5-year Average Net Income Total Assets 5-year Average EBITDA Sales Number of Employees Median Risk Premium

Risk Premium 12.34% 13.10% 12.72% 12.93% 12.21% 12.57% 12.6% (rounded)

3. Industry adjustment factor. You might consider applying this adjustment as listed in the SBBI Yearbook since the Size Study contains no comparable data. SBM is in the SIC code 58, Eating and Drinking Places. The industry risk premia for that industry was developed using the full-information beta with contributions to that beta from eighty companies is 2.07%. For purposes of this example, we are including this risk adjustment and adding it as part of the company-specific risk premium. 4. Company-specific risk premium. SBM is considerably smaller than the average of the smallest 10% of NYSE stocks, and our analyst perceives that the subject company is riskier than the average for the companies included in the industry adjustment. Although the assessment is somewhat subjective, our analyst recommends adding a company-specific risk factor of 3.0% because of risk factors identified as unique to this company. Substituting the preceding information in Formula 7.5, we have:

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Cost of Capital

(Formula 7.5) EðRi Þ ¼ 4:6% þ 12:6 þ ð2:07%Þ þ 3:0% ¼ 18:13%; rounded to 18% The indicated cost of capital for SBM is approximately 18%. Some analysts prefer to present these calculations in tabular form, as shown. Build-up Cost of Equity Capital for SBM Using Duff & Phelps Size Study Data Risk-free rate Risk premium (ERP plus size premium) Industry risk premium Company-specific premium SBM indicated cost of equity capital

4.6% 12.6% 2.07% 3.0% 18% (rounded)

If we were using the Capital Asset Pricing Model (CAPM) (the subject of Chapter 8), a portion of the size premium and probably the entire industry portion of the specific risk premium would be captured in the ‘‘beta’’ factor, which is the difference between CAPM and the straight build-up method. Of course, if these build-up method figures were presented in a formal valuation report, each of the numbers in the calculation would be footnoted as to its source, and each would be supported by a narrative explanation.

SUMMARY The build-up model for estimating the cost of equity capital has five components: A risk-free rate A general equity risk premium A small company premium A company-specific risk adjustment (which can be either positive or negative, depending on the risk comparisons between the subject company and others from which the size premium was derived) 5. Possibly, an industry adjustment factor 1. 2. 3. 4.

These factors are summarized schematically in Exhibit 7.2. In a sense, the build-up method is a version of the Capital Asset Pricing Model without specifically incorporating systematic risk.

Exhibit 7.2

Summary of Development of Equity Discount Rate Using Build-up Method

Risk-free rate* + Equity risk premium + Small company premium +/ Specific risk

20-year, 5-year, or 30-day Treasury yield as of valuation date Expected equity risk premium corresponding to risk-free rate Small stock premium or premium over return expected by CAPM Specific risk difference in subject company relative to companies from which above data are drawn.

*The ‘‘risk-free’’ rate actually has one element of risk: maturity risk (sometimes called interest rate or horizon risk), the risk that the value of the bond will fluctuate with changes in the general level of interest rates.

Chapter 8

Capital Asset Pricing Model

Concept of Market or Systematic Risk Background of the Capital Asset Pricing Model Market or Systematic and Unique or Unsystematic Risks Using Beta to Estimate Expected Rate of Return Expanding CAPM to Incorporate Size Premium and Specific Risk Firm Size Phenomenon Company-Specific Risk Factor Expanded CAPM Cost of Capital Formula Examples of the CAPM Example of CAPM Method Using Morningstar Data Example of CAPM method using Duff & Phelps Size Study Data Assumptions Underlying the Capital Asset Pricing Model Summary

CONCEPT OF MARKET OR SYSTEMATIC RISK For more than 30 years, financial theorists generally have favored the notion that using the Capital Asset Pricing Model (CAPM) as the preferred method to estimate the cost of equity capital. In spite of many criticisms, it is still one of the most widely used models for estimating the cost of equity capital, especially for larger companies.* The primary difference between the CAPM and the build-up model presented in Chapter 7 is the introduction of market or systematic risk for a specific stock as a modifier to the general equity risk premium. Market risk is measured by a factor called beta. Beta measures the sensitivity of excess total returns (total returns over the risk-free rate of returns) on any individual security or portfolio of securities to the total excess returns on some measure of the market, such as the Standard & Poor’s (S & P) 500 Index or the New York Stock Exchange (NYSE) Composite Index. Chapter 10 discusses methods for estimating beta. Note at this point, however, that beta is measured by reference to total stock returns, which have two components: 1. Dividends 2. Change in market price Because closely held companies divisions and reporting units have no market price, their betas cannot be measured directly. Thus, to use the CAPM to estimate the cost of capital for a closely held

* Chapter 8 draws heavily on Shannon P. Pratt, Valuing a Business: The Analysis and Appraisal of Closely Held Companies, 5th ed. (New York: McGraw-Hill, 2008).

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Cost of Capital

company division or reporting unit, it is necessary to estimate a proxy beta for that business. This usually is accomplished by using an average or median beta for the industry group or by selecting specific guideline public companies and using some composite, such as the average or median, of their betas. CAPM is one of several mechanisms to estimate the cost of equity capital. All other things being equal, the cost of capital for any given company is the same whether you arrive at it by CAPM or by the build-up method. CAPM, however, generally requires public companies from which to develop betas. For some industries, especially those characterized by many small companies, public companies on which to base an estimate of beta simply do not exist.

BACKGROUND OF THE CAPITAL ASSET PRICING MODEL The Capital Asset Pricing Model is part of a larger body of economic theory known as capital market theory (CMT). CMT also includes security analysis and portfolio management theory, a normative theory that describes how investors should behave in selecting common stocks for their portfolios, under a given set of assumptions. In contrast, the CAPM is a positive theory, meaning it describes the market relationships that will result if investors behave in the manner prescribed by portfolio theory. The CAPM is a conceptual cornerstone of modern capital market theory. Its relevance to business valuations and capital budgeting is that businesses, business interests, and business investments are a subset of the investment opportunities available in the total capital market; thus, the determination of the prices of businesses theoretically should be subject to the same economic forces and relationships that determine the prices of other investment assets.

MARKET OR SYSTEMATIC AND UNIQUE OR UNSYSTEMATIC RISKS In Chapter 5 we defined risk conceptually as the degree of uncertainty regarding the realization of future economic income. Capital market theory divides risk into two components (other than maturity risk): market or systematic risk and unique or unsystematic risk. Stated in nontechnical terms, market risk or systematic risk (also known as undiversifiable risk) is the uncertainty of future returns owing to the sensitivity of the return on the subject investment to variability in the returns for a composite measure of marketable investments. Unique or unsystematic risk (also known as diversifiable risk, residual risk, or specific risk) is a function of the characteristics of the industry, the individual company, and the type of investment interest and is unrelated to variation of returns in the market as a whole. To the extent that the industry as a whole is sensitive to market movements, that portion of the industry’s risk would be captured in beta, the measure of market risk. Company-specific characteristics may include, for example, management’s ability to weather changing economic conditions, relations between labor and management, the possibility of strikes, the success or failure of a particular marketing program, or any other factor specific to the company. Total risk depends on both systematic and unsystematic factors. A fundamental assumption of the CAPM is that the risk premium portion of a security’s expected return is a function of that security’s market risk. That is because capital market theory assumes that investors hold, or have the ability to hold, common stocks in large, well-diversified portfolios. Under that assumption, investors will not require compensation (i.e., a higher return) for the unsystematic risk because they can easily diversify it away. Therefore, the only risk pertinent to a study of capital asset pricing theory is market risk. As one well-known corporate finance text puts it: ‘‘The crucial

Using Beta to Estimate Expected Rate of Return

81

distinction between diversifiable and nondiversifiable risks is the main idea underlying the capital asset pricing model.’’1

USING BETA TO ESTIMATE EXPECTED RATE OF RETURN The CAPM leads to the conclusion that a security’s equity risk premium (the required excess rate of return for a security over and above the risk-free rate) is a linear function of the security’s beta. This linear function is described in this univariate linear regression formula: (Formula 8.1) EðRi Þ ¼ R f þ BðRPm Þ where: E(Ri) ¼ Expected return (cost of capital) for an individual security Rf ¼ Rate of return available on a risk-free security (as of the valuation date) B ¼ Beta RPm ¼ Equity risk premium for the market as a whole (or, by definition, the equity risk premium for a security with a beta of 1.0) The preceding linear relationship is shown schematically in Exhibit 8.1, which presents the security market line (SML), a schematic portrayal of the expected return-beta relationship. According to CAPM theory, if the combination of an analyst’s expected rate of return on a given security and its risk, as measured by beta, places it below the security market line, such as security X in Exhibit 8.1, the analyst would consider that security (e.g., common stock) mispriced. It would be mispriced in the sense that the analyst’s expected return on that security is less than it would be if the security were correctly priced, assuming fully efficient capital markets. To put the security in equilibrium according to that analyst’s expectations, the price of the security must decline, allowing the rate of return to increase until it is just sufficient to compensate the investor for bearing the security’s risk. In theory, all common stocks in the market, in equilibrium, adjust in price until the consensus expected rate of return on each is sufficient to compensate investors for holding them. In that situation, the market risk/expected rate of return characteristics of all those securities will place them on the security market line. As Exhibit 8.1 shows, the beta for the market as a whole is 1.0. Therefore, from a numerical standpoint, beta has the following interpretations:

Beta > 1.0

Beta ¼ 1.0

1

When market rates of return move up or down, the rates of return for the subject tend to move in the same direction and with greater magnitude. For example, for a stock with no dividend, if the market is up 10%, the price of a stock with a beta of 1.2 would be expected to be up 12%. If the market is down 10%, the price of the same stock would be expected to be down 12%. Many hightech companies are good examples of stocks with high betas. Fluctuations in rates of return for the subject tend to equal fluctuations in rates of return for the market.

Richard A. Brealey, Stewart C. Myers, and Franklin Allen, Principles of Corporate Finance, 8th ed. (Boston: Irwin McGrawHill, 2006), 958.

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Cost of Capital

Beta < 1.0

Negative beta (rare)

When market rates of return move up or down, rates of return for the subject tend to move up or down, but to a lesser extent. For example, for a stock with no dividend, if the market is up 10%, the price of a stock with a beta of .8 would be expected to be up 8%. The classic example of a lowbeta stock would be a utility that has not diversified into riskier activities. Rates of return for the subject tend to move in the opposite direction from changes in rates of return for the market. Stocks with negative betas are rare. A few gold-mining companies have had negative betas. Another example would be an investment company whose investment policy was to take short positions. It would have a negative beta.

To illustrate, using Formula 8.1 as part of the process of estimating a company’s cost of equity capital, consider stocks of average size, publicly traded companies i, j, and k, with betas of 0.8, 1.0, and 1.2, respectively; a risk-free rate in the market (Rf) of 4.6% (0.046) at the valuation date; and a market equity risk premium (RPm) of 5% (0.05). For company i, which is less sensitive to market movements than the average company, we can substitute in Formula 8.1 in this way: (Formula 8.2) EðRi Þ ¼ 0:046 þ 0:8ð0:05Þ ¼ 0:046 þ 0:04 ¼ 0:086 Thus, the indicated cost of equity capital for company i is estimated to be 8.6% because it is less risky, in terms of market risk, than the average stock on the market.

Expected Rate of Return

E(Ri)

Security Market Line 0.166 0.15 E(Rm)

X

0.134 Rf

0.8

Exhibit 8.1

1.0

1.2 Beta

Security Market Line

Source: Shannon P. Pratt, Valuing a Business: The Analysis and Appraisal of Closely Held Companies, 5th ed. (New York: The McGraw-Hill Companies, Inc., 2008). Reprinted with permission. All rights reserved. E(Ri) = Expected return for the individual security E(Rm) = Expected return on the market Rf = Risk-free rate available as of the valuation date In a market in perfect equilibrium, all securities would fall on the security market line. The security X is mispriced, with a return less than it would be on the security market line.

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For company j, which has average sensitivity to market movements, we can substitute in (Formula 8.1) in this way: (Formula 8.3) EðRi Þ ¼ 0:046 þ 1:0ð0:05Þ ¼ 0:046 þ 0:05 ¼ 0:096 The indicated cost of equity capital for company j is estimated to be 9.6%, the estimated cost of capital for the average stock, because its market risk is equal to the average of the market as a whole. For company k, which has greater-than-average sensitivity to market movements, we can substitute in Formula 8.1 as shown: (Formula 8.4) EðRi Þ ¼ 0:046 þ 1:2ð0:05Þ ¼ 0:046 þ 0:06 ¼ 0:106 Thus, the indicated cost of equity capital for company k is estimated to be 10.6% because it is riskier, in terms of market risk, than the average stock on the market. Note that in the preceding pure formulation of the CAPM, the required rate of return for a given stock is composed of only three factors: 1. The risk-free rate 2. The market’s general equity risk premium of the subject security 3. The stock’s volatility to the market, the beta See Chapter 9 for a discussion of the market’s general equity risk premium.

EXPANDING CAPM TO INCORPORATE SIZE PREMIUM AND SPECIFIC RISK FIRM SIZE PHENOMENON Many empirical studies performed since CAPM was originally developed have indicated that realized total returns on smaller companies have been substantially greater over a long period of time than the original formulation of the CAPM (as given in Formula 8.1) would have predicted. Morningstar comments on this phenomenon: One of the most remarkable discoveries of modern finance is that of a relationship between firm size and return. The relationship cuts across the entire size spectrum but is most evident among smaller companies, which have higher returns on average than larger ones. . . . The firm size phenomenon is remarkable in several ways. First, the greater risk of small stocks does not, in the context of the capital asset pricing model (CAPM), fully account for their higher returns over the long term. In the CAPM, only systematic or beta risk is rewarded; small company stocks have had returns in excess of those implied by their betas. Second, the calendar annual return differences between small and large companies are serially correlated. This suggests that past annual returns may be of some value in predicting future annual returns. Such serial correlation, or autocorrelation, is practically unknown in the market for large stocks and in most other equity markets but is evident in the size premia.2 2

SBBI, Valuation Edition 2007 Yearbook (Chicago: Morningstar, 2007), 129, 134.

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There are currently two widely used sources of size premium data: SBBI Yearbook and the Duff & Phelps Size Study. The size effect and those sources are the subjects of Chapter 12. COMPANY-SPECIFIC RISK FACTOR The notion that the only component of risk that investors care about is market or systematic risk is based on the assumption that all unique or unsystematic risk can be eliminated by holding a perfectly diversified portfolio of risky assets that will, by definition, have a beta of 1.0. Without addressing the validity of that assumption for the public markets here, it is obviously not feasible for investors in closely held companies to hold such a perfectly diversified portfolio that would eliminate all unique risk. Therefore, for the cost of capital for closely held companies, even when using the CAPM, we have to consider whether there may be other risk elements that neither the beta factor (market risk factor) nor the size premium fully accounts for. If so, an adjustment to the discount rate for unique risk would be appropriate. Just as in the build-up model, the ‘‘specific risk’’ factor could be negative if the analyst concluded that the subject company was less risky than the average of the other companies from which the proxy estimates for the other elements of the cost of equity capital were drawn. For example, a company could have a well-protected, above-average price for its products as a result of a strong trademark, resulting in significantly less earnings volatility than experienced by its competitors.

EXPANDED CAPM COST OF CAPITAL FORMULA If we expand CAPM to also reflect the size effect and company specific risk, we can expand the cost of equity capital formula to add these two factors: (Formula 8.5) EðRi Þ ¼ R f þ BðRPm Þ þ RPs þ RPu where: E(Ri) ¼ Expected rate of return on security i Rf ¼ Rate of return available on a risk-free security as of the valuation date B ¼ Beta RPm ¼ General equity risk premium for the market RPs ¼ Risk premium for small size RPu ¼ Risk premium attributable to the specific company (u stands for unique or unsystematic risk)

EXAMPLES OF A CAPM MODEL The next examples use two sources of size premium data: Morningstar and the Duff & Phelps Study. EXAMPLE OF CAPM METHOD USING MORNINGSTAR DATA To put some numbers into Formula 8.5, we will make five assumptions about Unique Computer Systems (UCS), a fictional specialty manufacturer in the computer industry with publicly traded stock:

Examples of a CAPM Model

85

1. Risk-free rate. As of the valuation date, the yield to maturity on 20-year U.S. government bonds is 4.6%. 2. Beta. The UCS beta is 1.6. 3. Equity risk premium. We will use the results of the research on the expected equity risk premium discussed in Chapter 9 and use estimate of 5.0% for this example. 4. Size premium. The SBBI Valuation Edition 2006 Yearbook shows that the size premium for microcap stocks (the size premium for this size firm in excess of the risk captured in CAPM through beta) is 3.95%. (We will assume here that this is on the borderline between Morningstar’s ninth and tenth size deciles and use the micro-cap size premium.) 5. Company-specific risk factor. Because of special risk factors, the analyst has estimated that there should be an additional specific risk factor of 1.0%. Substituting this information in Formula 8.5, we have: (Formula 8.6) EðRi Þ ¼ 4:6 þ 1:6ð5:0Þ þ 3:95 þ 1:0 ¼ 4:6 þ 8:0 þ 3:95 þ 1:0 ¼ 17:55% Thus, the indicated cost of equity capital for UCS is estimated to be 18% (rounded). Some analysts prefer to present the preceding calculations in tabular form:

Risk-free Rate Equity risk premium: General equity risk premium Beta Size premium Company-specific risk premium UCS cost of equity capital

4.6% 5.01.6 8.0% 3.95 1.0 18% (rounded)

EXAMPLE OF A CAPM METHOD USING DUFF & PHELPS SIZE STUDY DATA The Size Study sorts companies by eight size measures, breaking the NYSE universe of companies into 25 size-ranked categories or portfolios and adding AMEX- and Nasdaq-listed companies to each category based upon their respective size measures. Again, using Formula 8.5, we assume: 1. Risk-free rate. As of the valuation date, the yield to maturity on 20-year U.S. government bonds is 4.6%. 2. Beta. The UCS beta is 1.6. 3. Equity risk premium. We will use the results of the research on the expected equity risk premium discussed in Chapter 9 and use an RPm estimate of 5.0% for this example. 4. Size premium. The Duff & Phelps Risk Premium Report Size Study (premium over CAPM) indicates that the size premia for UCS are:

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Cost of Capital

Size as Measured by

Size Premium

Market Value of Common Equity Book Value of Common Equity 5-year Average Net Income Market Value of Invested Capital Total Assets 5-year Average EBITDA Sales Number of Employees Median Size Premium

5.03% 4.34% 5.21% 4.17% 4.58% 4.75% 4.11% 5.63% 4.7% (rounded)

5. Company-specific risk factor. Because of special risk factors, the analyst has estimated that there should be an additional specific risk factor of 1.0%. Substituting this information in Formula 8.5, we have: (Formula 8.7) EðRi Þ ¼ 4:6 þ 1:6ð5:0Þ þ 4:7 þ 1:0 ¼ 4:6 þ 8:0 þ 4:7 þ 1:0 ¼ 18:30% Thus, the indicated cost of equity capital for UCS is estimated to be 18% (rounded). Some analysts prefer to present the preceding calculations in tabular form: Risk-free rate Equity risk premium: General equity risk premium Beta Small stock size premium Specific risk premium UCS cost of equity capital

4.6% 5.01.6 8.0% 4.7 1.0 18% (rounded)

Of course, if this information were presented in a formal valuation report, each of the numbers would be footnoted as to its source, and each would be supported by narrative explanation.

ASSUMPTIONS UNDERLYING THE CAPITAL ASSET PRICING MODEL Eight assumptions underlie the CAPM: Investors are risk averse. Rational investors seek to hold efficient portfolios (i.e., portfolios that are fully diversified). All investors have identical investment time horizons (i.e., expected holding periods). All investors have identical expectations about such variables as expected rates of return and how capitalization rates are generated. 5. There are no transaction costs. 1. 2. 3. 4.

Summary

87

6. There are no investment-related taxes. (However, there may be corporate income taxes.) 7. The rate received from lending money is the same as the cost of borrowing money. 8. The market has perfect divisibility and liquidity (i.e., investors can readily buy or sell any desired fractional interest). Obviously, the extent to which these assumptions are or are not met in the real world will have a bearing on the validity of the CAPM for the valuation of any company and particularly closely held businesses, business interests, or investment projects. For example, while the perfect divisibility and liquidity assumption approximates reality for publicly traded stocks, the same is not true for closely held companies. This is one reason why the company-specific, nonsystematic risk factor may be rewarded in expected returns for closely held companies, even if it is not for public companies. The CAPM, like most economic models, offers a theoretical framework for how certain relationships would exist subject to certain assumptions. Although not all assumptions are met in the real world, the CAPM provides a reasonable framework for estimation of the cost of capital. Other models are discussed in later chapters.

SUMMARY The Capital Asset Pricing Model expands on the build-up model by introducing the beta coefficient, an estimate of market risk or systematic risk, the sensitivity of returns for the subject company stock to returns for the market. The CAPM has several underlying assumptions, which may be met to a greater or lesser extent for the market as a whole or for any particular company or investment. While some question its usefulness given its underlying assumptions, CAPM is widely used today.3 But CAPM has been attacked because beta (discussed in Chapter 11) has not been found to be a very reliable measure of risk in practice and because its underlying assumptions may not in fact hold true in practice. Therefore, practitioners in all fields must understand its usefulness and its limitations. Exhibit 8.2 is a schematic summary of using the CAPM to estimate the cost of equity capital.

Exhibit 8.2

Capital Asset Pricing Model Method of Estimating Equity Discount Rate

Risk-free rate* + Equity risk premiumy + Size premium Specific risk *

y

3

20-year, 5-year, or 30-day Treasury yield as of valuation date Expected equity risk premium corresponding to risk-free rate. In CAPM, multiplied by beta. Premium over that predicted by beta Specific risk difference in subject company relative to companies from which above data are drawn

The ‘‘risk-free’’ rate actually has one element of risk: maturity risk (sometimes called interest risk or horizon risk)—the risk that the value of the bond will fluctuate with changes in the general level of interest rates. Short-term estimate matched to 30-day risk-free rate; mid-term estimate matched to 5-year risk-free rate; long-term estimate matched to 20-year risk-free rate. Such data available from Morningstar. The equity risk premium could also be estimated by other models, as discussed in Chapter 9, Equity Risk Premium.

John R. Graham, and Campbell R. Harvey, ‘‘The Theory and Practice of Corporate Finance,’’ Journal of Financial Economics (May 2001): 187–243, survey corporate executives and find that 75% of firms use the CAPM to estimate the cost of equity capital; 34% use CAPM with additional adjustment factors (such as premium for international operations), 39% use historical realized returns (combining market and unique risks), and 15% input cost of equity capital from a discounted cash flow model. Many firms use more than one method.

Chapter 9

Equity Risk Premium

Introduction Defining the Equity Risk Premium Estimating the ERP Nominal or Real? Which Risk-free Rate to Use in Estimating the ERP Matching Risk-free Rate with ERP Measuring the Average Period of the Expected Cash Flows Realized Risk Premium (ex Post) Approach Measuring Realized Risk Premiums Historical Stock and Bond Returns Summarizing Realized Risk Premium Data What Periodicity of Past Measurement? Selecting a Sample Period Is Bias Introduced by Using the Arithmetic Average in Estimating ERP? Comparing Investor Expectations to Realized Risk Premiums Changes in Economics that Caused Unexpectedly Large Realized Risk Premiums Forward-Looking (ex Ante) Approaches Bottom-up Approaches Top-down Approaches Surveys Other Sources of ERP Estimates Unconditional versus Conditional ERP Summary Appendix 9A

INTRODUCTION The equity risk premium (ERP) (often interchangeably referred to as the market risk premium) is defined as the extra return (over the expected yield on risk-free securities) that investors expect to receive from an investment in a diversified portfolio of common stocks.

The authors wish to acknowledge the contribution of David King, CFA, former colleague of Mr. Grabowski, to the discussion contained herein. This chapter is an update and expansion to prior work of these authors; see Roger Grabowski and David King, ‘‘Equity Risk Premium,’’ in The Handbook of Business Valuation and Intellectual Property Analysis (New York: McGraw-Hill, 2004), 3–29; ‘‘Equity Risk Premium: What Valuation Consultants Need to Know About Current Research—2005 Update,’’ Valuation Strategies (September/October 2005); and Roger Grabowski, ‘‘Equity Risk Premium: 2006 Update,’’ Business Valuation Review (Summer 2006). We also wish to thank David Turney and Aaron Reddington for the calculation assistance they provided.

89

90

Cost of Capital

Estimating the ERP is one of the most important decisions you must make in developing a discount rate. For example, the effect of a decision that the appropriate ERP is 4% instead of 8% in the Capital Asset Pricing Model (CAPM) will generally have a greater impact on the concluded discount rate than alternative theories of the proper measure of other components, for example, beta. One academic study looked at sources of error in estimating expected rates of return over time and concluded: We find that the great majority of the error in estimating the cost of capital is found in the risk premium estimate, and relatively small errors are due to the risk measure, or beta. This suggests that analysts should improve estimation procedures for market risk premiums, which are commonly based on historical averages.1

In ranking what matters and what does not matter in estimating the cost of equity capital, another author categorizes the choice of the ERP as a ‘‘high impact decision,’’ likely to make a difference of more than two percentage points and could make a difference of more than four points.2 Three driving forces behind the discussions that have evolved on ERP include: 1. What returns can be expected from investments by retirement plans in publicly traded common stocks by retirement plans? 2. What expected returns are being priced in the observed values of publicly traded common stocks? 3. What is the appropriate cost of capital to use in discounting future cash flows of a company or a project to their present value equivalent? Because of the importance of the ERP estimate and the fact that we find many practitioners confused about estimating ERP, we report on recent studies and report on ERP estimates at the beginning of 2007. We conclude with our recommended ERP.

DEFINING THE EQUITY RISK PREMIUM The ERP (or notational RPm) is defined as: where:

RPm ¼ Rm R f

RPm ¼ the equity risk premium Rm ¼ the expected return on a fully diversified portfolio of equity securities Rf ¼ the rate of return expected on a risk-free security What is referred to as the ERP means, in practice, a general equity risk premium using as a proxy for the ‘‘market’’ either the Standard & Poor’s (S&P) 500 or the New York Stock Exchange (NYSE) composite stock index. ERP is a forward-looking concept. It is an expectation as of the valuation date for which no market quotes are observable. In this chapter, we are addressing returns of publicly traded stocks. Those returns establish a beginning benchmark for closely held investments. 1

2

Wayne Ferson and Dennis Locke, ‘‘Estimating the Cost of Capital through Time: An Analysis of the Sources of Error,’’ Management Science (April 1998): 485–500. Seth Armitage, The Cost of Capital: Intermediate Theory (Cambridge: Cambridge University Press, 2005), 319–320.

Which Risk-free Rate to Use in Estimating the ERP

91

ESTIMATING THE ERP While you can observe premiums realized over time by referring to historical data (i.e., realized return approach or ex post approach), such realized premiums do not represent the ERP expected in prior periods, nor do they represent the current ERP. Rather, realized premiums may, at best, represent only a sample from prior periods of what may have been the expected ERP. To the extent that realized premiums on the average equate to expected premiums in prior periods, such samples may be representative of current expectations. But to the extent that events that are not expected to reoccur caused realized returns to differ from prior expectations, such samples should be adjusted to remove the effects of these nonrecurring events. Such adjustments are needed to improve the predictive power of the sample. Alternatively, you can directly derive implied forward-looking estimates for the ERP from data on the underlying expectations of growth in corporate earnings and dividends or from projections of specific analysts as to dividends and future stock prices (ex ante approach).3 The goal of either approach is to estimate the true expected ERP as of the valuation date. Even then the expected ERP can be thought of in terms of a normal or unconditional ERP and a conditional ERP based on current prospects.4 We address issues involving the conditional ERP later. There is no one universally accepted standard for estimating ERP. A wide variety of premiums are used in practice and recommended by academics and financial advisors. NOMINAL OR REAL? Both the expected return on a fully diversified portfolio of equity securities and the rate of return expected on a risk-free security can be stated in nominal (including expected inflation) or real terms (expected inflation removed). ERP should not be affected by inflation. If both returns are expressed in nominal terms, the difference in essence removes the expected inflation; if both returns are expressed in real terms, inflation has been removed, but the difference remains the same. But ex post realized returns will be affected by differences between expected inflation and realized inflation.

WHICH RISK-FREE RATE TO USE IN ESTIMATING THE ERP Any estimate of ERP must be made in relation to a risk-free security. That is, the expected return on a fully diversified portfolio of equity securities must be measured in its relationship to the rate of return expected on a risk-free security. The selection of an appropriate risk-free security with which to base the ERP estimate is a function of the expected holding period for the investment to which the discount rate (rate of return) is to apply. For example, if you were estimating the equity return on a highly liquid investment and the expected holding period were potentially short-term, a U.S. government short-term bond (e.g., Treasury or T-bill) may be an appropriate instrument to use in benchmarking the ERP estimate. Alternatively, if you were estimating the equity return on a long-term investment, such as the valuation of a business where the value can be equated to the present value of a series of future cash flows over many years, then the yield on a long-term U.S. government bond may be the more appropriate instrument in benchmarking the ERP estimate.

3 4

See, for example, Eugene F. Fama and Kenneth R. French, ‘‘The Equity Premium.’’ Journal of Finance (April 2002): 637–659. Robert Arnott, ‘‘Historical Results,’’ Equity Risk Premium Forum, AIMR (November 8, 2001): 27.

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Cost of Capital

Common academic practice in empirical studies of rates of return realized on portfolios of stocks in excess of a risk-free rate is to benchmark stock returns against realized monthly returns of ‘‘riskfree’’ 90-day T-bills or one-year government bonds. A T-bill rate is the purest risk-free base rate because it contains essentially no maturity risk. If inflation is high, it does reflect the inflation component, but it contains little compensation for inflation uncertainty. Problems in using such a risk-free security as a benchmark are that (1) T-bill rates may not reflect market-determined investor return requirements on long-term investments due to central bank actions affecting the short-term interest rates, and (2) rates on short-term securities tend to be more volatile than yields on longer maturities. Long-term government bonds are free of default risk but are not ‘‘risk-free.’’ Long-term government bonds are sensitive to future interest fluctuations. Investors are not sure of the purchasing power of the dollars they will receive upon maturity or the reinvestment rate that will be available to them to reinvest the interest payments received over the life of the bond. As a result, the long-term empirical evidence is that returns on long-term government bonds on the average exceed the returns on T-bills.5 The long-term premium of government bond returns in excess of the average expected interest rates on T-bills (average of future forward rates) is commonly referred to as the horizon premium. The horizon premium compensates the investor for the maturity risk of the bond. The horizon premium equals the added return expected on the average on long-term bonds due to inflation and interest rate risk. As interest rates change unexpectedly in the future, the bond price will vary. That is, bonds are subject to market risk due to unexpected changes in interest rates. The horizon premium compensates investors for that market risk. MATCHING RISK-FREE RATE WITH ERP In theory, when determining the risk-free rate and the matching ERP you should be matching the risk-free security and the ERP with the period in which the investment cash flows are expected. For example (where b is a risk measure for the investment): Short-term cash flows: Current T-bill rate þ b ðRPm over T-billsÞ Cash flows expected in: Year 1 : 1-year government bond rate þ b ðRPm over1-year bondsÞ Year 2 : 2-year forward rate on government bonds þ b ðRPm over 2-year bondsÞ Year 3 : 3-year forward rate on government bonds þ b ðRPmÞ over3-year bondsÞ; and so on Cash flows expected in the long-term : Current long-term government bond rate þ b ðRPm over long-term government bondsÞ MEASURING THE AVERAGE PERIOD OF THE EXPECTED CASH FLOWS Can one measure the ‘‘average’’ period of expected cash flows and use an average maturity period for the risk-free security and the ERP? One measure of the length of planning horizon over which cash flows are expected is the duration of cash flows. We introduced the concept of duration in Chapter 6 as a measure of the effective time period over which you receive cash flows from bonds. In a similar manner, you can calculate the expected duration of any stream of expected cash flows for any project. For valuation of a ‘‘going-concern’’ business, for example, assume you expect the cash flow in the first year following the valuation date of $1 million to increase at an average 5

When short-term interest rates exceed long-term rates, the yield curve is ‘‘inverted.’’

Realized Risk Premium (ex Post ) Approach

93

compound rate of 4% per annum. Assume a discount rate of 15%. If you project cash flows each year for 100 years, the calculated duration of the cash flows is approximately 10.5 years.6 In practice, few discount each cash flow using a matched maturity risk-free rate and ERP estimate. In valuing going-concern businesses and long-term investments made by businesses, practitioners generally use long-term government bonds as the risk-free security and estimate the ERP in relation to long-term government bonds. This convention both represents a realistic, simplifying assumption and is consistent with the CAPM.7 If the expected cash flows are risky and follow a random walk, but the risk-free rate and the ERP are expected to be constant over time, then the risk-adjusted discount rate for discounting the risky cash flows is constant as well. Most business investments have long durations and suffer from a reinvestment risk comparable to that of long-term government bonds. As such, the use of long-term government bonds and an ERP estimated relative to long-term bonds more closely matches the investment horizon and risks confronting business managers in capital budgeting decisions and valuators in valuation problems than reference to T-bills. Therefore, in the remainder of this chapter we have translated all estimates of ERP to estimates relative to long-term government bonds.

REALIZED RISK PREMIUM (ex POST ) APPROACH While academics and practitioners agree that ERP is a forward-looking concept, many practitioners use historical data only to estimate the ERP under the assumption that historical data are a valid proxy for current investor expectations. In the realized risk premium approach, the estimate of the ERP is the risk premium (realized return on stocks in excess of the risk-less rate) that investors have, on the average, realized over some historical holding period (realized risk premium). The underlying theory is that the past provides a reasonable indicator of how the market will behave in the future and investors’ expectations are influenced by the historical performance of the market. If period returns on stocks (e.g., monthly stock returns) are not correlated (e.g., this month’s stock returns are not predictable based on last month’s returns) and if expected stock returns are stable through time, then the arithmetic average of historical stock returns provides an unbiased estimate of expected future stock returns. Similarly, the arithmetic average of realized risk premiums provides an unbiased estimate of expected future risk premiums (the ERP). A more indirect justification for use of the realized risk premium approach is the contention that, for whatever reason, securities in the past have been priced in such a way as to earn the returns observed. By using an estimated cost of equity capital incorporating the average of realized risk premiums in applying the income approach to valuation, you may to some extent replicate this level of pricing.

MEASURING REALIZED RISK PREMIUMS The measure of the risk-free rate is not controversial once the proper duration (long term versus short term) of the investment has been estimated since the expected yield to maturity on appropriate

6

½ð1;000;000 1Þ=ð1:15Þ þ ð1;000;000 1:04 2Þ=ð1:15Þ2 þ ð1;000;000 1:042 3Þ=ð1:15Þ3 . . . ½ð1;000;000 1Þ=ð1:15Þ þ ð1;000;000 1:04Þ=ð1:15Þ2 þ ð1;000;000 1:042 Þ=ð1:15Þ3 . . .

7

¼ 10:5ðroundedÞ

Carmelo Giaccotto, ‘‘Discounting Mean Reverting Cash Flows with the Capital Asset Pricing Model,’’ The Financial Review (May 2007): 247–265. This is true for both the textbook CAPM of Sharpe and Linter and the extension of the textbook CAPM, the intertermporal CAPM of Merton.

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Cost of Capital

government securities is directly observable in the marketplace. Differences in approach to estimating the ERP then hinge on the measure of expected return on equity securities. In applying the realized risk premium approach, the analyst selects the number of years of historical return data to include in the average. One school of thought holds that the future is best estimated using a very long horizon of past returns. Another school of thought holds that the future is best measured by the (relatively) recent past. These differences in opinion result in disagreement as to the number of years to include in the average. HISTORICAL STOCK AND BOND RETURNS The highest-quality data are available for periods beginning in 1926 (the year that the forerunner of the current S&P 500 was first published) from the Center of Research in Security Prices (CRSP) at the University of Chicago. The SBBI Yearbook contains summaries of returns on United States stocks and bonds derived from that data.8 The reported returns include the effects from the reinvestment of dividends. Returns on common stocks have been assembled by various sources and with various qualities for earlier periods. Good stock market data are available back to 1872, and less reliable data are available from various sources back to the end of the eighteenth century. (In the earliest period, the market consisted almost entirely of bank stocks, and by the mid-nineteenth century, the market was dominated by railroad stocks.9) Data for government bond yield data have also been assembled for these periods. Exhibit 9.1 presents the realized average annual risk premium for stocks assembled from various sources for alternative periods through 2006. We measure the realized risk premium by comparing the stock market returns realized during the period to the income return on long-term government bonds (or yield to maturity for the years before 1926). While some may question looking at averages including early periods for estimating today’s ERP, what is striking is that the largest arithmetic average of one-year returns is the 81 years from 1926 to 2006. Why use the income return on long term government bonds? The income return in each period represented the expected yield on the bonds at the time of the investment. An investor makes a decision to invest in the stock market today by comparing the expected return from that investment to the rate of return today on a benchmark security (in this case the long-term government bond). While the investor did not know the stock market return when one invested at the beginning of each year, he or she did know the rate of interest promised on long-term government bonds. To try to match the expectations at the beginning of each year, we measure historical stock market returns on an expectation that history will repeat itself over the expected return on bonds in each year. 8 9

Stocks, Bonds, Bills and Inflation (SBBI) Valuation Edition 2007 Yearbook (Chicago: Morningstar, 2007). See Lawrence Fisher and James Lorie, ‘‘Rates of Return on Investments in Common Stocks,’’ Journal of Business 37, no. 1 (1964); J. W. Wilson and C. P. Jones, ‘‘A Comparison of Annual Stock Market Returns: 1871–1925 with 1926–1985,’’ Journal of Business 60, no. 2 (1987): 239–258; G. W. Schwert, ‘‘Indexes of Common Stock Returns from 1802 to 1987,’’ Journal of Business 63, no. 3 (1990): 399–425; Roger G. Ibbotson and Gary P. Brinson, Global Investing: The Professional’s Guide to the World Capital Markets (New York: McGraw-Hill, 1993); J. W. Wilson and C. P. Jones, ‘‘An Analysis of the S&P 500 Index and Cowles’s Extensions: Price Indexes and Stock Returns, 1870–1999,’’ Journal of Business 75, no. 3 (2002): 505–533; S. H. Wright, ‘‘Measures of Stock Market Value and Returns for the US Nonfinancial Corporate Sector, 1900–2000,’’ Working paper, February 1, 2002; W. Goetzmann, R. Ibbotson, and L. Peng, ‘‘A New Historical Database for NYSE 1915 to 1925: Performance and Predictability,’’ Journal of Financial Markets 4 (2001): 1–32; E. Dimson, P. Marsh, and M. Staunton, Triumph of the Optimists: 101 Years of Global Investment Returns (Princeton, NJ: Princeton University Press, 2002) with annual updates of their Global Returns database available at http://corporate.morningstar.com/ib; W. Goetzmann and R. Ibbotson, ‘‘History and the Equity Risk Premium,’’ Yale ICF Working Paper No. 05-04, April 6, 2005.

Realized Risk Premium (ex Post ) Approach Exhibit 9.1

95

Historical Realized Premiums: Stock Market Returns Treasury Bonds

Period 20 years (1987–2006) 30 years (1977–2006) 40 years (1967–2006) 50 years (1957–2006) 81 years (1926–2006)** 107 years (1900–2006) 135 years (1872–2006) 209 years (1798–2006)

Arithmetic Average

Standard Error*

Geometric Average

6.4% 5.8% 4.8% 5.2% 7.1% 6.8% 5.9% 5.1%

3.7% 2.8% 2.6% 2.3% 2.2% 1.9% 1.6% 1.2%

5.2% 4.7% 3.6% 3.9% 5.2% 4.9% 4.3% 3.6%

*

Calculated as standard deviation of realized excess returns divided by square root N, number of years in sample. SBBI Valuation Edition 2007 Yearbook. Source: Data compiled from R. Ibbotson and G. Brinson, Global Investing (New York: McGraw-Hill, 1993); W. Schwert, ‘‘Indexes of U.S. Stock Prices from 1802 to 1987,’’ Journal of Business, 1990; S. Homer and R. Sylla, A History of Interest Rates, 3rd ed. (Piscataway, NJ: Rutgers University Press, 1991); and SBBI, 2007 Yearbook (Chicago: Morningstar, 2007). **

The realized risk premiums vary year to year, and the estimate of the true ERP resulting from this sampling is subject to a degree of error. We display the standard errors of estimate for each period in Exhibit 9.1. The standard error of estimate allows you to measure the likely accuracy of using the realized risk premium as the estimate of ERP. That statistic indicates the estimated range within which the true ERP falls (i.e., assuming normality, the true ERP can be expected to fall within two standard errors with a 95% level of confidence).

SUMMARIZING REALIZED RISK PREMIUM DATA The summarized data in Exhibit 9.1 represent the arithmetic and geometric averages of realized risk premiums for one-year returns. That is, the dollars invested including reinvested dividends are reallocated to available investments annually and the return is calculated for each year. The arithmetic average is the mean of the annual returns. The geometric average is the single compound return that equates the initial investment with the ending investment assuming annual reallocation of investment dollars and reinvestment of dividends. For example, assume this series of stock prices (assuming no dividends): Period 1 2 3

Stock price

Period Return

$10 $20 $10

100% 50%

The arithmetic average of period returns equals (100% þ 50%)/2 ¼ 25% while the geometric average equals (1 þ r1)(1 þ r2)1/2 1 ¼ (1 ¼ 1.00 1 .5)1/2 1 ¼ 0. Realized return premiums measured using the geometric (compound) averages are always less than those using the arithmetic average. The geometric mean is the lower boundary of the arithmetic mean, and the two are equal in the unique situation that every observation is identical to every other observation. Further, the more variable the period returns, the greater the difference between the arithmetic and geometric averages of those returns. This is simply the result of the mathematics of a series that has experienced deviations.

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The choice between which average to use is a matter of disagreement among practitioners. The arithmetic average receives the most support in the literature,10 though other authors recommend a geometric average.11 The use of the arithmetic average relies on the assumption that (1) market returns are serially independent (not correlated) and (2) the distribution of market returns is stable (not time-varying). Under these assumptions, an arithmetic average gives an unbiased estimate of expected future returns assuming expected conditions in the future are similar to conditions during the observation period. Moreover, the more observations available, the more accurate will be the estimate. . . .the arithmetic mean equates the expected future value of investment with its present value . This property makes the arithmetic mean the correct return to use as the discount rate or cost of capital.12 . . .the geometric mean measures changes in wealth over more than one period on a buy and hold (with dividends reinvested) strategy. . . . The arithmetic mean would provide a better measure of typical performance over a single historical period.13

WHAT PERIODICITY OF PAST MEASUREMENT? But even if we agree that stock returns are serially independent, the arithmetic average of realized risk premiums based on one-year returns may not be the best estimate of future returns. Textbook models of stock returns (e.g., CAPM) are generally single-period models that estimate returns over unspecified investment horizons. For example, assume that the investment horizon equals two years. Then in using realized returns to estimate expected returns, you need to calculate realized returns over two-year periods (i.e., the geometric average over consecutive two-year periods) and then calculate the arithmetic average of the two-year geometric averages to arrive at the unbiased estimate of future returns. For example, assume that the realized one-year returns are: Year 1 ¼ 10% Year 2 ¼ 25% Year 3 ¼ 15% The geometric averages of the two-year holding periods are: ð1:10 1:25Þ1=2 1 ¼ 17:3% ð1:25 0:85Þ1=2 1 ¼ 3:1% The arithmetic average of typical two-year periods is therefore: ð17:3 þ 3:1Þ ¼ 10:2% 2

10

11

12 13

See, e.g., Paul Kaplan, ‘‘Why the Expected Rate of Return Is an Arithmetic Mean,’’ Business Valuation Review (September 1995); SBBI Valuation Edition 2002 Yearbook, 71–73; Mark Kritzman, ‘‘What Practitioners Need to Know about Future Value,’’ Financial Analysts Journal (May/June 1994): 12–15; Zvi Bodie, Alex Kane, and Alan J. Marcus, Investments (1989): 720–723. See, for example, Aswath Damodaran, Investment Valuation:Tools and Techniques for Determining the Value of Any Asset, 2nd ed. (Hoboken, N.J.: John Wiley & Sons, 2002), 161–162. Roger Ibbotson and Rex Sinquefeld, Stocks, Bonds, Bills and Inflation: Historical Returns (1926–1987) (1989), 127. Willard T. Carleton and Josef Lakonishok ‘‘Risk and Returns on Equity: the Use and Misuse of Historical Estimates,’’ Financial Analysts Journal 41, no. 1 (1985): 39.


97

The issue then becomes what is the appropriate interval over which average realized returns should be measured (1-year periods as in the case of the returns reported in the SBBI Yearbook; 2-year periods; 20-year periods)? When you are valuing businesses, should you compare returns over periods greater than one year? The most likely answer is yes. Practitioners have adopted the use of interest rates on long-term government bonds, typically 20-year bonds, as the appropriate long-term benchmark risk-free rate when valuing businesses. It follows then that a longer investment horizon of, say, 20 years is the appropriate period over which you should calculate realized returns. As the investment horizon increases, the arithmetic average of realized investment returns decreases asymptotically to the geometric average of the entire series. While Morningstar only reports on the arithmetic average of one-year returns, we calculated the realized risk premiums for various investment horizons using the data from 1926 to 2006 as shown in the next table.14 Arithmetic Average of 1

1-year returns 2-year returns2 3-year returns3 4-year returns4 5-year returns5 81-year returns (geometric average)1

Realized Risk Premium 7.1% 6.1% 5.8% 5.5% 5.3% 5.2%

1

SBBI Valuation Edition 2007 Yearbook. Excluding investment period beginning 2006. 3 Excluding investment periods beginning 2005 and 2006. 4 Excluding investment periods beginning 2004, 2005, and 2006. 5 Excluding investment periods beginning 2003, 2004, 2005, and 2006. Source: Compiled from data in Stocks, Bonds, Bills, and Inflation 2007 Yearbook. Copyright # 2007 Morningstar, Inc. All rights reserved. Used with permission. 2

Assuming that you have an investment horizon longer than one year, you can conclude that the realized risk premium that provides the ‘‘best estimate’’ of the ERP is likely between the arithmetic average of one-year returns and the geometric average of the entire series. In one recent study, the authors show that compounding the arithmetic average of historical oneyear returns as a forecaster of cumulative future returns results in estimates of cumulative returns that overstate the future cumulative returns that investors are likely to realize. This is due to the fact that distributions of stock market returns are skewed. The authors show that use of the geometric mean of historical one-year returns result in estimates of cumulative returns that more approximate the median of cumulative returns (50% if investors will realize more than the median cumulative return and 50% will realize less than the median return). They demonstrate that the difference between the median of forecasted cumulative returns obtained from compounding the arithmetic average versus the geometric average of one-year historical returns increases as the expected investment horizon increases.15 14

15

The equity risk premium of each investment horizon was calculated by taking equity returns (S&P 500) less the bond returns (U.S. Long-term Government Bond Income Return) for the respective periods. We calculated a series of rolling returns, one for stocks and another for bonds, for each investment horizon. We then took the arithmetic average of each series of rolling returns for the respective investment horizon. For example, the 2-year return, for equities and bonds, is the arithmetic average of a series of 2-year rolling returns from 1926 to 2006. We performed the same calculation for each investment horizon. We then subtract the bond return from the equity return to estimate the equity risk premium for each investment horizon. Eric Hughson, Michael Stutzer, and Chris Yung, ‘‘The Misuse of Expected Returns,’’ Financial Analysts Journal (November/ December 2006): 88–96.

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Cost of Capital

SELECTING A SAMPLE PERIOD The average realized risk premium is sensitive to the period chosen for the average. While the selection of 1926 as a starting point corresponds to the initial publishing of the forerunner to the current S&P 500, that date is arbitrary. Regarding the historical time period over which equity risk should be calculated, Morningstar offers two observations16: 1. Reasons to focus on recent history: The recent past may be most relevant to an investor. Return patterns may change over time. The longer period includes ‘‘major events’’ (e.g., World War II, the Depression) that have not repeated for over 50 years. 2. Reasons to focus on long-term history:

Long-term historical returns have shown surprising stability. Short-term observations may lead to illogical forecasts. Focusing on the recent past ignores dramatic historical events and their impact on market returns. We do not know what major events lie ahead. Law of large numbers: More observations lead to a more accurate estimate.

But the average calculated using 1926 return data as a beginning point may be too heavily influenced by the unusually low interest rates during the 1930s to mid-1950s. For example, the average yield on long-term government bonds was only 2.3% during the 1940s (the lowest decade on record) and under 3% in each year from 1934 through 1955. Yields on government bonds exceeded 4% for most of the nineteenth century and have been consistently higher since the 1960s. The years 1942 through 1951 were a period of artificial stability in U.S. government bond interest rates. In April 1942, the Federal Reserve publicly committed itself to maintaining an interest rate ceiling on government debt, both long term and short term, to support the financing of World War II. After World War II, the Fed continued maintaining an interest rate ceiling fearing return to the high unemployment of the Great Depression. But postwar inflationary pressures caused the Treasury and the Fed to reach an accord announced March 4, 1951, freeing the Fed of its obligation of pegging interest rates. Including this period in calculating realized returns is analogous to valuing airline stocks today by looking at prices of airline stocks before deregulation. Some observers have suggested that the period, which includes the 1930s, 1940s, and the immediate post–World War II boom years, may have exhibited an unusually high average realized return premium. The 1930s exhibited extreme volatility while the 1940s and early 1950s saw a combination of record low interest rates and rapid economic growth that led the stock market to outperform Treasury bonds by a wide margin. The low real rates on bonds may have contributed to higher equity returns in the immediate postwar period. Since firms finance a large part of their capital investment with bonds, the real cost of obtaining such funds increased returns to shareholders. It may not be a coincidence that the highest 30-year average equity return occurred in a period marked by very low real returns on bonds. As real returns on fixedincome assets have risen in the last decade, the equity premium appears to be returning to the 2% to 3% norm that existed before the postwar surge.17

16 17

SBBI Valuation Edition 2007 Yearbook (Chicago: Morningstar, 2007), 129, 134. Jeremy Siegel, Stocks for the Long Run (New York: McGraw-Hill, 1994), 20.

Realized Risk Premium (ex Post ) Approach Exhibit 9.2

99

Realized Equity Risk Premiums over Treasury Bond Income Returns

Nominal (i.e., without inflation removed) Arithmetic Average Geometric Average Standard Deviations Stock Market Annual Returns Long-term Treasury Income Returns Long-term Treasury Total Returns Ratio of Equity to Bond Total Return Volatility

1926–1955 10.5% 7.5%

1956–2006 5.1% 3.9%

25.3% 0.5% 4.7% 5.4

16.5% 2.4% 10.9% 1.5

Source: Compiled from data in Stocks, Bonds, Bills and Inflation 2007 Yearbook. Copyright # 2007 Morningstar, Inc. All rights reserved. Used with permission. For more information on other Morningstar publications, please visit global.morningstar.com/ DataPublications. Calculated (or Derived) based on CRSP1 data, #2006 Center for Research in Security Prices (CRSP1), Graduate School of Business, The University of Chicago.

If we disaggregate the 81 years reported in the SBBI Yearbook into two subperiods, the first covering the periods before and after the mid-1950s, we get the comparative figures for stock and bond returns shown in Exhibit 9.2. The period since the mid-1950s has been characterized by a more stable stock market and a more volatile bond market compared to the earlier period. Interest rates, as reflected in Long-term Government Bond Income Return statistics as summarized in the SBBI Yearbook, have become more volatile in the later period. The effect is amplified in the volatility of Long-term Government Bond Total Returns as summarized in the SBBI Yearbook, which include the capital gains and losses associated with interest rate fluctuations. From these data, we can conclude that the relative risk of stocks versus bonds has narrowed; based on this reduced relative risk, we would conclude that the ERP is likely lower today. As a result, we question the validity of using the arithmetic average of one-year returns since 1926 as the basis for estimating today’s ERP. Evidence since 1871 clearly supports the premise that the difference between stock yields and bond yields is a function of the long-run difference in volatility between these two securities.18 And if you examine the volatility in stock returns (as measured by rolling 10-year average standard deviation of real stock returns), you find that the volatility beginning in 1929 dramatically increased and that the volatility since the mid-1950s has returned to prior levels.19 This also suggests that the arithmetic average realized risk premiums reported for the entire data series since 1926 as reported in the SBBI Yearbook likely overstate expected returns. Using historical data may also tend to overstate expected returns given the increasing opportunities for international diversification. International diversification lowers the volatility of investors’ portfolios, which in theory should lower the required return on the average asset in the portfolio. This would lower the expected return on U.S. government securities generally and hence would suggest a lower ERP on a forward-looking basis than indicated by historical data. Several authors have studied the influence of increased globalization, and their results suggest that costs of capital for companies operating in the international markets have decreased.20

18

19

20

Clifford S. Asness, ‘‘Stocks versus Bonds: Explaining the Equity Risk Premium,’’ Financial Analysts Journal (March/April 2000): 96–113. Laurence Booth, ‘‘Estimating the Equity Risk Premium and Equity Costs: New Ways of Looking at Old Data,’’ Journal of Applied Corporate Finance (Spring 1999):100–112 and ‘‘The Capital Asset Pricing Model þ Equity Risk Premiums and the Privately-Held Business,’’ 1998 CICBV/ASA Joint Business Valuation Conference (September 1998): 23. See, e.g., Kate Phylaktis and Lichuan Xia, ‘‘Sources of Firm’s Industry and Country Effects in Emerging Markets,’’ Journal of International Money and Finance (2005): 459–475; and Gikas Hardouvelis, Dimitrious Malliartopulos, and Richard Priestly, ‘‘The Impact of Globalization on the Equity Cost of Capital,’’ Working paper, May 9, 2004.

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Cost of Capital

If the average expected risk premium has changed through time, then averages of realized risk premiums using the longest available data become questionable. A shorter-run horizon may give a better estimate if changes in economic conditions have created a different expected return environment than that of more remote past periods. Why not use the average realized return over the past 20year period? A drawback of using averages over shorter periods is that they are susceptible to large errors in estimating the true ERP due to high volatility of annual stock returns. Also, the average of the realized premiums over the past 20 years may be biased high due to the general downward movement of interest rates since 1981. While we can only observe historical realized returns in the stock market, we can observe both expected returns (yield to maturity) and realized returns in the bond market. Prior to the mid-1950s, the difference between the yield at issue and the realized returns was small since bond yields and therefore bond prices did not fluctuate very much. Beginning in the mid-1950s until 1981, bond yields trended upward, causing bond prices to generally decrease. Realized bond returns were generally lower than returns expected when the bonds were issued (as the holder experienced a capital loss if sold before maturity). Beginning in 1981, bond yields trended downward, causing bond prices to generally increase. Realized bond returns were generally higher than returns expected when the bonds were issued (as the holder experienced a capital gain if sold before maturity). If we choose the period during which to measure realized premiums beginning from the late 1950s/early 1960s to today, we will be including a complete interest rate cycle.21 Even if we use long-term observations, the volatility of annual stock returns will be high. Assuming that the 81-year average gives an unbiased estimate, still a 95% confidence interval for the unobserved true ERP spans a range of approximately 3.0% to 11.5%.22 IS BIAS INTRODUCED BY USING THE ARITHMETIC AVERAGE IN ESTIMATING ERP? The issue of bias is important from two different vantage points when using an ERP estimate derived from the arithmetic average of realized risk premium data: 1. In predicting the compound return you might expect for an investment in stocks, will you get an answer that is biased? (i.e., will measurement error be introduced simply due to the mathematics?) 2. In discounting expected cash flows where you develop a cost of equity capital estimate using that ERP estimate, will you get an answer that is biased? Even if you accept the arithmetic average of annual realized risk premiums as an unbiased estimate of expected annual risk premium (i.e., investment horizon equals one year), it is a somewhat stronger assumption to compound this annual average over multiple periods (i.e., investment horizon equals n years); you are assuming that the estimate of the expected single-period return is accurate (in other words, that the estimate has no allowance for error). If you introduce measurement error and compound the estimated annual return over multiple periods, you will get a biased estimate of the true expected future value. This upward bias occurs even if the single-period arithmetic average itself is an unbiased estimate. The fact that you get an expected upward bias in future investment results if you project future returns using an arithmetic average is important if you are estimating the returns 21 22

Booth, ‘‘Estimating the Equity Risk Premium and Equity Costs.’’ Calculated as two standard errors around the average; 7.1% Aþ/ (2 2.2%).


101

you might expect to realize when investing funds for future retirement. This is the subject of much discussion in the pension investment literature. The statistical properties of this problem are such that you get a different answer if, instead of focusing on unbiased expected future values, you seek instead an unbiased estimate of the present value discount factor. One proposed correction that focuses on present value factors finds that the adjustment from the arithmetic average is small even when discounting over fairly long periods.23 Moreover, the bias is toward discount rates that are too high rather than too low. Most of the value in a discounted cash flow analysis typically is derived from cash flows over the first 10 years, which limits potential bias in an overall present value calculation.24 Is bias introduced by using the arithmetic average in compounding expected cash flows? If we reverse the discounting process and compound the expected cash flows, do we get an unbiased estimate of an investor’s expected cumulative wealth? If the expected returns are not correlated, then the arithmetic average of realized risk premiums is an unbiased estimator of the mathematical expected return per period. If we compound those returns will the result equal the amount a typical investor would expect (i.e., the median—50% of the time the investor’s wealth will be less than the amount and 50% of the time the investor’s wealth will be greater than the amount)? Compounding rates of returns estimated using the arithmetic average of realized risk premiums will in fact result in an expected cumulative wealth that exceeds the median. The result is biased high and the difference grows larger as the investment horizon increases.25 A number of academic studies have suggested that U.S. stock returns are not serially independent but rather have exhibited negative serial correlation.26 One recent study suggests that if stock returns have negative serial correlation, then the best estimate of expected returns would lie somewhere between the arithmetic and geometric averages, moving closer to the geometric average as the degree of negative correlation increases and the projection period lengthens.27 But another study has shown that if the rates of return are not independent but display even a small amount of negative serial correlation, then the degree of bias in cumulative wealth is reduced substantially. This also is the case if the rates of return are independent but the risky expected cash flows are mean reverting. The result is that the cumulative wealth will be slightly greater than that expected by the typical investor (i.e., the median) even over a long investment horizon.28 All in all, using the arithmetic average of realized risk premiums as an estimate of ERP does not appear to introduce serious bias in estimating expected cumulative wealth. COMPARING INVESTOR EXPECTATIONS TO REALIZED RISK PREMIUMS Much has recently been written comparing the realized returns as reported in sources such as the SBBI Yearbook and the ERP that must have been expected by investors, given the underlying economics of publicly traded companies (e.g., expected growth in earnings or expected growth in 23

24 25

26

27

28

Ian Cooper, ‘‘Arithmetic versus Geometric Mean Estimators: Setting Discount Rates for Capital Budgeting,’’ European Financial Management (July 2001): 157–167. See Appendix 9A for a more complete explanation of the bias issue. Eric Hughson, Michael Stutzer, and Chris Yung, ‘‘The Misuse of Expected Returns,’’ Financial Analyst Journal (November/ December 2006): 88–96. Eugene F. Fama and Kenneth R. French, ‘‘Dividend Yields and Expected Stock Returns,’’ Journal of Financial Economics (October 1988): 3–25; Andrew Lo and Craig McKinlay, ‘‘Stock Market Prices Do Not Follow Random Walks,’’ Review of Financial Studies 1 (1988): 41–46; James Poterba and Lawrence Summers, ‘‘Mean Reversion in Stock Prices: Evidence and Implications,’’ Journal of Financial Economics (October 1988): 27–59. Daniel C. Indro and Wayne Y. Lee, ‘‘Biases in Arithmetic and Geometric Averages as Estimates of Long-Run Expected Returns and Risk Premia,’’ Financial Management (Winter 1997): 81–90. Giaccotto, The Financial Review (2007).

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Cost of Capital

dividends) and the underlying economics of the economy (e.g., expected growth in gross domestic product [GDP]). Such studies conclude that investors could not have expected as large an ERP as the equity risk premiums actually realized. A sampling of those studies follows.

Robert Arnott and Peter Bernstein conclude that the long-run normal ERP is approximately 4.5% on an arithmetic average basis (for the period studied, 1926 to 2001).29 They believe that the historical realized premium exceeded the expected premium because (1) the expected ERP in 1926 was above the long-term average, making 1926 a better-than-average starting point for the realized returns, and (2) important nonrecurring developments occurred that were not anticipated by investors (such as rising valuation multiples, survivor bias of the U.S. economy, and regulatory reform).30 Eugene Fama and Kenneth French examine the unconditional expected stock returns from fundamentals, estimated as the sum of the average dividend yield and the average growth rate of dividends or earnings derived from studying historical observed relationships for 1872 to 2000. They conclude that investors (during the period they studied, 1951 to 2000) should have expected an ERP lower than the actual realized risk premium. Their calculations indicate expected ERP of 2.6% (based on dividend growth rate fundamentals) or 3.6% (based on earnings growth rate fundamentals).31 Fama and French believe that the greater premium actually realized during those years was due to an unanticipated decline in the discount rate: [T]he bias-adjusted expected return estimates for 1951 to 2000 from fundamentals are a lot lower (more than 2.6% per year) than bias-adjusted estimates from realized returns. Based on this and other evidence, our message is that the unconditional expected equity premium of the last 50 years is probably far below the realized premium.32

Elroy Dimson, Paul Marsh, and Mike Staunton studied the realized equity returns and equity premiums for 17 countries (including the United States) from 1900 to the end of 2006.33 These authors report that the realized risk premiums have been 6.6% on an arithmetic basis (4.6% on a geometric basis) for the United States (in excess of the total return on government bonds). Dimson, Marsh, and Staunton observe larger equity returns earned in the second half of the twentieth century compared to the first half due to (1) corporate cash flows growing faster than

29

30 31

32 33

Robert D. Arnott and Peter L. Bernstein, ‘‘What Risk Premium is Normal?’’ Financial Analysts Journal (March/April 2002): 64–85. Arnott and Bernstein estimate that a ‘‘normal’’ equity risk premium equals 2.4% (geometric average). One method of converting to the geometric average from an arithmetic average is to assume the returns are independently log-normally distributed over time. Then the arithmetic and geometric averages approximately follow the relationship: Arithmetic average of returns for the period ¼ Geometric average of returns for the period plus (variance of returns for the period/2). In this case we get: 2.4% þ (.0412/2) ¼ 4.5% approximately. During the period 1926 to 2001, the arithmetic average realized premium (relative to Treasury bonds) was 7.4%. The difference is therefore 7.4% minus 4.5%, or approximately 3%. Ibid. Eugene F. Fama and Kenneth R. French, ‘‘The Equity Premium,’’ Journal of Finance (April 2002): 637–659. Fama and French estimate that the expected ERP using dividend growth rates was approximately 3.83% (after correcting for bias in the observed data) and using earnings growth rates was approximately 4.78% (after correcting for bias in the observed data) (arithmetic averages compared to six-month commercial paper rates). Subtracting a difference between the return on government bonds versus bills of 1.19% for the period of the study gives indicated premiums over long-term government bonds of approximately 2.6% and 3.6% (arithmetic average). Ibid, 658. Elroy Dimson, Paul Marsh, and Mike Staunton, ‘‘Global Evidence on the Equity Premium,’’ Journal of Applied Corporate Finance (Summer 2003): 27–38; ‘‘The Worldwide Equity Premium: A Smaller Puzzle’’ EFA 2006 Zurich Meetings Paper, April 7, 2006; Global Investment Returns Yearbook 2007 (London: ABN-AMRO/London Business School, February 2007).


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investors anticipated (fueled by rapid technological change and unprecedented growth in productivity and efficiency), (2) transaction and monitoring costs falling over the course of the century, (3) inflation rates generally declining over the final two decades of the century and the resulting increase in real interest rates, and (4) required rates of return on equity declining due to diminished business and investment risks. They conclude that the observed increase in the overall price-to-dividend ratio during the century is attributable to the long-term decrease in the required risk premium and that the decrease will most likely not continue into the future. They also conclude that a downward adjustment in the ERP compared to the realized risk premiums due to the increase in price/dividend ratio is reasonable. Removing the historical increase in the price/dividend ratio results in an adjusted realized risk premium (relative to bonds) of approximately 5.9% on an arithmetic basis (3.9% on a geometric basis) for the United States.34 This is before converting the adjusted realized risk premium to their estimate of a forward ERP (by adjusting the historical average dividend yield to today’s dividend yield).

Roger Ibbotson and Peng Chen report on a study in which they estimated forward-looking longterm sustainable equity returns and expected ERPs since 1926. They first analyzed historical equity returns by decomposing returns into factors including inflation, earnings, dividends, priceto-earnings ratio, dividend-payout ratio, book values, return on equity, and GDP per capita (the fundamental building blocks of ‘‘supply side’’ equity returns). They forecast the ERP through supply side models built from historical data. These authors determine that the long-term ERP that could have been expected given the underlying economics was less than the realized premium.35 In the most recent update to this study reported in the SBBI Yearbook, the long-term ERP since 1926 that could have been expected given the underlying economics (the supply side model estimate) was approximately 6.3% on an arithmetic basis (4.3% on a geometric basis) compared to the realized risk premium of 7.1% on an arithmetic basis (5.0% on a geometric basis). The greater-than-expected realized risk premiums were caused by an unexpected increase in market multiples relative to economic fundamentals (i.e., decline in the discount rates) for the market as a whole. This resulted in an extra return of 0.63% per annum (due to the price to earnings multiple in 1926 of 10.2 increasing to a price to earnings multiple of 17.4 in 2006). They do not anticipate that market multiples will continue to increase in future periods.36

William Goetzmann and Roger Ibbotson, commenting on the supply side approach of estimating expected risk premiums, note: These forecasts tend to give somewhat lower forecasts than historical risk premiums, primarily because part of the total returns of the stock market have come from price-earnings ratio expansion. This expansion is not predicted to continue indefinitely, and should logically be removed from the expected risk premium.’’37

34

35

36 37

Based on Grabowski’s converting premium over total returns on bonds as reported by Dimson, Marsh, and Staunton, removing the impact of the growth in price-dividend ratios from the geometric average historical premium and converting to an approximate arithmetic average. Roger G. Ibbotson and Peng Chen, ‘‘Long-Run Stock Market Returns: Participating in the Real Economy,’’ Yale ICF Working Paper No. 00-44, March 2002, Financial Analysts Journal (January/February 2003): 88–98; Charles P. Jones and Jack W. Wilson, ‘‘Using the Supply Side Approach to Understand and Estimate Stock Returns,’’ Working paper, June 6, 2006. SBBI Valuation Edition 2007. Goetzmann and Ibbotson, ‘‘History and the Equity Risk Premium,’’ 8.

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Cost of Capital

The greater than expected historical realized equity returns were caused by an unexpected increase in market multiples and a decline in discount rates relative to economic fundamentals. Each of these studies attempts to improve the estimate of the true ERP by removing the effects of changes in underlying economics that caused the realized risk premiums to differ from the ERP investors expected. However, even after adjusting for such unexpected changes, the realized risk premiums still are only estimates subject to statistical error. This potential for error reduces the reliability of claiming the resulting estimate is the true ERP. For example, in the study performed by Fama and French already discussed, those authors provide estimates of the ERP investors should have expected for the period 1951 to 2000 with confidence intervals. As is common, their study considers one variable at a time. They studied the relationship of underlying economic factors (growth in earnings and dividends) to realized risk premiums in years before 1951 and then asked what risk premium should have been expected given the underlying economic fundamentals in the years 1951 to 2000, if the relationships observed in prior years are assumed to continue. That is, based on the average observed relationship of dividend growth to return on equity capital during the periods 1872 through 1951 and then updated annually through 2000, they estimated the average return on equity (and volatility of the estimates) that should have been expected during 1951 through 2000 and subtracted the average risk-free rate. The Fama and French mean estimate of the equity risk premium that could have been expected based on dividend growth rate fundamentals is approximately 2.6% with a confidence interval (based on two standard errors), indicating that the average true ERP was between 0.1% and 3.8%.38 Similarly, their mean estimate of the equity risk premium that could have been expected based on earnings growth rate fundamentals is approximately 3.6% with a confidence interval (based on two standard errors) indicating that the average true ERP was between (effectively) zero and 7.4%. It is possible to consider multiple variables simultaneously and simulate an expected ERP that must have been expected to result in the realized return. For example, if you consider such variables as dividend yield, equity returns, volatility of equity returns, and average realized equity premium, what ERP might have been anticipated by investors that resulted in the realized returns observed? It is possible that such a joint simulation may result in a more accurate estimate of the true ERP during a period.

38

R. Glen Donaldson, Mark J. Kamstra, and Lisa A. Kramer have conducted such a joint simulation covering the period 1952 through 2004 using Monte Carlo simulation techniques with the parameters of the probability distributions of the variables derived from actual data observed during the period. They incorporate numerous variables but find average dividend yield, average equity return, and equity return volatility particularly informative in determining the ERP that must have been anticipated. They compare joint multivariate distributions of their simulated data with observed data and find that there was only a small range of anticipated ERP that could have yielded the outcomes during their study period: high average observed equity returns; the observed standard deviation of equity returns; the observed low dividend yield (which in part results due to the fact that firms increasingly distributed cash to shareholders via share repurchases instead of via dividends); high realized risk premium. Their results indicate that the average ERP that must have been expected during 1952 through 2004 is centered very close to 3.1% (arithmetic average premium over long-term government bonds) with a confidence interval (two standard errors) of 50 basis points. That is, they estimate

Based on Grabowski’s adjustment for bias reported in the Fama and French study and conversion of their results into the equivalent premium over long-term government bonds.


105

that the average true ERP was between 2.6% and 3.6%.39 Obviously their estimates are based on the assumed growth rates and the data-generating processes they hypothesize. We asked Morningstar to provide us with their supply side ERP estimates for the same period as used by Donaldson et al. (1952–2004) to compare the results. The supply side ERP estimates for 1952 to 2004 that could have been expected given the underlying economics were approximately 4.9% on an arithmetic basis (3.4% on a geometric basis) compared to the realized risk premium of 6.4% on an arithmetic basis (4.8% on a geometric basis).

CHANGES IN ECONOMICS THAT CAUSED UNEXPECTEDLY LARGE REALIZED RISK PREMIUMS Has there been a change in the relative volatility of market returns? Scott Mayfield found evidence of a structural shift in the relative volatility of market returns in 1940. His premise is that if the decrease in market risk was not fully anticipated, then stock prices during the subsequent period would be bid up and realized returns will not be representative of the ERP. He estimates that when looking at expectations following the structural shift in market volatility, the ERP (the risk premium over longterm government bonds that could have been expected for the period he studied, 1940 to 1997) was approximately 2.7%.40 McGrattan and Prescott find that the value of the stock market relative to the GDP in 2000 was nearly twice as large as in 1962.41 They determined that the marginal income tax rate declined (the marginal tax rate on corporate distributions averaged 43% in the 1955 to 1962 period and averaged only 17% in the 1987 to 2000 period). The regulatory environment also changed. Equity investments could not be held ‘‘tax free’’ in 1962. But by 2000, equity investments could be held ‘‘tax deferred’’ in defined benefit and contribution pension plans and in individual retirement accounts. The decrease in income tax rates on corporate distributions and the inflow of retirement plan investment capital into equity investments combined to lower discount rates and increase market multiples (i.e., lower capitalization rate) relative to economic fundamentals. Assuming that investors did not expect such changes, the true ERP during this period has been less than the realized risk premiums calculated as the arithmetic average of excess returns realized since 1926. Further, assuming that the likelihood of changes in such factors being repeated are remote and investors do not expect another such decline in discount rates, the true ERP as of today can also be expected to be less than the average realized risk premium.

39

40

41

R. Glen Donaldson, Mark J. Kamstra and Lisa A. Kramer, ‘‘Estimating the Ex Ante Equity Premium,’’ Working paper, November 2006, and ‘‘Stare Down the Barrel and Center the Crosshairs: Targeting the Ex Ante Equity Premium,’’ Working paper 2003-4, Federal Reserve Bank of Atlanta, January 2003. E. Scott Mayfield, ‘‘Estimating the Market Risk Premium,’’ Working paper, October 1999. See also: Chang-Jin Kim, James C. Morley, and Charles R. Nelson, ‘‘The Structural Break in the Equity Premium,’’ Journal of Business & Economic Statistics (April 2005): 181–191, in which they find evidence of a structural break that likely occurred in the early 1940s and appears to be driven by a reduction in the general level and persistence of market volatility; Lubos Pastor and Robert F. Stambaugh, ‘‘The Equity Premium and Structural Breaks,’’ Journal of Finance (August 2001): 1207–1239, study the equity risk premium from 1834 through 1999 and find several ‘‘structural breaks’’ (changes in volatility) in 1929 (increase compared to historic), 1941 (returning to historic levels), and 1992 (further reduced volatility). They find that the ERP compared to T bills (or equivalent) fluctuated between 3.9% to 6.0% over the period from January 1834 through June 1999. Ellen R. McGrattan and Edward C. Prescott, ‘‘Is the Market Overvalued?’’ Federal Reserve Bank of Minneapolis Quarterly Review 24 (2000): 20–24; ‘‘Taxes, Regulations and Asset Prices,’’ Working paper, Federal Reserve Bank of Minneapolis, July 2001.

106

Cost of Capital

FORWARD-LOOKING (ex Ante) APPROACHES Forward-looking approaches estimate the ERP by subtracting the current risk-free rate from the implied expected return for the stock market. Forward-looking approaches can be categorized into three groups based on the approach taken: 1. Fundamental information. This approach uses information such as earnings or dividends to estimate a bottom-up rate of return for a number of companies. An expected rate of return for an individual company can be implied by solving for the present value discount rate that equates the current market price of a stock with the present value of expected future dividends, for example. A bottom-up implied ERP begins with the averaging of the implied rates of return (weighted by market value) for a large number of individual companies and then subtracting the government bond rate. The bottom-up approach attempts to directly measure investor’s expectations concerning the overall market by using forecasts of the rate of return on publicly traded companies. 2. Top-down. This approach uses relationships across publicly traded companies over time between real stock returns, price/earnings ratios, earnings growth, and dividend yields. An estimate of the real rate of equity returns is developed from current economic observations applied to the historical relationships. Subtracting the current rate of interest provides an estimate of the expected ERP implied by the historical relationships. 3. Opinions. This approach relies on opinions of investors and financial professionals through surveys of their views on the prospects of the overall market. BOTTOM-UP APPROACHES This section presents estimates of ERP from three sources of bottom-up data. Merrill Lynch publishes bottom-up expected return estimates for the S&P 500 Stock Index derived from averaging return estimates for stocks in the S&P 500. While Merrill Lynch does not cover every company in the S&P 500 index, it does cover a high percentage of the companies as measured in market value terms. Merrill Lynch uses a multistage dividend discount model (DDM) to calculate expected returns for several hundred companies using projections from its own securities analysts. The resulting data are published monthly in the Merrill Lynch publication Quantitative Profiles. The Merrill Lynch expected return estimates have indicated an implied ERP ranging from 3% to 6.7% in recent years, with an average over the last 15 years of approximately 4.9%. The expected premium was approximately 5.1% at the end of 2006. In a DDM, the analyst first projects future company dividends. Merrill Lynch then calculates the internal rate of return that sets the current market price equal to the present value of the expected future dividends. If the projections correspond to the expectations of the market, then Merrill Lynch has estimated the rate at which the market is discounting these dividends in pricing the stock. The DDM is a standard method for calculating the expected return on a security.42 The theory assumes that the value of a stock is the present value of all future dividends. If a company is not currently paying dividends, the theory holds that it must be investing in projects today that will lead to dividends in the future. A number of consulting firms reportedly are using Merrill Lynch DDM estimates to develop discount rates. One author comments on the Merrill Lynch data:

42

See, for example, Sidney Cottle, Roger F. Murray, and Frank E. Block, Graham & Dodd’s Security Analysis, 5th ed. (New York: McGraw-Hill, 1988), 565–568.

Forward-Looking (ex Ante) Approaches

107

Two potential problems arise when using data from organizations like Merrill Lynch. First, what we really want is investor’s expectations, and not those of security analysts. However. . .several studies have proved beyond much doubt that investors, on the average, form their own expectations on the basis of professional analysts’ forecasts . The second problem is that there are many professional forecasters besides Merrill Lynch, and, at any given time, their forecasts of future market returns are generally somewhat different. . . .However, we have followed the forecasts of several of the larger organizations over a period of years, and we have rarely found them to differ by more than [plus or minus] 0.3 percentage points from one another.43

Although expected rates of return would be underestimated if the effects of share repurchases are not adequately considered, personnel from Merrill Lynch have indicated that their analysts take share repurchases into account by increasing long-term growth rates in earnings per share. If the effect is not completely modeled, the Merrill Lynch estimates may be biased downward. It is also possible that the DDM may understate expected returns to the extent that expected dividends are measured based on earnings from assets in place and understate future growth opportunities. But this is most likely a larger problem for the analysis of smaller companies than for the large companies that predominate the S&P 500 Index in market value terms. Value Line projections can be used to produce estimates of expected returns on the market. Value Line routinely makes ‘‘high’’ and ‘‘low’’ projections of price appreciation over a three- to five-year horizon for over 1,500 companies. Value Line uses these price projections to calculate estimates of total returns, making adjustments for expected dividend income. The high and low total return estimates are published each week in the Value Line Investment Survey. Midpoint total return estimates are published in Value Line Investment Survey for Windows CD database. There is some evidence that the Value Line analysts’ projections, at least for earnings growth, tend to be biased high.44 Implied ERP estimates developed from Value Line data have been more volatile than the Merrill Lynch DDM models. Recent implied ERP estimates have ranged from slightly below zero at the end of 1999 to more than 12% at the end of 2002, with the average implied ERP over the past 15 years being 5.3%. The most recent implied ERP was 5.0% as of the end of 2006. We believe that Value Line’s estimates of future prices are ‘‘sticky’’ (i.e., they tend to change slowly), with the result that the expected premium appears to rise after a bear market and fall after a bull market. The Cost of Capital Yearbook published by Morningstar annually reports the implied rates of return for a large number of companies derived from both a single-stage DDM and a three-stage DDM (with quarterly updates reported in their Cost of Capital Quarterly).45 Expected growth rates in dividends are derived from analysts’ estimates as reported in the Institutional Broker’s Estimate System (I/B/E/S) Consensus Estimates database. The Cost of Capital Yearbook reports statistics for large composite groups of companies, and from these statistics you can derive an ERP for the overall market. Implied ERP estimates derived from the reported three-stage DDM rates of return have ranged from 4.9% to 8.0% since publication commenced in 1994, with the implied ERP at the beginning of 2007 being 7.5%. Several academic studies have employed consensus forecasts of long-run earnings per share growth as a proxy for projected dividends in a DDM. One study extracted ex ante estimates of the ERP from several versions of the CAPM.46 The results suggest that the ERP varies over the business cycle; it is lowest in periods of business expansion and greatest in periods of recession. The ERP 43

44

45 46

Eugene Brigham and Louis Gapenski, Financial Management: Theory and Practice, 5th ed. (Fortworth, TX: Dryden Press 1988), 227. David T. Doran, ‘‘Forecasting Error of Value Line Weekly Forecasts,’’ Journal of Business Forecasting (Winter 1993–94): 22–26. See, for example, Cost of Capital Yearbook 2007 (Chicago: Morningstar, 2007). Fabio Fornari, ‘‘The Size of the Equity Premium,’’ Working paper, January 2002.

108 Exhibit 9.3

Cost of Capital Implied ERP Estimates

Merrill Lynch Value Line: 3- to 5-year horizon Cost of Capital Yearbook

Range

Period

Mean

As of Early 2007

3.0% to 6.7% 1.0% to 12.3% 4.9% to 8.0%

1988–2006 1988–2006 1994–2006

4.9% 5.7% 6.4%

5.1%* 5.0% 7.5%

*

As of the end of January 2007.

appears to be positively correlated with long-term bond yields (increasing as bond yields increase) and with the default premium (increasing as the differential between Aaa- and Baa-rated bond yields increases). Studies have indicated that analysts’ earnings forecasts (such as those reported by I/B/E/S and First Call) are biased high.47 These biases lead to high implied estimates of ERP. Exhibit 9.3 summarizes three forward-looking implied ERP estimates published over the past several years. TOP-DOWN APPROACHES Various researchers have published estimates of the expected ERP based on their analyses of the historical relationship of such variables as earnings growth, stock market levels in terms of price to earnings ratios and dividend yields, changes in interest rates, and real stock returns. They apply the observed relationships to the state of the current economic variables and stock market levels and project future real returns on stocks. By subtracting the current interest rates, you obtain an estimate of the expected ERP. A sampling of those studies follows.

Jeremy Siegel studied the link between real equity returns, price/earnings ratio, real growth, replacement cost of capital invested, and market value of capital. He estimates that the long-run price/earnings ratio will settle between 20 and 25 and that the real (before inflation) average compound future total equity return will be approximately 5%. This converts to an expected geometric ERP of 2% (equivalent to approximately 4% arithmetic average).48

Bradford Cornell studied the relationship between growth in real domestic product and earnings and dividends. He estimates that under any reasonable underlying assumptions about inflation, equity risk premiums cannot be more than 3% (geometric average) because the earnings growth rate is constrained unconditionally in the long run by the real growth rate in the economy, which has been in the range of 1.5% to 3.0%.49

47

48 49

James Claus and Jacob Thomas, ‘‘The Equity Premia as Low as Three Percent? Evidence from Analysts’ Earnings Forecasts for Domestic and International Stock Markets,’’ Journal of Finance (October 2001): 1629–1666; Alon Brav, Reuven Lehavy, and Roni Michaely, ‘‘Using Expectations to Test Asset Pricing Models,’’ Financial Management (Autumn 2005): 5–37; Sundaresh Ramnath, Steve Rock, and Philip Stone, ‘‘Value Line and I/B/E/S Earnings Forecast,’’ International Journal of Forecasting (January 2005): 185–198. Those authors report the results of projected earnings amounts rather than growth rates (they use the I/ B/E/S long-term growth rate to project the EPS four years into the future), and compare this with the actual EPS four years in the future. The results indicate that I/B/E/S mean forecast error in year 4 EPS is negative. This can be translated into a preliminary typical growth rate adjustment for, say, a projected 15% growth rate as follows: ((1.15^4)(1 .0545))^.25 1 ¼ 13.4%, implying a ratio of actual to forecast of .134/.15 ¼ .89. This would imply that equity risk premium forecasts using analyst forecasts are biased high; Roberto Bianchini, Stefano Bonini, and Laura Zanetti, ‘‘Target Price Accuracy in Equity Research,’’ Working paper, January 2006. Jeremy Siegel, ‘‘Historical Results: Discussion,’’ Equity Risk Premium Forum, AIMR (November 8, 2001): 30–34. Bradford Cornell, ‘‘Historical Results: Discussion,’’ Equity Risk Premium Forum, AIMR (November 8, 2001): 38–41.

Forward-Looking (ex Ante) Approaches

109

Given the level of the S&P 500 in early 2001 (the time of his study), his analysis indicates a current conditional estimate of the ERP of at most 3.5% on a geometric basis (equivalent to approximately a 5.5% arithmetic average).

Elroy Dimson, Paul Marsh, and Mike Staunton studied the realized equity returns and historical equity premiums for 17 countries (including the United States) from 1900 to the end of 2006.50 Assuming that the standard deviation of annual returns on equity will approximately equal the historical standard deviation, their analysis indicates an estimate of ERP in early 2007 of 3.9% on a geometric basis (equivalent to approximately a 5.9% arithmetic average) versus U.S. government bonds. The authors note that: Further adjustments should almost certainly be made to historical risk premiums to reflect long-term changes in capital market conditions. Since, in most countries corporate cash flows historically exceeded investors’ expectations, a further downward adjustment is in order.

They conclude that a further downward adjustment of approximately 50 to 100 basis points in the expected ERP at the beginning of 2007 is plausible. Adjusting the realized risk premium for the increase in price-to-dividend ratio which resulted from a decrease in the discount rate and dividend yield to current levels, the resulting estimated ERP at the beginning of 2007 for the United States is in the range of 4.9% to 5.4% (arithmetic average over U.S. government bonds; 2.9% to 3.4% geometric average over U.S. government bonds).51

Roger Ibbotson and Peng Chen prepared a forecast of ERP based on the contribution of earnings growth to price/earnings ratio growth and on growth in per capita GDP. Their supply side estimate of ERP at the beginning of 2007 is about 6.3% (arithmetic average relative to government bonds; 4.3% on a geometric basis).52 These forecasts tend to give somewhat lower forecasts than historical risk premiums, primarily because part of the total returns of the stock market have come from price-earnings ratio expansion. This expansion is not predicted to continue indefinitely, and should logically be removed from the expected risk premium.53

SURVEYS

50

51

52 53

54

Ivo Welch surveyed over 500 finance and economics professors at leading universities and found that, for long-term investments, the median forecast ERP (premium over T-bills) was 5%, with the interquartile range of 4% to 7%.54 Adjusting for the horizon premium embedded in government bonds versus T-bills, these results translate to a median forecast ERP (premium over government bonds) of 3.6%, with an interquartile range of 2.6% to 5.6%. Dimson, Marsh, and Staunton, ‘‘Global Evidence on the Equity Premium’’; ‘‘The Worldwide Equity Premium: A Smaller Puzzle’’ EFA 2006 Zurich Meetings Paper, April 7, 2006; Global Investment Returns Yearbook 2007. Based on converting the premium over total returns on bonds as reported by Dimson, Marsh, and Staunton, removing the impact of the growth in price-dividend ratios from the geometric average historical premium, reducing the historical average dividend yield to a current dividend yield, and converting to an approximate arithmetic average. SBBI Valuation Edition 2007 Yearbook. Goetzmann and Ibbotson, ‘‘History and the Equity Risk Premium,’’ 8. Also note that the supply side estimate of ERP for the period 1952 to 2004 discussed earlier is 4.9% arithmetic average (4.8% geometric average). Ivo Welch, ‘‘The Equity Premium Consensus Forecast Revisited,’’ Cowles Foundation Discussion Paper no. 1325, September 2001.

110

Cost of Capital

John Graham and Campbell Harvey report the results from quarterly surveys of chief financial officers of U.S. corporations conducted from mid-2000 to the end of 2006. The current survey attracts about 400 respondents (10% from companies with less than $10 million in revenue; 50% from companies with less than $500 million in revenue; 40% are private companies). That study reports that the range of ERP given a 10-year investment horizon was 2.5% to 4.7% (premium over 10-year government bonds), with the most recent survey concluding 3.3%.55

Greenwich Associates publishes an annual survey of several hundred pension plan officers concerning their expected returns for the S&P 500 Index for a five-year holding period. Using the survey and converting the results to an ERP estimate, it generally indicated an expected premium over long-term U.S. government bonds of between 2% and 4%. Their most recent survey indicated an expected premium over long-term U.S. government bonds of 3.4%.56

OTHER SOURCES OF ERP ESTIMATES A list of published opinions and guidelines on ERP. These are not the only sources but represent a cross section of opinion on the subject.

Principals of Corporate Finance, 8th ed., takes no official position on the exact ERP. But the authors believe a range of 5% to 8% premium over T-bills is reasonable for the United States (equivalent to a premium over government bonds of approximately 3.5% to 6.5%). They warn that ‘‘out of this debate only one firm conclusion emerges: Do not trust anyone who claims to know what returns investors expect.’’57

Valuation: Measuring and Managing the Value of Companies, 4th ed., recommends an ERP of 4.5% to 5.5%.58 The authors use a forward-looking model to estimate real expected market returns for 1962 through 2002 averaging 7.0%. Subtracting the real return on U.S. government inflation-protected bonds (TIPS), they estimate the risk premium. The authors conclude on their assessment of the research and evidence: Although many in the finance profession disagree about how to measure the (ERP), we believe 4.5 to 5.5% is the appropriate range. Historical estimates found in most textbooks (and locked in the minds of many), which often report numbers near 8%, are too high for valuation purposes because they compare the market risk premium versus short-term bonds, use only 75 years of data, and are biased by the historical strength of the U.S. market.59

55

56 57

58

59

John R. Graham and Campbell R. Harvey, ‘‘Expectations of Equity Risk Premia, Volatility and Asymmetry from a Corporate Finance Perspective,’’ Working paper, National Bureau of Economic Research, July 2003, updated quarterly by Duke CFO Outlook Survey, www.cfosurvey.org; ‘‘The Equity Risk Premium in January 2007: Evidence from the Global CFO Outlook Survey,’’ working paper reporting on the autumn 2006 survey, January 2007. Graham and Harvey believe the results represent a geometric average expected return. Grabowski estimated the arithmetic average equivalent ¼ geometric average risk premium estimate ¼ (standard deviation of risk premium estimates)2/2. The survey question answered is ‘‘On November 10, 2006 the annual yield on 10-year treasury bonds was 4.6%. Over the next 10 years, I expect the average annual S&P 500 return will be:___%.’’ This question leads us to question whether the replies represent the expected compound return over the next 10 years (geometric average) or the annual average return (arithmetic average). Greenwich Associates, ‘‘Market Trends, Actuarial Assumptions, Funding, and Solvency Ratios’’ (Fall 2006). Richard Brealey, Stuart Myers, and Franklin Allen, Principals of Corporate Finance, 8th ed. (Boston: Irwin McGraw-Hill, 2006), 154. Tim Koller, Marc Goedhart and David Wessels, Valuation: Measuring and Managing the Value of Companies, 4th ed. (New York: John Wiley & Sons, 2005), 305–306. Ibid., 306.

Other Sources of ERP Estimates

111

Damodaran on Valuation, 2nd ed., concludes that the most relevant realized return is the geometric average realized return versus government bonds 4.84% (geometric average realized premium 1926 through 2004 over government bonds) while the average implied (forwardlooking approach using expected dividends and expected dividend growth) EPR is only about 4% as of January 2006 (premium over government bonds).60 The author notes that the average implied ERP has been about 4% over the past 40 years.61 He uses 4% in most of his valuation examples.

The Equity Risk Premium concludes that ‘‘reasonable forward-looking ranges for the future equity risk premiums in the long run are 3.5% to 5.5% over treasury bonds. . . .’’62

Creating Shareholder Value, revised and updated, recommends that the premium should be based on expected rates of return rather than average historical rates. This approach is crucial because with the increased volatility of interest rates over the past two decades the relative risk of bonds increased, thereby lowering risk premiums to a range of 3 to 5%.63

60

61 62 63 64 65 66 67

68 69

Graham and Dodd’s Security Analysis uses an ‘‘equity risk premium’’ of 2.75% over the yield on Aaa industrial bonds for valuing the aggregate S&P 400 Index that approximates a 10-year historical average.64 This translates to a premium of approximately 3% over long-term government bonds. The authors reproduce the opinion of one security analyst who recommended a premium over the S&P Composite Bond yield of 3.5% to 5.5% in 1978 and 3.0% to 3.5% in 1983;65 this translates to premiums of approximately 4.5% to 7% in 1978 and 4% to 6% in 1983 over longterm government bonds. Stocks for the Long Run, concludes that ‘‘as real returns on fixed-income assets have risen in the last decade, the equity premium appears to be returning to the 2% to 3% norm that existed before the postwar surge.’’66 The author updates his views to the beginning of 2006 and concludes that projected equity returns of 3.5% to 4.5% (equivalent arithmetic average return) over government bonds ‘‘will still give ample rewards for investors willing to tolerate the short-term risks of stocks.’’67 The Quest for Value recommends a 6% premium based on a long-run geometric average difference between the total returns on stocks and bonds.68 Financial Statement Analysis and Security Valuation notes that ‘‘the truth is that the equity risk premium is a speculative number.’’ The author uses 5% in his examples but notes the wide range of estimates.69

Aswath Damodaran, Damodaran on Valuation: Security Analysis for Investment and Corporate Finance, 2nd ed. (New York: John Wiley & Sons, 2006), 41 and 48. Ibid., 47. Bradford Cornell, Equity Risk Premium: The Long-Run Future of the Stock Market (New York: John Wiley & Sons, 1999), 201. Alfred Rappaport, Creating Shareholder Value, rev. and updated (New York: The Free Press, 1997), 39. See Cottle, Murray, and Block, Graham & Dodd’s Security Analysis, 573. Ibid., 83–85. Siegel, Stocks for the Long Run, 20. Siegel, ‘‘Perspectives on the Equity Risk Premium.’’ Grabowski converted Siegel’s conclusion in terms of geometric average return (p. 70) compared to government bonds. G. Bennett Stewart, The Quest for Value (New York: HarperCollins, 1991), 436–438. Stephen H. Penman, Financial Statement Analysis and Security Valuation, 3rd ed. (New York: McGraw-Hill, 2007), 476.

112

Cost of Capital

Equity Premium: Historical, Expected, Required and Implied recommends that ‘‘an additional 4% (over government bonds) compensates the additional risk of a diversified portfolio.’’70 In the Duff & Phelps Risk Premium Report 2007, the historical realized equity premium for the period of 1963 through 2006 was 4.9%.71

UNCONDITIONAL VERSUS CONDITIONAL ERP The evidence presented represents a long-term average or unconditional estimate of the ERP. That is, what is a reasonable range of ERP that can be expected over an entire business cycle? Where in this range is the current ERP? Research has shown that ERP is cyclical during the business cycle. We use the term conditional ERP to mean the ERP that reflects current market conditions. For example, when the economy is near or in recession (and reflected in recent relatively low returns on stocks), the conditional ERP is more likely at the higher end of the range. When the economy improves (with expectations of improvements reflected in recent increasing stock returns), the conditional ERP moves toward the midpoint of the range. When the economy is near its peak (and reflected in recent relatively high stock returns), the conditional ERP is more likely at the lower end of the range. The issue of predicting future returns on the S&P 500 is the subject of much research, which generally has centered on the power of various models to predict future returns on the S&P 500 and the resultant equity premium given current prospects as measured by observed relationships. For example, Amit Goyal and Ivo Welch test a range of variables that have been held to predict ERP: dividend-to-price ratios, dividend yields, price/earnings ratios, interest rates, inflation rates, and consumption-based macroeconomic ratios. They find that the models are unstable when used to predict the resulting equity risk premium in periods not included in the sample periods. They find that ‘‘most models not only cannot beat the unconditional benchmark, but also outright underperform it.’’72 Others have disputed their results, finding that predictive power is small, but economically meaningful, or their results are really the result of poor predictability of, say, dividend growth.73 But research suggests that only models allowing explicitly for time varying factors succeed in maintaining their predictive power across periods of time.74 But the predictability of the ERP most impacts short-term investment. As the focus of this book is valuation of businesses and investments by businesses, the conditional ERP is of less importance, and we can fall back on using the long-term, unconditional ERP in developing discount rates.

SUMMARY The results presented in this chapter do not point to a single estimate of ERP. They point to a conclusion that the current ERP is in a range that is consistent with the principle that investor’s 70 71

72

73

74

Pablo Fernandez, ‘‘Equity Premium: Historical, Expected, Required and Implied,’’ Working paper, February 18, 2007, 28. See Chapter 12 for a discussion of the realized risk premiums observed for the period covered by that study. Duff & Phelps Risk Premium Report 2007. Available at www.corporate.morningstar.com/ib or www.bvresources.com. Amit Goyal and Ivo Welch, ‘‘A Comprehensive Look at the Empirical Performance of Equity Premium Prediction,’’ Working paper, January 11, 2006. John Y. Campbell and Samuel B. Thompson, ‘‘Predicting the Equity Premium Out of Sample: Can Anything Beat the Historical Average?’’ HIER Discussion Paper No. 2084 (July 2005); John H. Cochrane, ‘‘The Dog That Did Not Bark: A Defense of Return Predictability,’’ Working paper, January 30, 2006. Thomas Dangl, Michael Halling, and Otto Randl, ‘‘Equity Return Prediction: Are Coefficients Time Varying?’’ Working paper, April 2006.

Summary

113

expectations are not homogeneous. Different investors have different cash flow expectations and future assessments of the risk that those cash flows will be realized. You can think of this in terms of the dividend discount model; numerous combinations of expected future cash flows and discount rates equate to the existing price.75 Estimating the ERP is one of the most important issues when you estimate the cost of capital of a subject business or project. You need to consider a variety of alternative sources, including examining realized returns over various periods and employing forward-looking estimates such as those implied from projections of future prices, dividends, and earnings. What is a reasonable estimate of ERP in 2007? While giving consideration to long-run historical arithmetic average of realized risk premiums, these authors conclude that the post-1925 historical arithmetic average of one-year realized premiums as reported in the SBBI Yearbook results in an expected ERP estimate that is too high. We come to that conclusion based on the works of various researchers (e.g., Dimson, Marsh, and Staunton; Goetzmann and Ibbotson) and current market expectations (e.g., survey of chief financial officers). Some practitioners express dismay over the necessity of considering a forward ERP since that would require changing their current ‘‘cookbook’’ practice of relying exclusively on the post-1925 historical arithmetic average of one-year realized premiums reported in the SBBI Yearbook as their estimate of the ERP. Our reply is that valuation is a forward-looking concept, not an exercise in mechanical application of formulas. Correct valuation requires applying value drivers reflected in today’s market pricing. You need to mimic the market. In our experience, you often cannot match current market pricing for equities using the post-1925 historical arithmetic average of one-year realized premiums as the basis for developing discount rates. The entire valuation process is based on applying reasoned judgment to the evidence derived from economic, financial, and other information and arriving at a well-reasoned opinion of value. Estimating the ERP is no different. After considering the evidence, a reasonable long-term estimate of the normal or unconditional ERP as of the beginning of 2007 should be in the range of 3.5% to 6%. For examples in this book, the authors have concluded that an estimate of ERP of 5% is consistent with the research presented here and we use 5% in the examples. While we present data and calculations elsewhere in this book using data through the end of 2005 and earlier, we do that to help the reader understand the methodology. Since the choice of ERP is so important, in this chapter we present data that are as up to date as possible as we were preparing the text.

75

Fernandez, ‘‘Equity Premium: Historical, Expected, Required and Implied.’’

Appendix 9A

Bias Issues in Compounding and Discounting

Bias in Compounding Bias in Discounting

BIAS IN COMPOUNDING In predicting the compound return, you might expect for an investment in stocks using an ERP estimate derived from the arithmetic average of realized risk premium data that you will get an answer that is biased (i.e., will measurement error be introduced simply due to the mathematics). Comparing future values that result from compounding an investment at an erroneous ‘‘too high’’ rate of return with results from compounding an investment at an equally erroneous ‘‘too low’’ estimated rate of return, the estimated future value in the too-high case will be further from the true expected future value than the estimate in the too-low case. This is simply a function of the mathematics of compounding. Averaging across these possibilities, the compounded future values derived from arithmetic averages will be too high in general. For example, assume that the true expected annual return on stocks for the next 10-year holding period equals 10%. The true expected future value in 10 years will then equal (1.10)10 ¼ 2.5937. The true expected return is not observable; historical data are compiled in an attempt to estimate the true expected return. While the estimation process of compiling an arithmetic average of historical returns results in an unbiased estimate, the estimate will be either too high or too low. Assume that there is a 50/50 chance of choosing an estimated future return that is either too high or too low. If the estimate is too high (e.g., the estimate is that the future return will be 12%), the estimated future value will equal (1.12)10 ¼ 3.1058. Alternatively if the estimate is too low (e.g., the estimate is that the future return will be 8%), the estimated future value will equal (1.08)10 ¼ 2.1589. The average of these two estimates equals 2.6324, which is greater than the true expected return of 2.5937. Using the arithmetic average of historical returns or realized risk premiums with error (and we know there will always be error) as the estimate of the true expected return results in too high a compounded future return on average. Several authors have studied biases that may arise in multiperiod compounding when the single-period estimate of expected return is subject to measurement error.1 Proposals in the academic literature for a correction of this bias (for predicting future values) involve downward adjustments in the arithmetic average of single-period realized returns. These 1

Michael E. Blume, ‘‘Unbiased Estimators of Long-Run Expected Growth Rates,’’ Journal of the American Statistical Association (September 1974): 634–638; Ian Cooper, ‘‘Arithmetic Versus Geometric Mean Estimators: Setting Discount Rates for Capital Budgeting,’’ European Financial Management (July 2001): 157–167; Eric Jacquier, Alex Kane, and Alan J. Marcus, ‘‘Optimal Forecasts of Long-Term Returns and Asset Allocation: Geometric, Arithmetic, or Other Means?’’ Working paper, October 31, 2002.

114

Bias in Discounting

115

adjustments increase as the length of the investment horizon increases. One proposed correction has the expected rate of return falling to the geometric average rate of return if the investment horizon is as long as the time horizon over which the historical averages are measured.2 While corrections for the measurement error problem in the arithmetic average of annual realized returns may be material for compounding over several decades, the proposed corrections for near-term compounding are minor. You should always use the geometric average of historical data (i.e., stock returns, earnings before interest, taxes, depreciation, and amortization [EBITDA], etc.) for projections. For example, you should use the geometric average of realized risk premiums in projecting future value of a portfolio of stocks, not the arithmetic average. You should use the geometric average of historical growth in EBITDA to project future EBITDA, not the arithmetic average.3

BIAS IN DISCOUNTING In discounting expected cash flows where you develop a cost of equity capital estimate using an ERP estimate derived from the arithmetic average of realized risk premium data, will you get an answer that is biased? The statistical properties of this problem are such that you get a different answer if, instead of focusing on unbiased expected future values, you seek instead an unbiased estimate of the present value discount factor.4 A proposed correction that focuses on present value factors finds that the adjustment from the arithmetic average is small even when discounting over fairly long periods.5 Moreover, the bias is towards discount rates that are too low rather than too high. For example, assume that the true expected annual return on stocks for the next 10-year holding period equals 10% and that rate of return represents the correct risk-adjusted return to use in discounting a stream of future cash flows. The correct discount factor to use in determining the present value of cash flows expected 10 years in the future will then equal (1.10)10 ¼ 0.3855. Again the true expected return is not observable, and historical data are compiled in an attempt to estimate the true expected return. While the estimation process of compiling an arithmetic average of historical returns results in an unbiased estimate, the estimate will be either too high or too low. Assume that there is a 50/50 chance of choosing an estimated future return that is either too high or too low. If the estimate is too high (e.g., the estimate is that the future return will be 12%), the discount factor will equal (1.12)10 ¼ 0.3220. Alternatively, if the estimate is too low (e.g., the estimate is that the future return will be 8%), the discount factor will equal (1.08)10 ¼ 0.4632. The average of these two estimates equals 0.3926 (the arithmetic average results in an equivalent of a 9.8% rate of return), which results in a larger present value than had you used the correct discount factor of 0.3855 (i.e., the equivalent rate of return of 9.8% is too low compared to the true rate of return of 10%). Using the arithmetic average of historical realized premiums with error as the estimate of the true ERP results in an estimated rate of return that is too low. But the error in most practical valuations is minimal. The arithmetic average of historical realized premiums can be used as one estimate of the ERP without introducing mathematical bias.

2 3 4

5

Jacquier, Kane, and Marcus, ‘‘Optimal Forecasts of Long-Term Returns and Asset Allocations.’’ Pablo Fernandez, ‘‘80 Common Errors in Company Valuation,’’ Working paper, May 12, 2004: 12. When there is measurement error in expected returns, the unbiased estimate of the present value discount factor is not equal to the inverse of the unbiased estimate of future value. The bias in the arithmetic average for discounting runs in the direction opposite that of the bias for future values (i.e., the bias causes an underestimate of the true compounded discount rate rather than an overestimate). Cooper, ‘‘Arithmetic Versus Geometric Mean Estimators.’’

Chapter 10

Beta: Differing Definitions and Estimates

Introduction Estimation of Equity Beta Differences in Estimation of Equity Beta Length of the Sample or Look-Back Period Frequency of Return Measurement Choice of Market Index Choice of Risk-free Rate Levered and Unlevered Equity Betas Formulas for Unlevering and Levering Equity Betas Choosing Among Unlevering or Levering Formulas Adjusting Asset Beta Estimates for Differences in Operating Leverage Adjusting Asset Beta Estimates for Excess Cash and Investments Modified Betas: Adjusted, Smoothed, and Lagged Adjusted Beta Incorporates Industry Norm Smoothed beta ‘‘Sum Beta’’ Incorporates Lag Effect ‘‘Full-Information’’ Equity Beta Peer Group Equity Beta Fundamental Equity Beta Equity Beta Estimation Research Estimation of Debt Beta Other Beta Considerations Summary Appendix 10A Appendix 10B

INTRODUCTION Betas for equity capital are used as a modifier to the equity risk premium in the context of the Capital Asset Pricing Model (CAPM). Beta is the sole risk measure of equity capital of the textbook CAPM. The combination of equity beta times the ERP equals the estimated market risk premium. The concept of beta as a risk measure can be extended to debt capital. If equity capital is bearing all of the risk of the variability of operating income, then the debt capital is bearing no market risk and its beta is zero. But as the level of debt financing of the firm increases and the credit rating decreases, debt capital is also bearing market risk. That market risk can likewise be measured in terms of a beta.

117

118

Cost of Capital

This chapter explores some widely used methods in the estimation and applications of betas for equity capital and debt capital. Beta estimates are derived from data on publicly traded securities. If one is valuing a closely held company or a non-public, division or reporting unit, for example, one is using the beta estimate of publicly traded securities as a proxy for the non-public business.

ESTIMATION OF EQUITY BETA Market or systematic risk is measured in CAPM by a factor called beta. Beta is a function of the expected relationship between the return on an individual security (or portfolio of securities) and the return on the market. The market is generally measured by a broad market index, such as the Standard & Poor’s (S&P) 500 Index. The broad market index is a proxy for the broad economy. The beta is theoretically the expected sensitivity of the individual security to changes in the economy and, similar to the equity risk premium (ERP), beta is a forward-looking concept. The sensitivity of individual security returns is the sensitivity of the company to cash flow risks and discount rate risk. It represents the sensitivity of changing expectation about expected cash flows of the company relative to changing expectations about expected cash flows of the economy as a whole (i.e., the market), changing expectations for the ERP.1 Existing techniques for estimating beta generally use historical data over a sample or ‘‘look-back’’ period and assume that the future will be sufficiently similar to this past period to justify extrapolation of betas calculated using historical data. Research shows betas are time-varying (i.e., sensitive to market changes as the economy changes; beta differs during improving economic conditions compared to during declining economic conditions). Using a historical method based on a sample period may not perform well when economic conditions are changing. The current and expected future economic conditions may differ from the economic conditions during the look-back period. Therefore, the beta estimated using the data for the look-back period will not reflect the future. Theorists prefer to estimate beta by comparing the excess returns on an individual security relative to the excess returns on the market index. By excess return, we mean the total return (which includes both dividends and capital gains and losses) over and above the return available on a risk-free investment (e.g., U.S. government securities). For a publicly traded stock, you can estimate beta via regression (ordinary least squares [OLS] regression), regressing the (excess) returns on the individual security ðR R f Þ against the (excess) returns on the market ðRm R f Þ during the look-back period. The resulting slope of the best-fit line is the beta estimate. Formula 10.1 shows the regression formula. (Formula 10.1) ðR R f Þ ¼ a þ B ðRm R f Þ þ e where: R ¼ Historical return for publicly traded stock Rf ¼ Risk-free rate a ¼ Regression constant B ¼ Estimated beta based on historical data Rm ¼ Historical return on market portfolio e ¼ Regression error term 1

John Y. Campbell and Jianping Mei, ‘‘Where Do Betas Come From? Asset Price Dynamics and the Sources of Systematic Risk,’’ The Review of Financial Studies 6 No. 3 (1993): 567–592.

Estimation of Equity Beta

119

Morningstar uses excess returns in all its computations. Practitioners and some financial data services calculate betas using total returns instead of excess returns. The OLS regression using total return is: (Formula 10.2) R ¼ a þ B Rm þ e where variables are defined as above. However, comparisons of measurements using excess returns or total returns show that, as a practical matter, it makes little difference. Beta equals the covariance of the returns for subject security to the returns for the market (the S&P Index) relative to the variance in the returns for the market during the sampling or look-back period. An example of calculating betas using total returns is shown in Exhibit 10.1. The look-back period in this example is 120 months. An example of a beta estimate using the OLS regression method for a look-back period of 60 months of total returns for J.B. Hunt Transport Services, Inc. (as of December 2005) is displayed in Exhibit 10.2. Because beta is an expected sensitivity, any estimation using historical methods is subject to error. How useful are the results of the regression in estimating the relationship between the returns on a stock and the returns on the market? Or, how close to the true beta is the estimated beta? Exhibit 10.1 Month End, t [a] 1/89 2/89 3/89 10/98 11/98 12/98 Sum Average Beta ¼

Illustrative Example of One Common Method for the Calculation of Beta Return on Security A [b]

Return on S&P Index [c]

Calculated Covariance [d]

0.041 (0.007) 0.052

0.069 (0.029) 0.021

0.00211 0.00045 0.00043

0.00325 0.00168 0.00008

0.113 0.033 (0.016)

0.077 0.057 0.055

0.00709 0.00131 (0.00086)

0.00423 0.00203 0.00185

0.500 0.004

1.488 0.012

0.21060 0.00176 [f]

Calculated Variance [e]

0.26240 0.00219 [g]

CovarianceðSecurity A; S&P IndexÞ 0:00176 ¼ ¼ 0:80 Variance of S&P Index 0:00219

a. 10 years or 120 months. b. Returns based on end-of-month prices and dividend payments (versus quarterly or annually). c. Returns based on end-of-month S&P Index. d. Values in this column are calculated as: (Observed return on Security A Average return on Security A) (Observed return on S&P Index Average return on S&P Index) 0:00211 ¼ ½ð0:041 0:004Þ ð0:069 0:012Þ e. Values in this column are calculated as: (Observed return on S&P Index Average return on S&P Index) 0:00325 ¼ ð0:069 0:012Þ f. The average of this column is the covariance between Security A and the S&P Index. g. The average of this column is the variance of return on the S&P Index. Source: Shannon P. Pratt, Valuing a Business: The Analysis and Appraisal of Closely Held Companies, 5th ed. (New York: McGraw-Hill Companies, 2008), Chap. 9. Reprinted with permission. All rights reserved.

120

Cost of Capital

Exhibit 10.2

Example of Beta Estimation for J.B. Hunt Transport Service, Inc.

Ticker Symbol: JBHT SIC 4213: Trucking, Except Local Date: December, 2005 Company: J.B. Hunt Transport Services, Inc., together with its subsidiaries, provides full-load freight transportation services in the United States, Canada, and Mexico. Calculated OLS Beta Number of Months of Data Regression Coefficients R-Squared Intercept Beta T-Statistic Summary Statistics Average Return Standard Deviation Correlation Matrix Company Market

1.62 60 OLS 0.30 3.49% 1.624 4.95 Company 3.716% 12.742% Company 1.000 0.545

Average Monthly Volume (millions) Average Volume/Total Outstanding

Market 0.138% 4.275% Market 1.000

34.978 22.74%

Monthly Returns versus Market Index

40%

Company Returns

30% 20% 10% 0%

++ + ++ +

+

–10% +

–20%

++ + + + + + + ++ + +++ +++ + + + + + +++ ++ + + + ++ +++ + + + ++ ++ +

–30% –40% –50% –25%

+ –20%

–15%

–10%

0% 5% –5% S&P 500 Returns

+ Observed returns

10%

15%

20%

25%

Y = 3.49% + 1.62 X

Source: Calculated (or derived) based on Standard & Poor’s Compustat data. Calculations by Duff & Phelps LLC. Used with permission. All rights reserved.

Accuracy of the beta estimate can be described in statistical terms. Important statistics are:

T-statistic: Only indicates if the beta coefficient is different from zero (i.e., if t-statistic > x, beta differs from zero).

Standard error of estimate: Measures likelihood that true beta is measured by estimate of the beta made by regression.

See Appendix F for a discussion of the relationship between two sets of measures (the return on the subject security and the return on the market).

Differences in Estimation of Equity Betas

121

The beta estimate for the example in Exhibit 10.2 equals 1.624. The t-statistic in the example equals 4.95, indicating that the data provide a beta estimate that is statistically significant (i.e., different from zero). R2 equals 0.30. The standard error of estimate equals 0.328.2 That is, we have 95% confidence that the true beta is between 1.62 þ= (2)(0.328) or between .96 and 2.28. Because we cannot compute a beta directly for a division, reporting unit, or closely held company, we need to estimate a proxy beta for these businesses. We can either calculate beta estimates or go to reference sources to obtain beta estimates for guideline public companies or industries to use as a proxy beta for our subject business. In developing a proxy beta, you must consider the differences between the subject company and the possible guideline public companies. Also, you must be cautious of beta estimates using smaller public companies without an active market, as their betas tend to be underestimated using OLS beta estimates and by reference sources. Further, the more beta estimates drawn from guideline public companies you use as the basis for the beta estimate of the subject business, the better the accuracy because the standard error of estimation is reduced. Details on sources of beta estimates can be found in Appendix B.

DIFFERENCES IN ESTIMATION OF EQUITY BETAS Be aware that significant differences exist between betas for the same stock published by different financial reporting services. One of the implications of this fact is that betas for guideline companies used in a valuation should all come from the same source. If all betas for guideline companies are not available from a single source, the best solution probably is to use the source providing betas for the greatest number of guideline companies and not use betas given for the others. Otherwise an applesand-oranges mixture will result. Differences in the beta measurement derive from choices within four variables: 1. The length of the time period over which the historical returns are measured (i.e., the length of the look-back period) 2. The periodicity (frequency) of return measurement within that time period 3. The choice of an index to use as a market proxy 4. The risk-free rate above which the excess returns should be measured In addition to how these four variables are treated, adjustments can be made to recognize the beta’s tendency to adjust toward either the industry average beta or the market portfolio beta (1.0). These adjustments are discussed later in this chapter. LENGTH OF THE SAMPLE OR LOOK-BACK PERIOD Most services that calculate beta use a two- to five-year sample measurement or look-back period. Five years is the most common historical period on which the forward estimate is based. This balances the use of a long history with the likelihood that betas are changing and betas estimated with ‘‘older’’ data may not be representative of future betas. The Morningstar Beta Book uses a 60-month look-back period for most stocks but includes a beta based on as few as 36 months if data are available for only this length of time. The Beta Book is published semiannually in hard copy and contains beta information on public companies. You can also get beta information by company on the Morningstar Web site (www.Morningstar.com). The example in Exhibit 10.1 uses a look-back period of 120 months. The beta estimate for J.B. Hunt displayed in Exhibit 10.2 uses a look-back period of 60 months. 2

Standard error of estimate of the beta coefficient ¼ beta=t-statistic.

122

Cost of Capital

But if the company characteristics change during the sampling period (e.g., major divestiture or acquisition), it may be more appropriate to use a shorter period. However, as the sampling period used is reduced, the accuracy of the estimate is generally reduced. FREQUENCY OF RETURN MEASUREMENT Returns for the publicly traded stock and the market returns may be measured on a daily, weekly, monthly, quarterly, or annual basis. Monthly is the most common frequency, although Value Line uses five years of weekly data. CHOICE OF MARKET INDEX The market index used in calculating beta could be any of these or, in some cases, another index:

Standard & Poor’s (S&P) 500 Index

New York Stock Exchange (NYSE) Composite Index NYSE and American Stock Exchange (AMEX) Index

NYSE, AMEX, and over-the-counter (OTC) Index Value Line Index

For an index to be representative of the market, it must be market-capitalization weighted. That is, the weight for each company in the index is determined by the market value of its equity. The sizes of the companies in the S&P 500 Index are so great that the index comprises about 70% of the total capitalization of all of the stocks constituting the combined indexes listed here. Furthermore, the broader market indexes listed correlate almost perfectly with the S&P 500 Index. As a result, it generally does not make a great deal of difference which index is used. Morningstar uses the S&P 500 in its calculations for the Cost of Capital Yearbook and the Beta Book. But the beta estimate for a specific company may underestimate that company’s true beta if the market index used during the look-back period is overweighted by a specific industry. The theory is that the market index should reflect the overall economy. But at times the market value for a particular segment of the economy will ‘‘take over’’ the market index (e.g., technology stocks in the late 1990s). For example, the risks for basic manufacturing companies appeared to have gone down in the late 1990s. Prior to the run-up in prices of technology stocks, basic manufacturing companies represented significant weight in the stock indices. Changes in the returns on basic manufacturing stocks were highly correlated to the changes in the stock indices. As technology stocks began to dominate the indices, the returns on the stocks of basic manufacturing companies were significantly less correlated with returns in the market indices, making it appear that their risks had been reduced. The underlying risks of basic manufacturing companies had not changed. But their observed betas then looked low compared to their long-term average betas. At times when one segment takes over the market index, alternative, longer look-back periods or alternative beta measurements, such as fundamental betas (discussed later in this chapter), may be more representative of the risks of the companies in other segments. CHOICE OF RISK-FREE RATE To avoid the maturity risk (interest rate risk) inherent in long-term bonds, the risk-free rate used to compute excess returns generally is either the Treasury bill (T-bill) rate or the interest yield from U.S. government bonds. Morningstar uses the 30-day T-bill rate in its calculations for the Cost of Capital Yearbook and the Beta Book. Differences in the choice of risk-free rate will cause differences in the beta estimates.

Levered and Unlevered Equity Betas

123

LEVERED AND UNLEVERED EQUITY BETAS Published and calculated betas for public stocks typically reflect the capital structure of each respective company at market values. These betas sometimes are referred to as levered betas, betas reflecting the leverage in the company’s capital structure. Levered betas incorporate two risk factors that bear on systematic risk: business (or operating) risk and financial (or capital structure) risk. Removing the effect of financial leverage leaves the effect of business risk only. The unlevered beta is often called an asset beta. Asset beta is the beta that would be expected were the company financed only with equity capital. When a firm’s beta estimate is measured based on observed historical total returns (as most beta estimates are), its measurement necessarily includes volatility related to the company’s financial risk. In particular, the equity of companies with higher levels of debt is riskier than the equity of companies with less leverage (all else being equal). If the leverage of the division, reporting unit, or closely held company subject to valuation differs significantly from the leverage of the guideline public companies selected for analysis, or if the debt levels of the guideline public companies differ significantly from one another, it typically is desirable to remove the effect that leverage has on the betas before using them as a proxy to estimate the beta of the subject company. This adjustment for leverage differences is performed in three steps: Step 1. Compute an unlevered beta for each of the guideline public companies. An unlevered beta is the beta a company would have if it had no debt. Step 2. Decide where the risk would fall for the subject company relative to the guideline companies, assuming all had 100% equity capital structures. Step 3. Lever the beta for the subject company based on one or more assumed capital structures (i.e., relever the beta). The result will be a market-derived beta specifically adjusted for the degree of financial leverage of the subject company. If the relevered beta is used to estimate the market value of a company on a controlling basis, and if it is anticipated that the actual capital structure will be adjusted to the proportions of debt and equity in the assumed capital structure, then only one assumed capital structure is necessary. However, if the amount of debt in the subject capital structure will not be adjusted, an iterative process may be required. The initial assumed capital structure for the subject will influence the cost of equity, which will, in turn, influence the relative proportions of debt and equity at market value. It may be necessary to try several assumed capital structures until one of them produces an estimate of equity value that actually results in the assumed capital structure. We discuss the iterative process in Chapter 17. This process of unlevering and relevering betas to an assumed capital structure is based on the assumption that the subject business interest has the ability to change the capital structure of the subject company. In the case of the valuation of a minority ownership interest, for example, the subject business interest may not have that ability and the existing capital structure should likely be the one assumed. FORMULAS FOR UNLEVERING AND LEVERING EQUITY BETAS The general relationship of the various formulas for unlevering and levering betas can be defined in terms of the next equations: Value of Levered Firm ¼ Value of Levered Assets or alternatively; Value of the Levered Firm ¼ Value of the Unlevered Assets þ Present Value of Tax Shield

124

Cost of Capital

Value of Levered Firm = Value of Levered Assets Capital Assets Value of Levered Assets

Value of Debt Capital minus Value of Tax Shield plus Value of Equity Capital

In this formulation, cost of debt capital is measured after the tax affect (kd) as the value of the tax deduction on interest payment reduces the cost of debt capital. Value of the Levered Firm = Value of the Unlevered Assets + Present Value of Tax Shield Assets

Capital

Value of Unlevered Assets

Value of Debt Capital

plus

plus

Value of Tax Shield

Value of Equity Capital

In this formulation, the cost of debt capital is measured prior to the tax effect (kd(pt)) as the value of the tax deduction on the interest payments equals the value of the tax shield.

Exhibit 10.3

Value of a Levered Firm

The value of the tax shield equals the present value of the expected tax deductions on interest payments for the debt capital financing. Graphically the relationship is shown in Exhibit 10.3. The values of debt capital and equity capital are expressed as market values. Various authors have proposed alternative formulas for unlevering and relevering betas. These formulas are generally functions of the risk of realizing the tax savings resulting from the tax deductions from the interest expense of the debt component of the capital structure. For example, if the guideline public company is losing money, has tax-loss carryforwards from prior-period losses, or is marginally profitable, the tax savings from current interest payments will not be recognized in the current period; in essence the cost of debt is greater by the loss or deferral of the income tax savings. In Appendix 10A we present a discussion and examples for these formulas:

Hamada formulas

Miles-Ezzell formulas Harris-Pringle formulas


Exhibit 10.4

125

Summary of Examples

Hamada Miles-Ezzell Harris-Pringle Practitioners’

Practitioners’ method formulas

Fernandez formulas

BU

BL

0.95 0.876 0.87 0.84

1.85 1.92 1.95 2.25

These formulas can be modified for the effects of warrants, employee stock options, and convertible debt.3 CHOOSING AMONG UNLEVERING OR LEVERING FORMULAS Each of these formulas assumes that there are no cost-negative effects to leverage (other than interest expense). That is, they assume that there are no negative impacts on the operations of the business from the amount of debt in the capital structure. We discuss such costs in Chapter 15. Exhibit 10.4 summarizes the beta estimates from applying the various formulas as shown in Exhibits 10A.1–10A.4. The guideline public company in each example had a published (levered) beta of 1.2. The Hamada formula, compared to the other formulas, assumes that more of the total risk was business risk rather than financial risk. That is, the Hamada formula understates the benefit from the tax shield for the example guideline public company with highly rated debt because it assumes that debt is constant. Upon relevering the concluded asset beta of the subject company of 0.90, the practitioners’ method formula results in the greatest increase in total risk due to the increased financial risk of the subject company compared to the guideline public company as it assumes the least benefit from the tax shield. Exhibit 10.5 shows an example of the different relevered betas and discount rates of common equity capital you get when applying Formulas 10A.2, 10A.4, 10A.6, 10A.8, and 10A.10. The choice of unlevering and relevering formula is important. The examples in Exhibit 10.5 indicate the impact on the resulting cost of equity capital estimates based on the underlying risk of realizing the tax shield. The less likely that tax deductions from interest payments will be realized in the periods in which interest is paid, the more risky is the leverage and the greater will be the resulting cost of equity capital. For example, if you are deriving a beta estimate for a subject business using guideline public company beta estimates and one or more of the guideline public companies carry a large amount of debt financing, the unlevered beta estimate will be overestimated if you use the Hamada formula Formula 10A.1. Assume that the higher-leveraged guideline public company likely cannot currently benefit from tax deductions on its interest expense. Its levered (observed) beta estimate equals 2.8 and its debt to capital ratio (at market value weights) equals 75%. Unlevering this beta estimate using Formula 10A.1, the Hamada formula, we get: BU ¼

3

2:8 ¼ 1:0 1 þ ð1 0:40Þð0:75=0:25Þ

Phillip R. Daves and Michael C. Ehrhardt, ‘‘Convertible Securities, Employee Stock Options, and the Cost of Equity,’’ Working paper, June 7, 2004.

126

Cost of Capital

Exhibit 10.5 Example of Applying Formulas for Relevering Beta Assume that for the subject company: Concluded asset beta for subject firm: 1.15 Tax rate: 0.30 Capital structure: 50% debt (market value of debt equity $25 million), 50% equity (market value of equity $25 million) Interest rate on debt: 8% Beta of debt capital: 0.20 Risk-free rate: 6% ERP: 5%

Using the Hamada formula (10A.2) we get: BL ¼ 1:15ð1 þ ð1 0:30Þ0:50=0:50Þ ¼ 1:15ð1 þ 0:70ð1ÞÞ ¼ 1:15ð1:7Þ ¼ 1:955 Discount rate for common equity: ke ¼ 0:06 þ 1:955 0:05 ¼ 15:78%

Using the Miles-Ezzell Formula (10A.4) we get: BL ¼ 1:15 þ

0:50 ð0:30 0:08Þ ð1:15 0:20Þ 1 0:50 ð1 þ 0:08Þ

¼ 1:15 þ 1 0:95 ð1 :022Þ ¼ 2:079 Discount rate for common equity: ke ¼ 0:06 þ 2:079 0:05 ¼ 16:40%

Using the Harris-Pringle Formula (10A.6) we get: BL ¼ 1:15 þ ð0:50=0:50Þð1:15 0:20Þ ¼ 1:15 þ 0:95 ¼ 2:10 Discount rate for common equity: ke ¼ 0:06 þ 2:10 0:05 ¼ 16:50%

Using the practitioners’ method formula (10A.8) we get: BL ¼ 1:15ð1 þ ð0:50=0:50ÞÞ

Discount rate for common equity:

¼ 1:15 þ 2 ¼ 3:15 ke ¼ 0:06 þ 3:15 0:05 ¼ 21:75%


127

Using the Fernandez formula (10A.10) we get: BL ¼ 1:15 þ ð0:50=0:50Þð1 0:30Þð1:15 0:20Þ ¼ 1:15 þ 1 0:70 0:95 ¼ 1:15 þ 0:665 ¼ 1:815 Discount rate for common equity:

ke ¼ 0:06 þ 1:815 0:05 ¼ 15:08% ke estimated using CAPM without regard to any size premium or company-specific risk premiums.

But if we use Formula 10A.7, the practitioners’ method formula, we get: BU ¼ :25 2:8 ¼ 0:70 Using the Hamada formula, 10A.1 results in an estimated asset beta that is too high because the formula implies that the value of the tax shield on the observed beta is too great. With respect to levered versus unlevered betas, the capital structure of companies often can change significantly over the measurement period of the beta. For example, a beta often is estimated using five years of returns in which, for the majority of time, a company was unleveraged. If at the end of the five-year period the company has become highly leveraged, the levered betas computed would incorporate very little leverage. Yet in unlevering the beta, the analyst would incorporate the current level of high leverage. Thus the unlevered beta could be highly underestimated. The reverse effect applies for a company that reduces its outstanding debt during the beta estimation period. There is no specific method of correcting for this other than accounting for capital structure changes when unlevering the beta. A reasonable approach might be to determine the average leverage for the company during the beta measurement period rather than the leverage at the end of the estimation period. The practitioner must apply judgment in unlevering guideline public company betas and relevering betas for subject businesses. Authors have concluded that of the formulas presented, the Miles-Ezzell and Harris-Pringle formulas are the most consistent if the assumption is that the firm will maintain a constant debt-to-equity ratio based on market value weights.4 The Fernandez formulas are the most consistent if the assumption is that the firm will maintain a fixed book value leverage ratio.5 Exhibit 10.6 outlines our guidance as to the formulas to apply if the assumption is that the firm will maintain a constant debt-to-equity ratio based on market value weights. ADJUSTING ASSET BETA ESTIMATES FOR DIFFERENCES IN OPERATING LEVERAGE Applying the unlevering formula to levered betas of guideline public companies adjusts for the effect of financial risk only and provides an estimate of business risk. (See Chapter 5 for a discussion of business risk.) But the operating leverage of the guideline companies may differ from that of the 4

5

Andre Farber, Roland Gillet, and Ariane Szafarz, ‘‘A General Formula for the WACC,’’ International Journal of Business (Spring 2006): 211–218; Enrique R. Arzac and Lawrence R. Glosten, ‘‘A Reconsideration of Tax Shield Valuation,’’ European Financial Management (2005): 458. Pablo Fernandez, ‘‘Levered and Unlevered Beta,’’ Working paper, April 20, 2006, 1.

128

Cost of Capital

Exhibit 10.6

Guidance for Applying Levering and Relevering Formulas

Formulas for Unlevering Returns of Guideline Public Companies: o Public debt (1) o

Debt not public Credit rated AAA to A-(2)

o

Debt not public Credit rated BBB+ to BBB-(3)

o

Debt not public Credit rated lower than BBB-(4) Formulas for Relevering Betas of Subject Company: Public debt (1)

o o

Debt not public Credit rated AAA to A-

o

Debt not public Credit rated BBB þ to BBB-

o

Debt not public Credit rated lower than BBB-

Formula 10A.3 or Formula 10A.5 Formula 10A.3 Assume Bd ¼ 0 Formula 10A.3 or Assume Bd ¼ 020 Formula 10A.7

Formula 10A.4 or Formula 10A.6 Formula 10A.4 Assume Bd ¼ 0 Formula 10A.4 or 10A.6 Assume Bd ¼ 030 Formula 10A.8

(1) Can calculate beta of debt. (2) AAA is the highest investment-grade S&P debt rating; Aaa is the equivalent Moody’s rating. The A-S&P debt rating is equivalent to the A3 Moody’s rating. (3) BBB- is the lowest investment-grade S&P debt rating; Baa3 is the equivalent Moody’s rating. It indicates adequate-payment capacity. (4) Ratings below the S&P debt rating BBB- and equivalent Moody’s rating Baa3 are speculative grade.

subject division, reporting unit, or closely held company. We can think of fixed operating costs in much the same way as interest expense of debt capital and apply the unlevering formulas to remove the effects of fixed expenses from the asset beta estimates. This ‘‘unlevered’’ asset beta can be thought of as an operating beta. We can then adapt the operating beta for the operating leverage of the subject company. We can use a variation of the Harris-Pringle formula to remove the effects of operating leverage, where the weight in the operating expense structure of fixed costs is equivalent to the weight of debt in the capital structure and the weight in the operating expense structure of variable costs is equivalent to the weight of equity in the capital structure. (Formula 10.3) Bop ¼

Bu ð1 þ F=VÞ

where: Bop ¼ Operating beta or beta with effects of fixed operating expense’s removed F ¼ Fixed operating costs (without regard to costs of financing) V ¼ Variable operating costs Once the operating leverage of the subject business is analyzed, then we can relever the operating beta to arrive at an estimated asset beta for the subject business. Formula 10.4 can be used for estimating the asset beta from the operating beta.

Levered and Unlevered Equity Betas Exhibit 10.7

129

Example of Computing Operating Beta and Recomputing Asset Betas

Example 1 Assume that for guideline public company A: Levered (published) beta: 1.2 Capital structure: 30% debt, 70% equity Operating cost structure: 75% fixed, 25% variable Beta of debt capital: Zero Using Formula 10A.5 we get: Using Formula 10.3 we get:

Bu ¼ 0:84 0:84 ð1 þ 0:75=0:25Þ 0:84 ¼ ð1 þ 3Þ 0:84 ¼ 4 ¼ 0:21

Bop ¼

Assume that you make the previous calculation for all guideline companies, the median operating beta was 0.40, and you believe the riskiness of the subject company is about equal to the median of the guideline companies. The next step is to estimate the asset beta for your subject company. Example 2 Assume for the subject company: Operating beta: 0.40 Operating cost structure: 25% fixed, 75% variable Using Formula 10.4 we get:

0:25 BU ¼ 0:40 þ 1 þ 0:75 ¼ 0:53

You can then relever the asset beta given the appropriate debt to equity structure, tax rate, and beta of debt capital for the subject business.

(Formula 10.4) BU ¼ Bop ð1 þ F=VÞ An example of the adjustment is shown in Exhibit 10.7.

ADJUSTING ASSET BETA ESTIMATES FOR EXCESS CASH AND INVESTMENTS The assets of the guideline public companies used in estimating beta often include excess cash and marketable securities. If you do not take into account the excess cash and marketable securities, you can arrive at an incorrect estimate of the asset beta for the operating business, which often is a guideline public company. This will lead to an incorrect estimate of the beta for the subject company. After unlevering the beta for the guideline public companies, you adjust the unlevered beta estimates for any excess cash or marketable securities held by each guideline public company. This adjustment is

130

Cost of Capital

Exhibit 10.8

Examples of Adjusting Asset Beta Estimates for Excess Cash and Investments

Assume that for guideline public company A: Levered (published) beta: 1.2 Tax rate: 0.40 Capital structure: market value of debt $500, market value of equity $1,000 Interest rate on debt: 10% Beta of debt capital: Zero Excess cash and investments: $300 Using Formula 10A.3 we get: $1;000 1:2 þ $500 0½1 ð0:4 0:1Þ=ð1 þ 0:1Þ $1;000 þ $500½1 ð0:4 0:1Þ=ð1 þ 0:1Þ $1;200 ¼ $1;000 þ $500½0:9721 $1;200 ¼ $1;000 þ $486:05 $1;200 ¼ $1;486:05 ¼ 0:808

BU ¼

Overall asset beta ¼ ½Asset beta for operations ðoperating assets=total assetsÞ

þ ½Asset beta for surplus assets ðsurplus assets=total assetsÞ Since the market value of invested capital equals $1,500 and excess cash and investments equals $300, operating assets equals $1,200. Assuming the excess investments are held in low-risk securities (i.e., shorter-term U.S. government bonds), the beta for surplus cash and investments will generally near zero and the second part of the equation equals zero. For guideline public company A, therefore, we have:

0:808 ¼ ½Asset beta of operations ð$1;200=$1;500Þ þ ½Zero Solving for the asset beta of operations, we have:

Asset beta for operations ¼ 0:808=ð$1;200=$1;500Þ ¼ 1:01 The adjusted asset beta of the operating business is 1.01.

based on the principle that the beta of the overall company is the market-value weighted average of the businesses or assets (including excess cash) comprising the overall firm. An example of the adjustment is shown in Exhibit 10.8.

MODIFIED BETAS: ADJUSTED, SMOOTHED, AND LAGGED Several research studies have provided significant support for two interesting hypotheses regarding betas: 1. Tendency toward industry or market average. Over time, a company’s beta tends toward its industry’s average beta. The higher the standard error in the regression used to calculate the beta, the greater the tendency to move toward the industry average.

Modified Betas: Adjusted, Smoothed, and Lagged

131

2. Lag effect. For all but the largest companies, the prices of individual stocks tend to react in part to movements in the overall market with a lag. The smaller the company, the greater the lag in the price reaction. Recognizing these phenomena, Paul D. Kaplan, himself a participant in some of the relevant studies, introduced new methodologies in the first 1997 Beta Book to reflect this latest research. He called it the ‘‘sum beta’’ because it averaged more than one month’s beta.6 But Morningstar stopped presenting the sum beta starting with the second 2001 edition because they did not know whether anyone was using it. ADJUSTED BETA INCORPORATES INDUSTRY NORM The adjusted beta is computed by a rather sophisticated technique called Vasicek shrinkage.7 The general idea is that betas with the highest statistical standard errors are adjusted toward the industry average more than are betas with lower standard errors. Because high-beta stocks also tend to have the highest standard errors in their betas, they tend to be subject to the most adjustment toward their industry average. This is the adjustment used in the Morningstar Beta Book. This adjusted beta is labeled Morningstar Beta in the Beta Book. SMOOTHED BETA An alternative adjustment that is used by Bloomberg and Value Line adjusts the historical beta to a ‘‘forward’’ estimated beta by averaging the historical beta estimate by two-thirds and the market beta of 1.0 by one-third. This adjustment is based on the assumption that over time, betas gravitate toward the market beta of 1.0. This is a mechanical adjustment and does not indicate that any adjustment to the data used in calculating the historical beta estimate was made. ‘‘SUM BETA’’ INCORPORATES LAG EFFECT A sum beta consists of a multiple regression of a stock’s current month’s excess returns over the 30day T-bill rate on the market’s current month’s excess returns and on the market’s previous month’s excess returns, and then a summing of the coefficients. This helps to capture more fully the lagged effect of comovement in a company’s returns with returns on the market (systematic risk).8 Because of the lag in all but the largest companies’ sensitivity to movements in the overall market, traditional betas tend to understate systematic risk. As the first 2006 edition of the Beta Book explains it, ‘‘Because of non-synchronous price reactions, the traditional betas estimated by ordinary least squares are biased down for all but the largest companies.’’9 Exhibit 10.9 shows the differences between OLS betas and sum betas for the companies comprising the Center for Research in Security Prices (CRSP) deciles. Exhibit 10.10 displays the differences in beta estimates by size of company for a sampling of industries.

6 7

8

9

Former Ibbotson Associates vice president and economist, now vice president, Quantitative Research, Morningstar, Inc. The formula, used in the Morningstar Beta Book, was first suggested by Oldrich A. Vasicek, ‘‘A Note on Using Cross-Sectional Information in Bayesian Estimation of Security Prices,’’ Journal of Finance (1973). The company beta and the peer group (industry) beta are weighted. The greater the statistical confidence in the company beta, the greater the weight on the company beta relative to the peer group beta. The sum beta estimates conform to the expectation that betas are higher for lower capitalization stocks. Research also shows that sum betas are positively related to subsequent realized returns over a long period of time; see Roger G. Ibbotson, Paul D. Kaplan, and James D. Peterson, ‘‘Estimates of Small-Stock Betas Are Much Too Low,’’ Journal of Portfolio Management (Summer 1997): 104–111. Morningstar, Beta Book, 2006 ed. (Chicago: Morningstar, 2006). The second 2001 edition discontinued presenting sum betas.

132

Cost of Capital

Exhibit 10.9

Comparison of OLS Betas and Sum Betas by Company Size

CRSP Market Value-Based Deciles Decile 1 2 3 4 5 6 7 8 9 10 Mid-Cap 3–5 Low-Cap 6–8 Micro-Cap 9–10

60 Months Ending December 2006

Largest

OLS Beta

Sum Beta

Difference

$ 398,907 17,292 7,913 4,221 2,812 1,985 1,343 956 639 321

0.96 0.96 1.09 1.08 1.08 1.15 1.23 1.26 1.29 1.17

0.95 1.11 1.31 1.40 1.39 1.46 1.64 1.75 1.84 1.70

0.01 0.15 0.22 0.32 0.31 0.31 0.41 0.48 0.54 0.53

7,913 1,985 639

1.08 1.21 1.24

1.36 1.59 1.79

0.27 0.39 0.54

}

Not much difference for larger companies

}

Difference is material for smaller companies

Source: Calculated (or derived) based on CRSP1 data, # 2006 Center for Research in Security Prices (CRSP1), Graduate School of Business, The University of Chicago, and Standard & Poor’s Compustat data. Calculations by Duff & Phelps LLC. Used with permission. All rights reserved.

The research suggests that this understatement of systematic risk by the traditional beta measurements accounts in part, but certainly not wholly, for the fact that small stocks achieve excess returns over their apparent CAPM required returns (where the market equity risk premium is adjusted for beta). The formula for the sum beta is: ðRn R f ;n Þ ¼ a þ Bn ðRm;n R f ;n Þ þ Bn1 ðRm;n1 R f ;n1 Þ þ e (Formula 10.5) Sum beta ¼ Bn þ Bn1 where: Rn ¼ Return on individual security subject stock in current month Rf,n ¼ Risk-free rate in current month a ¼ Regression constant Bn ¼ Estimated market coefficient based on sensitivity to excess returns on market portfolio in current month Rm ¼ Historical return on market portfolio ðRm;n R f ;n Þ ¼ Excess return on the market portfolio in the current month Bn1 ¼ Estimated lagged market coefficient based on sensitivity to excess returns on market portfolio last month ðRm;n1 R f ;n1 Þ ¼ Excess return on market portfolio last month e ¼ Regression error term The 2006 SBBI Valuation Edition has a table (Table 7–10) titled ‘‘Long-term Return in Excess of CAPM for Decile Portfolios of the NYSE/AMEX/NASDAQ, with Sum Beta,’’10 which is included as Exhibit 13.1 in Chapter 13 in this book. The table shows that the returns in excess of CAPM are much lower than for the OLS betas, reflecting the superiority of sum betas over OLS betas. Graph 7–5 in the 2006 SBBI Valuation Edition on the same page shows how much closer the portfolios track the 10

SBBI Valuation Edition 2006 Yearbook (Chicago: Morningstar, 2006), 143.

Modified Betas: Adjusted, Smoothed, and Lagged Exhibit 10.10

133

Comparison of OLS Betas and Sum Betas for Different Industries

Data as of December 2006 Median Count

OLS Beta

Sum Beta

Healthcare (SIC 80) All Companies Over $1 Billion* Under $200 Million*

117 23 48

0.69 0.30 0.74

1.19 0.70 1.29

Publishing (SIC 27) All Companies Over $1 Billion Under $200 Million

82 20 58

0.74 0.62 0.71

0.91 0.62 0.90

Petroleum & Natural Gas (SIC 1311) All Companies 84 Over $1 Billion 35 Under $200 Million 15

0.70 0.61 1.03

0.80 0.74 0.78

Computer Software (SIC 7372) All Companies Over $1 Billion Under $200 Million

360 47 217

1.85 1.87 1.66

2.32 1.99 2.40

Auto Parts (SIC 3714) All Companies Over $1 Billion Under $200 Million

60 12 26

0.90 1.02 0.55

1.40 1.35 1.18

Pharmaceutical (SIC 2834) All Companies Over $1 Billion Under $200 Million

278 54 119

1.27 0.85 1.31

1.68 0.94 1.87

*Market value of equity as of December, 2006.

Source: Compiled from Standard & Poor’s Compustat data. Calculations by Duff & Phelps LLC. Used with permission. All rights reserved.

Security Market Line, except for the tenth decile.11 If sum betas are used, the size effect (realized returns in excess of those predicted by CAPM) is greatly reduced. Sum betas for individual stocks can be calculated using Microsoft Excel and 61 months of return data, which is available from several sources, such as Compustat. Thus, even though the sum betas have been removed from the Beta Book, some analysts prefer to calculate their own sum betas for a peer group of public companies (which they use as a proxy for the beta of their subject private company in the context of CAPM), and thus make a smaller adjustment for the size effect. The theory is that this corrects for the larger size effect that is principally due to a misspecification of beta when using traditional OLS betas for the smaller companies. Exhibit 10.11 shows the sum beta estimate for J.B. Hunt. The sum beta estimate equals 1.979 (21.8% over the OLS beta estimate). While some may consider J.B. Hunt a large company, its market value of equity ($3,482 million) plus debt capital ($124 million) only ranks the company as a ‘‘midcap’’ firm (as of December 31, 2005). For smaller firms the difference can be even greater. Exhibit 10.12 shows the OLS and sum beta calculation for Martin Transport, Ltd. It had $393 million (as of December 31, 2005) market value of 11

Ibid.

134 Exhibit 10.11

Cost of Capital Example of Beta Estimation for J.B. Hunt Transport Services, Inc.

Ticker Symbol: JBHT SIC 4213: Trucking, Except Local Date: December 2005 Company: J.B. Hunt Transport Services, Inc., together with its subsidiaries, provides full-load freight transportation services in the United States, Canada, and Mexico. Calculate OLS Beta Calculate Sum Beta Number of Months of Data Regression Coefficients R-Squared Intercept Market Coefficient Market Lag Coefficient Beta T-Statistic Summary Statistics Average Return Standard Deviation

1.62 1.98 60 OLS 0.30 3.49%

1.624 4.95 Company 3.716% 12.742%

Correlation Matrix Company Market Market Lag

Company 1.000 0.545 0.162

Sum Beta 0.31 3.44% 1.599 0.380 1.979 Market 0.138% 4.275%

Market Lag 0.145% 4.275%

Market

Market Lag

1.000 0.065

1.00


Exhibit 10.12

Example of Beta Estimation for Marten Transport. Ltd.

Ticker Symbol: MRT SIC 4213: Trucking, Except Local Date: December 2005 Company: Marten Transport, Ltd., operates as a temperature-sensitive truckload carrier in the United States and Canada. Calculated OLS Beta Calculated Sum Beta Number of Months of Data Regression Coefficients R-Squared Intercept Market Market Lag Beta T-Statistic Summary Statistics Average Return Standard Deviation Correlation Matrix Company Market Market Lag

0.23 1.00 60 OLS 0.01 3.13% 0.230 0.230 0.89 Company 3.161% 8.991% Company 1.000 0.109 0.395

Sum Beta 0.16 3.02% 0.177 0.819 0.996 Market 0.138% 4.275% Market 1.000 0.065

Market Lag 0.145% 4.275% Market Lag

1.00


Peer Group Equity Beta

135

equity plus $50 million debt capital. Its OLS beta estimate is equal to 0.230 and its sum beta estimate is equal to 0.996 (a 333% difference). Appendix 10.B provides an example of estimating beta using the OLS beta and sum beta methods.

‘‘FULL INFORMATION’’ EQUITY BETA Betas for individual companies can be unreliable. There may be many divisions of the largest companies in the industry, making ‘‘pure play’’ (e.g., 75% of revenue from single Standard Industrial Classification [SIC] code) beta estimation difficult. Morningstar’s Beta Book includes industry betas calculated using full-information methodology (the full-information beta is seen as RIi in Appendix 10B and Chapter 19).12 After identifying all companies with segment sales in an industry, Morningstar calculates the beta. They then run a multiple regression with betas as the dependent variables (applying a weight to each beta based on its relative market capitalization to the industry market capitalization) and sales of the segments of each of the companies in the industry as the independent variable. That is, they are measuring the relative impact on the betas of companies in an industry based on the relative sales each company has within the industry. Measuring the impact on betas using segment sales data may present a problem in that the market weights profits, not sales. This procedure can overweight the relative importance of business segments with high sales and low profits. Appendix 10.B provides an example of estimating beta using the full-information methodology.

PEER GROUP EQUITY BETA Morningstar’s Beta Book also includes a peer group beta by industry. The peer group beta is calculated using the full-information betas by industry and weighting them for the subject company based on the sales by segments of the subject company. Exhibit 10.13 shows an example of calculating a peer group beta. Exhibit 10.14 is an excerpt from Morningstar’s second 2006 edition Beta Book (which is published twice annually). Note that it includes (1) traditional least squares regression beta (labeled raw beta, both levered and unlevered); (2) peer group beta; (3) adjusted beta (labeled Morningstar Beta, both levered and unlevered); and (4) the Fama-French 3-factor models. Chapter 19 on using Morningstar data shows an entire sample page from the 2006 edition Beta Book.

Exhibit 10.13

Example of Calculating the Peer Group Beta

Segment reporting lists ‘‘sales’’ in three different two-digit SIC codes.

SIC Code 1221 6794 4953

12

Industry Composite Beta

Company Sales in Industry ($ millions)

% of Company Sales in Industry

Sales-Weighted Beta Component

1.10 1.16 1.56

113.80 1.30 36.10

75.30% 0.84% 23.86%

.83 .01 .37

Totals

151.20

100.00%

1.21

Paul D. Kaplan and James D. Peterson, ‘‘Full Information Betas,’’ Financial Management (Summer 1998): 85–93.

136 Exhibit 10.14

Cost of Capital Excerpt from Second 2006 Edition Beta Book CAPM: Ordinary Least Squares Fama-French Three-Factor Model Levered Raw

Unlevered Pr Grp Ibbotson Raw Ibbotson FF

FF SMB SMB HML HML

FF

Beta t-Stat R-Sqr Beta APPA AP PHARMA INC 1.13 1.94 0.06 0.67

Beta 1.04

Beta 1.13

Beta 1.04

Beta t-Stat Prem t-Stat Prem t-Stat R-Sqr 1.01 1.64 3.02 3.65 2.24 2.46 0.08

APAT APA ENTERPRISES 2.46 3.98 0.21 0.63 INC

2.07

2.26

1.88

1.73 3.09 11.89 15.72 0.62 0.75 0.44

APAC APAC CUSTOMER 0.72 1.19 0.02 1.67 SERVICES INC

0.92

0.72

0.92

0.35 0.56 1.72 2.04 8.54

APA APACHE CORP

0.18 0.71 0.01 0.63

0.20

0.16

0.18

0.15 0.57 1.65 4.62 2.39 6.11 0.06

AIV APARTMENT INVT & MGMT-CLA

0.34 1.94 0.06 0.40

0.34

0.20

0.12

0.28 1.49 1.04 4.16 0.07 0.27 0.09

9.28 0.11

Source: Ibbotson Associates’ Beta Book, First 2006 Edition. Copyright # 2006 Ibbotson Associates, Used with permission. All rights reserved. (Morningstar, Inc. acquired Ibbotson in 2006.) To purchase copies of The Beta Book, or for more information on other Morningstar publications, please visit Global.Morningstar.com/DataPublications. Calculated (or derived) based on CRSP1 data, # 2006 Center for Research in Security Prices (CRSP1), Graduate School of Business, The University of Chicago.

FUNDAMENTAL EQUITY BETA As an alternative to using betas estimated from market information, you can estimate a fundamental beta, the sensitivity of subject company’s operating characteristics to changes in operating characteristics of the industry or market (segment or whole). Various studies have measured fundamental betas for publicly traded companies.13 For example, you can calculate a fundamental beta for a division, reporting unit, or closely held company by regressing quarterly changes in subject company operating profit to quarterly changes in the S&P 500 operating profit or to the appropriate S&P industry operating profit. Using operating profit for both the subject company and the S&P 500 should yield beta estimate equivalent to unleveled asset betas. Exhibit 10.15 displays fundamental betas for the sectors of the S&P 500 Index. The fundamental beta estimates can be useful during periods of large downward stock price adjustments when reliable beta estimates are difficult to derive. A source of fundamental beta estimates is Barra (www.mscibarra.com). Barra ‘‘predicted’’ betas are, in essence, historical OLS betas (calculated by regressing the log of 60 months of excess returns to log of excess returns of S&P 500) adjusted to be forward estimates. Barra actively cleans stock return data to ensure that stock splits, time gaps between trades, and other price inconsistencies are correctly accounted for. But historical betas do not recognize fundamental changes in a company’s operations during the prior 60 months and may be influenced by specific events that are unlikely to be repeated. Barra predicted betas are derived from a fundamental risk model. Risk factors are reestimated monthly and reflect changes in companies’ underlying risk structures in a timely manner. Barra uses 13

See, for example, Carolyn M. Callahan and Roseanne M. Mohr, ‘‘The Determinants of Systematic Risk: A Synthesis,’’ Financial Review (May 1989); Kee H. Chung, ‘‘The Impact of Demand Volatility and Leverage on the Systematic Risk of Common Stocks,’’ Journal of Business Finance and Accounting (1989); Aswath Damodaran, Investment Valuation, 2nd ed. (New York: John Wiley & Sons, 2002), 58–59.

Fundamental Equity Beta Exhibit 10.15

137

Fundamental Betas and OLS Betas by Sector Fundamental Beta

Consumer Discretionary Consumer Staples Energy Financials Healthcare Industrials Information Technology Materials Telecom Services Utilities

2.3 0.3 0.6 0.4 0.1 0.5 1.6* 0.8 2.2 1.5

Betas shown were calculated based on the relative quarterly change for the given sector compared to the change in the S&P 500 as a whole. The quarterly data used were from the time period from March 31, 1997 to December 31, 2006. The fundamental betas were calculated based on the operating earnings for each sector and the operating earnings of the S&P 500 as a whole. These are unlevered asset betas. *Excludes fourth quarter of 2001 with negative operating earnings considered abnormal.


company risk factors (company characteristics) plus industry risk exposures in developing their predicted betas. These risk factors are: Company Risk Factors Variability in markets—predictor of volatility of stock based on behavior of stock’s options; measures stocks’ overall volatility and response to market

Success compared to historical earnings growth information (i.e., analysts’ earnings estimates) ‘‘Size’’ based on log of market capitalization and log of total assets

Trading activity and number of analysts following stock Growth: Historical and expected future

Earnings-to-price ratio

Book-to-price ratio Earnings variability

Financial leverage Foreign income: Sensitivity to currency exchange rate changes

Labor intensity: Labor costs versus capital costs

Dividend yield ‘‘Low-cap’’ characteristics (extension of size model based on market capitalization)

Industry Risk Exposure Company categorized into up to 6 of 55 industry groups

Historical stock returns correlated with company risk factors, and these relationships are used to estimate company betas conditional on company characteristics Industry seems to be a dominant factor

138

Cost of Capital

Exhibit 10.16

Comparison of Barra Historical (OLS) Predicted Betas to Sum Betas Market Weighted Averages

Equal Weighted Averages

Medians

Largest Company Barra Barra Sum Barra Barra Sum Barra Barra Sum Decile Mkt Cap Count Historical Predicted Beta Historical Predicted Beta Historical Predicted Beta 1 2 3 4 5 6 7 8 9 10

$398,907 17,292 7,913 4,215 2,812 1,975 1,343 953 639 321

176 189 197 198 203 275 487 331 662 1,696

0.99 1.14 1.16 1.11 1.20 1.22 1.19 1.34 1.27 1.27

0.98 1.10 1.11 1.14 1.16 1.17 1.15 1.18 1.14 1.10

0.98 1.29 1.38 1.37 1.46 1.47 1.59 1.75 1.71 1.76

0.99 1.14 1.15 1.12 1.21 1.21 1.19 1.33 1.28 1.21

1.01 1.10 1.10 1.13 1.16 1.17 1.15 1.18 1.14 1.12

1.04 1.27 1.39 1.37 1.46 1.46 1.59 1.75 1.72 1.72

0.84 0.93 0.93 0.94 1.02 1.05 0.95 1.05 1.02 0.95

0.97 1.06 1.07 1.11 1.14 1.17 1.11 1.14 1.13 1.09

0.86 1.07 1.11 1.10 1.18 1.24 1.31 1.39 1.46 1.36

Source: Calculations by Duff & Phelps LLC. Used with permission. All rights reserved.

The predicted betas are based on Barra’s proprietary model. How do Barra predicted betas fare with small companies? Barra tends to report small predicted betas for small companies. Exhibit 10.16 shows a comparison of market capitalization data as of September 30, 2006, and Barra predicted betas as of December 2006 to historical beta estimates. We believe that since Barra bases its predicted beta estimate on OLS beta estimates, it may miss the lag effect on returns for smaller-company stocks captured by the sum beta. This is likely due to Barra’s focus on larger-company stocks.

EQUITY BETA ESTIMATION RESEARCH There continues to be much research on improving beta estimation techniques. For example, in one study the authors found that OLS beta estimates are subject to misestimation due to the small fraction of exceptionally large or small returns, called outliers, that are not predictable.14 They found that outliers occur more frequently with small companies. They recommend using weighted least squares estimation where outliers are discarded based on their impact on residual (error in the fit). Using a computational algorithm in a statistical modeling system S-Plus (MathSoft, 1999), they test the difference in predicting future or true beta. The authors found that when data do not contain influential outliers, OLS beta is the most precise estimate of true beta. But when influential outliers are present, OLS beta is an exceedingly poor estimate of true beta and the beta estimate is improved by removing outliers from the sample period. In another study, the authors extract ‘‘forward-looking’’ beta estimates from option pricing data on the Dow Jones 30.15 They use option models to estimate the implied volatility of the stocks and the covariance of the individual stocks with the market. They find that forward-looking betas extracted from data on liquid options often outperform historical market beta in following periods. They find that 180 days of historical excess returns provides the ‘‘best’’ estimate of forward-looking betas.

14

15

R.D. Martin and T.T. Simin, ‘‘Outlier-Resistant Estimates of Beta,’’ Financial Analysts Journal (September/October 2003): 56–69. Peter F. Christoffersen, Kris Jacobs, and Gregory Vainberg, ‘‘Forward-Looking Betas,’’ Working paper, February 27, 2007.

Estimation of Debt Beta

139

ESTIMATION OF DEBT BETA The risk of debt capital can be measured by the beta of the debt capital (in cases where the debt capital is publicly traded). The Bd can be measured in a manner identical to measuring the BL of equity. For a public company, a regression of returns provides an estimate of Bd. That estimate indicates how the market views the riskiness of the debt capital as the stock market changes (or a proxy for the economy). That risk is a function of the amount of debt capital in the capital structure, the variability of earnings before interest, taxes, depreciation, and amortization (EBITDA), the level of and variability of EBITDA/Sales, and so on. These are fundamental risks that the interest (and principal) will or can be paid when due. Research has found that long-run betas on debt capital have been in the range of 0.30 to 0.40.16 Betas of debt generally correlate with credit ratings. Exhibit 10.17 shows an example of the relationship between bond ratings and estimated betas of debt. The general formula for estimating the beta for debt (e.g., traded bonds) is:17 (Formula 10.6) Rd ¼ a þ Bd ðRm R f ð1 tÞÞ þ e

Exhibit 10.17 [A] Rating AAA AA A+ A A BBB BB B+ B B CCC

Estimated Beta of Debt Based on Credit Ratings [B] Default Spread (Over RFR) 0.53% 0.73 0.81 0.88 0.95 1.36 2.80 3.15 3.95 4.55 9.90

[C] Cost of Debt

[D] Beta of Debt

5.44% 5.64 5.72 5.79 5.86 6.27 7.71 8.06 8.86 9.46 14.81

0.05 0.07 0.08 0.09 0.10 0.14 0.28 0.32 0.40 0.46 0.99

½C ¼ ½B þ R f of 4.91% To estimate the beta of debt, we used the default spread at each level of debt, and assumed that half the risk is market risk ½D ¼ ð½B=5:0%Þ 0:5; where ERP ¼ 5:0% Note: This is an update of beta estimates that appear in Aswath Damodaran, Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2002), 413. Source: BondsOnline; FT Interactive Data; Bloomberg; calculations by Duff & Phelps LLC. Used with permission. All rights reserved.

16

17

See, for example, Tim Koller, Marc Goedhart, and David Wessels, Valuation: Measuring and Managing the Value of Companies, 4th ed. (Hoboken, NJ: John Wiley & Sons, 2005), 32. Simon Benninga, Financial Modeling, 2nd ed. (Cambridge: MIT Press, 2000), 414.

140

Cost of Capital

where: Rd ¼ Rate of return on subject debt (e.g., bond) capital a ¼ Regression constant Bd ¼ Estimated beta for debt capital based on historical data Rm ¼ Historical rate of return on the ‘‘market’’ t ¼ Marginal corporate tax rate e ¼ Regression error term Research has shown that longer-maturity bonds appear to have greater market risk and often have betas more closely resembling those of small-capitalization stocks compared to shorter-maturity bonds.18

OTHER BETA CONSIDERATIONS ‘‘Top-down’’ beta estimate for a publicly traded company comes from a regression of excess returns of the company’s stock to the excess returns of a market portfolio. Alternatively, a ‘‘bottom-up’’ beta can be estimated by:

Identifying the businesses in which the company operates

Identifying the unlevered (asset) betas of other companies in these businesses

Taking a weighted average of these unlevered betas, where the weights are based on the relative operating income Re-levering using the subject company’s debt/equity ratio

Bottom-up beta will give you a better estimate of the true beta when:

The standard error of the beta from the regression is high and the top-down beta for the subject company is very different from the average of the bottom-up betas for the businesses. Averaging across regression betas reduces standard error. Standard error pffiffiffi of average beta ¼ average standard deviation of individual company beta estimates= n. The subject company has reorganized or restructured itself substantially during the period of the regression.

And of course, you need to use bottom-up beta when the subject company is a division, reporting unit, or closely held business. The beta of a company after a merger is the market-value weighted average of the betas of the companies involved in the merger. The beta of an overall company is the market-value weighted average of the businesses (i.e., divisions and/or projects) or assets (operating assets and excess cash and investments) comprising the overall firm.

18

Tao-Hsien Dolly King and Kenneth Khang, ‘‘On the Importance of Systematic Risk Factors in Explaining the Cross-Section of Corporate Bond Yield Spreads,’’ Journal of Banking & Finance (2005): 3149.

Summary

141

SUMMARY An equity beta is a measure of the sensitivity of the movement in returns on a particular stock to movements in returns on some measure of the market. As such, beta measures market or systematic risk. In cost of capital estimation, beta is used as a modifier to the general equity risk premium in using the Capital Asset Pricing Model. There are many variations on the way betas are estimated by different sources of published betas. Thus, a beta for a stock estimated by one source may be very different from a beta estimated for the same stock by another source. Modern research is attempting to improve betas. Two such improvements implemented are the adjusted beta, which blends the individual stock beta with the industry beta, and the lagged beta, also called the sum beta, which blends the beta for the stock and the market during a concurrent time period with a beta regressed on the market’s previous period returns. These two adjustments both help to reduce outliers, thus perhaps making the betas based on observed historical data a little more representative of future expectations. The size premium in excess of CAPM is much lower using sum betas. However, betas are not very stable over time, especially for individual securities. Here are some recommendations from the authors. First we recommend graphing the returns over the sample or look-back period for any guideline public company you will be using in developing a beta estimate (time along the x-axis, returns or excess return along the y-axis). Similarly, graph the returns for the S&P 500 Index. You can then examine any changes in the relative pattern of returns over time. This will alert you to investigate if an underlying change has occurred in the public company (e.g., a merger, change in relative expectations about the company, etc.). Then you should investigate any changes. If the underlying fundamentals of the company have changed, that will indicate that a more recent period should be used in developing a beta estimate. This will often require calculating your own beta estimate. We recommend using sum beta calculations (whether you are using a pure-play or full-information beta methodology) for smaller public companies. We calculate OLS and sum beta estimates for all comparable public companies we are investigating for use in developing a better estimate. If the estimates differ, we gravitate to using the sum beta estimate. We recommend initially unlevering all the calculated beta estimates for the guideline public companies. Differences in leverage (both financial and operating leverage) are important differences in risk. For example, empirical evidence indicates that stock return volatility generally rises when stock prices decrease and stock return volatility generally falls when stock prices rise. One study found that approximately 85% of this change in stock return volatility is due to financial leverage and 15% is due to operating leverage.19 Care needs to be exercised in choosing the formula for unlevering betas. Generally use either the Miles-Ezzell, Harris-Pringle, or Fernandez formulas for unlevering public guideline company betas. The widely used Hamada formulas are generally inconsistent with capital structure theory and practice. The practitioners’ method formulas should be used only for companies with low debt ratings. Examine the differences in operating leverage (ratio of fixed operating costs to variable operating costs) among the guideline public companies and compare to the subject company. If significantly different, calculate the operating betas for each and adjust the unlevered beta for the subject company accordingly. Rank the companies by size characteristics (e.g., sales) other than market capitalization. Generally, do large companies have lower unlevered betas than smaller companies? If the unlevered betas for the smallest companies (even derived from sum beta methodology) are greater than the unlevered 19

Hazem Daouk and David Ng, ‘‘Is Unlevered Firm Volatility Asymmetric?’’ AFA 2007 Chicago meetings, January 11, 2007.

142

Cost of Capital

betas for larger companies, examine the reasons why the business risk of these smallest companies appears to be less than that of larger companies. They may be thinly traded, and conventional beta estimation methods may not be providing reliable beta estimates. Estimate the appropriate unlevered beta for the subject business (company, division, reporting unit, function within the firm) and relever that estimate based on the characteristics of the subject business (e.g., using its debt capacity, etc.). Finally, compare your estimated relevered beta with industry betas (e.g., Morningstar Cost of Capital Yearbook). Are differences sensible, given the differences between the subject business and the typical company comprising the industry statistics? Betas are an important element in estimating the cost of equity capital. The process of estimating beta requires considerable diligence, effort, and judgment on the part of the analyst.

Appendix 10A

Formulas and Examples for Unlevering and Levering Equity Betas

Hamada Formulas Miles-Ezzell Formulas Harris-Pringle Formulas Practitioners’ Method Formulas Capital Structure Weights Fernandez Formulas

HAMADA FORMULAS The Hamada formulas are commonly cited formulas for unlevering and levering equity beta estimates.1 The Hamada formula for unlevering beta is shown as Formula 10A.1. This is the formula used by Morningstar to unlever betas in its Beta Book. (Formula 10A.1) Bu ¼

BL 1 þ ð1 tÞWd =We

where: Bu ¼ Beta unlevered BL ¼ Beta levered t ¼ Tax rate for the company Wd ¼ Percent debt in the capital structure We ¼ Percent equity in the capital structure The companion Hamada formula for relevering beta is Formula 10A.2. (Formula 10A.2) BL ¼ Bu ð1 þ ð1 tÞWd =We Þ where the definitions of the variables are the same as in Formula 10A.1. 1

Robert S. Hamada, ‘‘The Effect of the Firm’s Capital Structure on the Systematic Risk of Common Stocks,’’ Journal of Finance (May 1972): 435–452.

143

144

Cost of Capital

The Hamada formulas are consistent with the theory that:

The discount rate used to calculate the tax shield equals the cost of debt capital (i.e., the tax shield has the same risk as debt).

Debt capital has negligible risk that interest payments and principal repayments will not be made when owed, which implies that tax deductions on the interest expense will be realized in the period in which the interest is paid (i.e., beta of debt capital equals zero). Value of the tax shield is proportionate to the value of the market value of debt capital (i.e., value of tax shield ¼ t Wd).

But the Hamada formulas are based on Modigliani and Miller’s formulation of the tax shield values for constant debt. The formulas are not correct if the assumption is that debt capital remains at a constant percentage of equity capital (equivalent to debt increasing in proportion to net cash flow to the firm in every period).2 The formulas are equivalent to assuming a steadily decreasing ratio of debt to equity value if the company’s cash flows are increasing. The formulas are often wrongly assumed to hold in general. An example of applying the Hamada formula is shown in Exhibit 10A.1.

MILES-EZZELL FORMULAS The Miles-Ezzell formulas are alternative formulas for unlevering and levering equity betas which assume that there is risk in the timely realization of the tax deductions for interest payments on debt capital.3 The Miles-Ezzell formula for unlevering beta is shown in Formula 10A.3. (Formula 10A.3) BU ¼

Me BL þ Md Bd ½1 ðt kdð ptÞ Þ=ð1 þ kdð ptÞ Þ Me þ Md ½1 ðt kdð ptÞ Þ=ð1 þ kdð ptÞ Þ

where: BU ¼ Unlevered beta of equity capital BL ¼ Levered beta of equity capital Me ¼ Market value of equity capital (stock) Md ¼ Market value of debt capital Bd ¼ Beta of debt capital t ¼ Tax rate for the company kd(pt) ¼ Cost of debt prior to tax affect We discuss the beta of debt capital in Chapter 10. The companion Miles-Ezzel formula for relevering beta is Formula 10A.4. (Formula 10A.4) " # ðt kdð ptÞ Þ Wd B L ¼ BU þ ðBU Bd Þ 1 We ð1 þ kdð ptÞ Þ where the definitions of the variables are the same as in Formulas 10A.1 and 10A.3. 2

3

Enrique R. Arzac and Lawrence R. Glosten, ‘‘A Reconsideration of Tax Shield Valuation,’’ European Financial Management (2005): 453–461. James A. Miles and John R. Ezzell, ‘‘The Weighted Average Cost of Capital, Perfect Capital Markets and Project Life: A Clarification,’’ Journal of Financial and Quantitative Analysis (1980): 719–730

Miles-Ezzell Formulas

Exhibit 10A.1

145

Computing Unlevered and Relevered Betas Using Hamada Formulas

Example 1 Assume that for guideline public company A: Levered (published) beta: 1.2 Tax rate: 0.40 Capital structure: 30% debt, 70% equity Using Formula 10A.1 we get: Bu ¼ ¼

1:2 1 þ ð1 0:40Þ0:30=0:70 1:2 1 þ 0:60ð0:429Þ

1:2 1:257 ¼ 0:95 ¼

Assume you made the previous calculation for all the guideline public companies, the median unlevered beta was 0.90, and you believe the riskiness of your subject company, on an unlevered basis, is about equal to the median for the guideline public companies. The next step is to relever the beta for your subject company based on its tax rate and one or more assumed capital structures. Example 2 Assume for the subject company: Unlevered beta: 0.90 Tax rate: 0.30 Capital structure: 60% debt, 40% equity Using Formula 10A.2 we get: BL ¼ 0:90ð1 þ ð1 0:30Þ0:60=0:40Þ ¼ 0:90ð1 þ 0:70ð1:5ÞÞ ¼ 0:90ð2:05Þ ¼ 1:85 Source: Shannon P. Pratt, Valuing a Business: The Analysis and Appraisal of Closely Held Companies, 5th ed. (New York: McGraw-Hill, 2007), Chapter 9. All rights reserved. Used with permission.

The Miles-Ezzell formulas are consistent with the theory that:

The discount rate used to calculate the tax shield equals the cost of debt capital (i.e., the tax shield has the same risk as debt) during the first year and the discount rate used to calculate the tax shield thereafter equals the cost of equity calculated using the asset beta of the firm (i.e., the risk of the tax shield after the first year is comparable to the risk of the operating cash flows). That is, the risk of realizing the tax deductions is greater than assumed in the Hamada formulas.

Debt capital bears the risk of variability of operating net cash flow in that interest payments and principal repayments may not be made when owed, which implies that tax deductions on the interest expense may not be realized in the period in which the interest is paid (i.e., beta of debt capital may be greater than zero).

146

Cost of Capital

Exhibit 10A.2 Computing Unlevered and Relevered Betas Using Miles-Ezzell Formulas Example 1 Assume that for guideline public company A: Levered (published) beta: 1.2 Tax rate: 0.40 Capital structure: 30% debt (market value of $15 million), 70% equity (market value of $35 million) Interest rate on debt: 7.5% Beta of debt capital: 0.10 Using Formula 10A.3 we get: BU ¼ ¼

$35m 1:2 þ $15m 0:10½1 ð0:4 0:075Þ=ð1 þ 0:075Þ $35m þ $15m½1 ð0:4 0:075Þ=ð1 þ 0:075Þ $42m þ $15m 0:10½0:972 $35m þ $15m½1 0:0279

$42m þ $1:458m $35m þ $14:5815 $43:458m ¼ $49:5815m ¼

¼ 0:876 Assume that you make the previous calculations for all guideline companies, the median unlevered beta was 0.90, and you believe the riskiness of your subject company, on an unlevered basis, is about equal to the median of the guideline companies. The next step is to relever the beta for your subject company tax rate and one or more assumed capital structures. Example 2 Assume for the subject company: Unlevered beta: 0.90 Tax rate: 0.30 Capital structure: 60% debt, 40% equity Interest rate on debt: 9.0% Beta of debt capital: 0.20 Using Formula 10A.4 we get: BL ¼ 0:90 þ

0:60 ð0:40 0:090Þ ð0:90 0:20Þ 1 0:40 ð1 þ 0:090Þ

¼ 0:90 þ 1:5 0:70 ð1 :033Þ ¼ 1:92

Market value of debt capital remains at a constant percentage of equity capital, which is equivalent to saying that debt increases in proportion to the net cash flow of the firm (net cash flow to invested capital) in every period.

An example of applying the Miles-Ezzell formulas is shown in Exhibit 10A.2. We begin with Example 1, where Bd, the beta of debt capital, equals 0.10 (i.e., the debt is highly rated and there is little risk to the debt capital that interest and principal will not be repaid when due for the guideline public company). But there is risk to the company that tax deductions on interest expense will not result in tax savings in the same period as the interest is paid in future years.

Harris-Pringle Formulas

147

In Example 2 we relever the beta, taking into account that the risk of debt capital is not negligible because the ratio of debt capital to equity capital is greater than that of the guideline public company in Example 1. We assume beta of debt capital Bd ¼ 0.20 (i.e., the debt is lower-rated than the debt of the guideline public company in Example 1).

HARRIS-PRINGLE FORMULAS The Harris-Pringle formulas are alternative formulas for unlevering and levering equity beta estimates that assume the tax shield is even more risky.4 Formula 10A.5 shows the Harris-Pringle formula for unlevering beta. (Formula 10A.5) Wd B L þ Bd We BU ¼ Wd 1þ We where the definitions of the variables are the same as in Formula 10A.6. The companion Harris-Pringle formula for relevering beta is Formula 10A.6. (Formula 10A.6) Wd BL ¼ BU þ ðBU Bd Þ We where the definitions of the variables are the same as in Formulas 10A.1 and 10A.3. The Harris-Pringle formulas are consistent with the theory that:

The discount rate used to calculate the tax shield equals the cost of equity calculated using the asset beta of the firm (i.e., the risk of the tax shield is comparable to the risk of the operating cash flows). That is, the risk of realizing the tax deductions is greater than assumed in the Hamada and Miles-Ezzell formulas. Debt capital bears the risk of variability of operating net cash flow in that interest payments and principal repayments may not be made when owed, which implies that tax deductions on the interest expense may not be realized in the period in which the interest is paid (i.e., beta of debt capital may be greater than zero) Market value of debt capital remains at a constant percentage of equity capital, which is equivalent to saying that debt increases in proportion to the net cash flow of the firm (net cash flow to invested capital) in every period.

An example of applying the Harris-Pringle formulas is shown in Exhibit 10A.3. We begin with Example 1, where Bd, the beta of debt capital, equals 0.10 (i.e., the debt is highly rated and there is little risk that interest and principal will not be repaid when due for the guideline public company). But there is risk to the company that tax deductions on interest expense will not result in tax savings in the same period as the interest is paid in future years. In Example 2, we relever the beta, taking into account that the risk of debt capital is not negligible because the ratio of debt capital to equity capital is greater than that of the guideline public company in Example 1. The debt is lower rated than the debt of the guideline public company in Example 1. 4

R. S. Harris and J. J. Pringle, ‘‘Risk-Adjusted Discount Rates—Extensions from the Average Risk Case,’’ Journal of Financial Research (Fall 1985): 237–244.

148

Cost of Capital

Exhibit 10A.3 Computing Unlevered and Relevered Betas Using Harris-Pringle Formulas Example 1 Assume that for guideline public company A: Levered (published) beta: 1.2 Tax rate: 0.40 Capital structure: 30% debt (market value of $15 million), 70% equity (market value of $35 million) Beta of debt capital: 0.10 Using Formula 10A.5 we get:

0:30 1:2 þ 0:10 0:70 BU ¼ 0:30 1þ 0:70 ¼ 0:87

Assume that you make the previous calculations for all guideline companies, the median unlevered beta was 0.90, and you believe the riskiness of your subject company, on an unlevered basis, is about equal to the median of the guideline companies. The next step is to relever the beta for your subject company tax rate and one or more assumed capital structures. Example 2 Assume for the subject company: Unlevered beta: 0.90 Tax rate: 0.30 Capital structure: 60% debt, 40% equity Beta of debt capital: 0.20 Using Formula 10A.6 we get: BL ¼ 0:90 þ

0:60 ð0:90 0:20Þ 0:40

¼ 1:95

PRACTITIONERS’ METHOD An alternative formulation often used by consultants and investment banks is referred to as the practitioners’ method. In this formula, no certainty for the tax deduction of interest payments is assumed. It has also been called the conventional relationship.5 The practitioners’ method formula for unlevering beta is shown in Formula 10A.7 (Formula 10A.7) BL Bu ¼ Wd 1þ We where the definitions of the variables are the same as in Formula 10A.1. The companion practitioners’ method formula for relevering beta is Formula 10A.8. 5

Tim Ogier, John Rugman, and Lucinda Spicer, The Real Cost of Capital (New York: Financial Times Prentice-Hall, 2004), 49.

Practitioners’ Method

149

(Formula 10A.8)

BL ¼ BU

Wd 1þ We

where the definitions of the variables are the same as in Formula 10A.1. The practitioners’ method formulas are consistent with the theory that:

The discount rate used to calculate the tax shield equals the cost of equity calculated using the asset beta of the firm (i.e., the risk of the tax shield is comparable to the risk of the operating cash flows). That is, the risk of realizing the tax deductions is greater than assumed in the Hamada and Miles-Ezzell. Market value of debt capital remains at a constant percentage of equity capital, which is equivalent to saying that debt increases in proportion to the net cash flow of the firm (net cash flow to invested capital) in every period.

This formula assumes the least benefit from tax deductions on interest payments and may be looked on as indirectly introducing costs of leverage beyond interest expense. An example of applying the practitioners’ method formulas is shown in Exhibit 10A.4. In Example 2, we relever the beta.

Exhibit 10A.4

Computing Unlevered and Relevered Betas Using Practitioners’ Method Formulas

Example 1 Assume that for guideline public company A: Levered (published) beta: 1.2 Capital structure: 30% debt (market value of $15 million), 70% equity (market value of $35 million) Using Formula 10A.7 we get: 1:2 BU ¼ 0:30 1þ 0:70 ¼ 0:84 Assume that you make the previous calculations for all guideline companies, the median unlevered beta was 0.90, and you believe the riskiness of your subject company, on an unlevered basis, is about equal to the median of the guideline companies. The next step is to relever the beta for your subject company tax rate and one or more assumed capital structures. Example 2 Assume for the subject company: Unlevered beta: 0.90 Capital structure: 60% debt, 40% equity Using Formula 10A.8 we get:

0:60 BL ¼ 0:90 1 þ 0:40 ¼ 2:25

150

Cost of Capital

CAPITAL STRUCTURE WEIGHTS Each of the formulas discussed—Hamada, Miles-Ezzell, Harris-Pringle, and practitioners’ method— are based on measuring debt capital and equity capital at market values. But in relevering the beta for a division, reporting unit, or closely held business, we do not know the market value of equity capital until we are completed with the valuation. Appendix 17A discusses the use of the iterative process using the Capital Asset Pricing Method (CAPM) for estimating the cost of equity capital, including the calculation of a relevered beta, assuming a constant capital structure in future years (i.e., debt capital changes in proportion to changes in the net cash flows to the firm). Appendix 17B discusses the use of the iterative process using CAPM for estimating the cost of equity capital, including the calculation of a relevered beta, assuming a varying capital structure in future years.

FERNANDEZ FORMULAS The levering and unlevering Fernandez formulas are useful in cases when it is assumed that the company maintains a fixed book value leverage ratio (ratio of debt to book value of equity remains constant).6 Chapter 17 discusses the use of market value versus book value weights. Formula 10.A.9 is the Fernandez formula for unlevering beta. (Formula 10A.9) Wd ð1 tÞBd BL þ W e Bu ¼ Wd ð1 tÞ 1þ We where the definitions of the variables are the same as in Formulas 10A.1 and 10A.3. The companion Fernandez formula for relevering beta is Formula 10A.10. (Formula 10A.10) Wd ð1 tÞðBU Bd Þ BL ¼ BU þ We where the definitions of the variables are the same as in Formulas 10A.1 and 10A.3. The Fernandez formula is consistent with the theory that:

Debt capital is proportionate to equity book value, and the increase in assets is proportionate to increases in net cash flow. Debt capital bears the risk of variability of operating net cash flow in that interest payments and principal repayments may not be made when owed, which implies that tax deductions on the interest expense may not be realized in the period in which the interest is paid (i.e., beta of debt capital may be greater then zero).

Formulas 10A.9 and 10A.10 are identical to Formulas 10A.1 and Formulas 10A.2 when Bd equals zero (i.e., equity capital is bearing all of the risk of the firm). de Bodt and Levasseur offer an alternative formula to the Fernandez formulas, consistent with the theory that debt capital is proportionate to equity book value, and the increase in assets is proportionate to increases in net cash flow.7 6 7

Pablo Fernandez, ‘‘Levered and Unlevered Beta,’’ Working paper, April 20, 2006. Eric de Bodt, and Michel Levasseur, ‘‘A Short Note on the Hamada Formula,’’ Working paper, March 26, 2007.

Appendix 10B

Examples of Computing OLS Beta, Sum Beta, and Full-Information Beta Estimates David Ptashne, CFA Introduction Computing OLS and Sum Beta Estimates—An Example Computing Historical Return Data Computing OLS Beta Estimate Computing Sum Beta Estimate Computing Full-Information Beta Estimate— An Example

INTRODUCTION Two common methods of calculating beta estimates for a subject public company involve regressing returns for the subject public company against the returns of a benchmark market index over the same periods (also known as ordinary least squares regression or OLS estimate of beta) or lagged returns (sum beta estimate of beta). These public company beta estimates can also be used as proxy beta estimates for a particular division, reporting unit, or comparable closely held company. An alternative method for estimating a beta for a subject company involves selecting and analyzing many guideline public companies that report segment data for businesses that are comparable to all or part of the business operations of the subject company. This ‘‘full-information’’ methodology takes into account the influence on beta of each of the business segments.

COMPUTING OLS AND SUM BETA ESTIMATES—AN EXAMPLE Estimating OLS beta and sum beta for a public company (subject company) as of a specific date (subject date) can be performed in the general steps shown using Microsoft Excel and common market data that can be obtained from a variety of industry data sources, such as Bloomberg or Standard & Poor’s (S&P) Compustat or Capital IQ. For purposes of these examples only, the beta estimates are based on a 12-month ‘‘look-back’’ period and are computed using 13 observations of historical monthly data for OLS beta and 14 observations of historical monthly data for sum beta. Note that a 12-month look-back period was chosen for purposes of this example for simplicity. Ordinarily, we recommend computing OLS and sum beta estimates using a longer look-back period, such as 60 months, which would require 62 months of historical data to compute both estimates accurately. We wish to thank Nick Arens and Brendan Achariyakosol for their assistance in preparing these examples.

151

152

Cost of Capital

Exhibit 10B.1

Example of Return Data for J.B. Hunt Transport Service, Inc. and S&P 500 Subject Company Return Data A Closing Price

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Dec05 Nov05 Oct05 Sep05 Aug05 Jul05 Jun05 May05 Apr05 Mar05 Feb05 Jan05 Dec04 Nov04 Oct04

22.640 22.390 19.410 19.010 18.070 19.630 19.230 20.080 19.545 21.885 23.595 22.060 22.425 20.100 20.430

B Dividends/ Share 0.060

0.060

0.060

0.060 0.015

Market Index Return Data

C Total Return

D S&P500 Index

E S&P500 Dividends

F Index Return

G Lagged Return

H Lead Return I

1.12% 15.66% 2.10% 5.20% 7.95% 2.39% 4.23% 2.74% 10.42% 7.25% 6.96% 1.36% 11.57% 1.54% 10.02%

1248.29 1249.48 1207.01 1228.81 1220.33 1234.18 1191.33 1191.50 1156.85 1180.59 1203.60 1181.27 1211.92 1173.82 1130.20

1.64 3.14 1.30 1.40 2.60 1.43 1.88 2.13 1.36 1.72 2.52 1.11 1.83 2.10 1.41

0.04% 3.78% 1.67% 0.81% 0.91% 3.72% 0.14% 3.18% 1.90% 1.77% 2.10% 2.44% 3.40% 4.04% 1.53%

3.78% 1.67% 0.81% 0.91% 3.72% 0.14% 3.18% 1.90% 1.77% 2.10% 2.44% 3.40% 4.04% 1.53% 1.08%

0.04% 3.78% 1.67% 0.81% 0.91% 3.72% 0.14% 3.18% 1.90% 1.77% 2.10% 2.44% 3.40% 4.04%

Note: Total return ¼ (this month’s price þ current dividend)=(last month’s price) 1; use price only if no dividend information available. Index return ¼ (this month’s price þ current dividend)=(last month’s price) 1. Source: Compiled from Standard & Poor’s Compustat data. Calculations by Duff & Phelps LLC. Used with permission. All rights reserved.

COMPUTING HISTORICAL RETURN DATA Exhibit 10B.1 presents the basic return data that must be calculated for the subject company and market index prior to computing the OLS and sum beta estimates. In this example, the subject company is J.B. Hunt Transport Services, Inc. (J.B. Hunt) and the subject market benchmark is the S&P 500 Index. This is the same company used in Exhibits 10.2 and 10.11 in Chapter 10. For simplicity, this example assumes that each beta estimate is to be computed based on a 12-month look-back period. The steps to obtain the required historical total return data for the subject company and benchmark index are:1 1. Column A. Obtain historical month-end closing prices (adjusted for splits, etc.) for your subject company for N þ 2 months, where N is the number of months in your look-back period. In this example, since our look-back period (N) is 12 months, we have obtained N þ 2, or 14 months, of historical data. 2. Column B. Obtain historical monthly cash dividends for your subject company for N þ 1 months. 3. Column C. Compute total monthly return for your subject company, which is defined as (current month’s price þ current month’s dividend)=(last month’s price) less 1. 4. Column D. Obtain historical month-end closing prices for your selected benchmark index for N þ 2 months. 5. Column E. Obtain historical monthly cash dividends for your benchmark index for N þ 1 months. 1

These steps assume the use of Microsoft Excel. Note that specific formulas entered into Excel to re-create this example might be slightly different, depending on placement of historical return data on your worksheet.

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6. Column F. Compute total monthly returns for your benchmark index, which is defined as (current month’s price þ current month’s dividend)=(last month’s price) less 1. 7. Column G. Compute the lagged return of the selected benchmark index. The lagged return is defined as (previous month’s price þ previous month’s dividend)=(price from 2 months ago). Compare columns F and G. Note that the lagged return G for the current month is simply the index return F from the previous month. This computation of lagged return will be used in the calculation of sum beta. COMPUTING OLS BETA ESTIMATE OLS beta can be computed in Excel in a single cell using this formula: OLS beta ¼ CovarðCompany; MarketÞ=VarpðMarketÞ where: Covar ¼ Covariance function in Excel, which returns the covariance (the average of the products of deviations for each data point pair) of two arrays. Company ¼ Array of the subject company’s total returns for months 0 to –11 for a 12-month look-back period. In Excel, it would be the range of cells that includes (C0:C-11). Market ¼ Array of benchmark index total returns for months 0 to –11 for a 12–month lookback period. In Excel, it would be the range of cells that includes (F0:F-11). Varp ¼ Variance function in Excel, which returns the variance of a user-defined population. If you were to follow the example exactly, the resulting OLS beta estimate would equal 2.309 for the 12–month look-back period. By following the same procedures, an OLS beta estimate for J.B. Hunt as of the subject date using the recommended 60-month look-back period was computed to be 1.624 (Exhibit 10.11). COMPUTING SUM BETA ESTIMATE Sum beta can be computed in Excel in three steps. Step 1. Compute the market coefficient in Excel in a separate cell using this formula: Market coefficient ¼ þ ðVarpðLaggedÞ CovarðMarket; CompanyÞCovarðMarket; LaggedÞ CovarðCompany; LaggedÞÞ=ðVarpðMarketÞ VarpðLaggedÞ CovarðMarket; LaggedÞ^ 2Þ where: Varp ¼ Variance function in Excel, which returns the variance of a user-defined population. Lagged ¼ Array of lagged total returns for months 0 to –11 for a 12-month look-back period. In Excel, it would be the range of cells that includes (G0:G-11). Covar ¼ Covariance function in Excel, which returns the covariance (the average of the products of deviations for each data point pair) of two arrays. Market ¼ Array of benchmark index total returns for months 0 to –11 for a 12-month lookback period. In Excel, it would be the range of cells that includes (F0:F-11).

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Company ¼ Array of subject company total returns for months 0 to –11 for a 12-month lookback period. In Excel, it would be the range of cells that includes (C0:C-11). Step 2. Compute the market lagged coefficient in Excel in a separate cell using this formula: Market lagged coefficient ¼ þ VarpðMarketÞ CovarðCompany; LaggedÞ CovarðMarket; LaggedÞ CovarðCompany; MarketÞ=ðVarpðMarketÞ

VarpðLaggedÞ CovarðMarket; LaggedÞ^ 2Þ

where: Varp ¼ Variance function in Excel, which returns the variance of a user-defined population. Market ¼ Array of benchmark index total returns for months 0 to –11 for a 12-month lookback period. In Excel, it would be the range of cells that includes (F0:F-11). Covar ¼ Covariance function in Excel, which returns the covariance (the average of the products of deviations for each data point pair) of two arrays. Company ¼ Array of subject company total returns for months 0 to –11 for a 12-month lookback period. In Excel, it would be the range of cells that includes (C0:C-11). Lagged ¼ Array of lagged total returns for months 0 to –11 for a 12-month look-back period. In Excel, it would be the range of cells that includes (G0:G-11). Step 3. Add the value computed in Step 1 to the value computed in Step 2. This is the sum beta estimate. If you were to follow the example exactly, the resulting sum beta estimate would equal 1.739 for the 12-month look-back period. By following the same procedures, a sum beta estimate for J.B. Hunt as of the subject date using the recommended 60-month look-back period was computed to be 1.979 (Exhibit 10.11).

COMPUTING FULL-INFORMATION BETA ESTIMATE—AN EXAMPLE A full-information beta estimate as of a specific date can be calculated in the general steps described using Excel and market data of guideline public companies obtained from industry data sources, such as Standard & Poor’s Compustat or Capital IQ or from the public filings of the selected guideline companies. For purposes of this example, we are attempting to estimate a full-information beta for Exxon Mobil Corp. (Exxon), which operates in the Oil and Gas industry. We further distinguished the businesses of Exxon for this example as upstream operations such as exploration (Upstream), downstream operations such as refining (Downstream), Chemicals, and Other. The Other segment was used as a reservoir for all sales and operating income that were not attributable to the Upstream, Downstream, or Chemical segments, such as corporate headquarters, pipelines, and finance; a wellselected group of guideline public yet non–pure play companies should represent businesses accounting for the bulk of the business of the subject company. We have gathered selected segment-level data for 19 guideline companies, including sales and operating income information for FY2006. Our guideline companies were selected because each report segment-level results for a segment of its operations that is comparable to one or more of the main business segments of Exxon excluding

Computing Full-Information Beta Estimate—An Example Exhibit 10B.2

155

Business Segment Data for Exxon

EXXON MOBIL CORP TICKER: XOM SIC: 2911 GICS: 10102010 FISCAL YEAR ENDED: December 2006 Segment Business Segments SIC Codes U.S. Upstream Non-U.S. Upstream U.S. Downstream Non-U.S. Downstream U.S. Chemicals Non-U.S. Chemicals Corporate & Financing

Segment % of Segment % of Segment % of Segment % of Segment % of Sales Total Oper Inc Total Depr Total Car Exp Total Assets Total

1,311 1,311 2,911 2,911

1,321 1,321 NA NA

6,054 26,821 93,437 205,020

2,911 2,911 NA

2,824 13,273 2,824 NA 37

1.66 5,168 13.08 1,263 11.06 1,942 12.56 21,119 9.64 7.34 21,062 53.32 6,482 56.78 9,735 62.96 75,090 34.29 25.57 4,250 10.76 632 5.54 718 4.64 16,740 7.64 56.10 4,204 10.64 1,605 14.06 1,757 I1.36 47,694 21.78 3.63 5.70 0.01

1,360 3,022 434

3.44 7.65 1.10

427 473 534

3.74 4.14 4.68

257 384 669

1.66 7,652 3.49 2.48 11,885 5.43 4.33 38,835 17.73


Other (i.e., Upstream, Downstream, or Chemicals). Note that in our example, the Other segment only accounted for 5.5% of sales and 6.16% of operating income for the group and 0.01% of sales and 1.1% of operating income for Exxon. Our list of guideline companies is not intended to be an exhaustive list of guideline companies for Exxon but rather was selected for demonstrative purposes. We are using data for the 19 guideline companies to estimate the beta for Exxon. We will then compare the full-information beta estimate with the OLS beta estimate for Exxon. In order to estimate a full-information beta, you must first aggregate the reported segment data for the subject company into the four identified segments. This is accomplished by the analyst with the assistance of the Standard Industrial Classification (SIC) codes assigned to each of the companies’ segments as provided by compustat. An example of this raw data for Exxon is shown in Exhibit 10B.2. Note that this information provides segment data for sales, operating income, depreciation, capital expenditures, and assets. For purposes of calculating this example’s full-information beta estimate, we will compare the estimate using sales and operating income as the weighting factors. That is, we will weight the influence of differences in segment sales and segment operating income in the betas of the 19 guideline public companies. The SIC codes and corresponding segments that were applicable in our example were identified to be: SIC Code (starting with):

Segment

131 & 132 291 282

Upstream Operations Downstream Operations Chemicals

Notice in Exhibit 10B.2 that two SIC codes are provided for some of the segments and none is provided for other segments. Compustat often assigns two SIC codes to a single segment;

156 Exhibit 10B.3

Cost of Capital Segment Operating Income Segment Operating Income

EXXON MOBIL CORP ANADARKO PETROLEUM CORP CANADIAN NATURAL RESOURCES CHESAPEAKE ENERGY CORP CHEVRON CORP CONOCOPHILLIPS DEVON ENERGY CORP DOW CHEMICAL DU PONT (E I) DE NEMOURS DUKE ENERGY CORP EL PASO CORP HESS CORP IMPERIAL OIL LTD MARATHON OIL CORP MURPHY OIL CORP OCCIDENTAL PETROLEUM CORP ROHM AND HAAS CO SUNCOR ENERGY INC TESORO CORP WILLIAMS COS INC

Upstream

Downstream

Chemicals

Other

Total Segments

26,230 5,370 2,745 3,192 13,142 10,324 4,496 – – 569 640 1,763 2,661 2,019 616 7,239 – 3,114 – 530

8,454 – – – 3,973 4,481 – – – – – 390 784 2,795 105 – – 328 1,476 –

4,382 – – – 539 – – 4,893 2,296 – – – 188 – – 901 649 – – –

434 (483) 35 147 – 745 – 510 1,987 3,360 1,110 (237) (114) – (83) (239) 106 – (159) 840

4 1 2 2 3 3 1 2 2 2 2 2 3 2 2 2 2 2 1 2


therefore, in some instances it is necessary to determine which SIC code and thus which segment label ‘‘best’’ defines the sales and operating income for that segment. In instances where two SIC codes fell into the same segment label—for example, US Upstream—since both 131 and 132 correspond with upstream operations, that segment is clearly labeled as an Upstream segment. However, in some segments, such as US Chemicals, the two SIC codes listed are 291 and 282, which correspond to Downstream Operations and Chemicals, respectively. In these instances, it is necessary to determine a single segment in which to classify the revenue and operating income. Based on the segment name, this is clearly more closely aligned to the Chemicals segment, and so we assigned it to Chemicals. Finally, notice that the business segment named Corporate & Financing has no SIC code assigned to it; in this instance, we determined that this should be categorized into the Other segment. Once all of the companies’ business segments were appropriately assigned into our four segment categories, we organized these data into a chart as shown in Exhibit 10B.3. (While this analysis was completed separately with sales and operating income data, for brevity we show only operating income results in the exhibit.) Using these amounts, we then created a segment weighting for each company. The segment weighting and the OLS beta estimates for each guideline public company (using a look-back period of 60 months) are displayed in Exhibit 10B.4. Although we are estimating the beta for Exxon using the other 19 guideline public companies, we display Exxon’s beta estimate in this exhibit for comparison purposes. The data in Exhibit 10B.4 are then used to run the regression necessary to estimate the fullinformation beta for the subject company (i.e., Exxon) with operating income weights.

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157

Segment Operating Income Weights and OLS Beta Estimates

Exhibit 10B.4

Segment Operating Income Weights

EXXON MOBIL CORP ANADARKO PETROLEUM CORP CANADIAN NATURAL RESOURCES CHESAPEAKE ENERGY CORP CHEVRON CORP CONOCOPH1LLIPS DEVON ENERGY CORP DOW CHEMICAL DU PONT (E I) DE NEMOURS DUKE ENERGY CORP EL PASO CORP HESS CORP IMPERIAL OIL LTD MARATHON OIL CORP MURPHY OIL CORP OCCIDENTAL PETROLEUM CORP ROHM AND HAAS CO SUNCOR ENERGY INC TESORO CORP WILLIAMS COS INC

OLS Beta

Upstream

Downstream

Chemicals

Other

0.763 0.623 0.316 0.596 0.743 0.642 0.562 1.066 1.072 1.185 2.219 0.458 0.291 0.560 0.418 0.498 0.992 0.371 1.723 2.726

66.4% 109.9% 98.7% 95.6% 74.4% 66.4% 100.0% 0.0% 0.0% 14.5% 36.6% 92.0% 75.6% 41.9% 96.5% 91.6% 0.0% 90.5% 0.0% 38.7%

21.4% 0.0% 0.0% 0.0% 22.5% 28.8% 0.0% 0.0% 0.0% 0.0% 0.0% 20.4% 22.3% 58.1% 16.5% 0.0% 0.0% 9.5% 112.1% 0.0%

11.1% 0.0% 0.0% 0.0% 3.1% 0.0% 0.0% 90.6% 53.6% 0.0% 0.0% 0.0% 5.3% 0.0% 0.0% 11.4% 86.0% 0.0% 0.0% 0.0%

1.1% 9.9% 1.3% 4.4% 0.0% 4.8% 0.0% 9.4% 46.4% 85.5% 63.4% 12.4% 3.2% 0.0% 13.0% 3.0% 14.0% 0.0% 12.1% 61.3%


Full-Information Regression Results Using Operating Income Weights

Exhibit 10B.5 Summary Output

Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

0.927 0.860 0.765 0.466 19

Anova

Regression Residual Total

df

SS

MS

F

Significance F

4 15 19

20.030 3.259 23.290

5.008 0.217

23.047

0.000

Coefficients Standard Error t Stat P-Value Lower 95% Upper 95% Intercept Upstream Downstream Chemicals Other

0 0.491 1.446 0.707 2.359

0.158 0.360 0.352 0.360

3.106 4.016 2.007 6.551

0.007 0.001 0.063 0.000

0.154 0.678 0.044 1.591

0.827 2.213 1.459 3.126


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Cost of Capital

In order to run the regression in Excel, we utilized the Regression function found under Tools ! Data Analysis. The Y Variable Range is the column of OLS beta estimates for the 19 guideline public companies, and the X Variable Range is the four columns of segment weights. In the regression tool, we then select ‘‘Labels’’ (to show the labels in the output), ‘‘Constant is Zero’’ (to force the intercept of the regression line through the origin), and a 95% confidence level. The regression output for operating income weights is shown in Exhibit 10B.5. According to these results, the divisional beta for the segment Upstream, for example, is 0.491 with a 95% confidence interval of 0.154 to 0.827. These results also show that the R-square value of the regression is 0.860. Using Formula 10B.1, the formula for full-information beta: (Formula 10B.1) RIiL ¼

n X ðWs BLs Þ 1

where: RIiL ¼ Full-information levered beta estimate of the subject company Ws ¼ Weight of each segment of the subject company BLs ¼ Levered beta estimate of each segment from the regression n ¼ Number of segments

Exhibit 10B.6

Full-Information Regression Results Using Sales Weights

Summary Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

0.941 0.885 0.795 0.422 19

Anova

Regression Residual Total

df

SS

MS

F

Significance F

4 15 19

20.614 2.676 23.290

5.153 0.178

28.890

0.000

Coefficients Standard Error t Stat P-Value Lower 95% Upper 95% Intercept Upstream Downstream Chemicals Other

0 0.384 0.739 0.625 2.087

0.205 0.195 0.348 0.249

1.874 3.794 1.796 8.378

0.081 0.002 0.093 0.000

0.053 0.324 0.177 1.556

0.821 1.154 1.367 2.618


Computing Full-Information Beta Estimate—An Example

159

we then calculate the full-information beta estimate for Exxon to be: BExxon ¼ ð0:491 0:664Þ þ ð1:446 0:214Þ þ ð0:707 0:111Þ þ ð2:359 0:011Þ ¼ 0:740 This full-information beta estimate of 0.740 closely approximates the OLS beta estimate for Exxon of 0.763 (difference of 3.1%). Similarly, we calculated the full-information beta estimate using sales weights. The regression results are displayed in Exhibit 10B.6. We then calculate the full-information beta estimate for Exxon to be: BExxon ¼ ð0:384 0:090Þ þ ð0:739 0:817Þ þ ð0:625 0:093Þ þ ð2:087 0:000Þ ¼ 0:696 This yields a full-information beta estimate of 0.696. This estimate compares with the OLS beta estimate for Exxon of 0.763 (difference of 8.8%). Why is the full-information beta estimate using operating income weights more accurate than using sales weights? Stock returns are driven by profits, not revenues. In the case of Exxon, the segment operating margin (operating income/sales) differed across segments. The Upstream segment represented 9% of the sales but 66.4% of the operating income (operating margin of 79.8%) while the Downstream segment represented 81.7% of the sales but only 21.4% of the operating income (operating margin of 2.8%).

Chapter 11

Criticism of CAPM and Beta versus Other Risk Measures

Introduction CAPM Assumptions and Beta as a Risk Measure Problems with CAPM Assumptions Is Beta a Reliable Risk Measure? Variations of CAPM Risk Measures Unique or Unsystematic Risk Total Risk Adjusting the Textbook CAPM Downside Risk Value at Risk Adjusting Beta for Risk of Company Size and Company-Specific Risk Duration Yield Spreads Fundamental Risk Summary Appendix 11A

INTRODUCTION Even though the Capital Asset Pricing Model (CAPM) is the most widely used method of estimating cost of equity capital, the accuracy and predictive power of beta as the sole measure of risk has increasingly come under attack. As a result, alternative measures of risk have been suggested. That is, despite its wide adoption, academics and practitioners alike have questioned the usefulness of CAPM in accurately estimating the cost of equity capital and beta as a reliable measure of risk. This chapter explores the criticisms and alternative measures of risk, and the resulting methods used to estimate the cost of equity capital.

CAPM ASSUMPTIONS AND BETA AS A RISK MEASURE Harry Markowitz, father of modern portfolio theory, organized the concepts and methodology of portfolio selection using statistical techniques.1 His goal was to help investors choose portfolios that were mean-variance efficient; that is, to choose stocks that minimize expected portfolio variance

1

Harry M. Markowitz, ‘‘Portfolio Selection,’’ Journal of Finance (March 1952): 77–91.

161

162

Cost of Capital

given an expected return, or alternatively, choose stocks that maximize an expected return given expected portfolio variance. Markowitz decided on variance as a risk measure because variance was ‘‘cheaper’’ to calculate, given the computing power at the time, application of the formula for portfolio was straightforward, and variance was a familiar concept. But Markowitz found that other measures of portfolio risk resulted in ‘‘better’’ portfolios; that is, portfolios with lower risk given an expected return. William Sharpe2 and John Lintner3 extended the Markowitz model by introducing assumptions of (1) complete agreement among investors on the joint probability distribution of asset returns from time ‘‘t–1’’ to time ‘‘t’’ (and its true probability distribution) (2) and unrestricted risk-free borrowing and lending. The results were that the only risk measure that mattered was beta. Beta measures expected market or systematic risk in the CAPM. The eight assumptions underlying the CAPM are: 1. Investors are risk averse. 2. Rational investors seek to hold efficient portfolios and, as a result, the portfolios they hold are fully diversified. 3. All investors have identical investment time horizons (i.e., expected holding periods). 4. All investors have identical expectations about such variables as expected rates of return and how capitalization rates are generated. 5. There are no transaction costs. 6. There are no investment-related taxes. (However, there may be corporate income taxes.) 7. The rate received from lending money is the same as the cost of borrowing money. 8. The market has perfect divisibility and liquidity (i.e., investors can readily buy or sell any desired fractional interest). These assumptions, combined with the assumption that security returns are normally distributed, result in beta being the correct risk measure. Because risk of an individual security is considered only in relation to other securities in the portfolio, all investors will choose to hold the market portfolio, M. Obviously, the extent to which these assumptions are not met in the real world will have a bearing on the validity of the CAPM. Traditional CAPM theory predicts that only market (or systematic) risk should be priced in equilibrium. But the inability to hold a market portfolio or choose not to hold a market portfolio will force investors to consider more than market risk.

PROBLEMS WITH CAPM ASSUMPTIONS A central conclusion of the textbook CAPM is that every investor holds the identical market portfolio stemming from two assumptions: homogenous expectations and no transaction costs. But many investors hold small portfolios.

2

3

William F. Sharpe, ‘‘Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk,’’ Journal of Finance (September 1964): 425–442. John Lintner, ‘‘The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets,’’ Review of Economics and Statistics (February 1965): 13–37.

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163

Market evidence indicates that individual investors do not wish to hold the market portfolio. Investors are willing to pay fees and expenses to hold nonindexed mutual funds.4 And holding a diversified portfolio is more difficult today than in the past. For example, the number of stocks needed to have a well diversified portfolio has increased due to the increase in unique risk (idiosyncratic risk or residual volatility of the portfolio).5 How diversifiable is unique or unsystematic risk? In one study, researchers compared the number of securities in a portfolio and the remaining idiosyncratic risk. Their empirical results demonstrate that even very large portfolios have substantial firm-specific risk. Failure to hold any portfolio except the market portfolio exposes an investor to the risk of experiencing firm-specific shocks. They conclude that since firm-specific risk is not easily diversifiable, then firm-specific risk may be ‘‘priced’’ (i.e., drive returns). Arguments that claim little added diversification is gained beyond, say, 30 or 50 stocks are erroneous.6 Another study concludes that investors need many more stocks to diversify and reduce their risk. In fact, investors need at least 164 stocks to have at most a 1% chance of underperforming U.S. government bonds.7 If there is no unrestricted risk-free borrowing and unrestricted short sales of risky assets are not allowed, then the market portfolio almost surely is not efficient, so the CAPM risk-return relationship does not hold. Further, research has shown that the unconditional security return distribution is not normal. Therefore, mean and variance of returns alone are insufficient to characterize return distributions completely. But despite its drawbacks, CAPM, when properly applied, may be a useful benchmark discount rate, even for investments in closely held businesses. It provides a benchmark measure of expected risk versus expected return. Given the problems with the underlying assumptions of CAPM, you must understand problems and benefits/issues of using alternative cost of capital methodologies and alternative risk measures. In fact, multiple methods of estimating the cost of equity capital with the conclusion drawn from the range of methods may be better than relying on a single methodology.

IS BETA A RELIABLE RISK MEASURE? Beta is a forward-looking concept similar to equity risk premium (ERP). The true beta must be estimated. Existing techniques for estimating beta generally use historical data and assume that the future will be sufficiently similar to the past to justify extrapolation of betas calculated using historical data. A series of studies have studied the predictive power of beta. That is, do ‘‘high-beta’’ stocks earn higher returns in future periods? (The theory implies that with a high beta, the market perceives the

4

5

6 7

See, for example, William N. Goetzmann and Alok Kumar, ‘‘Why Do Individual Investors Hold Underdiversified Portfolios?’’ Working paper, November 2004. The authors study diversification decisions of 60,000 individual investors during 1991 to 1996 and find that the majority are underdiversified with greatest underdiversification in retirement accounts; and Theodore Day, Yi Wang, and Yexiao Xu, ‘‘Investigating Underperformance by Mutual Fund Portfolios,’’ Working paper, May 2001; the authors demonstrate that the portfolios of equity mutual funds are not mean-variance efficient with respect to their holdings. John Y. Campbell, Martin Lettau, Burton G. Malkiel, and Yexiao Xu, ‘‘Have Individual Stocks Become More Volatile? An Empirical Exploration of Idiosyncratic Risk,’’ Journal of Finance (February 2001): 1–43; the authors demonstrate that a welldiversified portfolio needs at least 40 stocks in recent decades due to increasing trends in idiosyncratic volatility. James A. Bennett and Richard W. Sias, ‘‘How Diversifiable Is Firm-Specific Risk?’’ Working paper, February 2007. Dale L. Domian, David A. Louton, and Marie D. Racine, ‘‘Diversification in Portfolios of Individual Stocks: 100 Stocks Are Not Enough,’’ Working paper, April 4, 2006; In press The Financial Review.

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Cost of Capital

investment to be more risky). Similarly, do ‘‘low-beta’’ stocks earn lower returns in future periods? (The theory implies that the lower the beta, the market perceives the investment to be less risky.) For example, in one study, the author found that realized returns on stocks with high earnings-tomarket price were greater than predicted by beta and the realized returns on stocks with low earningsto-market price were lower than predicted.8 In another study, that author documents that the average realized returns on small stocks are greater than predicted by CAPM (i.e., the size effect).9 In another study, the author found that companies with high debt-to-market value of equity ratio earn too high a return relative to their betas.10 In two studies the respective authors investigated the relationship between average return and ratio of book-value-to-market-value of stocks. They found that returns on stocks with high book-value-to-market-value ratios had greater average realized returns than implied by their betas and returns on stocks with low book-value-to-market-value ratios realized lower average realized returns than implied by their betas. These studies imply that ratios involving stock prices have information about expected returns missed by betas. A stock’s price depends on expected cash flows and on expected returns that discount expected cash flows to present value.11 Eugene Fama and Kenneth French (FF) published two studies critical of beta. In one study they state: The efficiency of the market portfolio implies that (a) expected returns on securities are a positive linear function of their market betas (the slope in the regression of a security’s return on the market’s return), and (b) market betas suffice to describe the cross-section of expected returns.

They observed that the relation between market beta and average return is flat.12 In a follow-on study, they find that problems with CAPM using U.S. data show up in the same way in the stock returns of 12 non-U.S. major markets.13,14 Further, the CAPM cost of equity estimates for high-beta stocks are too high and estimates for low-beta stocks are too low relative to historical returns. And finally, CAPM cost of equity estimates for high book-value-to-market-value stocks (so-called value stocks) are too low and estimates for low book-value-to-market-value stocks (so-called growth stocks) are too high (relative to historical returns). The implications of their work are if market betas do not suffice to explain expected returns, the market portfolio is not efficient then CAPM has potentially fatal problems. They believe that their results point to the need for an asset pricing model not dependent on beta alone because beta as traditionally measured is not a complete description of an asset’s risk.15 FF go on to introduce another cost of equity capital model, the Fama-French 3-factor model based on an empirical study confirming 8

Sanjay Basu, ‘‘Investment Performance of Common Stocks in Relation to Their Price-Earnings Ratios,’’ Journal of Finance (1977): 129–156. 9 Rolf W. Banz, ‘‘The Relationship between Return and Market Value of Common Stocks,’’ Journal of Financial Economics (March 1981): 3–18. 10 Laxmi Chad Bhandari, ‘‘Debt/Equity Ratio and Expected Common Stock Returns: Empirical Evidence,’’ Journal of Finance (June 1988): 507–528. 11 See Dennis W. Stattman, ‘‘Book Value and Stock Returns,’’ The Chicago MBA: A Journal of Selected Papers 4 (1980): 25–45; and Ronald Lanstein, Kenneth Reid, and Barr Rosenburg, ‘‘Persuasive Evidence of Market Inefficiency,’’ Journal of Portfolio Management 11, no. 3 (1985): 9–17. 12 Eugene Fama and Kenneth French, ‘‘The Cross-Section of Expected Stock Returns,’’ Journal of Finance (June 1992): 427–486. 13 Eugene Fama and Kenneth French, ‘‘Value versus Growth: The International Evidence,’’ Journal of Finance (December 1998): 427–465. 14 One author has determined that a basic problem with these and other studies about beta is the result of using periodic returns, rather than continuously compounded returns. See Carl R. Schwinn, ‘‘The Predictable and Misleading Consequences When Using Periodic Returns in Traditional Tests of the Capital Asset Pricing Model,’’ Working paper, December 2006. 15 See Fama and French note 12 above.

Is Beta a Reliable Risk Measure?

165

that size, earnings-to-price, debt-to-equity, and book-value-to-market-value ratios all add to an explanation of realized returns provided by market betas. We discuss the FF 3-factor model in Chapter 16. After the FF 3-factor model was introduced, researchers discovered that this empirically based model was not very reliable either. In several studies researchers show that within portfolios formed on price ratios (e.g., book-value-to-market-value of stocks), stocks with higher expected cash flows have higher expected return, a measure not captured by FF 3-factor model or by CAPM.16 Still other studies found additional problems with the FF 3-factor model. A stock’s price is the present value of future cash flows discounted at the required return on the stock; therefore, given the same book-value-to-market-value ratio, expected return is positively related to cash flows. If two stocks have the same price, the one with higher expected cash flows must also have higher expected return. Given a particular book-value-to-market-value ratio, positive relation between expected profitability and expected return is a direct prediction of valuation theory. But the FF 3-factor model does not indicate which stocks with the same book-value-to-market-value ratio are expected to have higher returns.17 Other researchers found that traditional CAPM works over the long run, but not after 1963 (i.e., on the average, stocks with higher betas realized higher returns and stocks with lower betas realized lower returns). The book-value-to-market-value relationship better explains differences in returns after 1963 (though after 1980 its explanatory power is almost zero). They estimate CAPM with timevarying betas (dependent on economic conditions), constant market risk premium, and constant market volatility. They find that time-varying betas explain the book-value-to-market-value effect except for medium-size stocks. They question if the post-1963 problem with beta is just a small-samplesize’’ issue.18 Alternatively, could it be that the problem with CAPM and beta is the use of historical excess returns? Could it be that expectations can change so quickly that standard statistical methods for estimating beta using historical excess returns are just not sensitive enough to measure those expectations? In another study of CAPM and beta, the authors use expected returns from two sources (Value Line data for the period 1975 to 2001 and expected returns based on sell-side analysts as reported by First Call for the period 1997 to 2001) rather than historic realized returns to compare to beta estimates. They find that stocks’ expected returns for the periods using both sources were positively related to their estimated betas. Further they found that investors expected higher rates of return on small-cap stocks and on average received higher returns. That is, they found that the expected return on small (market value) stocks was greater than for large (market value) stocks after taking into account differences in beta estimates, consistent with the size effect.19 In still another study, researchers used information embedded in the prices of individual stock options and index options to compute forward-looking, or option implied, beta estimates. They 16

17

18 19

Richard Frankel and Charles M.C. Lee, ‘‘Accounting Valuation, Market Expectation, and Cross-Sectional Stock Returns,’’ Journal of Accounting and Economics (June 1998): 283–319; Patricia M. Dechow, Amy P. Hutton, and Richard G. Sloan, ‘‘An Empirical Assessment of the Residual Income Valuation Model,’’ Journal of Accounting and Economics (January 1999): 1–34; Joseph D. Piotroski, ‘‘Value Investing: The Use of Historical Financial Statement Information to Separate Winners from Losers,’’ Journal of Accounting Research 38 (Supplement 2000): 1–41. John Y. Campbell and Robert J. Shiller, ‘‘The Dividend—Price Ratio and Expectations of Future Dividends and Discount Factors,’’ Review of Financial Studies (May 1989): 195–228; Tuomo Vuolteenaho, ‘‘What Drives Firm Level Stock Returns,’’ Journal of Finance (February 2002): 233–264. Andrew Ang and Joseph Chen, ‘‘CAPM over the Long-Run: 1926–2001,’’ Working paper, October 2002. Brav, Lehavy, and Michaely, ‘‘Using Expectations to Test Asset Pricing Models,’’ Financial Management (Autumn 2005): 5–37.

166

Cost of Capital

compared their forward-looking beta estimates to historical-based beta estimates. They determined that the forward-looking beta estimates have better predictive power (i.e., high-beta stocks earn greater returns in future periods, etc.) than the best performing historical-based beta estimates in about half the cases. In total, the forward-looking beta estimates explain about 22% of the variation in the returns across the securities studied.20 Other researchers have shown that stock returns are not normally distributed; that problem alone creates the situation where beta is not the sole measure of risk.21 In summary, these studies show that beta alone is not a reliable measure of risk and realized future returns (at least not using betas drawn from historical excess returns). Yet CAPM and beta persist even today as the most widely used method of estimating the cost of equity capital. As one commentator said: In spite of the lack of empirical support, the CAPM is still the preferred model for classroom use in MBA and other managerial finance courses. In a way it reminds us of cartoon characters like Wile E. Coyote who have the ability to come back to original shape after being blown to pieces or hammered out of shape.22

VARIATIONS OF CAPM RISK MEASURES Could the market be measuring risk using a risk measure other than or in addition to beta? Could the market be measuring risk using some combination of total risk (market or systematic plus unique or unsystematic risk)? UNIQUE OR UNSYSTEMATIC RISK Unsystematic volatility of returns on stocks has increased over the past forty years. This increase has been linked to an increase in the fundamental volatility of firms’ earnings, cash flows, and sales.23 But is unsystematic risk priced by the market? That is, do firms with higher unsystematic risk earn higher returns (possibly the theory of higher-beta stocks earning higher returns)? One set of researchers confirm FF’s findings for a later time period, finding that the relationship of realized returns and beta measured using historical excess returns is essentially flat for the period 1963 to 1994. They confirm FF’s finding that size (as measured by market value) is a better proxy for risk than beta. While the volatility of the market as a whole during the 1980s and early 1990s was lower than earlier decades, volatility of individual stocks has been rising over time. The residual

20

21

22

23

Peter Christoffersen, Kris Jacobs, and Gregory Vainberg, ‘‘Forward-Looking Betas,’’ Working paper, August 4, 2006, rev. March 16, 2007. Fang and Lai in ‘‘Co-Kurtosis and Capital Asset Pricing,’’ Financial Review (May 1997): 293–307, derive a four-moment CAPM and show that systematic variance, systematic skewness, and systematic kurtosis contribute to the risk premium, not just beta; Arditti in ‘‘Risk and the Required Return on Equity,’’ Journal of Finance (March 1967): 19–36, demonstrates that skewness and kurtosis cannot be diversified away by increasing the size of the portfolios. Jagannathan and Wang, ‘‘The Conditional CAPM and the Cross-Section of Expected Returns,’’ Journal of Finance (August 1996): 3–53. Paul J. Irvine and Jeffrey Pontiff, ‘‘Idiosyncratic Return, Cash Flows, and Product Market Competition,’’ Working paper, April 2005.

Variations of CAPM Risk Measures

167

risk or idiosyncratic volatility of individual stocks is strongly related to size of company. They hypothesize that the size effect found by FF may be reflecting idiosyncratic volatility.24 In a later study, these same researchers study the impact on returns if some investors cannot hold the market portfolio. In those circumstances, idiosyncratic volatility (unsystematic risk) is useful in explaining expected returns across stocks. They find that stock returns directly covary with idiosyncratic risk. They find that the intercept (a) and beta estimated from CAPM are highly negatively correlated, reducing the explanatory power of beta. Idiosyncratic volatility is greater for small-size (small-market-value stocks) portfolios of stocks than for large-size (large-market-value stocks) portfolios. They do find that stocks with high betas tend to have high idiosyncratic risk. They recommend including an idiosyncratic risk measure.25 And in a subsequent study these same researchers use two methods to estimate idiosyncratic risk. Large stocks (as measured by market value) rather than small stocks appear to play an important role in the increasing idiosyncratic volatility of individual stocks because of increases in holdings by institutions that do not hold diversified portfolios. Stocks with greater expected growth have greater idiosyncratic volatility.26 What could be the tie between idiosyncratic volatility and small firms? Two other researchers examine averages of idiosyncratic risk in portfolios of firms grouped by market value and length of listing (proxy for age) using data from August 1963 to December 2001. They find that idiosyncratic volatilities of small firms (market capitalization below the median market capitalization of all issues: approximately 3% of total market capitalization in 1962–1969 and 1% in 2000–2001) are positive predictors of stock returns (and are unlike volatilities of bigger, older, and newer firms). They find that ‘‘size’’ is a significant predictor of returns primarily because it is a proxy for entrepreneurial risk. Entrepreneurial risk exists because investors with proprietary business income are significant stockholders and appear to invest in smaller, entrepreneurial companies. The authors find that entrepreneurial risk (measured by covariance with proprietary business income) is correlated with the positive effect of size on expected future returns. Entrepreneurial risk is an economically important determinant of variations in expected equity returns.27 In a further study using data from the London Stock Exchange from 1979 through 2003, researchers found that valuing idiosyncratic volatility of small companies (measured by market capitalization) is a predictor of returns for small-capitalization stocks (and insignificant as a predictor of returns for larger-capitalization stocks).28 In another study, the authors found no statistical significance of idiosyncratic volatility in predicting stock prices across both small and large company (by market capitalization) stocks.29 TOTAL RISK Shannon Pratt called his doctoral dissertation ‘‘The Relationship between Risk and the Rate of Return for Common Stocks.’’ He used the standard deviation of 12 quarters of past returns as his measure of risk. He divided the stocks into quintiles based on the values of the risk measures. It turned out 24 25 26

27

28

29

Burton G. Malkiel and Yexiao Xu, ‘‘Risk and Return Revisited,’’ Journal of Portfolio Management (Spring 1997): 9–14. Burton G. Malkiel and Yexiao Xu, ‘‘Idiosyncratic Risk and Security Returns,’’ Working paper (July 2000). Burton G. Malkiel and Yexiao Xu, ‘‘Investigating the Behavior of Idiosyncratic Volatility,’’ Journal of Business (October 2003): 613–644. David P. Brown and Miguel A. Ferreira, ‘‘Information in the Idiosyncratic Volatility of Small Firms and Entrepreneurial Risk,’’ Working paper (December 2005). Timotheos Angelidis and Nikolaos Tessaromatis, ‘‘Equity Returns and Idiosyncratic Volatility: UK Evidence,’’ Working paper, June 2, 2005. Note: idiosyncratic risk measured a residual from FF 3-factor model. Turan G. Bali and Nusret Cakici, ‘‘Idiosyncratic Volatility and the Cross-Section of Expected Returns,’’ Working paper, July 2006. Note: idiosyncratic risk measured a residual from FF 3-factor model.

168

Cost of Capital

Annual Rates of Return

Figure 1 Average Annual Rates of Return for Stock Portfolios of Different Risk Grades (Annual Rates Derived from Geometric Mean IPRs) .175 .170 .165 .160 .155 .150 .145 .140 .135 .130 .125 .120 .115 .110 .105 .100 0.95

1929–1960

A

1-Year Holding Periods 3-Year Holding Periods 5-Year Holding Periods 7-Year Holding Periods

B

C

D

E

Annual Rates of Return

Figure 2 Average Annual Rates of Return for Stock Portfolios of Different Risk Grades (Annual Rates Derived from Geometric Mean IPRs)

Exhibit 11.1

.175 .170 .165 .160 .155 .150 .145 .140 .135 .130 .125 .120 .115 .110 .105 .100 0.95

1931–1960

1-Year Holding Periods 3-Year Holding Periods 5-Year Holding Periods 7-Year Holding Periods A

B

C

D

E

Relationship between Risk and the Rate of Return for Common Stocks

Source: E. Bruce Fredrikson, Frontiers of Investment Analysis, 2nd ed. (Scranton, PA: Intext Educational Publishers, 1971), 345. Used with permission. All rights reserved.

to be a very good risk measure in the sense that the cross-sectional standard deviations of future oneyear returns rose dramatically with each quintile. The returns rose through the fourth quintile and dropped off a little for the fifth quintile. The results are shown schematically in Exhibit 11.1

Variations of CAPM Risk Measures

169

Other researchers have also found that total risk matters, not just market risk.30 To the extent that undiversified investors have an impact in the market, this should be reflected in pricing of the overall stock market. They cite another study in which those researchers found that over 45% of the net worth of investors with closely held businesses consists of private equity, with more than 70% concentrated in a single firm; these investors are not diversified.31 They then study the relation between average stock total risk and market return over time and find a significant positive relation between average stock variance and return on the market. If you view equity and debt as contingent claims on assets of a company, as the volatility of the assets increases, the value of the equity goes up at the expense of the debt holders. They conclude that total risk, including unique or idiosyncratic risk (volatility of returns), drives forecastability of the stock market. ADJUSTING THE TEXTBOOK CAPM Some authors acknowledge that it is appropriate to adjust textbook CAPM but only when considering the rate of return appropriate for an undiversified investor. This will result in a set of rates of return for diversified investors (those for whom in theory only beta risk matters) and another set of rates of return for undiversified investors (those for whom idiosyncratic risk matters). But the theory of cost of capital is that it should be a function of beta or market risk. The studies are mixed as to whether unsystematic risk is priced by the market. One study finding no statistical relationship between unsystematic risk and expected returns measured idiosyncratic risk in terms of residuals from the FF 3-factor model, not the textbook CAPM.32 The FF 3-factor model controls for size and other differences among the firms. Another study though finds a strong X link between implied idiosyncratic volatility derived from options and future returns.33 We, therefore, believe that it is appropriate to adjust cost of capital for smaller companies when using CAPM to estimate the cost of equity capital. One method of adjusting is adding a size premium (we discuss this adjustment in Chapter 12). An alternative method would be to adjust the beta estimate to account for risks not measured by the risk of the investment rather than to adjust for the characteristics of particular investors. How can a company estimate its cost of capital if it needs to ‘‘guess’’ if its investors are diversified? The studies we have just cited suggest that investors in general are much less diversified than predicted by the textbook CAPM. Other studies suggest that at least for small companies (measured by market capitalization), returns are a function of more than beta. One such method of adjusting beta is called total beta, beta adjusted for the total risk of the firm. The R2 of the regression of excess returns used to estimate beta regression measures proportion of risk that is market risk and can be used to adjust beta.34 For example, assume the average guideline public company beta ¼ 1.10, the correlation of the regression used to estimate beta ¼ .33 (i.e., R not R2; risk-free rate ¼ 6%; RPm equal 5%). We can calculate total beta as: Total beta ¼ 1:10=0:33 ¼ 3:30

30 31 32

33

34

Amit Goyal and Pedro Santa-Clara, ‘‘Idiosyncratic Risk Matters!’’ Journal of Finance (June 2003): 975–1008. Tobias J. Moskowitz and Annette Vissing-Jorgensen, ‘‘The Private Equity Premium Puzzle,’’ Working paper, November 2000. Turan G. Bali and Nusret Cakici, ‘‘Idiosyncratic Volatility and the Cross-Section of Expected Returns,’’ Working paper, July 2006. Dean Diavatopoulos, James S. Doran, and David R. Peterson, ‘‘Implied Idiosyncratic Volatility and the Cross-Section of Stock Returns,’’ Working paper, April 8, 2007. Aswath Damodaran, Damodaran on Valuation, 2nd ed. (New York: John Wiley & Sons, 2006): 58–59.

170

Cost of Capital

Total Cost of Equity ¼ 6% þ 3:30 ð5:0%Þ ¼ 22:5% ðnot CAPM 11:5%Þ: Total beta includes any risk premium for company size and company-specific risk premium inherent in the guideline public companies used in the analysis. DOWNSIDE RISK Conduct a survey of the ‘‘man on the street’’ and the common concept of risk is loss below some threshold. Several concepts of risk emerge:

Downside frequency. How often investment is likely to fall below threshold over specified time horizon

Average downside. Average shortfall when returns fall below threshold Semivariance. Variance on downside—combination of downside frequency and average downside35

If the distribution of security returns is not normally distributed, is the market measuring downside risk instead of equally weighting upside and downside risk? Two researchers compared the mean-variance (MV) CAPM with regular beta as risk measure to mean-semivariance (MS) CAPM with downside beta as risk measure. Downside beta measures comovement of returns with the market portfolio in falling markets. They found that return distributions are not normal and, as a result, MV CAPM and MS CAPM give different results. They found that:

Downside betas for low-beta portfolios are greater than regular beta; regular beta understates risk of low-beta stocks.

Downside betas for high-beta portfolios are smaller than the regular beta; regular beta overstates risk of high-beta stocks.

They found that combining downside risk and time variation works best in explaining stock returns. Further they found:

The role of downside betas is more pronounced during ‘‘bad times’’ (periods of low stock prices and resulting high dividend yield for the market).

Investors fear negative stock returns during ‘‘bad times’’ most. Downside beta better explains observed returns during those times.

Even then they found a residual size effect not fully explained by MS CAPM. That is, returns on small-cap companies are not fully explained by MS CAPM downside beta. MS CAPM assumes perfect capital markets and ignores transaction costs and market liquidity—issues that affect returns of small-cap companies the most.36 More researchers are now finding empirical results implying that the market prices stocks based on their downside risk. For example, in another study, the authors found that stocks that covary with the market when the market declines have high average return, which is consistent with investors 35

36

Philip S. Fortuna ‘‘Old and New Perspectives on Equity Risk,’’ Practical Issues in Equity Analysis (CFA Institute AIMR) (February 2000): 37–45. Thierry Post and Pim vanVliet, ‘‘Conditional Downside Risk and the CAPM,’’ Working paper, June 2004.

Variations of CAPM Risk Measures Exhibit 11.2

171

Comparison of OLS Betas, Sum Betas, and Downside Betas Different for Industries Median Count

OLS Beta

Sum Beta

Downside Beta

Healthcare (SIC 80) All Companies Over $1 Billion* Under $200 Million*

117 23 48

0.69 0.30 0.74

1.19 0.70 1.29

1.14 0.77 1.33

Publishing (SIC 27) All Companies Over $1 Billion Under $200 Million

82 20 58

0.74 0.62 0.71

0.91 0.62 0.90

0.75 0.64 1.42

Petroleum & Natural Gas (SIC 1311) All Companies Over $1 Billion Under $200 Million

84 35 15

0.70 0.61 1.03

0.80 0.74 0.78

1.00 0.64 1.42

360 47 217

1.85 1.87 1.66

2.32 1.99 2.40

2.20 1.91 2.23

60 12 26

0.90 1.02 0.55

1.40 1.35 1.18

1.22 1.03 1.22

278 54 119

1.27 0.85 1.31

1.68 0.94 1.87

1.76 1.11 1.87

Computer Software (SIC 7372) All Companies Over $1 Billion Under $200 Million Auto Parts (SIC 3714) All Companies Over $1 Billion Under $200 Million Pharmaceutical (SIC 2834) All Companies Over $1 Billion Under $200 Million

*Market value of equity as of December, 2006. Source: Compiled from Standard & Poor’s Compustat data. Calculations by Duff & Phelps LLC. Used with permission. All rights reserved.

placing greater weight on downside risk than on upside gains. Downside risk not simply measured by market beta. They found that pricing of downside risk is not subsumed by coskewness or liquidity risk either. They found that past downside beta is a good predictor of future covariation with down market movements.37 Exhibit 11.2 repeats the data from ordinary least squares (OLS) beta and sum beta from Exhibit 10.9 and adds downside beta estimates. Exhibit 11.2 compares OLS betas, sum betas, and downside betas for different industries. We show an example of calculating downside beta in Appendix 11A. Semi-variance is an alternative downside risk measure. It’s the ratio of semivariance of the individual security (variance on downside) to the semivariance of the market.38 VALUE AT RISK Value at risk (VaR) is a statistical measure of downside risk. VaR measures the largest percentage of the portfolio value that you might lose over a given time period, to a given degree of certainty, based 37 38

Andrew Ang, Joseph Chen, and Yuhang Xing, ‘‘Downside Risk,’’ Review of Economic Studies 19, no. 4 (2006): 1191–1139. Javier Estrada, ‘‘Downside Risk in Practice,’’ Journal of Applied Corporate Finance (Winter 2006): 117–125.

172

Cost of Capital

on historical average return and variability. For example, for a given asset held over the next six months, you might be 95% sure that the asset value will fall by no more than 15%. Value at risk has become widely used since the 1994 introduction of the J.P. Morgan RiskMetrics1 system, which provides the data required to compute VaR for a variety of financial instruments. New Federal Reserve Board rules require banks to compute the VaR of all their assets, and this total firm-wide VaR determines the bank’s capital requirements. ADJUSTING BETA FOR RISK OF COMPANY SIZE AND COMPANY-SPECIFIC RISK Assume that you use the expanded CAPM (Formula 8.5) to estimate the cost of equity capital. You now need to estimate the effect of expanding the CAPM model in, say, an option pricing model. The effect of adding risk premiums for company size and specific company risk is equivalent to increasing the variability of returns, as risk is no longer measured by beta alone. We can adjust beta in this way to get an expanded beta or Be (beta adjusted from the expanded CAPM): (Formula 11.1) EðRÞ ¼ R f þ ðBL RPM Þ þ RPs þ RPu EðRÞ R f ¼ ðBL RPM Þ þ RPs þ RPu Be ¼

ðEðRÞ R f Þ RPs RPu RPM RPM RPM

where: E(R) ¼ Expected rate of return Rf ¼ Rate of return on a risk-free security BL ¼ Levered beta for (equity) capital RPm ¼ Risk premium for the ‘‘market’’ RPs ¼ Risk premium for ‘‘small’’ stocks RPu ¼ Risk premium for company-specific or unsystematic risk attributable to the specific company Be ¼ Expanded beta (equity) Expanded beta incorporates both the small-company risk and company-specific risk. Assuming that any further traditional nondiversified risk is small, we can then derive variance of returns for use in option pricing models as follows: s 2 ¼ B2L s 2M þ s 2e Substituting expanded beta Be for B and assuming s 2e is close to zero, we get: s 2 ¼ Be s 2M where: s2 ¼ Variance of returns for subject company stock BL ¼ Levered beta (equity) s 2M ¼ Variance of the returns on the market portfolio (e.g., S&P 500) s 2e ¼ Variance of error terms Be ¼ Expanded beta (equity) s2 takes into account the entirety of the effect of all risk factors used in the expanded CAPM.

Duration

173

DURATION Is the length of time one expects to receive cash flows a good risk measure? That is, if one expects cash flows in early years, is that a less risky company or project than one where expected cash flows do not develop until years later? Researches have developed a measure of implied duration based on traditional measure of bond duration (see Chapter 6). They project future cash distributions for common equity using simple forecasting models based on historical financial data for 10 years and spreading the remaining market value implicit in observed stock price as a level perpetuity thereafter. They find:

Stock return volatility and betas are both increasing as equity duration increases. Book value-to-market value factor may be interpreted as a noisy duration factor. Stocks categorized as value stocks generally have shorter equity duration than stocks categorized as value stocks.39

But duration can be combined with the more commonly used risk measure variability to better quantify uncertainty as to both the amounts and the timing of expected economic income. Further, investors may not have the same risk tolerance. This does not mean that one should not measure the risk of the investment in determining the cost of equity capital. It simply means that there are different pools of investors with different risk tolerances; one pool may prefer longer term investments with greater absolute risk and another pool may prefer shorter term investments with lesser absolute risk. This is consistent with the so-called ‘‘clientele’’ explanation of investing; investments with different risks attract a different clientele of investors. It may be appropriate to think in terms of measuring the cost of equity capital in terms of the clientele attracted to investments with certain risk characteristics. Assuming that investment returns each future year are approximately normally distributed, the standard deviation of the expected value of the investment increases as the duration of the net cash flows increase, but the per-annum risk of the investment decreases because the marginal risk of an investment declines as a function of the square root of time. Risk, as measured by standard deviation, increases at a declining rate over time. For example, assume that a project with an initial investment of $100 (time ¼ zero to N). Assume that the present value of future net cash flows increase over time such that measuring the net present value of cash flows at the end of the first year (time ¼ 1 to N) equals $120. That is, the value of the investment increased $20. Assume that the standard deviation of the present value equals $10. That is, there is an approximate 2/3 chance that the present value of net cash flows from time ¼ 1 to N will be between $110 and $130. Assume that the value is expected to increase by $20 measured each year in the future (e.g., the value of net cash flows measured from the end of year 2 ¼ $140) and the standard deviation of that expected present value is $10. At the end of 5 years, the expected value is $200 (¼ $100 þ 5 years $20) with a standard deviation of $22.36 (¼ $10 H5). The normalized per-annum risk of the investment equals $4.47 (¼ $22.36/5 years). At the end of 25 years, the expected value of the investment is $600 (¼ $100 ¼ 25 years $20) with a standard deviation of $50 (¼ $10 H25). But the normalized per annum risk of the investment has decreased to $2 (¼ $50/25 years). The risk premium (i.e., cost of equity capital measured as the rate of return per annum) should be less for the longer-term investment compared to the shorter-term investment. 39

Patricia M. Dechow, Richard G. Sloan, and Mark T. Soliman, ‘‘Implied Equity Duration: A New Measure of Equity Risk,’’ Review of Accounting Studies (June 2004): 197–228.

174

Cost of Capital

If one assumes that there is a pool of investors with short-term investment horizons, those investors will likely invest in short-term investments (short duration) with low absolute risk. On the other hand, the pool of investors with long-term investment horizons may be attracted to longer-term investments with greater absolute risk but lower per annum risk; investors with longer-term investment horizons have an increased appetite for risk (as measured only by variance) knowing that over time the annualized variance is less. One study looks at the types of industries that are populated by companies owned by investors with generally shorter-term investment horizons (firms controlled by short-term institutional investors with ‘‘professional’’ management) and compares industry characteristics with the types of industries that are populated by companies controlled and managed by founding families (so-called ‘‘family firms) that attract long-term institutional investors.40 The authors find that as the cyclical nature of industries increases, the greater percentage of companies in the industry that are family firms also increases. Such firms can invest in (and create value from investing in) longer-term projects with lower per-annum returns because the appropriate cost of capital (measured as return per annum) is less. On the other hand, firms with short-term oriented investors and management can only invest in projects with higher per annum returns because the appropriate cost of capital (measured as return per annum) is greater.

YIELD SPREADS Can you use a company’s bond rating and the yield spread among bond ratings to directly estimate a company’s cost of equity capital? A company’s bond rating reflects its size and company-specific risk. In one study, the authors constructed an ex ante measure of expected equity return based on data from bond yield spreads (after adjusting bond yields for default risk, ratings transition risk, and tax spreads [differences in yields due to taxation of interest] between corporate bonds and U.S. government bonds). Their approach is based on the premise that providers of both debt capital and equity capital have contingent claims on the same set of assets; therefore, they must share the same risk factors that govern covariance between the underlying firm’s business risk (asset risk) and the economy. They found that beta plays a significant role in explaining variation in expected returns among firms (even after controlling for size and book-value-to-market-value ratio) and that the yield spread is highly correlated with systematic risk. They also found the company size premium statistically significant.41

FUNDAMENTAL RISK The Duff & Phelps Risk Study is discussed in Chapter 14. That research correlates realized equity returns (and historical realized risk premiums) directly with measures of company risk derived from accounting information. The measures of company risk derived from accounting information may also be called ‘‘fundamental’’ or ‘‘accounting’’ measures of company risk to distinguish these risk measures from a stock market-based measure of equity risk such as beta. 40

41

Thomas Zellweger, ‘‘Time Horizon, Costs of Equity Capital, and Generic Investment Strategies of Firms,’’ Family Business Review 20 No.1 (March 2007): 1–15. Murillo Campello, Long Chen, and Lu Zhang, ‘‘Expected Returns, Yield Spreads, and Asset Pricing Tests,’’ Working paper, January 2004.

Summary

175

The Risk Study examines three separate measures of risk: 1. Operating margin (the lower the operating margin, the greater the risk) 2. Coefficient of variation in operating margin (the greater the coefficient of variation, the greater the risk) 3. Coefficient of variation in return on equity (the greater the coefficient of variation, the greater the risk) Other authors also study fundamental risk. For example, in one recent study, the authors identified four cash-flow-related factors for explaining returns: earnings yield, capital investment, changes in profitability, and growth opportunities. The cash-flow-related factors plus any change in the discount rate form a full set of information associated with returns. Their model explains approximately 17% of variation in annual stock returns with earnings yield and changes in profitability being the most important factors and the change in the discount rate over time the least important factor.42 In another study, the authors derived a simplified covariance risk adjustment based on accounting variables. They use covariance of excess firm return on equity (ROE) (residual or abnormal earnings is equal to income minus a charge for the use of capital measured by the beginning book value times the cost of capital) with market excess ROE (developing a fundamental or accounting beta) and ROE of company size as measured by market capitalization and book-value-to-market-value factors as accounting-based risk measures to estimate the covariance risk of the firm. They find that valuation errors are reduced (i.e., expected returns are more accurately measured) compared to CAPM and FF 3-factor models.43

SUMMARY The conclusion that can be reached by studying the research reviewed in this chapter is that textbook CAPM and its sole risk measure, beta, while theoretically appealing and useful tools for understanding risk, are not reliable measures alone for measuring the cost of equity capital for many firms. This fact has caused academics and practitioners alike to look beyond the textbook CAPM. As the authors of one paper stated: While the CAPM ‘‘has been the model on which most finance theory and practice was built . . . tests of CAPM by F-F (1992) revealed that the model is no longer able to explain the cross-section of asset returns.’’44

Accurately measuring risk is the subject of continued research. We conclude that beta does not fully measure the risk of most securities, especially securities of smaller companies. We recommend that analysts use other risk measures beyond just beta, particularly for smaller companies. We recommend that the analyst use multiple estimates of risk (for example, OLS beta, sum beta, downside beta), compare the results and use judgment to decide on which estimate best represents the risk of the subject company. Mechanical use of beta estimates from a published service often leads to erroneous estimates of the cost of equity capital. 42

43

44

Peter F. Chen and Guochang Zhang, ‘‘How Do Accounting Variables Explain Stock Price Movements? Theory and Evidence,’’ Working paper, June 2006. Alexander Nekrasov and Pervin K. Shroff, ‘‘Fundamentals-Based Risk Measurements in Valuation,’’ Working paper, January 2007. Qing Li, Maria Vassalou, and Yuhang Xing, ‘‘An Investment-Growth Asset Pricing Model,’’ AFA 2002 Atlanta Meetings, March 7, 2001.

Appendix 11A

Example of Computing Downside Beta Estimates David Ptashne, CFA Introduction Computing Downside Beta Estimates

INTRODUCTION This appendix is a continuation of Appendix 10B, which presents examples of how to calculate historical returns, ordinary least squares (OLS) beta, and sum beta. Here we present an example of how to compute downside beta for a guideline public company. Similar to OLS beta and sum beta, this guideline public company downside beta estimate can be used as a proxy beta estimate for a division, reporting unit, or closely held company.

COMPUTING DOWNSIDE BETA ESTIMATES Estimating downside beta for a public company (subject company) as of a specific date (subject date) can be performed in the general steps shown using Microsoft Excel and common market data that can be obtained from industry data providers, such as Standard and Poor’s (S&P) Compustat or Capital IQ. For purposes of this example, we have assumed that the downside beta estimate will be based on 12 months of observed returns, which is computed using 13 observations of historical monthly data (as discussed in Appendix 10B). Note that a 12-month look-back period was chosen for purposes of this example only. Ordinarily, we recommend computing this risk measure using a longer period, such as a 60-month look-back period, which would require 61 months of historical data. In this example, the subject company is J.B. Hunt Transport Services, Inc. (J.B. Hunt) and the subject market benchmark is the S&P 500 Index. Exhibit 11A.1 presents historical returns for our subject company over the look-back period. For more detail regarding the computation of historical returns, see Appendix 10B.2. The next example presents the computation of the downside beta with respect to the average total return over the look-back period. This downside beta can be computed as shown:1

We wish to thank Nick Arens for his assistance in preparing these examples. These steps assume the use of Microsoft Excel. Also note that specific formulas entered into Excel to recreate this example might be slightly different depending on where on your worksheet you place historical return data.

1

176

Computing Downside Beta Estimates Exhibit 11A.1

177

Example of Return Data for J.B. Hunt Transport Service Inc. and S&P 500 Subject Company Return Data A Closing Price

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Dec05 Nov05 Oct05 Sep05 Aug05 Jul05 Jun05 May05 Apr05 Mar05 Feb05 Jan05 Dec04 Nov04 Oct04

22.640 22.390 19.410 19.010 18.070 19.630 19.230 20.080 19.545 21.885 23.595 22.060 22.425 20.100 20.430

B Dividends/ Share 0.060

0.060

0.060

0.060 0.015

Market Index Return Data

C Total Return

D S&P500 Index

E S&P500 Dividends

1.12% 15.66% 2.10% 5.20% 7.95% 2.39% 4.23% 2.74% 10.42% 7.25% 6.96% 1.36% 11.57% 1.54% 10.02%

1,248.29 1,249.48 1,207.01 1,228.81 1,220.33 1,234.18 1,191.33 1,191.50 1,156.85 1,180.59 1,203.60 1,181.27 1,211.92 1,173.82 1,130.20

1.64 3.14 1.30 1.40 2.60 1.43 1.88 2.13 1.36 1.72 2.52 1.11 1.83 2.10 1.41

F Index Return

G Lagged Return

H Lead Return

0.04% 3.78% 1.67% 0.81% 0.91% 3.72% 0.14% 3.18% 1.90% 1.77% 2.10% 2.44% 3.40% 4.04% 1.53%

3.78% 1.67% 0.81% 0.91% 3.72% 0.14% 3.18% 1.90% 1.77% 2.10% 2.44% 3.40% 4.04% 1.53% 1.08%

0.04% 3.78% 1.67% 0.81% 0.91% 3.72% 0.14% 3.18% 1.90% 1.77% 2.10% 2.44% 3.40% 4.04%

Note: Total return ¼ (this month’s price þ current dividend)/(last month’s price)1; use price only if no dividend information available. Index return ¼ (this month’s price þ current dividend)/(last month’s price)1. Source: Compiled from Standard & Poor’s Compustat data. Calculations by Duff & Phelps LLC. Used with permission. All rights reserved.

Step 1. Compute the average return for the subject company over the look-back period. In this case, the formula is ‘¼Average(C0:C-11)’. Let us call the cell that contains the subject company’s average return value ‘‘XX’’ for purposes of this example. Step 2. Compute the average return for the benchmark index over the look-back period. In this case, the formula is ‘¼Average(F0:F-11)’. Let us call the cell that contains the benchmark index’s average return value ‘‘YY’’ for purposes of this example. Step 3. In a separate cell in Excel, input ‘¼Linest(If(C0:C-11<XX, C0:C-11-XX,0),If(F0:F11

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