(continued from front flap)
• Expanded
ROGER J. GRABOWSKI is managing director of Duff & Phelps LLC.
Roger 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. He testified in
the Northern Trust case, the first U.S. Tax Court
decision that adopted the discounted cash flow
method to value the stock of a closely held business with the discount rate based on the capital
asset pricing model. Grabowski authors the annual
Duff & Phelps Risk Premium Report.
“This book is the most incisive and exhaustive treatment of this critical subject to date.”
—From the Foreword by Stephen P. Lamb, Esq., Partner, Paul, Weiss, Rifkind, Wharton & Garrison LLP,
and former vice chancellor, Delaware Court of Chancery
“Cost of Capital, Fourth Edition treats both the theory and the practical applications from the view
of corporate management and investors. It contains in-depth guidance to assist corporate executives
and their staffs in estimating cost of capital like no other book does. This book will serve corporate
practitioners as a comprehensive reference book on this challenging topic in these most challenging
economic times.”
—Robert L. Parkinson Jr., Chairman and Chief Executive Office, Baxter International Inc., and former dean,
School of Business Administration and Graduate School of Business, Loyola University of Chicago
“Shannon Pratt and Roger Grabowski have consolidated information on both the theoretical framework
and the practical applications needed by corporate executives and their staffs in estimating cost of
capital in these ever-changing economic times. It provides guidance to assist corporate practitioners from
the corporate management point of view. For example, the discussions on measuring debt capacity is
especially timely in this changing credit market environment. The book serves corporate practitioners as a
solid reference.”
—Franco Baseotto, Executive Vice President, Chief Financial Officer, and Treasurer, Foster Wheeler AG
“When computing the cost of capital for a firm, it can be fairly said that for every rule, there are a
hundred exceptions. Shannon Pratt and Roger Grabowski should be credited with not only defining the
basic rules that govern the computation of the cost of capital, but also a road map to navigate through
the hundreds of exceptions. This belongs in every practitioner’s collection of must-have valuation books.”
—Aswath Damodaran, Professor, Stern School of Business, New York University
“Pratt and Grabowski have done it again. Just when you thought they couldn’t possibly do a better
job, they did. Cost of Capital, Fourth Edition is a terrific resource. It is without a doubt the most
comprehensive book on this subject today. What really distinguishes this book from other such texts is
the fact that it is easy to read—no small feat given the exhaustive and detailed research and complicated
subject matter. This book makes you think hard about all the alternative views out there and helps
move the valuation profession forward.”
—James R. Hitchner, CPA/ABV/CFF, ASA, Managing Director, Financial Valuation Advisors; CEO, Valuation
Products and Services; Editor in Chief, Financial Valuation and Litigation Expert; and President,
Financial Consulting Group
“The Fourth Edition of Cost of Capital continues to be a ‘one-stop shop’ for background and current
thinking on the development and uses of rates of return on capital. While it will have an appeal for a
wide variety of constituents, it should serve as required reading and as a reference volume for students
of finance and practitioners of business valuation. Readers will continue to find the volume to be a solid
foundation for continued debate and research on the topic for many years to come.”
—Anthony V. Aaron, Americas Leader, Quality and Risk Management,
Ernst & Young Transaction Advisory Services
Capital
SHANNON P. PRATT, CFA, FASA,
ARM, MCBA, CM&AA, referred to
as the father of business valuations,
is the author of several bestselling
Wiley business valuation books and a
sought-after speaker at business valuation industry
conferences. He is the managing owner of Shannon
Pratt Valuations, Inc., and has served as supervisory
analyst for over 3,000 business valuation engagements in forty years and as an expert witness in
numerous state and federal courts on contested
business valuations.
Fourth Edition
Applications and Examples
This definitive text is an indispensable reference
tool for professional valuation practitioners as well
as attorneys and judges, investment bankers, CFOs,
academicians and students, and CPAs.
Cost of Capital
Cost of
The landmark book corporate treasurers, business
appraisers, CPAs, and valuation experts have come
to rely on, Cost of Capital lays out the basic tools to
use immediately when estimating cost of capital or
when reviewing an estimate. This dynamic author
team also analyzes criticism of major models for
developing estimates of the cost of capital in use
today, and also presents procedures for a number
of alternative models.
Praise for
Cost of
Pratt
Grabowski
chapters on cost of capital for
distressed companies
• Expanded discussions on the Morningstar
SBBI data on supply-side equity risk
premium and size premium
• Updated chapter on the cost of capital in
transfer pricing related to the valuation
of intangible assets under the new
cost-sharing regulations
FOURTH
EDITION
with WEBSITE
Capital
Applications and Examples
FOURTH EDITION
C
ost of capital estimation has long been
recognized as one of the most critical elements in business valuation, capital budgeting, feasibility studies, and corporate finance
decisions and is also the most difficult procedure
to assess and perform. Now in its fourth edition,
Cost of Capital: Applications and Examples addresses the most controversial issues and problems in
estimating the cost of capital.
Cost of
Capital
Applications and
Examples
Renowned valuation experts and authors Shannon
Pratt and Roger Grabowski present both the theoretical development of cost of capital estimation
and its practical application to valuation, capital
budgeting, and forecasting of expected investment returns encountered in current practice.
In this learning text/handy reference, Pratt and
Grabowski deftly review and explore the theory
of what drives the cost of capital, the models currently in use to estimate cost of capital, and the
data available as inputs to the models to estimate
cost of capital.
In this thoroughly updated and comprehensive
fourth edition, Cost of Capital summarizes the
results and practical implications of the latest
research—much of which is gleaned from unpublished academic working papers—and includes
scores of formulas and elucidating examples
throughout to enhance readers’ insights.
Pratt and Grabowski have updated their text to
include a host of new material, including:
A new chapter reconciling various forms of the
income approach
• Expanded material on estimating the equity risk
premium, chronicling the impact of the crisis of
2008–2010 and its impact on the cost of
equity capital
• Expanded material on estimating the cost of
debt capital and the impact of deleveraging on
the debt capacity of businesses
• An updated chapter covering cost of capital
for financial reporting under SFAS 141R, 142,
and 144 (with full cross-referencing to the new
FASB Accounting Codification), with examples
of inferring rates of return for underlying assets
from cost of capital of reporting units
• Expanded chapters on risk measures and their
relationship to cost of capital and companyspecific risk
•
FOURTH EDITION
with WEBSITE
Shannon P. Pratt
Roger J. Grabowski
(continued on back flap)
PMS 280
PMS 319
GLOSSY
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Cost of
Capital
Applications and Examples
Fourth Edition
SHANNON P. PRATT
ROGER J. GRABOWSKI
John Wiley & Sons, Inc.
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Copyright # 2010 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 for technical support, please contact our
Customer Care Department within the United States at (800) 762-2974, outside the United States at
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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
www.wiley.com.
Library of Congress Cataloging-in-Publication Data
Pratt, Shannon P.
Cost of capital : application and examples / Shannon P. Pratt, Roger J. Grabowski. – 4th ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-47605-5 (cloth); ISBN 978-0-470-88656-4 (ebk); ISBN 978-0-470-88662-5 (ebk);
ISBN 978-0-470-88671-7 (ebk)
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 2010
2010012330
658.150 2–dc22
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
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Dedication
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 Susie Wilder
Daughter-in-law Barbara Brooks
Randall and Kenny
Son-in-law Tim Wilder
John, Calvin, and Meg
Portland, OR
Springfield, VA
Son Steve Pratt
Daughter Georgia Senor
Daughter-in-law Jenny Pratt
Addy and Zeph
Son-in-law Tom Senor
Elisa, Katie, and Graham
Portland, OR
Fayetteville, AR
Mary Ann
Son Roger Grabowski Jr.
Daughter Sarah Harte
Daughter-in-law Misako Takahashi
Son-in-law Michael Harte
Rob and Sayaka
Tokyo, Japan
Kevin and Rosemary
Evanston, IL
Daughter Julia Grabowski, MD
Son Paul Grabowski
Pittsburgh, PA
Daughter-in-law Melissa Ruiz, MD
Chicago, IL
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Contents
About the Authors
ix
Foreword
xix
Preface
xxi
Acknowledgments
xxvii
Notation System and Abbreviations Used in This Book
xxix
PART ONE
Cost of Capital Basics
CHAPTER 1
Defining Cost of Capital
3
CHAPTER 2
Introduction to Cost of Capital Applications: Valuation and Project Selection
10
CHAPTER 3
Net Cash Flow: Preferred Measure of Economic Income
16
CHAPTER 4
Discounting versus Capitalizing
26
CHAPTER 5
Relationship between Risk and the Cost of Capital
45
CHAPTER 6
Cost Components of a Business’s Capital Structure
61
PART TWO
Estimating the Cost of Equity Capital and the Overall Cost of Capital
CHAPTER 7
Build-up Method
87
v
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CONTENTS
CHAPTER 8
Capital Asset Pricing Model
103
CHAPTER 9
Equity Risk Premium
115
APPENDIX 9A
Realized Risk Premium Approach and Other Sources of ERP Estimates
146
CHAPTER 10
Beta: Differing Definitions and Estimates
159
CHAPTER 11
Unlevering and Levering Equity Betas
185
CHAPTER 12
Criticism of CAPM and Beta versus Other Risk Measures
208
CHAPTER 13
Size Effect
232
CHAPTER 14
Criticisms of the Size Effect
262
APPENDIX 14A
Other Data Issues Regarding the Size Effect
279
CHAPTER 15
Company-specific Risk
287
CHAPTER 16
Distressed Businesses
313
CHAPTER 17
Other Methods of Estimating the Cost of Equity Capital
347
CHAPTER 18
Weighted Average Cost of Capital
369
CHAPTER 19
Global Cost of Capital Models
403
CHAPTER 20
Using Morningstar Cost of Capital Data
429
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Contents
vii
PART THREE
Corporate Finance Officers: Using Cost of Capital Data
CHAPTER 21
Capital Budgeting and Feasibility Studies
461
CHAPTER 22
Cost of Capital for Divisions and Reporting Units
468
CHAPTER 23
Cost of Capital for Fair Value Reporting of Intangible Assets
497
CHAPTER 24
Cost of Capital in Evaluating Mergers and Acquisitions
518
CHAPTER 25
Cost of Capital in Transfer Pricing
531
CHAPTER 26
Central Role of Cost of Capital in Economic Value Added
558
PART FOUR
Other Cost of Capital Considerations
CHAPTER 27
Handling Discounts for Lack of Marketability and Liquidity for
Minority Interests in Operating Businesses
571
CHAPTER 28
The Private Company Discount for Operating Businesses
587
CHAPTER 29
Cost of Capital of Interests in Pass-through Entities
597
CHAPTER 30
Relationship between Risk and Returns in Venture Capital and Private
Equity Investments
611
CHAPTER 31
Minority versus Control Implications of Cost of Capital Data
624
CHAPTER 32
How Cost of Capital Relates to the Excess Earnings Method of Valuation
634
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CONTENTS
CHAPTER 33
Adjusting the Discount Rate to Alternative Economic Measures
642
CHAPTER 34
Estimating Net Cash Flows
647
PART FIVE
Advice to Practitioners
CHAPTER 35
Common Errors in Estimation and Use of Cost of Capital
669
CHAPTER 36
Dealing with Cost of Capital Issues
684
Appendix I Bibliography
697
Appendix II Data Resources
725
Appendix III International Glossary of Business Valuation Terms
738
Index
747
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About the Authors
Dr. Shannon P. Pratt, CFA, FASA, ARM, MCBA, ABAR, 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. He regularly reviews
business valuation reports for attorneys in litigation matters.
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.
P r o f e s s i o n a l A f fi l i a t i o n s
Dr. Pratt is an Accredited Senior Appraiser and Fellow (FASA), Certified in Business Valuation, of the American Society of Appraisers (their highest designation)
ix
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ABOUT THE AUTHORS
and is also accredited in Appraisal Review and Management (ARM). He is a
Chartered Financial Analyst (CFA), a Master Certified Business Appraiser
(MCBA) and Accredited in Business Appraisal Review (ABAR) by 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 a teacher of
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 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 ed. (New York: McGraw-Hill, 2008); co-author,
Valuing Small Businesses and Professional Practices, 3rd ed., with Robert
Schweihs and Robert Reilly (New York: McGraw-Hill, 1998); co-author, Guide
to Business Valuations, 20th ed., with Jay Fishman, Cliff Griffith, and Jim
Hitchner (Fort Worth, TX: Practitioners Publishing Company, 2010); co-author,
Standards of Value, with William Morrison and Jay Fishman (Hoboken, NJ:
John Wiley & Sons, 2007); co-author, Business Valuation and Taxes: Procedure, Law, and Perspective, 2nd ed., with Judge David Laro (Hoboken, NJ:
John Wiley & Sons, 2010); and author, Business Valuation Discounts and Premiums, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2008); Business Valuation
Body of Knowledge: Exam Review and Professional Reference, 2nd ed. (Hoboken,
NJ: John Wiley & Sons, 2003); The Market Approach to Valuing Businesses, 2nd
ed. (Hoboken, NJ: John Wiley & Sons, 2005); and The Lawyer’s Business Valuation Handbook, 2nd ed. (Chicago: American Bar Association, 2010). 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,
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About the Authors
xi
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 are 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); In re: Louisiana Riverboat Gaming Partnership, et al. Debtors
(valuation of business enterprise owning two riverboat casinos and feasibility of
plan of reorganization); ABC-NACO, 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).
Previous Experience
Mr. Grabowski was formerly managing director of the Standard & Poor’s Corporate Value Consulting practice and 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 BBA–Finance from Loyola University of Chicago and
completed all coursework in the doctoral program, Finance, at Northwestern University, Chicago.
P r o f e s s i o n a l A f fi l i a t i o n s
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. He serves as Editor of the
Business Valuation Review, the quarterly journal of the Business Valuation Committee of the American Society of Appraisers.
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ABOUT THE AUTHORS
Publications
Mr. Grabowski authors the annual Duff & Phelps Risk Premium Report. He lectures and publishes regularly. Recent articles include ‘‘The Cost of Capital,’’ Journal
of Business Valuation, the Canadian Institute of Chartered Business Valuators,
August 2009; ‘‘Problemas relacionados con el cálculo del coste de capital en el entorno actual: actualizaciòn,’’ co-authored with Mathias Schumacher, Análisis Financiero Internactional, Sumario No 137 Tercer trimestre 2009; ‘‘Cost of Capital
Estimation in the Current Distressed Environment,’’ The Journal of Applied
Research in Accounting and Finance, July 2009; ‘‘Cost of Capital in Valuation of
Stock by the Income Approach: Updated for an Economy in Crisis,’’ with Shannon
P. Pratt, Jahreskonferenz der NACVA, Bewertungs Praktiker, January 2009; ‘‘Problems with Cost of Capital Estimation in the Current Environment—2008 Update,’’
Business Valuation Review, Winter 2008 and Business Valuation E-Letter, February
2009; and ‘‘Cost of Capital in Valuation of Stock by the Income Approach: Updated
for Economy in Crisis,’’ The Value Examiner, January–February 2009.
He is the co-author Cost of Capital: Applications and Examples, 3rd ed., with
Shannon P. Pratt (Hoboken, NJ: John Wiley & Sons, 2008) and co-author 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.
Michael W. Barad is the Vice President of Morningstar’s Financial Communications
Business. Mr. Barad oversees all investor publishing, advisor communication materials, events, and Ibbotson valuation services.
Prior to Morningstar’s acquisition of Ibbotson Associates in 2006, Mr. Barad
was vice president of financial communications at Ibbotson. During his time at Ibbotson, he also served as the manager of valuation and legal services and senior editor of the SBBI Yearbooks.
Mr. Barad has published, spoken, and testified on such topics as the cost of capital, equity risk premium, and size premium. He earned his bachelor’s degree in
finance from the University of Illinois at Urbana-Champaign.
Mr. Barad co-authored Chapter 20 of Cost of Capital: Applications and Examples, 4th ed.
Joanne Fong, CFA, CPA, is a Senior Manager in the Transaction Advisory Services–
Valuation & Business Modeling practice in the Chicago office of Ernst & Young LLP.
Ms. Fong holds a Master of Business Administration and a Bachelor of Business
Administration, both from the University of Michigan, Ross School of Business.
Ms. Fong co-authored Chapter 7 of the Cost of Capital: Applications and Examples, 4th ed. Workbook and Technical Supplement.
William H. Frazier, ASA, 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 Government Relations Committee. He has participated as an appraiser
and/or expert witness in numerous U.S. Tax Court cases, including testimony in
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xiii
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 co-author 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 IRS Advisory
Council (IRSAC) and the Valuation Advisory Board of Trusts & Estates Journal.
Mr. Frazier contributed Chapter 8 of the Cost of Capital: Applications and Examples, 4th ed. Workbook and Technical Supplement and the companion Excel
worksheets that appear on the John Wiley & Sons web site.
Terry V. Grissom, PhD, CRE, MAI, serves on the faculty at the University of Washington. He just completed a faculty assignment at the University of Ulster, Built Environment Research Institute. He received his PhD in Business from the University of
Wisconsin, Madison, majoring in Real Estate and Urban Land Economics, with minors in Finance/Risk Management and Civil-Environmental Engineering. He
received an MS in Real Estate Appraisal and Investment Analysis, also from the
University of Wisconsin, and an MBA in Finance, Real Estate, and Urban Affairs
from Georgia State University. He did postdoctoral work at Texas A&M University
in Econometrics and Statistics.
Dr. Grissom was formerly Professor of Real Estate and Urban Land Economics
at Georgia State University, Atlanta, in the Robinson College of Business. Prior to
his tenure at GSU, he was Vice-President of Investment Research for Equitable Real
Estate Investment Management, an institutional investment advisory for pension
funds, insurance companies, and other financial institutions. From 1992 through
October 1994, he was the National Research Director for Price Waterhouse’s Financial Services Industry Practice.
Dr. Grissom has published more than 100 academic and professional articles,
monographs, and working papers in his career to this point. He has also authored,
co-authored, and edited four books concerning real estate appraisal and investment
analysis, market analysis, and real estate development and land economics. He has
also authored chapters in books on real estate development, investment analysis,
business and property valuation techniques, and education theory and practice for
both academics and practitioners and for both domestic and international
audiences.
Dr. Grissom co-authored Chapters 9 and 10 of the Cost of Capital: Applications
and Examples, 4th ed. Workbook and Technical Supplement.
James Harrington is an accomplished financial writer and analyst and is Vice President of Duff & Phelps, where he serves as a champion for execution of creative ideas
for thought leadership content, provides technical support on client engagements involving cost of capital and business valuation matters, and leads efforts for development of Duff & Phelps studies, surveys, and online content and tools.
Prior to joining Duff & Phelps in 2010, Mr. Harrington was the Director of Valuation Research in Morningstar’s Financial Communications Business, leading the group
that produces the widely used and cited Ibbotson SBBI Valuation Yearbook and Ibbotson SBBI Classic Yearbook, the Ibbotson Cost of Capital Yearbook, the Ibbotson Beta
Book, and various international and domestic reports. During his tenure at
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Morningstar, Mr. Harrington expanded and refined the suite of Ibbotson valuation and
cost of capital product offerings, and he spearheaded the effort to expand Morningstar’s investment in valuation research and online tools and applications.
Prior to Morningstar, Mr. Harrington 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 dockworkers 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 finance and economics from the University of Illinois at Chicago,
where he graduated at the top in his class.
Mr. Harrington co-authored Chapter 20 of Cost of Capital: Applications and
Examples, 4th ed.
Vinay Kapoor, PhD, is a Managing Director in the transfer pricing practice of Duff
& Phelps, LLC. He has more than 15 years of experience in providing transfer pricing economics and other quantitative consulting services, with a focus on the analysis of intangibles and complicated fact patterns. He has done significant work with
clients in the technology, health care, and manufacturing industries. He received his
PhD in economics with a concentration in finance and his MA and BA in economics
from Cornell University.
Mr. Kapoor co-authored Chapter 25 of Cost of Capital: Applications and Examples, 4th ed.
Glen N. Kernick is a Managing Director in the Silicon Valley office of Duff &
Phelps, LLC, and the Technology Industry Practice Leader. He has performed numerous valuations and financial analyses for more than 12 years for a variety of purposes, including financial reporting, tax, fairness opinions, litigation, and strategic
planning. He was formerly a Managing Director of the Standard & Poor’s Corporate Value Consulting practice and a Director at PricewaterhouseCoopers, LLP. Mr.
Kernick received an MBA from the University of Washington and a Bachelor of Arts
in Economics from the University of California, San Diego.
Mr. Kernick co-authored Chapter 23 of Cost of Capital: Applications and Examples, 4th ed.
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 Pricewaterhouse Coopers
LLP. He received a BS degree from Cornell University and an MBA from Long Island
University, C. W. Post Center.
Mr. MacCrate 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.
Mr. MacCrate co-authored Chapters 9 and 10 in the Cost of Capital: Applications and Examples, 4th ed. Workbook and Technical Supplement.
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xv
Harold G. Martin Jr., CPA/ABV/CFF, ASA, CFE, is the Principal-in-Charge of the
Business Valuation, Forensic, and Litigation Services Group for Keiter, Stephens,
Hurst, Gary & Shreaves, P.C., in Richmond and Charlottesville, Virginia. He has
more than 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 receiverships. He is an adjunct faculty member of
the College of William and Mary Mason Graduate School of Business and teaches
forensic accounting and valuation in the Master of Accounting program. He is also
a guest lecturer on valuation in the MBA program.
Prior to joining Keiter Stephens, he served as a Senior Manager in Management
Consulting Services for Price Waterhouse and as a Director in Financial Advisory
Services for Coopers & Lybrand. He currently serves as an instructor for the
American Institute of Certified Public Accountants National Business Valuation
School and ABV Exam Review Course and also as an editorial advisor and contributing author for the AICPA CPA Expert. He is a former member of the AICPA Business Valuation Committee, former editor of the AICPA ABV e-Alert, and a two-time
recipient of the AICPA Business Valuation Volunteer of the Year Award. He is a
frequent speaker and author on valuation topics and is a co-author of Financial Valuation: Applications and Models, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2006).
Mr. Martin received his AB degree in English in 1979 from the College of William
and Mary and his MBA degree in 1991 from Virginia Commonwealth University.
Mr. Martin contributed Chapter 10 of the companion Cost of Capital: Applications and Examples, 4th ed. Workbook and Technical Supplement and the companion Excel worksheets that appear on the John Wiley & Sons web site.
James Morris, PhD, AM, received his PhD in Finance from University of
California, Berkeley. He is a professor of finance at the University of Colorado at
Denver, where he teaches courses in business valuation, financial modeling, and
financial management, and he has also served on the finance faculties at the Wharton
School of University of Pennsylvania and at the University of Houston and taught
finance courses at business schools in England, France, and Germany.
Dr. Morris’s recent publications include Introduction to Financial Models for
Management and Planning with J. Daley (CRC Press, 2009); ‘‘Life and Death of
Businesses: A Review of Research on Firm Mortality,’’ Journal of Business Valuation and Economic Analysis (2009); ‘‘Firm Mortality and Business Valuation,’’ Valuation Strategies (September–October 2009); ‘‘The Iterative Process Using CAPM to
Calculate the Cost of Equity Component of the Weighted Average Cost of Capital
When Capital Structure is Changing,’’ Appendix 7.2 in Pratt and Grabowski, Cost
of Capital: Applications and Examples, 3rd ed. (Hoboken, NJ: John Wiley & Sons,
2008); ‘‘Growth in the Constant Growth Model,’’ Business Valuation Review
(Winter 2006); ‘‘Understanding the Minefield of Weighted Average Cost of Capital,’’ Business Valuation Review (Fall 2005); and ‘‘Reconciling the Equity and
Invested Capital Methods of Valuation When the Capital Structure is Changing,’’
Business Valuation Review (March 2004). In addition, his research articles have
been published in the Journal of Finance, Journal of Financial & Quantitative Analysis, Journal of Applied Psychology, Academy of Management Journal, and
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ABOUT THE AUTHORS
Management Science, among others. In addition to teaching, he provides valuation
services to the business community.
Dr. Morris contributed Chapter 6 of the Cost of Capital: Applications and
Examples, 4th ed. Workbook and Technical Supplement and the companion Excel
worksheets that appear on the John Wiley & Sons web site.
Carla Nunes, CFA, is a Director in the Office of Professional Practice of Duff &
Phelps, LLC, where she provides firmwide technical guidance on a variety of valuation, financial, and tax-reporting issues. Prior to this role, Ms. Nunes was part
of the Valuation Advisory Services business unit, focusing on the valuation needs
of consumer and industrial product firms, primarily for financial reporting or tax
purposes. Before she joined Duff & Phelps, she was a Manager in the Standard
& Poor’s Corporate Value Consulting practice and a Senior Associate at PricewaterhouseCoopers, LLP. She has 14 years of experience in providing valuation
and tax services.
Ms. Nunes has conducted numerous business and asset valuations for a variety
of purposes, including purchase price allocations, goodwill impairment testing,
mergers and acquisitions, corporate tax restructuring, and debt analyses. She has
substantial experience working with multinational companies, having addressed
complex tax, international cost of capital, and foreign exchange issues. She is also
one of Duff & Phelps’s experts in addressing valuation issues related to cost of capital and foreign exchange. She is a frequent instructor at Duff & Phelps’s annual newhire training event.
Ms. Nunes received her MBA in finance from the University of Rochester, completed coursework for a Masters of Taxation from Villanova University School of
Law, and received an Honors Degree in Business Administration from the Technical
University of Lisbon.
Ms. Nunes contributed Chapter 22 of Cost of Capital: Applications and Examples, 4th ed.
David M. Ptashne, CFA, is an Associate Director with Ceteris, a global economic
consulting firm that provides transfer pricing and business valuation services. Mr.
Ptashne has performed numerous valuation studies of businesses, interests in businesses, and intangible assets across various industries, including advertising and
communications, consumer products, technology, financial services, integrated oil
and gas, retail, and health care. He received a Bachelor of Science degree in Finance
with High Honors from the University of Illinois at Urbana-Champaign.
Mr. Ptashne contributed Chapters 2 and 4 of the Cost of Capital: Applications
and Examples, 4th ed. Workbook and Technical Supplement.
Gary Roland, CFA, CPA, is a Managing Director in the Office of Professional Practice, with 26 years of valuation experience, and resides in the Philadelphia office of
Duff & Phelps, LLC. The Office of Professional Practice provides technical interpretation and guidance on financial reporting valuation matters such as business combinations; intangible assets; and goodwill and asset impairments. He was formerly a
Director in the Standard & Poor’s Corporate Value Consulting practice and at PricewaterhouseCoopers, LLP. Mr. Roland received an MBA and a Bachelor of Sciences
in Engineering from the State University of New York at Buffalo.
Mr. Roland co-authored Chapter 23 of Cost of Capital: Applications and Examples, 4th ed.
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Mark Shirley, CPA/ABV/CFE, has earned advanced accreditations: Certified Valuation Analyst and Certified Forensic Financial Analyst. 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, his technical contributions have been published by Wiley Law
Publications, Aspen Legal Press, and in professional periodicals, including Valuation Examiner, BewertungsPraktiker Nr. (a German-language business valuation journal), Practical Accountant, CPA Litigation Services Counselor,
Gatekeeper Quarterly, 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 has been 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.
Mr. Shirley contributed Chapter 3 of the Cost of Capital: Applications and
Examples, 4th ed. Workbook and Technical Supplement and Appendix III of the Workbook and Technical Supplement which appears on the John Wiley & Sons web site.
David Turf, CFA, is a Managing Director in the Investment Banking Group of Duff
& Phelps, LLC. He is a member of the Transaction Opinions Practice and serves on
the Senior Review Committee. He has extensive experience executing engagements
for fairness opinions, solvency opinions, and valuation opinions.
Mr. Turf has advised boards of directors, management, trustees, and shareholders on a variety of corporate finance and valuation issues for mergers, acquisitions,
divestitures, ESOPs, financings, and other general corporate purposes. He has provided fairness opinions in a variety of transactions, including going-private, reverse
merger, restructuring, and related-party transactions. He has provided solvency
opinions in spin-offs, leveraged buyouts and recapitalizations, and other transactions. In addition, he has structured debt securities and issued commercially reasonable opinions for debt issuances in related-party transactions.
Prior to joining Duff & Phelps in 1997, he was an Assistant Vice President in the
commercial loan department of Corus Bank.
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ABOUT THE AUTHORS
Mr. Turf received an MBA with a focus on Finance and Organizational Behavior
from the J. L. Kellogg Graduate School of Management at Northwestern University,
where he graduated with distinction. He also received a BS in Finance from the
University of Illinois at Urbana-Champaign, where he graduated with highest honors.
Mr. Turf contributed Chapter 24 of Cost of Capital: Applications and Examples, 4th ed.
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Foreword
T
he discounted cash flow method is a valuation analysis often used by the
Delaware Court of Chancery. As many readers know, this method is highly
dependent upon the cost of capital used for the discount rate.
This book is the most incisive and exhaustive treatment of this critical subject to
date. Shannon Pratt and Roger Grabowski have clearly laid out the issues for the
practitioner in computing the cost of capital, relegating technical proofs and theoretical nuances to appendices. The authors definitively identify the issues on which
there is consensus as to the best practice, and for issues where there is still legitimate
controversy, they describe the alternatives and give the practitioner guidance as to
the proper choice, taking into account the facts and circumstances of a given case.
I especially like the chapter on using and (possibly) adjusting management’s projections that were made for some other purpose than litigation. Additionally, the
authors treat well such sticky issues as the interaction between company-specific risk
and the small stock premium, and whether to use a company-specific risk premium
and, if so, how to support it.
For specialized applications of cost of capital, the authors sought out some of the
leading practitioners in their respective fields. Moreover, those contributions are
carefully edited so that the book has a logical flow rather than containing a mere
collection of readings.
It is easy to say this is a book the Court of Chancery, and other courts, will rely
on heavily in future years. Corporate executives and their advisors would be well
advised to do so, too.
Stephen P. Lamb, Esquire
Partner, Paul, Weiss, Rifkind,
Wharton & Garrison;
Former Vice Chancellor,
Delaware Court of Chancery
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E1FPREF
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Preface
C
ost of capital is arguably the most important concept in all of finance. 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, allowing us to make informed
pricing decisions for purchases and sales and to compare one investment opportunity against another.
Cost of capital estimation is the pricing of risk. In the marketplace, betterinformed 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. And additional billions of dollars are at stake based on
the cost of capital in disputes.
Why did we undertake writing this book? 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.
We decided to do this update to the book because of the dramatic changes in
cost of capital resulting from the financial crisis that began in 2008 and the subsequent recession. Many of the commonly used methods for estimating the cost of
capital literally fell apart, providing faulty estimates just at a time when providing
more accurate cost of capital estimates became more important than ever before.
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,
and forecasting of expected investment returns 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.
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,
valuing intangible assets for financial reporting, and transfer pricing. In this text, the
reader can expect to learn about:
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PREFACE
&
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
&
&
&
&
&
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 equity
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 present and discuss the materials published by Morningstar, formerly Ibbotson Associates. We thoroughly cover the Duff & Phelps Risk Premium Report data
and how it can help in estimating cost of equity capital, particularly when beta estimates often indicate decreases in risk at the very time business risk is increasing.
Throughout the book, we summarize the results and practical implications of
the latest cost of capital research, much of which has been gleaned from unpublished
academic working papers.
WHAT’S NEW IN THIS EDITION
Equity Risk Premium
Based on empirical research on the magnitude of the equity risk premium, we conclude that the long-term equity risk premium is in the range of 4% to 6% rather than
above the 7% that many analysts have used in recent years. We follow the financial
crisis from 2008 through late 2009 and explain how the conditional equity risk premium increased during the depths of the crisis and has decreased subsequently.
New and Expanded Chapters
Given the impact of the financial crisis that began in 2008, we added a chapter
on distressed businesses. We expanded the chapter on the overall cost of capital
(or weighted average cost of capital) to examine the impact on cost of capital,
given the changes in the market perception of and the reduced availability of
debt financing. We also added a chapter on the cost of capital for fair value
reporting of intangible assets.
We expanded the chapters on company-specific risk. We updated the chapter on
cost of capital in transfer pricing to reflect the impact of cost of capital due to the
new Internal Revenue Service Cost Sharing Regulations.
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Preface
xxiii
We also expanded the chapters on common errors made in valuation and advice
from the authors about dealing with specific, often controversial, cost of capital
issues.
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.
Much of the research cited in this book is from working papers. To locate a
working paper, search online using any major search engine (e.g., Google or
Yahoo!) by author and title. An example: the first link provided using the
Google search engine when a search for Jennifer Lynne M. Altamuro, Rick
Johnston, Shail Pandit, and Haiwen (Helen) Zhang, ‘‘Operating Leases and
Credit Assessments’’ is ‘‘SSRN–Operating Leases and Credit Assessments. . . . ’’
This link will direct you to the Social Science Research Network, where the
article is downloadable.
AUDIENCES FOR THE BOOK
In addition to the traditional professional valuation practitioner, this book is designed to serve the needs of:
&
&
&
&
&
Corporate finance officers for pricing or evaluating mergers and acquisitions,
raising private or public equity, property taxation, and stakeholder disputes
Investment bankers for pricing public offerings, mergers and acquisitions, and
private equity financing
CPAs who deal with either valuation for financial reporting or client valuations
issues
Judges and attorneys 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
Academicians and students who wish to learn anywhere from the basic theory
to the latest research
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 appendixes present detailed expositions of the more complex procedures.
Finally, the book is comprehensively indexed to serve as a reference for specific
concepts and procedures within the general topic of cost of capital.
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PREFACE
PRACTICAL APPLICATIONS
In Part 1, we discuss the relationship of risk and return and the basic building blocks
of the later parts. In Part 2, we go through the commonly employed methods of
estimating cost of equity capital for public companies. These methodologies also
serve as the beginning point for estimating the cost of capital for closely held businesses. Part 3 is primarily addressed to corporate finance officers. It includes capital
budgeting, mergers and acquisitions, fair value reporting, transfer pricing, and economic value added. In Part 4, we address the factors and methodologies for adjusting the ‘‘as if public’’ cost of equity capital for the closely held business. Part 5
presents some specific insights of the authors.
Some authors claim that the differences between the market for public company
investments and private company investments are so great that one analyzing the
cost of equity capital for a closely held business should not consider starting with
the ‘‘as if public’’ cost of equity capital. We disagree. We do recognize that there are
significant differences in the pool of willing buyers and the risks of small and
medium-size closely held businesses. But we believe that the underlying principles of
analyzing and pricing business risk transcend the size of the business.
We also believe that reasonable tools are available to make appropriate adjustments for the differences between the ‘‘as if public’’ company and the closely held
company. For example, while there may be only a few public companies with risk
characteristics matching those of a particular closely held company, that problem
happens even when valuing smaller public companies. The company-specific risk
premium is intended to allow adjusting for differences in risks. Similarly, the added
risks of lack of marketability confronting the investor in the closely held business is
one commonly addressed by valuators beginning with the ‘‘as if public’’ cost of
equity capital or as if public value.
One author who has considered both sides of the issue summarizes the debate
as follows:
Should business appraisers use public market data to estimate the value of
private business interests? In my opinion the answer is ‘‘yes.’’ Given the current state of private capital market theory and practice, I am reluctant to
discard valuation methods simply because they rely on public market data.1
NEW WORKBOOK AND TECHNICAL SUPPLEMENT
We added a companion Workbook and Technical Supplement to further assist practitioners in better understanding how to estimate the cost of capital. Part 1 contains
the Technical Supplements to several chapters. Part 2 contains examples of specific
applications to private investment companies, real property, and real estate
businesses.
1
M. Mark Walker, ‘‘Are the Public and Private Capital Markets Worlds Apart?’’ Business
Appraisal Practice (Winter 2007/2008): 8–20.
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Preface
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Part 3 contains learning objectives, questions, and problems; Part 4 contains the
answers to the questions and solutions to the problems.
The Workbook and Technical Supplement companion John Wiley & Sons web
site contains several Excel worksheets to assist the user in implementing the methods
discussed.
The three Appendixes appear on the companion John Wiley & Sons web site.
Appendix I presents an excerpt of a report submitted to the U.S. Tax Court.
Appendix II presents a helpful tool for the practitioner, the ValuSource Valuation Software. Appendix III contains a comprehensive review of the statistics discussed in the Cost of Capital: Applications and Examples, 4th ed.
The web site includes PowerPoints covering the material for those who want to
adopt the book for a seminar.
NEW COST OF CAPITAL IN LITIGATION
Because of the positive feedback from readers of the chapter on cost of capital in the
courts in the last edition, we expanded the topics covered and added a new companion text, Cost of Capital in Litigation: Applications and Examples.
That text covers cost of capital basics from the view of a judge or attorney dealing with cost of capital issues. The book covers the underlying theories and decisions
in estate and gift matters; corporate restructuring and other federal tax matters, including transfer pricing; cost of capital issues in intellectual property and other damages disputes; bankruptcy cases; appraisal, oppression, and fairness cases; family
law matters; ad valorem taxation matters; and regulated industry matters. The book
also includes a chapter on questions to ask the business valuation expert on cost of
capital matters.
Finally, we have enjoyed the challenge of assembling the materials for the three
books. They reflect our collective 75 years of experience in doing valuations. We do
anticipate updating the books again, so please contact us 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, OR 97221
(503) 459-4700
www.shannonpratt.com
E-mail: shannon@shannonpratt.com
Roger J. Grabowski, ASA
Duff & Phelps, LLC
311 S. Wacker Drive, Suite 4200
Chicago, IL 60606
(312) 697-4720
www.duffandphelps.com
E-mail: roger.grabowski@duffandphelps.com
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Acknowledgments
T
his 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:
Bruce Bingham
Capstone Advisory Group LLC
New York, NY
Mark Lee
Eisner LLP
New York, NY
Stephen J. Bravo
Apogee Business Valuation
Framingham, MA
Dan McConaughy
Grobstein, Horwath LLP
Sherman Oaks, CA
James Budyak
Valuation Research Corp.
Milwaukee, WI
George Pushner
Duff & Phelps LLC
New York, NY
David Clarke
The Griffing Group
Oak Park, IL
Raymond Rath
PricewaterhouseCoopers LLC
Los Angeles, CA
Stan Deakin
Mosaic Capital LLC
Los Angeles, CA
Jeffrey Tarbell
Houlihan Lokey
San Francisco, CA
Donald A. Erickson
Erickson Partners, LLC
Dallas, TX
Terence Tchen
Houlihan Lokey
Los Angeles, CA
Aaron A. Gilcreast
PricewaterhouseCoopers LLC
Atlanta, GA
Marianna Todorova
Duff & Phelps LLC
New York, NY
Professor Joao Gomes
The Wharton School of the
University of Pennsylvania
Philadelphia, PA
Richard M. Wise
Wise, Blackman, LLP
Montreal (Quebec), Canada
In addition, we thank:
&
Dustin Snyder and Elizabeth Anderson for assistance with editing and research,
including updating of the bibliography; updating and shepherding the
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&
&
&
&
&
&
ACKNOWLEDGMENTS
manuscript among reviewers, contributors, authors, and publisher; typing; obtaining permissions; and other invaluable help.
David Turney, Nick Arens, William Suscott, and Katherine Nierman of Duff &
Phelps, LLC, for preparing numerous tables and calculations that appear
throughout the book.
Dr. Ashok Abbott for allowing us to include his research on the size premium
and liquidity in Chapters 14A and 27.
Joel M. Stern, G. Bennett Stewart III, and Donald H. Chew Jr. for contributing
Chapter 26 on economic value added.
David Fein of ValuSource for contributing Appendix II of the Workbook and
Technical Supplement on ValuSource Pro.
Noah Gordon of Shannon Pratt Valuations, Inc., for general editorial assistance.
For the granting of permissions, we would like to thank:
& Professor Edwin Burmeister, Duke University
& Business Valuation Resources, LLC
& Professors Elroy Dimson, Paul Marsh, and Mike Staunton, London School of
Economics
& Duff & Phelps, LLC
& FactSet Mergerstat, LLC
& FMV Opinions, Inc.
& Professor Arthur Korteweg, University of Chicago
& The McGraw-Hill Companies, Inc.
& Morningstar, Inc.
& National Association of Real Estate Investment Trusts
& Pluris Valuation Advisors, LLC
& 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 here for their assistance. Of course, any
errors herein are our responsibility.1
Shannon Pratt
Roger Grabowski
1
Any opinions presented in this book are those of the authors. The opinions of Mr.
Grabowski do not represent the official position of Duff & Phelps, LLC. This material is
offered for educational purposes with the understanding that neither the authors nor Duff &
Phelps, LLC, are engaged in rendering legal, accounting, or any other professional service
through presentation of this material. The information presented in this book has been
obtained with the greatest of care from sources believed to be reliable, but is not guaranteed
to be complete, accurate, or timely. The authors and Duff & Phelps LLC expressly disclaim
any liability, including incidental or consequential damages, arising from the use of this
material or any errors or omissions that may be contained in it.
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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, 2008).
VALUE AT A POINT IN TIME
Pn
P0
Pi
PV
PV b
PVkeu
PVts
PVdc
PVTSn
PVf
TVn
Me
Md
Mp
MVIC
BV
BVn
BVi
Fd
FV RU
FV NWCRU
FV ICRU
FV FARU
¼ Stock price in period n
¼ Stock price at valuation period
¼ Price per share for company i (seen elsewhere as PV)
¼ Present value
¼ Present value of net cash flows due to business operations before cost of
financing
¼ Present value of net cash flows using unlevered cost of equity capital, keu, as
the discount rate
¼ Present value of tax shield due to interest expense on debt capital
¼ Present value of net distress-related costs
¼ Present value of the tax shield as of time ¼ n
¼ Present value of invested capital
¼ Terminal value at time n
¼ Market value of equity capital (stock)
¼ Market value of debt capital
¼ Market value of preferred equity
¼ Market value of invested capital
¼ Enterprise value
¼ Me þ Md þ Mp
¼ Book value of net assets
¼ Book value of equity at time ¼ n
¼ Measure of book value (typically book value to market value) of stock of
company i
¼ Fair value of debt
¼ Fair value of reporting unit
¼ Fair value of net working capital of the reporting unit
¼ Fair value of invested capital of the reporting unit
¼ Fair value of fixed assets of the reporting unit
xxix
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XXX
FV IARU
FV UIVRU
FV dRU
FV eRU
FMV BE
FMV NWC
FMV FA
FMV IA
FMV UIV
FMV e
FMV e;n;up
FMV BE;n;down
FMV e;n;down
NOTATION SYSTEM AND ABBREVIATIONS USED IN THIS BOOK
¼ Fair value on intangible assets, identified and individually valued, of the
reporting unit
¼ Fair value of unidentified intangibles value (i.e., goodwill) of the reporting
unit
¼ Fair value of debt capital of the reporting unit
¼ Fair value of equity capital of the reporting unit
¼ Fair market value of the business enterprise
¼ Fair market value of net working capital
¼ Fair market value of fixed assets
¼ Fair market value on intangible assets
¼ Fair market value of unidentified intangibles value (i.e., goodwill)
¼ Fair market value of equity capital
¼ Fair market value of equity at time ¼ n assuming ‘‘up’’ scenario (value of BE
increases)
¼ Fair market value of business enterprise at time ¼ n assuming ‘‘down’’
scenario (value of BE decreases)
¼ Fair market value of equity at time ¼ n assuming ‘‘down’’ scenario (value of
BE decreases)
COST OF CAPITAL AND RATE OF RETURN VARIABLES
k
kc
ke
ke,local
ke,u.s.
kBV
keu
klocal
ki
kni
kðptÞ
kp
kd
kdðptÞ
kA
kTS
keRU
kNWCðptÞ
¼ Discount rate (generalized)
¼ Country cost of equity
¼ Discount rate for common equity capital (cost of common equity capital).
Unless otherwise stated, it generally is assumed that this discount rate is
applicable to net cash flow available to common equity.
¼ Discount rate for equity capital in local country for discounting expected
cash flows in local currency
¼ Discount rate for equity capital in the United States
¼ Rate of return on book value, retained portion of net income, usually estimated as ¼ NInþ1/BVn
¼ Cost of equity capital, unlevered (cost of equity capital assuming firm financed with all equity)
¼ Cost of equity capital in local country
¼ Discount rate for company i
¼ Discount rate for equity capital when net income rather than net cash flow is
the measure of economic income being discounted
¼ Discount rate applicable to pretax cash flows
¼ Discount rate for preferred equity capital
¼ Discount rate for debt (net of tax effect, 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 kdðptÞ ð1 tax rateÞ
¼ Cost of debt prior to tax effect
¼ Discount rate for the firm’s assets
¼ Rate of return used to present value tax savings due to deducting interest
expense on debt capital financing
¼ After tax rate of return on equity capital of reporting unit
¼ Rate of return for net working capital financed with debt capital (measured
before interest tax shield) and equity capital
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Notation System and Abbreviations Used in This Book
xxxi
¼ Rate of return for fixed assets financed with debt capital (measured before
interest tax shield) and equity capital
kdRU
¼ Rate of return on debt capital of the reporting unit net of tax effect
¼ kdðptÞRU ð1 tax rateÞ
¼ Rate of return for net working capital of the reporting unit financed with
kNWCRU
debt capital (return measured net of the tax effect on debt financing, if any)
and equity capital
kFARU
¼ Rate of return for fixed assets financed with debt capital (return measured net
of the tax effect on debt financing, if any) and equity capital
¼ Rate of return for identified and individually valued intangible assets fikIARU
nanced with debt capital (return measured net of the tax effect on debt
financing, if any) and equity capital
¼ Rate of return for unidentified intangibles value of the reporting unit financed
kUIVRU
with debt capital (return measured net of the tax effect on debt financing, if
any) and equity capital
kIAþUIV ðptÞ
¼ Pretax rate of return on all intangible assets, identified and individually valued, plus the unidentified intangible value financed with debt capital (measured before interest tax shield) and equity capital
c
¼ Capitalization rate
cðptÞ
¼ Capitalization rate on pretax cash flows (Note: For complex capital structures, there could be more than one class of capital in any of the preceding
categories, requiring expanded subscripts.)
D/P0
¼ Dividend yield on stock
¼ Downside risk in the local market (U.S. dollars)
DRj
DRw
¼ Downside risk in global (‘‘world’’) market (U.S. dollars)
R
¼ Rate of return
¼ Return on stock i
Ri
Rd
¼ Rate of return on subject debt (e.g., bond) capital
¼ Return on market portfolio in current month n
Rm,n
¼ Rate of return on a risk-free security
Rf
¼ Risk-free rate in current month n
Rf; n
Rf; local
¼ Return on the local country government’s (default-risk-free) paper
¼ U.S. risk-free rate
Rf; u:s:
Rlocal euro $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
(Rlocal euro
¼ Yield spread between government bonds issued by the local country versus
$issue Rf; u:s: )
U.S. government bonds
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)
¼ Risk premium for ‘‘small’’ stocks (usually average size of lowest quintile or
RPs
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)
¼ Risk premium for small size plus risk premium attributable to the specific
RPsþu
distressed company
RPmþsþu
¼ Risk premium for the ‘‘market’’ plus risk premium for size plus risk attributable to the specific company
kFAðptÞ
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NOTATION SYSTEM AND ABBREVIATIONS USED IN THIS BOOK
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
¼ Risk premium for the ith security
RPi
¼ Bi,s Si ¼ Risk premium for size of company i
RPi,s
¼ Bi,BV BVi ¼ Risk premium for book value of company i
RPi,BV
¼ Bi,u Ui ¼ Risk premium for unique or unsystematic risk of company i
RPi,u
RPlocal
¼ Equity risk premium in local country’s stock market
¼ Full-information levered beta estimate of the subject company
RIiL
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)
¼ Expected rate of return on security i
EðRi Þ
EðRdiv Þ
¼ Expected rate of return on dividend
E Rcapgains ¼ Expected rate of return on capital gains
¼ Expected rate of return on security i for undiversified investor j
E(Ri,j)
B
¼ Beta (a coefficient, usually used to modify a rate of return variable)
Bi
¼ Expected beta of the stock of company i
¼ Levered beta for (equity) capital
BL
BU
¼ Unlevered beta for (equity) capital
¼ Levered segment beta
BLS
¼ Beta for debt capital
Bd
¼ Beta of preferred capital
Bp
Be
¼ Beta (equity) expanded
¼ Operating beta (beta with effects of fixed operating expense removed)
Bop
¼ Beta of company i (F-F beta)
Bi
Bi,m
¼ Sensitivity of return of stock of company i to the market risk premium
or ERP
¼ Sensitivity of return of stock of company i to a measure of size, S, of
Bi,s
company i
Bi,BV
¼ Sensitivity of return of stock of company i to a measure of book value (typically measure of book-value-to-market-value) of stock of company i
¼ Sensitivity of return of stock of company i to a measure of unique or unBi,u
systematic risk of company i
Bn
¼ Estimated market coefficient based on sensitivity to excess returns on market
portfolio in current month
¼ Market risk of the subject company measured with respect to the local securiBlocal
ties market
Bw
¼ Market or systematic risk measured with respect to a ‘‘world’’ portfolio of
stocks
Bi1 . . . Bin ¼ Sensitivity of security i to each risk factor relative to the market average sensitivity to that factor
¼ True beta estimate for stock of company i based on relationship to excess
Bi0
returns on market portfolio of equity plus debt, ME þ MD
Bu.s. RPu.s. ¼ Risk premium appropriate for a U.S. company in similar industry as the
subject company in local country, expressed in U.S. dollar-denominated
returns
FI-Beta
¼ Full-information beta for industry
TBi
¼ Total beta for security i
¼ Country covariance with region
bcr
RPu
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Notation System and Abbreviations Used in This Book
bcw
Si
Ui
l
RP1 . . . RPn
si
SMBP
hi
HMLP
Fd
b
WACCðptÞ
WACCRU
WACCðptÞRU
s 2i
s 2m
s 2e
s
se
sA
sB
s rev
s BE
s local
s u:s:
s stock
s bond
si
sm
s i,m
s D2
s MEþMD2
r
dr
CCRlocal
l
t
h
Inflationlocal
Inflationu:s:
xxxiii
¼ Country covariance with world
¼ Measure of size of company i
¼ Measure of unique or unsystematic risk of company i
¼ A measure of individual stock’s liquidity
¼ Risk premium associated with risk factor 1 through n for the average asset in
the market (used in conjunction with arbitrage pricing theory)
¼ Small-minus-big coefficient in the Fama-French regression
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 (also shown as SMB)
¼ High-minus-low coefficient in the Fama-French regression
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 (also shown as HML)
¼ Face value of outstanding debt
¼ 1 Payout ratio ¼ retention ratio
¼ Weighted average cost of capital (before interest tax shield)
¼ Overall rate of return for the reporting unit
¼ Weighted average cost of capital for the reporting unit
¼ Before interest tax shield WACC of the reporting unit
¼ Variance of returns for security i
¼ Variance of the returns on the market portfolio (e.g., S&P 500)
¼ Variance of error terms
¼ Standard deviation
¼ Standard deviation of returns on firm’s common equity
¼ Standard deviation of returns on firm’s assets
¼ Standard deviation of operating cash flows of business before cost of
financing
¼ Standard deviation of revenues
¼ Standard deviation of value of business enterpise
¼ Volatility of subject (local) stock market
¼ Volatility of U.S. stock market
¼ Volatility of local country’s stock market
¼ Volatility of local country’s bond market
¼ Standard deviation of returns for security i
¼ Standard deviation of returns for the market portfolio (e.g., S&P 500)
¼ Variance of returns on the security, i, and the market, m
¼ Variance in excess returns on market of debt
¼ Variance in excess returns on market portfolio of equity plus debt, ME
þ MD
¼ Correlation coefficient between the returns on the security, i, and the
market, m
¼ Regional risk not included in RPw
¼ Country credit rating of local country
¼ Company’s exposure to the local country risk
¼ Tax rate (expressed as a percentage of pretax income)
¼ Holding period
¼ Expected rate of inflation in local country
¼ Expected rate of inflation in U.S.
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XXXIV
NOTATION SYSTEM AND ABBREVIATIONS USED IN THIS BOOK
INCOME VARIABLES
E
F
Fc
NI
NCIe,n
NCIf,n
CF
NCFe
NCFf
NCFue
D
De,n
Df,n
RIe,n
TS
EBT
EBIT
EBITDA
V
AEG
¼ Expected economic income (in a generalized sense; i.e., could be dividends,
any of several possible definitions of cash flows, net income, etc.)
¼ Fixed operating assets (without regard to costs of financing)
¼ Fixed operating costs of the business
¼ Net income (after entity-level taxes)
¼ Net comprehensive income to common equity in period n, which includes
income terms reported directly in the equity account rather than in the income statement
¼ Net comprehensive income to the firm in period n, which includes income
terms reported directly in the equity account rather than in the income
statement
¼ Cash flow for a specific period
¼ Net cash flow (free cash flow) to equity
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)
¼ Net cash flow to unlevered equity
¼ Dividends
¼ Distributions to common equity, net of new issues of common equity in period n
¼ Distributions to total capital, net of new issues of debt or equity capital in
period n
¼ Residual income for common equity capital
¼ Present value of tax savings due to deducting interest expense on debt capital
financing
¼ Earnings before taxes
¼ Earnings before interest and taxes
¼ Earnings before interest, taxes, depreciation, and amortization
¼ Variable operating costs
¼ Abnormal earnings growth
PERIODS OR VARIABLES IN A SERIES
i
n
0
py
¼ ith period or ith variable in a series (may be extended to the jth variable, the
kth variable, etc.)
¼ Number of periods or variables in a series, or the last number in a series
¼ Period 0, the base period, usually the latest year immediately preceding the
valuation date
¼ Partial year of first year following the valuation date
WEIGHTINGS
W
We
Wp
¼ Weight
¼ Weight of common equity in capital structure
¼ Me/(Me þ Md þ Mp)
¼ Weight of preferred equity in capital structure
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Notation System and Abbreviations Used in This Book
Wd
W dRU
Ws
W NWC
W NWCRU
W FA
W FARU
W IARU
W UIVRU
W TS
xxxv
¼ Mp =ðMe þ Md þ Mp Þ
¼ Weight of debt in capital structure
¼ Md/(Me þ Md þ Mp) (Note: For purposes of computing a weighted average
cost of capital [WACC], it is assumed that preceding weightings are at market value.)
¼ Weight of debt capital in capital structure of reporting unit
¼ Fair value of debt capital/FV RU
¼ Weight of segment data to total business (e.g., sales, operating income)
¼ Weight of net working capital in FMV BE
¼ FMV NWC =FMV BE
¼ Weight of net working capital in FV RU
¼ FV NWCRU =FV RU
¼ Weight of fixed assets in FMV BE
¼ FMV FA =FMV BE
¼ Weight of fixed assets in FV RU
¼ FV FARU =FV RU
¼ Weight of intangible assets in FV RU
¼ FV IARU =FV RU
¼ Weight of unidentified intangibles value FV RU
¼ FV UIVRU (i.e., goodwill)=FV RU
¼ Weight of TS in FMV BE
¼ TS=FMV BE
GROWTH
g
gi
gni
¼ Rate of growth in a variable (e.g., net cash flow)
¼ Dividend growth rate for company i
¼ Rate of growth in net income
MATHEMATICAL FUNCTIONS
S
\
X
G
a
e
ei
1
N (*)
D
¼ Sum of (add all the variables that follow)
¼ Product of (multiply together all the variables that follow)
¼ Mean average (the sum of the values of the variables divided by the number
of variables)
¼ Geometric mean (product of the values of the variables taken to the root of
the number of variables)
¼ Regression constant
¼ Regression error term
¼ Error term, difference between predicted return and realized return, Ri
¼ Infinity
¼ Cumulative normal density function (the area under the normal probability
distribution)
¼ Change in . . . (whatever follows)
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XXXVI
NOTATION SYSTEM AND ABBREVIATIONS USED IN THIS BOOK
NOTATION FOR REAL PROPERTY VALUATION (CHAPTER
9 OF WORKBOOK AND TECHNICAL SUPPLEMENT)
DSCR
EGIM
NOI; Ip
OER
PVp
ke
km
kp
cp
ce
cm
cn
cB
cL
cLF
cLH
A
P
1=Sn
Dp
SC %
PGI
PGIM
EGI
NIM
Fd =PV p
[1(Fd/PVp)]
MB
Mm
ML
MLF
MLH
Ip
IL
IB
Ie
Im
ILF
ILH
¼ Debt service coverage ratio
¼ Effective gross income multiplier
¼ Net operating income
¼ Operating expense rates
¼ Overall value or present value of the property
¼ Equity discount or yield rate (dividend plus appreciation)
¼ Mortgage interest rate
¼ Property yield discount rate
¼ Overall property capitalization rate
¼ Dividend to equity capitalization rate
¼ Mortgage capitalization rate or constant
¼ Terminal or residual or going-out capitalization rate
¼ Building capitalization rate
¼ Land capitalization rate
¼ Leased fee capitalization rate
¼ Leasehold capitalization rate
¼ Change in income and value (adjustment factor)
¼ Principal paid off over the holding period
¼ Sinking fund factor at the equity discount or yield rate ðke Þ
¼ Change in value over the holding period
¼ Cost of sale
¼ Potential gross income
¼ Potential gross income multiplier
¼ Effective gross income
¼ Net income multiplier
¼ Face value of debt (loan amount outstanding) to value ratio
¼ Equity to value ratio
¼ Building value
¼ Mortgage value
¼ Land value
¼ Leased fee value
¼ Leasehold value
¼ Overall income to the property
¼ Residual income to the land
¼ Residual income to the building
¼ Equity income
¼ Mortgage income
¼ Income to the leased fee
¼ Income to the leasehold
ABBREVIATIONS
ERP
WACC
WARA
T-Bill
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)
¼ Weighted average cost of capital
¼ Weighted average return on assets
¼ U.S. government bill (usually 30-day, but can be up to one year)
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Notation System and Abbreviations Used in This Book
xxxvii
¼ Separate trading of registered interest and principal of securities
¼ Center for Research in Security Prices, at the University of Chicago Booth
School of Business
PIPE
¼ Private investment in public equity
SBBI
¼ Stocks, Bonds, Bills, and Inflation, published annually by Morningstar
(previously Ibbotson Associates) 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
PAT
¼ Profit after tax
¼ Net Income
RIe,n
¼ Residual income to equity
¼ Residual income for total capital
RIf,n
EVA
¼ Economic value added
DY
¼ Dividend yield
ROCE
¼ Return on common equity
RNOA
¼ Return on net operating assets
RPF
¼ Risk premium factor
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)]
SSP
¼ Small stock premium
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
NOA
¼ Net operating assets
RU
¼ Reporting unit
¼ Net working capital of the reporting unit
NWCRU
FARU
¼ Fixed assets of the reporting unit
¼ Intangible assets of the reporting unit
IARU
¼ Unidentified intangible value (i.e., goodwill) of the reporting unit
UIVRU
MP Synergies ¼ Market participant synergies resulting from the expectation of cash flow
enhancements achievable only through the combination with a market
participant
E
¼ Exit multiple
NICE
¼ Nonmarketable investment company evaluation
REIT
¼ Real estate investment trusts
VDM
¼ Value driver model
MV CAPM ¼ Mean-variance capital asset pricing model
STRIPS
CRSP
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XXXVIII
MS CAPM
VaR
CVaR
CV
CRP
NOTATION SYSTEM AND ABBREVIATIONS USED IN THIS BOOK
¼ Mean-semivariance capital asset pricing model
¼ Value at risk
¼ Conditional value at risk
¼ Coefficient of variation
¼ [(Rlocal euro $issue Rf,u.s.) (sstock/sbond)]
E1C01
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Page 1
PART
One
Cost of Capital Basics
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CHAPTER
1
Defining Cost of Capital
Introduction
Components of a 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
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
Standard of Value
Summary
INTRODUCTION
The 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 fund 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
3
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4
COST OF CAPITAL BASICS
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 a business, security, or project comes from the investment market. The cost of
capital is always an expected (or forward-looking) return. Thus, analysts and
would-be investors never actually observe the market’s views as to expected returns
at the time of their investment. However, we often form our views of the future by
analyzing historical market data.
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.
The valuation process is one of analysis of expected returns and pricing of risk.
The cost of capital is the return appropriate for the expected level of risk in the
expected returns. It is the price of risk. But often observed returns do not match
expected returns. That is the essence of risk. (See Chapter 5 for a more complete
discussion of risk.)
COMPONENTS OF A 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 capital (i.e., stock, partnership, limited liability company, or
other type of entity interests with preference features, such as seniority in receipt
of dividends or liquidation proceeds)
Common equity capital (i.e., stock, partnership, limited liability company,
or other type of entity 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 own 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.
1
Mike Kaufman, ‘‘Profitability and the Cost of Capital,’’ Chapter 8 of Robert Rachlin, ed.,
Handbook of Budgeting, 4th ed. (New York: John Wiley & Sons, 1999).
2
Ibbotson Associates, ‘‘What Is the Cost of Capital?’’ 1999 Cost of Capital Workshop,
Chicago: Ibbotson Associates, 1999.
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Defining Cost of Capital
5
On the liability side, it is the economic cost to the business 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 business’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
Simply and cogently stated, ‘‘The cost of equity is the rate of return investors
require on an equity investment in a firm.’’4
When we talk about the cost of ownership capital (e.g., the expected return to
an equity investor), we usually use the phrase cost of equity capital. When we talk
about the cost of capital to the business overall (e.g., the average cost of capital
for both equity ownership interests and debt interests), we commonly use the
phrases weighted average cost of capital (WACC), blended cost of capital, or overall cost of capital. In rate-making cases, this array is sometimes called the band of
investment.
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
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, and the marketplace is the pool of investors ‘‘pricing’’ the risk of a particular asset. Thus it
represents the consensus assessment of the pool of investors that are participants
in a particular market.
Allen, Brealey, and Myers state the same concept: ‘‘The true cost of capital
depends on the use to which that capital is put.’’7 They make the point that it would
be an error to evaluate a potential investment on the basis of a business’s overall
cost of capital if that investment were more or less risky than the business’s existing
3
Stocks, Bonds, Bills and Inflation Valuation Yearbook (Chicago: Morningstar, 2009), 21.
Aswath Damodaran, Investment Valuation: Tools and Techniques for Determining the
Value of Any Asset, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2002), 182.
5
Tim Koller, Marc Goedhart, and David Wessels, Valuation: Measuring and Managing the
Value of Companies, 4th ed. (Hoboken, NJ: John Wiley & Sons, 2005), 291.
6
Ibbotson Associates, ‘‘What Is the Cost of Capital?’’ 1999 Cost of Capital Workshop,
Chicago: Ibbotson Associates, 1999.
7
Richard A. Brealey, Stewart C. Myers, and Franklin Allen, Principles of Corporate Finance,
9th ed. (Boston: Irwin McGraw-Hill, 2008), 239.
4
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COST OF CAPITAL BASICS
business. ‘‘Each project should in principle be evaluated at its own opportunity cost
of capital.’’8
When a business 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 business 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 business’s overall cost
of capital, depending on the degree of risk of the prospective investment compared
with the business’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 value and is sometimes referred to as 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 three elements to
these expectations:
1. The risk-free rate, which includes:
& Rental rate. A real return for lending the funds risk-free, thus forgoing consumption for which the funds otherwise could be used.
& Inflation. The expected rate of inflation over the term of the risk-free investment.
& Maturity risk or investment rate risk. The risk that the investment’s principal
market value will rise or fall during the period to maturity as a function of
changes in the general level of interest rates.
2. Risk—the uncertainty as to when and how much cash flow or other economic
income will be received. (Risk is discussed more fully in Chapter 5.)
It is the combination of the two items comprising the risk-free rate 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 net cash flows. We convert the stream of expected economic benefits to its present value equivalent 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 would be willing to pay
today for the stream of expected economic income. In mathematical terms, the
8
9
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: A Guide for Managers and Investors, revised ed. (New York: Free Press, 1997).
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Defining Cost of Capital
7
cost of capital is the percentage rate of return that equates the stream of expected
economic income with its present cash value (see Chapter 4).
COST OF CAPITAL IS BASED ON MARKET 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, par value,
or carrying 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. The cost of capital is estimated from market data. These
data refer to expected returns relative to market prices. By applying the cost of
capital derived from market expectations to the expected net 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. Assuming inflationary expectations, the return an investor requires includes compensation for reduced purchasing power of the currency 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 the cost of capital in real terms rather
than in nominal terms and apply it to expected net cash flows expressed in real
terms. We discuss the problems with estimating cash flows and cost of capital in
real terms in Chapter 19.
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 is discounted back to its present value. A discount rate reflects both the time
value of money and risk. Therefore, in its totality it represents the cost of capital.
The sum of the discounted present values of each future period’s net 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.
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COST OF CAPITAL BASICS
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.
DISCOUNT RATE IS NOT THE SAME AS
CAPITALIZATION RATE
Because some practitioners and their clients confuse the terms, we point out here
that discount rate and capitalization rate are two distinctly different concepts.
Discount rate equates to cost of capital. It is a rate applied to all expected economic
income to convert the expected economic income stream to a present value.
A capitalization rate, however, is merely a divisor applied to one single element
of the economic income stream to estimate a present value. The only instance in
which the discount rate is equal to the capitalization rate is when each future
period’s economic income is equal (i.e., no growth), and the economic income is
expected to continue into perpetuity. One of the few examples would be a preferred
stock paying a fixed dividend amount per share into perpetuity.
The relationship between discount and capitalization rates is discussed in
Chapter 4.
STANDARD OF VALUE
Throughout this book, we discuss expected economic income and cost of capital in
the context of various definitions of the generic term value. The term has many
meanings. In this book, a standard of value is a definition of the type of value being
sought. The standard of value addresses the questions: ‘‘value to whom?’’ and
‘‘value under what circumstances?’’ We will identify the applicable standard of
value and its meaning when we are speaking about a particular application. But for
background, a quick summary here would be useful.10
Fair market value is the value standard used in many federal income tax
matters. But in transfer pricing matters under Internal Revenue Code Section
482, the standard of value is the arm’s length standard. The understanding of
these terms is based on the Internal Revenue Code, Treasury regulations, and
interpretations by various courts.
10
Definitions of fair market value, investment value, and intrinsic value are included in the
International Glossary of Business Valuation Terms, jointly developed by the American Institute of Certified Public Accountants, American Society of Appraisers, Canadian Institute
of Chartered Business Valuators, National Association of Certified Valuation Analysts, and
The Institute of Business Appraisers. For a more complete discussion, see Chapter 2 in
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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Defining Cost of Capital
9
Fair value is the standard of value used in financial reporting and is defined in
Financial Accounting Standards Board pronouncements. Fair value has a totally
different meaning in another context. Fair value is typically the applicable standard
of value in fairness and shareholder disputes and is defined by state statute and
court interpretations.
In the United States, the most widely recognized and accepted standard of value
related to real estate appraisals is market value.
Investment value is the specific value of an investment to a particular investor or
class of investors based on individual investment requirements. Intrinsic value
(sometimes called fundamental value) is the specific value of an investment based on
its perceived characteristics inherent in the investment but not based on the value to
any one investor or class of investors.
SUMMARY
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 not observable.
It is forward-looking, based on expected returns. Past returns, at best, provide
guidance as to what to expect in the future.
It is a function of the investment, not a particular investor. To make the discount rate a function of the particular investor’s perceptions implies investment
value rather than fair market value or fair value.
The base against which cost of capital is measured is market value.
It is usually measured in nominal terms, which includes the expected rate of
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.
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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, and Capital Budgeting Projects
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.
The basic steps in valuation and investment selection are:
1. Estimation of economic income
2. Estimation of the cost of capital
3. Use of the cost of capital to calculate present values
These steps are applicable to both the discounted cash flow (DCF) method and
the single-period capitalization method.
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
Throughout this book, we usually assume that the measure of economic income to
which the cost of capital will be applied is net cash flow (sometimes called free cash
10
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Introduction to Cost of Capital Applications: Valuation and Project Selection
11
flow). Net cash flow represents discretionary cash available to be paid out to stakeholders (providers of capital) of an entity (e.g., interest, debt payments, dividends,
withdrawals) without jeopardizing the projected ongoing operations of the entity.
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 upon which most financial
analysts today prefer to focus for both valuation and capital investment purposes.
Net cash flow represents money available to stakeholders, assuming the business
owned by the entity is a going concern and the entity is able to support the projected
operations. Net cash flow can also be used to evaluate liquidation scenarios.
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 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 economic income to a present value. This concept
is summarized succinctly by Brealey, Myers, and Allen: ‘‘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 the critical characteristics of a discount rate:
Definition: A discount rate is a yield rate used to convert anticipated future economic income (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 on the amount invested.
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 future period’s expected economic income. There are, however, cases
in which analysts might choose to estimate a discrete cost of capital to apply to the
expected economic income in each future period. Examples include cases where the
analyst anticipates a changing weighted average cost of capital because of a changing capital structure or cases where the risk characteristics of the economic income
change (e.g., the net cash flows in the early years are ‘‘guaranteed’’ due to contracts
with customers and are risky in later years).
1
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):
147–156.
2
Richard A. Brealey, Stewart C. Myers, and Franklin Allen, Principles of Corporate Finance,
9th ed. (Boston: Irwin McGraw-Hill, 2008), 18.
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COST OF CAPITAL BASICS
Notwithstanding, well-known author, professor, and consultant Dr. Alfred
Rappaport recommends using 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
This latter view is most widely used in practice.
PRESENT VALUE FORMULA
The use of the cost of capital to estimate present value thus requires two sets
of estimates:
1. The numerator. The expected amount of economic income (e.g., the net cash
flow) to be received from 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
Converting the concepts into a mathematical formula, we have the following,
which is the essence of using cost of capital to estimate present value.
(Formula 2.1)
PV ¼
where:
3
NCF1
NCF2
NCFn
þ
þ þ
ð1 þ kÞ ð1 þ kÞ2
ð1 þ kÞn
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
Alfred Rappaport, Creating Shareholder Value: A Guide for Managers and Investors,
revised ed. (New York: Free Press, 1997), 37.
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Introduction to Cost of Capital Applications: Valuation and Project Selection
13
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 and maturity as the subject bond is 10%.
Note three important implications of this scenario:
&
&
&
The issuing business’s embedded cost of capital (i.e., the historic rate of interest
at which the bond was issued) 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 risk rating associated with this bond was lowered between the date of issuance and the valuation date.
If the issuing business 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.
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:
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COST OF CAPITAL BASICS
(Formula 2.2)
PV ¼
¼
$80
$80
$80
$1; 000
þ
þ
þ
ð1 þ :10Þ ð1 þ :10Þ2 ð1 þ :10Þ3 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
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).
Formula 2.2 is sometimes presented in terms of present value factors, or multipliers, which would be presented as follows:
Period
1
2
3
Terminal Value
Cash Flow
$80
80
80
1,000
Factor Multiplier
.9091
.8264
.7513
.7513
Present Value
¼
¼
¼
¼
$72.73
66.12
60.11
751.30
$950.26
(The $.02 difference is due to rounding.)
APPLICATIONS TO BUSINESSES, BUSINESS INTERESTS,
AND CAPITAL BUDGETING PROJECTS
The same framework can be used to estimate the value of an equity interest in a
business or a business’s entire invested capital. One would project the cash flows
available to the interest to be valued and discount those cash flows to their present
value equivalent at a cost of capital (discount rate) that reflects the risk associated
with achieving the particular cash flows. Details of the procedures for valuing entire
businesses or interests in businesses and evaluating and pricing their risks 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 business’s overall risk, then the cost of capital by which that project is evaluated should
be commensurately higher or lower than the business’s overall cost of capital.
Although some businesses apply a single ‘‘hurdle rate’’ to all proposed projects
or investments, the consensus in corporate finance literature is that the rate by which
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Introduction to Cost of Capital Applications: Valuation and Project Selection
15
to evaluate any investment should be based on the risk of that investment, not on the
business’s borrowing cost or overall risk that drives its cost of capital. We agree with
this consensus. If the business invests in projects or assets that increase its overall
risk, then the business’s overall risk will increase marginally. When this increased
risk is recognized and reflected in the market, it will raise the business’s cost of capital. If the returns on the higher risk investment are less than the higher returns commensurate with this higher cost of capital, the result will be a decrease in the value of
the entity or interest (e.g., decrease in the stock price) and a loss in owners’ value.
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.
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CHAPTER
3
Net Cash Flow: Preferred Measure
of Economic Income
Introduction
Defining Net Cash Flow
Net Cash Flow to Common Equity Capital
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
Alternative Measure of Economic Income
Summary
Additional Reading
Technical Supplement Chapter 1: Alternative Net Cash Flow Definitions
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 measure 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 analyses?
Within the income approach, the most often used methods are the discounted
cash flow (DCF) method and the single-year capitalization method. The analyst
must choose the method that is most appropriate, given the facts and circumstances
surrounding the subject business. Once that decision is made, the analyst must
choose between two general frameworks: valuing net cash flows to common equity
capital or valuing net cash flows to the aggregate of invested capital.
When net cash flow to common equity is valued, the discount rate should be the
cost of equity capital. When net cash flow to invested capital is valued, the discount
rate should be the overall cost of capital (commonly referred to as the weighted average cost of capital, or WACC).
16
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Net Cash Flow: Preferred Measure of Economic Income
17
The reasons why the financial community tends to focus on net cash flow as
the preferred measure of economic income are both conceptual and empirical and
are explained further in the next section.
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 order to generate the projected 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 business’s expected cash flow generating capability in future years.
The net cash flow is available to be distributed to the investors or reinvested
in some incremental project not reflected in the net cash flows that have been
discounted. That reinvestment results in incremental value in future years.
Net cash flow is sometimes called free cash flow. It is also sometimes called
net free cash flow, although this phrase seems redundant. With finance terminology
being as ambiguous as it is, minor variations in the definitions of these terms
frequently arise, making it essential to clearly define the measure of income to be
employed in the valuation.
Net Cash Flow to Common Equity Capital
In valuing equity capital by discounting or capitalizing expected net cash flows (keeping
in mind the important difference between discounting and capitalizing, as discussed in
Chapters 1 and 4), net cash flow to equity (NCFe in our notation system) is defined as:
(Formula 3.1)
Net income to common equity (after income taxes)
Plus:
Noncash charges (e.g., depreciation, amortization, deferred revenues, and
deferred income taxes)
Minus: Capital expenditures (amount necessary to support projected revenues and
expenses)
Minus: Additions to net working capital (amount necessary to support projected
revenues)
Minus: Dividends on preferred equity capital
Plus:
Cash from increases in the preferred equity or debt components of the capital structure (amount necessary to support projected revenues)
Minus: Repayments of any debt components or retirement of any preferred components of the capital structure
Equals: Net cash flow to common equity capital
Capital expenditures are those amounts needed to match the revenue and expense
forecasts. That is, the capital expenditures are those amounts needed for replacement
of plant and/or equipment that are retired in the normal course of business, those
amounts needed for increases in capacity consistent with the projected revenue (e.g.,
increased number of machines, increased warehouse space), and those amounts
needed for replacement of existing plant and/or equipment consistent with projected
expenses (e.g., replacement of inefficient equipment with more efficient equipment).
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COST OF CAPITAL BASICS
Net working capital excludes (1) any excess cash and investments that are not
needed to support the level of business activity in the projected revenues and (2) any
debt classified as short-term that is a component of the capital structure (e.g., the
amount included in current liabilities for the current portion of long-term debt).
We discuss this more fully in Chapter 6.
Because we are only including amounts of investment in net working capital and
capital expenditures needed for the projected revenues and expenses included in
the projected net cash flows to be discounted, we can term these sustainable net
cash flows.
Net cash flow to equity is also called free cash flow to equity (FCFe).
Net Cash Flow to Invested Capital
In valuing the entire invested capital of a business or project by discounting or
capitalizing expected cash flows, net cash flow to invested capital or net cash
flow to the firm (NCFf in our notation system) is defined as:
(Formula 3.2)
Net income to common equity (after income taxes)
Plus:
Noncash charges (e.g., depreciation, amortization, deferred revenues, and
deferred income taxes)
Minus: Capital expenditures (amount necessary to support projected revenues and
expenses)
Minus: Additions to net working capital (amount necessary to support projected
revenues)
Plus:
Interest expense (net of the tax deduction resulting from interest as a taxdeductible expense)
Plus:
Dividends on preferred equity capital
Equals: Net cash flow to invested capital
The amounts of capital expenditures and additions to net working capital are
consistent with the projections of revenues and expenses and the amounts defined
previously (in the net cash flow to common equity capital).
In other words, NCFf adds back interest (tax-affected because interest is a taxdeductible expense) because invested capital includes the debt on which the interest
is paid. Interest is the payment to the debt component of the invested capital. It also
adds back dividends on preferred stock for the same reason (i.e., invested capital
includes the preferred capital on which the dividends are paid).
Net cash flow to invested capital is also called free cash flow to the firm (FCFf).
Occasionally, an analyst treats earnings before interest, taxes, depreciation, and
amortization (EBITDA) as if it were equivalent to net cash flow to invested capital.
This error may be a significant matter because the analyst has added back the noncash charges but ignored the requisite capital expenditures and additions to net
working capital necessary to sustain the business as projected.
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 overall cost of capital or WACC.
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19
Net Cash Flow: Preferred Measure of Economic Income
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 net cash flows. In the real world, it is far more
common for realized net cash flows to be below forecast than above, as we explain
later. A valuation that does not take this factor into account will overvalue a
business.
If the distribution of possible net cash flows in amount and likelihood in each
period is symmetrical above and below the most likely net cash flow in that period,
then the most likely net cash flow is equal to the probability-weighted net cash flow
(the mathematical expected value of the distribution). However, many times distributions of possible net cash flows for any given period are skewed. This is where
probability weighting comes into play. Exhibit 3.1 shows the calculation of the
probability-weighted expected values of projected net cash flows under a symmetrically distributed scenario (Scenario A) and under a skewed distribution scenario
(Scenario B). Exhibit 3.2 portrays the information in Exhibit 3.1 graphically.
In both scenario A and scenario B of Exhibit 3.1, the most likely net 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 $714. In scenario B, $714 is the figure that
should appear in the numerator of the discounted cash flow formula, not $1,000.
EXHIBIT 3.1 Example of Net Cash Flow Expectations
Scenario A—Symmetrical Net Cash Flow Expectation
Projected Net Cash Flows
$1,600.00
1,500.00
1,300.00
1,000.00
700.00
500.00
400.00
Probability of Occurrence
Probability-Weighted Value
0.01
0.09
0.20
0.40
0.20
0.09
0.01
100%
$16
135
260
400
140
45
4
$1,000
Scenario B—Skewed Net Cash Flow Expectation
Projected Net Cash Flows
$1,600.00
1,500.00
1,300.00
1,000.00
700.00
500.00
(100.00)
(600.00)
Probability of Occurrence
Probability-Weighted Value
0.01
0.02
0.05
0.35
0.25
0.20
0.10
0.02
100%
$16
30
65
350
175
100
(10)
(12)
$714
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COST OF CAPITAL BASICS
Scenario A: Symmetrical Cash Flow Expectation
Probabilitiy of Occurrence
0.45
0.4
Expected Value
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
$0
$200
$400
$600
$800
$1,000
$1,200
$1,400
$1,600
$1,800
Expected Net Cash Flow
Scenario B: Skewed Cash Flow Expectation
0.4
0.35
Probabilitiy of Occurrence
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Expected Value
0.3
0.25
0.2
0.15
0.1
0.05
($1,000)
($500)
0
$0
$500 $714 $1,000
$1,500
$2,000
Expected Net Cash Flow
EXHIBIT 3.2 Example of Net Cash Flow Expectations (Graphs of Data from Exhibit 3.1)
Most analysts do not have the benefit of receiving, or the time or information
to develop, a probability distribution for each year’s expected net cash flow (and
it is not a common practice to develop one). However, analysts should be aware of
the concept when deciding on the amount of each expected net cash flow to
be discounted.
Many analysts first think in terms of symmetrical distributions. But most
businesses have a maximum capacity to produce their services or goods in any
one year. For example, in scenario B, the business in any year may run up against
capacity constraints to increase revenues and net cash flows (except with short
spurts of multishift seven-day-per-week production). But on the downside, the
business is more likely to lose sales and experience reduced cash flows. So in any
one year, there is a great likelihood that the distribution of expected cash flows is
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Net Cash Flow: Preferred Measure of Economic Income
21
skewed. This does not mean that the net cash flows are identical in future years.
As investment is made to increase capacity, the entire distribution of expected net
cash flows can be ratcheted upwards. But in any year in the future, the distribution of possible net cash flow may be and probably is skewed. So even if one does
not receive or develop a probability distribution of net cash flows, the analyst
should be aware that there is often more downside risk than upside potential in
any year’s net cash flows for many businesses.
As we pointed out in Formula 2.1 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 book on forecasting, the analyst may need to facilitate the preparation of
expected net cash flows and/or test the reasonableness of the net cash flows projection provided. In Chapter 34, we discuss projections of expected future economic
benefits, focusing on net cash flows, and tools one can use in understanding the distribution of expected net cash flows.
WHY NET CASH FLOW IS THE PREFERRED MEASURE OF
ECONOMIC INCOME
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 for two
reasons.
1. Conceptual. Net cash flow provides amounts that are available to compensate
providers of capital for their investments in a discrete period of time. In a valuation context, it is important that the numerator of Formula 2.1 gives the most
accurate estimate of what the business expects to generate as a return on the
capital invested.
2. Empirical. It is the economic income measure that best matches discount rate
estimates.
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 is clearly stated in
Morningstar’s 2009 Stocks, Bonds, Bills and Inflation Valuation Yearbook:
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.
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COST OF CAPITAL BASICS
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 one uses the SBBI data or the data contained in the Duff & Phelps Risk
Premium Report to develop a common equity discount rate—using either the
build-up model (see Chapter 7) or the capital asset pricing model (CAPM)
(see Chapter 8)—the discount rate is applicable to net cash flow available to
the common equity investor because the SBBI and Duff & Phelps return data
have two components:
1. Dividends to the common stock
2. Changes in common stock prices
The investor receives the dividends, so their utilization is entirely at the investor’s discretion. To the extent that net cash flows are retained in the business,
they are assumed to be reinvested for the benefit of the common equity and
added to the value of the common equity. For actively traded stock, the investor’s realization of the change in stock price is equally discretionary because the
stocks are assumed to be 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).
Is accounting data useful in better estimating net cash flows? One study examines whether accounting variables explain stock price movements by assisting users
of accounting information to better forecast cash flows.2 The authors find that
changes in four accounting variables explain about 20% of the differences in stock
returns.
Another study concludes that the direct method of cash flow statements (compared with the more popular indirect method) is valuable to investors and improves
accuracy of forecasting future operating cash flows and earnings.3
While theoreticians and practitioners alike accept the primacy of net cash flows
in valuation,4 this premise is the subject of recent study.
1
Stocks, Bonds, Bills and Inflation Valuation Yearbook (Chicago: Morningstar, 2009),
13–14.
2
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.
3
Steven Orpurt and Yoonseok Zang, ‘‘Do Direct Cash Flow Disclosures Help Predict Future
Operating Cash Flows and Earnings?’’ Working paper, May 2009. Available at http://ssrn
.com/abstract=1319102.
4
See, for example, Steven J. Kaplan and Richard S. Ruback, ‘‘The Valuation of Cash Flow
Forecasts: An Empirical Analysis,’’ Journal of Finance 50(4) (September 1995): 1059–1093.
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Net Cash Flow: Preferred Measure of Economic Income
23
In one study, the authors analyze whether multiples for guideline public companies based on (1) earnings per share or (2) 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) and applied to the earning per share or the operating cash flow per share of the subject company result in
a more accurate estimate of the stock price of the subject public company. Their
results suggest that valuation multiples based on earnings forecasts provide better
valuations where consensus earnings forecasts of analysts are available.5
And finally another study finds that when investors are provided complete cash
flow data, stock prices fully reflect that data.6
ALTERNATIVE MEASURE OF ECONOMIC INCOME
An alternative formulation of economic income is residual income.7 Valuations
using residual income always yield the same value as does the discounted net cash
flow method when applied with consistent valuation assumptions. Residual income
represents the economic profit of the business after deducting the cost of capital.
We discuss residual income because it is called in the literature by various
names, such as economic profit, abnormal earnings,8 and Economic Value Added1
(EVA) (EVA is discussed in Chapter 26). The term abnormal earnings implies that
the business is expected to earn more than its cost of capital. Residual income can
be formulated as residual income to common equity capital or residual income to
total capital. This measure of economic profit is often used in measures of internal
business performance.
In the residual income model, the value of business has two components: the
current net book value of common equity plus the present value of future residual
income. The resultant value is equivalent to the net cash flow to common equity
model using consistent assumptions.
Residual income to common equity is based on the clean-surplus accounting
statement:
(Formula 3.3)
BV n ¼ BV n1 þ NCIe;n De;n
where:
5
BVn ¼ Book value of net assets
NCIe,n ¼ Net comprehensive income to common equity, which includes income terms reported directly in the equity account rather than in
the income statement
Jing Liu, Doron Nissim, and Jacob Thomas, ‘‘Is Cash Flow King in Valuations?’’ Financial
Analysts Journal 63(2) (March–April 2007): 56–68.
6
Keren Bar-Hava, Roni Ofer, and Oded Sarig, ‘‘New Tests of Market Efficiency Using Fully
Identifiable Equity Cash Flows,’’ Working paper, February 2007. Available at http://ssrn
.com/abstract=965242.
7
Stephen H. Penman, Financial Statement Analysis and Security Valuation, 3rd ed. (New
York: McGraw-Hill, 2007), Chapter 5.
8
J. Feltham and J. Ohlson, ‘‘Valuation and Clean Surplus Accounting for Operating and
Financing Activities,’’ Contemporary Accounting Research 11(2) (Spring 1995): 689–731.
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COST OF CAPITAL BASICS
De,n ¼ Distributions to common equity, net of new issues of common
equity
¼ NCIn ½BV n BV n1 Clean surplus accounting captures the concept that all changes in common
equity not involving common equity pass through the income statement. Comprehensive income differs from reported net income primarily because of foreign
currency translation adjustments, derivative accounting, and certain pension liability adjustments.
Residual income is the return on common equity (expressed in dollars) in excess
of the cost of equity capital as is shown in Formula 3.4.
(Formula 3.4)
RIe;n ¼ NCIe;n ½BV n1 ke where:
RIe,n ¼ Residual income for common equity capital
NCIe,n ¼ Net comprehensive income to common equity
BVn1 ¼ Book value of net assets at period n–1
ke ¼ Cost of equity capital
Residual income to total capital is based on the clean-surplus accounting
statement:
(Formula 3.5)
NOAn ¼ NOAn1 þ NCIf ; n Df ; n
where: NOA ¼ Net operating assets
¼ Total capital of the business
NCIf,n ¼ Net comprehensive income to the firm, which includes income
terms reported directly in the equity account rather than in the income statement
Df,n ¼ Distributions to total capital, net of new issues of debt or equity
capital
¼ NCIf ; n ½NOAn NOAn1 Residual income is the return on total capital (expressed in dollars) in excess of
the overall cost of capital (WACC), as is shown in Formula 3.6.
(Formula 3.6)
RIf ; n ¼ NCI f ; n ½NOAn1 WACC
where:
RIf,n ¼ Residual income for total capital
NCIf,n ¼ Net comprehensive income to total capital
NOA ¼ Net operating assets
WACC ¼ Overall cost of capital
In Chapter 4, we demonstrate the conditions for equality between valuations
using net cash flow and residual income.
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Net Cash Flow: Preferred Measure of Economic Income
25
SUMMARY
Net cash flow is the measure of economic income that most financial analysts prefer
to use today when discounting or capitalizing using the cost of capital for valuation
or project selection. In valuing cash flows to equity capital, the discount rate should
be the cost of equity capital. In 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
probability-weighted 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(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.
TECHNICAL SUPPLEMENT CHAPTER 1: ALTERNATIVE
NET CASH FLOW DEFINITIONS
We summarize alternative net cash flow definitions in the Cost of Capital: Applications and Examples, 4th ed. Workbook and Technical Supplement, Chapter 1.
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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
Equivalency of Capitalizing Residual Income
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
measures 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 economic income to the respective class(es) of capital, we focus on the economic income of just one single period,
usually the economic income expected in the first year immediately following the
valuation date. That amount represents the long-term sustainable base level of economic income or a base from which the level of economic income is expected to
26
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Discounting versus Capitalizing
grow or decline at a more or less constant rate. We then divide that single-year economic income 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 implied by taking the inverse of pricing multiples, for example, inverting
the price-to-earnings ratio.)
A common error is the use of a 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 (i.e., 0% growth), 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.
CAPITALIZATION FORMULA
Putting the capitalization concept into a formula, we have:
(Formula 4.1)
PV ¼
where:
NCF1
c
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 a dividend 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
rating 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 preferred
stock is issued in perpetuity and there is no prospect of 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.
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COST OF CAPITAL BASICS
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 into
perpetuity. This is the one unique situation in which the discount rate (cost of capital) equals the capitalization rate. The discount rate equals the capitalization rate
because no change (no increase, commonly termed growth, or decline) in the investor’s cash flow is expected. Few real-world investments are that simple.
Investors often are expecting some level of growth over time in the cash flows
available to pay dividends or distributions. 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 growth in cash flows for the
investment is stable and sustainable over a long period of time, then the discount
rate (cost of capital) can reasonably be converted to a capitalization rate.
The capitalization rate is a function of the discount rate. This raises the obvious
question: What is the functional relationship between the discount rate and the capitalization rate?
Assuming stable long-term growth in cash flows from the subject investment,
the capitalization rate equals the discount rate minus the expected long-term growth
rate. 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 from the
subject investment
The critical assumption in this formula is that the expected rate of increase
(growth) in the cash flow from the investment is relatively constant over the
long term (technically into perpetuity).
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Discounting versus Capitalizing
Caveat: As explained in Chapter 3, in estimating the net cash flow of a
business to capitalize, we subtract from net income investments such as capital
expenditures and additional net working capital needed to realize the projected
future revenues and expenses of the existing business (i.e., we are matching the
projected capital expenditure and net working capital investments with the projected revenues and expenses). In this formulation, we are valuing the existing
business as of a specific date. While we are not valuing currently unknown investments that may be made in future years from investing these net cash flows
in new business ventures, the underlying assumption inherent in the methodology is that any retained net cash flow is reinvested at the cost of capital. This is
further explained with an example in Chapter 34.
Now we know two essential things about using the cost of capital to estimate
present value using the capitalization method for a business, assuming relatively
stable long-term growth in the net cash flow:
1. Present value equals the next period’s expected net cash flow divided by the
capitalization rate.
2. The net cash flow capitalization rate is the discount rate (cost of capital) minus
the expected long-term rate of growth in the net cash flow. (Technically, growth
in this context means into 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 of more distant future years.) The growth in net
cash flow is sustainable because one has subtracted the investments needed to
realize the expected revenues and expenses.
We can combine these two relationships into a single formula as:
(Formula 4.4)
PV ¼
where:
NCF1
kg
PV ¼ Present value
NCF1 ¼ Net cash flow expected in period 1, the period immediately following the valuation date
k ¼ Discount rate (cost of capital)
g ¼ Expected long-term growth rate in net cash flow
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 long-term growth in net cash flow from year 1 to perpetuity is expected to be 3%.
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COST OF CAPITAL BASICS
Substituting numbers from the preceding assumptions into Formula 4.4 gives us:
(Formula 4.5)
$100
0:13 0:03
$100
¼
0:10
¼ $1; 000
PV ¼
In this example, the estimated value of the investment in the business is $1,000.
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 change in cash flow is estimated specifically and included in the numerator.
In capitalizing: Estimates of rates of changes in future cash flows are averaged
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
cash flow from the investment into 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
the next period’s estimate. The assumption is that net cash flows will grow evenly into
perpetuity from the period immediately preceding the valuation date. This constant
growth capitalization formula, commonly known as the Gordon Growth Model
(named for Professor Myron Gordon, who popularized this formulation1), as applied
to the net cash flow is as follows:
(Formula 4.6)
PV ¼
where:
1
NCF0 ð1 þ gÞ
kg
PV ¼ Present value
NCF0 ¼ Net cash flow in period 0, the period immediately preceding the
valuation date
Myron J. Gordon, The Investment, Financing, and Valuation of the Corporation
(Homewood, IL: R. D. Irwin, 1962).
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Discounting versus Capitalizing
k ¼ Discount rate (cost of capital)
g ¼ Expected sustainable long-term growth rate in net cash flow
Note that for this model to make economic sense, NCF0 must represent a normalized amount of net 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
net cash flow for period 0 but may reflect 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. Any valuation report prepared should
state the steps taken and the assumptions made in concluding that last year’s actual
results are the most reasonable base for expected net cash flow 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 the use of 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
¼ $1;030
PV ¼
In this example, the estimated value of the investment is $1,030. The relationship between this and the previous example is simple and straightforward. We
moved the receipt of the $100 back in time by one period, and the value of the
investment was increased by 3%, the growth rate. In a constant growth capitalization model, even assuming that all of the net 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 sustain the projected growth.
The investor in this example thus earns a total rate of return of 13%: 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 pure discounting or pure capitalizing methods
of valuing expected net cash flows.
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&
&
COST OF CAPITAL BASICS
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 into 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 expected 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: Determine a reasonable length of time for which discrete projections of
net cash flows can be made.
Step 2: Estimate specific expected net cash flows for each of the discrete projection periods.
Step 3: Estimate a sustainable long-term rate of growth in net cash flows from
the end of the discrete projection period forward.
Step 4: Use the constant growth model (Gordon Growth Model) (Formula 4.6)
to estimate the future value as of the end of the discrete projection period
(commonly referred to as the terminal or residual value).
Step 5: Discount each of the discrete net cash flows back to their present value at
the discount rate (cost of capital) for the number of periods until it is projected to be 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 discrete net cash flow).
Step 7: Sum the value derived from steps 5 and 6.
These steps can be summarized by the next formula, which assumes that net
cash flows are received at the end of each year:
(Formula 4.8)
NCF1
NCF2
NCFn
þ
þ þ
þ
PV ¼
2
ð1 þ kÞ ð1 þ kÞ
ð1 þ kÞn
NCFn ð1 þ gÞ
kg
ð1 þ kÞn
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Discounting versus Capitalizing
where: NCF1 . . . NCFn ¼ Net cash flow expected in each of the periods 1 through
n, n being the last period of the discrete net cash flow
projections
k ¼ Discount rate (cost of capital)
g ¼ Expected sustainable long-term 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 depends on how
many years or periods there will be variable change in net cash flows. The
residual period begins whenever the net cash flows begin growing at a constant
growth rate. Having said this, it is not uncommon for the discrete periods to be
as few as 3 years or as many as 10 years, while for cyclical businesses, the discrete period commonly corresponds to the number of years or periods until the
point is reached where the net cash flow represents an average base net cash
flow expected over an entire business cycle. 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, based on the business’s performance and industry and overall
economic expectations, 5% average growth in net cash flow appears to be a reasonable estimate of sustainable long-term growth.
3. The cost of capital for this investment is estimated to be 12%.
Substituting numbers derived from these assumptions into Formula 4.8
produces:
(Formula 4.9)
$100
$120
$140
þ
þ
þ
PV ¼
ð1 þ 0:12Þ ð1 þ 0:12Þ2 ð1 þ 0:12Þ3
$140ð1 þ 0:05Þ
0:12 0:05
ð1 þ 0:12Þ3
$147
$100
$120
$140
¼
þ
þ
þ 0:07
1:12 1:2544 1:4049 1:4049
¼ $89:30 þ $95:66 þ $99:65 þ
$2; 100
1:4049
¼ $89:30 þ $95:66 þ $99:65 þ $1; 494:77
¼ $1; 779:38
Thus, the estimated value of this investment is $1,779.
As in Chapter 2, the preceding formula is often presented in terms of present value factors (multipliers). In this case, the preceding would be presented
as follows:
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Period
1
2
3
Terminal Value
COST OF CAPITAL BASICS
Cash Flow
$100
120
140
2,100
Factor Multiplier
.8929
.7972
.7118
.7118
Present Value
¼
¼
¼
¼
$89.29
95.66
99.65
1,494.78
$1,779.38
A common error is to discount the terminal value for n + 1 periods instead of n
periods. The assumption we have made is that the nth period net 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 = 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 discrete projection period, the greater the impact of the terminal
value on the total present value. The length of the discrete projection period
should be the number of periods until the business is expected to reach a steady
state, that is, until the business is expected to reach a normalized level of
net cash flow that it can grow at a more or less constant percentage rate over a
long period of time. There is no fixed number of years for the discrete projection
period. There is no ‘‘magic’’ in using 5 years or 10 years for the discrete 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 capitalization rate is
negative and the model 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 in that case
would be a salvage value. For example, the license to operate the business may have
a finite life at which the operating business is liquidated.
Some practitioners use a market multiple, such as the industry average multiple of earnings before interest, income taxes, depreciation, and amortization
(EBITDA) to estimate a terminal value. We believe that use of a market-derived
multiple for calculation of the terminal value is not appropriate as it mixes elements of the market and income approaches and does not represent a true income approach. In addition to mixing valuation approaches, it is not clear that
a current average industry multiple reflects a long-term estimate of growth consistent with the sustainable long-term growth rate in net cash flows. If the
growth rate embedded in the multiple is inconsistent, utilizing this method will
either overvalue or undervalue the business. As an example, current multiples in
an industry in a rapid growth phase would probably include rates of growth for
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Discounting versus Capitalizing
a period of years in excess of sustainable long-term growth for the industry
upon maturity.
EQUIVALENCY OF DISCOUNTING AND
CAPITALIZING MODELS
If certain 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 net cash flow in period 1
of $100, growing into perpetuity at 3%. The cost of capital (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 net cash flow) at 10% gave us an estimated
present value of $1,000 ($100 0.10 = $1,000).
Let us take these same assumptions and put them into a discounting model. For
simplicity, we will use only 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
þ 0:13 0:03
ð1 þ 0:13Þ ð1 þ 0:13Þ2 ð1 þ 0:13Þ3
ð1:13Þ3
$109:27
$100
$103
$106:09
þ
þ
þ 0:10
1:13 1:2769
1:4429
1:4429
$1092:73
$88:50 þ $80:66 þ $75:53 þ
1:4429
$88:50 þ $80:66 þ $73:53 þ $757:31
$1;000
2
PV ¼
¼
¼
¼
¼
This example, showing the equivalency of using the cost of capital in either the
discounting or the capitalizing model, when certain key assumptions are met, demonstrates the point that capitalizing is really just a shorthand form of discounting.
Capitalization is often used when one believes that the current sustainable net cash
flow will grow at an average growth rate in the future or when one does not have
sufficient information to implement a discounting model but nevertheless feels comfortable that capitalizing a single year’s net cash flow will provide meaningful valuation results. Nevertheless, when using a capitalization of net cash flow model, the
analyst should consider whether the present value of net cash flows would be the
same if a full discounting model were used. If not, it may be propitious to review
and possibly adjust certain assumptions. If the discounting and capitalization of net
cash flow models produce different answers using the same cost of capital and the
same inputs, there may be some kind of an internal inconsistency.
Caveat: We have seen instances where analysts have used both a discounted
cash flow method and the single-year capitalization of net cash flow method in the
same valuation analysis and then weighed the two results in the reconciliation of
value. This is not correct. In developing an income approach, the analyst should
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COST OF CAPITAL BASICS
determine whether a multiperiod discounted cash flow is needed or whether the
abridged single-year capitalization method will suffice.
MIDYEAR CONVENTION
In our previous examples, we have assumed that net cash flows are received by investors at the end of each period. Even if a company realizes cash flows throughout
the year, this is a reasonable assumption in those cases in which investors receive
either contractual distributions or after an assessment by company management
that sufficient funds are available to make an end-of-period distribution to investors.
For many businesses or investments, however, it may be more reasonable to assume
that the cash flows are distributed more or less evenly throughout the year. For
example, many businesses make quarterly distributions. To accommodate this latter
assumption, we can modify our formulas for what we call the midyear convention.
Midyear Discounting Convention
The formula for midyear discounting requires 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 kÞ
0:5
þ
NCF2
1:5
ð1 þ kÞ
þ þ
NCFn
ð1 þ kÞn0:5
Midyear Capitalization Convention
Similarly, we can make a modification to the capitalization formula to reflect the
receipt of net cash flows more or less uniformly throughout the year. The modification to Formula 4.4, the capitalization formula, is handled by accelerating the returns by a half year in the numerator:2
(Formula 4.12)
PV ¼
NCF1 ð1 þ kÞ0:5
kg
Formula 4.12 is a mathematical equivalent of Formula 4.13.
2
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; and Jay B. Abrams and R. K. Hiatt, ‘‘The Bias
in Annual (Versus Monthly) Discounting Is Immaterial,’’ Business Valuation Review
(September 2003): 127–135.
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Discounting versus Capitalizing
(Formula 4.13)
NCFn ð1 þ gÞð1 þ kÞ0:5
NCF1
NCF2
NCFn
kg
þ
þ þ
þ
PV ¼
n
0:5
1:5
n0:5
ð 1 þ kÞ
ð1 þ kÞ
ð1 þ kÞ
ð1 þ kÞ
Midyear Convention in the Two-Stage Model
Combining discrete period discounting and capitalized terminal value into a twostage model as shown in Formula 4.8, the midyear convention two-stage equation
becomes:
(Formula 4.14)
NCFn ð1 þ gÞ
kg
PV ¼
þ
þ þ
þ
0:5
1:5
n0:5
ð1 þ kÞn0:5
ð1 þ kÞ
ð1 þ kÞ
ð1 þ kÞ
NCF1
NCF2
NCFn
Using the same assumptions as in Formula 4.9 (where the value using the yearend convention was $1,779) and using Formula 4.13, we get Formula 4.15:
(Formula 4.15)
$140ð1 þ 0:05Þð1 þ 0:12Þ0:5
$100
$120
$140
0:12 0:05
PV ¼
þ
þ
þ
0:5
1:5
ð1 þ 0:12Þ
ð1 þ 0:12Þ
ð1 þ 0:12Þ2:5
ð1 þ 0:12Þ3
$155:57
$100
$120
$140
þ
þ
þ 0:07
¼
1:0583 1:1853 1:3275 1:4049
$2;222:43
¼ $94:49 þ $101:24 þ $105:46 þ
1:4049
¼ $94:49 þ $101:24 þ $105:46 þ $1;581:91
¼ $1;883
In this case, using the midyear convention increased the value by
$104ð$1; 883 $1;779 ¼ $104Þ or 5:8%ð$104 $1; 779 ¼ 0:058Þ.
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.
Repeating Formula 4.14, we have:
(Formula 4.16)
NCFn ð1 þ gÞ
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
NCF1
NCF2
NCFn
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COST OF CAPITAL BASICS
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
ð1 þ 0:12Þ0:5 ð1 þ 0:12Þ1:5 ð1 þ 0:12Þ2:5
ð1 þ 0:12Þ2:5
$100
¼
$120
$140
$147
$100
$120
$140
þ
þ
þ 0:07
1:0583 1:1853 1:3275 1:3275
¼ $94:49 þ $101:24 þ $105:46 þ
$2;100
1:3275
¼ $94:49 þ $101:24 þ $105:46 þ $1;581:92
¼ $1;883
(Any difference is due to rounding.)
Note that using the midyear convention will always produce a greater value
when the annual projected net cash flows are the same (and positive), because of the
time value of money. The assumption underlying the midyear convention is that investors receive the net 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 byð1 þ kÞ0:5 .
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 that best represents the weighted average receipt of the
monthly net cash flows during the year. You can substitute n = 0.3 for n = 0.5 in the
midyear convention formula. The important point is that you need to understand
the timing of the net 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 a fiscal (financial reporting)
year; rather, the valuation date is in the middle of the fiscal 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 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.
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Discounting versus Capitalizing
Exhibit 4.1 presents the timeline of net cash projections valued in Formula 4.8
(net cash flows assumed to be realized at the end of each of the future years).
0
Year 1
2
3
4
^
^
^
^
EXHIBIT 4.1 Timeline of Net Cash Flows Equivalent to Formula 4.8
where: 0 ¼ Valuation Date
^ ¼ Point in year where net cash flows assumed to be realized
Exhibit 4.2 presents the timeline of net cash flow projections valued in Formula
4.13 (net cash flows assumed to be realized at the midpoint of each of the future
years or uniformly during those future years).
Year 1
0
^
2
^
3
4
^
^
EXHIBIT 4.2 Timeline of Net Cash Flows Equivalent to Formula 4.13
where: 0 ¼ Valuation Date
^ ¼ Point in year where net cash flows assumed to be realized
Now assume, for example, that the valuation date is at the end of the fifth
month of the current financial reporting year and that 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, 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 25 months after the valuation date, and each subsequent year’s net cash
flows are expected to be realized 12 months thereafter. Exhibit 4.3 presents the timeline of net cash flows expected in this example.
Partial Full
Year Year 1
0 ^
^
2
^
3
^
4
^
EXHIBIT 4.3 Timeline of Net Cash Flows Equivalent to Formula 4.16
where: 0 ¼ Valuation Date
^ ¼ Point in year where net cash flows assumed to be realized
Formula 4.18, a variation of Formula 4.14, displays the calculation of the present value of net cash flows where the first projection period is a partial year.
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COST OF CAPITAL BASICS
(Formula 4.18)
PV ¼
ðNCF1 pyÞ
py
ð1 þ kÞ 2
þ
NCF2
ð1 þ kÞpyþ0:5
þ þ
NCFn
ð1 þ kÞpyþðn0:5Þ
þ
NCFn ð1 þ gÞ
ðk gÞ
ð1 þ kÞpyþðn0:5Þ
where: py ¼ months of partial first year expressed as a decimal
7
In the example, the partial year represents 12
¼ 0:5833 of the first year, and the
partial year factor for the present value of the net cash flows in the partial first year
Þ
¼ 0:2917. That is, the first-period net cash flows are expected to be
equals ð70:5
12
received 0.2917 of a year following the valuation date (3.5 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 (0.5833 + 0.5) =
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 4.18 using the same
assumptions as in Formula 4.15 except for the partial first year, we get:
(Formula 4.19)
7
$100 $120
$140
12
PV ¼
þ
þ
7
7
0:2917
þ0:5
12
ð1 þ 0:12Þ
ð1 þ 0:12Þ
ð1 þ 0:12Þ 12þð20:5Þ
þ
¼
$140ð1 þ 0:05Þ
ð0:12 0:05Þ
ð1 þ 0:12Þ 12þð20:5Þ
7
$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
$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 $56:41 þ
¼ $1;931:55
EQUIVALENCY OF CAPITALIZING RESIDUAL INCOME
As we discussed in Chapter 3, the literature includes an alternative formulation of
the valuation of net cash flows based on residual income. The equivalent residual
income valuation to Formula 4.6 as applied to net cash flows to equity capital is:
(Formula 4.20)
where:
PV ¼ BV 0 þ RIe;1 =ðke gÞ
PV ¼ Present value
BV0 ¼ Book value (net asset value) for period 0, the period immediately preceding the valuation date
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Discounting versus Capitalizing
RIe,1 ¼ Residual income to common equity capital for period 1
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.
Applying Formula 3.4 and assuming the ke = 13%, we get the residual income
for common equity capital:
(Formula 4.21)
RIe;n ¼ NCIe;n ½BV n1 ke RIe;1 ¼ $127 ð$800 0:13Þ
¼ $127 $104
¼ $23
Now applying Formula 4.20 we get:
$23
ð0:13 0:03Þ
¼ $800 þ $230
¼ $1;030
PV ¼ $800 þ
This is the same result we obtained in Formula 4.7.
EXHIBIT 4.4 Example of Valuation Using Residual Income to Common Equity
For Company A We Have:
Income Statement
EBIT
Interest Expense
EBT
Taxes
NI
Year 1
Growth Rate
$228
16
$212
85
$127
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
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
Growth Rate
3%
3%
3%
3%
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COST OF CAPITAL BASICS
Using the clean-surplus accounting statement, Formula 3.3, we get:
(Formula 4.22)
NCF1 ¼ BV 0 þ NCIe;1 BV
¼ $800 þ $127 $824
¼ $103
where: NCF1 ¼ Net cash flow to common equity in period 1
NCIe,1 ¼ Net comprehensive income to common equity in period 1
BVn ¼ Book value of equity at time = zero (valuation date)
BV1 ¼ Book value of equity at time = 1
This is the same net cash flow we capitalized in Formula 4.7.
The abnormal earnings growth (AEG)–based valuation is equivalent to the residual income and net cash flows to equity capital models using consistent assumptions. Abnormal earnings growth is defined as:
(Formula 4.23)
AEG2 ¼ RIe;2 RIe;1
where: AEG ¼ Abnormal earnings growth
RIe,n ¼ Residual income to equity
The abnormal earnings growth–based valuation formula equivalent to Formula
4.6 is defined as:
(Formula 4.24)
PV ¼ ð1=ke Þ NCIe;1 þ AEG2 =ðke gÞ
where: NCIe,1 ¼ Net comprehensive income to common equity, which includes income terms reported directly in the equity account rather than in
the income statement and the variables are as defined in Formula
4.20, Formula 4.21, and Formula 4.23.
Exhibit 4.5 continues the example in Exhibit 4.4 for abnormal earnings growth.
EXHIBIT 4.5 Example of Valuation Using Abnormal Earnings Growth
RIe;2 ¼ RIe;1 1:03
¼ $23:69
AEG2 ¼ RIe;2 RIe;1
¼ $23:69 $23:00
¼ $0:69
Applying Formula 4.24 we get:
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Discounting versus Capitalizing
1
ð$23:69 $23Þ
$127 þ
0:13
ð0:13 0:03Þ
1
½$127 þ $6:9
¼
0:13
1
¼
½$133:9
0:13
¼ $1;030
PV ¼
This is the same result as we obtained in Formula 4.7.
We can 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, as well
as the comparable residual income and abnormal earnings models.3
Why would we use the residual income 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 business is earning returns in excess of its cost
of 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.
Although 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
income will be reduced if the carrying amounts of net assets overstate their values,
reducing the present value of residual income.
The indicated value resulting from application of the residual income model will
always be equivalent to a dividend discount valuation and a discounted cash flow
(DCF) valuation if we could forecast dividends and net cash flows for very long (infinite) horizons, or if we 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 at net asset (book) value. More generally, book value is not
sufficient, so one adds forecasts of residual income for the near term, component (2),
and speculation about the long term, component (3), to estimate the difference between indicated value and book value.4
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 income formulation of
valuation.
3
Ronald S. Longhofer, ‘‘The Residual Income Method of Business Valuation,’’ Business Valuation Review (June 2005): 65–70.
4
Stephen H. Penman, ‘‘Handling Valuation Models,’’ Journal of Applied Corporate Finance
(Spring 2006): 51.
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COST OF CAPITAL BASICS
SUMMARY
This chapter presented the mechanics of discounting and capitalizing and has defined the difference between a discount rate and a capitalization rate.
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 net 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. This is a mathematical truism if the long-term growth rate in net cash flows
is the same each period. In reality, the analyst must determine which net cash flow
model is most appropriate to use: either the multiyear DCF model or the single-year
capitalization of net cash flow model.
Because many businesses 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.
Many analysts apply the discounting and capitalization formulas that reflect
the implicit assumption that investors will realize their net cash flows at the end
of each year. This assumption often does match the average timing of the investors’ realization of net cash flows. 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 or, for seasonal companies, an appropriate variant of the
midyear convention.
In estimating the net cash flows to discount to present value, we deduct cash
outflows such as capital expenditures and additional net working capital needed to
realize the projected future revenues and expenses of the existing business investment (i.e., we are matching the projected capital expenditure and net working capital investments with the projected revenues and expenses). In this formulation, we
are valuing the existing business. Although we are not valuing currently unknown
investments that may be made in future years from investing these net cash flows in
new business ventures, the underlying assumption inherent in the methodology is
that any net cash flow retained is reinvested at the cost of capital. This is further
explained with an example in Chapter 34.
We also discussed valuation using residual income and concluded that, assuming we have consistent assumptions, the residual income method of valuation will
yield the same result as the net cash flow methods of valuation.
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CHAPTER
5
Relationship between Risk
and the Cost of Capital
Introduction
Defining Risk
How Risk Affects the Cost of Capital
Valuation of Risky Net Cash Flows
Risk Aversion versus Risk Neutrality
Market Returns Increase as Risk Increases by Asset Class
FASB’s Concepts Statement No. 7: Cash Flows and Present Value Discount Rates
Types of Risk
Maturity Risk
Market Risk
Unique Risk
Liquidity and Marketability Risk
Measuring Riskiness of Net Cash Flows
Summary
INTRODUCTION
The cost of capital for any given investment is a combination of two basic factors:
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. It 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 (e.g., risk of amount
and timing of net cash flows, and liquidity of the asset).
45
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COST OF CAPITAL BASICS
The generalized cost of capital relationship is:
(Formula 5.1)
EðRi Þ ¼ Rf þ RPi
where: E(Ri) ¼ Expected return of asset i
Rf ¼ Risk-free rate
RP ¼ Risk premium for asset i
Quantifying the amount by which risk affects the cost of capital for any particular business 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.1 The definition implies uncertainty as
to both the amounts and the timing of expected economic income. By expected
economic income, in a technical sense, we mean the expected value (i.e., mean or
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 most often based on the judgment of the 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 equate risk with uncertainty, consistent with most
related 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 business’s
net cash flows will still vary, but within the forecast of recession). ‘‘Uncertainty beyond risk’’ occurs when analysts have the possibility of an infinite number of subjective inputs (e.g., wide divergence of opinion among forecasters as to whether there
will be a recession next year).2
No matter how many probability distributions or Monte Carlo simulations are
used to create a financial forecast, all risk cannot be eliminated. Therefore, projected
net cash flows cannot be discounted at the risk-free rate.
1
David Laro and Shannon P. Pratt, Business Valuation and Federal Taxes: Procedure, Law,
and Perspective, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2010), Chapter 12.
2
Evan W. Anderson, Eric Ghysels, and Jennifer L. Juergens, ‘‘The Impact of Risk and Uncertainty on Expected Returns,’’ Working paper, June 22, 2009. Available at http://ssrn.com/
abstract=890621.
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Relationship between Risk and the Cost of Capital
HOW RISK AFFECTS THE COST OF CAPITAL
The cost of capital for any given investment is a combination of two basic factors: a
risk-free rate, Rf, and a premium for risk, 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 of expected cash flows. The greater the market’s
required rate of return, the lower the present value of the investment, and the lower the
market’s required rate of return, the greater the present value of the investment.
Risk is a major concern of 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 currency with less purchasing power in the future). 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 results from 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 requires a high rate of return to accept
uncertainty. Since uncertainty as to timing and amounts of future net cash flow is
greatest for equity investors, the high risk requires equity as a class of capital to
have the greatest cost of capital.
Valuation of Risky Net Cash Flows
In Chapter 3, we discussed measuring future net cash flows in terms of the mean 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 net cash flows over the life of an assumed
five-year business project.
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 Symmetrical Distributions
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COST OF CAPITAL BASICS
In each year, the net cash flows have the potential to vary. When viewed in
terms of the valuation date, these possible net cash flows (distributions of net cash
flows) 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 from market information how market participants
price risk as of the valuation date for an investment with a comparable distribution
of expected net cash flows. We need to first measure the risks 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?).
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 in
any number of years because of capacity constraints, pricing limitations due to competition, and so on, making such skewed distributions of expected future net cash flows
more representative of possible outcomes than symmetrical distributions.
For example, a business will be limited in expanding revenues because of plant
capacity constraints. Although business management can expand production by
temporarily going to multishift seven-day-per-week production schedules, eventually the business will reach a practical capacity constraint. At that point, the only
way to increase production is to replace machinery with more efficient machinery or
to add plant capacity (square feet of plant building and equipment). There is typically a delay before a business makes that commitment and the time the added plant
capacity is available to increase production.
But on the downside, if orders decrease, revenues can decrease rapidly. Expenses
are typically not completely variable. As revenues decrease, expenses decrease at a
slower rate, resulting in rapid decreases in net cash flows.
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.
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
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Relationship between Risk and the Cost of Capital
Risk Aversion versus Risk Neutrality
In Chapter 3, we discussed that one should be discounting or capitalizing the statistical expected value of net cash flows. Any one year’s distribution of possible net cash
flows can be thought of as a bundle of possible outcomes (sometimes termed contingent claims on the asset). The present value of this series of contingent claims can be
depicted in the following formula:
(Formula 5.2)
PV ¼
n
X
Eðcash f lowÞ
1
ð1 þ kÞn
n
If investors were risk neutral, the appropriate discount rate for estimating the
present value of the expected cash flows would be the risk-free rate.
What is risk neutral? Assume the investor is risk neutral and that the return on
the investment is expected to be received one year from the date of the investment.
Assume that the possible payoff in one year equals the expected value of the cash
flows. Investors would be satisfied with a payoff equal to the present value of
expected cash flows calculated at the risk-free rate because the expected cash flows
represent a fair bet. The investor pays an amount equal to the present value of the
expected net cash flows discounted at the risk-free rate (which takes into account
the time-value of money for the one-year period of the investment), and the investor
receives the opportunity to realize one of the possibilities of net cash outcomes. The
expected payoff is exactly equal to the possible net cash flow outcome multiplied by
the probability that the net cash flow outcome will occur.
But investors are not risk neutral; in the literature, investors are generally
assumed to be risk averse. Risk aversion is equivalent to paying more attention to
unpleasant outcomes, relative to their actual probability of occurrence. Generally,
investors are more concerned about losing an amount of money than about the
possibility of making the same amount of money. Exhibit 5.3 helps explain the concept of risk aversion.
Scenario A represents the expected net cash flow from a risk-free investment.
Assume that the investor could buy the investment today and be guaranteed $1,000
in one year. The expected net cash flow equals $1,000 in one year. An investor
would be willing to pay an amount for this investment opportunity and require a
return that compensates him for the time-value of money, that is, the rate of return
that compensates the investor for his preference of holding money today versus one
year hence (but with no risk of loss). Assuming a risk-free rate of, say, 5%, the present value of the expected net cash flow equals $952, and the market price could be
expected to be approximately $952.
Scenario B represents the expected cash flows from a risky investment. The
expected net cash flow in one year is again equal to $1,000, but there is a chance
that the net cash flows will be less than $1,000 or greater than $1,000. If the investor
were risk-neutral, he would be willing to pay $952 for the fair bet to earn more than
$1,000 (up to $1,500) or less than $1,000 (as little as $500). But investors are not
risk neutral. They want to be compensated for the chance that they could end up
with only $500. The investor would require a greater rate of return than in Scenario
A because investors are risk averse and want to be compensated for the risk by a
greater rate of return. Let’s assume that the market prices the investment
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COST OF CAPITAL BASICS
Probability of Occurrence
Scenario A: Certain Net Cash Flow
1.2
1
0.8
0.6
0.4
0.2
0
0
500
1,000
1,500
2,000
Expected Net Cash Flow
Probability of Occurrence
Scenario B: Risky Net Cash Flows
Probability of Occurrence
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0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
300
500
700
900
1,100 1,300
Expected Net Cash Flows
1,500
1,700
Scenario C: Riskier Net Cash Flows
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
–1,500 –1,000 –500 0
500 1,000 1,500 2,000 2,500 3,000 3,500
Expected Net Cash Flow
EXHIBIT 5.3 Valuation of Expected Net Cash Flows with Varying Distributions
opportunity in Scenario B at $870. This yields an expected rate of return equal to
approximately 15%.
Scenario C represents the expected net cash flows from an even riskier investment. The expected net cash flow in one year is again equal to $1,000, but there is
even a greater chance that the net cash flows will be less than $1,000 or greater than
$1,000. A risk-neutral investor would be willing to pay $952 for the fair bet to earn
more than $1,000 (up to $3,000) or less than $1,000 (a loss of $1,000). But as
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Relationship between Risk and the Cost of Capital
investors are not risk neutral, they want to be compensated for the chance that they
could end up losing $1,000. The investor would require a greater rate of return than
in Scenario B because investors are risk averse and want to be compensated for the
increased risk by an increased rate of return. Let’s assume that the market prices the
investment opportunity in Scenario C at $770, yielding an expected rate of return
equal to approximately 30%.
The appropriate discount rate for discounting risky net cash flows is not a riskfree rate of return. Would the market only demand the risk-free rate of return for
taking on the variability of the net cash flows? The answer is no. The market will
demand compensation (added return) for accepting the risk that the actual net cash
flows will differ from the expected net 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 net cash
flows, the greater the discount rate.3
Market Returns Increase as Risk Increases
by Asset Class
Because investors are risk averse, the market requires an increasing rate of return as
the risk of a bad outcome increases, even if the expected net cash flow is identical in
all three scenarios. How do we know the market demands and receives greater
returns for taking on greater risk? If one looks across asset classes at mean returns
and risk (as measured by the standard deviation of returns realized over time), one
observes that greater returns seem closely related to greater risk (see Exhibit 5.4).4
In fact, if one plotted the observed relationship of risk and returns over time (as
compiled in Exhibit 5.4), one observes a strong linear relationship between risk and
return, which is referred to as the capital market line. The capital market line is
EXHIBIT 5.4 Returns and Standard Deviation of Returns by Asset Class for 1926–2008
1926–2008
Large Company Stocks
Ibbotson Small Company Stocks
Mid-Cap Stocks
Low-Cap Stocks
Micro-Cap Stocks
Ibbotson Long-Term Corporate Bonds
Ibbotson Long-Term Government Bonds
Treasury Bills
Arithmetic Mean
Returns
Standard Deviation
of Returns
11.7%
16.4%
13.4%
14.9%
17.7%
6.2%
6.1%
3.8%
20.6%
33.0%
24.9%
29.4%
39.2%
8.4%
9.4%
3.1%
Source: Compiled from data in Stocks, Bonds, Bills, and Inflation 2009 Yearbook. Copyright
# 2009 Morningstar, Inc. All rights reserved. Used with permission.
3
If one converts the expected cash flows to their certainty equivalent cash flows, the risk-free
rate is the correct discount rate. See discussion on page 54.
4
William F. Sharpe, ‘‘Capital Asset Prices: A Theory of Market Equilibrium under Conditions
of Risk,’’ Journal of Finance (September 1964): 425–442. Updated by Duff & Phelps.
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COST OF CAPITAL BASICS
Std. Dev / Arithmetic Mean 1926–2008
y = 0.394x + 0.025
R 2= 0.942
20.0%
18.0%
16.0%
Arithmetic Mean
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14.0%
12.0%
10.0%
8.0%
6.0%
4.0%
2.0%
0.0%
0.0%
5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 45.0%
Standard Deviation
EXHIBIT 5.5 Capital Market Line—Empirical Estimate Based on Realized Returns by
Asset Class
defined as a line used in the capital asset pricing model that plots the rates of return
for efficient portfolios, depending on the rate of return and the level of risk (standard
deviation) for a particular portfolio.5 The empirical estimate of the capital market
line shows the market’s pricing of portfolios of assets over the period 1926 through
2008. As the risk (standard deviation of returns) increases, the realized return
increases as shown in Exhibit 5.5.
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,6 addresses issues surrounding the use of cash flow projections and present value techniques in accounting
measurement. Practitioners often read the statement especially for guidance on implementation of ASC 350, Intangibles—Goodwill and Other7. The guidance was
clarified in ASC 820, Fair Value Measurements and Disclosures8, specifically in ASC
820-10-55 (Section 55, ‘‘Implementation,’’ of Subtopic 820-10).
Two particular elements of Con 7 seem to generate 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.
5
The capital market line is different than the security market line (SML).
Con 7 now represents nonauthoritative guidance per Accounting Standards Codification
Topic 105, Subtopic 10, Section 5-3.
7
Prior to the Codification, FASB Statement No. 142, Goodwill and Other Intangible Assets.
8
Prior to the Codification, FASB Statement No. 157, Fair Value Measurements.
6
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Relationship between Risk and the Cost of Capital
Con 7 observes that when values are uncertain, accountants are trained to use
‘‘most likely’’ values or ‘‘best estimates.’’ Con 7 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.
In ASC 820, the FASB expanded the guidance on Con 7 to clarify that when
using an expected present value technique, the adjustment for risk may be reflected
in either the expected cash flows (the numerator) or in the discount rate (a riskadjusted discount rate). ‘‘Risk is an essential element in any present value technique.
Therefore, a fair value measurement, using present value, should include an adjustment for risk if market participants would include one in pricing the related asset or
liability.’’9
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 model [CAPM], betas, etc.) was developed for the so-called expected value method, not for the traditional method. 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 5.3)
PV ¼
Eðcash f lowsÞ
ð1 þ kÞ
where: k ¼ Risk-adjusted discount rate. Where k > risk-free rate of return (Rf).
E(cash flows) ¼ Expected cash flows.
In fact, 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 for equity; WACC for the business’s
overall discount rate).
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.
9
FASB Statement No. 157, Fair Value Measurements, Appendix C: Background Information
and Basis for Conclusions, par. C60. The Basis for Conclusions is, for the most part, not
included in the Codification.
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COST OF CAPITAL BASICS
(Formula 5.4)
PV ¼
½Eðcash f lowsÞ cash risk premium
1 þ Rf
The approach, though rarely used by practitioners, also is a present value
method. 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 Con 7’s Appendix A.
What Con 7 calls the traditional method versus what it calls the expected value
method is misleading from a finance perspective. The so-called traditional method
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?’’); or
It can be computed formulaically using capital market data as shown in
Formula 5.5:
(Formula 5.5)
Eðcash f lowÞ1 ðcash risk premiumÞ1
Eðcash f lowÞ1
Certainty Equivalent1
¼
¼
1 þ Rf
ð1 þ k Þ
1 þ Rf
Therefore, to get from the expected cash flow to its certainty equivalent, just
multiply the former by the ratio [(1 + Rf)/(1 + k)], where k is a risk-adjusted discount
rate that can be computed in the usual way. For example, k may be the WACC of
the particular division of the business, reflecting the risk of the net cash flows. One
can estimate the certainty equivalent as follows:
(Formula 5.6)
Eðcash f lowÞ1 1 þ Rf
¼ Certainty equivalent
ð1 þ WACCÞ
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Relationship between Risk and the Cost of Capital
55
Con 7 does not explain this, but it is part of widely available and accepted corporate finance theory.10 It is not controversial. It works for all the examples shown
here and for broad classes of distributions.
Appendix III of the Workbook and Technical Supplement expands on this brief
discussion and provides examples of certainty equivalent cash flows, risk neutral
payoffs (i.e., payoffs correlated to certainty equivalents) and risk neutral probabilities (i.e., probabilities correlated to certainty equivalents). One can only discount
risky cash flows using a risk-free rate if the cash flows are adjusted to their certainty
equivalent.
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 four components:11
1.
2.
3.
4.
Maturity risk
Market risk
Unique risk
Liquidity and marketability risk
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. 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 (i.e., an opportunity cost) to capital embedded in these
bonds.
The longer the maturity, the greater the susceptibility to changes 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.
10
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.
11
See Richard A. Brealey, Stewart C. Myers, and Franklin Allen, Principles of Corporate
Finance, 9th ed. (New York: McGraw-Hill, 2008), 188, for a discussion of maturity,
market, and unique risks.
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COST OF CAPITAL BASICS
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 many U.S. companies, the investment market as a whole is generally
limited to the U.S. equity markets, and typically returns are measured on either the
New York Stock Exchange (NYSE) Composite Index or the Standard & Poor’s
(S&P) 500 Index. For other multinational companies, the investment market may be
more appropriately considered the world equity markets with returns measured on
the S&P Global 1200 Index or one of the MSCI Global Standard indices.
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 increasingly shows that it may be difficult or nearly impossible to be fully
diversified. We discuss that research in Chapter 15.
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 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 and to general economic conditions as reflected in the broad stock market). Individual stocks have beta
risks (e.g., to general economic conditions as reflected in the broad stock market and
to the relative risks of large company stocks to small company stocks).
In the context of the CAPM, beta measures the expected sensitivity of changes
in returns of a security (issued by an individual company or a portfolio of companies
in an industry) to changes in returns of ‘‘the market.’’ The market proxy is often the
S&P 500 Index or the NYSE Composite Index. With regard to the beta risk of a
particular company’s securities, beta risk embodies both business (operating) risk
and financial risk of the company.
The size premium is a systematic risk factor and is an adjustment to pure
CAPM. Empirical evidence indicates that beta alone does not measure the risk of
smaller companies. We discuss the size premium in Chapters 13 and 14.
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/or 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.
Fully capturing unique risk in the discount rate requires analysis of the company in
comparison with other companies, which is discussed in Chapter 15. However,
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Relationship between Risk and the Cost of Capital
57
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 of 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 public 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 study supports the
rationality of stock prices where data on expected cash flows are available to
investors.12
But market inefficiency can and does occur for small public stocks, particularly for
smaller company stocks that do not have sufficient investor following such that their
prices do not react to changes in fundamentals in a timely fashion. 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 discuss problems with
textbook theories that fall short in such circumstances in Chapters 13, 14, and 15.
Liquidity and Marketability Risk
Discussions of capital market theory generally assume 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.
These risks, while listed here separately, are systematic risks in that the pricing of
these risks for a particular investment moves with the overall market pricing of
liquidity and marketability risks. The risk premiums for lack of liquidity and marketability can be embedded in the discount rate or as a separate adjustment from an
‘‘as if liquid’’ estimate of value. We specifically address issues pertaining to illiquidity risk and lack of marketability in Chapter 27 for minority interests and lack
of marketability in Chapter 28 for entire businesses.
MEASURING RISKINESS OF NET CASH FLOWS
All businesses are portfolios of operations and assets. The risk of the expected cash
flows can be thought of in terms of the risk of company operations and assets (business risk) and the risk of how it’s financed (financial risk).
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. Business risk 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. In understanding the variability of net cash flows from operations, one begins with the study of the variability of revenues. One then needs to study the degree
12
Aharon R. Ofer, Oded Sarig, and Keren Bar-Hava, ‘‘New Tests of Market Efficiency Using
Fully Identifiable Equity Cash Flows,’’ Working paper, February 2007. Available at http://
ssrn.com/abstract=965242.
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COST OF CAPITAL BASICS
to which costs vary as revenues vary. This leads one to classifying costs as either
variable costs or fixed costs. Variable costs are those that are dependent on the rate
of output or revenues of the business. Fixed costs occur regardless of the level of
output or revenue of the business. Business risk can be quantified in terms of variability of revenue in this way:13
(Formula 5.7)
sB ¼
where:
Fc
s rev
1þ
PV b
s B 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)
s rev ¼ Standard deviation of revenues derived from output
That is, as the level of fixed costs rises relative to total costs, the variability of
operating cash flows increases. All else being equal, a business 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.
Another way to look at a company’s risk is in terms of its assets. Any company
operations can be thought of as a portfolio of assets. 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).
However, we can depict the risk hierarchy of the asset mix of a business generally as
shown in Exhibit 5.6.
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 5.6 Risk of Business’s Asset Mix
13
Hazem Daouk and David Ng, ‘‘Is Unlevered Firm Volatility Asymmetric?’’ AFA 2007
Chicago Meetings, January 11, 2007.
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Relationship between Risk and the Cost of Capital
Generally, net working capital is 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 can typically be used in a variety of businesses and in producing a variety of products.
Further, if need be, the fixed assets can be sold to other businesses, but the proceeds
from any such sale are likely to vary more from their carrying values than are proceeds from, for example, net working capital. Some intangible assets may have a use
outside the subject business, but others may have little or no value outside the existing business. Their value is often dependent on the success of the specific operations
of the subject business.
The capital structure of the business adds another layer of risk, financial risk.
Financial risk is the added volatility providers of equity capital will experience because returns to debt holders and other preferred investors generally are fixed and
are senior to returns on common equity. The fixed costs from the financing increase
the volatility of returns on common equity. We can depict the risk hierarchy of the
components of the company capital structure generally as shown in Exhibit 5.7.
How can we estimate an appropriate risk premium for a business as a whole
and for its component assets? We can look at the capital structure of the business
and the business’s overall cost of capital as a mirror of the business and financial
risk. Think of the mix of business assets as the left-hand side of the balance sheet
and the overall capital structure as the right-hand side of the balance sheet. By determining the business’s overall cost of capital, we can then impute the overall return
required from business operations to provide investors (suppliers of capital to the
business) with their expected returns. In addition, by observing market returns investors have received in the past, we can impute implied returns expected by investors from investments in companies with similar business and financial risks. We are
imputing the risk of the investment (business assets) from the risks of the securities
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 5.7 Risks of the Components of the Business Capital Structure
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COST OF CAPITAL BASICS
used to supply the investment and the pricing of risk implied from the returns on
those securities.
This book is about measuring and pricing the risks of the assets and components
of the capital structure of a business. We discuss separating the measurement of
business risk and financial risk in Chapter 11.
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.
We observe a common error of discounting probability-weighted net cash flows
using the risk-free rate. 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 and/or industry as opposed to the market as a whole (i.e., the risk that remains after taking into
account the market risk). Unsystematic risk has received increased attention in recent years.
Risk affects 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.
In the upcoming chapters, we will discuss pricing risk.
When using the build-up method (Chapter 7), the CAPM (Chapter 8), or
another model such as the Fama-French three-factor model (FF) (Chapter 17), 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 public stock data to imply the cost of equity capital (e.g., the DCF
method discussed in Chapter 17), 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.
The cost of invested capital is a blending of the costs of each component, commonly referred to as the WACC. Chapter 6 discusses each component in the capital
structure, and Chapter 18 addresses the WACC.
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CHAPTER
6
Cost Components of a Business’s
Capital Structure
Introduction
Debt Capital
Estimating Current Market Yields on Debt
Duration
Tax Effect Lowers Cost of Debt
Leases Are Debt
Debt Guarantees
Risky Debt
Preferred Equity Capital
Convertible Debt and Convertible Preferred Equity Capital
Qualified Equity Issued under TARP
Employee Stock Options
Common Equity Capital
Obligations That Are Subtracted in a Valuation
Postretirement Obligations
Contingent Liabilities
Summary
INTRODUCTION
The capital structure of many businesses includes two or more components, each of
which has its own cost of capital. The capital structure often includes many components; such companies may be said to have a complex capital structure. The major
components commonly comprising a business’s capital structure are:
&
&
&
Debt capital
Preferred equity capital
Common equity capital
Similarly, a project being considered in a capital budgeting decision may be
financed by multiple components of capital.
61
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COST OF CAPITAL BASICS
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 business’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 18 shows the process of blending them into a business’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. Although there can be many controversies surrounding costs of fixedincome (debt or preferred) 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
Traditionally, only long-term liabilities are included in a capital structure. However,
many businesses, especially smaller closely held businesses, 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 include the short-term debt as
part of the debt component of the capital structure for the purpose of estimating the
business’s WACC. The debt component of the capital structure should include (1)
the current portion of long-term debt classified on the balance sheet as a short-term
liability and (2) short-term debt used as if it were long-term debt.
Estimating Current Market Yields on Debt
The business’s current interest expense is readily ascertainable from the footnotes to
the business’s financial statements (if the business has either audited or reviewed
statements or compiled statements with footnote information).
But if the interest rate the business is paying is not representative of a longterm, current market rate, then the analyst should estimate a current market rate
for that component of the business’s capital structure. The interest rate should be
consistent with the financial condition of the subject business, based on a comparative analysis of the subject business’s average ratios. If the business’s debt
has a debt rating, one can estimate the cost of debt using a yield curve analysis.
If the business’s debt is not formally rated, you must estimate a credit rating
(often termed a synthetic rating).
Standard & Poor’s publishes debt rating criteria along with the Standard &
Poor’s Bond Guide. Standard & Poor’s Global Credit Portal indicates median ratios
by rating. RatingsDirect 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
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Cost Components of a Business’s Capital Structure
where the investment would fit within the bond rating system and 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 Global Credit Portal.
EXHIBIT 6.1
Table 1.
Key Financial Ratios, Long-Term Debt
Key Financial Ratios, Long-Term Debt, Three-Year (2006–2008) Medians
Oper. income (bef. D&A)/revenues (%)
Return on capital (%)
EBIT interest coverage (x)
EBITDA interest coverage (x)
FFO/debt (%)
Free oper. cash flow/debt (%)
Disc. cash flow/debt (%)
Debt/EBITDA (x)
Debt/debt plus equity (%)
No. of companies
Table 2.
AA
A
BBB
BB
B
27.8
30.5
34.9
38.8
190.2
154.6
93.9
0.4
13.3
6
25.2
29.9
16.6
20.8
76.9
42.5
26.5
1
27.6
15
18.8
21.7
10.8
13.3
54
30.9
20.2
1.5
36.1
100
17.7
15.1
5.9
7.8
34.8
14
8.4
2.3
45.3
202
17.2
12.6
3.6
5.1
26.9
7.8
5.8
3
52.9
271
15.7
8.6
1.4
2.2
11.6
2.1
1
5.4
75.6
321
Key Utility Financial Ratios, Long-Term Debt, Three-Year (2006–2008) Medians
Oper. income (bef. D&A)/revenues (%)
Return on capital (%)
EBIT interest coverage (x)
EBITDA interest coverage (x)
FFO/debt (%)
Free oper. cash flow/debt (%)
Disc. cash flow/debt (%)
Debt/EBITDA (x)
Debt/debt plus equity (%)
No. of companies
Table 3.
AAA
AA
A
BBB
BB
B
15.9
10.1
4.3
6.4
23.9
1.9
9.2
3
47.5
6
22.1
9.1
3.4
4.8
19.8
3
9
3.6
53.2
49
23.6
8.3
2.9
4.3
17.9
2.9
7.7
4
57.1
116
25.1
7.9
2.1
2.9
12.7
5.7
8.7
5.3
61.7
11
37.2
9.8
1.4
2.4
14.4
3.6
0.1
4.8
59.2
8
Key Ratios
Formulas
EBIT interest coverage
EBITDA interest coverage
Funds from operations
(FFO)/total debt
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 (D&A)/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
(continued )
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COST OF CAPITAL BASICS
EXHIBIT 6.1
Table 3.
(Continued)
Key Ratios
Formulas
Free operating cash flow/
total debt
Total debt/total debt þ
equity
Return on capital
Total debt/EBITDA
FFO capital expenditures increase (or þ 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 þ current
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 noncurrent 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.
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.
y
Source: Standard & Poor’s RatingsDirect on the Global Credit Portal: 2008 Adjusted Key
U.S. and European Industrial and Utility Financial Ratios (New York: Standard & Poor’s, a
division of McGraw-Hill Companies, Inc.) copyright 2009: 2, 3. Used with permission. All
rights reserved.
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 yieldto-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.
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Cost Components of a Business’s Capital Structure
EXHIBIT 6.2
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
Sixþ
2.61%
4.77%
12.01%
2.90%
5.01%
12.58%
3.14%
5.47%
13.03%
3.42%
5.69%
13.05%
3.65%
6.04%
13.54%
4.82%
6.86%
14.40%
Sample Yield Curve Approach Based on Weighted Average Yields
1 Year
2 Year
3 Year
4 Year
5 Year
Over 5 Years
Total
(1)
Face Value
Yield
Weighted Average(1)
$180
166
45,978
108
48
8,400
$54,880
12.01%
12.58%
13.03%
13.05%
13.54%
14.40%
0.04%
0.04%
10.92%
0.03%
0.01%
2.20%
13.24%
= (Face Value Maturing in Year/Total Face Value) Yield
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. Typically, the
operating profits of small companies are more variable than those of large companies.1 It is generally recognized that volatility of past earnings reduces predictability
of future earnings.2 Research has found that the difference in yields among bonds is
a function of difference in cash flow volatility among the firms.3
Some companies have more than one class of debt, each with its own cost of
debt capital (e.g., senior, subordinate).
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 its 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 would probably be the appropriate yield, because it is likely to 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
1
See data on average coefficient of variation of operating margin in Exhibits 13.11 and 13.12.
John Graham, Campbell Harvey, and Shivaram Rajgopal, ‘‘The Economic Implications of
Corporate Financial Reporting,’’ Journal of Accounting and Economics 40 (2005): 3–73.
3
Kenneth R. Vetzal, Alan V. S. Douglas, and Alan Guoming Huang, ‘‘Cash Flow Volatility
and Corporate Bond Yield Spreads,’’ Working paper, February 2009. Available at http://
ssrn.com/abstract=1362167.
2
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COST OF CAPITAL BASICS
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 the only criterion to use in 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 one is matching nontraded debt instruments to traded debt instruments to obtain market observations of yields, one
needs to estimate the credit quality and the length of time over which cash flows are
expected to be received.
Duration
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:4
(Formula 6.1)
where:
n n Eðcash f lowÞ
P
n
ð 1 þ kÞ n
1
Duration ¼ n
P Eðcash f lowÞn
ð 1 þ kÞ n
1
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 business
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 more frequently.) The expected cash
flows are $100 per year for 9 years and $1,100 in year 10. Assume that the current
market rate of interest, given current interest rates and the risk of the issuing company, is now 15%.
&
4
The duration is the weighted present value of the cash flows, with the weights
being the number of years from the valuation date (i.e., 1, 2, 3, etc.). The
For an explanation of duration in the context of bond valuation, see, e.g., Richard A. Brealey,
Stewart C. Myers, and Franklin Allen, Principles of Corporate Finance, 9th ed. (Boston:
Irwin McGraw-Hill, 2008), 63–65; Aswath Damodaran, Investment Valuation: Tools and
Techniques for Determining the Value of Any Asset, 2nd ed. (Hoboken, NJ: John Wiley &
Sons, 2002), 891–892.
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Cost Components of a Business’s Capital Structure
EXHIBIT 6.3
(1)
Year
Example of Calculating Duration of a Bond
(2)
Expected
Cash Flow
(3)
Present
Value Factor
(4)
Present Value of
Expected Cash Flow
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
&
&
$86.96
75.61
65.75
57.18
49.72
43.23
37.59
32.69
28.43
271.90
$749.06
(5)
(5) ¼ (1) (4)
$
86.96
151.23
197.25
228.70
248.59
259.40
263.16
261.52
255.84
2,719.03
$4,671.68
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
from the debt instrument.
One can estimate the cost of debt for the subject business by matching market
yield for bonds with a comparable credit rating and a comparable duration to
the subject bond. A bond with a shorter duration is less risky than a bond with
the same credit rating and maturity but a longer duration.
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 business, the net cost of
debt to the business is the interest paid minus the tax savings resulting from the deductible interest payment. The value of the tax shield equals the present value of the
expected tax deductions on interest payments for the debt capital financing.
The cost of debt capital is measured prior to the tax affect (denoted as kd(pt) in
our notation system), as the value of the tax deduction on the interest payments
equals the value of the tax shield. Assuming that the tax deductions on interest can
be fully realized (i.e., save cash taxes) in the period in which they are paid and deducted, this pretax cost of debt and the value of the tax shield can be combined into
the after-tax cost of debt, which can be expressed (as one typically sees in textbooks)
by Formula 6.2:
(Formula 6.2)
kd ¼ kdðptÞ ð1 tÞ
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where:
COST OF CAPITAL BASICS
kd ¼ Discount rate for debt (the business’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)
We will discuss alternative formulations of the value of the tax shield based on
when the tax deductions on interest are likely to be realized in Chapter 18 and the
impact of debt financing on the risk of equity and debt in Chapter 11.
For decision-making purposes, corporate finance theoreticians generally recommend using the marginal tax rate (the rate of tax paid on the last incremental dollar
of taxable income) if that differs from the business’s effective tax rate.5 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.6
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.7
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 business. Operating lease payments are treated as
part of operating expenses but are really financing expenses. The stated operating
income, capital, profitability, and cash flow measures for businesses with operating
leases have to be adjusted when operating leases get categorized as financing
expenses.8
Financial Accounting Standards Board’s (FASB) Accounting Standards Codification (ASC) 840, Leases (formerly Statement of Financial Accounting Standards
No. 13—Accounting for Leases (October 1975)), requires footnote disclosure for
noncancelable long-term operating leases.9 The disclosure includes:
5
See, e.g., Richard A. Brealey, Stewart C. Myers, and Franklin Allen, Principles of Corporate
Finance, 9th ed. (Boston: Irwin McGraw-Hill, 2008), 488.
6
John.Graham@Duke.edu.
7
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.
8
Aswath Damodaran, ‘‘Leases, Debt and Value,’’ Journal of Applied Research in Accounting
and Finance (July 16, 2009): 3–29.
9
FAS No. 13, Accounting for Leases (November 1976): paragraphs 16, 122.
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Cost Components of a Business’s Capital Structure
&
&
&
&
&
&
&
69
Aggregate future minimum payments
Minimum payments for first five fiscal years
Aggregate future minimum sublease rentals
Historical rental expense
The use of operating leases has grown significantly. A recent study by the Security and Exchange Commission found that 77% of the sampled firms had operating leases and that public companies have $1.25 trillion in undiscounted
future cash obligations related to operating leases.10 In a recent study of the
credit default swap (CDS) market, the authors found that the price impact of
operating leases on debt spreads is larger than the price impact of on-balancesheet debt.11
The International Accounting Standards Board/Financial Accounting Standards
Board Lease Working Group proposes recording the fair value of the rights and
obligations of a lease at inception under the concept of a ‘‘right to use’’ model.
That is, the lessee has, upon contract signing, an unconditional right to use the
leased asset and should record the fair value of that right on the balance sheet.
This model assumes that all lease transactions are economically similar. This
new rule will not be ready until at least 2011.
As we are preparing this book, we learned that in their most recent deliberations, the Lease Working Group has chosen to exclude certain leases. The new
lease accounting rules will exclude lease contracts that are effectively purchases
(as such leases are already recorded as capital leases). And they have yet to decide on issues of materiality, as the Equipment Lease and Financing Association
estimates that 90% of leases (by count of leases) involve assets worth less than
$5 million and have lease terms of two to five years.
The Standard & Poor’s Ratings Services group routinely capitalizes operating
leases for purposes of calculating comparative ratios.12 An excerpt from their web
site describing their methodology follows.
Corporate Ratings Criteria 2006:
To improve financial ratio analysis, Standard & Poor’s uses a financial
model that capitalizes off–balance sheet 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. This technique is, on balance, superior to the
10
United States Securities and Exchange Commission, ‘‘Report and Recommendations Pursuant to Section 401c of the Sarbanes-Oxley Act of 2002 on Arrangements with Off-Balance
Sheet Implications, Special Purpose Entities, and Transparency of Filings of Issuers,’’ June
15, 2005. Available at http://www.sec.gov/news/studies/soxoffbalancerpt.pdf.
11
Sandro Andrade, Elaine Henry, and Dhananjay Nanda, ‘‘Leases, Off-Balance Sheet Leverage and the Pricing of Credit Risk,’’ Working paper, October 9, 2009. Available at http://
moya.bus.miami.edu/sandrade/Andrade_Henry_Nanda_03052010.pdf.
12
See also Brian Oak, ‘‘Off–Balance Sheet Leases: Capitalization and Ratings Implications:
Out of Sight but Not Out of Mind,’’ Moody’s Investors Service Global Credit Research
(October 1999).
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COST OF CAPITAL BASICS
alternative ‘‘factor method,’’ which multiplies annual lease expense by a
factor reflecting the average life of leased assets.
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 and the analysis resulting from capitalizing operating leases.13
One recent study finds that banks consider off–balance sheet obligations when
setting loan spreads, though adjusted financial ratios are relevant for banks’ assessments only in the absence of a credit rating.14
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.15
Debt Guarantees
The debt of closely held companies is often secured by personal guarantees of
one or more of the entrepreneurs with major ownership in the subject business.
When estimating the cost of debt for a closely held business, the analyst should
ascertain whether the debt is secured by personal guarantees. If so, this is an
13
This example is drawn from a report submitted by Roger Grabowski in a disputed matter.
Jennifer Lynne M. Altamuro, Rick Johnston, Shail Pandit, and Haiwen (Helen) Zhang,
‘‘Operating Leases and Credit Assessments,’’ Working paper, January 2009. Available at
http://ssrn.com/abstract=1115924.
15
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; Eugene A. Imhoff
Jr., Robert C. Lipe, and David W. Wright, ‘‘Operating Leases: Income Effects of Constructive Capitalization,’’ Accounting Horizons (June 1997): 12–32; Charles Mulford and Mark
Gram, ‘‘The Effects of Lease Capitalization on Various Financial Measures: An Analysis of
the Retail Industry,’’ Journal of Applied Research in Accounting and Finance 2(2) (2007):
3–13; Aswath Damodaran, Damodarn on Valuation: Security Analysis for Investment and
Corporate Finance, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2006), Appendix 1 and
Appendix 3.
14
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EXHIBIT 6.4
Example of Operating Lease Capitalization (2004)
Table 1 provides data that would typically appear in the financial statement disclosure.
Table 1 Lease Model Calculation
Reporting Year
Payment Period
2004
2003
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
Reported figures: Future minimum lease commitments (mil. $).
Source: Standard & Poor’s RatingsDirect on the Global Credit Portal: Operating Lease Analytical Model (New York: Standard & Poor’s, a division of McGraw-Hill Companies, Inc.)
copyright # 2009: 2. Used with permission. All rights reserved.
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
& Multiply the average (current and previous years) PV of operating leases by the interest
rate. In Table 2 we have ($336.5 þ $318.7)/2 ¼ $327.6.
& This figure is then added to the firm’s total interest expense.
Depreciation 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
2004
Total debt (reported)
Total interest (incl. capitalized
interest)
Implied interest rate
2003
659.4
36.2
664.9
40.2
5.5
5.6
2002
766.8
(continued )
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EXHIBIT 6.4
COST OF CAPITAL BASICS
(Continued )
Table 2 (Continued )
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
SG&A—Selling, general, and administrative expenses.
Source: Standard & Poor’s RatingsDirect on the Global Credit Portal: Operating Lease Analytical Model (New York: Standard & Poor’s, a division of McGraw-Hill Companies, Inc.)
copyright # 2009: 3. Used with permission. All rights reserved.
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 ()
EBITDA interest coverage ()
Return on capital (%)
Funds from oper./total debt (%)
Total debt/EBITDA ()
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
41.3
Source: Standard & Poor’s Global Credit Portal: Operating Lease Analytical Model (New
York: Standard & Poor’s, a division of McGraw-Hill Companies, Inc.) copyright # 2009: 3.
Used with permission. All rights reserved.
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 business’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 business debt without the
influence of a guarantor’s pledge of personal assets.
Similarly, lenders to joint ventures or foreign subsidiaries of larger companies
often require debt guarantees.
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EXHIBIT 6.5
Sample Capitalization of Operating 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 20xx, the Company purchased a
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, Exxon, 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, 20xx, future minimum obligations under non-cancelable operating
leases and warehousing agreements are as follows:
(Thousands of Dollars)
20xxþ1
20xxþ2
20xxþ3
20xxþ4a
20xxþ5a
Thereafter
Total Payments
Less: future minimum sublease income
$159,441
$147,674
$134,432
$107,610
$ 44,023
$390,376
$983,556
$110,855
Net Total Payments
$872,701
a
Excludes guaranteed residual payments, totaling $123,221,000
(20xxþ4) and $191,522,000 (20xxþ5) due at the end of the lease
term, which will be reduced by the fair market value of the leased
assets.
Source: Company 10k, December 20xx.
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:
(continued )
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COST OF CAPITAL BASICS
EXHIBIT 6.5
(Continued)
Present Value of Lease Payments @ 7%
$702,703
Future Sublease Income as % of Total Lease Payments
Estimated Value of Future Sublease Income
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 (20xx)
Market Value of Balance Sheet Debt1
Value of Operating Leases
Market Value of Debt Plus Operating Leases
11.3%
$79,405
$702,703
(79,405)
230,557
$853,855
1
$1,893,165
$2,075,200
853,855
$2,929,055
See Exhibit 18.15 for sample 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
Book1
9.4
24%
Adjusted2
9.0
33%
1
Using book value of debt and unadjusted EBITDA.
Using market value of debt plus operating leases and EBITDA adjusted for lease rent expense.
2
Debt guarantees can be analyzed as put options.16 The outstanding debt of the
business satisfies the following equation:
Value of Risky Debt ¼ Value of Risk-free Debt Value of Guarantee against Default
One can use the synthetic ratings methodology discussed in the earlier section on
debt capital to estimate the current implicit credit rating on business debt without the
guarantee. From the implicit credit rating, you can estimate the appropriate current
interest rate and the implicit market value of outstanding debt.17 One can estimate
the value of the guarantee as the difference between the value of the risky debt (interest rate priced at market) minus the value of an equivalent amount of risk-free debt.
One practitioner reports on surveying banks as to the impact of personal guarantees on interest rates charged to business customers with and without a personal
guarantee. He reports that the banks said that the interest rate with a personal guarantee would be 200 basis points less than the rate to the same business without personal guarantee.18
16
See, e.g., Zvi Bodie and Robert C. Merton, Finance (Upper Saddle River, NJ: Prentice Hall,
2000).
17
See, e.g., Richard A. Brealey, Stewart C. Myers, and Franklin Allen, Principles of Corporate
Finance, 9th ed. (Boston: Irwin McGraw–Hill, 2008), 655–656.
18
Surveys conducted by Trugman Valuation Associates, Inc.
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75
In the late 1990s, insurance companies offered guarantees on seller financing.
That is, when a business 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 business 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 to the
smaller closely held business, and all the business’s capital structure should be
discounted at the cost of equity.
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 business (or comparably rated debt
of other companies) in theory, the expected return on debt should reflect the
promised yield net of expected default loss. But the newly issued debt includes
both default loss and a systematic risk premium (debt beta, as discussed in
Chapter 10) that increases with the duration of the bonds. The expected default
loss (net of expected recovery) should not be included in the cost of debt when
estimating the market value of debt because the expected default loss of newly
issued debt is not part of the expected return of the subject business’s bonds.
Further, the expected default loss (net of expected recovery) occurs only when
the bonds actually default, and these costs are not known with certainty at the
time the newly issued bonds are priced. As a result, it is difficult to match the
correct discount rate for debt that has been outstanding for some time, which
has different default risk likelihoods than a newly issued bond.
Therefore, it is typically a better practice to price risky debt without regard to
the expected default loss and then subtract the expected default loss (net of expected
recovery) that is appropriate for the subject debt, given the credit rating of the subject business and the remaining duration of the debt.
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.19 One can look upon the value of equity as a call option on the business’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.20
Alternatively, if one considers risky debt as a combination of a safe bond
and a short position in a put option (i.e., the business has the option of defaulting when the value of the operations and assets declines to amounts below the
19
Ian A. Cooper and Sergei A. Davydenko, ‘‘Estimating the Cost of Risky Debt,’’ Journal of
Applied Corporate Finance (Summer 2007): 90–94.
20
Jens Hilscher, ‘‘Is the Corporate Bond Market Forward Looking?’’ European Central Bank
Working Paper Series No. 800, August 2007. Available at http://ssrn.com/abstract=1005120.
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COST OF CAPITAL BASICS
face value of the debts), then one can use volatility of publicly traded 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).21
PREFERRED EQUITY CAPITAL
If the capital structure includes preferred equity capital and it is publicly traded, 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 business’s capital structure. The yield on preferred stock is often less than the yield on comparably rated corporate bonds because dividends paid on preferred stock to corporate investors in the stock are not
taxed at the full corporate income tax rate.
Standard & Poor’s publishes preferred stock rating criteria along with the Standard & Poor’s Stock Guide. Using this publication, analysts can see where the business’s preferred stock would fit within the preferred stock rating system given the
financial metrics of the subject company, 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.22
CONVERTIBLE DEBT AND CONVERTIBLE PREFERRED
EQUITY CAPITAL
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 choose. 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
21
See Zvi Bodie and Robert C. Merton, Finance (Upper Saddle River, NJ: Prentice Hall,
2000), 92–94.
22
See, e.g., Aswath Damodaran, Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2002), 212–213;
and Marcelle Arak and L. Ann Martin, ‘‘Convertible Bonds: How Much Equity, How
Much Debt?’’ Financial Analysts Journal (March–April 2005): 44–49.
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Cost Components of a Business’s Capital Structure
77
analyzed first as debt or nonconvertible preferred equity and then the value is adjusted for the value of the warrants (long-term call options).23
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 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.24
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.25 Another study proposes use of an advanced binomial warrant (option) pricing model.26 Another study compares various models to
observed market prices.27
QUALIFIED EQUITY ISSUED UNDER TARP
On October 3, 2008, Congress passed and President Bush signed the Emergency Economic Stabilization Act of 2008, which established the Office of Financial Stability
within the U.S. Treasury and authorized the Troubled Asset Relief Program (TARP).
On October 14, 2008, the U.S. Treasury announced a voluntary TARP capital
purchase program (CPP) to encourage U.S. financial institutions to build capital to
increase the flow of financing to U.S. businesses and consumers and to support the
U.S. economy. This facility was intended to allow banking and life insurance organizations to apply for a preferred stock investment by the U.S. Treasury. The U.S.
Treasury also received warrants.
The key terms of the preferred stock and warrants are summarized as follows:28
23
Aswath Damodaran, Investment Valuation: Tools and Techniques for Determining the
Value of Any Asset, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2002), 806–914.
24
Igor Loncarski, Jenke ter Horst, and Chris Veld, ‘‘Why Do Companies Issue Convertible
Bonds? A Review of Theory and Empirical Evidence,’’ Working paper, October 9, 2005,
Available at http://ssrn.com/abstract=837184.
25
Ali Bora Yigitbasioglu and Naoufel El-Bachir, ‘‘Pricing Convertible Bonds by Simulation,’’
Working paper, May 2004. Available at http://ssrn.com/abstract=950213.
26
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. Available at http://ssrn.com/abstract=977819.
27
Yuriy Zabolotnyuk, Robert Jones, and Chris Veld, ‘‘An Empirical Comparison of Convertible Bond Valuation Models,’’ Working paper, October 15, 2009. Available at http://ssrn.
com/abstract=994805.
28
U.S. Department of the Treasury, ‘‘TARP Capital Purchase Program: Senior Preferred Stock
and Warrants,’’ Available at http://www.treas.gov/press/releases/reports/termsheet.pdf.
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COST OF CAPITAL BASICS
Senior Preferred (the ‘‘CPP Preferred Shares’’):
&
&
&
&
&
&
&
&
&
&
Term: Perpetual
Ranking: Senior to common stock and pari passu with existing preferred shares
other than those that rank junior to any existing preferred shares
Dividend: 5% per annum prior to fifth anniversary of the issue date
9% per annum subsequent to the fifth anniversary of the issue date
Cumulative
Liquidation preference: $1,000 per share
Redemption: Prior to the third anniversary of the issue date, not redeemable,
except with proceeds from an equity offering that results in aggregate gross proceeds of not less than 25% of the issue price of the CPP Preferred Shares.
Subsequent to the third anniversary of the issue date, redeemable at the
option of the institution. Redeemed at 100% of the issue price plus all accrued
and unpaid dividends in the case of cumulative CPP Preferred Shares.
Dividend Restrictions: If accrued and unpaid dividends are not fully paid on the
CPP Preferred Shares, no dividends may be declared or paid on junior preferred
shares, preferred shares ranking pari passu with the CPP Preferred Shares, or
common shares.
Common Dividend Limits: Prior to the third anniversary of the issue date (unless the CPP Preferred Shares have been redeemed in whole or the U.S. Treasury
has transferred all of the CPP Preferred Shares); U.S. Treasury consent required
for any increase in common dividends per share.
Repurchase Restrictions: Prior to the third anniversary of the issue date (unless
the CPP Preferred Shares have been redeemed in whole or the U.S. Treasury has
transferred all of the CPP Preferred Shares), the U.S. Treasury’s consent is
required for any share repurchase (other than (a) repurchases of the CPP Preferred Shares and (b) repurchase of junior preferred shares or common shares in
connection with any benefit plan consistent with past practice).
No repurchases of junior preferred shares, preferred shares ranking pari
passu with the CPP Preferred Shares, or common shares if prohibited under
‘‘Dividend Restrictions.’’
Voting: Nonvoting other than class voting rights on (a) authorization or issuance of shares ranking senior to the CPP Preferred Shares, (b) amendments to the rights of the CPP Preferred Shares, or (c) any transactions (e.g.
mergers, exchanges) that would adversely affect the rights of the CPP Preferred Shares.
Transferability: No restrictions on transfer. The institution is required to file
shelf registration statement covering the CPP Preferred Shares and grant the
U.S. Treasury piggyback registration rights for the CPP Preferred Shares.
Warrants (the ‘‘CPP Warrants’’):
&
&
Term: 10 years
Exercise Price: 20-day trailing average price of the institution’s common stock
prior to announcement of participation in the CPP.
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79
&
Number: Number of CPP Warrants issued such that: Exercise Price multiplied
by the number of CPP Warrants is equal to 15% of the liquidation preference of
the CPP Preferred Shares.
& Voting: The U.S. Treasury agrees not to exercise voting power of any shares
issued through exercise.
& Antidilution: Exercise price and number of shares issuable shall be subject to
adjustment for the following:
(i) Stock splits, subdivisions, reclassifications, or combinations
(ii) Certain issuances of common shares or convertible securities (prior to the
earlier of (a) the holder of the CPP Warrants is no longer the U.S. Treasury,
and (b) the third anniversary of the issue date)
(iii) Certain repurchases of common stock
(iv) Business combinations
(v) Other distributions
& Reduction: Number of shares of common stock underlying the CPP Warrants
shall be reduced by 50% in the event that the institution receives aggregate gross
proceeds of not less than 100% of the issue price of the CPP Preferred Shares
from one or more equity offerings on or prior to December 31, 2009.
& Substitution: In the event that the institution is no longer listed or traded on a
national securities exchange, the warrants are exchangeable, at the option of
the U.S. Treasury, for senior term debt or another security of the institution that
appropriately compensates the U.S. Treasury.
& Transferability: The U.S. Treasury may transfer one-half of the CPP Warrants
prior to the earlier of (a) December 31, 2009, and (b) the date on which the
institution has received aggregate gross proceeds of not less than 100% of the
issue price of the CPP Preferred Shares from one or more equity offerings. The
institution is required to file shelf registration statement covering the CPP Warrants and the common stock underlying the CPP Warrants and grant the U.S.
Treasury piggyback registration rights for the CPP Warrants and the common
stock underlying the CPP Warrants.
One should value the TARP investment as two securities.
For the CPP Preferred Shares, one needs to determine the market yield on the
preferred stock to determine the current cost of capital, not on the embedded cost
of the preferred, as the preferred shares are fully transferable (and their market
value will reflect the market yield), and if the institution decides to redeem the CPP
Preferred Shares, it will be required to pay market yields.
For the CPP Warrants, one needs to value the warrants using one or more
option pricing models.
The dividend restrictions while the CPP Preferred Shares are outstanding
will affect the market value of the other outstanding preferred stock and the
common stock.
Among the large bank holding companies that received investments under the
CPP, eight large banks repurchased their warrants in August 2009. Warrants
issued in conjunction with the CPP Preferred Shares for Capital One Financial
and JPMorgan Chase common shares were auctioned off in December 2009. The
JPMorgan Chase auction was the largest warrant auction in U.S. history. The
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COST OF CAPITAL BASICS
expiration dates are longer than other listed warrants. Based on the prices received
and the resulting implied volatility, it appears that investors either expected lower
volatility than the implied volatility embedded in other warrants or were willing to
purchase the warrants only at a significant discount to the expected volatility.29
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. If a business has and expects to continue to issue employee stock options,
they are part of the expected ongoing compensation expense.30
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.31 Valuation models have been proposed—binomial lattice models, modified Black-Scholes models, and so on—to incorporate the nuances of valuing employee stock options compared with traded
options.32
Finally, integral to pricing options is forecasting volatilities of the common
stock on which the option pricing models depend.33
COMMON EQUITY CAPITAL
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:
1. Dividends or distributions
2. Changes in market value (capital gains or losses)
29
Linus Wilson, ‘‘The Biggest Warrant Auction in U.S. History,’’ Working paper, December
20, 2009. Available at http://ssrn.com/abstract=1521335.
30
See, e.g., Aswath Damodaran, Damodaran on Valuation, 2nd ed. (Hoboken, NJ: John
Wiley & Sons, 2006), 72; and Investment Valuation, 2nd ed., 440–450.
31
See, e.g., Mark H. Lang, ‘‘Employee Stock Options and Equity Valuation,’’ Research Foundation of CFA Monograph, July 2, 2004.
32
Jak9sa Cvitanic, Zvi Wiener, and Fernando Zapatero, ‘‘Analytic Pricing of Employee Stock
Options,’’ Working paper, July 19, 2006. Available at http://ssrn.com/abstract¼612881.
33
George J. Jiang and Yisong S. Tian, ‘‘Volatility Forecasting and the Expensing of Stock
Options,’’ Working paper, March 14, 2006.
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Cost Components of a Business’s Capital Structure
81
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 theoretically 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)
Fama-French three-factor model (Chapter 17)
Arbitrage pricing theory (Chapter 17)
Market-derived capital pricing model (Chapter 17)
Yield-spread model (Chapter 17)
Or they derive an implied cost of equity capital from the current market price of
the common stock (for public companies) (Chapter 17).
OBLIGATIONS THAT ARE SUBTRACTED IN A VALUATION
In performing a valuation as of a specific date, one typically subtracts certain obligations that appear on (or off) the subject company balance sheet that are not considered part of the ongoing capital structure. Examples include postretirement
obligations and contingent liabilities.
These types of obligations are not typically considered providing an ongoing
source of capital to the company. But these sorts of liabilities need to be considered
in the debt rating of the outstanding debt of the subject company and the equity risk
as of the valuation date of the subject company.
Postretirement Obligations
Unfunded liabilities relating to defined benefit pension plans and retiree medical
plans are debtlike in nature.34 Employees become the equivalent of creditors of the
business 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, one must be cognizant of the relatively uncertain nature of accounting for postretirement obligations.
When assessing assumptions, one can focus on differences among companies’
disclosures.
The analysis requires that you compare the current value of the business’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 does not take into account
future benefit improvements, even if probable, unless provided for in the current
34
This section is drawn from Standard & Poor’s Corporate Credit Criteria 2006 (New York:
McGraw-Hill, 2006): 96–111.
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COST OF CAPITAL BASICS
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 business 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 business continues (a ‘‘going concern’’ scenario). Under ASC 715,35 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.36
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 yearend 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
PBO, end of year
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
2,000.0
60.0
140.0
100.0
300.0
2,000.0
1,300.0
100.0
300.0
900.0
1,100.0
Source: Standard & Poor’s RatingsDirect on the Global Credit Portal: Postretirement Obligations (New York: Standard & Poor’s, a division of McGraw-Hill Companies, Inc.) copyright
# 2009: 13–14. Used with permission. All rights reserved.
35
36
Prior to the Codification, FAS Statement No. 87, Employers’ Accounting for Pensions.
This section is drawn from Standard & Poor’s Corporate Credit Criteria 2006 (New York:
McGraw-Hill, 2006): 98.
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Cost Components of a Business’s Capital Structure
EXHIBIT 6.6
(Continued)
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 debtlike liabilities ¼
Adjusted equity ¼
Adjusted debt and debtlike 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)]
$1.0bil. þ (66 2/3% $1.1bil.) ¼ $1.733bil.
$600 mil. [66 2/3%
($1.1bil. $800
mil.)] ¼ $400 mil.
$1.733bil./($1.733bil. þ
$400mil.) ¼ 81.2%
$1.0bil./($1.0bil. þ
$600mil.) ¼ 62.5%
Source: Standard & Poor’s Global Credit Portal: Postretirement Obligations (New York:
Standard & Poor’s, a division of McGraw-Hill Companies, Inc.) copyright # 2009: 14. 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.
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.
Contingent Liabilities
Liabilities for either current or prior period issues, such as potential judgments or
settlements for ongoing litigation and proposed or potential adjustments to prior
period income taxes, are real liabilities that should be subtracted from the overall
business valuation as of the valuation date but are not considered part of the ongoing capital structure of the subject entity. These potential liabilities must be considered, regardless of whether they are recorded on the balance sheet.
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EXHIBIT 6.7
COST OF CAPITAL BASICS
Capital Structure Components
Short-term notes
Long-term debt
Capital leases
Operating leases
Off–balance sheet
financing
Preferred equity
Common equity
Additional paid-in
capital
Retained earnings
Employee stock
options
Warrants
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
Normally YES
YES
YES—all part of common equity
YES—all part of common equity
YES—all part of common equity
YES—all part of common equity
YES—all part of common equity
SUMMARY
The typical components of a business’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.
The cost of debt and preferred capital should reflect the expected costs of raising
future debt capital and preferred capital from external capital sources. These costs,
commonly termed flotation or transaction costs, reduce the actual proceeds received
by the firm. Some of these are direct out-of-pocket outlays, such as fees paid to
underwriters, legal expenses, and prospectus preparation costs. Because of this reduction in proceeds, the business’s required returns must be greater to compensate
for the additional costs. Flotation costs can be accounted for either by amortizing
the cost, thus reducing the net cash flow to discount, or by incorporating the cost
into the cost of capital. Since flotation costs typically are not applied to operating
cash flow, they must be incorporated into the cost of debt and preferred capital. The
greater the size of the expected debt and preferred stock offerings, the lower the flotation cost relative to the size of the offering.
Chapter 18 explains how to combine the costs of each of these components to
derive a business’s overall cost of capital, the weighted average cost of capital.
Whereas this chapter has addressed briefly the cost of each component, Part II
focuses primarily on the many ways to estimate the cost of equity capital.
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PART
Two
Estimating the Cost of
Equity Capital and the
Overall Cost of Capital
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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
Size 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 has
two primary components, with three subcomponents:
1. A risk-free rate
2. A premium for risk, including any or all of these subcomponents:
The authors want to thank David Turney of Duff & Phelps LLC for preparing materials for
this chapter.
87
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
&
&
&
A general equity risk premium
A small-company risk 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 such instability is greater than in the United States. We
discuss the cost of capital in developing economies in Chapter 19.
FORMULA FOR ESTIMATING THE COST OF EQUITY
CAPITAL BY THE BUILD-UP METHOD
Stating the preceding concept in a formula, the cost of equity capital can be estimated by the build-up method as:
(Formula 7.1)
EðRi Þ ¼ Rf þ RPm þ RPs 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 expected 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 risk premium rates 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
to match the expected timing of cash flows:
&
&
&
30 days
5 years
20 years
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Build-up Method
89
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
on U.S. government obligations that closely match the expected timing of investment cash flows, choose the U.S. government obligation that most closely matches
the expected timing of investment cash flows.)
To obtain a yield on long-term U.S. government bonds—for example, a 20-year
yield, which is commonly used as the default long-term U.S. 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/
releases/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 zero-coupon government STRIPS.1 Please
keep in mind that because long-term U.S. government bonds make interim interest
payments, their duration is less than their stated maturity. See Chapter 6 for a discussion of duration.
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
investment’s principal 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 these factors
(with the exception of expected inflation, which can be estimated based on Treasury
inflation-protected securities [TIPS]).
It is important to 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 net
cash flows, those future net 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.
One can estimate the long-term overall economic inflation forecast embedded in
the risk-free rate by taking the difference in yield between the risk-free security and
the yield on TIPS. While this long-term estimated inflation rate provides an overall
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. Available at http://ssrn.com/
abstract=249286.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
estimate, it is not necessarily equal to the estimated rate of inflation in the net cash
flows, which probably will vary year to year and will be specific to the circumstances
of the subject company.
W h y O n l y T h r e e S p e c i fi c M a t u r i t i e s ?
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 partially 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 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 Morningstar
data in the build-up method. The risk premiums for the build-up method in the Duff
& Phelps Studies combine a general equity risk premium and size premium in one
number, measured in terms of a premium over long-term (20-year) U.S. government
bonds.
Selecting the Best Risk-Free Maturity
In valuing going-concern businesses and long-term investments made by businesses,
practitioners generally use long-term U.S. government bonds as the risk-free security
and estimate the equity risk premium (ERP or notationally RPm) in relation to longterm U.S. 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 U.S. government bonds. As such,
the use of long-term U.S. government bonds and an ERP estimated relative to longterm bonds more closely matches the investment horizon and risks confronting
business managers making capital allocation decisions and valuators in applying
valuation methods.
Many financial analysts today 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. Analysts using the Morningstar data series
generally use 20-year U.S. government bond yields as their risk-free rate.
Some 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
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Build-up Method
EXHIBIT 7.1 Yields on 10-Year, 20-Year, and 30-Year
U.S. Government Bonds
Yields
Period
9/30/2009
8/31/2009
7/31/2009
6/30/2009
5/31/2009
4/30/2009
3/31/2009
2/28/2009
1/31/2009
12/31/2008
11/30/2008
10/31/2008
9/30/2008
8/31/2008
7/31/2008
6/30/2008
5/31/2008
4/30/2008
3/31/2008
2/29/2008
1/31/2008
12/31/2007
12/31/2006
12/31/2005
12/31/2004
12/31/2003
12/31/2002
12/31/2001
12/31/2000
10-Year
20-Year
30-Year
3.4
3.6
3.6
3.7
3.3
2.9
2.8
2.9
2.5
2.4
3.5
3.8
3.7
3.9
4.0
4.1
3.9
3.7
3.5
3.7
3.7
4.1
4.6
4.3
4.3
4.3
4.0
5.1
5.2
4.0
4.2
4.3
4.3
4.3
4.1
3.6
4.0
3.9
3.0
3.7
4.8
4.4
4.5
4.7
4.6
4.8
4.6
4.3
4.4
4.4
4.5
4.9
4.6
4.8
5.1
4.8
5.8
5.6
4.2
4.4
4.4
4.5
4.2
3.8
3.6
3.6
3.1
2.9
4.0
4.2
4.3
n/a
4.6
4.7
4.6
4.4
4.4
4.5
4.3
4.5
4.7
n/a
n/a
n/a
n/a
5.5
5.4
yields on 10-year, 20-year, and 30-year U.S. government bonds since year-end 2000
and monthly data for 2008 and 2009 through September.
Although the use of the 20-year U.S. government bond has historically been the
most widely used estimate of the risk-free rate, the assumption that this rate was
the best estimate of the risk-free rate began to change beginning in September 2008,
as the financial crisis started to unfold. Long-term U.S. government bond yields, the
typical benchmark used in cost of equity capital models, became abnormally low for
several months, resulting in unreasonably low estimates of the cost of equity capital
(if the analyst used historical realized risk premiums as an estimated equity risk premium) as of the important valuation date, December 31, 2008.2 Most analysts
would agree that the world economies were (and may still be, as of the date of this
2
Roger J. Grabowski, ‘‘Cost of Capital Estimation in the Current Distressed Environment,’’
Journal of Applied Research in Accounting and Finance (July 2009): 31–40.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
writing) in crisis. Financial crises are often accompanied by a flight to quality, such
that the nominal returns on ‘‘risk-free’’ securities fall dramatically for reasons other
than inflation expectations and thus, without adjustment, become less reliable as the
best indicator of the risk-free rate. Recent macroeconomic research suggests that
short-term inflation expectations remain fairly stable, and therefore the dramatic decline in the government bond yields in November and December 2008 was probably
not due to expected declines in expected long-term inflation.3 In fact, long-term (10year horizon) Consumer Price Index (CPI) expectations continued to be at 2.5% at
the end of 2008.4
Although short-term inflation expectations had decreased,5 many commentators were warning that long-term inflation would increase, not decrease, given the
projected U.S. budget deficit. Based on surveys of professional forecasters, yields on
long-term U.S. government bonds were also expected to increase.
Since the bottom at December 31, 2008, yields on 20-year (constant maturity)
U.S. government bonds have increased. For example, as of September 30, 2009, the
yield had increased to 4.1%. It appears that the flight to quality that drove yields on
U.S. government bonds to unreasonably low levels as of December 2008 has eased,
and yields on U.S. government bonds appear to have returned to more normalized
levels. According to Federal Reserve Chairman Bernanke in his prepared testimony
to the U.S. House of Representatives Budget Committee on June 3, 2009, regarding
recent increases in yields on longer-term government bonds and fixed-rate
mortgages:
These increases appear to reflect concerns about large federal deficits but
also other causes, including greater optimism about the economic outlook,
a reversal of flight-to-quality flows, and technical factors related to the
hedging of mortgage holdings.
Further evidence of the flight to quality and its impact on U.S. government interest
rates was the implied forward volatility (based on options on exchange traded funds or
ETFs) on 20-year U.S. government bonds in November and December of 2008. The
volatility had increased significantly (to approximately double the implied forward volatility in earlier months6), suggesting that the market was uncertain that the lower
yields (and correspondingly higher prices) in November and December of 2008 were
sustainable. (See Exhibit 7.2.) By September 2009, the implied forward volatility had
decreased but was still approximately 12% greater than the average for months leading up to the November–December flight to quality.
3
V. V. Chari, Lawrence Christiano, and Patrick J. Kehoe, ‘‘Facts and Myths about the Financial
Crisis of 2008,’’ Federal Reserve Bank of Minneapolis Research Department, Working paper
666, October 2008. Available at http://www.minneapolisfed.org/research/wp/wp666.pdf.
4
‘‘Survey of Professional Forecasters: Fourth Quarter 2008,’’ Federal Reserve Bank of
Philadelphia (November 17, 2008); ‘‘The Livingston Survey: December 2008,’’ Federal Reserve Bank of Philadelphia (December 9, 2008).
5
‘‘The Livingston Survey: June 2009,’’ Federal Reserve Bank of Philadelphia (June 9, 2009): 1.
6
Implied volatility for three-month options on iShares Lehman 20+year Treasury Bonds averaged 31.5% in November and December 2008, compared with an average of 15.0% during
the first 10 months of 2008. The implied volatility was nearly 16.8% in September 2009.
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Build-up Method
EXHIBIT 7.2 Implied Volatility
Ticker:
Description:
SPY
S&P 500 ETF
TLT
iShares Lehman 20þ Year Treasury Bond
Implied Volatility
As of:
12/31/2005
12/31/2006
12/31/2007
1/31/2008
2/29/2008
3/31/2008
4/30/2008
5/31/2008
6/30/2008
7/31/2008
8/31/2008
9/30/2008
10/31/2008
11/30/2008
12/31/2008
1/31/2009
2/28/2009
3/31/2009
4/30/2009
5/29/2009
6/30/2009
7/31/2009
8/31/2009
9/30/2009
(1)
(2)
(1)
30 Day
10.765
10.255
21.525
26.121
24.581
25.037
19.403
15.929
22.804
22.058
19.111
39.166
52.078
51.756
36.267
39.630
40.919
39.529
33.320
26.759
23.937
22.761
22.698
22.628
3 Month
Implied Volatility
(2)
12.655
11.023
22.604
23.983
24.925
24.590
19.977
18.885
22.508
21.838
21.246
31.297
46.356
48.393
37.567
38.683
39.475
39.385
33.163
28.109
25.276
24.480
25.424
23.015
30 Day
(1)
8.700
7.490
14.952
17.578
17.807
16.846
12.954
13.081
11.516
11.085
10.759
18.686
16.809
28.837
31.332
26.101
25.140
17.989
19.808
22.022
18.966
16.897
16.109
15.859
3 Month(2)
9.239
8.079
14.356
16.294
17.305
17.239
13.341
14.165
12.966
12.316
12.133
16.118
18.464
31.087
31.213
25.258
25.410
19.401
19.875
21.802
19.452
17.803
17.259
16.793
30 Day Implied Volatility.
3 Month Implied Volatility.
Source: Bloomberg. Compiled by Duff & Phelps LLC. Used with permission. All rights reserved.
In summary, the evidence suggests that the yield on U.S. government bonds represented an aberration as of December 31, 2008, overly influenced temporarily by
the flight to quality. In the examples in this book, we have chosen to use 4.5% as the
yield on U.S. 20-year government bonds as of December 31, 2008, as a proxy for a
more normalized risk-free rate of return. Other authors may offer alternative views
to this approach, but that is the convention we have adopted, and we believe it is
well supported by the evidence.7
More recently U.S. government bond yields have returned to more normal levels, though there is some evidence that these yields may still be artificially low even
as of September 30, 2009.
7
Aswath Damodaran, ‘‘What Is the Riskfree Rate? A Search for the Basic Building Block,’’
Stern School of Business Working paper, December 2008. Available at http://ssrn.com/
abstract=1317436.
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Sometimes analysts select a five-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 (particularly during periods of high inflation), but it
contains little compensation for inflation uncertainty. While there may be advantages to using these U.S. government securities to obtain the risk-free rate, for reasons already discussed, we prefer using the 20-year U.S. Treasury bond yield.
EQUITY RISK PREMIUM
For an equity investment, the return on the 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 distributions).
2. The capital gain or loss in the value of the investment. (For an active public
security, the gain or loss 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 riskier)
than the interest and maturity payments on U.S. government obligations. This difference in risk is well documented by much higher standard deviations (year-to-year volatility) in returns on the stock market compared with the standard deviation of yearto-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, including difficulties in estimating the equity risk premium in the recent economic crisis. In the examples in this book, the authors have chosen to use an equity risk premium estimate of
6%. Other authors have offered alternative views to this approach, but that is the
convention we have adopted, and we believe it is well supported by the evidence.8
SIZE PREMIUM
The size premium is an addition to the generalized ERP, as the ERP estimates
are based on expected returns for large company stocks (e.g., S&P 500). Studies
have provided evidence that the degree of risk, and corresponding cost of capital,
increase with the decreasing size of a 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 smaller company’s market risk. Chapter 13 discusses the results
of research on this phenomenon, as well as the data sources. Many practitioners use
8
Aswath Damodaran, ‘‘Equity Risk Premium (ERP): Determinants, Estimation and Implications—A Post-Crisis Update,’’ Stern School of Business Working paper, October 2009.
Available at http://ssrn.com/abstract=1492717.
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the small-company premium in the build-up method (difference between the realized
returns on small company stocks and large company stocks).
COMPANY-SPECIFIC RISK PREMIUM
The company-specific risk premium is also an addition to the generalized ERP.
To the extent that the subject company’s risk characteristics are greater or less than
the typical risk characteristics of the guideline public 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 a specific company. Such an adjustment
may be based on (but not necessarily limited to) analysis of the following 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
For example, as will be seen in Exhibits 13.7 and 13.8 from the Duff & Phelps studies, the smallest size group for which Duff & Phelps calculates an equity risk premium has an average of $111 million in market value of equity, $112 million in
sales, and so forth. If the subject company is smaller than these averages, most
observers believe that a further size premium adjustment is warranted, but there
have not yet been adequate empirical studies to quantify this adjustment. The Duff
& Phelps studies do provide regressions of the observed relationships between size
and returns for use in extrapolating the ERP to smaller firms. Alternatively, a conservative approach may be appropriate, perhaps adding 100 to 200 basis points to
the discount rate for a significantly smaller company and leaving any greater adjustments to be attributed to other specifically identifiable risk factors.
Incorporating an Industry Risk Factor into the
Build-up Method
The Ibbotson (R) Stocks, Bonds, Bills, and Inflation (R) (SBBI (R)) Valuation Edition 2006 Yearbook and subsequent editions present an expanded alternative buildup 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 Rf þ RPm þ RPs RPi 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)
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The industry in which the subject 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 100 to 200 basis point 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 15). These ‘‘industries’’ are based on Standard
Industrial Classification (SIC) codes. The industry premia were adjusted quarterly
through 2007 and are now adjusted twice each year. 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 free from the Morningstar web site: www.global.morningstar.com/us/
IRPCompanyList.
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. We believe it is better to have an industry risk factor
based on fewer guideline companies that actually are closely comparable to the subject company than many guideline companies, a number of which are only remotely
comparable to the subject company.
To aid this judgment, the analyst is likely to need to go to the 10-K filings for the
companies included. The description of the companies included gives the analyst a
much better picture of the similarities of the subject to the companies included in
the industry. 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. Our caution is to rely on segment profitability because stock returns are a function of profit, not revenue, and use of revenue segmenting may result
in overweighting the low-profit segment.
The SBBI formula for the RPi is as follows:
(Formula 7.3)
RPi ¼ ðFI-beta RPm Þ RPm
where:
RPi ¼ Industry risk premium
FI-beta ¼ Full-information beta for industry
RPm ¼ ERP estimate used in calculating RPi
SBBI Valuation Edition Yearbook uses the long-term, realized risk premiums
measured from 1926 through the most recent period. For example, as of the end of
2008, the realized risk premium equaled 6.5%. If one is going to use the RPi in
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97
conjunction with their own estimated ERP, one needs to adjust the RPi for the differences in the estimated equity premium.
For example, assume that the subject RPi from the SBBI Valuation Edition
equaled 0.57 percent,9 and assume that your current estimate of the ERP was
6.00% instead of the average realized risk premium of 6.50% for 1926–2008. We
can then determine an RPi for that SIC code consistent with the ERP of 6.00% as
follows:
(Formula 7.4)
New RPi ¼ SBBI RPi ðNew ERP estimate=SBBI historical risk premium estimateÞ
0:53% ¼ 0:57% ð6:0%=6:5%Þ
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 analyst perceives that the subject
company’s 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. This would be a factor to consider as part of a company-specific risk
premium.
Leverage
Leverage is clearly a factor that can be compared between the subject company and
its peers. For example, Exhibit 13.7 categorizes companies based on the market
value of equity for each size category. The smallest size category based on market
capitalization averages $111 million in market value of equity with a market value
capital structure of roughly 30% debt and 70% equity, at market value. When we
examine other measures of size, such as number of employees, and then at the average capital structure within the different size categories, we find that the average
capital structure is generally close to the average capital structure of roughly 30%
debt and 70% equity. If the subject company’s capital structure is significantly different from the average, upward or downward adjustment to the cost of equity relative to the capital structure of the average company in a size category would seem
warranted. For example, highly leveraged companies within a size category should
have higher equity costs of capital compared with companies with lower debt levels,
all else being equal. Similarly, a decrease in the cost of capital might be warranted if
the subject company’s capital structure has little or no debt. We discuss the risks of
leverage in the capital structure in Chapter 18.
9
‘‘SIC code 58, Eating and Drinking Places,’’ SBBI Valuation Edition 2009 Yearbook
(Chicago: Morningstar, 2009), 40.
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O t h e r C o m p a n y - s p e c i fi c F a c t o r s
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 litigation
Abnormal volatility of returns
Strengths or weaknesses of company management
A variety of other possible specific factors
Because the size premium already captures some of these listed factors, the analyst should exercise care in making any additional adjustments to avoid doublecounting. Adjustments may be appropriate for companies with unusual, unique risk
factors. Further, keep in mind that just because the guideline public companies to
which the analyst is comparing the subject company are public companies does not
mean that the smaller guideline companies do not share some of the same companyspecific risk factors as the subject company. In such cases, the risk characteristics
may already reflect some of these factors. The analyst should carefully read the public filings and, if available, investment analyst reports for the guideline public companies to understand their risks and determine whether these companies share
similar risks to the company being valued. As analysts often rely on summarized
databases to obtain information on guideline public companies, there is limited opportunity to assess whether a particular guideline company has a similar risk profile
to the company being valued. Therefore, to more accurately gauge comparability of
risks, one should carefully study each guideline company’s public filings, including a
review of management commentaries and risk factors.
Unfortunately, despite the widespread use by analysts and appraisers of a company-specific risk premium in a build-up (or CAPM) model, there is only limited
academic research on quantification of any company-specific risk premiums, and
the company-specific risk premium generally remains in the realm of the analyst’s
judgment. We discuss the research in Chapter 15.
EXAMPLE OF THE BUILD-UP METHOD USING
MORNINGSTAR DATA
Now that we have discussed the factors comprising the build-up model, we can substitute some numbers into the formula. In the first example, we calculate the cost of equity
capital using the build-up method for Shannon’s Bull Market (SBM), a closely held,
regional steakhouse chain with excellent food and drink that is noted for its friendly
service. To illustrate the use of the model, we first make five assumptions, which follow.
We use Morningstar data and apply that data to Formula 7.2:
EðRi Þ ¼ Rf þ RPm þ RPs RPi RPu
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1. Risk-free rate. We will use the proxy for the normalized 20-year U.S. government bond, which at the valuation date of December 31, 2009, was 4.5%.
2. Equity risk premium. We will use the results of the research on the expected
ERP discussed in Chapter 9 and use an RPm estimate of 6.0%.
3. Size premium. The SBBI Valuation Edition 2009 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 5.81%.10
4. Industry adjustment factor. SBM is in the SIC code 58, Eating and Drinking
Places. The industry risk premium for that industry, developed using the fullinformation beta with contributions to that beta from 63 companies, is 0.57%,
which was adjusted to 0.53% using Formula 7.4 for ERP estimate of 6.0%.
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. SBM has one key manager, Shannon, and
is heavily leveraged. Although the assessment is 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 7.5)
EðRi Þ ¼ 4:5% þ 6:0% þ 5:81% þ ð0:53%Þ þ 3:0%
¼ 18:78% ðrounded to 19%Þ
The indicated cost of equity capital for SBM is approximately 19%.
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
4.5%
6.0%
5.81%
0.53%
3.00%
19% (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 Rf þ RPm þ
RPs 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
10
Morningstar recommends using the size premium (return in excess of CAPM) analysis for
both the build-up and CAPM cost of equity estimates. Some analysts use the small stock
premium in this example instead. See Chapter 13 in this volume for more discussion.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
Duff & Phelps Risk Premium Report—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.
The formula then is modified to be:
(Formula 7.6)
EðRi Þ ¼ Rf þ RPmþs 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 13 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 proxy for the normalized 20-year U.S. government bond, which, at the valuation date of December 31, 2009, was 4.5%.
2. Risk premium. The Size Study exhibits for the build-up method combine the
equity risk premium and the size premium into a single premium. The Size Study
indicates that the risk premiums for the smallest companies are as shown in
the table from the smoothed average risk premium. We use only six of the eight
size measures listed in the report because SBM is closely held, and two of the size
measures are based on market capitalization. While these risk premiums are
those published in the Size Study exhibits, one can adjust these premiums to better estimate the expected equity risk premium at the valuation date. This is
explained in Chapter 13.
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
10.88%
11.74%
11.21%
11.42%
10.46%
10.64%
11.0% (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. The analyst will need to consider adjusting the ERP implicit in the industry adjustment
factor (as explained earlier) or adjust the risk premiums for the analyst’s estimated
ERP (as we explain in Chapter 15). SBM is in the SIC code 58, Eating and Drinking Places. The industry risk premiums for that industry were developed using the
full-information beta with contributions to that beta from 63 companies and was
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0.57%, adjusted for the ERP estimate of 6.0% it equals –0.53% (Formula 7.4).
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 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 7.6, we have Formula 7.7:
(Formula 7.7)
EðRi Þ ¼ 4:5% þ 11:0% þ ð0:53%Þ þ 3:0%
¼ 17:97% ðrounded to 18%Þ
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
4.5%
11.0%
0.53%
3.0%
18% (rounded)
If we were using the 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, 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 the following
components:
1.
2.
3.
4.
A risk-free rate
A general equity risk premium (ERP)
A size premium
A company-specific risk adjustment (which can be either positive or negative,
depending on the risk comparisons between the subject company and guideline
companies from which the size premium was derived)
5. Possibly, an industry adjustment factor
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These factors are summarized schematically in Exhibit 7.3. In a sense, the buildup method is a version of the CAPM without specifically incorporating systematic
risk.
EXHIBIT 7.3 Summary of Development of Equity Discount Rate Using Build-up Method
Risk-free rate
þ Equity risk premium
þ Size premium
þ/ Company-specific
risk premium
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 size premium (premium over return
predicted by CAPM)
Specific risk difference in subject company relative to guideline
companies from which these data are drawn.
The risk-free rate for 5-year and 20-year maturities 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.
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CHAPTER
8
Capital Asset Pricing Model
Introduction
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 Company-specific Risk
Firm Size Phenomenon
Company-specific Risk Factor
Expanded CAPM Cost of Capital Formula
Examples of a CAPM Model
Example of CAPM Method Using Morningstar Data
Example of a CAPM Method Using Duff & Phelps Size Study Data
Assumptions Underlying the Capital Asset Pricing Model
Summary
INTRODUCTION
The CAPM is the most widely used method for estimating the cost of equity capital.
For example, one survey found that 75% of firms use the CAPM to estimate the cost
of equity, 34% of firms use the CAPM with additional adjustment factors (with
common adjustments being adding additional premiums for such added risks as
the risks of operating in developing economies), 39% of firms use historical average
returns (which in essence combines systematic and unique risk), and 16% of firms
impute the cost of equity (see Chapter 17).1 The survey also found that many firms
use multiple methods.
As with any model, certain assumptions are made in developing CAPM, and
those assumptions also represent limitations. Despite its limiting assumptions,
CAPM helps explain the relationship of the risk among stocks and their expected
returns.
1
John R. Graham and Campbell R. Harvey, ‘‘The Theory and Practice of Corporate
Finance,’’ Journal of Financial Economics (May 2001): 187–243.
103
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CONCEPT OF MARKET OR SYSTEMATIC RISK
For more than 30 years, financial theorists generally have favored using the CAPM
as the preferred method to estimate the cost of equity capital. Despite its many criticisms, the CAPM is still one of the most widely used models for estimating the cost
of equity capital, especially for larger companies.2
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 rates of return) 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. 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 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 procedures to estimate the cost of equity capital. All
other things being equal, the cost of capital for any given company at any given
point in time, theoretically, is the same whether you arrive at it by CAPM, by the
build-up method, or by some other model. The cost of equity capital does not
change at a given point in time because of the method used to determine it. CAPM,
however, generally requires public companies from which to develop a proxy beta.
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
2
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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105
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 (other
than maturity risk) into two components: market or systematic risk and unique or
unsystematic risk. Stated in less technical 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 well-diversified portfolios. Under this 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 the pure
capital asset pricing theory is market risk. As one well-known corporate finance
text puts it: ‘‘[T]he crucial distinction between diversifiable and nondiversifiable
risks . . . is the main idea underlying the capital asset pricing model.’’3
USING BETA TO ESTIMATE EXPECTED RATE OF RETURN
CAPM assumes 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
3
Richard A. Brealey, Stewart C. Myers, and Franklin Allen, Principles of Corporate Finance,
9th ed. (Boston: Irwin McGraw-Hill, 2008), 967.
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security’s beta. This linear function is described in this univariate linear regression
formula:
(Formula 8.1)
EðRi Þ ¼ Rf 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
B ¼ Beta
RPm ¼ Equity risk premium (ERP) for the market as a whole
The preceding linear relationship is shown schematically in Exhibit 8.1, which
presents the security market line (SML), a graphical presentation 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 mispriced. It would be mispriced in the sense that the analyst’s expected
E(Ri)
Expected Rate of Return
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Security
Market Line
0.166
0.15 E(Rm)
X
0.134
Rf
0.8
1.0
1.2
Beta
EXHIBIT 8.1 Security Market Line
E(Ri) ¼ Expected return for the individual security
E(Rm) ¼ Expected return on the market
Rf ¼ 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, as its expected rate of return is less than it should be based on the
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.
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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. In theory,
beta equals:
(Formula 8.2)
Bi ¼
where:
covðRi ; Rm Þ
varðRm Þ
Bi ¼ Expected beta of the stock of company i
Ri ¼ Return on stock i
Rm ¼ Return on market portfolio
Cov(Ri,Rm) ¼ Expected covariance between the excess return (Ri-Rf) on
stock of company i and the excess market return (Rm-Rf)
Var(Rm) ¼ Expected variance of excess return on the overall stock
market
Covariance measures the degree to which the return on a particular security and
the overall market’s return move together. Covariance is not volatility. Covariance
is a measure of the two variables’ tendency to vary in the same direction and in the
same relative amounts.
The excess returns on a stock exhibit positive covariance with the excess returns
of the market if large values of one variable (excess returns on the subject stock) tend
to be associated with large values of the other variable (excess returns of the market)
or if small values of one variable (excess returns on the subject stock) tend to be
associated with small values of the other (excess returns on the subject stock)—
whether negative or positive.
The excess returns on a stock exhibit negative covariance with the excess
returns of the market if large values of one variable (excess returns on the subject
stock) tend to be associated with small values of the other variable (excess returns of the market) or if small values of one variable (excess returns on the subject stock) tend to be associated with large values of the other (excess returns on
the subject stock); negative covariance does not require that one value be negative while the other is positive.
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
When excess market returns move up or down, the excess returns 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 return in excess
of the risk-free rate increases by 10%, the excess return of the subject
(continued )
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Beta ¼ 1:0
Beta < 1:0
Negative beta
(rare)
ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
stock with a beta of 1.2 would be expected to increase by 12%. If the
market return in excess of the risk-free rate is down 10%, the excess
return of the subject stock would be expected to decline 12%. Many hightech companies are good examples of stocks with high betas.
Fluctuations in rates of return for the subject stock tend to equal
fluctuations in rates of return for the market.
When excess market returns move up or down, excess returns 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 return in excess of the risk-free rate
increases 10%, the excess return of the subject stock with a beta of .8
would be expected to increase 8%. The classic example of a low-beta stock
would be a utility that has not diversified into riskier activities.
Rates of return for the subject tend to move very little or 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
could 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 4.5% at the valuation date; and a expected ERP (RPm) of 6%.
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 8.3)
EðRi Þ ¼ 0:045 þ 0:8ð0:06Þ
¼ 0:045 þ 0:048
¼ 0:093
Thus, the indicated cost of equity capital for company i is estimated to be 9.3%
because its beta is only 0.8 and it is less risky, in terms of market risk, than the average stock on the market.
For company j, which has average sensitivity to market movements, we can substitute in (Formula 8.1) in this way:
(Formula 8.4)
E Rj ¼ 0:045 þ 1:0ð0:06Þ
¼ 0:045 þ 0:06
¼ 0:105
The indicated cost of equity capital for company j is estimated to be 10.5%, the
estimated cost of capital for the average stock, because its beta is 1.0 and 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:
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Capital Asset Pricing Model
(Formula 8.5)
EðRk Þ ¼ 0:045 þ 1:2ð0:06Þ
¼ 0:045 þ 0:072
¼ 0:117
Thus, the indicated cost of equity capital for company k is estimated to be
11.7% because its beta is 1.2 and it is riskier, in terms of market risk, than the average stock on the market.
Beta is a forward-looking concept. Beta is the expected covariance over the
expected variance. A common method of estimating beta is to use realized returns
on the stock of company, i, and the market, m, over a look-back period and run a
regression. The regression beta is only an estimate of the expected relationships.
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 ERP
3. The stock’s volatility relative to the market, the beta
See Chapter 9 for a discussion of the market’s general ERP.
EXPANDING CAPM TO INCORPORATE SIZE PREMIUM AND
COMPANY-SPECIFIC RISK
Firm Size Phenomenon
Many empirical studies performed since CAPM was originally developed have
found 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. The original size effect studies
measured size based on market capitalization of equity; later studies expanded the
definition of size to include accounting measures of size (e.g., net income). 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
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
market for large stocks and in most other equity markets but is evident
in the size premia.4
There are currently two widely used sources of size premium data: Morningstar’s SBBI Yearbook and the Duff & Phelps Risk Premium Report—Size Study.
The size effect and those sources are the subjects of Chapter 13.
C o m p a n y - s p e c i fi c R i s k F a c t o r
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.
Just as in the build-up model, the company-specific risk premium 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 wellprotected, above-average price for its products as a result of a strong trademark,
resulting in significantly less earnings volatility than that of its competitors.
We discuss the company-specific risk premium in Chapter 15.
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 8.6)
EðRi Þ ¼ Rf þ BðRPm Þ þ RPs 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 Market ERP
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’s SBBI Yearbook and the Duff & Phelps Size Study.
4
SBBI Valuation Edition 2009 Yearbook (Chicago: Morningstar, 2009), 89, 93.
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Example of CAPM Method Using Morningstar Data
To put some numbers into Formula 8.6, we will make five assumptions about
Unique Computer Systems (UCS), a fictional specialty manufacturer in the computer
industry with publicly traded common stock:
1. Risk-free rate. We will use the proxy for the normalized 20-year U.S. government bond, which at the valuation date of December 31, 2009, was 4.5%.
2. Beta. The UCS beta is 1.6.
3. Equity risk premium. We will use the results of the research on the expected
ERP discussed in Chapter 9 and use estimate of 6.0% for this example.
4. Size premium. The SBBI Valuation Edition 2009 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 3.74%. We will assume here that this is
on the borderline between Morningstar’s ninth and tenth size deciles and use the
microcap 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 8.6, we have:
(Formula 8.7)
EðRi Þ ¼ 4:5 þ 1:6ð6:0Þ þ 3:74 þ 1:0
¼ 4:5 þ 9:6 þ 3:74 þ 1:0
¼ 18:84%
Thus, the indicated cost of equity capital for UCS is estimated to be 19%
(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
4.5%
6.0%
6.0 1.6 ¼ 9.6%
1.6
3.74%
1.0%
19% (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 on their respective size measures.
Again, using Formula 8.6, we assume:
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
1. Risk-free rate. We will use the proxy for the normalized 20-year U.S. government bond, which at the valuation date of December 31, 2009, was 4.5%.
2. Beta. The UCS beta is 1.6.
3. Equity risk premium. We will use the results of the research on the expected
ERP discussed in Chapter 9 and use an RPm estimate of 6.0% for this example.
4. Size premium. The Duff & Phelps Risk Premium Report—Size Study (smoothed
premium over CAPM) indicates that the size premia for UCS (approximately in
the 24th portfolio for each size measure) are:
Size as Measured by
Size Premium
Market value of common equity
Book value of common equity
Five-year average net income
Market value of invested capital
Total assets
Five-year average EBITDA
Sales
Number of employees
Median size premium
5.77%
4.76%
5.46%
5.47%
5.01%
5.22%
4.78%
5.07%
5.1% (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 8.6, we have:
(Formula 8.8)
EðRi Þ ¼ 4:5 þ 1:6ð6:0Þ þ 5:1 þ 1:0
¼ 4:5 þ 9:6 þ 5:1 þ 1:0
¼ 20:20%
Thus, the indicated cost of equity capital for UCS is estimated to be 20%
(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
4.5%
6.0%
6.0 1.6 ¼ 9.6%
1.6
5.1%
1.0%
20% (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.
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113
ASSUMPTIONS UNDERLYING THE CAPITAL ASSET
PRICING MODEL
Eight assumptions underlie the CAPM:
1. Investors are risk averse.
2. Rational investors seek to hold efficient portfolios (i.e., portfolios that 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 discount 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).
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. The analyst may not find guideline public companies with risk factors that
match those of the closely held business. This may be particularly true for the
smaller closely held businesses. This is one reason why the company-specific risk
premium may be rewarded in expected returns for a particular closely held company, or while the perfect divisibility and liquidity assumption approximates reality
for public stocks, the same is not true for closely held companies. A discount for lack
of marketability (equivalent to an increase in the cost of capital) may also be appropriate for the closely held business. 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 CAPM expands on the build-up model by introducing the beta coefficient, an
estimate of market risk or systematic risk, the sensitivity of excess returns for the
subject company stock to excess 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. CAPM has
been attacked because beta (discussed in Chapter 11) has been found to not be a very
reliable measure of risk in practice and because its underlying assumptions may not
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
EXHIBIT 8.2 Capital Asset Pricing Model Method of Estimating Equity Discount Rate
Risk-free rate
þ beta Equity risk
premiumy
þ Size premium
Company-specific
risk premium
20-year, 5-year, or 30-day Treasury yield as of valuation date
Expected equity risk premium corresponding to risk-free rate,
multiplied by beta
Premium over that predicted by beta
Specific risk difference in subject company relative to guideline
companies from which beta and size premium are estimated
The ‘‘risk-free’’ rate for 5-year and 20-year maturities 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.
y
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 are available
from Morningstar. The equity risk premium could also be estimated by other models, as discussed in Chapter 9, ‘‘Equity Risk Premium.’’
hold true in practice. 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.
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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 the Risk-free Rate with the ERP
Measuring the Average Period of the Expected Cash Flows
Realized Risk Premium (ex Post) Approach
Selecting a Sample Period
WWII Interest Rate Bias
Has the Relationship between Stock and Bond Risk Changed?
Comparing Investor Expectations to Realized Risk Premiums
Changes in Economics That Caused Unexpectedly Large Realized Risk Premiums
Other Sources of ERP Estimates
Forward-Looking (ex Ante) Approaches
Bottom-up ERP Estimates
Top-down ERP Estimates
ERP Surveys
Long-Term Unconditional ERP Estimate
Conditional Estimate of ERP and Crisis of 2008–2010
Summary
Appendix 9A: Realized Risk Premium Approach and Other Sources of ERP Estimates
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
The authors wish to acknowledge the contribution of David King, CFA, to the discussion contained herein. This chapter is an update and expansion to prior work of their prior work; see,
for example, Roger Grabowski and David King, ‘‘Equity Risk Premium,’’ in Robert Reilly
and Robert P. Schweihs, eds., The Handbook of Business Valuation and Intellectual Property
Analysis (New York: McGraw-Hill, 2004): 3–29. We also wish to especially thank David
Turney, CFA, of Duff & Phelps LLC for the assistance he provided.
115
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
securities) that investors expect to receive from an investment in a diversified portfolio of common stocks.
Estimating the ERP is one of the most important decisions the analyst 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, such as 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 percentage points.2
Three driving forces behind the discussions that have evolved on ERP include:
1. What returns can be expected by retirement plans from investments in publicly
traded common stocks?
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 of the
long-term average or unconditional ERP. That is, what is a reasonable range of ERP
that can be expected over an entire business cycle?
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. We report on ERP estimates at the beginning of 2009 and through September 2009. That is, where in this range is the current ERP, given the crisis of
2008–2010?
We conclude with our recommended ERP.
1
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.
2
Seth Armitage, The Cost of Capital: Intermediate Theory (Cambridge: Cambridge University Press, 2005), 319–320.
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Equity Risk Premium
DEFINING THE EQUITY RISK PREMIUM
The ERP (or notational RPm) is defined as:
RPm ¼ Rm Rf
where: 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
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 directly
observable.
In this chapter, we are addressing returns of publicly traded stocks. Those returns establish a beginning benchmark for closely held investments.
Estimating the ERP
While an analyst can observe premiums realized over time by referring to historical
data (i.e., realized return approach or ex post approach), such realized premium
data 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 then 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 prior events that are not expected to recur 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 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 (i.e., the long-term average) and a conditional ERP based on current levels of the stock market and economy relative to the long-term average.4 We
address issues involving the conditional ERP later.
There is no one universally accepted methodology for estimating ERP. A wide
variety of premiums are used in practice and recommended by academics and financial advisors.
3
See, e.g., Eugene F. Fama and Kenneth R. French, ‘‘The Equity Premium,’’ Journal of Finance (April 2002): 637–659.
4
Robert Arnott, ‘‘Historical Results,’’ Equity Risk Premium Forum, AIMR (November 8,
2001): 27.
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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). 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. Thus, the resulting ERP should not be affected by inflation.
But ex post realized returns will be affected by differences between expected inflation and realized inflation and will differ from ERP estimates made in prior periods.
WHICH RISK-FREE RATE TO USE IN ESTIMATING THE ERP
Any estimate of the 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 on 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 was potentially short-term, a U.S.
government short-term bond (e.g., a 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 longterm U.S. government bond may be the more appropriate instrument in benchmarking the ERP estimate.
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 risk-free 90-day T-bills or one-year U.S. government
bonds. A T-bill rate is the purest risk-free base rate because it contains essentially no
maturity or default risk. If inflation is high, the T-bill 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 because
of central bank actions affecting the short-term interest rates, and (2) rates on shortterm securities tend to be more volatile than yields on longer maturities.
Long-term U.S. government bonds are generally considered free of default risk
but are not entirely risk-free. Bonds are sensitive to future interest rate 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
5
When short-term interest rates exceed long-term rates, the yield curve is said to be
‘‘inverted.’’
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Equity Risk Premium
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 long-term bonds on the average 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 the
market risk that their expectations of interest rates today, period by period
over the term of the bond, will in fact be wrong.
Matching the Risk-free Rate with the 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 over 1-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 ðRPm over 3-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 net 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, one 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 one expects the cash flow in the first year following the valuation
date of $1 million to increase at an average compound rate of 4% per annum. Assume a discount rate of 15%. If one projects cash flows each year for 100 years, the
calculated duration of the cash flows is approximately 10.5 years.6
2
6 ½ð1;000;0001Þ=ð1:15Þþð1;000;0001:042Þ=ð1:15Þ þð1;000;0001:04 3Þ=ð1:15Þ ...
2
3
½ð1;000;0001Þ=ð1:15Þþð1;000;0001:04Þ=ð1:15Þ2 þð1;000;0001:042 Þ=ð1:15Þ3 ...
¼ 1:5ðroundedÞ
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In practice, few analysts discount each year’s expected net cash flow estimate
using a matched maturity risk-free rate and ERP estimate. In valuing goingconcern businesses and long-term investments made by businesses, practitioners
generally use long-term U.S. government bonds as the risk-free security and estimate the ERP in relation to long-term U.S. 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 future changes are independent
of prior changes 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 U.S. government bonds and an ERP estimated relative to such bonds more closely matches the investment horizon and
risks confronting business managers in capital budgeting decisions, as well as
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 U.S. government bonds.
REALIZED RISK PREMIUM (EX POST) APPROACH
In this section, we are looking at estimates of the unconditional ERP using realized
risk premium data. While academics and practitioners agree that ERP is a forwardlooking concept, some practitioners, including taxing authorities and regulatory
bodies, use historical data only to estimate the ERP under the assumption that historical data are a valid proxy for current investor expectations (the ex post approach). They like the appearance of accuracy, and we do emphasize the word
appearance. There are alternative conventions one could use to summarize realized
risk premiums. Before one concludes on the accuracy of using realized risk premiums as an estimate of the ERP, one must consider the adjustments to the realized
risk premiums, which we discuss in this chapter.
In using the realized risk premiums, there are certain issues that one must
address:
&
&
&
&
Which risk-free rate should be used to measure the realized premiums?
Is the arithmetic average or geometric average the more accurate method of
summarizing realized return data over the sample period?
Should returns be measured over one-year holding periods or over longer holding periods?
Do we introduce bias by using arithmetic averages of realized risk premiums?
We discuss these issues in detail in Appendix 9A.
7
Carmelo Giaccotto, ‘‘Discounting Mean Reverting Cash Flows with the Capital Asset Pricing Model,’’ Financial Review (May 2007): 247–265. This is true for both the original
CAPM of Sharpe and Linter and the extension of the textbook CAPM, the intertemporal
CAPM of Merton.
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In the realized risk premium approach, the estimate of the ERP is the risk premium (realized return on stocks in excess of the risk-free 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, you may to some extent replicate this level of pricing.
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 otherwise arbitrary. Regarding
the historical time period over which equity risk should be calculated, Morningstar
offers two observations:8
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 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.
WWII Interest Rate Bias
In addition, the average realized returns 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. Some observers have suggested that the period
including the 1930s, 1940s, and the immediate post–World War II boom years may
have exhibited an unusually high average realized return premiums. The 1930s
exhibited extreme volatility, while the 1940s and early 1950s saw a combination of
8
SBBI Valuation Edition 2009 Yearbook (Chicago: Morningstar, 2009), 61–63.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
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 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.9
The years 1942 through 1951 reflected a period of artificial stability in U.S. government bond interest rates. During World War II, the U.S. Treasury decreed that
interest rates had to be kept at artificially low levels in order to reduce government
financing costs. This led to the Federal Reserve’s April 1942 public commitment to
maintain an interest rate ceiling on government debt, both long term and short term.
After World War II, the Fed continued maintaining an interest rate ceiling, due to
the Treasury’s pressure and, to a lesser extent, a fear of returning to the high unemployment levels 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.
The following table displays the income returns on long-term U.S. government
bonds for the years 1942 through 1951 (the return used by Morningstar in calculating the realized risk premiums) versus inflation:
Year
Income Return
Rate of Inflation
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
2.46%
2.44%
2.46%
2.34%
2.04%
2.13%
2.40%
2.25%
2.12%
2.38%
9.29%
3.16%
2.11%
2.25%
18.16%
9.01%
2.71%
1.80%
5.79%
5.87%
Source: Compiled from data in Stocks, Bonds,
Bills, and Inflation 2009 Yearbook. Copyright #
2009 Morningstar, Inc. All rights reserved. Used
with permission. Derived based on CRSP1 data,
# 2009 Center for Research in Security Prices
(CRSP1), University of Chicago Booth School of
Business.
9
Jeremy Siegel, Stocks for the Long Run (New York: McGraw-Hill, 1994), 20.
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During these 10 years, long-term U.S. government income returns averaged
2.3%, while inflation averaged 5.66%, indicating that the realized risk premiums
calculated for these years was biased high compared with a more normal risk-free
rate benchmark. To better understand the effect of the interest rate accord on the
realized risk premiums, Grabowski recalculated the realized risk premiums for 1926
through 2008 after normalizing the income return on long-term U.S. government
bonds for the years 1942 through 1951 to an amount at least equal to the annual
rate of inflation as reported in the SBBI Yearbook (except 1949, when inflation was
1.8%). Making that adjustment lowered the realized risk premium from the published 6.5% to 6.0% for 1926–2008. One can interpret the results as the realized
risk premium data reported in the SBBI Yearbook is biased high by 50 basis points
(0.50%). We will term this the WWII Interest Rate Agreement bias.
Has the Relationship between Stock and Bond
Risk Changed?
If we disaggregate the 83 years reported in the SBBI Yearbook into two subperiods,
the first covering the periods before and the second covering the periods after the
mid-1950s, we get the comparative figures for stock and bond returns shown in
Exhibit 9.1.
The period since the mid-1950s has been characterized by a more stable stock
market and a more volatile bond market than the earlier period. Interest rates, as
reflected in long-term U.S. 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 U.S. 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 probably 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.10 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 returned to prior levels until the crisis of 2008–2009.11 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.12
10
Clifford S. Asness, ‘‘Stocks versus Bonds: Explaining the Equity Risk Premium,’’ Financial
Analysts Journal (March–April 2000): 96–113.
11
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: Equity Risk Premiums and the Privately-Held Business,’’
1998 CICBV/ASA Joint Business Valuation Conference (September 1998): 23.
12
The Duff & Phelps Risk Premium Report uses data on returns since 1963.
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EXHIBIT 9.1
ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
Realized Equity Risk Premiums over Long-Term U.S. Government
Bond Returns
Nominal (i.e., without inflation removed)
Realized Equity Risk Premium:
Arithmetic Average
Geometric Average
Standard Deviations:
Stock Market Annual Returns
Long-Term U.S. Government Bond Income Returns
Long-Term U.S. Government Bond Total Returns
Ratio of Equity to Bond Total Return Volatility
1926–1955
1956–2008
10.5%
7.5%
4.2%
2.7%
25.3%
0.5%
4.7%
5.4
17.6%
2.4%
11.0%
1.6
Source: Compiled from data in Stocks, Bonds, Bills and Inflation 2009 Yearbook. Copyright
# 2009 Morningstar, Inc. All rights reserved. Used with permission. Calculated (or Derived)
based on CRSP1 data, # 2009 Center for Research in Security Prices (CRSP1), University of
Chicago Booth School of Business. Compiled by Duff & Phelps LLC.
In addition, using historical data over such long periods 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. 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.13
If the average expected risk premium has changed through time, then averages of
realized risk premiums using the longest available data become questionable. A shorterrun 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 20-year 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 (and is subject to a large standard error).
While we can observe only 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
13
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 6, 2004. Available at http://ssrn.com/
abstract=541203.
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125
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 or early 1960s to today, we
will be including a complete interest rate cycle.14
Even if we use long-term observations, the volatility of annual stock returns will
be high. Assuming that the 83-year average gives an unbiased estimate, still a 95%
confidence interval for the unobserved true ERP spans a range of approximately
1.9% to 11.1%.15
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 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.
&
&
14
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).16 They believe that the historical realized premium exceeded the
expected premium because (1) the expected ERP in 1926 was above the longterm average, making 1926 a better-than-average starting point for the realized
returns, and (2) important nonrecurring developments were not anticipated by
investors (such as rising valuation multiples, survivor bias of the U.S. economy,
and regulatory reform).17
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
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: Equity Risk Premiums and the Privately-Held Business,’’
1998 CICBV/ASA Joint Business Valuation Conference (September 1998): 23.
15
Calculated as two standard errors around the average; 6.5% þ/ (2 2.3%).
16
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%.
17
Robert D. Arnott and Peter L. Bernstein, ‘‘What Risk Premium Is Normal?’’ Financial Analysts Journal (March–April 2002): 64–85.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
historical observed relationships from 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).18 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 biasadjusted 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.19
&
18
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 2008.20 These authors report that the realized risk premiums
have been 5.9% on an arithmetic basis (3.8% 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 than in the first half due to (1) corporate
cash flows growing faster than 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 downward adjustments to the realized risk premiums due to the increase in price-to-dividend ratio and downward adjustments
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).
19
Eugene F. Fama and Kenneth R. French, ‘‘The Equity Premium,’’ Journal of Finance (April
2002): 658.
20
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;
Credit Suisse Global Investment Returns Sourcebook 2009 (London: Credit Suisse/London
Business School, 2009). They expanded their study to 19 countries in the Credit Suisse
Global Investment Returns Sourcebook 2010.
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&
127
to the historic average dividend yield to today’s dividend yield to arrive at a forward ERP are reasonable. One can estimate a range of likely forward ERP estimates by removing the increase in price-to-dividend ratio (making that sole
adjustment results in the high end of the range) and adjusting dividend yield to
current levels (making both adjustments results in the low end of the range).
Assuming that the standard deviation of annual returns on equity will approximately equal the historical standard deviation, their analysis indicates an
estimate of the U.S. ERP in early 2009 of 3.6%–4.6% arithmetic average
(1.4%–2.6% on a geometric basis) versus long-term U.S. government bonds.21
Roger Ibbotson and Peng Chen report on a study in which they estimated
forward-looking long-term sustainable equity returns and expected ERPs since
1926. They first analyzed historical equity returns by decomposing returns
into factors including inflation, earnings, dividends, price-to-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 by removing the
price-to-earnings ratio inflation. These authors determine that the long-term
ERP that could have been expected, given the underlying economics, was less
than the realized premium.22 In the 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
5.7%, calculated on an arithmetic average basis (3.6% on a geometric average
basis), compared with the realized risk premium of 6.5%, calculated on an
arithmetic average basis (4.5% on a geometric average basis). The greater-thanexpected 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 0.58% per
annum (due to the price-to-earnings multiple in 1926 of 10.2 increasing to a
price to earnings multiple of 19.28 in 2008).
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.23
21
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.
22
Roger G. Ibbotson and Peng Chen, ‘‘Long-Run Stock Market Returns: Participating in the
Real Economy,’’ 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. Available at http://ssrn.com/abstract=906104.
23
William N. Goetzmann and Roger G. Ibbotson, ‘‘History and the Equity Risk Premium,’’
Chapter 12 in Rajnish Mehra, Handbook of the Equity Risk Premium (Amsterdam: Elsevier, 2008), 522–523.
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So one can interpret that a forward estimate of the long-term ERP derived
from data in the SBBI Yearbook should be 5.7% (supply side model on an arithmetic average basis) minus the 0.50% WWII Interest Rate bias discussed earlier
or 5.2% for one-year holding period returns. If one were to assume investors
have a longer holding period of, say, five years, then the forward estimate of
ERP could be reduced by another 1.2% (6.5% for one-year holding periods minus 5.3% for five-year holding periods as discussed in Appendix 9A), and we
arrive at a forward ERP estimate derived from realized returns as low as 4%. So
a reasonable range of forward ERP estimates derived from the supply side
model adjusted may be 4% to 5.2%.
Morningstar publishes realized return data such that the analyst can calculate the realized returns over any period beginning in 1926 or later and ending in
any period, allowing the analyst to estimate ERP using a sample period of one’s
own choosing. But Morningstar only publishes supply side ERP estimates for
periods beginning in 1926.24
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. 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.
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%.25 Similarly, their mean estimate of the
24
25
SBBI Valuation Edition 2009 Yearbook (Chicago: Morningstar, 2009), 69.
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.
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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%.
We asked Morningstar to provide us with their supply side ERP estimates for
the same period as used by Fama and French (1951–2000) to compare the results.
The supply side ERP estimates for 1951 to 2000 that could have been expected given
the underlying economics were approximately 4.7% on an arithmetic basis (3.3%
on a geometric basis) compared with the realized risk premium of 7.6% on an arithmetic basis (6.0% 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 long-term government bonds that could have been expected for the period he
studied, 1940 to 1997) was approximately 2.7%.26
McGrattan and Prescott find that the value of the stock market relative to the
GDP in 2000 was nearly twice as large as in 1962.27 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
generally not be held ‘‘tax deferred’’ in 1962. But by 2000, equity investment could
26
E. Scott Mayfield, ‘‘Estimating the Market Risk Premium,’’ Working paper, October 1999.
Available at http://ssrn.com/abstract=195569. 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 probably occurred in the early 1940s and appears to be driven by a reduction in
the general level and persistence of market volatility; and Lubos Pastor and Robert F. Stambaugh, ‘‘The Equity Premium and Structural Breaks,’’ Journal of Finance (August 2001):
1207–1239, who study the equity risk premium from 1834 through 1999 and find several
‘‘structural breaks’’ (changes in volatility) in 1929 (increase compared to historical), 1941
(returning to historical levels), and 1992 (further reduced volatility). They find that the ERP
compared to T-bills (or equivalent) fluctuated between 3.9% and 6.0% over the period
January 1834 through June 1999.
27
Ellen R. McGrattan and Edward C. Prescott, ‘‘Is the Market Overvalued?’’ Federal Reserve
Bank of Minneapolis Quarterly Review 24 (Fall 2000): 20–24; Ellen R. McGrattan and
Edward C. Prescott, ‘‘Taxes, Regulations and Asset Prices,’’ Federal Reserve Bank of Minneapolis Working paper, July 2001. Available at http://ssrn.com/abstract=292522.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
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.
OTHER SOURCES OF ERP ESTIMATES
Appendix 9A contains is a list of authors’ opinions and guidelines on ERP.
FORWARD-LOOKING (EX ANTE) APPROACHES
Forward-looking (ex ante) approaches can be categorized into three groups based on
the approach taken:
1. Bottom-up implied ERP estimates. This category of approach uses expected
growth in 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 investors’ expectations concerning the
overall market by using forecasts of the rate of return on publicly traded
companies.
2. Top-down implied ERP estimates. This category of approach examines the 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 return 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. Surveys. This approach relies on opinions of investors and financial professionals through surveys of their views on the prospects of the overall market and
the return expected in excess of a risk-free benchmark.
Bottom-up Implied ERP Estimates
This section presents implied estimates of ERP from four sources that use bottomup data.
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131
1. 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.
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.28
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.
The Merrill Lynch expected return estimates have indicated an implied ERP
ranging from 3.0% to 6.7% for the 15 years 1993 to 2007, with an average of approximately 5.2%. At the end of 2008, their implied ERP was approximately 9.2%
measured against an abnormally low long-term U.S. government bond rate (3.03%)
and 7.7% measured against a normalized long-term U.S. government bond rate
(4.5%). We discuss the issue of the abnormally low risk-free rates at the end of
2009 later.
A number of consulting firms reportedly have used Merrill Lynch implied
ERP estimates to develop discount rates. One author comments on the Merrill
Lynch data:
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.29
28
See, e.g., Sidney Cottle, Roger F. Murray, and Frank E. Block, Graham & Dodd’s Security
Analysis, 5th ed. (New York: McGraw-Hill, 1988), 565–568.
29
Eugene Brigham and Louis Gapenski, Financial Management: Theory and Practice, 5th ed.
(Fort Worth, TX: Dryden Press, 1988), 227.
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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.
2. 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 more than 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.30
Implied ERP estimates developed from Value Line data have been more volatile
than the Merrill Lynch DDM models. We believe that Value Line’s estimates of
future earnings and dividend growth 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 Value Line expected return estimates have indicated an implied ERP ranging from –1.1% to 12.3% for the 15 years 1993 to 2007, with an average of approximately 5.4%. The implied ERP was approximately 20.4% at the end of 2008
measured against an abnormally low long-term U.S. government bond rate
(3.03%); measured against a normalized long-term U.S. government bond rate
(4.5%), the implied ERP was approximately 18.9%. This result probably occurred
because of delay in updating estimates; the recession has been deeper than most
analysts expected.
3. 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 singlestage DDM and a three-stage DDM (with quarterly updates reported in their Cost of
Capital Quarterly).31 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% from 1994 (the year publication commenced) through the beginning of 2008, with an average of 6.5%. The implied
ERP was approximately 8.7% at the beginning of 2009, measured against an
abnormally low long-term U.S. government bond rate (3.55% as of March
2009, the date of the 2009 Cost of Capital Yearbook); measured against a normalized long-term U.S. government bond rate (4.5%), the implied ERP was
approximately 7.8%.
30
David T. Doran, ‘‘Forecasting Error of Value Line Weekly Forecasts,’’ Journal of Business
Forecasting (Winter 1993–94): 22–26.
31
See, e.g., Cost of Capital Yearbook 2009 (Chicago: Morningstar, 2009).
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Equity Risk Premium
EXHIBIT 9.2
Implied ERP Estimates—Bottom-up Approach
As of Early 2009 versus
Risk-Free Rate
Merrill Lynch
Value Line: 3- to
5-year horizon
Cost of Capital
Yearbook
Damodaran
Range
Period
Mean
Actual
Normalized(1)
3.0% to 6.7%
1.1% to 12.3%
1993–2007
1993–2007
5.2%
5.4%
9.2%
20.4%
7.7%
18.9%
4.9% to 8.0%
1994–2008
6.5%
8.7%
7.8%
1.5% to 4.0%
1993–2007
2.9%
5.6%
4.1%
Note: Converted to equivalent over a 20-year U.S. government bond yield.
Using risk-free rate adjusted because actual interest rates lower than warranted due to flight
to quality as discussed.
(1)
4. Professor Aswath Damodaran calculates implied ERP estimates for the S&P
500 and now publishes his estimates on his web site. He uses a two-stage model,
projecting expected distributions (dividends and stock buybacks) based on an average of analyst estimates for earnings growth for individual firms comprising the S&P
500 for the first five years and the risk-free rate thereafter (since 1985). He solves for
the discount rate, which equates the expected distributions to the current level of the
S&P 500. He benchmarks his implied ERP estimates against 10-year U.S. government bonds.
The Damodaran expected return estimates have indicated an implied ERP ranging from 1.5% to 4.0% for the 15 years 1993 to 2007 with an average of approximately 2.9% (converted to an equivalent premium over 20-year U.S. government
bonds). The expected premium was approximately 5.6% at the beginning of 2008
(converted to an equivalent premium over 20-year U.S. government bonds) measured
against an abnormally low long-term U.S. government bond rate (3.03%) or 4.1%
(converted to an equivalent premium over 20-year U.S. government bonds) measured
against a more normalized yield on long-term U.S. government bonds (4.50%).32
Exhibit 9.2 summarizes four implied ERP estimates published over the past several
years. 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.33 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 appears to be positively correlated with longterm bond yields (increasing as bond yields increase) and with the default premium (increasing as the differential between Aaa- and Baa-rated bond yields increases). Another
study extracted ex ante estimates of the ERP from the residual income model.34
32
Aswath Damodaran, available at http://pages.stern.nyu.edu/adamodar/.
Fabio Fornari, ‘‘The Size of the Equity Premium,’’ Working paper, January 2002. Available
at http://ssrn.com/abstract=299906.
34
Tristan Fitzgerald, Stephen Gray, Jason Hall, Ravi Jeyaraj, ‘‘Unconstrained Estimates of the
Equity Risk Premium,’’ Working paper, February 2010. Available at http://ssrn.com/
abstract=1551748.
33
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Are Bottom-up ERP Estimates Accurate? Studies have indicated that analysts’ earnings forecasts (such as those reported by I/B/E/S and First Call) are biased high.35
These biases lead to high implied estimates of ERP. It is also possible that the implied ERP estimates may overstate expected returns because analyst earnings and
cash flow forecasts are prone to error, with the error increasing for firms with high
volatility of earnings.36
Top-down ERP Estimates
We have already summarized the forward ERP estimates of Dimson and colleagues
and Ibbotson and Peng based on their adjusting realized risk premiums for growth in
price-to-dividend ratios or price-to-earnings ratios that are not expected to continue.
Stephen Hassett has developed a model for estimating the implied ERP and the
estimated S&P 500 based on the current yield on long-term U.S. government bonds
and the risk premium factor (RPF). The RPF is the empirically derived relationship
between the risk-free rate, S&P 500 earnings, real interest rates, and real GDP
growth to the S&P 500 over time. The RPF appears to change only infrequently.37
The model can be used monthly or even daily to estimate the S&P 500 and the conditional ERP based on the current level of interest rates.
The formula is as follows:
S&P 500 ¼ S&P Earnings=f½Rf ð1 þ RPFÞ
½ðRf Real Interest Rate þ Long-Term GDP growthÞg
where the implied ERP ¼ Rf (1þ RPF).
35
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.
36
See, e.g., Ilia D. Dichev and Vicki Wei Tang, ‘‘Earnings Volatility and Earnings Predictability,’’ Journal of Accounting and Economics (forthcoming); Dan Givoly, Carla Hayn,
and Reuven Lehavy, ‘‘The Quality of Analysts’ Cash Flow Forecasts,’’ Working paper,
December 2008. Available at http://ssrn.com/abstract=1130907.
37
Stephen D. Hassett, ‘‘The RPF Model for Calculating the Equity Risk Premium and
Explaining the Value of the S&P with Two Variables,’’ Journal of Applied Corporate
Finance 22, 2 (Spring 2010): 118–130.
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Equity Risk Premium
EXHIBIT 9.3
Implied ERP Estimates—Top-down Approach
As of Early 2009
versus Risk-Free Rate
Hassett
Range
Period
Mean
Actual
Normalized
3.4% to 6.5%
1993–2007
5.2%
7.5%
5.7%
Implied ERP given S&P 500 ¼ 903 and 10-year U.S. government bond rate ¼ 2.25%
Note: Converted to equivalent over a 20-year U.S. government bond yield
Exhibit 9.3 summarizes his top-down implied ERP estimates converted to an
equivalent premium over 20-year U.S. government bonds.
Hassett attributes the significant increase in price-to-earnings ratio for the market since the 1980s to the decline in the risk-free rate. This implies that a long-term
increase in the risk-free rate will cause an increase in the ERP and cause the price-to
earnings multiple for the market to contract.
ERP Surveys
John Graham and Campbell Harvey report the results from quarterly surveys of
chief financial officers of U.S. corporations conducted from mid-2000 to early
2009.38 The current survey attracted 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). Exhibit 9.4 summarizes the implied
ERP (converted to an equivalent premium over 20-year government bonds), with the
most recent survey concluding 3.3% converted to an equivalent premium over 20year government bonds, the highest since 2001.39
EXHIBIT 9.4
Implied ERP Estimates—Survey Results
As of Early 2009
versus Risk-Free Rate
Graham & Harvey
Range
Period
Mean
Actual
Normalized
1.9% to 4.6%
3Q2000–4Q2008
3.0%
4.7%
4.4%
Note: Converted to equivalent over a 20-year U.S. government bond yield.
38
John R. Graham and Campbell R. Harvey, ‘‘Expectations of Equity Risk Premia, Volatility
and Asymmetry from a Corporate Finance Perspective,’’ National Bureau of Economic Research Working paper, July 2003; John R. Graham and Campbell R. Harvey, ‘‘The Equity
Risk Premium amid a Global Financial Crisis,’’ Working paper, May 2009, updated quarterly by Duke CFO Outlook Survey (www.cfosurvey.org).
39
Graham and Harvey believe the results represent a geometric average expected return. Grabowski estimated the arithmetic average equivalent ¼ geometric average risk premium estimate þ (standard deviation of risk premium estimates)2/2. The survey question answered is
‘‘On February 16, 2009 the annual yield on 10-year treasury bonds was 2.9%. Over the
next 10 years, I expect the average annual S&P 500 return will be _%.’’
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LONG-TERM UNCONDITIONAL ERP ESTIMATE
Exhibit 9.5 summarizes the study results and data discussed previously. The evidence presented in most of the studies 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? In comparing implied ERP estimates to
realized risk premiums, one should compare the implied estimates to the geometric
average of realized risk premiums, remembering that the implied estimates are forward-looking and the realized risk premiums are historical.40 Therefore, for Exhibit
9.5, we have displayed under ‘‘Geometric Average’’ the mean of the implied ERP
estimates for the period, and we also converted a mean geometric average to an
equivalent arithmetic average.41
EXHIBIT 9.5 Long-Term ERP Estimates Measured Relative to Long-Term
U.S. Government Bonds
Realized Risk
Premiums
Period
20 years (1989–2008)
30 years (1979–2008)
40 years (1969–2008)
50 years (1959–2008)
83 years (1926–2008)(1)
109 years (1900–2008)
137 years (1872–2008)
211 years (1798–2008)
Adjusted Realized Risk
Premiums Average for Period
Arnott and Bernstein
Fama and French
SBBI Realized
SBBI Supply Side
Dimson et al.
Ibbotson and Chen
Supply Side
Adjusted for WWII
Interest Rate bias
40
Arithmetic
Average
Geometric
Average
4.1%
5.0%
3.2%
3.8%
6.5% or 6.0%(2)
6.3%
5.6%
4.9%
2.2%
3.4%
1.6%
2.4%
4.5%
4.3%
3.8%
3.3%
Period
Arithmetic
Average
Geometric
Average
1926–2001
1951–2000
1951–2000
1951–2000
1900–2008
1926–2008
4.5%
2.6%–3.6%
7.6%
4.7%
3.6%–4.6%
5.7%(1)
2.4%
1.2%–2.2%
6.0%
3.3%
1.4%–2.6%
3.6%(1)
1926–2008
5.2%(2)
The authors confirmed this interpretation with both Roger Ibbotson and Aswath
Damodaran.
41
In making that adjustment, we used the following estimated relationship: arithmetic average equivalent ¼ geometric average risk premium estimate þ (standard deviation of risk
premium estimates)2/2. We used the standard deviation of realized risk premiums for the
50 years 1959–2008 of approximately 17% to arrive at an estimate of 1.4% to add to the
geometric averages to estimate the arithmetic average equivalents.
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Equity Risk Premium
Implied Risk Premiums
Bottom-up Estimates
Merrill Lynch
Value Line
Cost of Capital Yearbook
Damodaran
Top-down
Estimates
Hassett
Survey Estimates
Graham and Harvey
Period
1993–2008
1993–2008
1994–2008
1993–2008
Period
1993–2008
Period
2000–2009
Arithmetic
Average
Equivalent(4)
Implied ERP
Average(3)
6.8%
7.6%
7.9%
4.4%
5.4%
6.2%
6.5%
3.0%
Arithmetic
Average
Equivalent (4)
Implied ERP
Average(3)
6.6%
5.2%
Arithmetic
Average
Equivalent(4)
Implied ERP
Average(3)
4.30%
2.90%
(1)
SBBI Valuation Edition 2009 Yearbook.
Realized risk premiums adjusted to 6.0% by normalizing1942–1951 interest rates to correct for WWII interest rate bias; see discussion in Chapter 9.
(3)
Mean for period; implied ERP as of early 2009 measured against a normalized long-term
U.S. government bond rate.
(4)
Implied ERP estimates are equivalent to geometric averages of realized risk premiums. For
comparison purposes, Grabowski converted mean implied ERP for period to its arithmetic
average equivalent based on standard deviation of realized risk premiums over prior 50-year
period.
(2)
Based on the studies and the data presented, we conclude that a reasonable longterm estimate of the average or unconditional ERP is 3.5% to 6.0%.
CONDITIONAL ESTIMATE OF ERP AND THE CRISIS OF
2008–2010
Beginning in September 2008, the stock market and the economy started to tumble
into crisis. Where in this range is the current ERP, given the crisis of 2008–2010?
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 (as reflected in recent relatively
low returns on stocks), the conditional ERP is at the higher end of the range (e.g., at
December 31, 2008). 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.
If one views pricing of the stock market over the long term, one can see in
Exhibit 9.6 the long-term versus the short-term relationships. In scenario A, we see
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Scenario A: Conditional ERP Estimate at Peak of Stock Market Cycle
Returns of Large Company Stocks
08/09/2010
Time
Scenario B: Conditional ERP Estimate at Trough of Stock Market Cycle
Returns of Large Company Stocks
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Time
EXHIBIT 9.6
Relationship of Conditional ERP to Long-Term ERP
the long-term trend in the returns in large company stocks. This is equivalent to the
long-term ERP estimate over time. We all know that the stock market goes through
cycles. Stocks get bid up at times faster than the long-term average. In scenario A, we
see a depiction of one of those upward cycles when the returns increase faster than
the long-term average (‘‘above average’’). Assume we are estimating the conditional
ERP at the valuation date (indicated by the vertical line). The conditional ERP will
be less than the average for some time in order for the average over the long run to
return to the average (that is, because it was above the average for a period, it will be
below average to get back to the average).These above-average returns occurred
during the tech boom; assume our valuation date was at the peak of the tech boom,
and the conditional ERP at that point would be less than the average.
Similarly, in scenario B, we see a decline from the long-term average (e.g., last
half of 2008). Assume we are estimating the conditional ERP at the valuation date
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139
(indicated by the vertical line). The conditional ERP will be greater than the average
for some time in order for the average over the long run to return to the average
(that is, because it was below the average for a period, i.e., losses during 2008, it
will be above average to get back to the average).
As the stock market declined and the risk to the economy increased, implied
ERP estimates increased while realized risk premiums decreased. If one were estimating cost of equity capital using a method just like ‘‘normal times’’ (e.g., using
realized risk premiums), the estimate would be flawed.
The crisis of 2008–2010 and the resulting recession were (and at the writing of
this book still are) not ordinary times. If one simply added an estimate of the ERP
taken from commonly used sources used during normal economic times to the spot
yield on 20-year U.S. government bonds on December 31, 2008, one would have
arrived at an estimate of the cost of equity capital that was too low. As of December 2007, for example, the yield on 20-year U.S. government bonds equaled 4.5%,
and the Morningstar realized risk premium for 1926–2007 was 7.1%. But at December 2008, the yield on 20-year U.S. government bonds was 3.0%, and the Morningstar realized risk premium for 1926–2008 was 6.5%. So just at the time that
the risk in the economy increased to maybe the highest point, the base cost of
equity capital using realized risk premiums decreased from 11.6% (4.5% plus
7.1%) to 9.5% (3.0% plus 6.5%).
Let us relate this relationship to observations of implied volatilities of the stock
market and bond market, interest rates, and implied ERP estimates. For our comparison, we will use data on implied volatility on options for the S&P 500 and U.S.
government bonds and interest rates on constant maturity 20-year U.S. government
bonds (Exhibit 9.7).
Implied volatility is the market’s best guess of the future volatility over the term
of the option. When the crisis began to unfold (September 15, 2008, with Lehman
Brothers filing for bankruptcy), the stock market moved down, and fear enveloped
the financial markets. We can see that the monthly implied volatilities increased in
the S&P 500 and long-term U.S. government bond options, peaking in the October–
December 2008 period. At the same time, though, the interest rates on U.S. government bills and bonds declined to levels below those justified by the real rate of interest plus expected rates of inflation. This increased volatility in the expected interest
rates implies that the market questioned whether such low interest rates were sustainable. Exhibit 9.7 also displays the interest rates month to month.
As of January 1, 2008, for example, Damodaran’s implied ERP estimate was
approximately 4.37% and held rather steady through September 1, 2008, when the
implied ERP was approximately 4.3% and long-term interest rates were normal
(yields not driven down by the flight to quality).42
So anyone estimating the implied ERP at the end of December 2008 had to deal
with both the declining stock market (function of increased risk evidenced by the
increasing volatility of the S&P 500 options) and the declining long-term U.S. government interest rates. The question is: Do you measure the implied ERP against the
actual interest rates or against normalized interest rates? If one estimates the ERP
42
Aswath Damodaran, ‘‘Equity Risk Premiums: Determinants, Estimation and Implications—A Post-Crisis Update,’’ Stern School of Business Working paper, October 2009.
Available at http://ssrn.com/abstract=1492717.
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EXHIBIT 9.7 Implied Volatilities for Options on S&P 500 and Options on U.S. Government
Bonds and Interest Rates on Constant Maturity 20-Year U.S. Government Bonds
S&P 500 ETF
Implied Volatility
As of:
12/31/2005
12/31/2006
12/31/2007
1/31/2008
2/29/2008
3/31/2008
4/30/2008
5/31/2008
6/30/2008
7/31/2008
8/31/2008
9/30/2008
10/31/2008
11/30/2008
12/31/2008
1/31/2009
2/28/2009
3/31/2009
4/30/2009
5/29/2009
6/30/2009
7/31/2009
8/31/2009
9/30/2009
iShares Lehman 20+
Year Treasury Bond
Implied Volatility
30 Day(1)
3 Month(2)
30 Day(1)
3 Month(2)
20-Year Treasury
Bond Rate
10.765
10.255
21.525
26.121
24.581
25.037
19.403
15.929
22.804
22.058
19.111
39.166
52.078
51.756
36.267
39.630
40.919
39.529
33.320
26.759
23.937
22.761
22.698
22.628
12.655
11.023
22.604
23.983
24.925
24.590
19.977
18.885
22.508
21.838
21.246
31.297
46.356
48.393
37.567
38.683
39.475
39.385
33.163
28.109
25.276
24.480
25.424
23.015
8.700
7.490
14.952
17.578
17.807
16.846
12.954
13.081
11.516
11.085
10.759
18.686
16.809
28.837
31.332
26.101
25.140
17.989
19.808
22.022
18.966
16.897
16.109
15.859
9.239
8.079
14.356
16.294
17.305
17.239
13.341
14.165
12.966
12.316
12.133
16.118
18.464
31.087
31.213
25.258
25.410
19.401
19.875
21.802
19.452
17.803
17.259
16.793
4.6
4.9
4.5
4.4
4.4
4.3
4.6
4.8
4.6
4.7
4.5
4.4
4.8
3.7
3.0
3.9
4.0
3.6
4.1
4.3
4.3
4.3
4.2
4.0
(1)
(2)
30-day implied volatility.
3-month implied volatility.
Sources: Bloomberg and SBBI Valuation Edition 2009 Yearbook. Compiled by Duff &
Phelps LLC. Used with permission. All rights reserved.
against the actual interest rates, the conditional ERP will be greater simply by the
fact that that interest rates have declined. Comparing implied ERP estimates over
time and comparing implied ERP estimates with realized risk premiums becomes
difficult, as most prior periods did not have interest rates so dramatically influenced
by the flight to quality.
Exhibit 9.8 displays implied ERP estimates against the actual benchmark 20-year
U.S. government bond yield and against a normalized yield (adjusting the yields for
December 2008 through March 2009) based on both Merrill Lynch and Damodaran
bottom-up implied ERP estimates (converted to an equivalent premium over 20-year
U.S. government bonds) month to month for December 2008 through September
2009. Some would argue that the long-term U.S. government bond rate was still too
low after March 2009 due to a continuing flight to quality, as discussed in Chapter 7.
3.03%
3.94%
4.01%
3.55%
4.10%
4.32%
4.29%
4.30%
4.15%
4.03%
Bond Yield
Actual
4.50%
4.50%
4.50%
4.50%
4.10%
4.32%
4.29%
4.30%
4.15%
4.03%
Bond Yield
Normalized
9.17%
8.45%
9.18%
9.44%
8.49%
8.07%
8.10%
7.69%
7.83%
7.75%
Merrill Lynch ERP
Actual
7.70%
7.89%
8.69%
8.49%
8.49%
8.07%
8.10%
7.69%
7.83%
7.75%
Merrill Lynch ERP
Normalized
Source: Quantitative Profiles and www.damodaran.com and Duff & Phelps calculations.
Long-term government bond rate normalized at 4.5% for December 2008 through March 2009.
903.25
825.88
735.09
797.87
872.81
919.14
919.32
987.48
1020.62
1057.08
Dec 31, 2008
Jan 31, 2009
Feb 28, 2009
March 31, 2009
April 30, 2009
May 31, 2009
June 30, 2009
July 31, 2009
Aug 31, 2009
Sept 30, 2009
S&P 500
Estimate as of
5.61%
5.80%
6.69%
6.17%
5.38%
5.09%
5.10%
4.68%
4.55%
4.13%
Damodaran ERP
Actual
4.14%
5.24%
6.20%
5.22%
5.38%
5.09%
5.10%
4.68%
4.55%
4.13%
Damodaran ERP
Normalized
Implied ERP Estimates Benchmarked against Actual and Normalized 20-Year U.S. Government (constant maturity) Bond Yields
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EXHIBIT 9.8
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EXHIBIT 9.9
ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
Implied ERP Estimates as of December 2008 and September 2009
December 2008 Implied ERP versus
20-Year Risk-free (normalized) Rate
Merrill Lynch
Damodaran
Hassett
September 2009 Implied ERP versus
20-Year Risk-free (actual) Rate
As Estimated
Arithmetic Average
Equivalent
As Estimated
Arithmetic Average
Equivalent
7.70%
4.14%
5.70%
9.10%
5.54%
7.10%
7.75%
4.13%
4.20%
9.20%
5.50%
5.60%
Based on normalized long-term bond rate.
Hassett’s top-down RPF model did not explain the S&P 500 and ERP well as of
December 31, 2008, due to the abnormally low long-term U.S. government bond
rate. The model’s estimate of the S&P 500 approximated the actual S&P 500 and a
more reasonable ERP if one substituted a normalized long-term U.S. government
bond rate at that date.
The S&P 500 closed the quarter ending September 30, 2009, at 1,057. Hassett’s top-down RPF model predicted the S&P to be at 903 (5% below actual).43 Hassett’s implied ERP at September 30, 2009, equaled 4.2% against the
yield on 20-year U.S. government bonds.44 The model implies that the S&P is
fairly valued as of September 30, 2009, and the ERP estimate is consistent with
the level of the S&P 500 and risk-free interest rates. He also did not find any
significant increase in the ERP relative to a normalized risk-free rate during the
2008–2009 crisis.
Given that the implied ERP estimates are comparable to the geometric average of realized risk premiums, we convert the implied ERP estimates as of September 2009 to their arithmetic average equivalent and summarize those results
in Exhibit 9.9.45
43
These estimates were based on the following assumptions:
S&P Earnings ¼ 39.10 based on S&P’s estimate for the four quarters ending September 30,
2009
Rf ¼ 3.31% based on closing yields on 10-year U.S. government bonds
RPF ¼ 1.48
Real interest rate ¼ 2.0% based on average yields on 10-year U.S. government inflationprotected bonds (TIPS)
Long-term expected real growth in GDP ¼ 2.6%
Applying the formula we get: 903 ¼ 39.10/{.0331 (1 þ 1.48) [.0331 .02 þ .026]}.
44
RPF ¼ 1.48 relative to 10-year U.S. government bonds (September 30, 2009, yield ¼
3.31%) and yield difference between 10-year and 20-year U.S. government bonds ¼ .72%
(September 30, 2009, yield ¼ 4.03%). Implied ERP ¼ (1.48 3.31%) .72% ¼ 4.2%
(rounded).
45
In making that adjustment, we used the following estimated relationship: arithmetic average equivalent ¼ geometric average risk premium estimate þ (standard deviation of risk
premium estimates)2/2. We used the standard deviation of realized risk premiums for the
fifty years 1959–2008 of approximately 17% to arrive at an estimate of 1.4% to add to the
geometric averages to estimate the arithmetic average equivalents.
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Equity Risk Premium
143
S&P 500 Index Jan 1953 – September 30, 2009
8
7
Log of Index Close
6
5
4
3
2
1
2/
1
1/ 9 53
2/
1
1/ 956
2/
19
1/ 59
2/
1
1/ 9 62
2/
1
1/ 965
2/
19
1/ 68
2/
1
1/ 971
2/
19
1/ 74
2/
1
1/ 9 77
2/
1
1/ 9 80
2/
1
1/ 9 8 3
2/
19
1/ 86
2/
1
1/ 9 89
2/
1
1/ 9 92
2/
1
1/ 995
2/
1
1/ 998
2/
20
1/ 01
2/
2
1/ 004
2/
20
07
0
1/
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EXHIBIT 9.10
S&P 500 Index from January 1953 to September 30, 2009
As we are writing this book, it appears that we are nearing a more normalized
long-term S&P 500 index level and an ERP more in the middle of the long-term
range. See Exhibit 9.10.
But there is some evidence that the level of the S&P 500 is being driven by a continued below-normal level of risk-free interest rates. 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, Goyal and 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.’’46
Others have disputed their results, finding that predictive power is small but
economically meaningful, or that their results are really the result of poor predictability of, say, dividend growth.47 But research suggests that only models allowing
explicitly for time-varying factors succeed in maintaining their predictive power
across periods of time.48
46
Amit Goyal and Ivo Welch, ‘‘A Comprehensive Look at the Empirical Performance of
Equity Premium Prediction,’’ Working paper, January 11, 2006. Available at http://ssrn
.com/abstract=517667.
47
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). Available at http://ssrn.com/abstract=770953. John H. Cochrane, ‘‘The Dog That
Did Not Bark: A Defense of Return Predictability,’’ Working paper, January 30, 2006.
Available at http://ssrn.com/abstract=1212064.
48
Thomas Dangl, Michael Halling, and Otto Randl, ‘‘Equity Return Prediction: Are Coefficients
Time Varying?’’ Working paper, April 2006. Available at http://ssrn.com/abstract=887780.
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As the focus of this book is valuation of businesses and investments by businesses, the conditional ERP will generally be of less importance over time, and once
the worst of the crisis is behind us, 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 normal ERP is in a range that is consistent with the
principle that investors’ 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.49
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 the unconditional or long-range ERP? While
giving consideration to the 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 normal ERP estimate that is too high. In fact, the example of the decline in
the ERP estimate from December 2007 to December 2008, if one mechanically
applies these data, results in a nonsensical estimate of the cost of equity capital as of
December 31, 2008.
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 wellreasoned opinion of value. Estimating the ERP is no different.
After considering the evidence, a reasonable long-term estimate of the average
or unconditional ERP should be in the range of 3.5% to 6%. This estimate is consistent with the SBBI Yearbook supply side ERP estimate (5.7%) minus the WWII
Interest Rate bias (due to the interest rate accord from 1942 through 1951) or 5.2%.
49
Pablo Fernandez, ‘‘Equity Premium: Historical, Expected, Required and Implied,’’ Working paper, February 18, 2007: 28. Available at http://ssrn.com/abstract=933070.
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Equity Risk Premium
145
But for the conditional ERP as of December 31, 2008 (the date we are using for
most of our examples), we have concluded that given the risks in the economy as of
December 31, 2008, that the conditional ERP should be at the high end of the longterm range relative to normalized long-term U.S. government bond yields. Therefore, we are using 4.5% as the normalized long-term U.S. government bond yield
and an ERP of 6% in the examples. Even as of September 30, 2009, the conditional
ERP still is in the upper end of the long-term range, say 5% to 6%.
While we present data and calculations elsewhere in this book using data
through the end of 2008 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 as up-to-date as possible as we were preparing the text.
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APPENDIX
9A
Realized Risk Premium Approach and
Other Sources of ERP Estimates
Realized Risk Premium (ex Post) Approach
Measuring Realized Risk Premiums
Realized Historical Stock and Bond Returns
Summarizing Realized Risk Premium Data
What Periodicity of Past Measurement?
Is Bias Introduced by Using the Arithmetic Average in Estimating the ERP?
Bias in Compounding
Bias in Discounting
Other Sources of ERP Estimates
REALIZED RISK PREMIUM (EX POST) APPROACH
Here we discuss in detail the following issues in applying the realized risk premium
approach:
&
&
&
&
&
Which risk-free rate should be used to measure the realized premiums?
Which period should be used as the sample period?
Is the arithmetic average or geometric average the more accurate method of
summarizing realized return data over the sample period?
Should returns be measured over one-year holding periods or over longer holding periods?
Is bias introduced by using the arithmetic average of realized risk premiums?
Measuring Realized Risk Premiums
The measure of the risk-free rate has generally not been controversial once the
proper duration (long- term versus short-term) of the investment has been estimated,
since the expected yield to maturity on appropriate U.S. government securities is directly observable in the marketplace. However, the normal relationship fell apart
during the crisis of 2008–2009, as investors sold risky assets and moved funds to
The authors want to thank David Turney of Duff & Phelps LLC. for preparing materials for
this appendix.
146
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Realized Risk Premium Approach and Other Sources of ERP Estimates
147
what they perceived to be risk-free assets, U.S. obligations. We discussed this phenomenon in Chapter 7. Consequently, choice of the benchmark risk-free rate in
2008 and 2009 does affect the observed risk premium in those years. The difference
between realized returns on stocks for 2008 and 2009 compared with the actual average yields on U.S. government bonds will result in a realized premium that is biased high. That is, the difference will be greater than it would be were yields on U.S.
government bonds not unusually low.
Differences in one’s approach to estimating the ERP hinge even more 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 historical average.
Realized 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 U.S. stocks and bonds derived from that data.50
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. Reasonably good stock market return data are
available back to 1872, and less reliable data are available 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.51) Data for government bond yield data have also been assembled for
50
Stocks, Bonds, Bills and Inflation (SBBI) Valuation Edition 2009 Yearbook (Chicago: Morningstar, 2009).
51
See Lawrence Fisher and James Lorie, ‘‘Rates of Return on Investments in Common
Stocks,’’ Journal of Business 37(1) (1964); C. P. Jones and J. W. Wilson, ‘‘A Comparison of
Annual Stock Market Returns: 1871–1925 with 1926–1985,’’ Journal of Business 60(2)
(1987): 239–258; G. W. Schwert, ‘‘Indexes of Common Stock Returns from 1802 to
1987,’’ Journal of Business 63(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); C. P. Jones and J. W. Wilson, ‘‘An Analysis of the S&P 500 Index
and Cowles’s Extensions: Price Indexes and Stock Returns, 1870–1999,’’ Journal of Business 75(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.
Available at http://ssrn.com/abstract=298039. 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 available in Credit Suisse Global Investment
Returns Sourcebook (London: Credit Suisse/London Business School); W. Goetzmann and
R. Ibbotson, ‘‘History and the Equity Risk Premium,’’ Chapter 12 in Rajnish Mehra, Handbook of the Equity Risk Premium (Amsterdam: Elsevier, 2008), 522–523.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
EXHIBIT 9A.1 Realized Equity Risk Premiums: Stock Market Returns Minus U.S.
Government Bonds
Period
20 years (1989–2008)
30 years (1979–2008)
40 years (1969–2008)
50 years (1959–2008)
83 years (1926–2008)(2)
109 years (1900–2008)
137 years (1872–2008)
211 years (1798–2008)
Arithmetic Average
Standard Error(1)
Geometric Average
4.1%
5.0%
3.2%
3.8%
6.5%(3)
6.3%
5.6%
4.9%
4.4%
3.1%
2.8%
2.4%
2.3%
1.9%
1.6%
1.2%
2.2%
3.4%
1.6%
2.4%
4.5%
4.3%
3.8%
3.3%
(1)
Calculated as standard deviation of realized excess returns divided by square root of N,
number of years in sample.
(2)
SBBI Valuation Edition 2009 Yearbook.
(3)
Adjusted to 6.0% if 1942–1951 interest rates are normalized to correct for WWII Interest
Rate bias; see discussion in Chapter 9.
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 63 (July 1990): 399–426; S. Homer and R. Sylla, A History of Interest Rates, 3rd
ed. (Piscataway, NJ: Rutgers University Press, 1991); and SBBI, 2009 Yearbook (Chicago:
Morningstar, 2009). Compiled by Duff & Phelps LLC. Used with permission. All rights
reserved.
these periods. Exhibit 9A.1 presents the realized average annual risk premium for
stocks assembled from various sources for alternative periods through 2008.
We measure the realized risk premium by comparing the stock market returns
realized during the period to the income return on long-term U.S. government bonds
(or yield to maturity for the years before 1926).
While some may question the relevance of averages including early periods for
estimating today’s ERP, what is striking is that the largest arithmetic average of oneyear returns is the 83 years from 1926 to 2008.
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. Investors make 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 U.S. government bond). While the investors did
not know the stock market return when they invested at the beginning of each year,
they did know the rate of interest promised on long-term U.S. government bonds
when they were first issued. 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.
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 9A.1. The standard error of estimate
allows you to measure the likely accuracy of using the realized risk premium as the
estimate of the true ERP. That statistic indicates the estimated range within which
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Realized Risk Premium Approach and Other Sources of ERP Estimates
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 9A.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 periodic returns equals ð100% þ 50%Þ=2 ¼ 25%,
and the geometric average equals ð1 þ r1 Þð1 þ r2 Þ1=2 1 ¼ ð1 þ 1:00 1 :5Þ1=2
1 ¼ 0.
Realized risk 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. 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.
The choice between which average to use is a matter of disagreement among
practitioners. The arithmetic average receives the most support in the literature,52
though some authors recommend a geometric average.53 The use of the arithmetic
average relies on the assumptions that (1) market returns are serially independent
(not correlated) and (2) the distribution of market returns is stable (not timevarying). 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 the resulting arithmetic average.
52
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 (Chicago: Irwin Professional Publishing, 1989), 720–723.
53
See, e.g., Aswath Damodaran, Investment Valuation: Tools and Techniques for Determining the Value of Any Asset, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2002),
161–162.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
. . . 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.54
. . . 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.55
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, one needs 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 oneyear 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
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 one values
businesses, should one 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
54
Roger Ibbotson and Rex Sinquefeld, Stocks, Bonds, Bills and Inflation: Historical Returns
(1926–1987) (Chicago: Irwin Professional Publishing, 1989), 127.
55
Willard T. Carleton and Josef Lakonishok, ‘‘Risk and Returns on Equity: The Use and Misuse of Historical Estimates,’’ Financial Analysts Journal 41(1) (January–February 1985):
39.
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Realized Risk Premium Approach and Other Sources of ERP Estimates
EXHIBIT 9A.2
Realized Risk Premiums over Varying Holding Periods
Arithmetic Average of
Realized Risk Premium
1-year returns1
2-year returns2
3-year returns3
4-year returns4
5-year returns5
83-year returns (geometric average)1
6.5%
6.0%
5.8%
5.5%
5.3%
4.5%
1
SBBI Valuation Edition 2009 Yearbook.
Excluding investment period beginning 2008.
3
Excluding investment periods beginning 2007 and 2008.
4
Excluding investment periods beginning 2006, 2007, and 2008.
5
Excluding investment periods beginning 2005, 2006, 2007, and 2008.
2
Source: Compiled from data in Stocks, Bonds, Bills, and Inflation 2009 Yearbook. Copyright
2009 Morningstar, Inc. Compiled by Duff & Phelps LLC. All rights reserved. Used with
permission.
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 one 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 2008 as shown in Exhibit 9A.2.56
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 probably between the arithmetic average of one-year returns and the geometric average of the entire series.
In one recent study, the authors showed that compounding the arithmetic average of historical one-year returns as a forecaster of cumulative future returns resulted in estimates of cumulative returns that overstated 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 showed that use of the geometric
mean of historical one-year returns resulted in estimates of cumulative returns that
56
The realized risk premium of each investment horizon was calculated by taking equity returns (S&P 500) minus the bond returns (long-term U.S. 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 two-year
return, for equities and bonds, is the arithmetic average of a series of two-year rolling returns from 1926 to 2008. We performed the same calculation for each investment horizon.
We then subtracted the bond return from the equity return to estimate the equity risk premium for each investment horizon.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
better approximate the median of cumulative returns (50% of investors will realize
more than the median cumulative return, and 50% will realize less than the median
return). They demonstrated 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 increased as the expected investment horizon increased.57
A number of academic studies have suggested that U.S. stock returns are not
serially independent but rather have exhibited negative serial correlation.58 One recent study suggested that if stock returns have negative serial correlation, then the
best estimate of expected returns lies 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.59 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.60
While using the arithmetic average of realized risk premiums as an estimate of
the ERP in compounding (i.e., estimating future cumulative wealth) will likely result
in an estimate that is biased high, using the arithmetic average of realized risk premiums as an estimate of the ERP in discounting does not appear to introduce serious
bias. This is explained in detail in the next sections.
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)?
57
Eric Hughson, Michael Stutzer, and Chris Yung, ‘‘The Misuse of Expected Returns,’’
Financial Analysts Journal (November–December 2006): 88–96.
58
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(1)
(Spring 1988): 41–46; James Poterba and Lawrence Summers, ‘‘Mean Reversion in Stock
Prices: Evidence and Implications,’’ Journal of Financial Economics (October 1988):
27–59.
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.
60
Carmelo Giaccotto, ‘‘Discounting Mean Reverting Cash Flows with the Capital Asset Pricing Model,’’ Financial Review (May 2007): 247–265.
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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?
Bias in Compounding
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 where 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 bias grows larger as the investment horizon increases.61
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 bias is due to
measurement error introduced simply due to the mathematics. 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 you might
expect to realize when investing funds for future retirement. This is the subject of
much discussion in the pension investment literature.
In predicting the compound return derived from the arithmetic average of realized risk premium data, you will get an answer that is biased due to measurement
error introduced simply due to the mathematics as follows. 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. However, 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
61
Eric Hughson, Michael Stutzer, and Chris Yung, ‘‘The Misuse of Expected Returns,’’
Financial Analyst Journal (November–December 2006): 88–96.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
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.62
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 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.63 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 (e.g., stock returns, earnings before interest, taxes, depreciation, and amortization [EBITDA]) 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.64
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
62
Marshall 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. Available at http://
ssrn.com/abstract=353242.
63
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. Available at http://ssrn.com/abstract=353242.
64
Pablo Fernandez, ‘‘80 Common Errors in Company Valuation,’’ Working paper, May 12,
2004: 12. Available at http://ssrn.com/abstract=545546.
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Realized Risk Premium Approach and Other Sources of ERP Estimates
155
discount factor.65 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.66 Moreover, the bias is toward discount rates that are too
low rather than too high. 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.
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 realized risk premiums can be used as one estimate of
the ERP in discounting without introducing significant mathematical bias.
OTHER SOURCES OF ERP ESTIMATES
The following is a list of published opinions and guidelines on the ERP. These are
not the only sources but represent a cross section of opinion on the subject.
&
65
Principles of Corporate Finance, 9th 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 long-term
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 is 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).
66
Ian Cooper, ‘‘Arithmetic versus Geometric Mean Estimators: Setting Discount Rates for
Capital Budgeting,’’ European Financial Management (July 2001): 157–167.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
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.’’67
Valuation: Measuring and Managing the Value of Companies, 4th ed., recommends an ERP of 4.5% to 5.5%.68 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 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 shortterm bonds, use only 75 years of data, and are biased by the historical
strength of the U.S. market.69
&
&
67
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 10-year
U.S. government bonds), while the average implied (forward-looking approach
using expected dividends and expected dividend growth) ERP is only about 4%
as of January 2006 (premium over 10-year U.S. government bonds).70 The author notes that the average implied ERP has been about 4% over the past 40
years.71 He uses 4% in most of his valuation examples. He updated his estimate
of the ERP to a range of 5%–6% as of September 30, 2009.72
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.’’73
Richard Brealey, Stuart Myers, and Franklin Allen, Principles of Corporate Finance, 9th ed.
(Boston: Irwin McGraw-Hill, 2008), 180.
68
Tim Koller, Marc Goedhart, and David Wessels, Valuation: Measuring and Managing the
Value of Companies, 4th ed. (Hoboken, NJ: John Wiley & Sons, 2005), 305–306.
69
Tim Koller, Marc Goedhart, and David Wessels, Valuation: Measuring and Managing the
Value of Companies, 4th ed. (Hoboken, NJ: John Wiley & Sons, 2005), 306.
70
Aswath Damodaran, Damodaran on Valuation: Security Analysis for Investment and Corporate Finance, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2006), 41, 48.
71
Aswath Damodaran, Damodaran on Valuation: Security Analysis for Investment and Corporate Finance, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2006), 47.
72
Aswath Damodaran, ‘‘Equity Risk Premiums (ERP): Determinants, Estimation and Implications—A Post-Crisis Update,’’ Stern School of Business Working paper, October 2009,
67. Available at http://ssrn.com/abstract=1492717. see also, ‘‘Equity Risk Premiums
(ERP): Determinants, Estimation, and Implications—The 2010 Edition,’’ Stern School of
Business Working paper, February, 2010. Available at http://ssrn.com/abstract=1556382.
73
Bradford Cornell, Equity Risk Premium: The Long-Run Future of the Stock Market (New
York: John Wiley & Sons, 1999), 201.
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Realized Risk Premium Approach and Other Sources of ERP Estimates
&
157
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%.74
&
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&
&
&
&
74
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.75 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;76
this translates to premiums of approximately 4.5% to 7% in 1978 and 4% to
6% in 1983 over long-term 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.’’77 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.’’78
The Quest for Value recommends a 6% premium based on a long-run geometric
average difference between the total returns on stocks and bonds.79
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.80
‘‘Equity Premium: Historical, Expected, Required and Implied’’ recommends
that ‘‘an additional 4% (over government bonds) compensates the additional
risk of a diversified portfolio.’’81
‘‘Market Risk Premium Used in 2008: A Survey of More than 1,000 Professors’’
reports on a survey of more than 1,000 college professors in the United States,
Alfred Rappaport, Creating Shareholder Value, revised ed. (New York: Free Press, 1997),
39.
75
See Sidney Cottle, Roger F. Murray, and Frank E. Block, Graham & Dodd’s Security Analysis, 5th ed. (New York: McGraw–Hill, 1988), 573.
76
Sidney Cottle, Roger F. Murray, and Frank E. Block, Graham & Dodd’s Security Analysis,
5th ed. (New York: McGraw–Hill, 1988), 83–85.
77
Jeremy J. Siegel, Stocks for the Long Run (New York: McGraw-Hill, 1994), 20.
78
Jeremy J. Siegel, ‘‘Perspectives on the Equity Risk Premium,’’ Financial Analysts Journal
(November–December 2005): 61–73. Grabowski converted Siegel’s conclusion in terms of
geometric average return (p. 70) compared to government bonds.
79
G. Bennett Stewart, The Quest for Value (New York: HarperCollins, 1991), 436–438.
80
Stephen H. Penman, Financial Statement Analysis and Security Valuation, 3rd ed. (New
York: McGraw-Hill, 2007), 476.
81
Pablo Fernandez, ‘‘Equity Premium: Historical, Expected, Required and Implied,’’ Working paper, February 18, 2007, 28. Available at http://ssrn.com/abstract=933070.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
Europe, Canada, the United Kingdom, and Australia.82 The average ERP used
by 351 college professors in the United States was 6.5% in 2008.
‘‘The Equity Premium in 150 Textbooks’’ finds a wide variety of ERP estimates.
The five-year moving average observed in the various textbooks has declined
from an average of 8.4% in 1990 to 5.7% in 2008 and 2009.83
‘‘Market Risk Premium Used in 2010 by Professors: A Survey with 1,500
Answers’’ reports that the average (and median) ERP estimate in the United
States at the beginning of 2010 equaled 6.0%.84
‘‘Market Risk Premium Used in 2010 by Analysts and Companies: A Survey
with 2,400 Answers’’ reports that the average ERP estimate used by analysts in
the United States and Canada at the beginning of 2010 was 5.1% (median equal
to 5.0%) while the average ERP estimate used by companies in the United States
was 5.3% (median equal to 5.0%) at the beginning of 2010.85
Pablo Fernandez, ‘‘Market Risk Premium Used in 2008: A Survey of More Than 1,000 Professors,’’ Working paper, February 16, 2009. Available at http://www.iese.edu/research/
pdfs/DI-0784-E.pdf.
83
Pablo Fernandez, ‘‘The Equity Premium in 150 Textbooks,’’ Working paper, September 14,
2009. Available at http://ssrn.com/abstract=1473225.
84
Pablo Fernandez and Javier Del Campo Baonza, ‘‘Market Risk Premium Used in 2010 by
Professors: A Survey with 1,500 Answers,’’ Working paper, May 13, 2010. Available at
http://ssrn.com/abstract=1606563.
85
Pablo Fernandez and Javier Del Campo Baonza, ‘‘Market Risk Premium Used in 2010 by
Analysts and Companies: A Survey with 2,400 Answers,’’ Working paper, May 21, 2010. ?
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CHAPTER
10
Beta: Differing Definitions
and Estimates
Introduction
Estimation of Equity Beta
Differences in Estimation of Equity Betas
Length of the Sample or Look-Back Period
Frequency of Return Measurement
Choice of Market Index
Choice of Risk-free Rate
Choosing the Best Method
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 Betas
Other Beta Considerations
Summary
Technical Supplement Chapters 2 and 3
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 pure CAPM and this is the form of the CAPM most often shown
in textbooks. The combination of equity beta for the subject business multiplied by
the equity risk premium (ERP) for the market equals the estimated risk premium for
The authors would like to thank David Turney and William Susott of Duff & Phelps LLC for
preparing material for this chapter.
159
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the subject business. Equity betas increase with the risk of the business. For example,
the beta of a business with greater business (operating) risk will be greater than the
beta of a business with lesser business risk. Similarly, the beta of a business with
more debt in the capital structure will be greater than the beta of a business with
lesser financial risk.1
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 the debt beta equals zero. But as the
level of debt financing of the business increases and the credit rating decreases,
debt capital will also bear market risk. That market risk can likewise be measured
in terms of a beta.
This chapter explores some widely used methods in the estimation and applications of betas for equity capital and debt capital. Beta estimates are generally derived
from data on publicly traded securities. If one is valuing a closely held business or
a nonpublic division or reporting unit, for example, one is using the beta estimate of
publicly traded securities as a proxy for the nonpublic business.
Published and calculated beta estimates for public stocks typically reflect the
capital structure of each respective business at market values. The beta estimates are
typically made using realized returns for the subject business’s stock and the stock
market as a whole, and both reflect market values. These betas sometimes are
referred to as levered betas, since these beta estimates reflect the actual leverage in
the subject business’s capital structure. The adjustment for leverage differences is
called unlevering and levering beta estimates. We discuss that process in Chapter 11.
ESTIMATION OF EQUITY BETA
Market or systematic risk is measured in CAPM by 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. In the CAPM, beta should be the expected
beta. We typically use regression betas and other techniques to develop the expected
beta for use in the CAPM. Newer estimation techniques, which we discuss later, use
implied volatility derived from options to estimate expected betas. 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 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 business relative to changing expectations about expected
cash flows of the economy as a whole (i.e., the market) changing expectations for the
ERP.2
There are two general ways for estimating betas. The top-down beta estimate
for a public company comes from a regression of excess returns of the company’s
1
2
See Chapter 5 for discussion of business and financial risk.
John Y. Campbell and Jianping Mei, ‘‘Where Do Betas Come From? Asset Price Dynamics
and the Sources of Systematic Risk,’’ Review of Financial Studies 6(3) (1993): 567–592.
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stock to the excess returns of a market portfolio. Alternatively, a bottom-up beta can
be estimated by:
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&
Identifying the businesses in which the subject business operates
Identifying guideline public companies and estimating their levered betas
Unlevering the guideline public company beta estimates to get estimates of
unlevered (asset) betas
Taking a weighted average of these unlevered betas, where the weights are
based on the relative values (or operating income) of the businesses in which the
subject business operates
Relevering using an appropriate debt-to-equity ratio for the subject business
Of course, you need to use a bottom-up or proxy beta when the subject business
is a division, reporting unit, or closely held business.
The most widely used 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 that betas are time-varying (i.e., sensitive to market changes as the
economy changes; betas differ during improving economic conditions compared with
periods when economic conditions are declining). Using a historical method based on
a sample period may not provide a reliable indication of expected beta 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 may not reflect the future.
Academicians 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
Ri Rf against the excess returns on the market Rm Rf 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)
Ri Rf ¼ a þ B Rm Rf þ e
where: Ri ¼ Historical return for publicly traded stock, i
Rf ¼ Risk-free rate
a ¼ Regression constant
B ¼ Estimated beta based on historical data over the look-back period
Rm ¼ Historical return on market portfolio, m
e ¼ Regression error term
Morningstar uses excess returns in all its computations. Some practitioners and
other financial data services calculate betas using total returns for the subject
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security and for the market returns instead of excess returns. Some practitioners calculate returns for the subject security and for the market simply using changes in
price and ignoring dividend returns.
However, comparisons of beta estimates using excess returns or total returns
show that, as a practical matter, it makes little difference in the aggregate. If one
uses only changes in price and ignores dividend return, it will make some difference,
especially for a stock whose returns are predominantly comprised of dividends.
The OLS regression using total return is:
(Formula 10.2)
R i ¼ a þ B Rm þ e
where the variables are defined as in Formula 10.1
Modern portfolio theory and CAPM do not require linearity of the market line
or the use of a regression model to estimate beta. In the original CAPM formulation,
beta is an ex ante measure of risk. The use of regression analysis confuses the issue
because one has to assume that the error terms are uncorrelated with the market
portfolio.3 But regression analysis is the most widely used method of beta estimation, so the user needs to understand the methodology, including its strengths and
its weaknesses.
Beta equals the covariance of the returns for the subject security to the returns
for the market (e.g., the S&P 500) 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 TIBCO Software, Inc. (market capitalization equal to $932.2 million and debt equal to $56 million as of December 2008) 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?
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 the likelihood that true beta is measured
by estimate of the beta made by regression.
See the Cost of Capital: Applications and Examples 4th ed. Workbook and
Technical Supplement, Chapter 3 an interpretation of the example OLS regression
beta estimate for TIBCO Software.
3
William F. Sharpe, ‘‘Capital Asset Prices with and without Negative Holdings,’’ Journal of
Finance 46 (1991): 489–509.
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163
The beta estimate for the example in Exhibit 10.2 equals 1.765. The
t-statistic in the example equals 4.66, indicating that the data provide a beta
estimate that is statistically significant (i.e., different from zero). R2 equals
0.33. The standard error of estimate equals 0.331.4 That is, we have 95% confidence that the true beta is between 1.765 þ/ (2)(0.331) or between 1.10 and
2.43. The statistics in regressions are discussed more fully in the Review of Statistical Analyses in the Cost of Capital: Applications and Examples 4th ed.
Workbook and Technical Supplement, Appendix III.
Because we cannot compute a beta directly for a division, reporting unit, or
closely held business, we need to estimate a bottom-up or 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 business 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. The sum beta method of estimating betas helps
correct for the tendency for OLS methods to underestimate betas. We discuss the use
of the sum beta method of estimating beta later in this chapter. Further, the more
beta estimates drawn from guideline public companies of similar size as the subject
business 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 II.
DIFFERENCES IN ESTIMATION OF EQUITY BETAS
Be aware that significant differences exist among beta estimates 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. Assuming you are not calculating beta yourself, 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 from other sources for the others. Otherwise an apples-andoranges 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 are 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.
4
Standard error of estimate of the beta coefficient ¼ beta/t-statistic.
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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 Ibbotson 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 Ibbotson Beta Book is published only electronically beginning in 2010
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 TIBCO Software displayed in Exhibit 10.2 uses a look-back period of
60 months.
EXHIBIT 10.1 Illustrative Example of One Common Method for the Calculation of Beta
Month
End, t [a]
Return on
Security A [b]
Return on S&P
Index [c]
1/89
2/89
3/89
..
.
10/98
11/98
12/98
Sum
Average
0.041
(0.007)
0.052
0.069
(0.029)
0.021
0.113
0.033
(0.016)
0.500
0.004
0.077
0.057
0.055
1.488
0.012
Beta ¼
Calculated
Covariance [d]
0.00211
0.00045
0.00043
0.00709
0.00131
(0.00086)
0.21060
0.00176 [f]
Calculated
Variance [e]
0.00325
0.00168
0.00008
0.00423
0.00203
0.00185
0.26240
0.00219 [g]
Covariance ðSecurity 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 plus dividends.
d. Values in this column are calculated as:
Observed return on Security A Average return on Security A (Observed return on Security A Average return on Security A) (Observed return on S&P Index Average return
on S&P Index), or 0.00211 ¼ (0.041 0.004) (0.069 0.012)
e. Values in this column are calculated as:
(Observed return on S&P Index Average return on S&P Index), or
0.00325 ¼ (0.069 0.012)
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 with Alina Niculita, Valuing a Business: The Analysis and Appraisal
of Closely Held Companies, 5th ed. (New York: McGraw-Hill, 2008), Chapter 9. Reprinted
with permission. All rights reserved.
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EXHIBIT 10.2 Beta Estimation for TIBCO Software, Inc. Using OLS Regression
Ticker Symbol: TIBX
SIC 7372: Prepackaged software
Date of beta estimate: December 2008
Company: Provides infrastructure software solutions in the Americas, Europe, the Middle
East, Africa, Asia Pacific, and Japan
Calculated OLS Beta est.
Number of Months of Data
OLS Regression Results
R-Squared
Std error
Intercept
Beta
1.765
60-month look-back period
Summary Statistics
TIBX
Market
Average Return
Standard Deviation
2.451%
39.290%
1.343%
12.752%
Correlation Matrix
TIBX
Market
TIBX
Market
Average Monthly Volume (millions)
Average Volume/Total Outstanding
1.000
0.573
3.645
2.03%
0.33
0.095
0.40%
1.765
t-stat
0.328
5.322
Std error
1.22%
0.331
1.000
Annualized
Source: Calculated (or derived) based on Standard & Poor’s Capital IQ data. Calculations by
Duff & Phelps LLC. Used with permission. All rights reserved.
But if the business characteristics change during the sampling period (e.g., major
divestiture or acquisition, financial distress, cancelation of a significant contract), 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. The Bloomberg online service gives the user the choice of daily, weekly, monthly, or annual returns.
Choice of Market Index
Providers of beta estimates generally choose one of the well-known market indices
used in calculating beta:
&
&
Standard & Poor’s (S&P) 500 Index
New York Stock Exchange (NYSE) Composite Index
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&
&
&
ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
NYSE and American Stock Exchange (AMEX) Index
NYSE, AMEX, and over-the-counter (OTC) Index
Value Line Index
The Bloomberg online service gives the user the choice of a number of market
indices.
Use of a market-capitalization weighted index is the predominant method in
academic research and commercial services. In a market-capitalization weighted index, the weight for each company in the index is determined by the market value of
its equity. While the commonly held belief is that for an index to be representative of
the market, it must be market-capitalization weighted, there is little academic research evidence to support this recommendation.
In fact, in one study the authors find that estimating betas using five years of
monthly return data and returns on an equal-weighted market index (the Center for
Research in Security Prices [CRSP] equal-weighted index) rather than a marketcapitalization weighted index provides the beta estimates using a look-back method
that best matches future realized betas.5
The sizes of the companies in the S&P 500 Index are so great that the index comprises a large percentage of the total capitalization of all of the stocks constituting the
combined indexes listed here. Furthermore, the broader market indices 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 Ibbotson Cost of Capital Yearbook and the Ibbotson 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 or in developing
economies where one or two stocks dominate the stock market capitalization).
For example, if one computed beta estimates using historical returns over a
look-back period or obtained beta estimates from data sources, the risks for basic
manufacturing companies appeared to have gone down in the late 1990s because
the beta estimates of these companies decreased. Prior to the run-up in prices of
technology stocks, basic manufacturing companies represented significant weight
in the stock indices. Prior to the 1990s, 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 in fact changed. But their observed betas then
looked low compared with 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 segments
other than the industry that dominates the index.
5
Jan Batholdy and Paula Peare, ‘‘The Relative Efficiency of Beta Estimates,’’ Working paper,
March 2001. Available at http://ssrn.com/abstract=263745.
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167
Choice of Risk-free Rate
To avoid the maturity risk (interest rate risk) inherent in long-term bonds, the riskfree 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 Ibbotson Cost of Capital Yearbook and the
Ibbotson Beta Book. Differences in the choice of risk-free rate will cause differences
in the beta estimates.
Choosing the Best Method
With all of these choices to make, is there any research that provides guidance as to
the best method for calculating beta? One set of researchers examined the use of
different return frequencies to estimate beta, using different numbers of periods for
the look-back period and different market indices to determine if there were characteristics that provided more efficient beta estimates. That is, which beta estimates
provide the most accurate estimates of returns going forward? Though they tested
limited combinations of return frequencies and look-back periods, they found that
estimating betas using five years of monthly return data and returns on an equalweighted market index (the Center for Research in Security Prices [CRSP] equalweighted index) rather than a market capitalization weighted index provides the
most efficient beta estimates using a look-back method.6
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.
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. This does not imply that the
market is inefficient. Rather, the market for some stocks is more efficient than
for other stocks. Large companies are followed by numerous analysts and are
owned by numerous institutional investors. These stocks react to changes in the
economy or changes in the business (e.g., introduction of a new product, signing
of a new major contract) nearly instantaneously. Smaller companies’ stocks react at a slower rate.
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
6
Jan Batholdy and Paula Peare, ‘‘The Relative Efficiency of Beta Estimates,’’ Working paper,
March 2001. Available at http://ssrn.com/abstract=263745.
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averaged more than one month’s betas.7 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
One technique for adjusting beta estimates to industry averages is a rather sophisticated technique called Vasicek shrinkage.8 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 Ibbotson Beta Book, where this adjusted beta is labeled Ibbotson Beta.
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 that is applied to the raw betas of each guideline company
to arrive at adjusted raw levered betas that can be unlevered and then relevered 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 30-day 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).9
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 Ibbotson Beta Book explains it, ‘‘Because of non-synchronous
7
Former Ibbotson Associates vice president and economist, now vice president, Quantitative
Research, Morningstar, Inc.
8
The formula, used in the Ibbotson 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.
9
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.
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EXHIBIT 10.3 Comparison of OLS Betas and Sum Betas by Company Size, December 2007
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 2007
Largest
OLS Beta
Sum Beta
Difference
$ 472,519
20,235
9,207
5,013
3,423
2,412
1,633
1,129
723
363
9,207
2,412
723
0.95
1.02
1.21
1.23
1.26
1.37
1.48
1.56
1.56
1.43
1.23
1.46
1.50
0.93
1.07
1.30
1.40
1.42
1.46
1.61
1.71
1.77
1.77
1.36
1.58
1.77
0.02
0.05
0.09
0.17
0.16
0.09
0.13
0.15
0.21
0.34
0.13
0.12
0.27
Source: Calculated (or derived) based on CRSP1 data, # 2008 Center for Research in
Security Prices (CRSP1), University of Chicago Booth School of Business, and Standard &
Poor’s Compustat data. Calculations by Duff & Phelps LLC. Size of largest company in each
decile from 2008 SBBI Valuation Yearbook. Copyright # 2008 Morningstar, Inc. Used with
permission. All rights reserved.
price reactions, the traditional betas estimated by ordinary least squares are biased
down for all but the largest companies.’’10
Exhibit 10.3 shows the differences between OLS betas and sum betas for the
companies comprising the CRSP deciles as of December 2007, and Exhibit 10.4
shows the differences between OLS betas and sum betas for the companies comprising the CRSP deciles as of December 2008. We present both because as of December
2008, many companies that had been larger cap companies in earlier periods became smaller cap companies as their market capitalization shrank (the largest company in the 10th decile at December 2008, for example, had a market capitalization
40% less than the largest company in the 10th decile at December 2007). Of the
public companies included in the CRSP data base, 77.5% were low-cap or microcap at December 2008, compared with 70% at December 2007. We would expect
that as the market prices in 2009 increased, the relationship observed in 2007 will
probably be more representative of the relationship in future periods.
Exhibit 10.5 displays the differences in beta estimates by size of company
(measured by market value of equity) for a sampling of industries.
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).
10
Ibbotson Beta Book, 2006 ed. (Chicago: Morningstar, 2006). The second 2001 edition discontinued presenting sum betas.
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EXHIBIT 10.4 Comparison of OLS Betas and Sum Betas by Company Size, December 2008
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 2008
Largest
OLS Beta
Sum Beta
Difference
$ 465,652
18,503
7,360
4,225
2,786
1,849
1,197
753
453
219
7,360
1,849
453
0.92
1.16
1.25
1.21
1.21
1.30
1.30
1.33
1.32
1.32
1.23
1.30
1.32
0.91
1.19
1.28
1.26
1.28
1.32
1.33
1.31
1.33
1.49
1.28
1.33
1.39
0.01
0.03
0.03
0.05
0.08
0.03
0.04
0.01
0.02
0.17
0.05
0.02
0.08
Source: Calculated (or derived) based on CRSP1 data, # 2009 Center for Research in
Security Prices (CRSP1), University of Chicago Booth School of Business, and Standard &
Poor’s Compustat data. Calculations by Duff & Phelps LLC. Size of largest company in each
decile from 2009 SBBI Valuation Yearbook. Copyright # 2009 Morningstar, Inc. Used with
permission. All rights reserved.
EXHIBIT 10.5 Comparison of OLS Betas and Sum Betas for Different Industries
Data as of December 2008
Median
Count
OLS
Beta
Sum
Beta
Computer Software
(SIC 7372)
All Companies
Over $1 Billion
Under $200 Million
151
29
79
1.47
1.24
1.49
1.57
1.17
1.76
Auto Parts
(SIC 3714)
All Companies
Over $1 Billion
Under $200 Million
27
5
15
1.74
1.42
1.82
1.91
1.48
2.20
Healthcare
(SIC 80)
All Companies
Over $1 Billion
Under $200 Million
81
11
44
1.11
0.96
1.34
1.37
1.01
1.53
Publishing
(SIC 27)
All Companies
Over $1 Billion
Under $200 Million
39
8
19
1.30
1.03
1.41
1.57
1.28
1.85
Petroleum and Natural Gas
(SIC 1311)
All Companies
Over $1 Billion
Under $200 Million
152
41
76
1.50
1.16
1.78
1.86
1.46
2.30
Source: Compiled from Standard & Poor’s Capital IQ data. Calculations by Duff & Phelps
LLC. Used with permission. All rights reserved.
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171
The formula for the sum beta is:
(Formula
10.3) Rn Rf ;n ¼ a þ Bn Rm;n Rf ;n þ Bn1 Rm;n1 Rf ;n1 þ e
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
R
¼
Historical
return on market portfolio
m
Rm;n Rf ;n 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
Rm;n1 Rf ;n1 ¼ Excess return on market portfolio last month
e ¼ Regression error term
The 2009 SBBI Valuation Yearbook 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,’’11 which is included as Exhibit 14.1 in Chapter 14 in this book. The
table shows that the returns in excess of CAPM are much lower than for the OLS
betas (shown in Exhibit 13.1 in this book), reflecting the superiority of measuring
betas using the sum beta methodology compared with using OLS methodology for
estimating betas of small businesses. Graph 7–5 in the 2009 SBBI Valuation Yearbook on the same page shows how much closer the portfolios track the Security
Market Line, except for the tenth decile.12 If sum betas are used for smaller companies, the size effect (realized returns in excess of those predicted by CAPM) is greatly
reduced.
Sum betas for individual stocks can be calculated using spreadsheet software
such as Microsoft Excel and historical return data, which is available from several
sources, such as Standard & Poor’s Compustat. 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 business in the context of CAPM) and thus make
a smaller adjustment for the size effect. The theory is that the sum beta helps correct
for the larger size effect that is principally due to a misspecification of beta when
using traditional OLS betas for smaller companies.
Exhibit 10.6 shows the OLS and sum beta estimate for Ultimate Software
Group, Inc. Ultimate Software’s market value of equity ($356.8 million) plus debt
capital ($27 million) ranks the company as a micro-cap company (as of December
31, 2008). Its OLS beta estimate is equal to 1.69, and its sum beta estimate is equal
to 1.92 (a 14% difference).
11
12
2009 SBBI Valuation Yearbook (Chicago: Morningstar, 2009): 98.
2009 SBBI Valuation Yearbook (Chicago: Morningstar, 2009): 98.
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EXHIBIT 10.6 Beta Estimation for Ultimate Software Group Using OLS and Sum Beta
Regression Methods
Ticker Symbol: ULTI
SIC 7372: Prepackaged software
Date of beta estimates: December 2008
Company: Ultimate Software Group Inc. designs, markets, implements, and supports human
resources, payroll, and talent management solutions primarily in the United States and Canada
Calculated OLS Beta estimate
Calculated Sum Beta estimate
Number of Months of Data
1.69
1.92
60-month look-back
period
Summary Statistics
ULTI
Average Return
Standard Deviation
22.946%
42.541%
Correlation Matrix
ULTI
ULTI
Market
Market Lag
Average Monthly Volume
(millions)
Average Volume/Total
Outstanding
1.000
0.507
0.289
.214
Market
1.343%
12.752%
Market
1.000
0.380
Market
Lag
0.515%
12.963%
Market
Lag
1.00
0.88%
Annualized.
Source: Compiled from Standard & Poor’s IQ data. Calculations by Duff & Phelps LLC.
Used with permission. All rights reserved.
For smaller businesses, the difference can be even greater. Exhibit 10.7 shows
the OLS and sum beta calculation for THQ, Inc. It had $281 million (as of December 31, 2008) market value of equity plus $18 million debt capital. Its OLS beta
estimate is equal to 2.26 and its sum beta estimate is equal to 2.63 (a 17%
difference).
Cost of Capital: Applications and Examples 4th ed. Workbook and Technical
Supplement, Chapter 2, 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. Ideally, one would like to have a
sampling of betas from many ‘‘pure play’’ guideline public companies (e.g., companies with at least 75% of revenue from a single Standard Industrial Classification
[SIC] code) when estimating a bottom-up beta. There may be many divisions of the
largest companies in the industry, making pure play beta estimation difficult.
E1C10
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173
Beta: Differing Definitions and Estimates
EXHIBIT 10.7 Beta Estimation for THQ, Inc. Using OLS and Sum Beta Regression Methods
Ticker Symbol: THQI
SIC 7372: Prepackaged software
Date of beta estimates: December, 2008
Company: THQ, Inc. engages in the development, publishing and distribution of interactive
entertainment software for various game systems worldwide
Calculated OLS Beta estimate
Calculated Sum Beta estimate
Number of Months of Data
2.26
2.63
60-month look-back
period
THQI
Market
Market
Lag
8.305%
44.0%
1.343%
12.752%
0.515%
12.963%
THQI
Market
Market
Lag
1.000
0.654
0.400
1.287
1.000
0.380
Summary Statistics
Average Return
Standard Deviation
Correlation Matrix
THQI
Market
Market Lag
Average Monthly Volume
(millions)
Average Volume/Total
Outstanding
1.00
1.92%
Annualized.
Source: Compiled from Standard & Poor’s Capital IQ data. Calculations by Duff & Phelps
LLC. Used with permission. All rights reserved.
The Ibbotson Beta Book includes industry betas calculated using full-information methodology (the full-information beta is seen as FI-beta in Chapter 7).13 The
full-information approach is based on the premise that a business can be thought of
as a portfolio of assets. The full-information approach is designed to capture the
impact that the individual segments have upon the overall business beta. After identifying all companies with segment sales in an industry, Morningstar calculates a
beta estimate of those companies. 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.
13
Paul D. Kaplan and James D. Peterson, ‘‘Full-Information Betas,’’ Financial Management
(Summer 1998): 85–93.
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174
ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
EXHIBIT 10.8 Example of Calculating the Peer Group Beta
Segment reporting lists ‘‘sales’’ in 3 different two-digit SIC codes.
Industry
Composite
Company Sales in
% of Company
SIC
Beta
Industry ($ millions)
Sales in Industry
Code
1221
6794
4953
1.10
1.16
1.56
Totals
113.80
1.30
36.10
151.20
75.30%
0.84%
23.86%
100.00%
Sales-Weighted
Beta Component
.83
.01
.37
1.21
Applying market capitalization weights in the process tends to reduce the beta
estimates because large capitalization companies on the average have lower betas
than smaller companies. 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.
Cost of Capital: Applications and Examples 4th ed. Workbook and Technical
Supplement, Chapter 2, provides an example of estimating beta using the full-information methodology. One can either weight the betas by market capitalization or
equal-weighting. If the subject company is a smaller company, then the equal
weighting is likely to be a better methodology.
PEER GROUP EQUITY BETA
The Ibbotson 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 10.8 shows an example of calculating a peer group beta.
Exhibit 10.9 is an excerpt from the 2008 Ibbotson Beta Book (which is published only electronically beginning in 2010). 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 three-factor models. Exhibit 20.5 displays a sample
page from the 2009 edition of Ibbotson Beta Book.
The Beta Book does not provide cost of equity estimates for individual companies; however, it does provide practitioners with the statistics necessary
for calculating cost of equity under both the CAPM and the Fama-French
three-factor model.
FUNDAMENTAL EQUITY BETA
As an alternative to using betas estimated from realized stock and market returns, you can estimate a fundamental beta. Fundamental betas are typically
0.51
1.36
1.06
1.20
2.86
2.81
1.34
1.87
1.76
5.09
4.77
3.87
2.30
2.89
5.53
t-Stat
0.03
0.06
0.05
0.31
0.28
0.21
0.08
0.13
0.34
R-Sqr
1.43
0.61
0.94
1.43
1.44
1.33
1.43
1.71
1.52
Beta
Grp
Pr
0.62
1.12
1.08
2.02
1.07
1.21
2.04
2.31
2.01
Beta
Ibbotson
0.50
1.21
1.13
2.10
1.04
1.16
2.86
1.77
2.06
Beta
Raw
0.61
0.98
1.08
2.02
1.06
1.17
2.04
0.95
2.00
Beta
Ibbotson
Unlevered
1.53
0.27
0.10
1.67
4.41
3.78
2.10
1.73
3.47
5.99
t-Stat
FF
1.25
1.21
1.86
0.88
0.65
2.42
3.68
2.41
Beta
FF
2.88
4.99
8.05
12.84
1.00
1.30
1.18
8.99
8.46
12.74
1.73
6.16
3.22
6.38
1.05
2.38
1.56
2.41
t-Stat
SMB
Prem
SMB
0.41
0.88
5.67
4.31
3.77
5.59
5.44
9.63
1.53
4.90
4.76
3.44
8.80
10.57
t-Stat
HML
1.77
5.85
11.30
5.14
Prem
HML
FF Statistics
Fama-French Three-Factor Model
0.25
0.09
0.07
0.45
0.38
0.39
0.10
0.20
0.42
R-Sqr
FF
copies of the Ibbotson Beta Book, or for more information on other Morningstar publications, please visit global.morningstar.com/DataPublications. Calculated (or Derived) based on data
from CRSP US Stock Database and CRSP US Indices Database # 2009 Center for Research in Security Prices (CRSP1), University of Chicago Booth School of Business. Used with permission.
Source: 2009 Ibbotson1 Beta Book First Edition Copyright # 2009 Ibbotson Associates. All rights reserved. Used with permission. (Morningstar, Inc. acquired Ibbotson in 2006.) To purchase
Company with less than 60 months’ data (minimum 36 months).
Data through December 2008.
APPLIED SIGNAL TECHNOLOGY
3APNS
1.13
APPLIED MICRO CIRCUITS CORP
AIT
APPLIED NANOTECH HOLDINGS
APPLIED MATERIALS INC
DIGA
APPLIED NEUROSOLUTIONS INC
APPLIED INDUSTRIAL TECH INC
ARCI
AMAT
APPLIED ENERGETICS INC
CRA
AMCCD
2.10
APPLIANCE RECYCLING CTR AMER
ABI
2.07
APPLE INC
AAPL
Beta
Company
Ticker
Raw
Levered
CAPM: Ordinary Least Squares
08/26/2010
EXHIBIT 10.9 Partial Page View from the 2009 Ibbotson Beta Book First Edition (data through December 2008)
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175
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176
ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
Coefficient of Variation of Operating Margin to Beta
1.6
1.4
1.2
1.0
Beta
E1C10
0.8
0.6
0.4
0.2
0.0
–2.0
–1.5
–1.0
–0.5
0.0
Log of Median CV (Operating Income)
0.5
EXHIBIT 10.10 Relationship of Coefficient of Variation of Operating Margin to Beta
estimated using accounting relationships and can take many forms. For example,
one may estimate a fundamental beta by measuring the sensitivity of the subject
business’s operating earnings to changes in operating earnings of the industry in
which the business operates or to the market (segment or whole). One may be
able to relate accounting data over time (time series) for companies to market
betas or relate accounting data at a particular point in time across companies
(cross-sectional) to market betas. Various studies have measured fundamental
betas for publicly traded companies.14 For example, you can calculate a fundamental beta estimate for a division, reporting unit, or closely held business by
examining the relationship of the variability or coefficient of variation (standard
deviation of operating margin divided by the mean operating margin over the
same period) to observed beta estimates.15 Exhibit 10.10 shows the relationship
found in the Duff & Phelps Risk Premium Report—Risk Study, which will be
discussed in Chapter 15.
One source for 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)
14
See, e.g., 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: Tools and
Techniques for Determining the Value of Any Asset, 2nd ed. (Hoboken, NJ: John Wiley &
Sons, 2002), 58–59.
15
It can be shown that beta is a linear function of the coefficient of variation of firm or project
cash flows. See, e.g., Cleveland S. Patterson, ‘‘CV or Not CV? That Is the Question,’’
Accounting and Finance (May 1989): 103.
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Beta: Differing Definitions and Estimates
177
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 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
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 10.11 shows a comparison of
market capitalization data as of December 31, 2008, and Barra predicted betas as of
December 2008 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 probably due to Barra’s focus on largercompany stocks.
178
406,067
18,494
7,293
4,215
2,784
1,834
1,197
751
453
219
1
2
3
4
5
6
7
8
9
10
105
128
116
132
168
201
247
291
496
1890
Company
Count
0.84
1.02
1.23
1.17
1.20
1.26
1.25
1.33
1.43
1.43
Barra
Historical
0.89
1.02
1.18
1.14
1.14
1.18
1.21
1.23
1.30
1.34
Barra
Predicted
0.79
0.96
1.22
1.13
1.16
1.20
1.21
1.23
1.27
1.34
Sum
Beta
0.94
1.09
1.26
1.22
1.23
1.32
1.34
1.41
1.55
1.56
Barra
Historical
0.94
1.07
1.20
1.15
1.17
1.20
1.24
1.25
1.34
1.48
Barra
Predicted
Equal Weighted Avg
0.92
1.07
1.27
1.24
1.26
1.32
1.39
1.39
1.59
1.76
Sum
Beta
0.86
1.05
1.25
1.15
1.17
1.40
1.29
1.36
1.51
1.53
Barra
Historical
0.88
1.02
1.11
1.08
1.12
1.15
1.19
1.24
1.32
1.46
Barra
Predicted
Medians
0.90
1.04
1.22
1.11
1.17
1.37
1.36
1.33
1.55
1.70
Sum
Beta
Source: Data from S&P Research Insight and Capital IQ databases. Calculations by Duff & Phelps LLC. Used with permission. All rights reserved.
Largest
Mkt Cap
Decile
Market Weighted Averages
08/26/2010
EXHIBIT 10.11 Comparison of Barra Historical (OLS) Predicted Betas to Sum Betas
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Beta: Differing Definitions and Estimates
179
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.16 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 tested
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 show that the OLS regression estimator for beta is
based on the quadratic weighting scheme that tends to contradict the assumptions of
risk aversion, and the probability distributions of market returns tend to have fatter
tails than they would if the returns were normally distributed, making the OLS estimate of beta sensitive to extreme outliers. They explored the use of an absolute value
weighting of differences (e.g., least absolute deviations approach) instead of the OLS
quadratic weighting and found that the absolute value weighting scheme is likely to
result in better beta estimates.17
In another study, the authors extracted ‘‘forward-looking’’ beta estimates from
option pricing data on the Dow Jones 30.18 They used option models to estimate the
implied volatility of the stocks and the covariance of the individual stocks with the
market. They found that forward-looking betas extracted from implied volatilities
on traded options often better estimated the beta in the next period than beta estimates derived from historical data. They also found that 180 days of historical
excess returns provides the ‘‘best’’ estimate of forward-looking betas. Research is
continuing on constructing beta estimates from option-implied volatility and skewness of return distributions.19
ESTIMATION OF DEBT BETAS
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).
16
R. D. Martin and T. T. Simin, ‘‘Outlier-Resistant Estimates of Beta,’’ Financial Analysts
Journal (September–October 2003): 56–69.
17
H. Shalit and S. Yitzhaki, ‘‘Estimating Beta,’’ Review of Quantitative Finance and Accounting 18(2) (2002): 95–118.
18
Peter F. Christoffersen, Kris Jacobs, and Gregory Vainberg, ‘‘Forward-Looking Betas,’’
Working paper, May 2, 2008. Available at http://ssrn.com/abstract=891467.
19
Bo-Young Chang, Peter F. Christoffersen, Kris Jacobs, and Gregory Vainberg, ‘‘OptionImplied Measures of Equity Risk,’’ Working paper, June 1, 2009. Available at http://ssrn
.com/abstract=1416753.
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180
ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
EXHIBIT 10.12 Estimated Beta of Debt Based on Credit Ratings
2008
Moody’s
Rating
2007
Dec Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Aaa
Aa
A
Baa
Ba
B
Caa
Ca-D
0.00
0.00
0.00
0.07
0.23
0.32
0.51
1.14
0.00
0.00
0.00
0.08
0.25
0.36
0.65
1.37
0.00
0.00
0.00
0.09
0.25
0.36
0.65
1.38
0.00
0.00
0.00
0.06
0.23
0.35
0.58
1.10
0.00
0.00
0.00
0.02
0.24
0.37
0.60
1.07
0.00
0.00
0.00
0.03
0.23
0.39
0.56
1.22
0.00
0.00
0.00
0.05
0.24
0.40
0.58
1.24
0.00
0.00
0.00
0.05
0.24
0.40
0.57
1.24
0.11
0.11
0.20
0.14
0.32
0.50
0.68
1.30
0.08
0.12
0.26
0.34
0.51
0.63
0.95
1.11
0.03
0.08
0.20
0.30
0.52
0.68
1.00
1.38
0.03
0.09
0.21
0.31
0.53
0.70
1.01
1.50
0.00
0.00
0.00
0.06
0.24
0.34
0.63
1.29
2009
Moody’s
Rating
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Aaa
Aa
A
Baa
Ba
B
Caa
Ca-D
0.07
0.10
0.20
0.26
0.42
0.58
0.88
1.41
0.10
0.10
0.21
0.24
0.38
0.56
0.86
1.37
0.09
0.09
0.19
0.22
0.37
0.53
0.86
1.38
0.07
0.09
0.19
0.24
0.40
0.57
0.95
1.53
0.07
0.10
0.21
0.25
0.41
0.59
0.98
1.57
0.07
0.10
0.21
0.26
0.41
0.59
0.98
1.58
0.08
0.12
0.22
0.28
0.42
0.60
1.00
1.58
0.08
0.12
0.23
0.28
0.42
0.59
1.00
1.58
0.08
0.12
0.23
0.28
0.43
0.60
1.01
1.60
Source: Ibbotson Morningstar EnCorr database; calculations by Duff & Phelps LLC. Used
with permission. All rights reserved.
The risk implied by debt beta 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 and variability of EBITDA/sales; and so on. These are
fundamental risks that the interest (and principal) will or can be paid when due.
Betas of debt generally correlate with credit ratings. Exhibit 10.12 shows the
relationship between bond ratings and estimated betas of debt during 2008 and the
first nine months of 2009. One needs to estimate an approximate credit rating (synthetic credit rating) for the business debt that is not rated by Moody’s, Standard &
Poor’s, or Fitch. If the risk to the economy diminishes and we return to some semblance of normalcy, the observed betas by debt rating for periods before September
2008 can be used in conjunction with the unlevering and relevering formulas, which
require debt beta estimates, as we discuss in Chapter 11.
The general formula for estimating the beta for debt (e.g., traded bonds) is:20
(Formula 10.4)
R d ¼ a þ B d R m Rf ð1 t Þ þ e
where: Rd ¼ Rate of return on subject debt (e.g., bond) capital
a ¼ Regression constant
20
Simon Benninga, Financial Modeling, 2nd ed. (Cambridge, MA: MIT Press, 2000), 414.
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Beta: Differing Definitions and Estimates
181
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 bonds with longer periods to maturity appear to have
greater market risk than bonds with shorter maturities. These longer-maturity bonds
often have beta estimates more closely resembling the beta estimates of smallcapitalization stocks.21
OTHER BETA CONSIDERATIONS
A top-down beta estimate for a public company comes from a regression of excess
returns of the company’s stock to the excess returns of a market portfolio.
You need to use bottom-up beta when the subject business is a division, reporting unit, or closely held business. You can always use a bottom-up beta even to estimate the beta for a public company. The bottom-up beta approach 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 of average
pffiffiffi beta ¼ average standard deviation of individual company beta estimates/ n
The subject business has reorganized or restructured itself substantially during
the period of the regression
Assume the subject business had become distressed and had recently emerged
from restructuring its debt and an infusion of equity. Exhibit 10.13 presents an
example of an adjustment in pricing for a stock of this hypothetical company.
In period A, the company returns had essentially moved with the market. In period B, the company is distressed, and its stock is experiencing a downward repricing. During this period, the company’s returns are not correlated with the movement
of the overall market at all. In period C, the restructuring of the company and the
repricing of the company’s stock is complete, and the company’s returns are once
again moving more in tandem with market returns.
If one were to compute beta at time 1, which includes period A as the look-back
period, the beta estimate would reflect a normal relationship between the company’s
returns and the market’s returns. In fact, its beta estimate would be near 1. In contrast, computing a beta estimate at time 2, which includes period B (the period of the
company’s stock repricing) as the look-back period, would not yield a reliable
forward-looking beta estimate. In fact, it would yield a beta estimate lower than
21
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.
08/26/2010
Page 182
182
ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
Example Company versus Index
over Time
1.6000
Compound Return
E1C10
1.4000
1.2000
1.0000
0.8000
0.6000
0.4000
0.2000
0.0000
A
B
1
C
2
Time
Example
S&P 500
EXHIBIT 10.13 Relationship of Returns for Example Company
expected since the company’s return was negative in a period when the market’s
return was generally positive. This result is counterintuitive, given the company’s
downward repricing; that is, the operating risk of the company has not declined
over period B, and in fact, its operating risk was greatest during this period. Once
the restructuring of the company and the repricing of the company’s stock is complete, its normal relationship to the market will resume in period C.
To estimate beta at time 2 for the company, one should use a bottom-up beta
estimate because a top-down estimate will result in an erroneous beta estimate.
Using betas of guideline public companies for estimating a bottom-up beta has
been found to provide reasonably accurate estimates of the subject company. The
more guideline companies used in the sample size, the better the accuracy. The accuracy is also enhanced if the guideline public companies are reasonably close in size to
the subject company. When the guideline public companies are larger than the subject company, the beta estimate for the subject company is biased low, because of
the propensity of betas of larger companies to be smaller than the betas of smaller
companies.22 Use of the beta estimate derived from guideline public companies
larger than the subject company will generally result in too low an estimate of the
cost of equity capital. Hence, one needs to consider adjusting for the size effect, as
discussed in Chapter 13.
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 business.
22
Robert G. Bowman and Susan R. Bush, ‘‘Using Comparable Companies to Estimate the
Betas of Private Companies,’’ Journal of Applied Finance (forthcoming).
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183
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 CAPM.
There are many variations on the way betas are estimated by different sources
of published betas and by practitioners. 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.
Academic research is attempting to improve beta estimation methodology. 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.
Following are some of our recommendations.
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 or change in relative
expectations about the company). Then you should investigate any changes. If the
underlying fundamentals of the business have changed, a more recent period should
be used in developing a beta estimate. This will often require calculating your own
beta estimate, and we encourage practitioners to do so.
Second, we recommend using sum beta calculations (whether you are using a
pure-play or full-information beta methodology) for smaller public companies. We
calculate both OLS and sum beta estimates for all guideline public companies we are
investigating. We are looking for the best beta estimate. If the estimates differ, we
gravitate toward using the sum beta estimate.
Third, 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.23 Comparing unlevered betas
helps you understand the differences among the companies. We discuss unlevering in Chapter 11.
23
Hazem Daouk and David Ng, ‘‘Is Unlevered Firm Volatility Asymmetric?’’ AFA 2007
Chicago meetings, January 11, 2007.
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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.
TECHNICAL SUPPLEMENT CHAPTERS 2 AND 3
Cost of Capital: Applications and Examples 4th ed. Workbook and Technical Supplement, Chapter 2, provides examples of computing the OLS beta estimate, the
sum beta estimate, and the full-information beta estimate.
Cost of Capital: Applications and Examples 4th ed. Workbook and Technical
Supplement, Chapter 3, provides a discussion of interpreting regression statistics
when estimating beta using OLS estimation.
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CHAPTER
11
Unlevering and Levering Equity Betas
Introduction
Formulas for Unlevering and Levering Equity Betas
Hamada Formulas
Miles-Ezzell Formulas
Harris-Pringle Formulas
Practitioners’ Method
Capital Structure Weights
Fernandez Formulas
Adjusting Formulas for Other Components of Capital Structure
Choosing among Unlevering and Levering Formulas
Adjusting Asset Beta Estimates for Differences in Operating Leverage
Adjusting Asset Beta Estimates for Excess Cash and Investments
Unlevering Equity Volatility
Summary
Additional Reading
INTRODUCTION
Published and calculated betas for publicly traded 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 (i.e., unlevering the beta) 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
185
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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 18.
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 probably be the one assumed.
FORMULAS FOR UNLEVERING AND LEVERING
EQUITY BETAS
It is useful to begin with the definition of a business enterprise (enterprise value):
BE ¼ NWC þ FA þ IA þ UIV
where:
BE ¼ Business enterprise
NWC ¼ Net working capital
FA ¼ Fixed assets
IA ¼ Intangible assets
UIV ¼ Unidentified intangible value (i.e., goodwill)
There are two equivalent formulations in the literature for valuing a levered
business enterprise, as depicted in Exhibit 11.1. The values of debt and equity capital are market values.
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Value of Levered BE = Value of Levered Assets
Assets
Capital
Formulation 1
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
Formulation 2
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 11.1 Value of a Levered Business Enterprise (BE)
The tax shield is the reduction of the cost of debt capital due to the tax deductibility of interest expense on debt capital. In the first formulation, cost of debt capital is measured prior to the tax affect (kdðptÞ ) and then adjusted for the tax affect
after the tax affect (kd) as the value of the tax deduction on interest payment reduces
the cost of debt capital. This formulation typically uses as the discount rate the textbook weighted average cost of capital (WACC) (formula 18.3). It is applied to net
after-tax (but before interest) net cash flows of the business enterprise.
In the second formulation, the cost of debt capital is measured prior to the tax
affect (kdðptÞ ). The value of the tax deduction on the interest payments equals the
value of the tax shield.
In the first formulation, you attach value to the assets of the business based on
their being partially financed with debt capital. In the second formulation, you attach value to the assets of the business as if they were financed with all equity capital, and then the tax shield is valued separately.
In the second formulation, the tax savings due to interest deductions are
directly valued as a cash flow. Therefore, the discount rate is the weighted
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pre-interest-tax-shield cost of debt capital and the cost of equity capital components (pre-interest-tax-shield weighted average cost of capital). It is applied to
the net after-tax (but before interest) net cash flows of the firm and the cash
flows due to the tax shield.
The appropriate formulation is in part a function of the risk of realizing the
interest-tax-shield in the period when the interest is paid. 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.1 This risk is
captured both in the levered equity beta observed in the market and by the observed
debt beta. The debt beta captures the sensitivity recognized by the market to the risk
of the debt as business conditions improve or deteriorate. The greater the debt beta,
the more the market recognizes that the debt is sharing risk with the equity. If there
were no assumed risk sharing, then the observed debt beta would be zero.
In the next sections, we present a discussion and examples for these formulas:
&
&
&
&
&
Hamada formulas
Miles-Ezzell formulas
Harris-Pringle formulas
Practitioners’ method formulas
Fernandez formulas
These formulas can be modified for the effects of warrants, employee stock options, and convertible debt.2 There have been other formulas offered to explain the
relationship between leverage and equity risk.3
Hamada Formulas
The Hamada formulas are commonly cited formulas for unlevering and relevering
equity beta estimates.4 The Hamada formula for unlevering beta is shown as Formula
11.1. This is the formula used by Morningstar to unlever betas in its Beta Book.
1
The Worker, Homeownership and Business Assistance Act of 2009 increased the net operating loss (NOL) carryback provision from two years to five years for corporate losses incurred
in 2008 and 2009. This law is an expansion of the February 2009 law that extended NOL
carryback provisions for businesses with $15 million or less in annual revenues. This new
law expands and liberalizes the utilization of NOLs and is an election that is available to
almost all businesses (large or small).
2
Phillip R. Daves and Michael C. Ehrhardt, ‘‘Convertible Securities, Employee Stock Options, and the Cost of Equity,’’ Working paper, June 7, 2004. Available at http://ssrn.com/
abstract=990906.
3
For example, the Conine formula considers a debt beta. T. E. Conine Jr., ‘‘Corporate Debt
and Corporate Taxes: An Extension,’’ Journal of Finance (September 1980): 1033–1077.
4
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.
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(Formula 11.1)
BU ¼
BL
1 þ ð1 tÞðW d =W e Þ
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 11.2.
(Formula 11.2)
BL ¼ BU ð1 þ ð1 tÞðW d =W e ÞÞ
where the definitions of the variables are the same as in Formula 11.1.
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).
The formulas imply that tax deductions on the interest expense will be realized
in the periods in which the interest is paid.
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 W d ).
The amount of debt capital is fixed as of the valuation date and remains
constant.
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 increases in net cash flow to the firm in every period).5 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 11.2.
Miles-Ezzell Formulas
The Miles-Ezzell formulas are alternative formulas for unlevering and relevering
equity betas that assume there is risk in the timely realization of the tax deductions
5
Enrique R. Arzac and Lawrence R. Glosten, ‘‘A Reconsideration of Tax Shield Valuation,’’
European Financial Management (2005): 453–461.
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EXHIBIT 11.2 Computing Unlevered and Relevered Betas Using Hamada Formulas
EXAMPLE 1
Assume that for guideline public company A:
Levered (published) equity beta: 1.2
Tax rate: 40%
Capital structure: 30% debt capital, 70% equity capital
Using Formula 11.1 we get:
1:2
1 þ ð1 0:40Þ0:30=0:70
1:2
¼
1 þ 0:60ð0:429Þ
1:2
¼
1:257
¼ 0:95
BU 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 equity beta: 0.90
Tax rate: 30%
Capital structure: 60% debt capital, 40% equity capital
Using Formula 11.2 we get:
BL ¼
¼
¼
¼
0:90ð1 þ ð1 0:30Þ0:60=0:40Þ
0:90ð1 þ 0:70ð1:5ÞÞ
0:90ð2:05Þ
1:85
Source: Shannon P. Pratt, Valuing a Business: The Analysis and Appraisal of Closely Held
Companies, 5th ed. (New York: McGraw-Hill, 2008), Chapter 9. All rights reserved. Used
with permission.
for interest payments on debt capital.6 The Miles-Ezzell formula for unlevering beta
is shown in Formula 11.3.
(Formula 11.3)
Me BL þ Md Bd 1 t kdðptÞ = 1 þ kdðptÞ
BU ¼
Me þ Md 1 t kdðptÞ = 1 þ kdðptÞ
6
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.
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Unlevering and Levering Equity Betas
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
kdðptÞ ¼ Cost of debt capital at market rates prior to tax effect
We discuss the beta of debt capital in Chapter 10.
The companion Miles-Ezzel formula for relevering beta is Formula 11.4.
(Formula 11.4)
"
#
t kdðptÞ
BL ¼ BU þ ðW d =W e ÞðBU Bd Þ 1 1 þ kdðptÞ
where the definitions of the variables are the same as in Formulas 11.1 and 11.3.
The Miles-Ezzell formulas are consistent with the theory that:
&
&
&
The discount rate used to calculate the value of 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 value of 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 is
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).
Market value of debt capital remains at a constant percentage of equity
capital, which is equivalent to saying that debt increases in proportion to
increases in 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 11.3. We
begin with Example 1, where Bd, the beta of debt capital, equals 0.30 (i.e., the debt
is rated as Baa, and there is some risk to the debt capital that interest and principal
will not be repaid when due and that tax deductions on interest expense will not
result in tax savings in the same period as the interest is paid in future years for the
guideline public company).
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 0:60 (i.e., the debt is rated lower than the debt of the guideline public company
in Example 1).
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EXHIBIT 11.3 Computing Unlevered and Relevered Betas Using Miles-Ezzell Formulas
EXAMPLE 1
Assume that for guideline public company A:
Levered (published) equity beta: 1.2
Tax rate: 40%
Capital structure: 30% debt capital (market value of $15 million), 70% equity capital
(market value of $35 million)
Interest rate on debt: 7.5%
Beta of debt capital: 0.30
Using Formula 11.3 we get:
BU 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
¼
35m þ 14:5815
43:458m
¼
49:5815m
¼ 0:876
¼
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 equity beta: 0.90
Tax rate: 30%
Capital structure: 60% debt capital, 40% equity capital
Interest rate on debt capital: 15.0%
Beta of debt capital: 0.60
Using Formula 11.4 we get:
0:60
ð0:40 0:090Þ
BL ¼ 0:90 þ
ð0:90 0:20Þ 1 0:40
ð1 þ 0:090Þ
¼ 0:90 þ 1:5 0:70 ð1 :033Þ
¼ 1:92
Harris-Pringle Formulas
The Harris-Pringle formulas are alternative formulas for unlevering and levering
equity beta estimates that assume the tax shield is even riskier.7 Formula 11.5 shows
the Harris-Pringle formula for unlevering beta.
7
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.
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(Formula 11.5)
BU ¼
BL þ Bd ðW d =W e Þ
½1 þ ðW d =W e Þ
where: BU ¼ Unlevered beta of equity capital
BL ¼ Levered beta of equity capital
Bd ¼ Beta of debt capital
Wd ¼ Percent debt in the capital structure
We ¼ Percent equity in the capital structure
The companion Harris-Pringle formula for relevering beta is Formula 11.6.
(Formula 11.6)
BL ¼ BU þ ðBU Bd ÞðW d =W e Þ
where the definitions of the variables are the same as in Formulas 11.1 and 11.3.
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).
The 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 11.4.
We begin with Example 1, where Bd, the beta of debt capital, equals 0.30 (as in
Exhibit 11.3).
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 0.60 (i.e., the debt is lower-rated than the debt of the guideline public company
in Example 1).
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.8 The Practitioners’ method formula for unlevering
beta is shown in Formula 11.7
8
Tim Ogier, John Rugman, and Lucinda Spicer, The Real Cost of Capital (New York: Financial Times Prentice-Hall, 2004), 49.
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EXHIBIT 11.4 Computing Unlevered and Relevered Betas Using Harris-Pringle Formulas
EXAMPLE 1
Assume that for guideline public company A:
Levered (published) equity beta: 1.2
Tax rate: 40%
Capital structure: 30% debt capital (market value of $15 million), 70% equity capital (market value of $35 million)
Beta of debt capital: 0.30
Using Formula 11.5 we get:
0:30
1:2 þ 0:3
0:70
BU ¼ 0:30
1þ
0:70
¼ 0:93
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 the subject company tax rate and one or more assumed capital
structures.
EXAMPLE 2
Assume for the subject company:
Unlevered equity beta: 0.90
Tax rate: 30%
Capital structure: 60% debt capital, 40% equity capital
Beta of debt capital: 0.60
Using Formula 11.6 we get:
0:60
BL ¼ 0:90 þ
ð0:90 0:60Þ
0:40
¼ 1:35
(Formula 11.7)
BU ¼
BL
1 þ ðW d =W e Þ
where the definitions of the variables are the same as in Formula 11.1.
The companion Practitioners’ method formula for relevering beta is Formula 11.8.
(Formula 11.8)
BL ¼ BU ð1 þ ðW d =W e ÞÞ
where the definitions of the variables are the same as in Formula 11.1.
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
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EXHIBIT 11.5 Computing Unlevered and Relevered Betas Using Practitioners
Method Formulas
EXAMPLE 1
Assume that for guideline public company A:
Levered (published) equity beta: 1.2
Capital structure: 30% debt capital (market value of $15 million), 70% equity capital (market value of $35 million)
Using Formula 11.7 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 equity beta: 0.90
Capital structure: 60% debt capital, 40% equity capital
Using Formula 11.8 we get:
0:60
BL ¼ 0:90 1 þ
0:40
¼ 2:25
&
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.
The 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 11.5.
In Example 2, we relever the beta.
Capital Structure Weights
Each of the formulas discussed—Hamada, Miles-Ezzell, Harris-Pringle, and Practitioners’ method—is 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 have completed the
valuation.
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In Chapter 18, we discuss the use of the iterative process using the capital asset
pricing model (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). In
the Cost of Capital: Applications and Examples 4th ed. Workbook and Technical
Supplement, Chapter 5 presents a comprehensive example.
In Chapter 18, we also discuss 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. In the Cost of Capital: Applications and Examples 4th ed. Workbook and Technical Supplement, Chapter 6
presents a comprehensive example.
Fernandez Formulas
The unlevering and relevering 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).9 Chapter 18 discusses the use of market
value versus book value weights. Formula 11.9 is the Fernandez formula for unlevering beta.
(Formula 11.9)
BU ¼
BL þ ½ðW d =W e Þð1 tÞBd 1 þ ðW d =W e Þð1 tÞ
where the definitions of the variables are the same as in Formulas 11.1 and 11.3.
The companion Fernandez formula for relevering beta is Formula 11.10.
(Formula 11.10)
BL ¼ B U þ
Wd
ð1 tÞðBU Bd Þ
We
where the definitions of the variables are the same as in Formulas 11.1 and 11.3.
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 than
zero).
An example of applying Fernandez formulas is shown in Exhibit 11.6.
Formulas 11.9 and 11.10 are identical to Formulas 11.1 and 11.2 when Bd
equals zero (i.e., equity capital is bearing all of the risk of the firm).
9
Pablo Fernandez, ‘‘Levered and Unlevered Beta,’’ Working paper, April 20, 2006. Available
at http://ssrn.com/abstract=303170.
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197
EXHIBIT 11.6 Computing Unlevered and Relevered Betas Using Fernandez Formulas
EXAMPLE 1
Assume that for guideline public company A:
Levered (published) equity beta: 1.2
Tax rate: 40%
Capital structure: 30% debt capital (market value of $15 million), 70% equity capital
(market value of $35 million)
Interest rate on debt capital: 7.5%
Beta of debt capital: 0.30
Using Formula 11.9 we get:
0:30
1:2 þ
ð1 :40Þ:30
0:70
BU ¼
:30
ð1 :40Þ
1þ
:70
1:2 þ :077
1 þ :257
¼ 1:02
¼
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 equity beta: 0.90
Tax rate: 30%
Capital structure: 60% debt capital, 40% equity capital
Interest rate on debt capital: 15%
Beta of debt capital: 0.60
Using Formula 11.10 we get:
0:60
ð1 :30Þð:90 :60Þ
BL ¼ 0:90 þ
0:40
¼ 1:22
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.10
Adjusting Formulas for Other Components
of Capital Structure
The preceding formulas can be adjusted for other components of the capital structure, for example, preferred stock. One consideration is whether the company receives a tax deduction on its payment to investors (e.g., tax deduction on interest on
10
Eric de Bodt and Michel Levasseur, ‘‘A Short Note on the Hamada Formula,’’ Working
paper, March 26, 2007. Available at http://ssrn.com/abstract=976347.
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debt financing) or the company does not (e.g., no tax deduction on preferred dividends). As an example, the following expands the Miles-Ezzell formulas to include
preferred stock in the capital structure. For unlevering observed beta estimates, one
can use the following formula:
(Formula 11.11)
BU ¼ f½Me BL þðMp Bp Þ½1 kp =ð1þkp ÞþðMd Bd Þ½1ðtkdðptÞ Þ=ð1þkdðptÞ Þg
=Me þ Mp ½1 kp =ð1 þ kp Þ þ ½1 ðt kdðptÞ Þ=ð1 þ kdðptÞ Þ
where: Mp ¼ Market value of preferred capital
Bp ¼ Beta of preferred capital
kp ¼ Cost of preferred stock at market yield and the definitions of the other
variables are the same as in Formula 11.3.
A companion formula for relevering the beta estimate of the subject company is
Formula 11.12:
(Formula 11.12)
BL ¼ BU þ W p =W e ðBU Bp Þ½1 kp =ð1 þ kp Þ þ W d =W e ðBU Bd Þ
½1 ðt kdðptÞ Þ=ð1 þ kdðptÞ Þ
where: Wp ¼ Percent preferred capital in the capital structure and the definitions of
the other variables are the same as in Formulas 11.1, 11.3, and 11.11.
An example of applying the Miles-Ezzell formula including preferred stock is
shown in Exhibit 11.7.
CHOOSING AMONG UNLEVERING
AND LEVERING FORMULAS
Each of these formulas captures the risk of equity in different ways. For example, the
Hamada formula assumes that tax savings are fully realized as interest on debt is paid.
The other formulas capture the risk of debt and risk sharing in different ways. But
there may be additional negative impacts on the operations of the business from the
amount of debt in the capital structure. We discuss the cost of distress in Chapter 16.
Exhibit 11.8 summarizes the beta estimation, applying the various formulas as
shown in Exhibits 11.1–11.5.
The guideline public company in each example had a published (levered)
beta of 1.2. The levered beta was first unlevered (debt beta of the guideline
company debt ¼ 0.30).
The Hamada formulas, compared with the Miles-Ezzell, Harris-Pringle, and
Practitioners’ method formulas, assume that more of the total risk was business risk
rather than financial risk shared with the debt capital. That is, the Hamada formulas
assume that debt is constant. As cash flows in future periods are expected to increase, the unlevered or asset value of the business will increase, but debt will become a smaller and smaller percentage of the overall business value.
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199
EXHIBIT 11.7 Computing Unlevered and Relevered Betas Using Miles-Ezzell Formulas
Modified to Include Preferred Capital
EXAMPLE 1
Assume that for guideline public company A:
Levered (published) equity beta: 1.2
Tax rate: 40%
Capital structure: 25% debt capital (market value of $15 million), 16.67% preferred capital
(market value of $10 million), 58.33% equity capital (market value of $35 million)
Interest rate on debt capital: 7.5%
Yield on preferred capital: 9%
Beta of debt capital: 0.3
Beta of preferred capital: 0.4
Using Formula 11.11 we get:
BU ¼
½35 1:2 þ ½10 :5½1 :09=ð1 þ :09Þ þ ½15 :3½1 ð:4 :075Þ=ð1 þ :075Þ
35 þ 10½1 :09=ð1 þ :09Þ þ 15½1 ð:40 :075Þ=ð1 þ :075Þ
¼
42 þ 5½1 :08257 þ 4:5½1 :02791
35 þ 10½1 :08257 þ 15½1 :02791
¼
42 þ 4:5872 þ 4:3744
35 þ 9:1743 þ 14:5814
¼
50:9616
58:7557
¼ :8673
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 equity beta: 0.90
Tax rate: 30%
Capital structure: 50% debt capital, 10% preferred capital, and 40% equity capital
Interest rate on debt: 15%
Yield on preferred capital: 11%
Beta of debt capital: 0.60
Beta on preferred capital: 0.80
Using Formula 11.12 we get:
BL ¼ :9 þ
:10
:11
:50
:3 :15
ð:9 :8Þ 1 þ
ð:9 :6Þ 1 :40
ð1 þ :11Þ
:40
ð1 þ :15Þ
¼ :9 þ :25ð:1Þ½1 :0991 þ 1:25ð:3Þ½1 :0391
¼ :9 þ :0225 þ :3603
¼ 1:28
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EXHIBIT 11.8 Summary of Examples
Hamada
Miles-Ezzell
Harris-Pringle
Practitioners’
Fernandez
BU
BL
0.95
0.935
0.93
0.84
1.02
1.85
1.33
1.35
2.25
1.22
The concluded asset beta from the sample of guideline companies equaled
0.90. The debt of the subject company was greater, and the asset beta was
relevered with the greater debt of the subject company (the debt beta of the
subject company debt ¼ 0.60). Upon relevering, the Practitioners’ method
formula results in the greatest increase in total risk due to the increased financial risk of the subject company compared with the guideline public company,
as it assumes the least benefit from the tax shield.
The choice of unlevering and relevering formula is important. The examples
in Exhibit 11.8 indicate the impact on the resulting cost of equity capital estimates based on the risk sharing with the debt capital. The less likely that tax
deductions from interest payments will be realized in the periods in which interest is paid, the riskier the leverage and the greater the resulting cost of equity
capital. The choice of formula is closely tied to the formulation of the WACC
and the likelihood that the tax shield will be realized in the period interest is
paid. This is discussed further in Chapter 18.
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 11.1.
Assume that the higher-leveraged guideline public company probably 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 11.1, the Hamada formula, we get:
2:8
¼ 1:0
BU ¼
1 þ ð1 0:40Þð0:75=0:25Þ
But if we use Formula 11.7, the Practitioners’ method formula, we get:
BU ¼ :25 2:8 ¼ 0:70
Using the Hamada formula (11.1) 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,
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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.11 The Fernandez formulas are the most consistent if
the assumption is that the firm will maintain a fixed book value leverage ratio.12
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 or
operating risk. (See Chapter 5 for a discussion of business risk.) But the operating
leverage of the guideline companies may differ from that of the 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 of variable costs in the operating expense structure is equivalent to the weight of equity in the capital structure.
(Formula 11.13)
Bop ¼
11
BU
ð1 þ Fc =V c Þ
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.
12
Pablo Fernandez, ‘‘Levered and Unlevered Beta,’’ Working paper, April 20, 2006. Available
at http://ssrn.com/abstract=303170.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
where: Bop ¼ Operating beta or beta with effects of fixed operating expenses
removed
Fc ¼ Fixed operating costs (before costs of financing)
Vc 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 11.14 can be used for estimating the asset beta from the operating beta.
(Formula 11.14)
BU ¼ Bop ð1 þ Fc =V c Þ
An example of the adjustment is shown in Exhibit 11.9.
EXHIBIT 11.9 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 capital, 70% equity capital
Operating cost structure: 75% fixed, 25% variable
Beta of debt capital: Zero
Using the Practitioners’ method Formula 11.7:
BU ¼ 0:84
Using Formula 11.13 we get:
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 11.14 we get:
0:25
BU ¼ 0:40 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.
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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. 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 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. The formula is as follows:
(Formula 11.15)
BU ¼ ½Asset beta for operations ðOperating assets=Total assetsÞ
þ ½Asset beta for surplus assets ðSurplus assets=Total assetsÞ
where:
BU ¼ Overall company unlevered or asset beta
Asset beta for operations ¼ Unlevered beta for subject company operations without the impact of surplus assets
Operating assets ¼ Assets of subject company without surplus
assets
Total assets ¼ Total of operating plus surplus assets
Asset beta for surplus assets ¼ Unlevered beta for surplus assets
Surplus assets ¼ Assets that could be sold or distributed without impairing company operations
An example of the adjustment is shown in Exhibit 11.10.
EXHIBIT 11.10 Example of Adjusting Asset Beta Estimates for Excess Cash and Investments
Assume that for guideline public company A:
Levered (published) beta: 1.2
Tax rate: 40%
Capital structure: market value of debt capital $500, market value of equity capital $1,000
Interest rate on debt capital: 10%
Beta of debt capital: Zero
Excess cash and investments: $300
Using the Miles-Ezzell Formula 11.3 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 $500½0:9636
$1; 200
¼
$1;000 þ $481:80
$1;200
¼
$1;481:80
¼ 0:809
BU ¼
(continued )
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EXHIBIT 11.10 (Continued)
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, applying Formula
11.13 we get:
0:809 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 0:809=ð$1;200=$1;500Þ ¼ 1:01
The adjusted asset beta of the operating business is 1.01.
UNLEVERING EQUITY VOLATILITY
One method for valuing distressed businesses is an option pricing method (see Chapter 16). One input to the option pricing model is volatility (i.e., the volatility of the
unlevered business enterprise). If the subject company is public, one can derive the
volatility estimate of equity capital for use in, say, an option pricing model such as
the Black-Scholes formula, from the subject equity itself either by using the observed
volatility over a look-back period (just like one estimates betas) or by the implied
volatility embedded in the traded options, assuming the subject company has traded
options. But if one is estimating the volatility of equity of a closely held company,
one needs to develop a proxy estimate of the volatility of the equity of the subject
company by using the volatility estimates for guideline public companies. Then if
the capital structures of the guideline public companies significantly differ from
each other and/or the subject company has a capital structure that differs from that
of the guideline public companies, one needs to adjust the volatility estimate for the
differences in financial leverage. The process of unlevering volatility parallels the
process of unlevering betas.
Following the discussion in Chapter 5 that any company can be considered as a
portfolio of assets, one can think of the expected returns on the firm’s assets as the
weighted average of returns on these individual components. One can also consider
the volatility of the firm’s assets as a function of the volatility of the components of
the firm’s capital structure. Research shows that financial leverage has a large influence on equity volatility.13 That is, the expected return on the firm’s assets can be
estimated using Formula 11.16:
(Formula 11.16)
kA ¼ W d kdðptÞ þ W e ke
where:
13
kA ¼ Expected rate of return on the firm’s assets
¼ Discount rate for the firm’s assets
Jaewon Choi and Matthew Richardson, ‘‘The Volatility of Firm’s Assets,’’ Working paper,
March 14, 2009. Available at http://ssrn.com/abstract=1359368.
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205
Wd ¼ Percent debt in the capital structure
kd(pt) ¼ Expected rate of return on the firm’s debt capital =
¼ Cost of debt capital prior to tax effect
We ¼ Percent common equity in the capital structure
ke ¼ Expected rate of return on the firm’s common equity capital ¼ Cost
of common equity capital
and the expected volatility of the firm’s assets can be thought of as a function of the
expected volatilities of the equity and debt capital. Debt capital has volatility in its
returns as the value of the debt capital changes with changes in market interest rates
as the risk of the firm’s assets changes through the business cycle. The relationship
can be shown in Formula 11.17:14
(Formula 11.17)
s e ¼ s A þ s A ½1 2 ðkdðptÞ kA Þ=s 2A ðMd =Me Þ
where: s e ¼ Standard deviation of returns on firm’s common equity
s A ¼ Standard deviation of returns on firm’s assets
Md ¼ Market value of debt capital in the capital structure
Me ¼ Market value of common equity capital in the capital structure.
Given an estimate of the volatility of equity (s e) one can solve for the volatility
of the firm’s assets (s A).
SUMMARY
Care needs to be exercised in choosing the formula for unlevering betas. We generally recommend that practitioners use either the Miles-Ezzell, Harris-Pringle, or Fernandez formulas for unlevering guideline public 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
high leverage and 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 betas for larger companies,
examine 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.
14
Andrew A. Christie, ‘‘The Stochastic Behavior of Common Stock Variances: Value, Leverage and Interest Rate Effects,’’ Journal of Financial Economics 10 (1982): 411.
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EXHIBIT 11.11 Example of Applying Formulas for Relevering Beta
Assume that for the subject company:
Concluded unlevered equity (asset) beta for subject firm: 1.15
Tax rate: 30%
Capital structure: 50% debt (market value of debt $25 million), 50% equity (market value of
equity $25 million)
Interest rate on debt: 13%
Beta of debt capital: 0.40
Risk-free rate: 4.5%
ERP: 6%
&
Using the Hamada formula (11.2) we get:
BL ¼
¼
¼
¼
&
1:15ð1 þ ð1 0:30Þ0:50=0:50Þ
1:15ð1 þ 0:70ð1ÞÞ
1:15ð1:7Þ
1:955
Discount rate for common equity:
ke ¼ 0:045 þ 1:955 0:06
¼ 16:23%
&
Using the Miles-Ezzell Formula (11.4) we get:
0:50
ð0:30 0:13Þ
ð1:15 0:40Þ 1 0:50
ð1 þ 0:13Þ
¼ 1:15 þ 1 0:75 ð1 :035Þ
¼ 1:874
BL ¼ 1:15 þ
&
Discount rate for common equity:
ke ¼ 0:045 þ 1:874 0:06
¼ 15:74%
&
Using the Harris-Pringle Formula (11.6) we get:
BL ¼ 1:15 þ ð0:50=0:50Þð1:15 0:40Þ
¼ 1:15 þ 0:75
¼ 1:90
&
Discount rate for common equity:
ke ¼ 0:045 þ 1:90 0:06
¼ 15:90%
&
Using the Practitioners’ method formula (11.8) we get:
BL ¼ 1:15ð1 þ ð0:50=0:50ÞÞ
¼ 1:15ð2Þ
¼ 2:30
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&
Discount rate for common equity:
ke ¼ 0:045 þ 2:30 0:06
¼ 18:30%
&
Using the Fernandez formula (11.10) we get:
BL ¼
¼
¼
¼
&
1:15 þ ð0:50=0:50Þð1 0:30Þð1:15 0:40Þ
1:15 þ 1 0:70 0:75
1:15 þ 0:525
1:675
Discount rate for common equity:
ke ¼ 0:045 þ 1:675 0:06
¼ 14:55%
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). Finally, compare
your estimated relevered beta with industry betas (e.g., Ibbotson Cost of Capital
Yearbook). Are differences sensible, given the differences between the subject business and the typical company comprising the industry statistics? The choice of
unlevering and relevering formulas is important. Exhibit 11.11 shows an example of
the different relevered betas and discount rates of common equity capital, ke, you get
when applying Formulas 11.2, 11.4, 11.6, 11.8, and 11.10 (ke is estimated using
CAPM without regard to any size premium or company-specific risk premiums).
In this chapter we discussed various relevering formulas. But these formulas
probably underestimate the effect on beta due to distress when leverage is high. For
example, the Practitioners’ method formula for relevering beta (Formula 16) will
result in the largest increase in levered betas as debt increases, but the relationship
between leverage and the levered beta is linear. In fact, the correct relationship is
probably nonlinear. We discuss these relationships in Chapter 16 (discussing companies in distress) and Chapter 18 (discussing the WACC). We display the relationship
between beta risk (for equity and debt capital) as debt increases and the costs of
financial distress increase in Exhibit 16.8.
ADDITIONAL READING
Beaton, Neil, Stillian Ghaidarov, and William Brigida.‘‘Volatility Measurement and Its
Impact on Valuation for Early-Stage Companies.’’ Valuation Strategies (November–
December 2009).
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CHAPTER
12
Criticism of CAPM and Beta versus
Other Risk Measures
Introduction
CAPM Assumptions and Beta as a Risk Measure
Problems with CAPM Assumptions
Testing Asset Pricing Models
Testing Risk Factors Priced by the Market
Adjusting the Pure CAPM
Adjusting Beta for Risk of Company Size and Company-specific Risk
Risk Measures beyond Beta
Total Risk
Downside Risk
Value at Risk
Scenario-based Approach
Duration
Yield Spreads
Fundamental Risk
Summary
Technical Supplement Chapter 4—Example of Computing Downside Beta Estimates
INTRODUCTION
Even though the capital asset pricing model (CAPM) is the most widely used method
of estimating the 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 proposed and tested. 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 the use of beta as a reliable measure
of risk. This chapter explores these criticisms, alternative measures of risk, and the
resulting methods used to estimate the cost of equity capital.
The authors want to thank David Turney of Duff & Phelps LLC for preparing material for
this chapter.
208
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209
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 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 selection was straightforward, and variance was a familiar
concept. However, Markowitz found that other measures of portfolio risk resulted
in ‘‘better’’ 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) and (2) unrestricted risk-free borrowing and lending. The results were a
model where 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 the 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. However, the
1
Harry M. Markowitz, ‘‘Portfolio Selection,’’ Journal of Finance (March 1952): 77–91.
William F. Sharpe, ‘‘Capital Asset Prices: A Theory of Market Equilibrium under Conditions
of Risk,’’ Journal of Finance (September 1964): 425–442.
3
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.
2
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inability to hold a market portfolio or to choose not to hold a market portfolio will
force investors to consider more than market risk.
PROBLEMS WITH CAPM ASSUMPTIONS
A central assumption of the pure CAPM is that every investor holds the identical
market portfolio. This assumption follows from two other assumptions: homogeneous expectations and no transaction costs. However, many investors simply do
not hold widely diversified portfolios.
Market evidence indicates that individual investors do not wish to hold the market portfolio. In fact, investors are willing to pay fees and expenses to hold nonindexed mutual funds.4 Also, 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 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 concluded 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 distribution of security returns is not normal. Therefore, mean and variance of returns alone are insufficient
to characterize return distributions completely.
4
See, e.g., William N. Goetzmann and Alok Kumar, ‘‘Equity Portfolio Diversification,’’ Working paper, December 2004. Available at http://ssrn.com/abstract=627321. The authors studied
the diversification decisions of 60,000 individual investors during 1991 to 1996 and found
that the majority are underdiversified, with greatest underdiversification in retirement
accounts. See also Theodore Day, Yi Wang, and Yexiao Xu, ‘‘Investigating Underperformance
by Mutual Fund Portfolios,’’ Working paper, May 2001. Available at http://www.utdallas
.edu/~yexiaoxu/Mfd.PDF. The authors demonstrate that the portfolios of equity mutual funds
are not mean-variance efficient with respect to their holdings.
5
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 well-diversified portfolio
needs at least 40 stocks in recent decades due to increasing trends in idiosyncratic volatility.
6
James A. Bennett and Richard W. Sias, ‘‘How Diversifiable Is Firm-Specific Risk?’’ Working
paper, February 2007.
7
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,
Financial Review. Available at http://ssrn.com/abstract=906686.
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211
But despite its drawbacks, one author calls the CAPM ‘‘the first, most famous
and (so far) most widely used’’ model in asset pricing.8 CAPM, when properly applied, may provide a useful benchmark cost of equity capital, even for investments
in closely held businesses. It provides a benchmark measure of expected risk versus
expected return. However, given that many of the assumptions underlying CAPM
are not valid in real life, you must understand the issues and consider the benefits of
using alternative cost of capital methodologies and alternative risk measures. In fact,
using 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 method
such as CAPM.
TESTING ASSET PRICING MODELS
The CAPM is one of a series of what are called asset pricing models. The risk measure, beta, is a forward-looking concept similar to the equity risk premium (ERP).
The true beta must be estimated. Existing techniques for estimating beta generally
use historical data and assume that future stock returns will be sufficiently similar to
past stock returns to justify extrapolation of betas calculated using historical data.
A series of studies have examined the predictive power of beta. Such studies ask,
Do ‘‘high-beta’’ stocks earn higher returns in future periods? (The theory implies
that with a high beta, the market perceives the investment to be riskier.) Similarly,
do ‘‘low-beta’’ stocks earn lower returns in future periods? (The theory implies that
the lower the beta, the less risky the market perceives the investment to be.)
Many researchers are examining what factors can be identified that explain differences in realized stock returns. That is, what factors can we observe that explain
the realized return, Ri, for a stock i? Using publicly traded stocks (as those stocks
have returns that are most easily observed), researchers have tested the pure CAPM
to determine whether Ri is a function only of what is termed the index model:
(Formula 12.1)
where:
8
Ri Rf ¼ a þ Bi Rm Rf þ ei
Ri ¼ Realized return for stock of company i
Rf ¼ Risk-free rate of return
Rc Ra ¼ Realized return in excess of risk-free rate
a ¼ Coefficient on realized return in excess of risk-free rate when Bi
¼ zero
Bi ¼ Sensitivity of return of stock of company i to the market risk
premium or ERP
(Rm Rf) ¼ RPm or realized risk premiums used as an estimate of the ERP
ei ¼ error term, difference between predicted return and realized return.
Or rearranged, we get the formula expressed as a function of realized
returns, Ri:
Ri ¼ Rf þ ai þ Bi (Rm Rf) þ ei
John H. Cochrane, Asset Pricing, revised ed. (Princeton, NJ: Princeton University Press,
2005), 152.
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We can then identify the relationship of realized returns for stock i to the realized risk of those returns as follows:
Total
¼
Risk-free
þ
Unique
þ
Market
þ
Random Error
Return:
Ri
¼
Rf
þ
ai
þ
ei
Risk:
¼
s 2rf
þ
s 2a;i
þ
Bi (Rm Rf)
2
B2i Rm Rf
þ
s 2i
þ
s 2e;i
The resulting risk is the total risk of company i returns. Under the pure CAPM
theory, the market only prices a portion of total risk. Investors do not factor into
market prices unique or company-specific risk, because, under CAPM, investors
essentially reduce unique or company-specific risk to zero through diversification.
If the pure CAPM fully explains asset returns and we can find an accurate method to
estimate the forward looking beta, then one can use that relationship to estimate
E(Ri), the expected return (cost of capital) for an individual security, for the common equity of company i.
The following sections examine some of the research that has been done in testing whether CAPM holds; that is, on the average, do stocks with high betas earn
greater returns than stocks with low betas? The central issue is that if the pure
CAPM and beta do not explain a significant amount of the differences in stock returns, then should CAPM be the primary method for estimating the cost of equity
capital? If not, then we need to understand whether the failure of CAPM to explain
changes in returns is a result of poor beta estimates (we discussed beta estimation
methods in Chapter 10) or whether the market is pricing other factors in addition
to beta.
For example, Ri may be a function of various factors with Bi,j being the sensitivity of observed returns to a particular factor. Generalizing the possible relationships
we get the following formula:
(Formula 12.2)
Ri ¼ Rf þ Bi;m RPm þ Bi;s Si þ Bi;BV BV i þ Bi;u Ui þ þ ei
where:
Ri ¼ Realized return for stock of company i
Rf ¼ Risk-free rate of return
Bi,m ¼ Sensitivity of return of stock of company i to the
market risk premium or ERP
RPm ¼ ERP
Bi,s ¼ Sensitivity of return of stock of company i to a measure of size, S, of company i
Si ¼ Measure of size of company i
RPi,s ¼ Bi,s Si ¼ Risk premium for size of company i
Bi,BV ¼ Sensitivity of return of stock of company i to a measure of book value (typically measure of book-valueto-market-value of equity) of stock of company i
BVi ¼ Measure of book value (typically book-value-tomarket-value of equity) of stock of company i
RPi,BV ¼ Bi,BV BVi ¼ Risk premium for book value of company i
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Bi,u ¼ Sensitivity of return of stock of company i to a measure of unique or unsystematic risk of company i
Ui ¼ Measure of unique or unsystematic risk of company i
RPi,u ¼ Bi,u Ui ¼ Risk premium for unique or unsystematic risk of
company i
. . . ¼ Other factors
ei ¼ Error term, difference between predicted return and
realized return
The researchers are trying to understand what factors are priced by the market.
For example, if company size is a priced factor, then one will observe differences in
realized stock returns for different size firms. Pure CAPM holds that the only factors
that are priced are Rf and RPm and Bi,m and that these factors fully explain how the
returns on stock of company i differ, given the risk differences of company i from
other companies.
Once we better understand which factors have been priced by the market, we
can use the observed relationships to estimate the expected return for the stock of
company i, E(Ri) as follows:
(Formula 12.3)
EðRi Þ ¼ Rf þ Bi;m RPm þ Bi;s Si þ Bi;BV BV i þ Bi;u Ui þ where: E(Ri) ¼ Expected return for the stock of company i and the other variables
are defined as for Formula 12.2
We are not insinuating that the realized returns on company i stock have been
greater than the actual observed returns. However, if there are other factors that
explain stock market returns and one ignores a risk factor that is priced by the market, then one is underestimating the expected return of the stock of company i as if
company i stock was public and its risks were exposed to market pricing.
Testing Risk Factors Priced by the Market
Let us examine some of the research that has been performed to determine which
risk factors are priced by the market.
In one study, the researchers compared the expected returns using the pure
CAPM and beta estimates derived from varying look-back periods to realized
returns for the Dow Jones 30 companies during the period 1989 through 2008.
They found that using a beta estimate equal to 1.0, the weighted average beta for
the stock market as a whole, predicted individual stock market returns more accurately than calculated betas for those individual stocks.9
Researchers have also found that betas for individual stocks (and even industry
betas) are very unstable. The author of one study calculated beta estimates using
realized returns for 3,813 companies every day of 2-month periods using 60 months
9
Pablo Fernandez and Vincente Bermejo, ‘‘b ¼ 1 Does a Better Job Than Calculated Betas,’’
Working paper, May 19, 2009. Available at http://ssrn.com/abstract=1406923.
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of monthly returns for the look-back period. The sample included 450 of the S&P
500 companies at the time. The author also calculated industry beta estimates. The
use of a mechanical methodology resulted in betas that were counterintuitive.
For example, high-risk companies had smaller beta estimates derived from realized
returns than low-risk companies. In addition, the beta estimates were unstable, even
for the industry beta estimates.10
In theory, beta equals (repeating Formula 8.2):
(Formula 12.4)
Bi ¼
where:
CovðRi ; Rm Þ
VarðRm Þ
Bi ¼ Expected beta of the stock of company i
Cov(Ri,Rm) ¼ Expected covariance between the excess return (Ri-Rf) on security i and the excess market return (Rm-Rf)
Var(Rm) ¼ Expected variance of excess return on the overall stock market
Covariance measures the degree to which the return on a particular security and
the overall market’s return move together. Covariance is not volatility. Covariance
is a measure of the tendency to vary in the same way and in the same relative
amounts. But to really understand difficulties with estimating beta, we need to further examine the meaning of beta. Repeating Formulas 8.6 and 8.7, we get the
following:
(Formula 12.5)
r ¼ s i;m =½s i s m where:
r ¼ correlation coefficient between the returns on the security i and the
market, m
s i,m ¼ Covariance between returns on security i and the market, m
s i ¼ Standard deviation in returns on security i
s m ¼ Standard deviation in returns on the market, m
From this follows:
(Formula 12.6)
Bi ¼ r ½s i =s m Again, the correlation coefficient that matches the beta is the expected correlation coefficient, r, and the expected standard deviation of returns on the security of
company i and the expected standard deviation of returns on the market, m. Any
estimate of the correlation coefficient obtained by regressing realized returns, R, is
only an estimate of the expected correlation r. Similarly, the standard deviations of
realized returns on security i and the market, m, over a look-back period are only
estimates of the expected standard deviations of returns.
10
Pablo Fernandez, ‘‘Are Calculated Betas Worth for Anything?’’ Working paper, October
16, 2008. Available at http://ssrn.com/abstract=504565.
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What causes the beta relationship? Does beta come primarily from correlations
of stock returns with the market index (i.e., r, the true correlation coefficient), or
does beta come primarily from the relative return volatilities [s i/s m] or from other
source as well? As such, Bi, r, s i, and s m are estimates of expected beta, expected
correlation, and expected standard deviations. We often estimate these expected
parameters with statistics drawn from regressions of realized returns over look-back
periods. As the research shows, this estimation process using historical data may
cause estimation errors.
Research has also shown that volatility affects the accuracy of beta estimates. At
times when the market is highly volatile, beta estimates are less reliable, as are the
correlations of individual stock returns with returns on the market. The research
further shows that even though correlations break down in times of high market
volatility, volatilities generally move together. That is, when the market volatility
increases on the average, so does the volatility on individual stock returns. This
means that estimating betas during periods of high volatility of market returns will
generally provide less reliable estimates of beta than during periods of low volatility.
Are beta estimates drawn from realized return data becoming more or less reliable? In one study, the author found that while stock returns are most consistently and
strongly correlated with returns on the market for larger companies and while the
smallest companies’ beta estimates have become increasingly statistically significant in
recent years, beta estimates are not reliable during periods of high market volatility.11
Thus, reliability in making beta estimates is a function of the level of market volatility.
Beta reliability is also a function of reliability of firm financial statement information and the ability of the market to absorb and correctly interpret that information.
One study shows that the covariance of returns of a stock with the market is a function
of the covariance of the stock’s expected cash flows and the expected cash flows of the
market as a whole. This leads to the inference that the quality of accounting information and financial statement disclosure can have an impact on beta and cost of capital
estimation. The authors derive conditions under which an increase in information
quality will lead to a decline in the cost of capital.12 One can see that the movement of
debt capital ‘‘off balance sheet’’ to special purpose vehicles in recent years has made
the interpretation of company financial statements and the expected cash flows of a
company using special purpose vehicle financing methods more difficult to analyze.
In another study, the author found that realized returns on stocks with high
earnings-to-market-value of equity ratios were greater than predicted by beta and
that the realized returns on stocks with low earnings-to-market-value of equity
ratios were lower than predicted.13 In still another study, the author documented
that the average realized returns on small stocks were greater than predicted by
11
Daniel Suh, ‘‘The Correlations and Volatilities of Stock Returns: The CAPM Beta and the
Fama-French Factors,’’ Working paper, March 21, 2009. Available at http://ssrn.com/
abstract=1364567.
12
Richard Lambert, Christian Leuz, and Robert E. Verrecchia, ‘‘Accounting Information,
Disclosure, and the Cost of Capital,’’ Working paper, March 2006. Forthcoming in Journal
of Accounting Research. Available at http://ssrn.com/abstract=823504.
13
Sanjay Basu, ‘‘Investment Performance of Common Stocks in Relation to Their PriceEarnings Ratios: A Test of the Efficient Market Hypothesis,’’ Journal of Finance (June
1977): 129–156.
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CAPM (i.e., the size effect).14 In another study, the author found that companies
with high debt-to-market-value of equity ratios earned too high a return relative to
their betas.15 Further evidence that beta does not fully explain stock market returns
is evident in two studies in which the respective authors investigated the relationship
between average return and ratio of book-value-to-market-value of equity of stocks.
They found that returns on stocks with high book-value-to-market-value of equity
ratios had greater average realized returns than implied by their betas, and returns
on stocks with low book-value-to-market-value of equity ratios realized lower average realized returns than implied by their betas. These studies imply that ratios involving stock prices have information about expected returns that is missed by
betas. A stock’s price depends on expected cash flows and on expected returns that
discount expected cash flows to present value.16
Eugene Fama and Kenneth French (FF) published two studies critical of beta. In
one study they stated:
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.17 In a follow-on study, they found that problems with CAPM using U.S. data
show up in the same way in the stock returns of non-U.S. major markets.18,19
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. Finally,
CAPM cost of equity estimates for high book-value-to-market-value of equity stocks
(so-called value stocks) are too low, and estimates for low book-value-to-marketvalue of equity stocks (so-called growth stocks) are too high (relative to historical
returns). The implications of this work are if CAPM betas do not suffice to explain
expected returns, the market portfolio is not efficient. If this implication is true, then
CAPM has potentially fatal problems. These authors believe that their results point
14
Rolf W. Banz, ‘‘The Relationship between Return and Market Value of Common Stocks,’’
Journal of Financial Economics (March 1981): 3–18.
15
Laxmi Chand Bhandari, ‘‘Debt/Equity Ratio and Expected Common Stock Returns: Empirical Evidence,’’ Journal of Finance (June 1988): 507–528.
16
See Dennis W. Stattman, ‘‘Book Value and Stock Returns,’’ 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(3)
(1985): 9–17.
17
Eugene Fama and Kenneth French, ‘‘The Cross-Section of Expected Stock Returns,’’ Journal of Finance (June 1992): 427–486.
18
Eugene Fama and Kenneth French, ‘‘Value versus Growth: The International Evidence,’’
Journal of Finance (December 1998): 427–465.
19
One author demonstrated that the linear relationship between beta and the expected return
is obtained only by using continuously compounded returns. Tests of the CAPM that use,
say, monthly returns reflect the variance of returns but not the expected return as required
by an asset pricing model. Carl R. Schwinn, “The Measurement of Returns in Tests of the
CAPM,” Working paper, July 2010. Available at http://ssrn.com/abstract=1649478.
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217
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.20 Fama and
French go on to introduce another cost of equity capital model, the FF three-factor
model based on an empirical study confirming that size, earnings-to-price, debt-toequity, and book-value-to-market-value of equity ratios all add to an explanation of
realized returns provided by market betas. We discuss the FF three-factor model in
Chapter 2.
However, after the FF three-factor model was introduced, researchers discovered that this empirically based model did not reliably predict stock market returns
any better than CAPM. In several studies, researchers showed that within portfolios
formed on price ratios (e.g., book-value-to-market-value of equity ratios), stocks
with higher expected cash flows have had higher expected returns, a measure not
captured by the FF three-factor model or by CAPM.21
Other studies found additional problems with the FF three-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 of equity
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 of equity ratio, positive relation
between expected profitability and expected return is a direct prediction of valuation
theory. But the FF three-factor model does not indicate which stocks with the same
book-value-to-market-value of equity ratio are expected to have higher returns.22
Other researchers found that the pure CAPM works over the long run, but not
for asset pricing after 1963. That is, they found that, on the average, stocks with
higher betas realized higher returns and stocks with lower betas realized lower returns before 1963, but not after 1963. The book-value-to-market-value of equity
relationship better explains differences in returns after 1963 (though after 1980 its
explanatory power is almost zero). They estimated CAPM with time-varying betas
(dependent on economic conditions), constant market risk premium, and constant
market volatility. They found that time-varying betas explain the book-value-tomarket-value of equity effect except for stocks of medium-size companies. As a
result of their findings, they question if the post-1963 problem with beta is just a
small-sample-size issue.23
20
Eugene Fama and Kenneth French, ‘‘The Cross-Section of Expected Stock Returns,’’ Journal of Finance (June 1992): 427–486.
21
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.
22
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.
23
Andrew Ang and Joseph Chen, ‘‘CAPM over the Long-Run: 1926–2001,’’ Journal of
Empirical Finance (January 2007): 1–40.
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Alternatively, could it be that the problem with CAPM and beta is the use of
historical excess returns? Can it be that expectations regarding stock market returns
change so quickly that standard statistical methods for estimating beta using realized excess returns are just not sensitive enough to measure those expectations? In
one study, the researchers condition the beta estimate based on factors that seem to
predict the stage of business cycle in which the beta estimate is being made. For
example, they adjusted the beta estimate (derived from realized excess returns) by
the yield on T-bills, the yield spreads between 10-year and 1-year U.S. government
bonds, and the credit spread between 10-year AAA-rated corporate bonds and
10-year U.S. government bonds. They examined how realized betas changed as these
factors changed over time and applied the conditioning observations to the beta estimates. They were able to explain 60% of the differences in realized returns in the
period 1980 through 2004 using the adjusted beta versus only approximately 2%
using a static CAPM.24
In another study of the 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 historical realized returns to compare a basis for calculating beta estimates. They found 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 (market value) 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.25
In still another study, researchers used information embedded in the prices of
individual stock options and index options to compute forward-looking, or optionimplied, beta estimates. They compared their forward-looking beta estimates with
historically based beta estimates. They determined that the forward-looking beta
estimates had better predictive power (i.e., high-beta stocks earned greater returns
in future periods, etc.) than the best-performing historically based beta estimates in
about half the cases. In total, the forward-looking beta estimates explained about
22% of the variation in the returns across the securities studied.26
In another 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.
24
Devraj Basu and Alexander Stremme, ‘‘CAPM and Time-Varying Beta: The Cross Section
of Expected Returns,’’ Working paper, March 2007. Available at http://ssrn.com/
abstract=972255.
25
Alon Brav, Reuven Lehavy, and Roni Michaely, ‘‘Using Expectations to Test Asset Pricing
Models,’’ Financial Management (Autumn 2005): 5–37.
26
Peter Christoffersen, Kris Jacobs, and Gregory Vainberg, ‘‘Forward-Looking Betas,’’ Working paper, May 2, 2008. Available at http://ssrn.com/abstract=891467.
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These authors found that beta plays a significant role in explaining variation in
expected returns among firms (even after controlling for company size and bookvalue-to-market-value of equity ratio) and that the yield spread is highly correlated
with systematic risk. They found that previous research claiming beta is dead can be
a result of using realized returns rather than expected returns, as these returns often
differ. They found that the expected ERP implied by yield spreads is consistent with
the analysis of the ERP that should have been expected (see the Chapter 9 discussion
of Fama-French research) and less than the realized risk premiums during their test
period.
These same authors also found that the expected company size premium was
statistically significant. Their results showed that the expected company size premium moved countercyclically; investors seemed to perceive small companies as
riskier during business downturns.27
Other researchers have shown that stock returns are not normally
distributed—a finding that in and of itself demonstrates that beta cannot be the
sole measure of risk.28 The studies have found that distributions of stock returns
are skewed and have fatter tails than a normal distribution, with the lognormal
distribution having the best fit to observed returns for longer investment horizons.
Many critics of CAPM hold that the finding of nonnormalcy of returns alone
invalidates CAPM. While the assumption that returns are normally distributed is
a very crucial assumption to the derivation of CAPM, studies have found that the
utility loss or financial loss one may suffer by using a mean-variance efficiency
analysis are negligible.29 Levy shows that criticism based on rejection of the risk
aversion inherent in the classic mean-variance framework does not negate CAPM.
While some versions of expected investor utility require modifications to the classic choices of optimal mean-variance portfolios (the ‘‘efficient frontier’’), he found
that CAPM is theoretically intact. He goes on to show that CAPM cannot be
rejected on empirical grounds when ex ante rather than ex post estimates of beta
are employed.30
In summary, even though Levy concluded that CAPM cannot be rejected on
theoretical or empirical grounds, that conclusion does not negate the results of
empirical studies that show that beta alone is not a reliable measure of risk and realized future returns (at least not using betas drawn from realized 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:
27
Murillo Campello, Long Chen, and Lu Zhang, ‘‘Expected Returns, Yield Spreads, and
Asset Pricing Tests,’’ Working paper, January 2006. Available at http://ssrn.com/
abstract=491403.
28
Hsing Fang and Tsong-Yue 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; Fred 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.
29
Haim Levy, ‘‘The CAPM Is Alive and Well: A Review and Synthesis,’’ European Financial
Management 16(1) (2010): 43–71.
30
Haim Levy, ‘‘The CAPM Is Alive and Well: A Review and Synthesis,’’ European Financial
Management 16(1) (2010): 68.
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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.31
Adjusting the Pure CAPM
The prevailing theory regarding the cost of capital is that expected stock market returns are a function of beta and market risk. As we have shown, studies regarding
the reliability of this theory are mixed as to whether other factors, such as company
size, book-value-to-market-value of equity ratio, and unsystematic risk, are priced
by the market, but the preponderance of this research is that factors besides market
or systematic risk are priced by the market.
One study found no statistical relationship between unsystematic risk and
expected returns when unsystematic risk is measured in terms of residuals from the FF
three-factor model, not residuals from the index model estimate of the pure CAPM.32
The FF three-factor model controls for size and other differences among the firms.
Another study found a strong link between implied unsystematic volatility
derived from options (for companies with traded stock options) and future stock
returns for those same companies. Those authors point out that the problem with
most studies is that the studies measure unsystematic volatility by examining historical realized volatilities. These researchers find that historical realized volatilities
do not explain future returns of individual stocks when the pricing model includes
implied unsystematic volatility. Controlling for company characteristics such as
company size, company book-value-to-market-value of equity ratio of equity, and
liquidity of stock factors (short-sale constraints and monthly company open option
interest), they found that the market does in fact price company size, company
book-value-to-market-value of equity ratio, and implied forward unsystematic risk
of individual companies.33
Based on this research, we therefore believe that it is appropriate to adjust the
expected cost of equity capital for smaller companies when using CAPM to estimate
the cost of equity capital for the size effect (discussed in Chapter 13) and adjust the
expected cost of equity capital for unique or unsystematic risk factors recognized by
the market (discussed in Chapter 15).
Adjusting Beta for Risk of Company Size and
C o m p a n y - s p e c i fi c R i s k
Assume that you use the expanded CAPM (Formula 8.6) to estimate the cost of
equity capital. You now need to estimate the effect of expanding the CAPM model
31
Ravi Jagannathan and Zhenyu Wang, ‘‘The Conditional CAPM and the Cross-Section of
Expected Returns,’’ Journal of Finance (August 1996): 3–53.
32
Turan G. Bali and Nusret Cakici, ‘‘Idiosyncratic Volatility and the Cross-Section of
Expected Returns,’’ Working paper, July 2006. Available at http://ssrn.com/abstract=
886717.
33
Dean Diavatopoulos, James S. Doran, and David R. Peterson, ‘‘The Information Content in
Implied Idiosyncratic Volatility and the Cross-Section of Stock Returns: Evidence from the
Option Markets,’’ November 27, 2007; forthcoming in Journal of Futures Markets.
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221
in, say, an option pricing model. The effect of adding risk premiums for company
size and company-specific 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 12.7)
EðRÞ ¼ Rf þ ðBL RPM Þ þ RPs RPu
EðRÞ Rf ¼ ðBL RPM Þ þ RPs RPu
EðRÞ Rf
RPs
RPu
Be ¼
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:
(Formula 12.8)
s 2 ¼ B2L s 2M þ s 2e
where: 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
Substituting expanded beta Be for B and assuming s 2e is close to zero, we get:
(Formula 12.9)
s 2 ¼ Be s 2M
s 2 takes into account the entirety of the effect of all risk factors used in the
expanded CAPM.
RISK MEASURES BEYOND BETA
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 company size,
book value, unique or unsystematic risk, liquidity, and other factors?
Because of its prominence in the literature and in practical application, we discuss the size effect in Chapter 13. Similarly, we discuss unique or unsystematic risk
in Chapter 15.
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Total Risk
Shannon Pratt titled 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. Standard deviation turned out to be a very good risk
measure in the sense that the cross-sectional standard deviations of future one-year
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 12.1.
Other researchers have also found that total risk matters, and more than just
market risk or systematic risk is reflected in public market stock returns.34 To the
extent that undiversified investors, who, by definition, violate the assumptions
underlying CAPM, affect market pricing, then the impact resulting from the lack of
diversification should be reflected in pricing of the overall stock market. These
researchers have studied the relationship between average stock total risk and
market return over time and have found a significant positive relationship between
average stock return 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 concluded that total risk, including unique or idiosyncratic risk (volatility of returns),
drives the ability to forecast the stock market. We will discuss total risk and idiosyncratic risk further in Chapter 15.
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 a threshold
over a 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 the CAPM beta as the risk
measure to mean-semivariance (MS) CAPM with downside beta as the risk measure.
In their study, 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:
&
34
Downside betas for low-beta portfolios are greater than CAPM betas; CAPM
beta understates the risk of low-beta stocks.
Amit Goyal and Pedro Santa-Clara, ‘‘Idiosyncratic Risk Matters!’’ Journal of Finance (June
2003): 975–1008.
35
Philip S. Fortuna, ‘‘Old and New Perspectives on Equity Risk,’’ Practical Issues in Equity
Analysis, CFA Institute (AIMR) (February 2000): 37–45.
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Criticism of CAPM and Beta versus Other Risk Measures
Figure 1
Average Annual Rates of Return for Stock Portfolios of Different Risk Grades
(Annual Rates Derived from Geometric Mean IPRs)
Annual Rates of Return
08/26/2010
.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
Figure 2
Average Annual Rates of Return for Stock Portfolios of Different Risk Grades
(Annual Rates Derived from Geometric Mean IPRs)
Annual Rates of Return
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.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
EXHIBIT 12.1 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.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
&
Downside betas for high-beta portfolios are smaller than the CAPM betas;
CAPM beta overstates the 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 (market value) companies are not fully explained by the
MS CAPM downside beta. MS CAPM assumes perfect capital markets and ignores
transaction costs and market liquidity—issues that affect returns of small (market
value) 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 returns, which is consistent with investors placing greater weight
on downside risk than on upside gains. This downside risk is not fully reflected by
CAPM 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 12.2 repeats the data from ordinary least squares (OLS) beta and sum
beta from Exhibit 10.5 and adds downside beta estimates. Exhibit 12.2 compares
OLS betas, sum betas, and downside betas for different industries. We show an
example of calculating downside beta in Cost of Capital: Applications and Examples 4th ed. Workbook and Technical Supplement, Chapter 4.
Semivariance is an alternative downside risk measure. It is the ratio of semivariance of the individual security (variance on downside) to the semivariance of the
market.38 An important research finding is that semivariance is a meaningful alternative downside risk measure and should be considered in estimating the cost
of equity capital. Analysts using the CAPM should consider incorporating some
measure of semivariance in their beta estimates, as research has shown that betas
adjusted to reflect semivariance better explain stock returns. The inclusion of
this semivariance means capturing the ratio of semivariance of the individual
security (variance on downside) to the semivariance of the market in the cost of
equity capital calculation.
36
Thierry Post and Pim van Vliet, ‘‘Conditional Downside Risk and the CAPM,’’ Working
paper, June 2004. Available at http://ssrn.com/abstract=797286.
37
Andrew Ang, Joseph Chen, and Yuhang Xing, ‘‘Downside Risk,’’ Review of Economic
Studies 19(4) (March 2, 2006): 1191–1239.
38
Javier Estrada, ‘‘Downside Risk in Practice,’’ Journal of Applied Corporate Finance (Winter
2006): 117–125.
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Criticism of CAPM and Beta versus Other Risk Measures
EXHIBIT 12.2 Comparison of OLS Betas, Sum Betas, and Downside Betas
for Different Industries
Median
Data as of December 2008
Count
OLS
Beta
Sum
Beta
Downside
Beta
Computer Software
(SIC 7372)
All Companies
Over $1 Billion
Under $200 Million
151
29
79
1.47
1.24
1.49
1.57
1.17
1.76
1.69
1.20
1.89
Auto Parts (SIC 3714)
All Companies
Over $1 Billion
Under $200 Million
27
5
15
1.74
1.42
1.82
1.91
1.48
2.20
2.14
1.63
2.29
Healthcare (SIC 80)
All Companies
Over $1 Billion
Under $200 Million
81
11
44
1.11
0.96
1.34
1.37
1.01
1.53
1.49
1.20
1.83
Publishing (SIC 27)
All Companies
Over $1 Billion
Under $200 Million
39
8
19
1.30
1.03
1.41
1.57
1.28
1.85
1.53
1.20
1.75
Petroleum and Natural
Gas (SIC 1311)
All Companies
Over $1 Billion
Under $200 Million
152
41
76
1.50
1.16
1.78
1.86
1.46
2.30
1.87
1.49
2.28
Market value of equity as of December, 2008.
Source: Compiled from Standard & Poor’s Capital IQ data. Calculations by Duff & Phelps
LLC. Used with permission. All rights reserved.
Value at Risk
Value at risk (VaR) is a statistical measure of downside risk. VaR measures the largest percentage of the portfolio value that one might lose over a given time period, to
a given degree of certainty, based 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 firmwide VaR determines
one measure of a bank’s capital requirements.
Scenario-based Approach
In 1952, Harry Markowitz invented a framework for identifying those portfolios
with the highest return for a given risk level (or the lowest risk for a given return)
based upon risk, reward, and the correlation of the assets held within the portfolio.
Markowitz’s model is commonly referred to as mean-variance optimization and is
regarded as the basis of modern portfolio theory.
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Paul Kaplan and Sam Savage supplement Markowitz’s original work by uniting
it with the latest in economic theory, and new sophisticated statistical technologies
pioneered by Savage. The two have titled this enhanced model ‘‘Markowitz 2.0.’’39
Kaplan and Savage offer five technology-driven enhancements to the original
Markowitz mean-variance portfolio optimization framework:
&
&
&
&
&
The use of a scenario-based approach for describing return distributions rather
than employing standardized ‘‘bell-curves,’’ as is done in traditional meanvariance analysis. Kaplan and Savage point out that normalizing a distribution
by forcing it into a standardized curve can have the effect of minimizing the
probability of tail events—those events that are far to the left or far to the
right in a distribution. Employing the raw computational power available today to generate scenario-based (i.e., Monte Carlo simulation) distributions
produces more realistic return distributions, and allows for tail events to be
more accurately modeled. Tail events, such as the financial crisis of 2008 and
2009, are more probable than they would appear in a standardized lognormal
distribution.
Kaplan and Savage propose using geometric mean to describe investors’ reward
rather than arithmetic mean, as is done in traditional mean-variance analysis,
since ‘‘investors who plan on repeatedly reinvesting in the same strategy over an
indefinite period would seek the highest rate of growth for the portfolios as
measured by geometric mean.’’
The use of conditional value at risk (CVaR) rather than standard deviation to
describe investors’ risk. While standard deviation is a measure of the dispersion
of returns (both up and down), CVaR is a downside measure that focuses on
what investors can lose.
Kaplan and Savage replace the correlation matrix used with traditional meanvariance optimization with a scenario-based model.
Finally, Savage’s advances in the field of probability management are incorporated.40 One of the main disadvantages of scenario-based modeling is that it
requires huge amounts of data to be stored and manipulated. Savage has developed technology he has titled the Distribution String, or DISTTM, which compresses thousands of simulation trials into a single data element, cutting both
storage and processing time dramatically.
Duration
Is the length of time over which one expects to receive cash flows a good measure of
risk? That is, if one expects cash flows in early years, is that a less risky company or
project than one in which expected cash flows are not realized until more distant
future years?
Researchers developed a measure of implied duration based on traditional measures of bond duration (see Chapter 6). They project future cash distributions for
common equity using simple forecasting models based on historical financial data
39
40
2010 Ibbotson SBBI Classic Yearbook, Chapter 10, 121–125.
Sam Savage, The Flaw of Averages (Hoboken, NJ: John Wiley & Sons, 2009).
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for 10 years and spreading the remaining market value implicit in observed stock
price as a level perpetuity thereafter. They have found:
&
&
&
Stock return volatility and betas both increase as equity duration increases.
The book-value-to-market-value of equity factor may be interpreted as a noisy
duration factor.
Stocks categorized as value stocks generally have shorter equity duration than
stocks categorized as growth stocks.41
In addition to these studies, an even better method for quantifying uncertainty
as to both the amounts and the timing of expected economic income is to combine
the duration measure with the more commonly used risk measure, volatility. To
further refine this measurement, we think it appropriate to consider that not all investors have the same risk tolerance. This does not imply 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. Consequently, it may be appropriate to consider measuring the cost of equity capital in terms of the clientele attracted to investments as having certain risk characteristics.
In the interest of considering all relevant factors affecting total risk, we think
an important assumption to consider is that if investment returns in each future
year are approximately normally distributed, then the standard deviation of the
expected value of the investment increases as the duration of the net cash flows
increase, while 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. In
other words, risk as measured by standard deviation increases at a declining rate
over time.
For example, assume a project with an initial investment of $100 (time ¼ zero
to N). Assume the present value of future net cash flows increases 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 two-thirds 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 five years, the expected value
p is $200 (¼$100 þ 5 years $20)
with a standard deviation of $22.36 (¼$10 5). The normalized per-annum risk
of the investment equals $4.47 (¼$22.36/5 years).
41
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.
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At the end of 25 years, the expected value of the investment
p is $600 (¼ $100 ¼
25 years $20) with a standard deviation of $50 (¼$10 25). But the normalized per-annum risk of the investment has decreased to $2 (¼$50/25 years).
The conclusion we reach, which may seem counterintuitive, is that 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 than for the shorter-term investment.
Based on the clientele effect, if one assumes that there is a pool of investors with
short-term investment horizons, those investors are likely to 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 looked 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
compared 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. 42 The author found 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 risk relating to its size and company-specific risk.
In one study, the authors estimated the expected return on debt and equity
based on yield spreads. In their study, they looked at the differences in market yields
on bonds of different ratings. Using historical default rates on bonds, they estimated
expected default rates on bonds and a firm’s current cost of debt for use in its cost of
capital. They then estimated a market consensus risk premium by debt rating (i.e.,
the equity risk premium for specific ratings classes based on differences in leverage),
which can be used to estimate firm-specific cost of equity, given the subject company’s debt rating. The data on which they built their analyses are drawn from
1994–1999. The authors estimated equity risk premiums ranging from 3.1% for
42
Thomas Zellweger, ‘‘Time Horizon, Costs of Equity Capital, and Generic Investment Strategies of Firms,’’ Family Business Review 20(1) (March 2007): 1–15.
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229
AA-rated firms to 8.5% for B-rated companies over U.S. government bonds of comparable duration.43
These results can be used as a check for the cost of equity capital calculated
using other methods such as CAPM. For example, one can look at the current yield
on long-term U.S. government bonds as of the valuation date and add a risk premium to estimate the cost of equity capital. Assume the subject company debt was
rated B and the current yield on U.S. government bonds was 5%.44 The estimated
cost of equity capital for the subject company using this method therefore would be
5.0% plus 8.5% or 13.5%. The result embodies the company-specific risk of the
subject company, assuming that the company’s specific risk is captured in its debt
rating. In using this method, one should be sure to use current rather than historical
yields, as the yields on U.S. government bonds vary over time. This is particularly
true because these authors used data on pricing yields and default spreads for 1994–
1999, years that reflect default spreads at that time. To the extent that current yields
on rated debt are greater today than for that period, one may want to adjust the
reported results by increasing the recommended equity premiums by the difference
between current yields on, say, B-rated debt to the average yield on B-rated debt
during 1994–1999 (the spread on B-rated debt averaged 396 basis points over U.S.
government bond yields of bonds with comparable duration45). Also the analyst
needs to make sure the yield reflects underlying market factors and are not being
affected by the flight to quality.
Fundamental Risk
The Duff & Phelps Risk Study is discussed in Chapter 15. 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 stock
market–based measures of equity risk, such as beta.
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)
43
Ian Cooper and Sergei Davydenko, ‘‘Using Yield Spreads to Estimate Expected Returns on
Debt and Equity,’’ London Business School IFA Working Paper and EFA 2003 Annual
Conference Paper No. 901, December 2003. Available at http://ssrn.com/abstract=387380.
In another paper, Harjoat Bhamra, Lars-Alexander Kuehn, and llya Strebuaev, ‘‘The Levered Equity Risk Premium and Credit Spreads: A Unified Framework,’’ Working paper,
July 18, 2007, study the substantial empirical evidence that stock returns can be predicted
by credit spreads and that movement in stock-return volatility can explain movements in
credit spreads and explore the joint pricing of corporate bonds and stocks. Available at
http://ssrn.com/abstract=1016891.
44
The study found that the actual average years to maturity for all corporate bonds in the
database were approximately 8.5–9.0 years and an average duration of 6–6.75 years.
45
The study found that the actual average years to maturity for all corporate bonds in the
database were approximately 9.0 years and an average duration of 6.6 years for bonds
rated B.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
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. These
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 the most important factors
and the change in the discount rate over time the least important factor.46
In another study, the authors derived a simplified covariance risk adjustment
based on accounting variables. They used covariance of excess firm return on equity
(ROE) (residual earnings are 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 of equity factors as accounting-based risk measures to estimate the covariance risk of the firm.
They found that valuation errors are reduced (i.e., expected returns are more accurately measured) compared with the pure CAPM and FF three-factor models.47
SUMMARY
The conclusion that can be reached by studying the research reviewed in this chapter
is that pure 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 pure 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.’’48
While the CAPM may be theoretically valid and cannot be rejected if ex ante
rather than ex post parameters are employed, accurately measuring risk from observed data is the subject of continued research.
We conclude that beta alone 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 also recommend that analysts use multiple estimates of risk (for example, OLS beta, sum beta,
downside beta), compare the results, and use judgment to decide which estimate best
represents the risk of the subject company. Mechanical use of beta estimates from a
46
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.
47
Alexander Nekrasov and Pervin K. Shroff, ‘‘Fundamentals-Based Risk Measurements in
Valuation,’’ Working paper, January 2007. Available at http://ssrn.com/abstract=930729.
48
Qing Li, Maria Vassalou, and Yuhang Xing, ‘‘An Investment-Growth Asset Pricing
Model,’’ AFA 2002 Atlanta Meetings, March 7, 2001.
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Criticism of CAPM and Beta versus Other Risk Measures
231
published service, while seemingly easy to defend before a trier of fact, nonetheless
may lead to an erroneous estimate of the cost of equity capital.
TECHNICAL SUPPLEMENT CHAPTER 4—EXAMPLE OF
COMPUTING DOWNSIDE BETA ESTIMATES
The Cost of Capital: Applications and Examples 4th ed. Workbook and Technical
Supplement,Chapter 4, contains an example of computing downside beta estimates.
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CHAPTER
13
Size Effect
Introduction
Morningstar Studies
Using Morningstar Data in the Build-up Method
Using Morningstar Data in the CAPM Method
Duff & Phelps Studies
What Is ‘‘Size’’?
Description of the Data
Using the Duff & Phelps Risk Premium Report—Size Study in the Build-up Method
Using the Duff & Phelps Size Study in the CAPM Method
Estimating Size Premiums for Nonpublic Company
Summary
INTRODUCTION
In the chapters on the build-up model and the capital asset pricing model (CAPM),
we made reference to the size effect, based on the empirical observation that companies of smaller size are associated with greater risk and, therefore, have greater cost
of capital. While the size effect is a factor in other cost of capital methods—for
example, the Fama-French three-factor method (see Chapter 17)—in this chapter,
we discuss evidence of the size effect and its measurement in the context of the
build-up and CAPM methods.
The size effect is not without controversy. Here we first examine studies that
quantify the existence of the size effect. In Chapter 14, we examine the criticisms of
the size effect. The evidence that the size effect is a correction to the cost of capital
models is mostly applicable for smaller companies.
To help measure the size effect in terms of its impact on cost of equity capital,
this chapter presents empirical data from two independent sets of studies: the
Morningstar studies and the Duff & Phelps studies.
Both of these sets of studies use rate of return data developed at the University of Chicago Center for Research in Security Prices (CRSP) and document the
The authors want to thank David Turney of Duff and Phelps LLC for preparing material for
this chapter.
232
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Size Effect
233
empirical evidence that smaller firms have greater risk-adjusted equity returns
than large firms.1
MORNINGSTAR STUDIES
Morningstar, Inc., segregates New York Stock Exchange (NYSE) stock returns into
deciles by size, as measured by the aggregate market value of common equity, and
adds the returns on American Stock Exchange (AMEX) stocks and returns of NASDAQ stocks that fall into the respective size deciles as computed based on the NYSE.
The excess returns over the basic realized return for the market increase dramatically with decreasing size, as shown in Exhibit 13.1. This excess return is especially
noticeable for the smallest 10% of the companies.
Exhibit 13.2 shows the market capitalization by value of company equity of the
largest company in each of the respective decile groups as of September 30, 2008.
Morningstar also reports the results of changing the benchmark used to calculate the market portfolio from the Standard & Poor’s (S&P) 500 stock index to the
NYSE total value weighted index. Those results also support the relationship between size and realized return.
Morningstar examined alternative methods of calculating beta. For example,
Morningstar calculates betas based on excess annual returns. This method helps correct for certain problems associated with monthly data for smaller companies when
using more common methods of estimating beta. The ‘‘annual’’ betas are greater
for smaller companies than the betas derived using a monthly frequency of data.
Exhibit 13.3 displays the same analysis as Exhibit 13.1 except that annual betas
are used.
The annual betas are similar to betas calculated by Morningstar using the
sum beta method. As described in Chapter 10, the sum beta method is an alternative way of handling monthly data. This method can provide a better measure of
beta for small stocks by taking into account the lagged price reaction of stocks of
small companies to movements in the stock market. The data indicate that even
using the sum beta method, when applied to the CAPM, does not account for the
returns in excess of the risk-free rate historically found in small stocks. Notice
that the size premium for the 10th decile (the smallest companies) is smaller
using annual betas (size premium 4.43%) than monthly betas (size premium
5.81%) because of the higher beta. We discuss this more fully later when we
explore controversies surrounding the size premium.
More recently, Morningstar has divided the 10th decile into subcategories 10a
and 10b, with 10a being the top half of the decile and 10b the bottom half of the
decile (measured by market capitalization). Most recently, Morningstar further divided 10a into 10w and 10x and 10b into 10y and 10z. We present and discuss that
data and explore the difficulties of using market value to measure size and the problems with 10b in Chapter 14.
1
We have included in this chapter exhibits drawn from the Ibbotson SBBI 2009 Valuation
Yearbook and the Duff & Phelps Risk Premium Report 2009, both of which report data
through 2008. Other data presented herein also are displayed through 2008 for comparison
purposes.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
EXHIBIT 13.1 Returns in Excess of CAPM with S&P 500 Benchmark
Term Returns in Excess of CAPM Estimation for Decile Portfolios of the NYSE/AMEX/
NASDAQ, 1926–2008
Size Premium
(Return in
Arithmetic Realized Return
Estimated
Excess of
Mean
in Excess of
Return in Excess
CAPM)
Decile
Beta
Return
Riskless Ratey
of Riskless Ratez
1-Largest
2
3
4
5
6
7
8
9
10-Smallest
Mid-Cap, 3–5
Low-Cap, 6–8
Micro-Cap, 9–10
0.91
1.03
1.10
1.12
1.16
1.18
1.24
1.30
1.35
1.41
1.12
1.22
1.36
10.75%
12.51%
13.06%
13.45%
14.23%
14.48%
14.84%
15.95%
16.62%
20.13%
13.37%
14.86%
17.72%
5.56%
7.31%
7.87%
8.25%
9.03%
9.28%
9.65%
10.76%
11.42%
14.93%
8.18%
9.66%
12.52%
5.91%
6.69%
7.13%
7.28%
7.49%
7.65%
8.03%
8.41%
8.71%
9.12%
7.24%
7.92%
8.79%
0.36%
0.62%
0.74%
0.97%
1.54%
1.63%
1.62%
2.35%
2.71%
5.81%
0.94%
1.74%
3.74%
Betas are estimated from monthly portfolio total returns in excess of the 30-day U.S. Treasury bill total returns versus the S&P 500 total returns in excess of the 30-day U.S. Treasury
bill, January 1926–December 2008.
y
Historical riskless rate is measured by the 83-year arithmetic mean income return component
of 20-year U.S. government bonds (5.20%).
z
Calculated in the context of the CAPM by multiplying the equity risk premium by beta. The
equity risk premium is estimated by the arithmetic mean total return of the S&P 500
(11.67%) minus the arithmetic mean income return component of 20-year government bonds
(5.20%), 1926–2005.
Source: Ibbotson Stocks, Bonds, Bills, and Inflation1 2009 Valuation Yearbook. Copyright #
2009 Morningstar, Inc. All rights reserved. Used with permission. (Morningstar, Inc. acquired
Ibbotson Associates in 2006.) Calculated (or derived) based on CRSP1 data, # 2009 Center
for Research in Security Prices (CRSP1), University of Chicago Booth School of Business.
EXHIBIT 13.2 Size-Decile Portfolios of the NYSE/AMEX/NASDAQ, Largest
Company and Its Market Capitalization by Decile
Decile
1-Largest
2
3
4
5
6
7
Market Capitalization of Largest
Company (in thousands)
Company Name
$465,651,938
18,503,467
7,360,271
4,225,152
2,785,538
1,848,961
1,197,133
Exxon Mobil Corp.
Waste Mgmt Inc. Del
Reliant Energy
IMS Health Inc.
Family Dollar Stores Inc.
Bally Technologies Inc.
Temple Inland Inc.
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Size Effect
EXHIBIT 13.2 (Continued)
Decile
Market Capitalization of Largest
Company (in thousands)
8
9
10-Smallest
753,448
453,254
218,553
Company Name
Kronos Worldwide Inc.
SWS Group Inc.
Beazer Homes USA Inc.
Source: Ibbotson Stocks, Bonds, Bills, and Inflation1 2009 Valuation Yearbook.
Copyright # 2009 Morningstar, Inc. All rights reserved. Used with permission.
(Morningstar, Inc. acquired Ibbotson Associates in 2006.) Calculated (or
derived) based on CRSP1 data, # 2009 Center for Research in Security Prices
(CRSP1), University of Chicago Booth School of Business.
EXHIBIT 13.3 Returns in Excess of CAPM with S&P 500 Benchmark
Long-Term Returns in Excess of CAPM Estimation for Decile Portfolios of the NYSE/AMEX/
NASDAQ, with Annual Beta, 1926–2008
Realized
Estimated Size Premium
Arithmetic
Return in
Return in
(Return in
Mean
Excess of
Excess of
Excess of
Annual
Return
Riskless Ratey Riskless Ratez
CAPM)
Decile
Beta
1-Largest
2
3
4
5
6
7
8
9
10-Smallest
Mid-Cap, 3–5
Low-Cap, 6–8
Micro-Cap, 9–10
0.94
1.05
1.08
1.16
1.19
1.18
1.28
1.37
1.44
1.62
1.12
1.25
1.50
10.75%
12.51%
13.06%
13.45%
14.23%
14.48%
14.84%
15.95%
16.62%
20.13%
13.37%
14.86%
17.72%
5.56%
7.31%
7.87%
8.25%
9.03%
9.28%
9.65%
10.76%
11.42%
14.93%
8.18%
9.66%
12.52%
6.07%
6.67%
7.00%
7.50%
7.71%
7.64%
8.31%
8.89%
9.32%
10.50%
7.28%
8.10%
9.69%
0.51%
0.53%
0.86%
0.75%
1.32%
1.64%
1.33%
1.86%
2.10%
4.43%
0.90%
1.56%
2.83%
Betas are estimated from monthly portfolio total returns in excess of the 30-day U.S. Treasury bill total returns versus the S&P 500 total returns in excess of the 30-day U.S. Treasury
bill, January 1926–December 2008.
y
Historical riskless rate is measured by the 83-year arithmetic mean income return component
of 20-year U.S. government bonds (5.20%).
z
Calculated in the context of the CAPM by multiplying the equity risk premium by beta. The
equity risk premium is estimated by the arithmetic mean total return of the S&P 500
(11.67%) minus the arithmetic mean income return component of 20-year U.S. government
bonds (5.20%) from 1926 to 2008.
Source: Ibbotson Stocks, Bonds, Bills, and Inflation1 2009 Valuation Yearbook. Copyright
# 2009 Morningstar, Inc. All rights reserved. Used with permission. (Morningstar, Inc.
acquired Ibbotson Associates in 2006.) Calculated (or derived) based on CRSP1 data, #
2009 Center for Research in Security Prices (CRSP1), University of Chicago Booth School of
Business.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
From 1926 through 1981, Morningstar’s small-stock group was composed of
stocks making up the fifth quintile (i.e., ninth and tenth deciles) of the NYSE, ranked
by capitalization (price times number of shares outstanding). From 1982 forward,
the small-stock return series has been the total return achieved by the Dimensional
Fund Advisors (DFA) Small Company 9/10 (for 9th and 10th deciles) Fund. The
fund is a market-value-weighted index of the 9th and 10th deciles of the NYSE, plus
stocks listed on the AMEX and NASDAQ with the same or less capitalization than
the upper bound of the NYSE 9th decile.
The Morningstar data in the Ibbotson Stocks, Bonds, Bills, and Inflation (SBBI)
Valuation Yearbook show, for all size categories, both total realized returns in
excess of the risk-free rate and the size effect over and above CAPM (the latter
having already accounted for beta, which tends to be higher for smaller stocks),
so the data can be used either with a build-up method or with a CAPM method.
Morningstar also shows the average arithmetic mean return for each size category
and the arithmetic average return on the S&P 500 Index.
Using Morningstar Data in the Build-up Method
One can use the data to derive a small-company premium by subtracting the difference between the realized returns on small-company stocks and large-company
stocks for use in the build-up method (a procedure that Morningstar used to suggest). This small-company premium is not beta adjusted.
Exhibit 13.4 displays the small-stock premium, RPs, using data from 1926 to
2008 for the 10 deciles. The data can be used to estimate RPs in the build-up model,
Formula 13.1, which is the same as Formula 7.1.
EXHIBIT 13.4 Small-Company Premium Based on CRSP Decile Long-Term Total Returns for
Decile Portfolios at NYSE/AMEX/NASDAQ, 12/1926–12/2008
Decile Total
Returns for 80
Periods
1-Largest
2
3
4
5
6
7
8
9
10-Smallest
S&P 500
CRSP NYSE
Deciles 1–10
Geometric
Mean (%)
Arithmetic
Mean (%)
Standard
Deviation
(%)
SmallCompany
Premium (%)
Arithmetic
Mean/
Standard Dev.
8.9
10.07
10.44
10.35
10.91
10.89
10.83
11.02
11.10
12.47
9.62
9.50
10.75
12.51
13.06
13.45
14.23
14.48
14.84
15.95
16.62
20.13
11.67
11.46
19.48
22.33
23.89
26.13
26.90
27.59
29.82
34.44
36.70
44.95
20.57
20.05
0.92
0.84
1.39
1.78
2.56
2.81
3.17
4.28
4.95
8.46
0.552
0.560
0.547
0.515
0.529
0.525
0.498
0.463
0.453
0.448
0.567
0.572
Source: Compiled from CRSP1 data. Copyright # 2009 Center for Research in Security
Prices (CRSP1), University of Chicago Booth School of Business. Calculations by Duff &
Phelps LLC. Used with permission. All rights reserved.
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Size Effect
(Formula 13.1)
EðRi Þ ¼ Rf þ RPm þ RPs RPu
For example, if the subject company had a market capitalization that ranked it
in the eighth decile, RPs ¼ 4.3% (rounded).
In the build-up method, Morningstar now recommends starting with the return
in excess of the return predicted by CAPM (size premium) and then adding (or subtracting) an industry adjustment (which Morningstar’s Ibbotson SBBI Valuation
Yearbook presents for about 450 Standard Industrial Classification [SIC] codes)
instead of using a small-company premium (non–beta adjusted).
However, not all practitioners have endorsed this procedure. The following
quote from Michael Mattson (a former managing director of Ibbotson & Associates) is typical of the dissenting opinions:
I am not in total agreement with [Morningstar]’s contention that the only
size premium to use is the one that is ‘‘beta adjusted.’’ The problem in the
build-up approach is that we have no place for a beta, so the aspect of size
that is captured by a higher beta—an additional 0.4 over the market beta of
1.0 for 10th decile stocks—is not captured anywhere. Using the full size premium, as opposed to the beta-adjusted one, assumes that the small company
being valued has similar risk characteristics to the average 10th decile company—this may not be such a bad assumption for many of the smallest companies we value. Assuming that 10th decile companies are not in riskier
industries than companies in the other size groupings, then their higher beta
is due primarily to their size and the size effect is in both the beta and the
premium over the CAPM line.2
Further, consistent with the discussion on the equity risk premium (Chapter 9),
the arithmetic average realized small-company premium is simply a measurement of
what happened in the past. Some people interpret it as an indication of what might
be expected in the future. Again, as in the discussion of the equity risk premium, one
must choose the appropriate period to include in the sample years, and the sample
years should represent current expectations of investors.
If we examine the data in Exhibit 13.4 for the 10th decile, we see that the
standard deviation of returns is approximately 45%, and the small-company premium is 8.4% (for 1926–2008). But looking at a shorter period, say, the last
50 years in Exhibit 13.5 (1959 to 2008), we see that the standard deviation of
returns for the 10th decile is approximately 32%, and the small-company premium is only 4.7%.
The arithmetic average realized return and the standard deviation of realized
returns for the 10th decile and the derived small-company premiums for varying
periods are displayed in Exhibit 13.6.
As in the discussion on the equity risk premium, the realized returns from periods before the mid-1950s appear to bias upward the results of using any average for
2
Shannon P. Pratt, Cost of Capital: Estimation and Applications, 2nd ed. (Hoboken, NJ: John
Wiley & Sons, 2002), 183.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
EXHIBIT 13.5 Small-Company Premium Based on CRSP Decile Long-Term Total Returns for
Decile Portfolios at NYSE/AMEX/Nasdaq, 12/1958–12/2008
Decile Total
Returns for
50 Periods
Geometric
Mean (%)
Arithmetic
Mean (%)
Standard
Deviation
(%)
SmallCompany
Premium (%)
Arithmetic
Mean/
Standard Dev.
8.63
9.81
10.85
10.64
10.90
11.44
10.83
11.62
10.64
10.93
9.18
9.24
10.09
11.39
12.57
12.59
12.97
13.86
13.45
14.62
13.98
15.35
10.62
10.75
17.39
17.95
18.98
20.42
21.15
22.81
23.82
26.10
27.52
32.27
17.20
17.63
0.53
0.77
1.95
1.97
2.35
3.24
2.83
4.00
3.36
4.73
0.580
0.635
0.662
0.617
0.613
0.608
0.565
0.560
0.508
0.476
0.617
0.610
1-Largest
2
3
4
5
6
7
8
9
10-Smallest
S&P 500
CRSP NYSE
Deciles 1–10
Source: Compiled from CRSP1 data. Copyright # 2009 Center for Research in Security
Prices (CRSP1), University of Chicago Booth School of Business. Calculations by Duff &
Phelps LLC. Used with permission. All rights reserved.
the entire post-1925 period. The standard deviations of small-stock returns in recent
periods are consistently smaller than they were in the earlier period. These results
point to the conclusion that a realistic, current small-company premium is in the
range of 2% to 5%, and not 8þ%, for companies that would fall in the 10th decile
where size is measured by market capitalization. This conclusion parallels the earlier
conclusion that using realized risk premiums for the entire post-1925 period as an
estimate of the current equity risk premium results in an unrealistically high result
relative to current expectations.
EXHIBIT 13.6 Small-Company Premium Based on CRSP Decile Long-Term Total Returns for
the 10th-Decile Portfolios at NYSE/AMEX/NASDAQ for Various Time Periods
Period
1989–2008
1979–2008
1969–2008
1959–2008
1926–2008
Years
Arithmetic
Mean (%)
Standard
Deviation (%)
Small-Stock
Premium (%)
Arithmetic Mean/
Standard Dev.
20
30
40
50
83
13.13
14.06
12.55
15.35
20.13
30.37
27.34
30.24
32.27
44.95
2.78
1.53
1.95
4.73
8.46
0.432
0.514
0.415
0.476
0.448
Source: Compiled from CRSP1 data. Copyright # 2009 Center for Research in Security
Prices (CRSP1), University of Chicago Booth School of Business. Calculations by Duff &
Phelps LLC. Used with permission. All rights reserved.
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Size Effect
Using Morningstar Data in the CAPM Method
In using the data with a CAPM model, one would use the size premium over the
CAPM-indicated equity risk premium, recognizing that the beta has captured some
of the size effect. The size premium is also called the beta-adjusted size premium.
The size premium is an empirically observed correction to the CAPM. For the
CAPM, there would be no further adjustment (except for a possible company-specific adjustment), because beta presumably would reflect any industry effects.
We get Formula 13.2, which is the same as Formula 8.5:
(Formula 13.2)
EðRi Þ ¼ Rf þ BðRPm Þ þ RPs RPu
Using the data in Exhibit 13.1 or Exhibit 13.3, and assuming that the subject
company was ranked in the eighth decile by market capitalization, we would get:
RPs ¼ 2:4% ðfrom Exhibit 13:1; roundedÞ or RPs
¼ 1:9% ðfrom Exhibit 13:3; roundedÞ
As previously mentioned, in the build-up method, the applicable procedure is
less clear-cut. But Morningstar now recommends starting with the return in excess
of CAPM for both the build-up and CAPM methods.
Does one adjust the Morningstar size premium data if one estimates the equity
risk premium to be smaller than the realized risk premium using data from 1926
through the most recent year? For example, assume the valuation date is December
31, 2008, and one concludes that the most reasonable estimate of the ERP is less
than the arithmetic average of the realized risk premiums for the period 1926 to
2008: 6.5% (total return on the S&P 500 [11.67%] minus income return component on long-term government bonds [5.20%]).3 One concludes that those realized
risk premiums were influenced by economic factors not expected to recur (e.g., decrease in the cost of equity over time as income tax rates decreased), and the arithmetic average is too high an estimate of the current ERP. Based on that analysis, one
decides to use the estimate of the ERP equal to Morningstar’s supply-side estimate
(5.7% arithmetic average for the period 1926 to 2008) as one’s estimated ERP.4
One multiplies the ERP by beta and then adds the size premium.
One commentator has suggested that in such an example, the Morningstar size
premiums (premiums in excess of that predicted by the CAPM) should be increased
by the difference between the historical realized risk premium and the supply-side
risk premium (6.5% minus 5.7% in this example). This is not correct. If one believes
that economic factors not expected to recur caused the returns on the broad market
to be higher than one would have expected, then the returns of stocks comprising all
deciles were probably influenced by the same factors.
Further discussion of the use of the Morningstar small-stock data is included in
Chapter 20.
3
Arithmetic average of realized risk premiums for 83 years (1926–2008). SBBI 2009 Valuation Yearbook (Chicago: Morningstar, 2009), 94.
4
Supply-side equity risk premium (arithmetic average) (1926–2008), Tables 5–6; SBBI 2009
Valuation Yearbook (Chicago: Morningstar, 2009), 69.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
DUFF & PHELPS STUDIES
Beginning in 1990, Roger Grabowski began closely studying the returns by size of
companies as reported in the SBBI Yearbooks. He was interested in understanding
whether the stock market recognized the differences in risk of different size companies where size was measured by alternative accounting or fundamental measures
instead of only by market price of the company stock (as it is in the SBBI Yearbooks
for the 10 deciles).
Grabowski, working with his then colleague David King, engaged CRSP to
build a database combining stock prices, number of shares, and dividend data by
company from the CRSP database with accounting and other data from the Standard & Poor’s Compustat database and designed software to analyze the data.
Thereafter, they published a series of articles reporting their findings.5 The Duff &
Phelps Risk Premium Report—Size Study annually updates this research.6
What Is ‘‘Size’’?
Traditionally, researchers have used market value of equity as a measure of size in
conducting historical rate of return research. For instance, this is the basis of the
small-company return series published in the SBBI Yearbooks. But there are various
reasons for seeking alternative measures of size.
First, it has been pointed out in the financial literature that researchers may unwittingly introduce a bias when ranking companies by ‘‘market value.’’7 Market
value of a company’s equity is not just a function of size; it is also a function of the
discount rate. Therefore, some companies will not be risky because they are small,
but instead will be small (low market value of equity) because they are risky (high
discount rate). Choosing a measure of size other than market value of equity helps
isolate the effects that are purely due to small size.
Also, the market value of equity is an imperfect measure of the size of a company’s operations. Companies with large sales or operating income may have a
small market value of equity if they are highly leveraged.
The use of fundamental accounting measures of size (such as assets or net income) may have the practical applied benefit of removing the need to make a guesstimate of size when determining a discount rate. For example, such data eliminate
certain circularities that may arise in applying market value of equity size-based
adjustments (i.e., where size is measured by market value of equity and one needs to
5
David King, CFA, is National Technical Director of Valuation Services at Mesirow Financial
Consulting, LLC. The research began when both he and Roger Grabowski were at Price
Waterhouse, predecessor firm to PricewaterhouseCoopers.
6
This section is adapted from the Duff & Phelps Risk Premium Report 2009. Used with permission. The Risk Premium Report was published as the Standard & Poor’s Corporate
Value Consulting Risk Premium Report for reports from 2002 to 2004 and as the PricewaterhouseCoopers Risk Premium Reports and Price Waterhouse Risk Premium Reports for
years before 2002.
7
Jonathan B. Berk, ‘‘A Critique of Size Related Anomalies,’’ Review of Financial Studies 8(2)
(Summer 1995): 275–286.
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241
know size to choose the adjustment) to a discount rate for determining the market
value of a nonpublic business.
Because Grabowski and King were interested in understanding how the stock
market prices the risk of established companies based on their size, the Duff &
Phelps studies are limited to companies with a track record of profitable performance. The company selection process is designed to parallel the process used in selecting guideline public companies when an analyst determines guideline public
companies in applying the market approach. For example, assume that the analyst
identifies 10 possible guideline public companies that are in the same SIC code as
the subject profitable company. One criterion for selecting among the guideline public companies is to include only profitable companies. That same selection criterion
was used in developing the database for the Duff & Phelps studies.
The Duff & Phelps studies measure size using eight alternative measures of company size, including fundamental accounting characteristics such as sales and book
value. The data show a clear inverse relationship between size and historical rates of
return and realized premiums.
Description of the Data
The Duff & Phelps studies make use of the CRSP database, together with Standard
& Poor’s Compustat database. This causes the population of companies considered to be limited to firms that are covered by both databases. The Duff & Phelps
studies exclude American Depository Receipts (ADRs), nonoperating holding
companies, and financial services companies (Standard Industrial Classification
[SIC] code ¼ 6).8 The Duff & Phelps studies report historical returns for the
period 1963 (inception of the Compustat database) through the current year-end.9
For each year covered, the Duff & Phelps studies consider only financial data for
the fiscal year ending no later than September of the previous year. For example, in
allocating a company to a portfolio to calculate realized returns for calendar year
1995, they consider financial data through the latest fiscal year ending September
1994 or earlier (depending on when the company’s fiscal year ended). In this way,
the study ensures that returns in any year are calculated for companies for which
all information was known before the year 1995 began. For example, companies
included in 1963 are screened by looking at data for the latest fiscal year ending
September 1962 (or earlier) and the prior four fiscal years; companies included for
1964 are screened looking at data for the latest fiscal year ending September 1963
(or earlier) and the prior four fiscal years; and so on.
8
Some of the financial data used in the Duff & Phelps studies are difficult to apply to many
companies in the financial sector (e.g., ‘‘sales’’ at a commercial bank), and financial institutions support a much higher ratio of debt to equity than is normal in other industries. Also,
companies in the financial services sector were poorly represented during the early years of
the Compustat database.
9
Compustat data are available for some companies going back into the 1950s, but these earlier data are only back histories for companies that were added to Compustat in 1963 or
later. Grabowski and King begin with 1963 data to avoid the obvious selection bias that
would otherwise result.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
For each year since 1963, the universe of companies excludes companies lacking
five years of publicly traded price history (i.e., companies with recent initial public
offerings), companies with sales below $1 million in any of the previous five fiscal
years (i.e., start-up companies), and companies with a negative five-year-average
EBITDA (earnings before interest, taxes, depreciation, and amortization) for the
previous five fiscal years (unprofitable companies).
Companies that meet these criteria and therefore are included in the base
dataset have been traded for several years, have been selling at least a minimal
quantity of product, and have achieved some degree of positive cash flow from
operations. This screening was a response to the argument that the small-cap
universe may include a disproportionate number of recent initial public offerings,
high-technology companies, and start-up companies, and that these unseasoned
companies may be inherently riskier than companies with a track record of viable performance. The number of companies eliminated by these criteria varies
from year to year.
Once the companies just described were eliminated, companies with any one of
these characteristics were excluded from the base set of companies:
&
&
&
&
&
Identified by Compustat as in bankruptcy or in liquidation
With five-year-average net income available to common equity for the previous
five years less than zero (either in absolute terms or as a percentage of the book
value of common equity)
With five-year-average operating income for the previous five years (defined as
sales minus [cost of goods sold plus selling, general, and administrative expenses
plus depreciation expense]) less than zero (either in absolute terms or as a percentage of net sales)
With negative book value of equity at any of the previous five fiscal year-ends
With debt to total capital of more than 80% (with ‘‘debt’’ measured as preferred
stock at carrying value plus long-term debt, including current portion, and notes
payable in book value terms and with total capital measured as book value of
debt plus market value of equity)
These companies are considered the high-financial-risk companies that are discussed further in Chapter 16.
Segregating such high-financial-risk companies isolates the effects of high financial risk. Otherwise, the results might be biased for smaller companies to the extent
that highly leveraged and financially distressed companies tend to have both high
returns and low market values. It is possible to imagine financially distressed (or
highly risky) companies that lack any of the listed characteristics. It is also easy to
imagine companies that have one of these characteristics but that would not be considered financially distressed. The high-financial-risk companies are largely companies whose financial condition is significantly inferior to the average, financially
healthy public company.
The exclusion of companies based on historical financial performance does not
imply any unusual foresight on the part of investors in these portfolios. In forming
portfolios to calculate returns for a given year, companies are excluded on the basis
of performance during previous years (e.g., average net income for the five prior fiscal years) rather than current or future years.
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243
The Duff & Phelps studies exclude or segregate certain types of companies on
the basis of past financial performance in response to arguments that the inclusion
of such companies might introduce a bias in favor of the size effect to the extent that
such companies tend to have low market values. A critic unfamiliar with this history
might question whether we are introducing a bias by excluding such companies.10
Again, the procedures parallel procedures used in applying the guideline publicly traded company method of valuation. Assume one is valuing a profitable company. First one screens all potential guideline companies, say, by SIC code; then one
reviews the financial and operating data of the potential guideline companies and
excludes those companies that may be unprofitable; and so on. The financial data
used in this screening include only the data that were known or knowable as of the
valuation date (equivalent in this case to the beginning of the year for which the
return is calculated). One then develops multiples for companies that best compare
to the financial and operating data of the subject company. The screens used in the
Duff & Phelps studies are similar to the screening one uses in determining appropriate guideline public companies.
Ranking Companies by Size For the companies remaining in the base dataset, the
Duff & Phelps Size Study forms portfolios of securities based on relative size. Since
NASDAQ and AMEX companies are generally small relative to NYSE companies,
their addition to the dataset produces portfolios that are more heavily populated by
small-cap stocks. The portfolios are rebalanced annually; that is, the companies are
reranked and sorted at the beginning of each year. Portfolio rates of return were
calculated using an equal-weighted average of the companies in the portfolio (to understand the returns of the median or typical company included in each portfolio).
Correcting for ‘‘Delisting Bias’’ An article by Tyler Shumway provided evidence that
the CRSP database omits delisting returns for a large number of companies.11 These
returns are missing for the month in which a company is delisted from an exchange.
Shumway collected data for a large number of companies that had been delisted for
performance reasons (such as bankruptcy or insufficient capital). He found that investors incurred an average loss of about 30% after delisting. He further showed
that delisting for nonperformance reasons (such as mergers or changes of exchange)
tended to have a neutral impact in the month that the delisting occurred. Skeptics of
the small-stock phenomenon often dismiss these results, holding that the returns
of the small companies are biased high because that group of companies is most
influenced by this overestimate of the realized returns.
The Duff & Phelps studies incorporate the Shumway evidence into their rate of
return calculations. In calculating rates of return, they impute a 30% loss in the
month of delisting in all cases in which CRSP identified the reason for delisting as
performance related and in all cases in which the reason for delisting was unknown.
10
Grabowski and King report that they ran alternative analyses in which no company was
excluded or segregated on the basis of past history (i.e., using all available nonfinancial
companies), and the results are similar to those reported in the Risk Premium Report.
11
Tyler Shumway, ‘‘The Delisting Bias in CRSP Data,’’ Journal of Finance (March 1997):
327–340.
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Eight Measures of Size, 25 Size Categories The Size Study presents average realized
risk premiums for the period 1963 through the most recent year-end. As discussed in
Chapter 9, a longer-run average realized risk premium often is used as an indicator
of the expected equity risk premium of a typical investor.
To calculate realized risk premiums, the Size Study first calculates an average
rate of return for each portfolio over our sample period. Returns are based on dividend income plus capital appreciation and represent returns after corporate-level income taxes (but before owner-level taxes). Then the average income return earned
on long-term U.S. government bonds over the same period (using SBBI data) is subtracted to arrive at an average realized risk premium.
The eight Size Study exhibits for use in the build-up model are:
Measures of Equity Size
1. Market value of common equity
2. Book value of common equity
3. Five-year average net income before extraordinary items for previous five fiscal
years
Measures of Company Size
1.
2.
3.
4.
5.
Market value of invested capital (MVIC)
Total assets (as reported on the balance sheet)
Five-year average EBITDA for the previous five fiscal years
Sales
Number of employees
The exhibits of the Size Study include these statistics for each of 25 size
categories:
&
&
&
&
&
&
&
&
&
&
Average of the size measure (e.g., average number of employees) for the latest
year
Log (base-10) of the median of the size measure
Number of companies in each portfolio in the latest year
Beta estimate relative to the S&P 500 calculated using the sum beta method
applied to monthly returns for 1963 through the latest year
Standard deviation of annual realized equity returns for each portfolio since
1963
Geometric average realized equity return for each portfolio since 1963
Arithmetic average realized equity return for each portfolio since 1963
Arithmetic average realized risk premium (realized equity return over long-term
government bonds) since 1963 (labeled ‘‘arithmetic risk premium’’)
‘‘Smoothed’’ average realized risk premium (i.e., the fitted premium from a regression with the average realized risk premium as the dependent variable and
the logarithm of the size measure as the independent variable) (labeled
‘‘smoothed average risk premium’’)
Average carrying value of the sum of preferred stock plus long-term debt (including current portion) plus notes payable (‘‘debt’’) as a percent of MVIC since
1963 (labeled ‘‘average debt/MVIC’’)
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Size Effect
The Size Study presents the coefficients and other statistics from the regression
analysis of the average realized risk premiums (the regression results in the
smoothed average realized risk premiums).
We have included two of their Size Study exhibits for use in the build-up
method: Exhibit 13.7, in which size is measured by market value of common equity,
and Exhibit 13.8, in which size is measured by book value of common equity (exhibits reproduced herein are for years ending 2008).
Each of eight Size Study exhibits for use in the build-up method displays one line
of data for each of the 25 size-ranked categories or portfolios, plus a separate line
for the high-financial-risk portfolio.
Observations on the Data By whatever measure of size they use, the result is a clear
inverse relationship between size and historical risk premiums. When we sort by a
size measure other than market value, the relationship is slightly flattened. The average realized risk premiums for the smallest companies are generally lower when we
sort by criteria other than market value.
For the 25 size-ranked portfolios with an arithmetic equity risk premium in
excess of the average realized market premium (3.84% for 1963 through 2008 from
the SBBI series for large companies), the premium incorporates a non–beta-adjusted
size premium.
The historic average debt to MVIC ratio is approximately 30% for most size
categories for all of the sorting criteria. This suggests that differences in leverage do
not explain the small-company effect in the data for these years.
The 25 portfolios of the Size Study exclude companies with high leverage, categorized as the high-financial-risk companies. The leverage in the high-financial-risk
portfolio is significantly greater than that of any of the other portfolios. The return
data for the high-financial-risk companies are reported in separate exhibits and discussed in Chapter 17. Beginning with the Risk Premium Report 2010, the single-line
high-financial-risk portfolio returns will not be displayed in the exhibits with the
Size Study 25 portfolios.
Using the Duff & Phelps Risk Premium Report—
Size Study in the Build-up Method
As an alternative to the Formula 13.1 for the build-up method, EðRi Þ ¼ Rf þ RPm þ
RPs RPu , where we add a general equity risk premium for the ‘‘market’’ (equity
risk premium) and a risk premium for small size to the risk-free rate, we 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. The formula then is modified to be:
(Formula 13.3)
EðRi Þ ¼ Rf þ RPmþs 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
RPmþs ¼ Risk premium for the ‘‘market’’ plus risk premium for size
RPu ¼ Risk premium attributable to the specific company or to the industry
08/26/2010
EXHIBIT 13.7 Duff & Phelps Size Study (market value of common equity)
Source: 200902 CRSP1, Center for Research in Security Prices. University of Chicago Booth School of Business. Used with permission. All rights
reserved. www.crsp.chicagobooth.edu. Calculations by Duff & Phelps LLC. # Duff & Phelps, LLC.
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Source: 200902 CRSP1, Center for Research in Security Prices. University of Chicago Booth School of Business. Used with permission. All rights
reserved. www.crsp.chicagobooth.edu. Calculations by Duff & Phelps LLC. # Duff & Phelps, LLC.
08/26/2010
EXHIBIT 13.8 Duff & Phelps Size Study (book value of common equity)
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
The Size Study sorts companies by size, 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.
Examples These data can be used as an aid in formulating estimated required rates
of return using objective measures of the size for a subject company. The realized
risk premiums reported in the eight exhibits have not been adjusted to remove beta
risk. Therefore, they should not be multiplied by a CAPM beta or otherwise included in a CAPM analysis. If one is estimating the cost of equity capital for a public
company, one can determine all eight measures of size and estimate an appropriate
risk premium based on all eight. If one is estimating the cost of equity for a closely
held entity, one can use six measures of size (ignoring the measures of size based on
the market value of common equity and the MVIC) and estimate an appropriate risk
measure based on the six fundamental accounting measures of size.
A straightforward method of arriving at a discount rate using the build-up
method with the data presented in eight exhibits is to derive RPmþs for use in
Formula 13.3. The premiums, RPmþs , incorporate both the ERP and the size
premium. One could match the sales or total assets of the subject company with
the portfolios composed of companies of similar size. The smoothed realized
premiums of these portfolios can then be added to the yield on long-term U.S.
government bonds as of the valuation date to obtain benchmarks for the cost of
equity capital.
Assume the subject public company has these characteristics:
Market value of common equity
Book value of common equity
5-year average net income
Debt
Market value of invested capital
Total assets
5-year average EBITDA
Sales
Number of employees
$120 million
$100 million
$10 million
$60 million
$180 million
$300 million
$30 million
$250 million
200
The simplest approach is to use the eight exhibits (such as in Exhibit 13.7) and,
for each of the eight size categories, locate the portfolio whose size is most similar to
the subject company. For each guideline portfolio, the column labeled ‘‘Smoothed
Average Equity Risk Premium’’ gives an indicated historical realized premium over
the risk-free rate, RPmþs . Exhibit 13.9 shows the premiums indicated for our subject
company.
In deriving the average realized risk premiums reported in the exhibits, the Duff
& Phelps studies use the SBBI income return on long-term U.S. government bonds
as their measure of the historical risk-free rate; therefore, a 20-year U.S. government
bond yield is the most appropriate measure of the risk-free rate for use with the
reported premiums in developing an indicated cost of equity capital.
If one’s estimate of the ERP for the S&P 500 on a forward-looking basis were
materially different from the average historical realized premium since 1963, it may
be reasonable to assume that the other historical portfolio returns reported here
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Size Effect
EXHIBIT 13.9 Size-Adjusted Risk Premiums over Risk-free Rate: Using Guideline Portfolios
Company
Size
Market Value of Equity
Book Value of Equity
5-Year Average Net Income
Market Value of Invested Capital
Total Assets
5-Year Average EBITDA
Sales
Number of Employees
Mean Premium over Risk-free Rate, RPmþs
Median Premium over Risk-free Rate, RPmþs
(1)
$120 mil.
$100 mil.
$10 mil.
$180 mil.
$300 mil.
$30 mil.
$250 mil.
200
Exhibit
13.7
13.8
(1)
(1)
(1)
(1)
(1)
(1)
Guideline
Portfolio
25
25
24
25
24
24
24
25
RPmþs
12.4%
10.9%
10.5%
12.0%
10.0%
10.2%
9.6%
10.6%
10.8%
10.6%
From additional exhibits provided in the Risk Premium Report.
Source: Duff & Phelps Risk Premium Report 2009, Copyright # 2009. Used with permission.
All rights reserved.
would differ on a forward-looking basis by approximately a similar differential. For
example, at the end of 2008, the average realized premium since 1963 for large company stocks equaled 3.84% (see the bottom of Exhibit 13.7). This is the historic
market risk premium, RPm, inherent in the Size Study exhibits for use in the buildup method as of that date. The risk premiums displayed in the Size Study exhibits for
the build-up method equal RPmþs , as shown in Formula 13.3 (RPm plus RPs). Assume that one’s estimate of the ERP at the end of 2008 is equal to 6%. That difference (2.2% ¼ 6% minus 3.84%) can be added to the average risk premium, RPmþs ,
for the portfolio (observed or ‘‘smoothed’’) that matches the size of the subject company to arrive at an adjusted forward-looking risk premium for the subject company
(matching one’s forward-looking ERP estimate). Then this forward-looking risk premium can be added to the risk-free rate as of the valuation date to estimate an appropriate cost of equity capital for the subject company.
Assume that the risk-free rate as of the valuation date equals 4.5%. The premiums would indicate the cost of equity capital ranging from 16.3% (4.5% risk-free
rate plus 9.6% risk premium from Exhibit 13.10 plus 2.2% adjustment for ERP
estimate) to 19.1% (4.5% risk-free rate plus 12.4% risk premium from Exhibit
13.10 plus 2.2% adjustment for ERP estimate), with a median of 17.3% (4.5%
risk-free rate plus 10.6% median risk premium from Exhibit 13.10 plus 2.2% adjustment for ERP estimate). This estimate of the cost of equity capital is before consideration of RPu, the risk premium attributable to the specific company or to the
industry.
As an alternative, one can estimate premiums using the regression equations
that underlie the smoothed premium calculations. To estimate a premium, one multiplies the logarithm (log base-10) of the size measure by the slope coefficient and
adds the constant term, as described. In practice, this approach generally produces
results that are very similar to those of the guideline portfolio approach presented
earlier (unless we are extrapolating to a company that is much smaller than the average size for the 25th portfolio).
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
EXHIBIT 13.10 Size-Adjusted Risk Premiums over Risk-free Rate: Using
Regression Equations
Market Value of
Equity
Book Value of Equity
5-Year Average Net
Income
Market Value of
Invested Capital
Total Assets
5-Year Average
EBITDA
Sales
Number of Employees
Mean Premium over
Risk-free Rate,
RPmþs
Median Premium over
Risk-free Rate,
RPmþs
(1)
Company
Size
Exhibit
Constant
Term
Slope
Term
log
(Size)
RPmþs
$120 mil.
13.7
19.601%
3.542%
2.079
12.2%
$100 mil.
$10 mil.
13.8
15.965%
(1)
(1)
2.863%
2.000
1.000
10.2%
10.5%
$180 mil.
(1)
(1)
(1)
2.2255
11.7%
$300 mil.
$30 mil.
(1)
(1)
(1)
(1)
(1)
(1)
2.477
1.477
10.2%
10.4%
$250 mil.
200
(1)
(1)
(1)
(1)
(1)
(1)
2.398
2.301
9.8%
10.8%
10.7%
(1)
10.4%
From additional exhibits provided in the Risk Premium Report.
Source: Duff & Phelps Risk Premium Report 2009, Copyright # 2009. Used with
permission. All rights reserved.
Exhibit 13.10 illustrates use of the regression equations from Exhibits 13.7 and
13.8 for our example subject company.
Practical Application of the Data The smoothed average realized risk premium is
the most appropriate indicator for most of the portfolio groups. At the largest
size and smallest size ends of the range, the average realized risk premiums tend
to jump off the smoothed line, particularly for the portfolios ranked by size as
measured by market value (market value of equity and market value of invested
capital). For the largest companies (the first portfolio), the observed historical
relationship flattens out, and the smoothed premium may be an inappropriate
indicator. Note, however, that a pronounced jump exists in the premium in the
smallest 4% of companies.12
Sometimes one must estimate the required rate of return for a company that is
significantly smaller than the average size of even the smallest of the 25 portfolios. In
12
This fact is of interest to many business valuators, since this jump occurs in a size category
in which, as a practical matter, many more valuation assignments are performed. For seven
of the eight size measures, the actual premium for the smallest group was greater than the
smoothed premium, generally by a considerable margin. For the smallest companies (portfolio 25), the smoothed average premium is likely the more conservative indicator of the
size premium and provides a basis for extrapolation.
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Size Effect
such cases, it may be appropriate to extrapolate the smoothed average premium to
smaller sizes using the slope and constant terms from the regression relationships
that we use in deriving the smoothed premiums. In so doing, we must be careful to
remember that the logarithmic relationship is base-10 and that the financial size data
are in millions of dollars, such that the log of $10 million is log(10), not log
(10,000,000).
Also, as a general rule, one should be cautious about extrapolating a statistical
relationship far beyond the range of the data used in the statistical analysis. We are
most comfortable with extrapolations for companies with size characteristics that
are within the range of companies comprising the 25th portfolio (as reported in the
Risk Premium Report). For example, the smallest company reported (the 4th percentile of companies) in the 25th portfolio of Exhibit 13.8 has a book value equal to
$9.104 million.13 We discuss the size of the companies included in the 25th portfolio
in Chapter 14.
In any extrapolation, one may find that the size of the subject company is
equal to or greater than the smallest size of the companies included in the 25th
portfolio (e.g., sales) and smaller when ranking by other size measures (e.g., fiveyear average income). One can then include the size measure for sales, for example, and exclude the size measure for five-year average net income. One should
not use those size measures for which the subject company’s size is equal to zero
or negative.
A brief example will illustrate the use of the regression equations in estimating
an equity risk premium. Assume a company has book value of $50 million. If we
insert this figure into the regression relationship reported in Exhibit 13.8, we obtain
the following estimate of RPmþs :
Smoothed Premium ¼ 15:965% 2:863% log ð50Þ
¼ 15:965% 2:863% ð1:699Þ ¼ 11:10%
Use of a portfolio’s average realized rate of return to calculate a cost of equity
capital is based (in part) on the implicit assumption that the risks of the subject company are quantitatively similar to the risks of the average company in the subject
portfolio. If the risks of the subject company differ materially from the average company in the subject portfolio, then an appropriate discount rate may be lower (or
higher) than a return derived from the average realized risk premium for a given
portfolio. We have included two of the exhibits displaying various risk measures for
each portfolio. Exhibit 13.11 displays various risk measures for portfolios where
companies were ranked by size by market value of equity, and Exhibit 13.12 displays various risk measures for portfolios where companies were ranked by size by
book value of equity.
These exhibits can be useful in identifying material differences between the
expected returns of a subject company of a given size and the characteristics of the
companies comprising the portfolio.
13
See, e.g., Alfred Zeiler, ‘‘Can Duff & Phelps Be Applied to the Very Small Company,’’ Business Appraisal Practice (2nd Quarter 2009): 7–17; Paul French and Jason Rae, ‘‘The Litigating Valuation Analyst and the Duff and Phelps Risk Premium Report Size Study,’’
National Litigation Consultants’ Review 9(2) (August 2009): 7–13.
252
3.26
1,347
111
331
515
697
838
977
1,172
2.04
2.52
2.71
2.84
2.92
2.99
3.07
3.13
3.19
1,808
1,558
3.32
3.37
3.43
3.47
3.53
3.60
3.68
3.75
3.85
3.97
4.09
4.21
4.33
4.56
5.11
Log
of
Size
2,086
2,346
2,675
2,933
3,418
3,948
4,775
5,597
7,150
9,399
12,369
16,126
21,569
36,587
127,995
Average
Mkt Value
($mils.)
354
146
117
81
54
69
51
54
51
51
51
47
40
37
44
43
41
40
37
41
36
35
35
34
40
Number
of Firms
14.6%
10.7%
10.0%
8.7%
10.2%
9.1%
8.1%
8.4%
9.5%
8.2%
6.5%
7.8%
6.6%
5.7%
7.8%
5.7%
7.1%
5.4%
6.3%
5.6%
4.3%
4.6%
3.1%
4.2%
3.9%
Average
Risk
Premium
29.41%
26.47%
26.14%
26.11%
26.38%
25.28%
24.91%
24.68%
24.06%
24.32%
25.14%
24.53%
24.10%
24.08%
22.99%
24.40%
23.52%
24.03%
25.13%
24.37%
24.47%
23.85%
22.87%
21.09%
14.76%
Average
Debt to
MVIC
41.7%
36.0%
35.4%
35.3%
35.8%
33.8%
33.2%
32.8%
31.7%
32.1%
33.6%
32.5%
31.8%
31.7%
29.8%
32.3%
30.8%
31.6%
33.6%
32.2%
32.4%
31.3%
29.7%
26.7%
17.3%
Average Debt to
Market Value of
Equity
10.3%
7.9%
7.4%
6.4%
7.5%
6.8%
6.1%
6.3%
7.2%
6.2%
4.9%
5.9%
5.0%
4.3%
6.0%
4.3%
5.4%
4.1%
4.7%
4.2%
3.3%
3.5%
2.4%
3.3%
3.3%
Average
Unlevered
Risk Premium
1.29
1.27
1.23
1.26
1.25
1.26
1.20
1.19
1.19
1.15
1.16
1.12
1.09
1.12
1.15
1.07
1.11
1.04
1.03
1.04
0.97
0.98
0.97
0.92
0.86
Beta (Sum
Beta) Since
’63
0.97
0.99
0.96
0.98
0.97
0.99
0.95
0.95
0.95
0.92
0.92
0.90
0.88
0.90
0.93
0.86
0.89
0.84
0.82
0.83
0.78
0.79
0.79
0.77
0.76
Average
Unlevered
Beta
6.1%
7.7%
8.0%
8.4%
8.7%
9.1%
9.6%
9.7%
10.3%
11.0%
11.2%
10.9%
11.3%
12.2%
12.4%
11.8%
12.5%
13.0%
12.8%
13.1%
12.4%
13.0%
13.1%
13.3%
15.8%
Average
Operating
Margin
41.8%
29.5%
26.9%
25.3%
22.8%
23.5%
20.7%
20.0%
18.2%
16.7%
17.2%
15.7%
15.4%
14.6%
14.8%
14.7%
14.1%
13.2%
14.1%
13.0%
13.2%
12.1%
11.1%
10.8%
9.7%
Average CV
(Operating
Margin)
56.2%
38.6%
35.0%
34.1%
30.7%
31.2%
26.9%
26.7%
25.3%
24.4%
23.8%
22.0%
21.9%
20.7%
21.8%
22.8%
21.3%
21.8%
21.4%
19.4%
21.0%
20.5%
18.4%
19.1%
15.1%
Average
CV
(ROE)
Source: Compiled from data from the Center for Research in Security Prices.
# 2009 CRSP1, Center for Research in Security Prices. University of Chicago Booth School of Business. Used with permission. All rights reserved. www.crsp.chicagobooth.edu. Calculations by
Duff & Phelps LLC.
CV(X) ¼ Standard deviation of X divided by mean of X, calculated over 5 fiscal years.
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Portfolio
Rank by
Size
Portfolio Statistics for 1963–2008
08/26/2010
Portfolio Statistics for 2008
Companies Ranked by Market Value of Equity: Comparative Risk Characteristics Data for Year Ending December 31, 2008
EXHIBIT 13.11 Duff & Phelps Study: Comparative Risk Statistics
E1C13
Page 252
60
162
235
312
382
430
482
553
840
736
825
923
1,029
1,157
1,368
1,551
1,739
2,016
2,447
3,055
4,184
5,622
7,877
11,465
37,502
Average
Book Value
($mils.)
1.77
2.21
2.37
2.49
2.58
2.63
2.68
2.74
2.81
2.87
2.92
2.97
3.01
3.06
3.14
3.19
3.24
3.30
3.39
3.49
3.62
3.75
3.90
4.06
4.57
Log
of
Size
391
394
142
112
84
61
56
45
59
49
46
44
42
45
44
36
35
39
38
33
36
33
35
34
38
Number
of Firms
12.0%
11.1%
8.5%
9.1%
9.3%
9.6%
6.9%
7.6%
7.0%
7.4%
8.1%
8.0%
7.3%
7.3%
5.5%
6.1%
6.7%
5.6%
5.6%
5.0%
5.5%
4.8%
6.4%
4.6%
4.3%
Average
Risk
Premium
25.71%
25.67%
25.53%
24.70%
25.42%
26.05%
26.53%
25.11%
24.26%
25.73%
24.82%
25.32%
25.90%
27.11%
26.88%
26.43%
25.49%
25.81%
26.21%
27.39%
27.52%
29.87%
30.55%
29.42%
24.52%
Average
Debt to
MVIC
34.6%
34.5%
34.3%
32.8%
34.1%
35.2%
36.1%
33.5%
32.0%
34.6%
33.0%
33.9%
34.9%
37.2%
36.8%
35.9%
34.2%
34.8%
35.5%
37.7%
38.0%
42.6%
44.0%
41.7%
32.5%
Average Debt to
Market Value of
Equity
253
8.9%
8.2%
6.3%
6.8%
6.9%
7.1%
5.0%
5.7%
5.3%
5.5%
6.1%
6.0%
5.4%
5.4%
4.0%
4.5%
5.0%
4.2%
4.1%
3.6%
4.0%
3.4%
4.5%
3.2%
3.3%
Average
Unlevered
Risk Premium
1.30
1.26
1.24
1.21
1.21
1.23
1.20
1.18
1.18
1.17
1.10
1.11
1.09
1.06
1.07
1.07
1.05
1.08
1.04
1.01
1.01
0.92
0.90
0.85
0.81
Beta (Sum
Beta) Since
’63
1.02
0.99
0.97
0.96
0.96
0.96
0.94
0.93
0.95
0.92
0.88
0.88
0.86
0.83
0.84
0.84
0.84
0.85
0.82
0.79
0.79
0.71
0.69
0.66
0.66
Average
Unlevered
Beta
7.9%
8.7%
9.3%
9.1%
9.4%
10.1%
11.0%
11.0%
11.5%
11.5%
11.7%
12.5%
11.7%
11.6%
13.2%
12.4%
12.5%
13.1%
12.6%
13.0%
12.4%
12.3%
12.3%
13.4%
13.2%
Average
Operating
Margin
36.9%
25.3%
23.9%
23.5%
21.9%
20.5%
18.4%
17.9%
16.6%
17.6%
15.8%
15.3%
15.3%
14.9%
14.5%
14.6%
14.5%
14.0%
13.5%
13.1%
13.2%
12.4%
12.0%
13.0%
12.5%
Average CV
(Operating
Margin)
52.2%
36.0%
34.6%
31.4%
31.5%
30.2%
28.3%
26.2%
24.7%
25.2%
23.0%
22.9%
23.3%
22.0%
23.4%
21.8%
21.5%
23.5%
22.6%
23.1%
22.5%
22.0%
20.5%
18.6%
20.2%
Average
CV
(ROE)
Source: Compiled from data from the Center for Research in Security Prices. # 2009 CRSP1, Center for Research in Security Prices. University of Chicago Booth School of Business. Used with
permission. All rights reserved. www.crsp.chicagobooth.edu. Calculations by Duff & Phelps LLC.
CV(X) ¼ Standard deviation of X divided by mean of X, calculated over 5 fiscal years.
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
Portfolio
Rank by
Size
Portfolio Statistics for 1963–2008
08/26/2010
Portfolio Statistics for 2008
Companies Ranked by Book Value of Equity: Comparative Risk Characteristics Data for Year Ending December 31, 2008
EXHIBIT 13.12 Duff & Phelps Study: Comparative Risk Statistics
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254
ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
Material differences between the expected returns for a subject company of a
given size and the stocks comprising the portfolio may arise due to differences in:
&
Leverage (the average debt/MVIC and the average levered portfolio beta [the
CAPM risk measure] of the portfolios are displayed)
For example, we can adjust the observed premiums over the risk-free rate for
differences in financial leverage between the average companies comprising the portfolio and the subject company. The subject company here has a debt/MVIC ¼ $60/
$180 ¼ 33%, which is slightly more leverage than the average of the companies
comprising portfolio 25 of Exhibit 13.7 (29.41%).
But assume that the subject company had no debt in its capital structure. The
Size Study displays unlevered average levered risk premium where the average debt
to equity (D/E) ratio of the portfolio is based on the average debt to MVIC for the
portfolio since 1963 and the income tax rate is the estimated federal income tax rate
plus effective state income tax rate for the companies comprising the portfolio
companies.
The Size Study presents unlevered average realized risk premiums for each of the
eight size measures in exhibits displaying various risk measures for each size category (see, e.g., Exhibits 13.11 and 13.12). For example, looking at Exhibit 13.11,
the unlevered average realized risk premium for portfolio 25 equals 10.3%. This
compares to the average levered realized risk premium of 14.6% (rounded but not
smoothed) reported in Exhibits 13.7 and 13.12.
These unlevered realized risk premiums represent the rates of return on a debtfree basis; the unlevered realized risk premiums can be used for estimating required
rates of return for companies with no debt. The unlevered realized risk premiums
displayed in Exhibits 13.11 and 13.12 are informative in that they generally indicate
that the market views operations of smaller companies to be riskier than the operations of larger companies (i.e., unlevered risk premiums increase as size decreases).
&
Operating risks (the average unlevered portfolio sum betas, the average operating margin,14 and the average coefficient of variation of operating margin15 for
the portfolios are displayed)
One can compare fundamental risk represented by the operating margins of the
subject company to the average operating margin of the companies included in the
portfolios. For example, were the subject company ranked in the 24th portfolio
based on its market value of equity, one can compare the operating margin of the
subject company to the average operating margin of companies included in the 24th
portfolio. Looking at Exhibit 13.11, we see that the average operating margin of the
companies in the 24th portfolio is 7.7%. Were the average operating margin of the
subject company less than the average of the portfolio companies, one can conclude
that the subject company is probably riskier than the average company of its size.
14
Operating margin ¼ [sales minus (cost of goods sold plus selling, general, and administrative expenses plus depreciation expense)] divided by sales.
15
Coefficient of operating margin ¼ standard deviation of operating margin over five years
divided by the mean operating margin over those same five years.
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Size Effect
255
Further, we can compare the coefficient of variation of operating margin of
the subject company to the average coefficient of variation of operating margin
of companies included in the 24th portfolio. Looking at Exhibit 13.11, we see
that the average coefficient of variation of operating margin of the companies in
the 24th portfolio is 29.5%. Were the average coefficient of operating margin
of the subject company greater than the average of the portfolio companies,
one can conclude the subject company is probably riskier than the average company of its size.
&
Other fundamental risk factors inherent in the business, such as dependence on
a single supplier, limited number of customers, etc.
Using the Duff & Phelps Size Study in the CAPM Method
Using the same dataset and similar methodology, Duff & Phelps computes premiums over CAPM. (Recall that beta captures some, but not all, of the size premium.)
These can be used with the CAPM. We have included two exhibits: Exhibit 13.13,
where size is measured by market value of common equity, and Exhibit 13.14,
where size is measured by book value of common equity (exhibits reproduced herein
are for years ending 2008).
In the context of the CAPM, the greater betas of the smaller companies explain
some but not all of the higher average returns in these size-ranked portfolios. An
example of the calculation of ‘‘Return in Excess of CAPM’’ will illustrate the
method. The next example uses data for portfolio 19 of companies ranked by book
value of equity from Exhibit 13.14:
1. Portfolio beta ¼ 1.20 (column 4 of Exhibit 13.14, portfolio 19)
2. Arithmetic average realized risk premium ¼ 3.84% (third line from bottom of
Exhibit 13.14, Arithmetic Average Risk Premium for SBBI series for Large
Companies for 1963–2008)
3. Indicated CAPM premium (1.20 3.84%) ¼ 4.61%
4. Arithmetic average U.S. government bond income return ¼ 7.04% (bottom line
of Exhibit 13.14, SBBI Long-Term Government Bond Income Returns for
1963–2008)
5. Indicated CAPM return (4.61% þ 7.04%) ¼ 11.65%
6. Arithmetic average realized return ¼ 13.91% (column 5 of Exhibit 13.14, portfolio 19)
7. RPs ¼ Return in excess of CAPM (13.91% 11.65%) ¼ 2.26% (column 8 of
Exhibit 13.14, portfolio 19) (difference due to rounding)
The return in excess of CAPM is often called the size premium or beta-adjusted
size premium. The size premium is an empirically observed correction to the CAPM.
This return in excess of CAPM of 2.26% compares to a premium over the overall
market of 3.03% (line 6 minus line 4 minus line 2) without regard to beta. The Size
Study exhibits report betas calculated using the sum beta method applied to monthly
portfolio return data. This method yields greater betas for smaller companies (and
smaller size premiums) than would be obtained using the ordinary least squares
method.
08/26/2010
EXHIBIT 13.13 Duff & Phelps Size Study (market value of common equity)
Source: 200902 CRSP1, Center for Research in Security Prices. University of Chicago Booth School of Business. Used with permission. All rights
reserved. www.crsp.chicagobooth.edu. Calculations by Duff & Phelps LLC. # Duff & Phelps, LLC.
E1C13
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256
Source: 200902 CRSP1, Center for Research in Security Prices. University of Chicago Booth School of Business. Used with permission. All rights
reserved. www.crsp.chicagobooth.edu. Calculations by Duff & Phelps LLC. # Duff & Phelps, LLC.
08/26/2010
EXHIBIT 13.14 Duff & Phelps Size Study (book value of common equity)
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257
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258
ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
Eight Measures of Size, 25 Size Categories The eight Size Study size measures for use in
the CAPM method are the same as those presented earlier for use in the build-up
method.
The eight exhibits for use in CAPM report these statistics for each of 25 size
categories:
1. Average of the size criteria (e.g., average number of employees) for the latest
year
2. Log (base-10) of the median of the size measure
3. Beta relative to the S&P 500 calculated using the sum beta method applied to
monthly returns for 1963 through the latest year
4. Arithmetic average realized equity return since 1963
5. Arithmetic average realized risk premium (realized equity return over long-term
U.S. government bonds) since 1963 (labeled ‘‘arithmetic risk premium’’)
6. Indicated CAPM premium calculated as the beta of the portfolio multiplied by
the arithmetic average realized market risk premium since 1963
7. Premium over CAPM calculated by subtracting the indicated CAPM premium
from the arithmetic risk premium
8. Smoothed premium over CAPM: the fitted premium from a regression with the
historical ‘‘premium over CAPM’’ as the dependent variable and the logarithm
(base-10) of the size measure as the independent variable
The premium over CAPM data should not be multiplied by beta. Rather, it is
the basis for RPs (in Formula 13.2) and is added to CAPM (Formula 8.1).
By whatever measure of size they use, the result is a clear inverse relationship
between the size and the size premium.
Examples Continuing with the same subject company used in Exhibits 13.9 and
13.10, the simplest approach is to find the smoothed premium over CAPM of the
guideline portfolios in a manner similar to that described for the Size Study data in
EXHIBIT 13.15 Premiums over CAPM: Using Guideline Portfolios
Company
Size
Market Value of Equity
Book Value of Equity
5-Year Average Net Income
Market Value of Invested Capital
Total Assets
5-Year Average EBITDA
Sales
Number of Employees
Mean Premium over CAPM, RPs
Median premium over CAPM, RPs
(1)
$120 mil.
$100 mil.
$10 mil.
$180 mil.
$300 mil.
$30 mil.
$250 mil.
200
Exhibit
13.14
13.15
(1)
(1)
(1)
(1)
(1)
(1)
Guideline
Portfolio
Premium
over CAPM
25
25
24
25
24
24
24
25
7.1%
5.7%
5.5%
6.7%
5.0%
5.2%
4.8%
5.9%
5.7%
5.6%
From additional exhibits provided in the Risk Premium Report.
Source: Duff & Phelps Risk Premium Report 2009, Copyright # 2009. Used with permission.
All rights reserved.
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259
Size Effect
EXHIBIT 13.16 Premiums over CAPM: Using Regression Equations
Market Value of
Equity
Book Value of
Equity
5-Year Average
Net Income
Market Value of
Invested Capital
Total Assets
5-Year Average
EBITDA
Sales
Number of
Employees
Mean Premium
over CAPM, RPs
Median Premium
over CAPM, RPs
(1)
Company
Size
Exhibit
Constant
Term
Slope
Term
log
(Size)
Premium
over CAPM
$120 mil.
13.14
13.059%
2.894%
2.079
7.0%
$100 mil.
13.15
9.353%
2.080%
2.000
5.2%
$10 mil.
(1)
(1)
(1)
1.000
5.5%
$180 mil.
(1)
(1)
(1)
2.255
6.4%
$300 mil.
$30 mil.
(1)
(1)
(1)
(1)
(1)
(1)
2.477
1.477
5.2%
5.4%
$250 mil.
200
(1)
(1)
(1)
(1)
(1)
(1)
2.398
2.301
4.9%
6.0%
5.7%
5.4%
From additional exhibits provided in the Risk Premium Report.
Source: Duff & Phelps Risk Premium Report 2009, Copyright # 2009. Used with permission.
All rights reserved.
the build-up method. Exhibit 13.15 illustrates this approach for the subject
company.
If the indicated CAPM estimate before the size adjustment, EðRi Þ ¼ Rf þ
BðRPm Þ, is, for example, 11.0%, then the size premiums, RPs, indicate a cost of
equity capital ranging from 15.8% to 18.1%, with a median of 16.6% before consideration of RPu, risk premium attributable to the specific company.
As an alternative, one can use the regression equations reported in exhibits to
estimate premiums over CAPM. Exhibit 13.16 illustrates the results for the subject
company.
Practical Application of the Data The exhibits report levered and unlevered portfolio
betas (for example, see Exhibits 13.12 and 13.13) where the average debt to equity
(D/E) ratio of the portfolio is based on the average debt to MVIC for the portfolio
since 1963, and the income tax rate is the estimated federal income tax rate plus
effective state income tax rate for the companies comprising the portfolio
companies.
The exhibits display unlevered portfolio betas for each of the eight size measures. For example, in Exhibits 13.12 and 13.13, the unlevered portfolio beta for
portfolio 24 equals 0.99 (Exhibit 13.12). This compares to the levered portfolio
beta of 1.27 reported in Exhibits 13.12 and 13.14.
Unlevered betas are often called asset betas in the literature, as they are intended to represent the risk of the operations of the business with the risk of
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
financial leverage removed. The unlevered betas displayed in Exhibits 13.12 and
13.13 are informative in that they generally indicate that the market views the
operations of smaller companies to be riskier than the operations of larger companies (i.e., unlevered betas increase as size decreases). While the unlevered
portfolio betas are informative, they would not generally be appropriate to use
in estimating the beta of a subject company as they represent betas calculated
since 1963. The convention for estimating the beta appropriate for a subject
company is generally to use data for a more recent look-back period (e.g., last
60 months excess returns).
The unlevering formulas used in the exhibits for unlevering the average realized
risk premiums and portfolio betas for size categories 1 through 25 assume that the
business risk is fully borne by the equity capital; that is, the variability of operating
cash flows has a negligible effect on the risk of the debt capital. As a first approximation, this assumption appears reasonable for most of the companies comprising size
categories 1 through 25 because of the modest debt to equity levels.
ESTIMATING SIZE PREMIUMS FOR
NONPUBLIC COMPANY
The Morningstar size premiums are based on measuring size using market value of
equity. Two of the Duff & Phelps size measures are also based on measuring size
based on market value (market value of equity and invested capital).
When one is valuing a nonpublic business, one does not know the market value
until the end of the valuation process. One can either use the size premiums from
the Duff & Phelps Size Study where size is not based on market value (e.g., size
measured based on net income) or one should use an iterative process. The steps in
the iterative process can be summarized in this way if one is discounting the net cash
flow to common equity:
Step 1: Estimate the market value of senior securities (debt and preferred equity)
and hold that dollar amount fixed throughout the process.
Step 2: Make a first estimate of the market value of common equity and the
market value weight for the capital structure. For example, if the initial estimate of the market value of common equity equals book value, then calculate the cost of equity capital based upon the relationship of the market
value of the senior securities to book value of common equity (e.g., for the
levering of beta) and the size premium assuming book value of equity equals
the market value of equity.
Step 3: Project (a) the net cash flows available to common equity and (b) the
projected growth rate for either a discounted valuation model (Formula
2.1) or a capitalization valuation model (Formula 4.4).
Step 4: Using the first-approximation cost of equity capital from Step 2 and the
projected cash flows from Step 3, compute a first approximation of market
value of common equity.
Step 5: Compute the capital structure weights and the size premium using the
common equity value from Step 4.
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Size Effect
261
Step 6: Repeat the process starting with Step 2, using the approximation of the
market value of common equity form Step 4, until the computed market
value of common equity comes reasonably close to the capital structure
weight and appropriate size premium where size is measured at market
value of equity.
SUMMARY
The Duff & Phelps data cover the years 1963 through the present, as compared with
1926 through the present for the Morningstar data.
Two results of the Size Study seem strikingly significant:
1. In spite of the different time period, the Duff & Phelps results corroborate the
Morningstar results that the size effect is empirically observed.
2. The results are significantly similar for all eight measures of company size.
Although the market value of common equity has both the highest degree of
statistical significance and the steepest slope when one regresses average returns
against size, all size measures show a high degree of statistical significance. This is
quite convenient in the context of valuing private companies, since it enables the
analyst to start with a known size measure rather than an estimated market value of
equity, which is the value being sought.
In evaluating the correct size premium for a nonpublic company, one must
either use a non market based measure of size (e.g., net income) and a size premium
drawn from the Duff & Phelps Size Study or use the iterative process to estimate the
market value of common equity and a size premium drawn from either Morningstar
or the Duff and Phelps Size Study.
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CHAPTER
14
Criticisms of the Size Effect
Introduction
Is the Size Effect the Result of Incorrectly Measuring Betas?
Composition of the Smallest Decile
Data Issues
Risks of Small Companies
Should the Cost of Equity Capital Use a Changing Size Premium?
Relationship of Size and Measures of Risk
Relationship of Size and Liquidity
Summary
Appendix 14A—Other Data Issues Regarding the Size Effect
INTRODUCTION
In Chapter 13, we discussed two sets of independent studies that document and
quantify the size effect: Morningstar studies and Duff & Phelps studies.
The size effect, though, is not without controversy, and various commentators question its validity. In fact, some commentators contend that the historical
data are so flawed that practitioners can dismiss all research results that support
the size effect. For example, is it simply the result of not measuring beta correctly? Are there simply market anomalies that cause the size effect to appear?
Is size just a proxy for one or more factors correlated with size, so one should
directly use those factors to measure risk rather than size? Is the size effect hidden because of unexpected events?
IS THE SIZE EFFECT THE RESULT OF INCORRECTLY
MEASURING BETAS?
Several authors have investigated problems with measuring beta. If beta is underestimated, the size premium will be observed, and the equity discount rate estimated
The authors would like to thank Ashok Bhardwaj Abbott for allowing us to reproduce
research results for the appendix to this chapter. We would also like to thank David Turney
and Nick Arens of Duff & Phelps LLC for preparing material for this chapter.
262
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263
using Formula 8.1 will be underestimated. The size premium can correct for this
underestimation. For example, two papers investigated the problem with underestimating betas for troubled firms (which tend to populate the smaller deciles where
size is measured by the market value of equity).1 The market value of equity gets bid
down for a troubled company, and the troubled company’s stock may trade like a
call option. In the 2004 study, the author estimated that betas (measured by the ordinary least squares [OLS] method) for troubled companies were underestimated by
more than 20% when the bankruptcy risk is at least 20%. This would cause the size
premium to be overestimated in, for example, Exhibit 13.1 for the 10th decile,
where betas are estimated using the OLS method of monthly excess returns and size
is measured by market value of equity.
Morningstar publishes size premium statistics where size is measured by market
value of equity and betas are estimated using the sum beta method. The sum beta
method is an alternative way of handling monthly data, essentially averaging betas
for two or more months. This method can provide a better measure of beta for small
stocks by taking into account the lagged price reaction of stocks of small companies
to movements in the stock market. The data indicate that even using the sum beta
method, when applied to the capital asset pricing model (CAPM), beta does not
account for the returns in excess of the risk-free rate historically found in small company stocks. If you use the sum beta method of estimating beta, you need to use the
size premiums based on sum beta, not Exhibit 13.1. Exhibit 14.1 displays the Morningstar size premium data using the sum beta method.
Morningstar also calculates size premium data using annual betas (see Exhibit
13.3) with size measured by market value of equity. Size premium calculated using
annual betas (as displayed in Exhibit 13.3) or sum beta (as displayed in Exhibit 14.1)
should be less plagued by the overestimation problem due to incorrectly measuring
beta (see Chapter 10 section on sum beta). But the troubled company issue still
plagues the 10th decile.
One needs to match the source of the size premium with the type of beta estimate one makes. One should use the Morningstar size premiums derived using
the OLS method (Exhibit 13.1) when estimating the subject company beta using the
OLS method, and one should use the Morningstar size premiums derived using the
sum beta method (Exhibit 14.1) when estimating the subject beta using the sum beta
method. But in applying the CAPM particularly for a small business, we are looking
for the most accurate estimate, not the most expedient one. If one uses an OLS beta
for a small company by multiplying the OLS beta times ERP estimate and adding
OLS-based size premium (Exhibit 13.1), one probably will not arrive at as accurate
an estimate of the cost of equity capital as by multiplying a sum beta times ERP
estimate and adding a sum beta-based size premium. One should be using the most
accurate estimate of beta and the most accurate measure of the appropriate size
premium.
1
Carlos A. Mello-e-Souza, ‘‘Bankruptcy Happens: A Study of the Mechanics of Distressed
Driven CAPM Anomalies,’’ Working paper, January 25, 2002, and ‘‘Limited Liability, the
CAPM and Speculative Grade Firms: A Monte Carlo Experiment,’’ Working paper, August
18, 2004. Available at http://ssrn.com/abstract=294804.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
EXHIBIT 14.1 Size Premium Using Sum Betas
Long-Term Returns in Excess of CAPM for Decile Portfolios of the NYSE/AMEX/
NASDAQ, with Sum Beta 1926–2008
Arithmetic
Estimated Return
Size Premium (Return in
Mean
in Excess of
Return in Excess Excess of
Sum
Return
Riskless Ratey
of Riskless Ratez CAPM)
Decile
Beta
1-Largest
2
3
4
5
6
7
8
9
10-Smallest
Mid-Cap, 3–5
Low-Cap, 6–8
Micro-Cap, 9–10
0.91
1.05
1.13
1.20
1.24
1.30
1.38
1.50
1.55
1.71
1.17
1.36
1.60
10.75%
12.51%
13.06%
13.45%
14.23%
14.48%
14.84%
15.95%
16.62%
20.13%
13.37%
14.86%
17.72%
5.56%
7.31%
7.87%
8.25%
9.03%
9.28%
9.65%
10.76%
11.42%
14.93%
8.18%
9.66%
12.52%
5.91%
6.82%
7.34%
7.76%
8.00%
8.39%
8.93%
9.68%
10.06%
11.06%
7.58%
8.82%
10.34%
0.35%
0.49%
0.53%
0.48%
1.03%
0.88%
0.71%
1.08%
1.37%
3.87%
0.59%
0.83%
2.18%
Betas are estimated from monthly portfolio total returns in excess of the 30-day U.S. Treasury bill total return versus the S&P 500 Index total returns in excess of the 30-day U.S. Treasury bill, January 1926–December 2008.
y
Historical riskless rate is measured by the 83-year arithmetic mean income return component
of 20-year U.S. government bonds (5.20%).
z
Calculated in the context of the CAPM by multiplying the equity risk premium by beta. The
equity risk premium is estimated by the arithmetic mean total return of the S&P 500
(11.67%) minus the arithmetic mean income return component of 20-year U.S. government
bonds (5.20%) from 1926 to 2008.
Source: Ibbotson Stocks, Bonds, Bills, and Inflation1 2009 Valuation Yearbook. Copyright #
2009 Morningstar, Inc. All rights reserved. Used with permission. (Morningstar, Inc. acquired
Ibbotson in 2006.) Calculated (or derived) based on CRSP1 data, Copyright # 2006 Center
for Research in Security Prices (CRSP1), University of Chicago Booth School of Business.
COMPOSITION OF THE SMALLEST DECILE
Morningstar also divides the 10th decile into subdeciles 10a and 10b, with 10a being
the top half of the 10th decile and 10b the bottom half of the 10th decile. Starting
with the 2010 SBBI Valuation Yearbook, Morningstar will provide more detailed
size premium for small cap companies. The 10th decile will be further split; subdecile 10a will be split into 10w and 10x, and subdecile 10b will be split into 10y
and 10z.
A Morningstar supplemental report split the subdeciles 10a into 10w and
10x and 10b into 10y and 10z for the period 1926–2008 (measured by market
capitalization) and reported the size premium. The reported size premiums for
subdecile 10y were 8.69% and for 10z, 11.45%, compared with the reported
size premium for 10b of 9.53%.
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EXHIBIT 14.2 Returns in Excess of CAPM—10th-Decile Split
Long-Term Returns in Excess of CAPM Estimation for Decile Portfolios of the NYSE/AMEX/
NASDAQ, with 10th Decile Split, 1926–2008
Arithmetic Realized Return Estimated Return
Size Premium
Mean
in Excess of
in Excess of
(Return in Excess
Return
Riskless Ratey
Riskless Ratez
of CAPM)
Decile
Beta
1-Largest
2
3
4
5
6
7
8
9
10a
10b-Smallest
0.91
1.03
1.10
1.12
1.16
1.18
1.24
1.30
1.35
1.42
1.38
10.75%
12.51%
13.06%
13.45%
14.23%
14.48%
14.84%
15.95%
16.62%
18.49%
23.68%
5.56%
7.31%
7.87%
8.25%
9.03%
9.28%
9.65%
10.76%
11.42%
13.29%
18.48%
5.91%
6.69%
7.13%
7.28%
7.49%
7.65%
8.03%
8.41%
8.71%
9.19%
8.95%
0.36%
0.62%
0.74%
0.97%
1.54%
1.63%
1.62%
2.35%
2.71%
4.11%
9.53%
Betas are estimated from monthly portfolio total returns in excess of the 30-day U.S. Treasury bill total return versus the S&P 500 total returns in excess of the 30-day U.S. Treasury
bill, January 1926–December 2008.
y
Historical riskless rate is measured by the 83-year arithmetic mean income return component
of 20-year U.S. government bonds (5.20%).
z
Calculated in the context of the CAPM by multiplying the equity risk premium by beta. The
equity risk premium is estimated by the arithmetic mean total return of the S&P 500
(11.67%) minus the arithmetic mean income return component of 20-year U.S. government
bonds (5.20%) for 1926–2008.
For mid-, low-, and micro-cap data, see Exhibit 13.1.
Source: Stocks, Bonds, Bills, and Inflation1 2009 Valuation Yearbook and 2009 Ibbotson
SBBI Valuation Supplement. Copyright # 2009 Morningstar, Inc. All rights reserved. Used
with permission. (Morningstar, Inc. acquired Ibbotson in 2006.) Calculated (or derived)
based on CRSP1 data, Copyright # 2009 Center for Research in Security Prices (CRSP1),
University of Chicago Booth School of Business.
Comparing Exhibits 13.1 and 14.2 (both calculated using OLS beta estimates)
shows the dramatic difference between the smallest 5% of companies and the next
smallest 5%. The size premium for the 10th decile from Exhibit 13.1 equals 5.81%,
and the size premiums for subdeciles 10a and 10b from Exhibit 14.2 equal 4.11%
and 9.53%, respectively.
What kind of companies populate subdeciles 10b and its top and bottom
halves, 10y and 10z? Morningstar includes all companies with no exclusion of
speculative (e.g., start-up companies) or distressed companies whose market capitalization is small because they are speculative or distressed. The inclusion of
speculative or distressed companies in the database is the basis for criticism of
the size effect.2 Exhibit 14.3 displays information on the type of companies that
2
Jonathan B. Berk, ‘‘A Critique of Size Related Anomalies,’’ Review of Financial Studies 8(2)
(Summer 1995): 225–286.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
EXHIBIT 14.3 Breakdown of Decile 10y Companies as of September 30, 2008: Market Value
between $74.9 and $136.5 Million
($ millions)
Size:
Market Value of Equity
Book Value of Equity
MVIC
Total Assets
$132
$117
$100
$86
$77
$246
$113
$67
$35
$6
$461
$146
$102
$79
$35
$828
$229
$123
$63
$20
95th Percentile
75th Percentile
Median
25th Percentile
5th Percentile
MVIC ¼ Market Value of Equity þ Book Value of Preferred Stock þ Book Value of Debt.
Profitability:
Sales
5-Yr Avg Net Income
before Extra Ordinary
5-Yr Avg
EBITDA
Latest Fiscal Year
Return on Book Equity
95th Percentile
75th Percentile
Median
25th Percentile
5th Percentile
$779
$204
$86
$29
$zero
$14
$5
$0
$10
$36
$51
$17
$7
$3
$21
36%
10%
0%
26%
121%
Measures of Risk for NYSE þ AMEX þ NASDAQ companies: ($ millions)
Size:
95th Percentile
75th Percentile
Median
25th Percentile
5th Percentile
Market Value of Equity
OLS Beta
Sum Beta
$132
$117
$100
$96
$77
2.72
2.01
1.53
0.96
0.36
3.31
2.34
1.73
1.09
0.34
Market Value of Equity
OLS Beta
Sum Beta
$132
$122
$104
$87
$80
2.19
1.92
1.59
0.85
0.27
2.39
2.06
1.53
0.91
0.27
Measures of Risk for NYSE companies only: ($ millions)
Size:
95th Percentile
75th Percentile
Median
25th Percentile
5th Percentile
Source: Compiled from Standard & Poor’s Capital IQ. Beta calculations use data from
Standard & Poor’s Research Insight. Calculations by Duff & Phelps LLC. Used with
permission. All rights reserved.
are included in decile 10y (New York Stock Exchange [NYSE], American Stock
Exchange [AMEX], and NASDAQ companies) and Exhibit 14.4 displays information on the type of companies that are included in decile 10z (NYSE, AMEX,
and NASDAQ companies).
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EXHIBIT 14.4 Breakdown of Decile 10z Companies as of September 30, 2008: Market Value
between $1.575 and $74.9 Million
($ millions)
Size:
Market Value of Equity
Book Value of Equity
MVIC
Total Assets
$68
$50
$32
$17
$6
$111
$47
$24
$11
$2
$203
$56
$31
$15
$zero
$373
$93
$46
$24
$7
95th Percentile
75th Percentile
Median
25th Percentile
5th Percentile
MVIC ¼ Market Value of Equity þ Book Value of Preferred Stock þ Book Value of Debt
Profitability:
Sales
5-Yr Avg Net Income
before Extra Ordinary
5-Yr Avg
EBITDA
Latest Fiscal Year
Return on Book Equity
95th Percentile
75th Percentile
Median
25th Percentile
5th Percentile
$325
$86
$37
$13
$zero
$6
$2
$1
$7
$31
$23
$5
$1
$3
$20
31%
7%
8%
47%
160%
Measures of Risk for NYSE þ AMEX þ NASDAQ companies: ($ millions)
Size:
95th Percentile
75th Percentile
Median
25th Percentile
5th Percentile
Market Value of Equity
OLS Beta
Sum Beta
$68
$50
$32
$17
$6
2.74
1.86
1.38
0.88
0.18
3.35
2.22
1.65
1.07
0.28
Market Value of Equity
OLS Beta
Sum Beta
$71
$67
$50
$41
$29
2.64
2.07
1.53
0.94
0.89
2.75
2.20
1.48
1.29
0.97
Measures of Risk for NYSE companies only: ($ millions)
Size:
95th Percentile
75th Percentile
Median
25th Percentile
5th Percentile
Source: Compiled from Standard & Poor’s Capital IQ. Beta calculations use data from
Standard & Poor’s Research Insight. Calculations by Duff & Phelps LLC. Used with
permission. All rights reserved.
From these data we can conclude:
&
Betas used for calculating the size premium for decile 10z (OLS method) generally
understate the beta estimates and overstate the size premium. (See comparison of
OLS betas and sum betas in Measures of Risk section of Exhibit 14.4 for all
companies.)
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
&
Decile 10y is populated by many large (see companies as measured by Total
Assets section of Exhibit 14.3 in 95th percentile) but highly leveraged companies with small-market capitalizations that probably do not match the characteristics of financially healthy but small companies.
There are companies with no sales included in the data (e.g., speculative start-ups).
Stocks of the troubled companies included in the data (companies with negative
returns on the latest fiscal year book value) probably are trading like call options
(unlimited upside, limited downside). Even if you were to use the sum beta
method, the beta estimates would likely be underestimated and the size premium overstated.
&
&
Before one uses the size premium data for 10b or its top and bottom halves, 10y
and 10z, one needs to determine if the mix of companies that comprise the subdeciles are indeed comparable to the subject company.
The Duff & Phelps studies screen out speculative start-ups, distressed (i.e.,
bankrupt), and high-financial-risk companies. The studies measure beta using the
sum beta method. This methodology was chosen to counter the criticism of the size
effect by some that the size premium is a function of the high rates of return for
speculative companies and distressed companies in the data set.
Duff & Phelps still observe the size effect for a more recent period (since 1963),
where size is measured by eight size measures, including six that are not market capitalization based. Exhibit 14.5 shows the breakdown of companies in portfolio 25
(the smallest companies). If the subject company is not highly levered, the companies
in portfolio 25 may be more comparable to a small subject company, and therefore
the size premium data for portfolio 25 may be more appropriate to use for your
subject company.
EXHIBIT 14.5 Breakdown of Portfolio 25 Companies as of December 31, 2008
5th Percentile
25th Percentile
50th Percentile
75th Percentile
95th Percentile
5th Percentile
25th Percentile
50th Percentile
75th Percentile
95th Percentile
Market Value
of Equity
Book Value
of Equity
5-Year Average
Net Income
Market Value of
Invested Capital
$17.135
51.001
97.977
167.623
226.405
$9.104
26.860
56.311
90.991
116.216
$0.448
1.539
3.203
4.927
6.770
$21.744
64.396
133.210
217.506
311.617
Total
Assets
5-Year
Average
EBITDA
Sales
Number of
Employees
$16.752
51.318
116.807
187.369
269.360
$1.679
5.250
10.666
18.611
26.911
$19.318
52.213
104.237
163.140
224.170
34
123
236
359
497
Source: Calculated (or derived) based on CRSP1 data, # 2008 Center for Research in Security
Prices (CRSP1), University of Chicago Booth School of Business, and Standard & Poor’s
Compustat data. Calculations by Duff & Phelps LLC. Used with permission. All rights reserved.
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269
DATA ISSUES
Critics of the size effect point out various issues with the data used, resulting in
anomalies that people mistakenly have observed as the size effect. These data issues
are seasonality, bid/ask bounce bias, transaction costs, and time-varying risk factors.
We present a discussion of these issues in Appendix 14A.
Some analysts have commented on the fact that small companies have not outperformed large companies consistently, particularly after 1982. The only reason for
breaking the data between pre- and post-1982 periods is that Morningstar changed
the methodology for calculating its small-company index returns in 1982.3
Through 1981, the Morningstar small-company series was calculated using the
returns on a synthetic portfolio constructed from Center for Research in Security
Prices (CRSP) data for stocks in the smallest quintile of the NYSE (ranked by market
value). This is ‘‘synthetic’’ in the sense that it is not based on the returns of an actual
fund. Actually, it is retrospectively calculated from a database of stock returns making assumptions about portfolio balance and reinvestment. From 1982 onward,
Morningstar measured the small-company returns using the actual returns on the
Dimensional Fund Advisors Small Company 9–10 Fund (DFA). The DFA returns
are net of transaction costs and free of the delisting bias just discussed.
For some time after 1982, small companies did not, on average, outperform
larger stocks (as measured by the Morningstar large-company returns). This observation sometimes is cited to cast doubt on the integrity of the pre-1982 smallcompany data. Some analysts contend that the small-stock effect disappeared after
1981 because Morningstar switched from the ‘‘biased’’ CRSP data to the
‘‘unbiased’’ DFA returns. This argument does not withstand scrutiny. If you want to
illustrate the extent to which a bias is eliminated by using the DFA returns, it is not
logical to compare the post-1981 DFA premium to a pre-1982 premium derived
from CRSP data. Rather, you would more appropriately compare the DFA returns
to CRSP returns over the same period.
Exhibit 14.6 presents size premium data for the CRSP deciles for various recent
periods. These size premiums are calculated relative to the Morningstar income
returns on long-term U.S. government bonds using annual betas.
As shown, while the size premium has varied in magnitude, it still exists in the
most recent 20-year period (even after 1989). Some authors’ claims that it did not
exist after 1980 is just due to the specific period chosen (as it did not appear in the
numbers in the period 1979–2008). But then again, it does not appear in the period
1969–2008 either, while it appeared in the period 1959–2008. This is evidence that
the size effect as an aggregate effect is cyclical. That cyclicality is part of the risk of
small companies; if they always earned more than large companies, they would not
be riskier in the aggregate.
Finally, the fact that the Duff & Phelps studies observe the size effect when
(1) size is measured by six size measures that are not market capitalization based
and (2) speculative and distressed companies are excluded, counters critics that the
size effect is a function of high risk, high discount rate companies and not company
size.
3
David King, ‘‘Do Data Biases Cause the Small Stock Premium?’’ Business Valuation
Review. (June 2003): 56–61.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
EXHIBIT 14.6 Returns in Excess of CAPM with S&P 500 Benchmark
Table 1. Long-Term Returns in Excess of CAPM Estimation for Decile Portfolios of
the NYSE/AMEX/NASDAQ, 1989–2008
Decile
Beta
Arithmetic
Mean
Return
1-Largest
2
3
4
5
6
7
8
9
10-Smallest
0.99
1.00
1.08
1.03
1.05
1.08
1.06
1.09
1.03
0.92
10.38%
11.29%
11.06%
10.94%
10.91%
11.66%
11.35%
11.85%
12.21%
13.13%
Realized
Return in
Excess of
Risk-free
Ratey
Estimated
Return in
Excess of
Risk-free
Ratez
4.17%
5.08%
4.85%
4.73%
4.70%
5.45%
5.14%
5.64%
6.00%
6.92%
4.10%
4.15%
4.46%
4.28%
4.34%
4.46%
4.38%
4.49%
4.28%
3.80%
Standard
Size Premium Deviation
(Return in of Realized
Excess
Excess of
Return
CAPM)
0.07%
0.93%
0.39%
0.45%
0.36%
0.99%
0.76%
1.25%
1.72%
3.12%
20.43%
19.53%
20.55%
19.62%
20.02%
22.67%
21.65%
23.29%
25.81%
30.35%
Table 2. Long-Term Returns in Excess of CAPM Estimation for Decile Portfolios of
the NYSE/AMEX/NASDAQ, 1979–2008
Decile
Beta
Arithmetic
Mean
Return
1-Largest
2
3
4
5
6
7
8
9
10-Smallest
0.98
1.02
1.07
1.06
1.05
1.09
1.09
1.12
1.07
0.97
12.17%
13.59%
13.60%
13.71%
13.63%
14.53%
13.91%
14.55%
14.38%
14.06%
Realized
Return in
Excess of
Risk-free
Ratey
Estimated
Return in
Excess of
Risk-free
Ratez
4.59%
6.01%
6.02%
6.13%
6.05%
6.96%
6.34%
6.97%
6.80%
6.48%
4.86%
5.04%
5.30%
5.24%
5.20%
4.37%
5.41%
5.53%
5.31%
4.81%
Standard
Size Premium Deviation
(Return in of Realized
Excess
Excess of
Return
CAPM)
0.27%
0.97%
0.72%
0.89%
0.85%
1.59%
0.93%
1.44%
1.49%
1.67%
17.89%
17.44%
17.97%
17.88%
18.28%
20.62%
20.27%
22.10%
23.64%
27.27%
Table 3. Long-Term Returns in Excess of CAPM Estimation for Decile Portfolios of
the NYSE/AMEX/NASDAQ, 1969–2008
Decile
Beta
Arithmetic
Mean
Return
1-Largest
2
0.98
1.04
10.19%
11.26%
Realized
Return in
Excess of
Risk-free
Ratey
Estimated
Return in
Excess of
Risk-free
Ratez
2.77%
3.83%
3.11%
3.30%
Standard
Size Premium Deviation
(Return in of Realized
Excess of
Excess
CAPM)
Return
0.34%
0.53%
18.05%
18.76%
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EXHIBIT 14.6 (Continued)
Table 3. (Continued)
Decile
Beta
Arithmetic
Mean
Return
3
4
5
6
7
8
9
10-Smallest
1.10
1.10
1.10
1.12
1.15
1.18
1.15
1.08
12.13%
12.03%
12.12%
12.93%
12.38%
12.92%
12.18%
12.55%
Realized
Return in
Excess of
Risk-free
Ratey
Estimated
Return in
Excess of
Risk-free
Ratez
4.71%
4.61%
4.70%
5.51%
4.96%
5.50%
4.76%
5.13%
3.48%
3.50%
3.49%
3.56%
3.66%
3.75%
3.67%
3.43%
Standard
Size Premium Deviation
(Return in of Realized
Excess
Excess of
Return
CAPM)
1.23%
1.11%
1.21%
1.95%
1.30%
1.75%
1.09%
1.70%
19.61%
20.82%
20.95%
22.85%
23.50%
25.22%
26.54%
30.01%
Table 4. Long-Term Returns in Excess of CAPM Estimation for Decile Portfolios of
the NYSE/AMEX/NASDAQ, 1959–2008
Decile
Beta
Arithmetic
Mean
Return
1-Largest
2
3
4
5
6
7
8
9
10-Smallest
0.98
1.04
1.09
1.10
1.10
1.13
1.16
1.18
1.15
1.08
10.09%
11.39%
12.57%
12.59%
12.97%
13.86%
13.45%
14.62%
13.98%
15.35%
Realized
Return in
Excess of
Risk-free
Ratey
Estimated
Return in
Excess of
Risk-free
Ratez
3.30%
4.59%
5.78%
5.79%
6.17%
7.06%
6.66%
7.82%
7.18%
8.56%
3.74%
3.96%
4.18%
4.21%
4.19%
4.31%
4.43%
4.52%
4.39%
4.12%
Standard
Size Premium Deviation
(Return in of Realized
Excess
Excess of
Return
CAPM)
0.44%
0.63%
1.60%
1.58%
1.98%
2.75%
2.23%
3.30%
2.79%
4.44%
17.05%
17.58%
18.66%
20.12%
20.82%
22.51%
23.58%
25.94%
27.39%
32.30%
Betas are estimated from monthly portfolio total returns in excess of the 30-day U.S. Treasury bill total return versus the S&P 500 total returns in excess of the 30-day U.S. Treasury
bill, January 1959–December 2008, January 1969–December 2008, January 1979–December
2008, and January 1989–December 2008.
y
Historical risk-free rate is measured by the arithmetic mean income return component of
20-year U.S. government bonds for each period.
z
Calculated in the context of the CAPM by multiplying the equity risk premium by beta.
The equity risk premium is estimated by the arithmetic mean total return of the S&P
500 minus the arithmetic mean income return component of 20-year U.S. government
bonds for each period.
Source: Calculated (or derived) based on CRSP1 data, Copyright # 2009 Center for
Research in Security Prices (CRSP1), University of Chicago Booth School of Business.
Calculations performed by Duff & Phelps LLC. Used with permission. All rights reserved.
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RISKS OF SMALL COMPANIES
Traditionally, small companies are believed to have higher required rates of return
than large companies because small companies inherently are riskier. One study
found that analysts and investors have difficulty evaluating small, little-known companies and estimating traditional quantitative measures of risk. This ambiguity adds
to the risk of investment and the return required to attract investors.4
However, this leaves the question of why small-stock returns have not consistently outperformed large-company stocks for various periods. The data suggest
alternative views. Readers of the SBBI Yearbooks have long been aware that the
small-stock premium tends to move in cycles, with periods of negative premiums
followed by periods of high premiums. Periods in which small firms have outperformed large firms have generally coincided with periods of economic growth. At
least one study contends that the variability in the size effect over time is predictable
since large firms generally outperform small firms in adverse economic conditions.
Credit conditions are exceedingly important for all firms, but especially for small
firms. Small firms generally are at a disadvantage when it comes to financing, and
suppliers of debt capital are less likely to lend to small firms in periods of adverse
economic conditions.5 For this reason, analysts should not be astonished to find
small-company stocks underperforming for lengthy periods of time. But even then,
factors affecting profitability change over time. For example, since the late 1990s,
many companies have faced a perceived lack of pricing power. In this type of environment, small firms are likely to be at a disadvantage.6
Exhibit 14.7 plots the annual return premium for the returns of the CRSP 10th
decile compared to the 1st decile from 1982 through 2008. The overall pattern since
1982 resembles the sort of cycles seen from 1926 to 1981. In this sense, small stocks
have performed in recent years much as they have always performed.
Some analysts claim that the historical average size premium is greatly reduced
if you exclude the period 1974 through 1983. During that time, small stocks outperformed large stocks by an extraordinary margin. It makes little sense to exclude
a 10-year period from the calculation of a historical average merely because its average premium was higher than that of any other 10-year period.
Advocates of the size effect can find satisfaction in the erratic performance of
small-cap stocks. If you believe that small stocks are riskier than large stocks, then it
follows that small stocks should not always outperform large stocks in all periods.
This is true even though the expected returns are higher for small stocks. By analogy,
4
R. Olsen and G. Troughton, ‘‘Are Risk Premium Anomalies Caused by Ambiguity?’’ Financial Analysts Journal (March–April 2000): 24–31.
5
Ching-Chih Lu, ‘‘The Size Premium in the Long Run,’’ Working paper, December 2009, 28,
reports on a study he conducted comparing the average market values of common equity
between companies with investment-grade credit ratings and those with non-investmentgrade credit ratings for the period 1994–2008. He found that the companies with better
credit ratings were 9 to 10 times larger than the size of companies with poorer credit ratings.
Available at http://ssrn.com/abstract=1368705.
6
Satya Dev Pradhuman, Small-Cap Dynamics: Insights, Analysis, and Models (New York:
Bloomberg Press, 2000), 23–28.
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273
40
0
-40
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
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EXHIBIT 14.7 Small-Stock Premium, 1982–2008, Small-Company Minus Large-Company
Returns (Size Measured by Market Capitalization of Common Equity)
Source: Calculated (or derived) based on CRSP1 data, Copyright # 2009 Center for
Research in Security Prices (CRSP1), University of Chicago Booth School of Business. Calculations by Duff & Phelps LLC. Used with permission. All rights reserved.
bond returns occasionally outperform stock returns, yet few would contend that
over time the expected return on bonds is greater than the expected return on stocks.
One market observer has written: ‘‘An important question that is not answered
by the doubters of the small stock effect is why smaller capitalization stocks have had
performance cycles at all.’’7 The explanations of data bias that have been offered by
the doubters are not such as would give rise to the small-stock cycles that can be
observed in the historical data. For instance, stock delistings may follow cyclical business conditions, giving some cyclicality to the delisting bias. Nonetheless, the smallstock premium is not strongly correlated with the business cycle. For example, during
bull markets, small stocks sometimes outperform and sometimes underperform larger
stocks. Moreover, the delisting effect is much too small to account for the wide
swings that are evident over time in the small-stock premium. It is even more difficult
to imagine how transaction costs could give rise to the observed multiyear cycles
because these transaction costs are incurred with every trade, every day.
Some analysts analyze small firms as equivalent to scaled-down large firms.
Practitioners know that small firms have risk characteristics that differ from those
of large firms. One study frames the differences in terms of options.8 Potential competitors can more easily enter the ‘‘real’’ market (market for the goods and/
or services offered to customers) of the small firm and ‘‘take’’ the value that the
small firm has built. Large companies have more resources to better adjust to competition and avoid distress in economic slowdowns. Small firms undertake less
research and development and spend less on advertising than large firms, giving
them less control over product demand and potential competition. Small firms have
fewer resources to fend off competition and redirect themselves after changes in the
market occur. Those authors describe the value of the firm as (1) the value of assets
in place plus (2) the present value of future growth options minus (3) an unwritten
call option (a ‘‘real’’ option) on the business or the value that can be taken by
7
Richard Bernstein, Style Investing: Unique Insights into Equity Management (New York:
John Wiley & Sons, 1995), 142.
8
M. S. Long and J. Zhang, ‘‘Growth Options, Unwritten Call Discounts and Valuing Small
Firms,’’ EFA 2004 Maastricht Meetings Paper No. 4057, March 2004. Available at http://
ssrn.com/abstract=556203.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
potential competition that can enter the market and destroy value. The value of this
unwritten call option increases with volatility. But even though company size and
variance of returns are highly correlated, the authors find that this unwritten call
option against the value of the small firm is greater than predicted by variance alone.
Those authors found a size premium and an economic basis for its existence.
The Duff & Phelps Risk Premium Report—Risk Study finds that as company
size decreases, measures of risk calculated from financial statement data generally
increase and that the market demands a greater rate of return as company risks
increase; hence the cost of equity capital for small firms is greater.
In a recent study, Hou and van Dijk determined that realized stock returns are a
very noisy measure of expected returns and that realized returns can deviate significantly from expected returns over prolonged periods of time. In fact, realized returns
are a function of expected returns plus shocks to cash flows (actual cash flows over
time may differ significantly from expected cash flows as of the date of investment)
plus shocks to discount rates (actual discount rates over time may differ significantly
from market consensus discount rates as of the date of investment).9
These authors found that the apparent disappearance of the size effect after the
early 1980s was due to cash flow shocks. Realized returns for small companies were
generally less than expected because of negative cash flow shocks. Cash flows were generally less than expected for small companies. On the other hand, realized returns for
large companies were generally greater than expected because of positive cash flow
shocks. Cash flows were generally greater than expected for large companies. What
caused the cash flow shocks? The number of newly public firms in the United States
increased dramatically in the 1980s and 1990s compared with prior periods. Simultaneously, the profitability and survival rate of the newly public firms generally declined
from the profitability and survival rates for firms that became public in prior years.
The authors showed that these shocks to profitability and cash flows caused the
observed size premiums to be negligible in the 1980s and 1990s. They adjusted the
realized returns for the cash flow shocks, and the result was that returns of small
firms on a pro forma basis exceeded the returns of large firms by approximately
10% per annum, consistent with the size premium in prior periods.
SHOULD THE COST OF EQUITY CAPITAL USE A CHANGING
SIZE PREMIUM?
Fama and French studied the composition of firms that have resulted in the greater
than expected returns observed for smaller firms.10 They studied the migration of
small (measured by market value) firms during the periods from 1927 to 1963 and
1963 to 2005 and found that a small percentage of successful small firms, with their
market capitalization increasing due to their success, resulted in the preponderance
of above-average return observed. Fama and French point out that when stocks are
9
Kewei Hou and Mathijs A. van Dijk, ‘‘Profitability Shocks and the Size Effect in the CrossSection of Expected Stock Returns,’’ Working paper, January 14, 2010. Available at http://
ssrn.com/abstract=1536804.
10
Eugene Fama and Kenneth French, ‘‘Migration,’’ Financial Analysts Journal (May–June
2007): 48–58.
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allocated to portfolios in one year, one does not know which stocks will change in
size (a small company becoming larger due to its success). If stock prices are rational, the stock price set in one year is the best forecast of (1) the probability of changing size (i.e., succeeding) during the following year and (2) the stock price observed
as a result of the success. The size (and market value) premium in average returns
‘‘are the result of rational risks of concern to investors.’’11
In another study, Lu again follows the composition of the firms that result in the
size premium.12 This author concludes that despite periods in which the small-stock
premium goes away, over the long term, there is evidence that the size effect is real,
given a longer sample period than many critiques of the size effect use in their studies. He does find that the size premium did disappear during the 1980s and the early
1990s, but it was intact in most other periods.13
While no one can estimate which small firms will become larger firms as of a
valuation date, the author finds that generally the successful firms will become large
firms after two years. Small companies cannot keep their return premium once they
become larger companies; otherwise, the small-stock premium will become a largestock premium.
The author recognizes that practitioners who do add a size premium in developing
the cost of equity capital typically do so for the entire projection period in their discounted cash flow valuations, even when the projections indicate that the expected net
cash flows (and the resulting value in future periods) will cause the company to become
a size that would not warrant a size premium in future years. Although the cost of
equity capital for larger companies can typically be well explained by the CAPM or by
one of the other asset pricing models (see Chapter 17), the author concludes that not
including a size premium overvalues the currently small company. The author contends that once the projected size of the subject company increases such that it qualifies
as a big company, the size premium should no longer be included in the cost of equity
capital. The author endorses using a time-varying and value-varying size premium.
The author finds that the size premium is more pronounced in periods of economic expansion, when yield differences between high-grade and low-grade bonds
is large, and during bear markets. In these periods, investors generally move out of
small company stocks into large company stocks due to the perception that large
companies are less risky. Prices of small company stocks are bid down compared to
large company stocks, causing returns in future periods to be greater for small company stocks than for big company stocks.
We recognize that many practitioners value small companies that may never be
expected to move from being a small company to becoming a large company. But
for other valuations, the analyst needs to study the progression of expected net cash
flows (and implied values in future periods) contained in the projections for the subject business and decide whether the recommendation of Lu to use a time-varying
size premium is appropriate.
11
Eugene Fama and Kenneth French, ‘‘Migration,’’ Financial Analysts Journal (May–June
2007): 57. Available at http://ssrn.com/abstract=556203.
12
Ching-Chih Lu, ‘‘The Size Premium in the Long Run,’’ Working paper, December 2009.
Available at http://ssrn.com/abstract=1368705.
13
Ching-Chih Lu, ‘‘The Size Premium in the Long Run,’’ Working paper, December 2009, 8.
Available at http://ssrn.com/abstract=1368705.
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RELATIONSHIP OF SIZE AND MEASURES OF RISK
It has been pointed out in the financial literature that researchers may be mixing a
‘‘size’’ effect with a ‘‘risk’’ effect when measuring company size by ‘‘market value.’’
Market value is not just a function of size; it is also a function of the discount rate.
Therefore, some companies will not be risky (high discount rate) because they are
small but instead will be small (low market value) because they are risky. The Duff
& Phelps Risk Study goes further in documenting indicators of risk in portfolios of
stocks of small companies. It also goes beyond size and beta and investigates
the relation between equity returns and fundamental risk measures drawn from
company financial statements.
In Chapter 15 on company-specific risk, we discuss the Risk Study.
In one study, the authors determined that after controlling for beta risk, the
size effect disappears in ‘‘up markets’’ but appears in ‘‘down markets.’’14 The size
effect in down markets appears to account for the size effect across both market
conditions. As recessions deepen in down markets, the assets of small companies
become riskier, causing investors to require a premium for investing in the small
companies. The authors determine that the size premium is correlated to the size
of residuals (from beta estimates using regressions of historical returns), casting
doubt on market efficiency.
RELATIONSHIP OF SIZE AND LIQUIDITY
Liquidity affects the cost of capital. The generalized cost of capital relationship was
shown in Formula 5.1, which we repeat here:
(Formula 14.1)
EðRi Þ ¼ Rf þ RPi
where: E(Ri) ¼ Expected return of security i
Rf ¼ Risk-free rate
RPi ¼ Risk premium for security i
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. But the degree of
liquidity is one of the risk factors for all investments. Any discussion of a liquidity
premium, therefore, would be incomplete without accounting for underlying stock
risks before considering relative liquidity.
Stocks of small companies generally do not have the same level of liquidity as
large-company stocks. This is a function, first and foremost, of the mix of shareholders. Many institutional investors do not own stocks in small companies
because they have too much money to invest. Were they to invest as little as 1%
14
Jungshik Hur and Vivek Sharma, ‘‘Stock Market Returns and Size Premium,’’ Working
paper, March 2007.
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Criticisms of the Size Effect
277
of their available funds in small companies, they would be likely to ‘‘control’’ the
company. Institutional investors generally want liquidity to move into and out of
positions in a single firm. Therefore, one does not see the breadth of investors
investing in small company stocks. Is the size premium simply the result of differences in liquidity? If one is valuing a small company, that company will never
have the same breadth of shareholders as a large company, and whatever impact
the relative illiquidity of small companies has on the cost of capital will carry
over to any small company.
A study by Chan and Ibbotson looked at the relationship of liquidity and size
and determined that while they are intermingled, the size premium is separate from
and affects the cost of capital, regardless of the relative illiquidity of small companies.15 They show that when one divides companies by liquidity first, small companies with low liquidity still earn a return that exceeds that of large companies with
low liquidity, supporting the size premium.16
Another recent study by Abbott assesses the absolute contribution for each
factor individually, as well as in combination with other factors, to form an
estimate of the combined contribution of the factors considered in the model.
His study investigates the relative importance of the size and liquidity risk factors. The author used the FF three-factor model (explained in Chapter 17) as
the underlying cost of equity capital model and added to that model a liquidity
premium factor. His results are similar to those of Chan and Ibbotson. The
study results are presented in Appendix 14A.
SUMMARY
Despite many criticisms of the size effect, it continues to be observed in data sources
that utilize the CAPM methodology. Further, observation of the size effect is consistent with an expansion of the pure CAPM. This chapter shows that if equilibrium
capital asset prices are determined by a segmented model,17 small-firm effect is fully
explained in a segmented market equilibrium, which explains why it persists for
many years and in many countries. Since empirical evidence supports market segmentation, the abnormal returns are explained without the need to assume the existence of systematic statistical errors, market inefficiency, and so on. Studies have
shown the limitations of beta as a sole measure of risk. The size premium is an
empirically derived correction to the pure CAPM.
The validity of the size effect also has received recognition by academics and the
courts. For example, in a recent Delaware Chancery Court decision, both experts
15
Zhiwu Chan and Roger Ibbotson, ‘‘Liquidity and Valuation,’’ 2009 Ibbotson SBBI Valuation Yearbook (2009), 108–109.
16
Zhiwu Chan and Roger Ibbotson, ‘‘Liquidity and Valuation,’’ 2009 Ibbotson SBBI Valuation Yearbook (2009), 109, Table 7-17.
17
As suggested by Haim Levy, ‘‘Equilibrium in an Imperfect Market: A Constraint on the
Number of Securities in a Portfolio,’’ American Economic Review (September 1978): 643–
658; and Robert C. Merton, ‘‘A Simple Model of Capital Market Equilibrium with Incomplete Information,’’ Journal of Finance (July 1987): 483–510.
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applied a size premium in estimating the cost of equity capital.18 One of the experts
was a noted academic. The issue was not whether it was appropriate to apply a size
premium; rather, the issue was how to measure the size premium. Further applying a
size premium in estimating the cost of equity capital is not just applicable for valuing
a minority interest in a company, as the standard of fair value in the dispute was
proportionate value of the entire company.
18
In re Sunbelt Beverage Cor. Shareholder Litigation, Consol. C.A. No. 16089-CC (January
5, 2010).
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APPENDIX
14A
Other Data Issues Regarding
the Size Effect
Seasonality
Bid/Ask Bounce Bias
Delisting Bias
Transaction Costs
Risk Factors Are Time-Varying
Seasonality
The ‘‘January effect’’ is the empirical observation that rates of return for small
stocks have on the average tended to be higher in January than in the other months
of the year. The existence of a January effect, however, does not present a challenge
to the small-stock effect. This is true unless it can be established that the effect is the
result of a bias in the measurement of returns. Some academics have speculated that
the January effect may be due to a bias related to tax-loss selling.
Investors who have experienced a loss on a security may be motivated to sell
their shares shortly before the end of December. An investor makes such a sale in
order to realize the loss for income tax purposes. This tendency creates a preponderance of sell orders for such shares at year-end. If this is true, then (1) there may be
some temporary downward pressure on prices of these stocks, and (2) the year-end
closing prices are likely to be at the bid rather than at the ask price. The prices of
these stocks will then appear to recover in January when trading returns to a more
balanced mix of buy and sell orders (i.e., more trading at the ask price).
This appendix draws on a number of works: David King, ‘‘Do Data Biases Cause the Small
Stock Premium?’’ Business Valuation Review (June 2003): 56–61; Roger Grabowski and
David King, ‘‘Equity Risk Premium,’’ in The Handbook of Business Valuation and Intellectual Property Analysis Robert Reilly and Robert P. Schweihs, eds. (New York: McGraw-Hill,
2004), 3–29; and Mathijs A. van Dijk, ‘‘Is Size Dead? A Review of the Size Effect in Equity
Returns,’’ Working paper, March 6, 2007. Available at http://ssrn.com/abstract=879282.
The authors wish to thank Mr. King and Professor Van Dijk for their contributions to this
important topic. The authors wish to thank Ashok Bhardwaj Abbott for allowing us to reproduce research results for this appendix.
279
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Such ‘‘loser’’ stocks will have temporarily depressed stock prices. This creates
the tendency for such companies to be pushed down in the rankings when size is
measured by market value. At the same time, ‘‘winner’’ stocks may be pushed up in
the rankings when size is measured by market value. Thus, portfolios composed of
small-market-value companies tend to have more losers in December, with the
returns in January distorted by the tax-loss selling.
A recent study finds that the January returns are smaller after 1963–1979 but
have reverted to levels that appear before that period.19 More important, they find
that trading volume for small companies in January does not differ from other
months. They conclude that the January effect continues.
This argument vanishes if you use a measure other than market value (e.g., net
income, total assets, or sales) to measure size; the size effect is evident in the Duff &
Phelps Size Study using size measures other than market capitalization.
Bid/Ask Bounce Bias
There is an argument that the existence of bid/ask spread adds a bias to all stock
returns. Bid/ask spreads may add a bias particularly to portfolios of less liquid (generally smaller) companies that have larger bid/ask spreads. This bias results because
the movement from a bid to an ask price creates a measured rate of return that is
greater in percentage value than a movement from the same ask price to the same
bid price. Since trades occur randomly at either the bid or the ask, a small bias can
creep into measured returns.
Most studies of the small-size effect (such as those by Morningstar and Duff
& Phelps) use the Center for Research in Security Prices (CRSP) database, which
generally uses the closing price to measure rates of return. The closing price is
either a bid or an ask. In cases where there were no trades on a given day (the
most illiquid stocks with the greatest bid/ask spread), CRSP uses the average of
the bid and ask price. This procedure automatically ameliorates the bias to some
extent. But for thinly traded stocks, the ask is often a phantom price at which
holders would like to sell. Market participants may not be offering stock, especially if they do not have long positions in the stock. This probably makes bids
more realistic than asks.
This bias can be most pronounced if you measure rates of return on a daily
basis. Morningstar and Duff & Phelps calculate returns monthly at the portfolio
level. Then they compound the portfolio returns for each 12 months of the year to
get that year’s annual return. This procedure further mitigates much of the possible
bid/ask bounce bias.
The bid/ask bias has only a trivial impact on the observed small-stock effect.
Average bid/ask spreads are less than 4% of underlying stock price for the smallest decile of the New York Stock Exchange (NYSE). Spreads of even 4% would
give rise to biases in measured returns that are, at most, only a few basis points.
This assumes that annual returns are being compounded from monthly portfolio
results, as in the Duff & Phelps Size Study. However, the size effect is observed
19
Kathryn E. Easterday, Pradyot K. Sen, and Jens A. Stephan, ‘‘The Persistence of the Small
Firm/January Effect: Is It Consistent with Investors’ Learning and Arbitrage Efforts?’’
Working paper, June 2007. Available at http://ssrn.com/abstract=1166149.
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281
even for midsize public companies—companies for which the bid/ask spread
averages less than 1.5%.
Some analysts have suggested that using the geometric average of realized
returns would correct for the bid/ask bounce bias. However, this argument is spurious. The difference between the higher arithmetic average and the lower geometric
average does not arise from the bid/ask bounce. Geometric averages are always less
than arithmetic averages due simply to the principles of mathematics.
Delisting Bias
A possible delisting bias exists in many studies that have used the CRSP database. The delisting bias may be due to the fact that CRSP in many (but not all)
cases is missing prices for the period immediately after a stock is delisted from an
exchange. This problem is not caused by a bias in the CRSP data per se because
the database explicitly flags all instances of missing returns. The possible bias
occurs in how these missing returns are handled when you calculate average
returns for portfolios of companies. There are procedures for handling this issue
effectively. When these procedures are used appropriately, the size effect still
exists after the adjustment is made.
Does delisting bias explain away the size effect? The evidence from the Duff &
Phelps Size Study suggests otherwise. The Duff & Phelps studies have adjusted for
the delisting bias in annual updates published since 1998. In fact, the adjustment for
delisting makes little difference in the Duff & Phelps study results. In other words,
the size effect is still present after making the delisting adjustment because companies with a history of losses (or with certain other indicators of poor financial performance) are placed in a separate high-financial-risk portfolio. Such companies are
not included in any of the Duff & Phelps size-ranked portfolios. Companies with
poor financial performance are much more likely to incur a performance-related
delisting than are profitable companies. When Grabowski and King first started
adjusting for the delisting bias, the average return on the high-financial-risk portfolio declined by about 150 basis points. However, the delisting adjustment did not
materially affect the average returns on the size-ranked portfolios.
Moreover, CRSP completed a multiyear project of filling in the missing
delisting data. The evidence from the CRSP white paper on the subject confirms
that the delisting bias has been greatly exaggerated.20 First, CRSP now has
returns for the large majority of performance-related delists on the NYSE, American Stock Exchange (AMEX), and NASDAQ. The average performance-related
delisting across this population reflected a loss of about 22%. Thus, the 30% loss
assumed for missing returns in the Duff & Phelps studies now appears overstated. Second, CRSP compared returns on the CRSP capitalization-based portfolios under alternate assumptions about missing delisting returns. For the 10th
decile of the NYSE/AMEX/NASDAQ population, the average bias created by
ignoring the missing delisting returns is at most about 20 basis points (0.2%) on
a compound market-weighted basis for the period 1926 to 2005. This assumes
the extreme case that companies with missing delisting returns incur a 100%
20
Center for Research in Security Prices, CRSP Delisting Returns (Chicago: Center for
Research in Security Prices, University of Chicago, 2001).
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
loss. If you assume only a 30% loss, the ‘‘bias’’ virtually disappears. This is
important, because Morningstar uses the CRSP capitalization-based portfolios in
deriving the size premiums. Accordingly, analysts can safely conclude that there
is little bias in the Morningstar data.
Transaction Costs
The capital asset pricing model (CAPM) abstracts from the influence of liquidity on
transaction costs. Some analysts have suggested that the size effect should be set
aside because various studies have ignored transaction costs in measuring rates of
return. The analysts point out that small stocks often have higher transaction costs
than large stocks. In addition, the historical size premium can be greatly reduced if
one makes certain assumptions about transaction costs and holding periods. However, in a discounted cash flow analysis, analysts typically use projected cash flows
that do not make any adjustment for an investor’s hypothetical transaction costs. It
may be that small stocks are priced in a way that increases the rates of return so as to
reward investors for the costs of executing a transaction. If so, it would be a distortion to express the discount rate on a net-of-transaction-cost basis while the cash
flow projections are on a before-transaction-cost basis.
Academic studies support the hypotheses that illiquidity is a factor in pricing
and returns of stocks and that small firms are more sensitive to market liquidity, but
the illiquidity factor does not capture the size effect completely. Moreover, any reasonable adjustment for transaction costs should recognize that investors can mitigate these costs on an annual basis by holding their stocks for a longer period. In
fact, investors in small companies tend to have longer holding periods than investors
in large companies.
For the study, Abbott built a database of daily observations spanning all listed
securities for the NYSE, AMEX, and NASDAQ for January 1993 to December 2008
obtained from CRSP. This time period was selected since the reporting of trading
volume was standardized across the three equity markets only in June 1992. The
trading volume reported by NASDAQ before June 1992 was the aggregate of volume reported by all dealers in the security, leading to inflated counts as dealers and
market makers reported each buy and sell transaction separately. Risk-free rates,
market returns, and Fama French factors for size and market-value-to-book-value
of equity ratio for each month were obtained.21 The securities ineligible for continued listing were deleted from the sample to avoid any delisting bias. Further, if a
security traded at a price below $1 during a given month, the firm was dropped
from the sample for that month to ensure consistency.
Daily market values were calculated for each firm by multiplying the closing
price for the day times the number of shares outstanding. The Amivest ratio, a measure of price pressure introduced by Amihud,22 was calculated for each trading day
by dividing the daily absolute return by the dollar volume traded. Trading cost was
estimated as the difference between the daily holding return (closing price to closing
21
Data set provided by Professor Kenneth French at http://mba.tuck.dartmouth.edu/pages/
faculty/ken.french/data_library.html.
22
Yakov Amihud, ‘‘Illiquidity and Stock Returns: Cross-Section and Time-Series Effects,’’
Journal of Financial Markets 5 (2002): 31–56
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Other Data Issues Regarding the Size Effect
EXHIBIT 14A.1 Size and Trading Cost Impact
Market Value of Equity Portfolio
Average Daily Trading Cost
1- Largest companies
2
3
4
5
6
7
8
9
10-Smallest companies
0.75489%
1.07736%
1.33369%
1.67466%
2.05954%
2.50398%
3.16594%
4.13995%
5.57523%
9.67356%
Source: Calculations by Ashok Abbott. # Ashok Bhardwaj Abbott, 2010.
price) and the daily trading return (ask price previous day to bid price current day).
The author calculated l (lambda),23 a measure of an individual stock’s liquidity,
with higher levels signifying that the current order flow in the market can absorb
larger volumes of trading without affecting prices.
Ten portfolios were constructed at the end of each month based on the average
market value of equity (see Exhibit 14A.1) and the mean l (liquidity) (see Exhibit
14A.2). Returns in the subsequent month were used to calculate the liquidity risk
premium factor by subtracting the returns of the three portfolios with the greatest
liquidity from the returns of the three portfolios with the least liquidity.
As expected, he finds significant negative relationships between the size of the
companies as measured by market value and trading cost/price impact measures.
Stocks of larger firms can be traded at a lower cost and are subject to less price pressure. Trading costs and price pressure (measured as the Amivest ratio) both decline
as the portfolios contain larger stocks (see Exhibit 14A.1).
EXHIBIT 14A.2 Liquidity and Trading Cost Impact
Liquidity (l) Portfolio
1-Most liquid companies
2
3
4
5
6
7
8
9
10- Least liquid companies
Average Daily Trading Cost
1.48241%
1.82615%
2.02649%
2.15579%
2.28703%
2.47802%
2.73914%
3.03041%
3.73256%
5.60277%
Source: Calculations by Ashok Abbott. # Ashok Bhardwaj Abbott, 2010.
23
Ashok Abbott, ‘‘Estimated Holding Period for Listed Securities,’’ Valuation Strategies 84
(September–October 2004).
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EXHIBIT 14A.3 Size and Expected Liquidation Period for 50% of Stock (days)
Market Value of Equity Portfolio
Liquidation Period for 50% of Stock (days)
1- Largest companies
2
3
4
5
6
7
8
9
10- Smallest companies
524
799
745
839
840
998
1117
1484
1299
1493
Source: Calculations by Ashok Abbott. # Ashok Bhardwaj Abbott, 2010.
A similar relationship is seen between liquidity and cost of trading/price impact
(see Exhibit 14A.2). As stocks become more liquid, trading costs and price impact
both decline (measured by Amivest ratio), as suggested by theory.
At first glance, there appears to be a commonality between size based on market
value and liquidity levels. The expected average liquidation period for 50% of the
stock, calculated as just described, appears to decline as the market value increases
(see Exhibit 14A.3).
However, the relationship between size as measured by market value of equity
and liquidity as measured by l is not stable across time (see Exhibit 14A.4). The
level of liquidity changes over time, and the relative liquidity differentials across
EXHIBIT 14A.4 Liquidity Changes across Time and Size Portfolios
Year
Liquidation Period for 50% of Stock
(days) for Largest Companies
Liquidation Period for 50% of Stock
(days) for Smallest Companies
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
851
1187
745
668
491
520
563
497
439
402
456
430
325
304
187
148
3058
2836
2029
1698
1384
1006
1092
1085
1743
1720
1581
994
885
979
759
1010
Source: Calculations by Ashok Abbott. # Ashok Bhardwaj Abbott, 2010.
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Other Data Issues Regarding the Size Effect
EXHIBIT 14A.5 Changes across Time for Monthly Size
and Liquidity Premiums
Year
Liquidity Premium
Size Premium
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
0.78%
0.69%
0.07%
1.38%
1.89%
0.24%
3.04%
4.90%
1.77%
3.46%
0.49%
0.78%
0.32%
0.43%
0.23%
0.08%
0.35%
0.12%
0.46%
0.12%
0.30%
1.80%
1.09%
0.02%
1.67%
0.38%
1.70%
0.42%
0.12%
0.08%
0.68%
0.62%
Source: Calculations by Ashok Abbott. # Ashok Bhardwaj Abbott, 2010.
portfolios based on market value of equity also change. Therefore, size and liquidity
effects are not substitutes for each other.
If we examine the multifactor model size premiums and the liquidity premium
calculated as before, the size premiums exhibit some interesting properties. Size premiums are calculated from buy and hold returns. When we incorporate trading costs
in our analysis, the realizable returns from smaller company stocks are much less
than buy and hold returns. The cost of liquidity is considerable and increases as
the market value of the firm declines. Research reports that the prevailing level of
liquidity has a significant role in explaining considerable differences in returns
across different asset classes. As events driving the crisis of 2008–2009 have shown,
episodic illiquidity when prevailing market conditions change can result in extreme
changes in realizable prices. Therefore, it is important to consider prevailing market
conditions specific to the time period of estimation while valuing assets and applying
any premiums or discounts (see Exhibit 14A.5).
Exhibit 14A.6 presents results for contribution of the size and liquidity premium
when used together to estimate the returns in excess of the risk-free rate for the portfolio of the smallest market capitalization companies and the portfolio of the largest
market capitalization companies, sorted by liquidity, l. As expected, Abbott found
that the size premium contribution to the cost of equity capital on smaller company
stocks is larger and positive. However, the liquidity premium contribution uniformly declines as the stocks become more liquid and frequently turns large and negative for more liquid stocks in each size portfolio, such that small company stocks
that are very liquid in fact generate smaller returns than large company stocks that
are less liquid. This result is similar to the reported results in the SBBI Yearbook
analysis of the effect of liquidity on realized rates of return, which show that very
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EXHIBIT 14A.6 Contribution of Market Risk, Size (as Measured by Market Value of Equity),
and Liquidity Premiums to Realized Excess Returns (Portfolio 1 and 10 Results Displayed)
Market Value
of Equity
Portfolioa
1
1
1
1
1
10
10
10
10
10
Liquidity
Rank l
Quintileb
1
2
3
4
5
1
2
3
4
5
Intercept
Market
Risk
Premium
Size
Premium
Liquidity
Premium
Stock
Excess
Returns
1.18%
2.07%
1.70%
0.02%
0.25%
4.90%
1.63%
0.30%
0.99%
1.63%
8.16%
7.45%
6.84%
6.02%
5.57%
4.26%
5.63%
5.89%
6.11%
5.82%
1.37%
0.39%
0.03%
0.08%
0.15%
5.29%
6.49%
6.42%
6.79%
5.61%
3.12%
0.43%
1.86%
2.09%
2.04%
1.57%
0.11%
0.90%
1.68%
2.42%
7.59%
6.20%
7.04%
8.05%
7.52%
3.08%
10.38%
12.91%
13.60%
12.21%
a
Portfolios are ranked from largest companies in Portfolio 1 (by market capitalization of
equity) to the smallest companies in Portfolio 10 (by market capitalization of equity).
b
Companies ranked by liquidity as measured by l within each size ranked portfolio. Quintile 1
contains most liquid companies within the portfolio; quintile 5 contains the least liquid.
Source: Calculations by Ashok Abbott. The full results are available from the author.
#Ashok Bhardwaj Abbott, 2010.
liquid small stocks may at times be generating returns lower than equally liquid
larger stocks.24
This analysis shows that the impact of liquidity is systematic, significant, and
incremental to the role of size in explaining realized excess returns on stocks. The
author found that across all size portfolios, stocks with the greatest degree of liquidity generate the smallest returns.
Risk Factors Are Time-Varying
Many treat the CAPM as if beta estimates are constant over time. But considerable
research indicates that betas vary considerably over time. Academic studies that allow for time-varying betas and other risk factors generally can explain some, but not
all, of the size premium empirically observed.
24
SBBI 2009 Valuation Yearbook, 108–109.
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CHAPTER
15
Company-specific Risk
Introduction
Matching Fundamental Risk and Return—Duff & Phelps Risk Study
Relationship of Measures of Risk from Company Financial Statements and Returns
Using the Duff & Phelps Risk Study in the Build-up Method
Market Pricing of Company-specific Risk
Research on Unique or Unsystematic Risk
Total Beta and the Butler-Pinkerton Interpretation
Cost to Cure
Other Company-specific Factors
Summary
INTRODUCTION
Company-specific risk adjustments are intended to account for company-specific
factors affecting a company’s competitive position in the industry or unique characteristics that would cause investors to view that company’s risk differently than the
average risk characteristics of the pure play guideline public companies to which it
would be compared. Practitioners often identify company-specific characteristics
that they believe would cause investors to view the cost of capital that should be
applied to the expected cash flows of the subject company to differ from the cost of
capital investors would apply to the expected cash flows of those pure plays.
If there were sufficient pure play guideline public companies that had the same
risk characteristics as the subject company such that the market priced those risk
characteristics, there would be no need to make company-specific risk adjustments.
However, there are often insufficient or even no pure play guideline public companies with risk characteristics matching those of the subject company. In those cases
in which it is appropriate, company-specific risk adjustments can either add to the
cost of equity capital or reduce the cost of equity capital.
According to the pure capital asset pricing model (CAPM), unanticipated events
relating specifically to a company’s risks affect a company’s expected future cash
flows and should not be a component of a company’s cost of equity capital.
The authors want to thank David Turney and Katherine Nierman of Duff & Phelps LLC for
preparing material for this chapter.
287
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As previously stated, the cost of equity capital under pure CAPM is based on a company’s systematic risk, not a company’s specific risks.
Brealey, Myers, and Allen critique the use of the company-specific risk
adjustment:
Managers often add fudge factors to discount rates to offset worries. . . .
This sort of adjustment makes us nervous . . . the need for a discount rate
adjustment usually arises because managers fail to give bad outcomes their
due weight in cash flow forecasts. The managers then try to offset that mistake by adding a fudge factor to the discount rate.1
The proper estimation of beta or other systematic risk factors (e.g., downside
beta as discussed in Chapter 12) should help the practitioner better match the risk
and return as priced by investors with the appropriate risk and return for the subject
division, reporting unit, or closely held business. In Chapter 12, we discussed research that shows that investors are much less diversified than expected, even after
consideration of the efforts of investment advisors urging them to diversify. Further,
many do not hold the market portfolio as predicted by pure CAPM. Based on these
findings, it is reasonable to assume that investor rates of return expectations are
influenced by company-specific risk factors.
Many analysts are able to express qualitative reasons for company-specific risk
adjustments but rarely can provide data relating those qualitative factors to actual
measurements in expected return. In this chapter, we discuss research that shows
that company-specific risk is priced by the market and characteristics of companies
that cause the market to price company-specific risk.
Another company-specific risk issue is distress. We discuss issues surrounding
distress in Chapter 16.
MATCHING FUNDAMENTAL RISK AND RETURN—DUFF &
PHELPS RISK STUDY
Practitioners typically have quantified the relationship between risk and expected
return only by measuring risk in terms of beta and size. While company size is a risk
factor in and of itself, Grabowski and King, original co-authors of what is now the
Duff & Phelps Risk Premium Report—Risk Study, were interested in understanding
whether the stock market recognized risk as measured by fundamental or accounting information.
They used a database combining stock prices, number of shares, and dividend
data by company from the Center for Research in Security Prices (CRSP) database,
with accounting and other data from the Standard & Poor’s Compustat database, to
analyze fundamental risk. Thereafter, Grabowski and King published a series of
articles reporting their findings. That research relates 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
1
Richard A. Brealey, Stewart C. Myers, and Franklin Allen, Principles of Corporate Finance, 8th ed. (Boston: Irwin McGraw-Hill, 2006), 225.
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Company-specific Risk
289
accounting information may also be called fundamental or accounting measures of
company risk to distinguish them from a stock market–based measure of equity risk
such as beta. The Duff & Phelps Risk Premium Report—Risk Study annually
updates this research.2
Because Grabowski and King were interested in understanding how the stock
market prices the risk of established companies, the Risk Study is limited to companies with a track record of profitable performance. The company selection process is
designed to parallel the process used in selecting guideline public companies when
an analyst determines guideline public companies in applying the market approach.
For example, assume that the analyst identifies 10 possible guideline public companies that are in the same SIC code as the subject profitable company. One criterion
for selecting among the guideline public companies is to include only profitable companies. That same selection criterion was used in developing the database for the
Risk Study.
They use three alternative measures of company 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.)
The data show a clear empirical relationship between risk measures and historical rates of return and realized premiums for profitable companies. The relationship
for each risk measure is divided into 25 risk-ranked portfolios, each portfolio with a
different risk and return.
The Duff & Phelps studies exclude certain high-financial-risk companies from
the base set of companies. The 25 portfolios of the Risk Study exclude those companies with high leverage, categorized as the high-financial-risk companies. The leverage of the high-financial-risk companies is significantly greater than that of any of
the other portfolios. The return data for the high-financial-risk companies are
reported in separate exhibits and discussed in Chapter 16. Beginning with the Risk
Premium Report 2010, the single-line high-financial-risk portfolio returns will not
be displayed in the exhibits with the Risk Study 25 portfolios.
To calculate realized risk premiums, Duff & Phelps first calculates an average
rate of return for each portfolio over the sample period. Returns are based on dividend income plus capital appreciation and represent returns after corporate-level income taxes (but before owner-level taxes). They then subtract the average income
return earned on long-term U.S. government bonds over the same period (using
SBBI data) to arrive at an average realized risk premium.
The Duff & Phelps Risk Study finds that as company size decreases, measures of
risk calculated from financial statement data generally increase and that the market
2
This section is adapted from the Duff & Phelps Risk Premium Report 2009. Used with
permission. The Risk Premium Report was published as the Standard & Poor’s Corporate
Value Consulting Risk Premium Report for reports titled 2002 to 2004 and as the PricewaterhouseCoopers Risk Premium Reports and Price Waterhouse Risk Premium Reports
for years before 2002.
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demands a greater rate of return as company risks increase; hence the cost of equity
capital for riskier firms is greater.
Relationship of Measures of Risk from Company
Financial Statements and Returns
The Risk Study documents indicators of risk in portfolios of stocks of the same companies as the Size Study. It investigates the relation between equity returns and the
three fundamental risk measures previously mentioned.
A variety of academic studies have examined the relationship between financial
statement data and various aspects of business risk.3 Research has shown that measures of earnings volatility can be useful in explaining credit ratings, predicting
bankruptcy, and explaining the CAPM beta.
The Risk Study exhibits document the relationship between the three measures
of fundamental risk and realized rates of return. Exhibit 15.1 displays the relationship between operating margin and rates of return.
Two of the risk measures are defined in terms of the coefficient of variation. The
coefficient of variation is the standard deviation divided by the mean and measures
volatility relative to the average value of the variable under consideration. Use of the
coefficient of variation normalizes for differences in the magnitude of the subject
variables.
The Risk Study shows that the lower the operating margin, on average, the
greater the return. The Risk Study shows that the greater the coefficient of variation
of operating margins, on average, the greater the return. The Risk Study shows that
the greater the coefficient of variation of rates of return on equity capital, on average, the greater the return.
The Duff & Phelps study also documents the relationship of size and risk. For
example, Exhibits 13.11 and 13.12 in Chapter 13 of this book display the relationship between two measures of size and the three measures of company risk. The
exhibits present average risk measures for each of the size-ranked portfolios of companies that were used in the Size Study (e.g., Exhibits 13.7 and 13.8 in Chapter 13).
While size may be considered a proxy for risk, the Risk Study investigates risk as
represented by information in company financial statements. The results reported
herein suggest a positive relationship; that is, the greater the risk as measured by
historical accounting information, the greater the rate of return earned by equity
investors. In addition, the Risk Study does document that size is correlated with
these fundamental risk measures.
Exhibit 13.11 displays 25 portfolios with size measured by market value of
equity as displayed in Exhibit 13.7. Exhibit 13.11 shows, for each portfolio, the average historical realized premium since 1963. Also shown are five measures of risk
corresponding to each portfolio:
1. Beta (calculated using the sum beta method applied to monthly returns for 1963
through the latest year)
3
A survey of the academic research can be found in Gerald White, Ashwinpaul Sondi, and
Haim Fried, The Analysis and Use of Financial Statements, 3rd ed. (Hoboken, NJ: John
Wiley & Sons, 2003), Chapter 18.
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EXHIBIT 15.1 Duff & Phelps Risk Study
Source: Compiled from data from Center for Research in Security Prices. # 200902 CRSP1 Graduate School of Business, The University of Chicago
used with permission. All rights reserved. www.crsp.chicagobooth.edu. Calculations by Duff & Phelps LLC. # Duff & Phelps, LLC.
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2.
3.
4.
5.
ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
Unlevered sum beta
Average operating margin (since 1963)
Average coefficient of variation of operating margin (since 1963)
Average coefficient of variation of return on book equity (since 1963)
We see that the beta (both levered and unlevered) of the portfolios
decreases (as expected) as market value of equity increases and that the average
operating margin increases as market value of equity increases. We also see that
the average coefficient of variation of operating margin and of variation of
return on book equity decreases as market value of equity increases. We see
that generally the three fundamental measures of risk display increasing risk as
size decreases, as the historical unlevered equity risk premium increases, and as
the unlevered beta increases.
When company size is measured by sales or by number of employees, the
Risk Study indicates that there is little differentiation in operating margin across
the companies of various sizes. But the coefficient of variation of both operating
margin and return on book equity indicate increasing risk as size decreases, as
with other size measures.
Why not just use measures of size as the measure of risk? First, certain measures of size (such as market value of equity) may be imperfect measures of the
risk of a company’s operations. For example, a company with a large and stable
operating margin may have a small and unstable market value of equity if it is
highly leveraged. In this case, the risk of the underlying operations is low while
the risk to equity is high.
Second, while small size may indicate greater risk, some small companies have
been able to maintain near economic monopolies by holding a geographic or market niche such that their risk is less than indicated by their size. Alternatively,
while larger size (e.g., as measured by sales) may indicate less risk, some companies may be riskier than the average of companies with similar sales. For example,
assume the subject company was expecting to emerge from reorganization following bankruptcy. The risk premium appropriate for this company may be more
accurately imputed from the pro forma operating profit (after removing nonrecurring expenses incurred during the bankruptcy) than from its size as measured
by sales (i.e., the subject company may be riskier than companies with similar sales
volume).
Use of fundamental accounting measures of risk allows one to assess the risk of
the subject company directly. For example, if one observes that the appropriate risk
premium for the subject company when measuring risk by one or more fundamental
risk measures is greater than the risk premium based on size measures, this may be
an appropriate measure of a company’s specific risk.
Using the Duff & Phelps Risk Study in the Build-up Method
As an alternative to Formula 7.1 for the build-up method, EðRi Þ ¼ Rf þ RPm þ
RPs RPu , one can use the Risk Study to develop a risk premium for the subject company that measures risk in terms of the total effect of market risk, size
premium, and risk attributable to the specific company. The formula then is
modified to be:
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Company-specific Risk
(Formula 15.1)
EðRi Þ ¼ Rf þ RPmþsþu
where: RPm+s+u ¼ Risk premium for the ‘‘market’’ plus risk premium for size plus
risk attributable to the specific company
The risk premiums in the Risk Study are used in the build-up method and include the market risk premium and the combined subject company-specific risk premiums based on the size and profitability of the subject company. Other companyspecific risk premiums may be applicable.
Three Measures of Fundamental Risk, 25 Risk Categories The Risk Study exhibits
(e.g., Exhibit 15.1) report average statistics for the period since 1963. For example,
in Exhibit 15.1, the statistics on returns are for the period 1963 through 2008. To
estimate realized premiums, the Risk Study uses the same methodology to develop
the database as in the Size Study (see Chapter 13). The Risk Study exhibits present
summary data for companies ranked by various measures of risk. The measures are:
&
&
&
Operating margin (operating income divided by sales; operating income is defined as sales minus [cost of goods sold plus selling, general, and administrative
expenses plus depreciation expense]) calculated as the mean operating income
for the five prior years divided by the mean sales for the five prior years. For
example, see Exhibit 15.1.
Coefficient of variation of operating margin calculated as the standard deviation
of operating margin over the prior five years divided by the mean operating margin for the same years, where operating margin is operating income as defined
previously divided by sales.
Coefficient of variation of return on book value of equity calculated as the standard deviation of return on book equity for the prior five years divided by the
mean return on book equity for the same years (where return on book equity is
net income before extraordinary items minus preferred dividends divided by
book value of common equity).
The Risk Study exhibits include these statistics:
&
&
&
&
The median of the risk measure for the latest year (e.g., the median average operating margin for the latest five years before 2008). The reported average risk
statistics in, for example, Exhibit 15.1 are calculated for portfolios grouped
according to risk, independent of the size of the companies, and are not averages
since 1963. They are not the same numbers as reported in, for example,
Exhibit 13.11. Exhibit 13.11 reports statistics calculated for portfolios of companies grouped according to size and are averages since 1963.
Log (base-10) of the median of the risk measure (use of logs indicates that
changes in risk measures for portfolio to portfolio is a percentage difference).
The number of companies in each portfolio in the latest year.
Beta relative to the S&P 500 calculated using the sum beta method applied to
monthly returns for 1963 through the latest year.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
&
Standard deviation of historical annual equity returns
Geometric average historical equity return since 1963
Arithmetic average historical equity return since 1963
Arithmetic average realized risk premium (historical equity return over longterm U.S. government bonds) since 1963 (labeled ‘‘arithmetic risk premium’’)
‘‘Smoothed’’ average realized premium (i.e., the fitted premium from a regression with the average historical realized premium as the dependent variable and
the logarithm of the average risk measure as the independent variable) (labeled
‘‘smoothed average risk premium’’)
Average carrying value of preferred stock plus long-term debt (including current
portion) plus notes payable (‘‘debt’’) as a percent of market value of invested
capital (MVIC) since 1963 (labeled ‘‘average debt/MVIC’’)
&
&
&
&
&
Each exhibit shows one line of data for each of the 25 risk-ranked portfolios.
The high-financial-risk statistics are drawn only from companies for which the ranking criterion (e.g., five-year-average operating margin) is available. We discuss
the results for the high-financial-risk portfolio of companies in Chapter 16 and
Exhibit 16.7.
For comparative purposes, the exhibits include average returns from SBBI series
for large companies, small companies, and long-term government bond income
returns for the period 1963 through the latest year.
Exhibit 15.2 displays the observed relationships for the three risk measures
and the risk premiums. By each measure of risk covered in the Risk Study, the
result is a clear relationship between risk and historical equity returns. The portfolios of companies with higher risk have yielded higher rates of return. In the first
graph, one sees that as the median operating profit margin increases (less risk), the
returns decrease. In the second graph, one sees that as the variability in the operating profit margin increases (more risk), the returns increase. In the third graph,
one sees that as the variability in the return on equity increases (more risk), the
returns increase.
Examples The data in the Risk Study can be used as an aid in formulating estimated
cost of equity capital using objective measures of the risk, including elements of
company-specific risk of a subject company.
In the build-up method, we want to determine a premium over the risk-free rate.
The simplest approach is to use exhibits for each of the three risk characteristics and
locate the portfolio whose risk is most similar to the subject company. For each
guideline portfolio, the column labeled ‘‘smoothed average risk premium’’ gives an
indicated historical realized premium over the risk-free rate, RPm+s+u.
One can match, say, the operating margin of the subject company with the portfolio of stocks with a similar average operating margin. The smoothed premium for
this portfolio can then be added to the yield on long-term U.S. government bonds as
of the valuation date, resulting in a benchmark required rate of return. The
smoothed average premium is a more appropriate indicator than the actual historical observation for most of the portfolio groups. Exhibits 15.3 and 15.4 illustrate
the application of this method for a hypothetical company.
Exhibit 15.3 shows, for a hypothetical company, the calculation of the
mean (average) and standard deviation over the last five fiscal years of
08/26/2010
EXHIBIT 15.2 Duff & Phelps Risk Study: Risk Premiums for Use in Build-up Method
Source: Derived from data from the Center for Research in Security Prices. # CRSP1, Center for Research in Security Prices.
University of Chicago Booth School of Business used with permission. All rights reserved. www.crsp.chicagobooth.edu. Calculations by Duff & Phelps, LLC.
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296
2.3%
14.6%
15.8%
$900
$150
16.7%
4.6%
13.3%
34.7%
$820
$110
13.4%
$710
$80
11.3%
2007
$800
$120
15.0%
2007
$630
$90
14.3%
2006
$850
$130
15.3%
2006
Source: Duff & Phelps Risk Premium Report 2009. Copyright 2009. Used with permission. All rights reserved.
Book Value
Net Income before Extraordinary Items
Return on Book Equity (ROE)
Standard Deviation of ROE
Average ROE
Coefficient of Variation
2008
Example 2: Coefficient of Variation of Return on Book Value of Equity:
(Standard Deviation of ROE)/(Average of ROE)
Net Sales
Operating Income
Operating Margin
Standard Deviation of Op. Margin
Average Operating Margin
Coefficient of Variation
2008
$540
$40
7.4%
2005
$750
$80
10.7%
2005
Example 1: Coefficient of Variation of Operating Margin: (Standard Deviation of Operating Margin)/(Average Operating Margin)
$500
$100
20.0%
2004
$900
$140
15.6%
2004
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EXHIBIT 15.3 Example of Calculating Risk Measures
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From Exhibit D-2 provided in the Risk Premium Report.
From Exhibit D-3 provided in the Risk Premium Report.
14.6%
15.8%
34.7%
Company
Indicator
15.1
15.3(1)
15.3(2)
Exhibit
8
14
13
Guideline
Portfolio
Source: Duff & Phelps Risk Premium Report 2009. Copyright 2009. Used with permission. All rights reserved.
(2)
(1)
Operating Margin
CV(Operating Margin)
CV(ROE)
Mean Premium over Risk-free Rate, RPm+s+u
Median Premium over Risk-free Rate, RPm+s+u
Risk Premiums over Risk-free Rate: Using Guideline Portfolios
7.2%
8.1%
8.1%
7.8%
8.1%
RPm+s+u
08/26/2010
EXHIBIT 15.4 Example of Estimating Risk Premiums1
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
operating margin and return on book value of equity (ROE). The ratio of the
standard deviation to the mean is the coefficient of variation. These risk metrics
can be used in conjunction with the exhibits in the Risk Study to estimate a
premium over the risk-free rate.
Exhibit 15.4 illustrates the procedure of estimating risk premiums.
In deriving the average realized risk premiums reported in their exhibits, the
Duff & Phelps studies use the SBBI income return on long-term U.S. government
bonds as their measure of the historical risk-free rate; therefore, a 20-year U.S. government bond yield is the most appropriate measure of the risk-free rate for use with
the reported premiums in developing an indicated cost of equity capital.
If one’s estimate of the ERP for the S&P 500 on a forward-looking basis
were materially different from the average historical realized premium since
1963, it may be reasonable to assume that the other historical portfolio returns
reported here would differ on a forward-looking basis by approximately a similar
differential. For example, at the end of 2008, the average realized premium since
1963 for large company stocks equaled 3.84% (see the bottom of Exhibit 15.1).
This is the historic market risk premium, RPm, inherent in the Risk Study exhibits for use in the build-up method as of that date. The risk premiums displayed in
the Size Study exhibits for the build-up method equal RPm+s+u, as shown in Formula 15.1 (RPm plus RPs plus RPu).
Assume that one’s estimate of the ERP at the end of 2008 is equal to 6%
rather than the realized risk premium for the market since 1963 of 3.84%. That
is, one’s forward-looking ERP is greater than the historical risk premium since
1963. That difference (2.2% ¼ 6% minus 3.84%) can be added to the average
risk premium, RPm+s+u, for the portfolio (observed or smoothed) that matches
the risk of the subject company to arrive at an adjusted forward-looking risk
premium for the subject company (matching the forward-looking ERP estimate).
Then this forward-looking risk premium can be added to the risk-free rate as of
the valuation date to estimate an appropriate cost of equity capital for the subject company.
Let us use the data from Exhibit 15.4 to estimate the cost of equity capital.
Assume a risk-free rate as of the valuation date of 4.5%. The Risk Study would
indicate the cost of equity capital ranging from 13.9% (4.5% risk-free rate plus
7.2% risk premium from Exhibit 15.4 plus 2.2% adjustment for the difference
between the estimated ERP of 6% and the realized risk premium of 3.84% for
the period 1963 through 2008) to 14.8% (4.5% risk-free rate plus 8.1% risk
premium from Exhibit 15.4 plus 2.2% adjustment for ERP estimate).
This result is before consideration of and further estimate of any additional RPu,
the risk premium attributable to other specific company factors.
Other Considerations The historical average debt/MVIC ratio does not appear
to be strongly correlated with either the level or the volatility of the operating
margin. This suggests that leverage does not explain the greater returns of the
riskier portfolios.
The companies that are riskier according to accounting information (operating margins and coefficients of variation) have also exhibited greater risk
according to stock market–based risk statistics (betas and standard deviations
of annual returns).
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299
The Risk Study data should not be used in isolation from other considerations about the subject company, its industry, and the general economic environment. For instance, a wholesale distributor might have thin operating margins
compared with the average company on the New York Stock Exchange, yet those
margins might exhibit unusually low variation due to a particularly strong position in a stable market niche. Alternatively, a company’s variation of operating
income (calculated in the manner used in the study) might be uncharacteristically
high due to an unusual event several years in the past. Appropriate knowledge of
the company and its industry would give useful guidance in reconciling the historical realized premiums reported in the Risk Study and the historical realized
premiums reported in the Size Study for portfolios of companies ranked by size.
As already stated, size can be an important consideration in determining an appropriate cost of equity capital.
The use of a portfolio’s average realized rate of return to calculate a discount
rate is based (in part) on the implicit assumption that the risks of the subject company are quantitatively similar to the risks of the average company in the subject
portfolio. If the risks of the subject company differ materially from the average company in the subject portfolio, then an appropriate discount rate may be lower (or
higher) than a return derived from the average premium for a given portfolio. The
data reported in the exhibits where risk statistics are reported for each size category
(e.g., Exhibits 13.11 and 13.12) may be helpful in making such a determination.
MARKET PRICING OF COMPANY-SPECIFIC RISK
Researchers often study the factors priced by the market in order to better understand the market’s pricing of the relationship of risk and return. For example, Ri
may be a function of various factors with Bi,j being the sensitivity of observed returns to a particular factor. Generalizing the possible relationships we can repeat
Formula 12.2:
(Formula 15.2)
Ri ¼ Rf þ Bi;m RPm þ Bi;S Si þ Bi;BV BV i þ Bi;u Ui þ . . . þ ei
where:
Ri ¼ Realized return for stock i
Rf ¼ Risk-free rate of return
Bi,m ¼ Sensitivity of return of stock i to the market risk premium or ERP
RPm ¼ ERP
Bi,s ¼ Sensitivity of return of stock i to a measure of size, S, of company i
Si ¼ Measure of size of company i
RPi,s ¼ Bi,s Si ¼ Risk premium for size of company i
Bi,BV ¼ Sensitivity of return of stock i to a measure of book value (typically,
measure of book value to market value) of stock of company i
BVi ¼ Measure of book value (or book value to market value) of stock of
company i
RPi,BV ¼ Bi,BV BVi ¼ Risk premium for book value of company i
Bi,u ¼ Sensitivity of return of stock i to a measure of unique or unsystematic risk of company i
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
Ui ¼ Measure of unique or unsystematic risk of company i
RPi,u ¼ Bi,u Ui ¼ Risk premium for unique or unsystematic risk of company i
. . . ¼ Other factors
ei ¼ Error term, difference between predicted return and realized return.
Unsystematic risk (often called idiosyncratic risk in the academic literature) is
generally defined by researchers as the realized returns that remain unexplained by
the specified characteristics. In Formula 15.2, the unsystematic risk would be
defined as the error term, es . The question for researchers is ‘‘do expected returns
vary as the error terms change’’ in prior periods? Not being able to quantify
expected returns, researchers ask if realized returns in period n are greater for companies with larger error terms in period n–1. If so, what characteristics cause the
error terms to be greater or lesser for a given company?
Research on Unique or Unsystematic Risk
Unsystematic volatility of returns on stocks has increased over the past 40 years compared to volatility explained by market risk. This increase has been linked to an increase in the fundamental volatility of firms’ earnings, cash flows, and sales.4 But is
unsystematic risk priced by the market? That is, do firms with greater unsystematic risk
earn higher returns (possibly the theory of higher-beta stocks earning higher returns)?
Studies suggest that at least for small companies (size measured by market capitalization), returns are a function of more than simply beta. The studies on unsystematic risk
generally show that the market appears to price unsystematic risk for small firms, or at
least unsystematic risk and firm size as measured by market value of equity are interrelated.5 What could be the tie between unsystematic volatility and small firms?
Two researchers examined unsystematic risk in portfolios of firms grouped by
market value of equity and length of public listing (used as a proxy for age) using
data from August 1963 to December 2001. They found that unsystematic 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) were positive predictors of stock returns (and are unlike volatilities of
bigger, older, and newer firms). They found that size is a significant predictor of
returns primarily because it is a proxy for entrepreneurial risk.6
4
5
6
Paul J. Irvine and Jeffrey Pontiff, ‘‘Idiosyncratic Return Volatility, Cash Flows, and Product Market Competition,’’ Working paper, March 2005, Available at http://ssrn.com/
abstract=1359528.
For example, see 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, May 2004, Available at http://ssrn.
com/abstract=255303.
Shivaram Rajgopal and Mohan Venkatachalam, ‘‘Financial Reporting Quality and Idiosycratic Return Volatility over the Last Four Decades,’’ Working paper, September 29, 2008,
Available
at
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=650081&rec=1&
srcabs=896102.
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301
In a further study using data from the London Stock Exchange from 1979
through 2003, researchers found that valuing unsystematic volatility of small companies (measured by market capitalization) is a predictor of returns for stocks of
small companies (and insignificant as a predictor of returns for stocks of large companies).7 In still another study, the authors found that time-varying unsystematic or
company-specific risk can explain the difference in returns among stocks in the
United States and in the United Kingdom over time.8
However, not all studies have been supportive of the theory that the market
prices unsystematic volatility. For example, in another study the authors found no
statistical significance of unsystematic volatility in predicting stock prices across
both small and large company stocks (measured by market capitalization).9
The same study found no statistical relationship between unsystematic risk and
realized returns. That study measured unsystematic risk in terms of residuals from
the FF three-factor model, not the pure CAPM.10 The FF three-factor model controls
for size and other differences among the firms.
But another study found a strong link between implied unsystematic volatility derived from options (for companies with traded stock options) and future
stock returns for those same companies. Those authors point out that the problem with most studies is that they measure unsystematic volatility by examining
historical realized volatilities. These researchers found that historical realized
volatilities do not explain future returns of individual stocks when the pricing
model includes implied unsystematic volatility. They found that the market
prices the following factors: company size, relative book-value-to-market-value
of equity, and implied forward unsystematic risk of individual companies. They
found that companies with greater implied forward unsystematic risk realized
greater stock returns and companies with lower implied forward unsystematic
risk realized smaller stock returns.11
The studies we cite in Chapter 12 suggest that investors in general are much less
diversified than predicted by the pure CAPM, and that the difference from the theoretical world assumed by the pure CAPM could cause more than market (systematic) risk to be priced by the market. While researchers investigating whether
unsystematic risk is priced by the market generally state that such a phenomenon
would be consistent with lack of perfect diversification, the majority of studies that
7
8
9
10
11
Timotheos Angelidis and Nikolaos Tessaromatis, ‘‘Equity Returns and Idiosyncratic Volatility: UK Evidence,’’ Working paper, June 2, 2005, Available at http://ssrn.com/abstract=
733906. Note: Idiosyncratic risk is measured as residual from FF three-factor model.
Xiafei Li, Chris Brooks, and Joelle Miffre, ‘‘The Value Premium and Time-Varying Volatility,’’ Working paper, March 13, 2009, Available at http://ssrn.com/abstract=983905.
Turan G. Bali and Nusret Cakici, ‘‘Idiosyncratic Volatility and the Cross-Section of
Expected Returns,’’ Journal of Financial and Quantitative Analysis (forthcoming). Note:
Idiosyncratic risk is measured as residual from FF three-factor model.
Turan G. Bali and Nusret Cakici, ‘‘Idiosyncratic Volatility and the Cross-Section of
Expected Returns,’’ Journal of Financial and Quantitative Analysis (forthcoming).
Dean Diavatopoulos, James S. Doran, and David R. Peterson, ‘‘The Information Content
in Implied Idiosyncratic Volatility and the Cross-Section of Stock Returns: Evidence from
the Option Markets,’’ Journal of Futures Markets 28 (November 2008): 1013–1039.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
actually investigate the causes for unsystematic risk to be priced by the market look
at the informational content of unsystematic risk, not the level of diversification, to
explain why the market prices company-specific risk.
In one study, the authors investigated the inefficiency of the stock market in
pricing companies that are small and less visible (i.e., followed by few or no
analysts) such that they can be considered neglected by investors.12 They found that
stock market prices of these firms adjust to news only slowly. They also found that
their stock prices are volatile. For the most neglected companies, the authors found
that company-specific risk is priced by the market. This finding is separate from,
though partly related to, the size and lack of liquidity effects.13
In another study, the authors found that there is a relationship between information quality, beta estimation, and the cost of capital. As information quality
improves, the cost of capital decreases.14 Other studies examine the relationship
between firm-specific information, unsystematic risk, and the cost of capital.15
These studies found that unsystematic risk and the cost of capital vary with the quality of information.
Is there meaningful information about risk of an investment in the error term of
the regression used to estimate beta? That is, if one looks at the error in estimating
beta over a look-back period, is there information in the magnitude of the errors?
One study found that firms with large beta estimation errors are characterized
by low-quality earnings, low persistence of earnings, low predictability of earnings,
and high volatility of returns. Firms with large beta estimation errors are fundamentally weak. The results of the study support the view that the reliability of the beta
estimate is an indicator of the uncertainty found by investors. This uncertainty is
caused by investors receiving low-quality information and/or fundamental weakness
in cash flows, making it more difficult for investors to evaluate firm information.
This leads to high firm-specific uncertainty associated with firm fundamentals.16
Further, the amount of firm-specific uncertainty about fundamentals is a crucial
determinant of the level of the reliability of the beta estimate.17
12
13
14
15
16
17
Kewei Hou and Tobias Moskowitz, ‘‘Market Frictions, Price Delay, and the Cross-Section
of Expected Returns,’’ Review of Financial Studies 18(3) (2005): 981–1020.
The lack of visibility to investors has been identified as an important factor leading public
companies to becoming closely held. See Hamid Mehran and Stavros Peristiani, ‘‘Financial
Visibility and the Decision to Go Private,’’ Review of Financial Studies 23(2) (2000): 519–
547.
Chris Armstrong, Sneehal Banerjee, and Carlos Corona, ‘‘Information Quality and the
Cross-Section of Expected Returns,’’ Working paper, November 2009, Available at http://
ssrn.com/abstract=1300100.
Philip G. Berger, Huafeng Chen, and Feng Li, ‘‘Firm Specific Information and Cost of
Equity Capital,’’ Working paper, January 6, 2006, Available at http://ssrn.com/
abstract=906152.
Siew Hong Teoh, Yong Yang, and Yinglei Zhang, ‘‘R-Square and Market Efficiency,’’
Working paper, July 30, 2009, Available at http://ssrn.com/abstract=926948.
Siew Hong Teoh, Yong Yang, and Yinglei Zhang, ‘‘R-Square and Market Efficiency,’’
Working paper, July 30, 2009, 6, Available at http://ssrn.com/abstract=926948.
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In another study, the authors found that errors in earnings forecasts play an important role in the pricing of unsystematic risk and how that relationship changes
through the business cycle. Firms tend to underestimate the growth rates in earnings
during the expansion phase of the business cycle and tend to overestimate the
growth rate in earnings during recessions. The tendency is for the underestimation
to be more frequent and small while the overestimates are infrequent and large. This
is borne out with the finding that firms with high volatility or high unsystematic
risk realized greater returns following good news and realized low returns following
bad news.18
However, we must also consider the results of what may be the most complete study to date. The authors, Paul Brockman, Maria Gabriela Schutte, and
Wayne Yu, studied unsystematic risk premiums using observations from individual stocks in 44 stock markets from 1980 to 2007.19 This study differs from
most of the prior studies because they looked at unsystematic risk of individual
company stocks, not portfolios of stocks. They found that stocks with greater
unsystematic risk realize greater returns.
The authors eliminated stocks with infrequent trades and the smallest companies so the measurement of unsystematic risk is not biased high by zero return observations. They found that unsystematic risk is time-varying. They use advanced
models to account for the conditional unsystematic volatility given prior period unsystematic volatilities.20 They measured the unsystematic risk difference across individual firm stocks, taking into account market risk, company size, book-value-tomarket-value of equity ratio, and certain trading and liquidity variables. They found
that as expected unsystematic risk increases, expected returns increase.
The authors then investigated whether the unsystematic risk premium is greater
in certain countries. After controlling for volatility of firm cash flows, they found
that the unsystematic risk effect increases in countries and at times when trading
costs and information costs are high.
These quantitative studies complement a qualitative assessment of the
strengths, weaknesses, opportunities, and threats of the subject company compared to its peers by matching the subject company to the guideline public companies with comparable (not identical) strengths, weaknesses, opportunities, and
threats relative to their peers.
For example, in one study the authors found that companies that are unionized
generally experience higher costs of capital. They found that powerful unions can
18
19
20
See for example, X. Frank Zhang, ‘‘Information Uncertainty and Stock Returns,’’ Journal
of Finance 61(2) (February 2006): 105–137; and Tony Berrada and Julien Hugonnier,
‘‘Incomplete Information, Idiosyncratic Volatility and Stock Returns,’’ Working paper,
January 9, 2009. Available at http://ssrn.com/abstract=1326840.
Paul Brockman, Maria Gabriela Schutte, and Wayne Yu, ‘‘Is Idiosyncratic Risk Priced?
The International Evidence,’’ Working paper, July 2009. Available at http://ssrn.com/
abstract=1364530. Note: Idiosyncratic risk is measured as residual from an extension of a
FF three-factor model.
The authors use an exponential general autoregressive conditional heteroskedasticity
model that considers the current period error term to be a function of previous time period
error terms. Autoregressive conditional heteroskedasticity models allow the joint modeling
of variances and expected returns.
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negatively influence a firm’s flexibility by making wages sticky and layoffs costly and
can make firm restructuring (e.g., plant closings) more costly.21
We conclude that the study of the market’s pricing of unsystematic risk is still in
its development stages22 and the reasons why the market prices unsystematic risk are
not certain.23 But we also conclude that there is sufficient evidence that companyspecific risk factors are priced by the market.
Total Beta and the Butler-Pinkerton Interpretation
Over the past few years, the topic of total beta has received considerable attention.
For example, it was a major topic of discussion at the 2009 Advanced Business Valuation Conference of the American Society of Appraisers. We believe that there is
much confusion about the topic. Accordingly, we have tried to organize the discussion and highlight the issues.
Total Beta and Total Risk Some authors have postulated that it is appropriate to adjust pure CAPM when considering the rate of return appropriate for an undiversified
investor.
The studies we summarized in Chapter 12 indicated that investors in general are
less diversified than predicted by the pure CAPM, suggesting that this deviation from
the assumptions of the pure CAPM could cause the market to price unsystematic
(company-specific) risk. Studies discussed previously suggest that at least for small
companies (size measured by market capitalization), rates of return are a function of
more than beta and include the pricing of company-specific risk factors.
One method for quantifying the risk taken on by an undiversified investor that
has been promulgated by some authors is called total beta. Total beta is an alternative risk measure equal to the standard deviation of total returns expected for a stock
divided by the standard deviation of total returns expected for the market portfolio.
Practitioners promulgating total beta generally use the standard deviation of realized returns over a look-back period as an estimate of expected future returns for
the subject stock and the market portfolio.24
21
22
23
24
Huafeng (Jason) Chen, Marcin Kacperczyk, and Hernan Ortiz-Molina, ‘‘Labor Unions,
Operating Flexibility, and the Cost of Equity,’’ Journal of Financial and Quantitative
Analysis (forthcoming).
One study questions whether all of the results finding that idiosyncratic risk is priced are
simply the result of the occurrence of zero returns, which causes unsystematic risks to be
larger. If so, the observed market pricing is really a function of liquidity. See Yufeng Han
and David Lesmond, ‘‘Idiosyncratic Volatility and Liquidity Costs,’’ Working paper,
March 18, 2009. Available at http://ssrn.com/abstract=1363888.
Stocks with high unsystematic risk appear to earn a greater return in the same month, and
then that greater return reverses in the following month. See Fangjian Fu, ‘‘Idiosyncratic
Risk and the Cross-Section of Expected Returns,’’ Journal of Financial Economics (January 2009): 24–37.
For example, see Aswath Damodaran, Damodaran on Valuation: Security Analysis for Investment and Corporate Finance, 2nd ed. (New York: John Wiley & Sons, 2006): 58–59;
Peter Butler and Keith Pinkerton, ‘‘Company-Specific Risk—A Different Paradigm: A New
Benchmark,’’ Business Valuation Review (Spring 2006): 22–28.
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Company-specific Risk
One author describes the analysis as measuring the opportunity cost to a
‘‘venturer’’ (the investor who holds an illiquid asset and cannot freely allocate
wealth in a classic CAPM fashion between the market portfolio and the risk-free
asset; such a venturer could be an entrepreneur in a start-up firm, an employee
holding stock or stock options in the employer firm, or an investor in an asset
with high transaction costs); that author contrasts this analysis with that of a
classic ‘‘investor’’ (who holds a diversified portfolio of assets). 25 The venturer
must then assess whether the opportunity cost for holding such an investment
adequately compensates the venturer vis-a-vis that same venturer making an
investment in a classic diversified investment.
For example, assume that the standard deviation of the excess returns for a pure
play guideline public company ¼ 39.621% (annualized standard deviation of
returns over a look-back period equal to 60 months) and the standard deviation of
the excess returns on the market portfolio over that same look-back period equals
12.860% (annualized standard deviation of returns).26 We can calculate total beta:
(Formula 15.3)
TBi ¼ ðs i =s m Þ
where: TBi ¼ Total beta for security i
s i ¼ Standard deviation of returns for security i
s m ¼ Standard deviation of returns for the market
Applying the sample data we get:
TB ¼ 0:39621=0:12860 ¼ 3:081
One can also estimate total beta in a less direct fashion by taking a beta estimate
calculated over a look-back period divided by R, the correlation of the regression
used to estimate beta (i.e., R, not R2).
Some proponents of total beta believe it can be used to estimate the opportunity
cost for an undiversified investor as follows:
(Formula 15.4)
E Ri;j ¼ Rf þ TBi RPm
where: E(Ri,j) ¼ Expected rate of return on security i for undiversified investor j
Rf ¼ Rate of return available on a risk-free security as of the valuation
date
TBi ¼ ðs i =s m Þ
RPm ¼ General equity risk premium for the market
25
26
Gerald Garvey, ‘‘What Is an Acceptable Rate of Return for an Undiversified Investor?’’
Working paper, September 2001. Available at http://ssrn.com/abstract=281432.
Data from Exhibit 10.2 for TIBCO Software (Symbol: TIBX).
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Two proponents of total beta for use in estimating the cost of equity capital for
closely held businesses, Butler and Pinkerton (Butler-Pinkerton), contend that total
beta is a ‘‘pure measure of the relative volatility’’ between an individual asset and
the market and that, unlike the market risk measure, beta, does not require consideration of the correlated relative volatility (i.e., sensitivity) of the subject stock to the
market as a whole.27
But consider that the variability of any stock is in part a function of its market
risk. The measures total beta and beta are related. If we examine the relationship of
return and risk as implied from the empirical index model (Formula 12.1 discussed
in Chapter 12), we get the following relationships of realized returns for stock i to
the realized risk of those returns:
(Formula 15.5)
Total Return ¼ Risk-free rate þ Return due to unique risk
þ Return due to Market risk þ Random error
Return : Ri ¼ Rf
þ
þ Bi Rm Rf
Risk : s 2i ¼ s 2rf
þ
B2i
þ
2
R m Rf
ai
þ
ei
þ
s 2e;i
s 2a;i
The risk of stock i in the index model (the empirical form of CAPM) is its total
variance of returns.
Substituting the definition for the risk of stock i from Formula 15.5 and the
definition of beta (Formula 8.2), we get the following expanded definition of
total beta:
(Formula 15.6)
h
2 i:5
TBi ¼ s 2rf þ s 2a;i þ s 2e;i þ ðcovðRi ; Rm Þ=varðRm ÞÞ2 Rm Rf
=s m
Formula 15.6 implies that unless the covariance between the returns on stock i
and the market equals zero, total beta is a function of beta.28
The issues confronting an analyst considering adopting the total beta approach
for estimating total cost of equity capital are:
&
27
28
This interpretation of total beta as the risk measure in estimating total returns is
based on the premise that most owners of private businesses are completely
undiversified and, therefore, the cost of equity capital of the private business
should include that extra amount due to the owner being undiversified. This
leads to the unreasonable position that there are at least two costs of capital for
Peter Butler and Keith Pinkerton, ‘‘There Is a New ‘Beta’ in Town and It’s Not Called
Total Beta for Nothing!’’ Business Valuation Update 15(3) (March 2009): 7, 10.
Larry Kasper, ‘‘Anomalous Findings from the Butler Pinkerton Model for Company Specific Risk Premiums,’’ Presented at the 2009 Advanced Business Valuation Conference of
the American Society of Appraisers: 18.
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&
307
a business—the cost of capital for investors who are the pool of likely buyers
who are likely to be diversified (for whom in theory only market or beta risk
matters) and the cost of equity capital to the current owner who is completely
undiversified (for whom both market risk and unsystematic risk matter).29
How can a company estimate its cost of capital if it needs to guess if the pool of
likely buyers is diversified?
Using total beta to estimate the cost of equity capital determines investment
value (the value to a particular investor), not fair market value or fair value for
financial reporting. The total cost of equity derived from total beta may not be
consistent with the definitions of fair market value or fair value.30
Businesses and interests in businesses (any asset) sell in various markets made up of
pools of likely buyers. The marginal investors in the pool of likely buyers set the
market price. No market, other than possibly the pool of buyers for the smallest
businesses, is comprised of fully undiversified investors. As the more diversified
buyer is likely to pay a higher price, the value of the business and business interests
in most cases must be greater than their value determined using total beta.
Of course, if one were estimating the risk for a very small company where the pool
of willing buyers were probably less diversified investors, it is doubtful that any public
company would be a reasonable guideline company for such small companies.
Risk of an investment and its fair market value must be developed based on the
risks (and pricing) perceived by investors who comprise the pool of likely buyers for
the subject asset—not based on the diversification or nondiversification of the current owner.31 As we noted in Chapter 1, ‘‘The cost of capital is a function of the
29
30
31
Larry Kasper, ‘‘Fallacies of the Butler-Pinkerton Model and the Diversification Argument,’’ Value Examiner (January–February 2010): 8–20.
For example, the standard of value for U.S. federal income tax purposes is fair market
value, which has been framed over the years by countless court cases. Fair market value is
defined as the price that a willing buyer would pay a willing seller, both having reasonable
knowledge of all of the relevant facts and neither being under compulsion to buy or to sell.
See United States v. Cartwright, 411 U.S. 546, 551, 36 L. Ed. 2d 528, 93 S. Ct. 1713
(1973); [17] 1.170A-1(c)(2), Income Tax Regs. Some important implications are as follows: (1) The willing buyer and the willing seller are hypothetical persons, rather than specific individuals or entities, and the peculiar characteristics of these hypothetical persons
are not necessarily the same as the individual characteristics of an actual seller or an actual
buyer. See Estate of Bright v. United States, 658 F.2d 999, 1005-1006 (5th Cir. 1981); (2)
the hypothetical willing buyer and willing seller are presumed to be dedicated to achieving
the maximum economic advantage. See Estate of Newhouse v. Commissioner, supra at
218; (3) the hypothetical sale should not be constructed in a vacuum isolated from actual
facts that affect value. See Estate of Andrews v. Commissioner, supra at 956; (4) the fair
market value of property should reflect the highest and best use to which the property
could be put on the date of valuation. See Stanley Works & Subs. v. Commissioner, 87 T.
C. 389, 400 (1986).
One may want to analyze the impact on the cost of equity capital (change in beta) as the
possible diversification of the pool of willing buyers varies. See Tony van Zijl, ‘‘Beta Loss.
Beta Quotient: Comment,’’ Journal of Portfolio Management 11(4) (Summer 1985):
75–78.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
investment, not the investor.’’32 The cost of capital should reflect the risk of the
investment, not the cost of funds to a particular investor.
But assume that one uses total beta to estimate the cost of equity capital for a
small company for which the pool of willing buyers are generally less diversified.
Does the total cost of equity capital estimate using total beta include an embedded
discount for lack of marketability (discussed in Chapter 27 for partial interests and
Chapter 28 for entire companies) because total beta purports to capture total risk?
This conclusion would be consistent with the conclusions of one paper.33 Perhaps
the value implied using total beta for the cost of equity capital should not be reduced
by the discount for lack of marketability.
Total Beta and Inferring Company-specific Risk From the total cost of equity, Formula 15.4, Butler and Pinkerton (Butler-Pinkerton) contend that one can infer the
company-specific risk premium for a closely held business using Formula 15.4 in
conjunction with Formula 8.5, the expanded CAPM formula, and get the following:
(Formula 15.7)
RPu ¼ ½TBi Bi ðRPm Þ RPS
where: RPS ¼ Risk premium for small company size
RPu ¼ Risk premium attributable to the specific company risk factors
(u stands for unique or unsystematic risk)
RPu is a residual percentage. According to Butler-Pinkerton, all of the variances
in the rates of return not attributed to either the beta or the size premium are attributable to company-specific risk.
For example, assuming that Rf ¼ 4.5%, RPm ¼ 6.0%, and TBi ¼ 3.081, we get:
E Ri;j ¼ 4:5% þ 3:081 6:0% ¼ 23% ðroundedÞ:
Further assuming that Bi ¼ 1.77 and RPs ¼ 1.62% (premium over CAPM from
7th decile of Exhibit 13.1, which we are using as the size premium, given the market
capitalization of equity of the subject company), we get:
RPu ¼ ð3:081 1:77Þ 6:0 1:62% ¼ 6:25%
As the cost of equity capital using beta and the size premium equals 16.7%,34
RPu adds 37% to the expected return derived from CAPM plus a size premium.
32
33
34
Roger Ibbotson, Ibbotson Associates Cost of Capital Workshop, 1999.
McConaughy and Covrig, ‘‘Owners’ Lack of Diversification and Cost of Equity Capital
for Closely Held Firm,’’ Business Valuation Review (Winter 2007).
Using formula E(Ri) ¼ Rf þ Bi (RPm) þ RPS, we get: 4.5% þ 1.77 6% þ 1.62 ¼ 16.7%
(rounded).
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Does such a measure capture the factors that cause company-specific risk? If one
thoroughly analyzes the risk factors of guideline public companies, the estimate of
company-specific risk premium should reflect the market’s pricing of these risks.35
However, as in any use of guideline public companies as proxies, the ability to determine a reasonable RPu is dependent on the availability of similar companies and on
the thoroughness of the analysis.
The issues confronting an analyst considering adopting the Butler-Pinkerton
approach for estimating company-specific risk are:
&
&
&
&
As RPu is a residual, it reflects the impact of many risk factors specific to each
public company. One must analyze each public company’s returns over an
extended period to determine the impact of these factors. There may be too
many factors or too few guideline companies as potential proxy companies to
do a meaningful analysis.
As RPu is an aggregate, the risk premium associated with specific risk factors
that the subject company has in common with the guideline company can only
be determined subjectively.
The calculated RPu is a function of the beta estimate. Beta estimates using lookback methods are subject to estimation error, as we explained in Chapter 10.
Therefore, company-specific risk estimates derived from beta estimates are also
subject to estimation error. Ascribing beta estimation error to company-specific
risk estimates confuses the company-specific risk estimate. Is the measure beta
estimation error, company-specific risk, or lack of diversification risk?
Using the total beta of an investment and deriving an estimate for companyspecific risk based on the relationship RPu ¼ ½TB B RPm RPs quantifies
two risks simultaneously: company-specific risk and lack of diversification risk
of the venturer.
The central problem of using the total beta of guideline public companies or the
Butler-Pinkerton interpretation to estimate company-specific risk and develop the
cost of equity capital for a closely held company is that both are based on a myriad
of unknown factors with unknown effects on the cost of equity capital.
As we discussed earlier, researchers do find that public stock returns reflect unsystematic risk as well as systematic risk. Empirical studies of company-specific risk,
RPu, based their analyses of error terms, ei, using formulas similar to Formula 15.2:
Ri ¼ Rf þ Bi;m RPm þ Bi;S Si þ Bi;BV BV i þ Bi;u Ui þ . . . þ ei
That is, RPu is independent of [b RPm]. Researchers define residuals of the
regression equation as unsystematic risk and then look for specific factors that
explain the magnitude of associated rates return based on the magnitude of the error
terms, ei. The explained unsystematic risk is independent of the systematic risk.
These results represent the market’s pricing of company-specific risk. These
35
Larry Kasper, ‘‘The Butler Pinkerton Model for Company-Specific Risk Premium—
Critique,’’ Business Valuation Review (Winter 2008): 233–243; ‘‘Total Beta: The Missing
Piece of the Cost of Capital Puzzle—A Reply,’’ Valuation Strategies (November–December
2009): 12–19, 48.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
observations can be consistent with the pure CAPM for investors who are not fully
diversified.
No researcher studying whether the market prices unsystematic risk or the magnitude of that pricing has used EðRi Þ ¼ Rf þ ½Tb RPm . Just because total beta
represents a relative relationship of risk does not mean it provides a functional relationship of the market’s pricing of expected return.36 To the extent that Formula
15.8 is used to estimate company-specific risk, it combines the company-specific risk
and the risk to a completely nondiversified investor, not necessarily the market’s
pricing of company-specific risk either for publicly traded companies or for closely
held companies.
In conclusion, as one practitioner summarized:
Professors Damodaran and Tofallis, Mr. Butler and Mr. Pinkerton are performing a service for business valuation. They are helping move the debate
from questioning the use of the CSRP to determining its size. Nevertheless,
the Butler-Pinkerton Model is not the answer. We need better quantitative
techniques in determining CSRP’s components and their magnitude.37
COST TO CURE
One way to account for company-specific risk is to estimate the cost to cure that
risk. For example, if a company-specific risk is reliant on a key salesperson for a
large amount of sales, then the cost of buying life insurance sufficient to reimburse
the company for the possible loss of that person due to death is one measure of the
cost to cure that risk. Another company-specific risk that may be accounted for by
estimating the cost to cure is potential environmental cleanup costs. One can estimate the probability-weighted costs of remediation, given the possibility that a
cleanup will be required and the possible timing of a required cleanup.
Applying an adjustment for company-specific risk using the cost to cure is completely consistent with capital market theories, as these are adjustments to the
expected cash flows.
OTHER COMPANY-SPECIFIC FACTORS
Other factors specific to a particular company that affect risk could include, for
example:
&
&
&
&
36
37
Concentration of customer base
Key person dependence
Key supplier dependence
Abnormal present or pending competition
See the discussion in Sarah von Helfenstein, ‘‘Revisiting Total Beta,’’ Business Valuation
Review (forthcoming).
M. Mark Lee, ‘‘Determining the Company Specific Risk Premium: Beta, Total Beta and
the Butler-Pinkerton Calculator,’’ Business Valuation Review (forthcoming).
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&
&
&
311
Pending regulatory changes
Pending lawsuits
A wide variety of other possible specific factors
Investors in closely held businesses also face a degree of information risk different from public companies. Public companies file audited financial statements and
disclosures with the Securities and Exchange Commission. Closely held businesses
may not have audited financial statements and may lack good internal controls
needed to properly report financial results.
Because the size premium tends to reflect some factors of this type, analysts
should adjust further only for specific items that are truly unique to the subject company. Analysts must be careful not to include any adjustment for risk factors that
may be included in other adjustments.
SUMMARY
Quantifying company-specific risk is one of the most controversial and elusive areas
of business valuation. As Chancellor Strine stated:
Much more heretical to CAPM, however, the build-up method typically
incorporates heavy dollops of what is called ‘‘company-specific risk,’’ the
very sort of unsystematic risk that the CAPM believes is not rewarded by
the capital markets and should not be considered in calculating a cost of
capital. The calculation of a company specific risk is highly subjective and
often is justified as a way of taking into account competitive and other factors that endanger the subject company’s ability to achieve its projected
cash flows. In other words, it is often a back-door method of reducing estimated cash flows rather than adjusting them directly.
To judges, the company specific risk premium often seems like the
device experts employ to bring their final results into line with their clients’
objectives, when other valuation inputs fail to do the trick. . . .
[Petitioners’ expert’s] own analysis also contains a subjective specific
risk premium of 2%, the quantification of which cannot be explained by
reference to objective factors. . . . 38
We have identified several sources of data that will assist the analyst in this most
difficult task. The Duff & Phelps Risk Study provides quantitative data for one to
analyze the company-specific risk of the subject company. Users of cost of capital
data should make themselves aware of updates of this and similar studies to incorporate the latest current quantitative data on measuring company-specific risk in cost
of capital estimates, whether using build-up models, CAPM, or other cost of equity
models.
Closely held businesses often suffer from poor information quality. For example, investors may find that the financial statements are not audited or even reviewed
38
Delaware Open MRI Radiology Associates, P.A. v. Howard B. Kessler et al. (Court of
Chancery of State of Delaware, Cons C.A. No. 275-N).
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
or, in fact, may not even exist. This poor information quality will cause the cost of
capital to be greater due to these company-specific risk factors.
Finally, we caution that the analyst should avoid a double counting because the
size premium may already reflect some of the company-specific risks. Thus the company-specific risk premium should be used judiciously.
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CHAPTER
16
Distressed Businesses
Introduction
What Is Distress?
Valuing Firms in Distress
Changing Capital Structure
Adjusted Present Value
Option Valuation
Bankruptcy Prediction Models
Accounting-Ratio-based Models
Market-based Models
Comparing the Models
Cost of Capital for Distressed Firms
WACC Considerations
Cost of Equity Capital Considerations
Valuing Companies Emerging from Bankruptcy
Duff & Phelps Risk Study—High-Financial-Risk Companies
Additional Information on Company Risk
Relevering Beta for a Highly Leveraged Company
Cost of Distress
Summary
Additional Readings
Technical Supplement Chapter 7: Cost of Capital and the Valuation of Worthless Stock
INTRODUCTION
The standard cost of equity capital models (e.g., CAPM) and standard application of a discounted cash flow analysis assume the business continues as a going
concern. However, a company may be distressed. What is distress? While there is
no universal definition, one paper defines an industry as distressed if the median
sales growth of pure play firms in an industry is negative and the median stock
return is 30%.1
1
T. C. Opler and S. Titman, ‘‘Financial Distress and Corporate Performance,’’ Journal of
Finance 49(3) (July 1994): 1015–1040.
313
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The distressed company’s underlying operating business may be struggling
from operational or economic distress. Optimal distress may manifest itself as a
loss of competitiveness, sales revenue may be declining, market share may have
been lost, or costs of labor and materials may be increasing while the business
is unable to increase prices, causing margins to deteriorate. Financial distress
may be present if a company’s debt level may be too high relative to its current
operating earnings (financial distress).
When the company’s debt level is too high, it may suffer commercial costs
of financial distress. For example, suppliers may lose confidence and cease offering short-term, favorable payment terms and may eventually require payment
on delivery or even in advance; customers may panic and switch in whole or in
part to safer suppliers; the best employees may leave for other jobs; and management spends inordinate time working with creditors rather than on business
operations. In addition, there are the cash costs of legal and advisory services
exploring restructuring options.
The value of the business in the case of either operational or financial distress may be negatively affected, and such distress will probably increase the
cost of capital.
We are also hampered in quantifying the true impact of financial distress on
the cost of equity capital because the commonly used formulas for unlevering and
relevering betas (and, therefore, capturing the effect of financial risk) are based on
modest levels of debt financing and, therefore, cannot adequately capture the impact
on the cost of equity capital as levels of debt and distress increase.
In this chapter, we first discuss the valuation approaches commonly employed in
valuing companies in distress.
Valuations of distressed companies occur in two general cases. First, an analyst
may be asked to value the common equity capital of a company that exhibits signs of
distress. If public, the common stock price will probably have started to decline relative to the market over some period of time. In these circumstances, the analyst
needs to consider the cost of capital, given the existing capital structure.
Second, the analyst may be asked to value the underlying business or businesses
owned by a company that is near declaring or has already declared bankruptcy. In
these circumstances, the analyst needs to consider the cost of capital without regard
to the existing capital structure. But the business may still be suffering from distress,
even though the analyst is ignoring the pre-bankruptcy-filing capital structure.
We discuss both valuation exercises in our book Cost of Capital in Litigation
(John Wiley & Sons).
WHAT IS DISTRESS?
Analysts often categorize distress into financial distress and operational distress.
A company whose equity and debt values reflect the potential or probability
of default or liquidation scenarios is considered to be operating under financial
distress. Financial distress is typically a result of a high debt burden, coupled
with difficulties in accessing capital markets. Investment decisions become distorted because of debt overhang, including distressed asset fire sales. The equity
and debt market values should reflect the analyst’s views and weighting of going
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315
concern and default scenarios. Default scenarios could include, for example, the
inability to pay current interest expense obligations or the inability to refinance
current debt obligations, resulting in the need to sell a portion of operating
assets. Rating downgrades, non-investment-grade debt, and high market yields
on debt are all indicators that the market is weighing the potential impact of
distress scenarios. Management spends much of its time talking to creditors
and to legal and financial advisors about reorganization and refinancing plans,
instead of running the business. A company does not need to be in or near
bankruptcy to be considered under financial distress. Financial distress can also
lead to operational distress.
Operational distress typically occurs in periods of significant economic downturn. Other nonrecurring events may also lead to operational distress, such as the
loss of a major lawsuit or a regulatory injunction, for example. While this is not an
exhaustive list, the following situations may be indicators of operational distress:
&
&
&
&
&
&
The company is unable to pay its suppliers on a timely basis, potentially leading
to supply shortages or disruptions.
The refusal by certain suppliers to service the company, again causing supply
disruptions.
Manufacturing facilities are operating at a significantly low level of capacity
utilization.
High employee turnover, leading to operational disruptions.
Impaired ability to do business due to customers’ concerns for parts, service and
warranty interruptions, or cancellations if the firm files for bankruptcy.
The loss of key customers due to concerns about supply reliability, in terms of
both quality and delivery times.
VALUING FIRMS IN DISTRESS
Assume that the subject company is distressed and its capital structure contains too
much debt. What is the value of the common equity?2 There are at least three widely
used methods for valuing common equity in cases where the company is in distress
(i.e., given the existing capital structure):
&
&
&
Value the enterprise with a changing capital structure over time
Value the enterprise using the adjusted present value (APV) method
Value equity as an option on the business enterprise
Changing Capital Structure
Applying this method, one values the enterprise and subtracts the debt outstanding
as of the balance sheet date. In valuing the enterprise with a changing capital structure over time, you begin with the terminal value and work in reverse. One analyzes
normalized net cash flows that could be expected, assuming that the company is able
2
For purposes of these discussions, we will assume that the company capital structure consists
of one class of interest-bearing debt and common equity.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
to survive and is able to pay off its outstanding debt over time, reducing its distress
(e.g., the terminal period net cash flows reflecting normalization of business operations and/or an amount reflective of industry average profit margins and industry
average debt to equity capital structure). This results in a terminal value for the business as if it were no longer in distress.
During the transition period from the current distressed state of operations to
normalized operations (a period that varies, depending on the level of current distress and economic and industry conditions), one projects detailed cash flows. The
cost of capital components change over time, as does the weighted average of
the overall cost of capital. During the near-term years, the business may not be able
to realize the tax deductions from interest expense, and the WACC during those
years should be calculated without tax affecting the cost of the debt component.
&
&
The cost of debt capital is reduced as debt is paid down and the credit rating
improves.
The cost of equity capital is reduced as financial distress is reduced.
Chapter 6 of the Cost of Capital: Applications and Examples, 4th ed. Workbook and Technical Supplement displays and discusses a comprehensive example of
estimating the value of the enterprise with a changing capital structure, including the
impact on the cost of equity capital as leverage changes over time.
Adjusted Present Value
The APV method is discussed in Chapter 18. The general formulation of the APV
method of valuation is:3
(Formula 16.1)
PV ¼ Present Value of Unlevered Business Enterprise
þ Present Value of Benefits Net of the Costs of Debt Financing
þ Other Adjustments
The net cash flows of the unlevered business enterprise (assuming no debt) are
discounted at the unlevered cost of equity capital, keu, which is calculated using the
following formula (assuming we are basing our discount rate on CAPM):
(Formula 16.2)
keu ¼ Rf þ BU ðRPm Þ þ RPs RPu
where:
3
keu ¼ Cost of unlevered equity capital
Rf ¼ Rate of return available on a risk-free security as of the valuation date
BU ¼ Unlevered beta
RPm ¼ General equity risk premium for the market
Marianne DeMario and Anthony Fazzone, ‘‘The Adjusted Present Value: An Alternative
Approach to the Effect of Debt on Business Value,’’ Business Valuation Update (December
2006): 1–4.
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RPs ¼ Risk premium for small size
RPu ¼ Risk premium attributable to the specific company (u stands for
unique or unsystematic risk)
The APV method has been touted as more flexible than a traditional DCF analysis in that it can be applied when debt capital is not assumed to be a constant percentage of the enterprise value, as is the assumption underlying the commonly used
formulation of the WACC. For example, if the assumption that the amount of debt
at valuation date is paid down over a scheduled repayment process, the APV method
can easily accommodate that analysis. The formulation of the APV method for valuation of a distressed company is:
(Formula 16.3)
PV ¼ PV keu þ PV ts PV dc
where:
PV ¼ Present value of net cash flows
PV keu ¼ Present value of net cash flows using unlevered cost of equity capital,
keu, as the discount rate
PVts ¼ Present value of tax shield due to interest expense on debt capital
PVdc ¼ Present value of net distress-related costs
The tax shield is the present value of the income tax savings due to the deduction of interest expense. Typically in a distressed situation, the interest expense on
outstanding debt exceeds the taxable income in the current and near-term years.
The present value of the tax shield should reflect the timing when tax deductions are
first being realized in future years. The net distress-related costs should reflect the
negative impact on the operations of the business (e.g., payments to retain personnel, cost of paying cash upon delivery of goods instead of payment in accordance
with regular trade terms) and should reflect the net tax savings due to the carryback
and carryforward of net operating loss deductions. We discuss the quantification of
the costs of distress later.
Option Valuation
There are a number of widely used and accepted approaches to estimate the fair
market value of equity in a highly leveraged capital structure. A probabilistic model
is the most appropriate valuation method when the expected payoff function is nonlinear, as is the case with highly leveraged equity. Types of probabilistic models
include the Black-Scholes option pricing model, probability weighted expected outcome, lattice, and Monte Carlo simulation. The selection of an appropriate valuation methodology is based on facts and circumstances.
Black and Scholes4 note that all ownership claims, such as common stock,
corporate bonds, or warrants, can be viewed as combinations of simple option contracts. For example, equity holders have the equivalent of an option to buy the assets
of the company, given they first repay the debt holders. The value of equity can be
4
Fischer Black and Myron Scholes, ‘‘The Pricing of Options and Corporate Liabilities,’’
Journal of Political Economy (May 1973): 637–654.
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Payoff
on equity
Value of
option
X
Business
Enterprise
Value
EXHIBIT 16.1 Value of a Call Option
viewed as the value of a call option on the company’s assets, with an exercise price
equal to the face value of debt.
The value of debt represents the risk-free right of debt holders to receive the
return of their lent monies and accrued interest, minus the value of default risk. In
other words, the value of debt can be viewed as a risk-free bond minus the value of
a put option on the assets.
The value of equity as a call option is the price a hypothetical buyer would pay
for the possibility that the value of the business enterprise, FMVBE,0, will exceed the
face value of debt (Fd or ‘‘X’’ on the graph in Exhibit 16.1) over a specified future
horizon. This can be depicted as in Exhibit 16.1.
The bold line represents the intrinsic value (or payoff) from a call option at
time T. When the business enterprise value, FMVBE,0, is less than the face value of
debt (i.e., FMVBE,0 < Fd), the equity holder will let the option expire worthless, that
is, default on the debt. When the BEV is greater than the face value of debt (i.e.,
FMVBE,0 > Fd), the equity holder will exercise the option, that is, repay the debt
holders and own the assets.
This same diagram also illustrates the value of a call option in relation to its
intrinsic value. Viewing equity as a call option is more significant when the FMVBE,0
is approximately equal to the face value of debt. When near the money, the value of
a call option is the farthest above intrinsic value and, therefore, the value of optionality is the greatest. When deep out of the money or deep in the money, the value of a
call option is closer to intrinsic value.
We can estimate the fair market value of equity as a call option based on a few
input assumptions. The option method indicates the fair market value of equity at
time 0 based on the asset volatility of the business enterprise. In Chapter 11, we
provide guidance on unlevering equity volatilities, and the unlevered equity volatility is equal to the unlevered volatility of the business assets. If the subject company is
public, the equity volatility can be estimated either from the observed volatility of
the subject company stock over a look-back period or from the implied volatility
from the subject company’s traded options. If the subject company is not public,
then the equity volatility can be estimated either from the observed volatilities of
guideline public (i.e., comparable) companies over a look-back period or from the
implied volatilities from the guideline companies’ traded options.
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The basic Black-Scholes call option equation is as follows:
(Formula 16.4)
R ðniÞ
FMV e;0 ¼ FMV BE;0 N ðd1 Þ Fd f
where:
N ðd2 Þ
FMVe,0 ¼ Fair market value of equity at time ¼ 0
FMVBE,0 ¼ Fair market value of business enterprise value at time ¼ 0
N () ¼ Cumulative normal density function (the area under the normal
probability distribution)
FMV BE;0
FMV BE;0
1 2
log
þ
s
log
ð
n
i
Þ
þ
R
þ Rf ðn iÞ
f
Fd
Fd
2
pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi
¼
d1 ¼
s ni
s ni
pffiffiffiffiffiffiffiffiffiffi
1
þ2s n i
Fd ¼ Face value of outstanding debt
Rf ¼ Risk-free rate
n i ¼ Time to maturity of debt or time to a liquidating event from
period i to period n
s BE ¼ Standard
deviation of the value of the business enterprise
pffiffiffiffiffiffiffiffiffiffi
d2 ¼ d 1 s n i
For a discussion of the cumulative normal density function, see the Cost of Capital: Applications and Examples 4th ed. Workbook and Technical Supplement,
Appendix III. We also display an example of using the Black-Scholes model in valuing a distressed business in the Cost of Capital: Applications and Examples 4th ed.
Workbook and Technical Supplement, Chapter 7.
BANKRUPTCY PREDICTION MODELS
Bankruptcy is the ultimate indication of distress. Bankruptcy prediction models can
assist the analyst in estimating the degree of severity of the business distress. There
are two broad categories of models for assessing the probability of bankruptcy:
accounting-ratio-based models and market-based models.
The most famous of the accounting-ratio-based models is the Altman z-score.5
The Altman model relies mostly on information obtained from company financial
statements.
Market-based models generally are built upon the work of Merton. These models generally use a company’s ratio of debt to equity and the estimate of the asset
volatility of the company to predict the probability of default on debt, often referred
to as the distance to default.
5
E. I. Altman, ‘‘Financial Ratios, Discriminant Analysis and the Prediction of Corporate
Bankruptcy,’’ Journal of Finance 23(4) (September 1968): 589–609; ‘‘Predicting Financial
Distress of Companies: Revisiting the Z-Score and Zeta Models’’ (July 2000); ‘‘Revisiting
Credit Scoring Models in a Basel 2 Environment,’’ Credit Ratings, Methodologies, Rationale
and Default Risk (Fall 2002).
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There are also hybrid models that combine accounting-based information and
market-based information.6
Accounting-Ratio-Based Models
The original z-score model was developed from a study of manufacturers, some of
which had gone bankrupt and others that remained going concerns. Altman
employed multiple discriminant analysis as his statistical tool to find the linear combination of characteristics that best discriminated between the bankrupt and nonbankrupt firms. From the original list of 22 possible ratios, 5 were selected as
providing the best combined prediction of bankruptcy. The following is the z-score
model for public companies:
(Formula 16.5)
z ¼ 1:2x1 þ 1:4x2 þ 3:3x3 þ 0:6x4 þ 0:999x5
where: z ¼ Overall index
x1 ¼ Working capital/total assets
x2 ¼ Retained earnings/total assets
x3 ¼ Earnings before interest and income taxes/total assets
x4 ¼ Market value of common equity/book value of total liabilities
x5 ¼ Sales/total assets
The ‘‘zones of discrimination’’ are as follows:
&
&
&
z > 2.99 ¼ safe zone
1.8 < z < 2.99 ¼ gray zone
z < 1.80 ¼ distress zone
Altman subsequently tested the z-score model. As stock prices increased, the
average z-score increased. But he still found that a z < 1.80 as a realistic cutoff of
the probability of bankruptcy.7 Altman’s tests have shown that the model is an accurate forecaster of bankruptcy up to two years prior to distress but that the accuracy
diminishes substantially as the lead time increases.
But closely held companies do not have observations of market value of common equity. Altman did an alternative analysis of the data, substituting book value
of equity for market value of equity. The formula for z0 -score is as follows:
6
For example, Ming-Yuan Leon Li and Peter Miu, ‘‘A Hybrid Bankruptcy Prediction Model
with Dynamic Loadings on Accounting-Ratio-Based and Market-Based Information: A
Binary Quantile Regression Approach,’’ Working paper, April 2009. Available at http://
ssrn.com/abstract=1506656.
7
Ming-Yuan Leon Li and Peter Miu, ‘‘A Hybrid Bankruptcy Prediction Model with Dynamic
Loadings on Accounting-Ratio-Based and Market-Based Information: A Binary Quantile
Regression Approach,’’ 17. Working paper, April 2009. Available at http://ssrn.com/
abstract=1506656.
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(Formula 16.6)
z0 ¼ 0:717x1 þ 0:847x2 þ 3:107x3 þ 0:420x4 þ 0:998x5
where: z0 ¼ Overall index
x1 ¼ Net working capital/total assets
x2 ¼ Retained earnings/total assets
x3 ¼ Earnings before interest and income taxes/total assets
x4 ¼ Book value of common equity/book value of total liabilities
x5 ¼ Sales/total assets
The zones of discrimination are as follows:
&
&
&
z0 > 2.90 ¼ safe zone
1.23 < z0 < 2.90 ¼ gray zone
z0 < 1.23 ¼ distress zone
To adapt the model to nonmanufacturers, Altman did an alternative analysis
of the data without variable x5, sales/total assets. The formula for z00 -score is
as follows:
(Formula 16.7)
z00 ¼ 6:56x1 þ 3:26x2 þ 6:72x3 þ 1:05x4
where: z00 ¼ Overall index
x1 ¼ Net working capital/total assets
x2 ¼ Retained earnings/total assets
x3 ¼ Earnings before interest and income taxes/total assets
x4 ¼ Book value of common equity/book value of total liabilities
The zones of discrimination are as follows:
&
&
&
z00 > 2.60 ¼ safe zone
1.1 < z00 < 2.60 ¼ gray zone
z00 < 1.1 ¼ distress zone
Altman later collaborated in developing a new model, the Zeta1 model,
designed to increase the accuracy of predicting bankruptcy up to five years before
bankruptcy happens.8 That model is proprietary and therefore we have not included
this formula in the text.
Another accounting-ratio-based model was developed by Ohlson.9 The
formula for the O-score is as follows:
8
Edward I. Altman, R. Haldeman, and P. Narayanan, ‘‘ZETA Analysis: A New Model to
Identify Bankruptcy Risk of Corporations,’’ Journal of Banking and Finance (June 1977).
9
James Ohlson, ‘‘Financial Ratios and Probabilistic Prediction of Bankruptcy,’’ Journal of
Accounting Research 19 (1980): 109–131.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
(Formula 16.8)
O-score ¼ 1:32 0:407o1 þ 6:03o2 1:43o3 þ 0:08o4
2:37o5 1:83o6 þ 0:285o7 1:72o8 0:52o9
where: o1 ¼ Total assets, inflation adjusted
o2 ¼ Total liabilities/total assets
o3 ¼ Net working capital/total assets
o4 ¼ Current liabilities/current assets
o5 ¼ Net income/total assets
o6 ¼ Earnings before interest, income taxes, depreciation, amortization/total
liabilities
o7 ¼ 1 if net income was negative for the last two years, zero otherwise
o8 ¼ 1 if book value of equity is negative, zero otherwise
o9 ¼ Change in net income (net income net income 1)/Absolute value
of (net income þ net income 1)
To transform the O-score into a bankruptcy probability, we use Formula 16.9:
(Formula 16.9)
Probability of default10 ¼ Exp½O-score=ð1 þ Exp½O-scoreÞ
A more recent study updated the coefficients of the Altman z-score model
and the Ohlson O-score model and tested the reliability of the updated models.
Those authors found that the original z-score model was superior to the updated
z-score model but that the updated O-score model was superior to the original
O-score model.11
Morningstar recently introduced the Morningstar Solvency Score, an accountingratio-based metric for predicting bankruptcy.12
Market-Based Models
The market-based models are generally derivations of the Black-Scholes option pricing model as adjusted by Merton. This class of market-based model is often called a
Black-Scholes-Merton model. The underlying assumption of the model is that all
available information is reflected in the stock price. Adjustments have been made
to the basic Black-Scholes-Merton model, for example, to account for the fact that
10
The function ex is called the exponential function, and its inverse is the natural logarithm,
or logarithm to base e. The number e is also commonly defined as the base of the natural
logarithm. See discussion in Appendix III of Cost of Capital: Applications and Examples,
4th ed. Workbook and Technical Supplement. The Appendix appears on the companion
John Wiley & Sons web site.
11
‘‘Stephen A. Hillegeist, Elizabeth Keating, Donald Cram, and Kyle Lundstedt, ‘‘Assessing
the Probability of Bankruptcy,’’ Review of Accounting Studies 9(1) (March 2004): 5–34.
12
Warren Miller, ‘‘Introducing the Morningstar Solvency Score, A Bankruptcy Prediction
Model,’’ Working paper, December 2009. Available at http://ssrn.com/abstract=1516762.
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dividends are accrued to owners of common equity, whereas in the option pricing
models no dividends are accrued.13
Under the market-based model, the probability of bankruptcy is the probability
that the market value of the assets of the firm (i.e., the business enterprise) will be
less than the face value of debt. The most common formulation of a market-based
model is called the distance to default. The general formulation of this model is
shown in Formula 16.10. The formula expresses the probability of the value of the
equity turning negative at time T measured from today, time ¼ 0. Assuming that the
value of the company’s business enterprise (value of all of its assets) at time ¼ T,
FMVBE,T, follows a normal distribution with a mean equal to its business enterprise
value (value of all of its assets) at time ¼ 0, FMVBE,0, and a standard deviation equal
to s, the distance to default can be estimated using Formula 16.10.
(Formula 16.10)
Probability FMV e;T <¼ 0 ¼ N ðd3 Þ
where: FMVe,T ¼ Fair market value of equity at time ¼ T
FMVe,0 ¼ Fair market value of equity at time ¼ 0
N (*) ¼ Cumulative
normaldensity
function
d3 ¼ FMV BE;0 Fd =s ¼ FMV e;0 =s
and the other variables as defined before.
Of course, the assumption that the value of the company’s enterprise follow a
normal distribution is probably flawed.
Moody’s has a proprietary model, referred to as the KMV Expected Default
Frequency credit measure, which provides a quantitative assessment of the credit
risk of publicly traded companies. It is an estimate of default probability using a
distance-to-default model. It is based on the relationships between the market value
of the company’s equity and the market value of its assets and the volatility of the
business’s assets and the volatility of the company’s equity.
Recent studies have looked at bond spreads and bond ratings as predictors of
default.14 Interestingly, bond spreads and bond ratings are not perfect substitutes,
as the correlation between spreads and ratings is only 0.45. These authors found
that adding credit spread to the market-based model is an important additional factor in improving the prediction of default. They also found that equity returns
increase with credit spreads.
Comparing the Models
Periodic studies are conducted examining whether accounting-ratio-based models or
market-based models have been superior in predicting bankruptcy.
13
Warren Miller, ‘‘Introducing the Morningstar Solvency Score, A Bankruptcy Prediction
Model,’’ Working paper, December 2009. Available at http://ssrn.com/abstract=1516762.
14
Deniz Anginer and Celim Yildizhan, ‘‘Is There a Distress Risk Anomaly? Corporate Bond
Spread as a Proxy for Default Risk,’’ Working paper, November 2009. Available at http://
ssrn.com/abstract=1344745.
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One study compared the Altman z-score model, the Ohlsen O-score model, and
a Black-Scholes-Merton market-based model, as well as other market-based data.
They found that the Black-Scholes-Merton model provides relatively more information than either of the accounting-ratio-based models used individually or together.
However, the accounting-ratio models do provide incremental information, improving on the predictive reliability of the Black-Scholes-Merton model. They also found
that added market information (e.g., size measured by market capitalization) can
add to the predictive reliability.15
One study found that accounting-ratio-based models have suffered a slight
decline in predictive accuracy, but when combined with market-based information,
the decline is very small.16
In another study, the authors examined the information provided by credit ratings and their ability to predict default. They found that a failure score model (a
linear combination of accounting-based and market-based measures of financial distress) is much better at predicting default up to two years before default than simple
reliance on credit ratings. There are wide variances in measures of distress for companies in any given credit rating category. Adding credit ratings to a failure score
model increases its accuracy.17
In another study, the authors found that the variation among firm credit ratings
is not well explained by either accounting-ratio-based or market-based models.18
Morningstar conducted two studies of bankruptcy prediction models. In the first
study, it found the market-based distance-to-default model to be superior as a
predictor of bankruptcy to the accounting-ratio-based Altman z-score, though the
distance-to-default model generated less stable ratings than the z-score.19 The
second study compared its Morningstar Solvency Score with the Altman z-score and
a distance-to-default model. They found the Solvency Score superior to the other
models within one year of bankruptcy.20
15
‘‘Stephen A. Hillegeist, Elizabeth Keating, Donald Cram, and Kyle Lundstedt, ‘‘Assessing
the Probability of Bankruptcy,’’ Review of Accounting Studies 9(1) (March 2004): 5–34.
16
William H. Beaver, Maureen McNichols, and Jung-Wu Rhie, ‘‘Have Financial Statements
Become Less Informative? Evidence from the Ability of Financial Ratios to Predict Bankruptcy,’’ Working paper, May 2008. Available at http://ssrn.com/abstract=634921.
17
Jens Hilscher and Mungo Wilson, ‘‘Credit Ratings and Credit Risk,’’ Working paper,
November 2009. Available at http://ssrn.com/abstract=1474863.
18
Alexander Charkou, Evgeny Chigrinov, and Toma Mchedlishvili, ‘‘Assessing Probability of
Bankruptcy: Comparing Accounting and Black-Scoles-Merton models’’ (Advanced Finance
Master Thesis, University of Gothenburg School of Business, Economics and Law, Spring
2006). This paper has an excellent summary of the models.
19
Warren Miller, ‘‘Comparing Models of Corporate Bankruptcy Prediction: Distance to
Default vs, z-score,’’ Working paper, July 2009. Available at http://ssrn.com/abstract=
1461704.
20
Warren Miller, ‘‘Introducing the Morningstar Solvency Score, A Bankruptcy Prediction
Metric,’’ Working paper, December 2009. Available at http://ssrn.com/abstract=1516762.
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325
COST OF CAPITAL FOR DISTRESSED FIRMS
In estimating cost of equity capital for distressed companies, the typically used
relationships do not hold, and the analyst must be aware of the issues making the
estimation process particularly difficult.
WACC Considerations
When a company is in distress, the valuation of its debt capital and the quantification of the present value of the tax shield become complex exercises. In
Chapter 6, we discussed considerations for estimating appropriate market yields
and valuing debt. For example, one needs to consider the type of debt instrument, the level of seniority, and the type of loan covenants or bond indentures of
the security being valued. Adding to this complexity, both bank loans and bonds
can be either privately placed or publicly traded. Secondary trading of bonds is a
well established and fairly active marketplace, while the secondary loan market is
not as liquid.
During recessions or financially distressed environments, we believe that many
public debt securities trade at a significant discount to par. Before the 2008–2010
credit crisis, certain entities were able to issue debt instruments featuring very few
covenants and other protections for bank lenders and bondholders, and at rates that
did not fully compensate the holders for that lack of protection.
During the 2008–2010 credit crisis, many publicly traded debt securities,
with disadvantageous terms to the holders, were trading at deep discounts
because of their subordinate position in the capital structure (or lack of protection) and not necessarily because of underlying operational difficulties within the
business itself. Further, uncertainty around the depth and length of the credit crisis and economic recession, with an accompanying flight to quality, led to significant liquidity constraints and deeply discounted pricing of debt instruments. As
we are writing this book, prices of debt securities have recovered significantly
since the depth of the crisis, thereby reducing credit spreads of investment-grade
and speculative instruments relative to government debt. However, the crisis
demonstrated that investors had not fully reflected these risks in the pricing of
these debt instruments.
This financial crisis has highlighted a number of issues, including the fact
that the traditional assumption of considering market value of debt equivalent to
its book or par value is no longer valid in many analyses. Any discussion of the
relationship between the book or par value of debt and market value of debt
should be structured around the formulations in the literature for valuing a
levered business enterprise as depicted in Exhibit 11.1, reproduced in part here
as Exhibit 16.2.
The tax shield is the reduction of the cost of debt capital due to the tax deductibility of interest expense on debt capital. In the first formulation, cost of debt capital is measured after the tax effect (kd) as the value of the tax deduction on interest
payment reduces the cost of debt capital. This formulation uses as the discount rate
the WACC. It is applied to net after-tax (but before interest) net cash flows of the
business enterprise.
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EXHIBIT 16.2 Value of a Levered Business Enterprise (BE)
Formulation 1
Value of Levered BE ¼ Value of Levered Assets
Formulation 2
Value of the Levered BE ¼ Value of the Unlevered Assets þ
Present Value of Tax Shield
In the second 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.
In the first formulation, you attach value to the assets of the business based on
their being partially financed with debt capital. In the second formulation, you
attach value to the assets of the business as if they were financed with all equity
capital, and then the tax shield is valued separately.
In the second formulation, the tax savings due to interest deductions are directly
valued as a cash flow. Therefore, the discount rate is the weighted pretax cost of
debt capital and the cost of equity capital components (pre-interest tax shield
weighted average cost of capital). It is applied to the net after-tax (but before interest) net cash flows of the firm and the cash flows due to the tax shield.
For example, during normal economic periods, the following relationship generally holds (referring to Exhibit 16.2):
$180 ¼ unlevered value of assets
þ
20 ¼ tax shield
$200
$100 ¼ debt at market
þ
100 ¼ equity at market
$200
Here the market value of debt equals the book value of debt (i.e., the contract
interest rate on debt equals the market interest rate on debt plus likelihood of collecting interest and principal when due is certain) and the tax shield equals the present value of tax savings due to interest deductions calculated at the pretax cost of
debt (hypothetically equal to approximately 20% of the par value of debt). Assume
that the company’s debt capacity indicated the debt was rated Baa and the interest
rate reflects that rating.
Now assume that we entered a period of distress (say the crisis of 2008–2010)
and the market value of debt and equity declined as follows:
$140 ¼ unlevered value of assets
þ
10 ¼ tax shield
$150
What happened?
$80 ¼ debt at market
þ
70 ¼ equity at market
$150
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As a result of entering a period of distress, expected cash flows for the business
decline. The value of the business without consideration of debt declined in the
hands of the current owner (that is the underlying basis that drives market values of
debt and equity). Cash flows in the near term are expected to decline and, in fact,
result in losses. The tax shield is reduced because tax savings due to interest expenses
are not going to be realized while the company is losing money (net of the impact of
tax loss carrybacks). The equity declined because the unlevered value of the assets
has declined (i.e., the expected cash flows have declined, and the variability of the
cash flows has increased, resulting in a higher discount rate and a lower present
value of the cash flows without regard to debt) and the present value of the tax
shield (a benefit to the equity) has also declined.
Bondholders in this scenario now realize that there is a greater risk of realizing
interest payments when they are due. They may still expect to ultimately receive
their $100 principal repayment in the future but not necessarily when contractually
due. In addition, bondholders now anticipate that there will be costs if bankruptcy
were to occur, even if they believe they will ultimately receive their $100 principal.
We can depict that scenario in present value terms as follows:
Market value of debt ¼ $80 ¼< $20 > þ$100
where the <$20> is the present value of: (1) the possible delay in receiving interest
payments when due, were bankruptcy to occur and (2) the costs of possible bankruptcy (even though the contractual $100 principal is ultimately expected to be
paid). The risk-adjusted discount rate equates the probability-weighted outcomes
with the market value of $80:
1. Outcome #1: Interest continues to be paid as contracted, and principal is repaid
when due.
2. Outcome #2: Interest is delayed and repaid with principal at a date after contractually due because the business generates lower expected operating cash
flows and, in the worst case, bankruptcy.
Assuming that the debt now is rated B- or lower, the interest rate has increased
and the market value of debt has decreased to a price below book or par value.
As we are writing this chapter, the increased risks to bonds due the crisis of
2008–2010 continues. For example, Fitch Ratings reported that after 103 noninvestment-grade companies defaulted on $79.7 billion of bonds in the first
half of 2009, 42 issuers defaulted on $36.8 billion of issuance from July to
November 2009. Despite the slower pace, the year-to-date default rate for 2009
rose to 13.6%. When defaults from December 2009 are added in, Fitch
expected the full-year default rate to be just short of its original forecast range
of 15% to 18%.
For 2010, Fitch projected high-yield defaults will continue to decline, to a range
of 6% to 7%, contrasted to the long-term average annual rate of 4.7%. Standard &
Poor’s lowered its 12-month-forward baseline projection to a similar range, largely
because of declines in funding costs for corporations, the reopening of the bond markets, and the abatement of volatility; S&P admited that it ‘‘had stated our expectations for a swath of defaults to occur in the first half of 2010. But now, we expect
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
many of the defaults might be postponed to later quarters beyond the 12-month
forecast horizon.’’21
We discussed in Chapter 11 the care that is required in choosing an appropriate
formula for unlevering observed equity betas. For companies in distress, the considerations are even greater. If one is using guideline public companies to estimate a
proxy asset beta for the subject company, for example, the unlevering formula must
match the risk of the debt of each guideline public company, as evidenced by the
debt beta of public debt with ratings comparable to that of the guideline public company. Debt betas greater than zero indicate the market’s assessment that there is
covariance between a company’s equity and debt and that the debt capital is sharing
risk with equity capital. Simply assuming that the cost of debt in the WACC equals
Formula 16.11 is not reasonable in today’s environment for many companies.
(Formula 16.11)
kd ¼ kdðptÞ 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)
We will discuss alternative formulations for the WACC in Chapter 18.
Cost of Equity Capital Considerations
Often the observations of returns on equity capital for public firms during periods of
distress do not represent expectations of investors. For example, as a firm begins to
realize distress, investors reassess the firm’s expected cash flows and the firm’s risk,
causing stock prices to adjust downward to reflect the new reality of the firm’s outlook. The result is that returns are in transition and do not reflect either the historical relationship to the market portfolio or the expected future relationship to the
market portfolio once the market fully adjusts to the effects of distress on the firm.
Assume the subject business had become distressed and had recently emerged from
restructuring its debt and an infusion of equity. Exhibit 16.3 presents an example of
an adjustment in pricing for a stock of this hypothetical company and the problem
one has in estimating the beta for a distressed company using look-back methods.
In period A, the company returns had essentially moved with the market. In period B, the company is distressed, and its stock is experiencing a downward repricing. During this period, the company’s returns are not correlated with the movement
of the overall market at all. In Period C, the restructuring of the company and the
repricing of the company’s stock is complete, and the company’s returns are once
again moving more in tandem with market returns.
If one were to compute beta at Time 1, which includes period A as the look-back
period, the beta estimate would reflect a normal relationship between the company’s
returns and the market’s returns. In fact, its beta estimate would be near 1. In contrast, computing a beta estimate at Time 2, which includes period B (the period of
the company’s stock repricing) as the look-back period, would not yield a reliable
21
Vincent Ryan, ‘‘Default Risk to Linger in 2010,’’ CFO.com (December 10, 2009).
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Example Company versus Index over Time
1.6000
1.4000
Compound Return
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1.2000
1.0000
0.8000
0.6000
0.4000
0.2000
0.0000
A
B
1
C
2
Time
Example
S&P 500
EXHIBIT 16.3 Relationship of Returns for Example Company
forward-looking beta estimate. In fact, it would yield a beta estimate lower than
expected, since the company’s return was negative in a period when the market’s
return was generally positive. This result is counterintuitive, given the company’s
downward repricing; that is, the distress of the company has not declined over
period B, and in fact, its distress was greatest during this period. Once the restructuring of the company and the repricing of the company’s stock is complete, its normal
relationship to the market will resume in period C.
Risk measures such as betas estimated using realized returns during such transition periods will underestimate the risk of the distressed firm and underestimate its
cost of equity capital. Further, once the stock price of a distressed firm ratchets
downward, it often trades more as an option than as a traditional equity security,
reducing the validity of beta estimation methods (such as ordinary least squares
[OLS]).22 Thus a top-down beta estimate will result in an erroneous beta estimate.23
To capture the greater cost of equity capital, one may substitute a forwardlooking measure of relative volatility for a historical-based estimate based on
observed yield spreads or observed volatility of traded options (see, for example, the
discussion in Chapter 12 of using yield spreads as a risk measure and the discussion
in Chapter 17 of the market-derived capital pricing model).
22
In ‘‘Limited Liability, the CAPM and Speculative Grade Firms: A Monte Carlo Experiment,’’ Working paper, August 18, 2004, Carlos A. Mello-e-Souza shows that limited liability allows equity to be valued as a ‘‘call option’’ within the CAPM framework. When
adjusting for measures of bankruptcy risk on beta estimation, he finds that when bankruptcy risk ¼ 5%, OLS beta is underestimated by 10%, and when bankruptcy risk ¼ 20%,
OLS beta is underestimated by 23%. Available at http://ssrn.com/abstract=589887.
23
For a good discussion of estimating betas for distressed companies, see Mathias Meitner,
‘‘Beware of the Beta Flip! Pitfalls in DCF-Valuations of Temporary and Sustainably
Distressed Companies and How to Avoid Them,’’ Working paper, September 1, 2009.
Available at http://ssrn.com/abstract=1466654.
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Any risk measure based on implied volatilities derived from options requires
that the firm have traded options. Alternative methods available are:
&
&
&
&
&
&
24
Estimating betas for the subject distressed firms from beta estimates for
guideline public companies not going through such a period of adjustment
(building a bottom-up estimate of beta), relevering beta, and adding a size
premium from the SBBI data and possibly a company-specific risk adjustment to account for the added risk due to distress (equivalent to subtracting
the cost of distress).
The SBBI size premium data discussed in Chapters 13 and 14 are based on
realized returns for companies based on market capitalization of equity
without removing troubled companies. The market capitalization for many
troubled companies is small; many troubled company stocks are trading like
options. The premiums in excess of the return predicted based on beta for,
say, the smallest subdecile 10z are influenced in part by the returns on these
distressed companies. That size premium reported for 10z can be used as a
guide for the total amount of premium (due both to size and companyspecific distress) that can be added in the adjusted CAPM derivation of the
cost of equity capital.
Estimate the cost of equity capital using the Fama-French three-factor (FF threefactor) formula (discussed in Chapter 17). The hi, the high-minus-low coefficient in the FF three-factor model, to the HMLP, expected high-minus-low
risk premium, is considered an indicator of financial distress. HMLP is estimated as the difference between the historical average annual returns on the
high book-value-to-market-value of equity and low book-value-to-market-value
of equity portfolios.
Estimating fundamental operating risk for the distressed firm and matching the
fundamental risk with observed market returns. The Duff & Phelps Risk Study
allows you to match the appropriate market returns with measures of operating
risk.
Using the relationship of fundamental risk and return for the companies comprising the Duff & Phelps Risk Study 25 portfolios, which exclude highfinancial-risk companies (see Chapter 14), you can assess whether the subject
company is a ‘‘good’’ company (i.e., good operating performance) with a ‘‘bad’’
balance sheet (i.e., too much debt) or a ‘‘bad’’ company’’ (i.e., poor operating
performance) with a bad balance sheet.24 Using this comparison, you can understand the appropriate risk premium for the operations without regard to financial distress.
You can use the risk-return relationship for the Duff & Phelps study highfinancial-risk portfolio to quantify an additional company-specific risk adjustment to account for the added risk due to financial distress (the study reports
a separate high-financial-risk portfolio, which we discuss here later). The difference between the risk-return relationships discussed in Chapter 14 and
those discussed in this chapter for the high-financial-risk portfolio is due to the
market’s assessment high financial risk.
American Institute of Certified Public Accountants, ‘‘Business Valuation in Bankruptcy (a
Nonauthoritative Guide),’’ AICPA Consulting Services Practice Aid, Draft, 2009.
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Studies of returns for distressed companies often produce puzzling results. Analysts rely on studies like those reported in the SBBI Yearbook to provide estimates
on realized returns. But for the most part, the companies included in the SBBI Yearbook results are not troubled companies. Academic researchers have expanded those
types of studies to investigate realized returns for distressed companies. Their results
generally indicate that just when the risk of the company is at its greatest, the realized returns for the distressed companies are less than the realized returns for companies not experiencing distress. This puzzle is easy to understand if one looks at
Exhibit 16.3. If one measures distress at Time 1 and instead of measuring returns
based on expected returns at Time 1, measures returns based on realized returns
during period B, one will see low realized returns.
One very recent study explains that this anomaly is due to the misspecification
of betas of distressed companies.25 In capital market theory, the market portfolio is
supposed to include all possible investment vehicles. Because of data constraints,
most studies limit the definition of market portfolio to equities (e.g., the S&P 500).
Analysts and data services estimate betas using returns on this all-equity market
portfolio. Using an equity-only proxy for the true investment market portfolio will
understate equity betas, and that underestimate is accentuated as the leverage of the
company increases.
The authors use a corrected CAPM to adjust stock returns and find that it can
explain what appear to be anomalous results of prior studies. Assume that one
expands the measure of market portfolio to include stocks and debt, the true beta
for the stock of company i relative to the market portfolio of equity, ME, combined
with the market portfolio of debt, MD, is equal to the following:
(Formula 16.12)
B0 i ¼ ðME =ME þ MD Þ s E2 =s MEþMD2 BL þ ðMD =ME þ MD Þs D2 =s MEþMD2 Bd
where:
25
0
B i ¼ True beta estimate for stock of company i based on relationship
to excess returns on market portfolio of equity plus debt, ME þ
MD
ME ¼ Market value of portfolio of equity (e.g., proxy might be market
value of NYSE plus ASE plus NASDAQ stocks)
MD ¼ Market value of debt capital
s E2 ¼ Variance in excess returns on market of stocks
s D2 ¼ Variance in excess returns on market of debt
s MEþMD 2 ¼ Variance in excess returns on market portfolio of equity plus
debt, ME þ MD
BL ¼ Beta estimate for stock of company i based on relationship to
market of stocks (as discussed in Chapter 10), levered beta
Bd ¼ Beta estimate for debt of company i based on relationship to
market of debt
Jing Chen, Loran Chollete, and Rina Ray, ‘‘Financial Distress and Idiosyncratic Volatility:
An Empirical Investigation,’’ Working paper, August 2009. Available at http://ssrn.com/
abstract=1524454.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
The authors conclude that the misspecification of true beta and the bias created
(the second term in Formula 16.12 is missing completely) lead to the anomalies
observed in empirical studies. As we saw in the data presented in Chapter 10 on
debt betas, the debt beta increases as the debt rating decreases. The increasing debt
beta offsets the otherwise underestimated equity beta for distressed companies.
Perhaps someone will take this research and make better beta estimates available to
practitioners.
VALUING COMPANIES EMERGING FROM BANKRUPTCY
Often analysts are asked to value the underlying business or businesses owned by a
company that is near declaring or has already declared bankruptcy. In these circumstances, the analyst needs to consider the cost of capital without regard to the existing capital structure. But the business may still be suffering from distress, even
though the analyst is ignoring the pre-bankruptcy-filing capital structure.
The analyst has no meaningful top-down beta estimate, so in using the CAPM,
for example, the analyst must build a bottom-up beta estimate. Even though the
subject business may have its debt reduced or eliminated soon, the effect of financial
distress may have affected its operations, and it may exit restructuring having a
profit margin and growth less than the average company operating in the industry.
Estimating the cost of equity capital in cases where the subject business is experiencing operating results that are inferior to the industry is difficult. One tool that is
available is to estimate the cost of equity capital based on fundamental risk. The
Duff and Phelps Risk Premium Report—Risk Study provides such a tool.
The Risk Study (discussed in Chapter 15) provides the analyst with the relationship of risk and return for companies whose financial risk is average. But
companies emerging from restructuring often have more debt than the industry
norm. The Risk Study of the high-financial-risk portfolio of companies provides
the analyst with the relationship of risk and return for companies whose financial risk is greater than average.
With regard to the appropriate debt, the analyst must closely examine the
changing levels of debt since the 2008–2009 crisis. The debt-to-equity ratios of the
mid-2000s often do not reflect the current debt-to-equity ratios, particularly with
regard to highly financially leveraged companies. We discuss the appropriate capital
structure in Chapter 18.
Duff & Phelps Risk Study—High-Financial-Risk Companies
Practitioners typically have been able to quantify the relationship between risk
and expected return only by measuring risk in terms of beta and size. While company size is a risk factor in and of itself, Grabowski and King, the original coauthors of the study, were interested in understanding if the stock market recognized risk as measured by fundamental or accounting information. The building
of the underlying database combining data by company from the Center for
Research in Security Prices (CRSP) database with accounting and other data
from the S&P’s Compustat database is explained in Chapter 13. They used a
database to analyze fundamental risk. That research correlates realized equity
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333
returns (and 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 them from a stock market–based measure of
equity risk such as beta. The Duff & Phelps Risk Premium Report—Risk Study
annually updates this research.
The Duff & Phelps studies separate companies in the dataset into high-financialrisk portfolios for companies with any one of these characteristics:
&
&
&
&
&
Identified by Compustat as in bankruptcy or in liquidation
With five-year-average net income available to common equity for the previous
five years less than zero (either in absolute terms or as a percentage of the book
value of common equity)
With five-year-average operating income for the previous five years (defined as
sales minus [cost of goods sold plus selling, general, and administrative expenses
plus depreciation expense]) less than zero (either in absolute terms or as a percentage of net sales)
With negative book value of equity at any of the previous five fiscal year-ends
With debt-to-total capital of more than 80% (with ‘‘debt’’ measured as preferred stock at carrying value plus long-term debt [including current portion]
and notes payable in book value terms and total capital measured as book value
of debt plus market value of equity)
Segregating such companies into a separate high-financial-risk portfolio isolates
the effects of high financial risk. Otherwise, the results might be biased by smaller
companies to the extent that highly leveraged and financially distressed companies
tend to have both high returns and low market values. It is possible to imagine financially distressed (or highly risky) companies that lack any of the listed characteristics. It is also easy to imagine companies that have one of these characteristics but
that would not be considered financially distressed. The resulting high-financial-risk
portfolio is composed largely of companies whose financial condition is significantly
inferior to the average financially healthy public company.
To calculate realized risk premiums, the Duff & Phelps studies first calculate an
average rate of return for each portfolio over the sample period. Returns are based
on dividend income plus capital appreciation and represent returns after corporatelevel income taxes (but before owner-level taxes). Then they subtract the average
income return earned on long-term U.S. government bonds over the same period
(using SBBI data) to arrive at an average realized risk premium.
The exclusion of companies from the base set and inclusion in the highfinancial-risk portfolio based on historical financial performance does not imply any
unusual foresight on the part of hypothetical investors in these portfolios. In forming
portfolios to calculate returns for a given year, they exclude companies from the
base set and include them in the high-financial-risk portfolio on the basis of performance during previous years (e.g., average net income for the five prior fiscal years),
rather than current or future years. For instance, to form portfolios for 1963,
they take into account the average net income for the five fiscal years preceding
September 1962. They repeat this procedure for each year from 1963 through the
latest available year.
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EXHIBIT 16.4 Companies Ranked by Z Score Equity Premiums for Use in the
Build-up Method
Realized Equity Risk Premium: Average since 1963
High-Financial-Risk Company Data for Year Ending December 31, 2008
Portfolio
Rank by
Z Score
Beta (Sum
Beta) since
1963
Standard
Deviation
of Returns
1.8 to
1.57
34.46%
2.99
< 1.8
1.70
43.29%
Large Stocks (SBBI data)
Small Stocks (SBBI data)
Long-Term Treasury Income (SBBI
data)
Geometric
Average
Return
Arithmetic
Average
Return
Arithmetic
Average
Risk
Premium
Average
Debt/
MVIC
13.15%
18.22%
11.18%
44.16%
14.44%
9.39%
13.07%
7.01%
21.41%
10.88%
15.96%
7.04%
14.37%
3.84%
8.92%
58.07%
Source: Calculations by # Duff and Phelps, LLC # 2009 CRSP1, Center for Research in
Security Prices. University of Chicago Booth School of Business used with permission. All
rights reserved. www.crsp.chicagobooth.edu.
For the companies in the high-financial-risk portfolio, Duff & Phelps forms
portfolios of securities based on relative risk as measured by Altman’s z-score.26
Altman has since offered improvements on the original z-score, but the original
z-score is still frequently calculated as a convenient metric that captures within a
single statistic a number of disparate financial ratios measuring liquidity, profitability, leverage, and asset turnover.
The Risk Study uses the z-score model, Formula 16.5, for public companies in
comparing the returns for companies in the high-financial- risk portfolio. The use of
the z-score here is not as a predictor of bankruptcy. Rather it is used to rank the risk
of companies in the high-financial-risk portfolio.
For each year, Duff & Phelps formed portfolios by sorting all of the companies in the high-financial-risk portfolio. They then calculated the z-score and
divided the companies into three portfolios: those companies with z-score greater
than 3.0 (not categorized as in distress), those companies with z-score between
1.8 and 2.99, and those companies with z-score less than 1.8. The portfolios
were rebalanced annually; that is, the companies were reranked and sorted at the
beginning of each year. Portfolio rates of return were calculated using an equalweighted average of the companies in the portfolio.
The results for the two portfolios indicating the companies are in distress for use
in the build-up method are shown in Exhibit 16.4 and the results for use in CAPM
are shown in Exhibit 16.5.
26
E. I. Altman, ‘‘Financial Ratios, Discriminant Analysis and the Prediction of Corporate
Bankruptcy,’’ Journal of Finance 23(4) (September 1968): 589–609; ‘‘Predicting Financial
Distress of Companies: Revisiting the Z-Score and Zeta Models’’ (July 2000); ‘‘Revisiting
Credit Scoring Models in a Basel 2 Environment,’’ Credit Ratings, Methodologies, Rationale and Default Risk (Fall 2002).
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EXHIBIT 16.5 Premiums over CAPM for Use in the CAPM
Realized Equity Risk Premium: Average since 1963
High-Financial-Risk Company Data for Year Ending December 31, 2008
Portfolio Rank
by Z Score
Beta (Sum Beta)
since 1963
1.8 to 2.99
1.57
< 1.8
1.70
Large Stocks (SBBI data)
Small Stocks (SBBI data)
Long-Term Treasury Income (SBBI
data)
Arithmetic
Average Return
Arithmetic Average
Risk Premium
Premium
over CAPM
18.22%
21.41%
10.88%
15.96%
7.04%
11.18%
14.37%
10.88%
15.96%
5.14%
7.84%
Source: Calculations by # Duff and Phelps, LLC # 2009 CRSP1, Center for Research in
Security Prices. University of Chicago Booth School of Business used with permission. All
rights reserved. www.crsp.chicagobooth.edu.
These data can be used as an aid in formulating estimated cost of equity capital
using objective measures of characteristics of a subject company.
The traditional z-score was developed using data for public companies, and one
of the statistics utilizes stock price. This creates problems for application of the data
to private companies. Altman developed a similar model using only the financial
statement data for private companies. If the subject company is not publicly traded,
then the analyst can calculate the z-score for a private company (the z0 -score) to
compare with the zones of discrimination as reported in Exhibits 16.4 or 16.5. The
formula for z0 -score is Formula 16.6.
Although the original companies used to develop the zones of discrimination for
the z-score and the z0 -score differed and are not strictly comparable, the realized
returns reported in Exhibits 16.4 and 16.5 can be useful to develop cost of equity
estimates based on the relative zones of discrimination. In applying either the z-score
or z0 -score equations, one should express the ratios in terms of their decimal equivalents (e.g., x1 ¼ net working capital/total assets ¼ 0.083).
Exhibit 16.6 provides information on the characteristics of the firms in the highfinancial-risk portfolios with z-scores in the gray zone and in the distress zone.
Using the Duff & Phelps Risk Study in the Build-up Method As an alternative to Formula 7.2 for the build-up method, EðRi Þ ¼ Rf þ RPm þ RPs RPu , you can use the
Risk Study to develop a risk premium for the subject company that measures risk in
terms of the total effect of market risk, size premium, and risk attributable to the
specific company. The formula then is modified to be:
(Formula 16.13)
EðRi Þ ¼ Rf þ RPmþsþu
where: RPmþsþu ¼ Risk premium for the ‘‘market’’ plus risk premium for size plus
risk attributable to the specific distressed company
336
129
216
1.8 to 2.99
< 1.8
$400.965
$337.355
Market
Value of
Equity
$140.112
$70.909
Book
Value of
Equity
$(3.776)
$(15.295)
5-Year
Average Net
Income
$653.705
$736.961
Market Value of
Invested Capital
$498.565
$703.060
Total
Assets
$31.776
$49.955
5-Year
Average
EBITDA
$418.9
$359.5
Sales
1,599
1,590
Number of
Employees
Source: Calculations by # Duff and Phelps, LLC # 2009 CRSP1, Center for Research in Security Prices. University of Chicago Booth School of
Business used with permission. All rights reserved. www.crsp.chicagobooth.edu.
Number
as of 2008
Portfolio
by Z Score
Portfolio Median
08/26/2010
EXHIBIT 16.6 Characteristics of Companies Comprising High-Financial-Risk Portfolios ($mils.)
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The realized risk premiums reported in Exhibit 16.4 for use in the build-up
method have not been adjusted to remove beta risk; therefore, they should not be
multiplied by a CAPM beta or otherwise included in a CAPM analysis. Use of these
exhibits is a four-step process.
&
&
&
&
One first determines if the subject company matches the characteristics of the
companies included in the Duff & Phelps study (e.g., the study excludes financial service companies and start-up companies). One then determines if the subject company better matches the characteristics of the base set of companies
(included in the 25 portfolios) or the high-financial-risk set of companies.
Second, assuming the subject company’s characteristics better matches the characteristics of the high-financial-risk portfolio of companies, one then calculates
the z-score or z0 -score for the subject company.
Third, if the z-score or z0 -score of the subject company indicates it is in the gray
zone or distress zone, one then matches the subject company with the companies
included in the portfolio most comparable to the subject company (e.g., the highfinancial-risk portfolio with z-score in the gray zone or in the distress zone).
Fourth, the premiums of these portfolios can then be added to the yield on longterm U.S. government bonds as of the valuation date to obtain benchmarks for
the cost of equity capital.
The realized return data reported herein for the high-financial-risk portfolios
has not been differentiated from any size effect. While the median size characteristics of the companies included in the z-score portfolios is reported in Exhibit 16.6,
the risk effect reported herein overlaps with the size effect documented in the Size
Study portion of the Risk Premium Report for the base set of companies. The
returns reported herein should be used instead of the returns reported in the Size
Study, not added to those returns.
If the z-score or z0 -score indicates that the subject company is in the safe zone,
one should consider whether the subject company is distressed. If one determines
that it is not distressed (even though it matched the characteristics for exclusion
from the base set of companies), the returns reported in the exhibits in the Risk
Premium Report for the 25 portfolios (e.g., Exhibits 13.7 and 13.8) may be more
appropriate for the subject company than the returns reported herein.
For example, the subject company may have a debt-to-total-capital ratio of
more than 80% (with debt measured in book value terms and total capital measured
as book value of debt plus market value of equity) and not be distressed. More
generally, an assessment that a company should be treated as distressed should be
based on an evaluation of the company’s current financial condition and circumstances. Such an assessment will generally involve more than a review of historical
financial statistics and ratios.
Use of a portfolio’s average realized rate of return to calculate a cost of equity
capital is based (in part) on the implicit assumption that the risks of the subject company are quantitatively similar to the risks of the average company in the subject
portfolio. If the risks of the subject company differ materially from the average company in the subject portfolio, then an appropriate cost of equity capital may be less
than (or greater than) than a return derived from the average realized risk premium
for a given portfolio. Material differences between the expected returns for a subject
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
company and a given portfolio of stocks may arise from differences in leverage (the
average debt/MVIC of the portfolios are displayed in Exhibit 16.3) or other fundamental risk factors.
The risk premiums reported here are realized averages since 1963. The average
realized risk premium over the same period for the SBBI large company stocks
(essentially the S&P 500) was 3.84%. If one’s estimate of the equity risk premium
(ERP) for the S&P 500 on a forward-looking basis was materially different from the
average realized risk premium since 1963, it may be reasonable to assume that the
other realized portfolio returns reported here would differ on a forward-looking
basis by approximately a similar differential. For example, assume that your current
estimate of the ERP was 6.0% (see Chapter 9). The difference between the 6.0%
estimated ERP and the average realized risk premium since 1963 of 3.84%, or
2.2%, can be added to the average equity risk premium for the z-score portfolio that
matches the z-score of the subject company to arrive at an adjusted forward-looking
risk premium for the subject company. This forward-looking risk premium can then
be added to the risk-free rate as of the valuation date to estimate an appropriate rate
of return for the subject company. As a caution, be aware that this reasoning does
not apply to the premiums over CAPM (Exhibit 16.4) since those premiums are
based on relative returns over the reported period.
Example We will show how the data reported here can be used to estimate the
required return on equity or discount rate for a hypothetical company. Assume the
subject company has the following characteristics:
Market value of equity
Book value of equity
Market value of invested capital
Total assets
Five-year average net income
Most recent year net income
Five-year average EBIT
Most recent year EBIT
Sales
Number of employees
Current assets
Current liabilities
Retained earnings
¼ $80 million
¼ $100 million
¼ $230 million
¼ $300 million
¼ $3.0 million
¼ $10 million
¼ $2.0 million
¼ $5.0 million
¼ $250 million
¼ 200
¼ $75 million
¼ $50 million
¼ $75 million
z-score ¼ 1:2 ð25=300Þ þ 1:4 ð75=300Þ þ 3:3 ð5:0=300Þ
þ 0:6 ð80=200Þ þ :999 ð250=300Þ
¼ 1:2 ð0:0833Þ þ 1:4 ð0:2500Þ þ 3:3 ð0:0167Þ
þ 0:6 ð0:4000Þ þ :999 ð0:8333Þ
¼ 1:4675
Because the five-year average net income ¼ $3.0 million and the five-year
average EBIT (operating income) ¼ $2.0 million, the subject company’s characteristics better match those companies included in the high-financial-risk portfolio.
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Distressed Businesses
If we are using a build-up method, we want to determine a premium over the
risk-free rate. The simplest approach is to turn to Exhibit 16.3 and locate the portfolio whose z-score is most similar to the subject company. Looking at Exhibit 16.3,
the premium indicated for our hypothetical company with a z-score in the distress
zone equals 14.4%. With a risk-free rate as of the valuation date of 4.5%, for example, the premiums would indicate the cost of equity capital of approximately 21.0%
(4.5% risk-free rate plus 14.4% risk premium from Exhibit 13.10 plus 2.2% adjustment for ERP estimate).
These estimated required rates of return on equity are derived from rates of
return for stocks of public companies. If the equity of the subject company is not
public, this cost of equity capital estimate should be adjusted either directly or
through application of a discount for lack of ready marketability for the relative
liquidity of shares of the publicly traded stock and in comparison to the shares of
the subject company.
Using the Duff & Phelps Risk Study in CAPM The cost of equity capital can be estimated by the CAPM method as follows:
(Formula 16.14)
EðRi Þ ¼ Rf þ BðRPm Þ þ RPsþu
where: EðRi Þ 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 estimate for the market
RPsþu ¼ Risk premium for small size plus risk premium attributable to the
specific distressed company
The premiums over CAPM reported in Exhibit 16.4 can be used in the context
of a CAPM analysis. Use of these exhibits is a four-step process.
&
&
&
&
One first determines if the subject company matches the characteristics of the
companies included in the Duff & Phelps study (e.g., the study excludes financial service companies and start-up companies). One then determines if the subject company better matches the characteristics of the base set of companies
(included in the 25 portfolios) or the high-financial-risk set of companies.
Second, assuming the subject company characteristics better match the characteristics of the high-financial-risk portfolio of companies, one then calculates the
z-score or z0 -score for the subject company.
Third, if the z-score or z0 -score of the subject company indicates it is in the gray
zone or distress zone, one then matches the subject company with the companies
included in the portfolio most comparable to the subject company (e.g., the highfinancial-risk portfolio with z-score in the gray zone or in the distress zone).
Fourth, the premiums over CAPM reported for these portfolios can then be
added to CAPM as of the valuation date to obtain benchmarks for the cost of
equity capital.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
That is, the premium should not be multiplied by beta but instead should be
added to the sum of the risk-free rate and the product of beta times the aggregate ERP.
One can use Exhibit 16.4 as the source for a combined risk premium for size
and a risk premium attributable to the specific risk of the subject company due to
the above-average risk characteristics of the companies in the portfolio. The premiums over CAPM data reported herein have not been differentiated for any size
effect. While the median size characteristics of the companies included in the
z-score portfolios are reported in Exhibit 16.5, the risk effect reported herein
overlaps with the size effect documented in the Size Study portion of the Risk
Premium Report for the base set of companies. The premiums over CAPM
reported herein should be used instead of the premiums over CAPM reported in
the Size Study, not added to those returns.
Again, if the z-score or z0 -score indicates that the subject company is in the safe
zone, one should consider whether the subject company is distressed. If one determines that it is not distressed (even though it matched the characteristics for exclusion from the base set of companies), the premiums over CAPM reported in the
exhibits in the Risk Premium Report for the 25 portfolios (such as Exhibits 13.14
and 13.15) may be more appropriate for the subject company than the premiums
over CAPM reported herein. For example, the subject company may have a debtto-total-capital ratio of more than 80% (with debt measured in book value terms
and total capital measured as book value of debt plus market value of equity) and
not be distressed.
Example One can adjust the cost of equity capital derived from the CAPM by
adding a high-financial-risk premium. The premiums can be measured using the
‘‘Premiums over CAPM’’ presented in Exhibit 16.5, which represents a highfinancial-risk premium (a combined risk premium for size and the specific risk
of the subject company due to the above-average risk characteristics of the companies in the portfolio).
The simplest approach is to turn to Exhibit 16.5 and locate the portfolio whose
z-score is most similar to the subject company. Looking at Exhibit 16.5, the premium over CAPM indicated for our hypothetical company with a z-score in the distress zone equals 7.84%. Assume that as of valuation date Rf ¼ 4.5%, B ¼ 1.75, and
RPm ¼ 6%, resulting in the indicated CAPM estimate before the size and risk adjustment equal to 15.0%, then the high-financial-risk premium over CAPM indicates a
cost of equity capital of approximately 22.8% (15% þ 7.84%).
These estimated required rates of return on equity are derived from rates of
return for stocks of publicly traded companies. If the equity of the subject company
is not publicly traded, this cost of equity capital estimate should be adjusted either
directly or through application of a discount for lack of ready marketability for the
relative liquidity of shares in publicly traded stock compared to the shares of the
subject company.
Additional Information on Company Risk
Grabowski and King previously published the results of research correlating realized
equity returns (and realized risk premiums) directly with measures of company risk
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EXHIBIT 16.7 Characteristics of Companies Included in the High-Financial-Risk
Portfolio (Median)
Portfolio
Rank by Z
Score
1.8 to 2.99
< 1.8
Average
Book
Value Number
($mils.) of Firms
351
1,072
129
216
Arithmetic
Average Average
Risk
Debt to
Premium
MVIC
12.5%
15.5%
44.94%
60.26%
Average
Debt to
Market
Value of
Equity
Beta
(Sum
Beta)
Since
1963
Average
Operating
Margin
81.6%
151.6%
1.57
1.70
2.0%
2.4%
Source: Calculations by # Duff and Phelps, LLC # 2009 CRSP1, Center for Research in
Security Prices. University of Chicago Booth School of Business used with permission. All
rights reserved. www.crsp.chicagobooth.edu.
derived from accounting information.27 These may also be called fundamental measures of company risk to distinguish these risk measures from a stock market–based
measure of equity risk such as beta. Research has shown that measures of earnings
volatility can be useful in explaining credit ratings, predicting bankruptcy, and
explaining the CAPM beta.
Exhibit 16.7 presents information on the debt of the companies in the highfinancial-risk portfolio, the beta,28 and the fundamental financial characteristic
‘‘operating margin’’ for portfolios formed by ranking public companies by z-score.29
In the Risk Study, Duff & Phelps also examines two other measures of risk
(coefficient of variation in operating margin and coefficient of variation in return on
equity), but they are unable to present comparable data because the denominators of
these ratios are often negative for companies in the high-financial-risk portfolio as a
result of either negative earnings or negative book value of equity, frequently resulting in meaningless statistics.
Another indication of the company-specific risk adjustment for companies
emerging from bankruptcy can be imputed from the rate of return, which equates
the market value of public companies that emerge from bankruptcy and the
expected cash flows. To the extent that the imputed cost of capital exceeds an industry average cost of capital, you can conclude that the market is adding a factor for
company-specific risk due to the greater risks of the companies that recently
emerged from bankruptcy. One such study indicates that such an imputed
27
‘‘New Evidence on Equity Returns and Company Risk,’’ Business Valuation Review
(September 1999; revised March 2000). These articles are available at www.appraisers.org.
28
Beta (calculated using the sum beta method applied to monthly returns for 1963 through
the latest year).
29
Average operating margin (since 1963). Operating margin is defined as operating income
divided by sales, and operating income is defined as sales minus (cost of goods sold plus
selling, general, and administrative expenses plus depreciation) calculated as the mean
operating income for the five prior years divided by the mean sales for the five prior years.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
company-specific risk adjustment is in the range of 3.0% to 4.0%.30 Companies
emerging from bankruptcy are generally riskier than peer companies because often
they are still burdened with too much debt, they may not have worked through all of
the problems that caused business distress, and they have a higher probability of
returning to bankruptcy than do companies that have never been bankrupt.
Any such company-specific risk premium is applicable during the time the company is in distress: This is the period when there is the added risk that plans to work
out of the distress situation may fail.
In a recent study, Altman and two other authors examined the level of distress
of firms that have emerged from bankruptcy.31 They reviewed studies of company
performance following emergence from bankruptcy and reported that approximately a third of the companies emerging from bankruptcy as public companies
experience some form of subsequent distressed restructuring. The study applies
the z00 -score model to firms that emerge from bankruptcy. They find that the z00 -score
at the time of emergence from bankruptcy was far worse for companies that
emerged from bankruptcy and then reentered bankruptcy than for companies
that emerged permanently. The model can be a tool in predicting the level of risk of
companies emerging from bankruptcy.
Some authors suggest looking at venture capital rates of return as a proxy for
distressed company rates of return. These are at best a poor proxy because most of
the rates observed are for newer ventures without a proven history in the market.
Distressed firms often have proven technologies, products, and/or services. Often
these firms’ problems are simply too much debt or poor execution by management.
These risks differ from those of most venture capital or buyout fund investments.
Relevering Beta for a Highly Leveraged Company
In Chapter 11, we discussed various relevering formulas. As has been stated, these
formulas probably underestimate the effect on beta due to distress. For example, the
Practitioners’ method formula for relevering beta (Formula 11.8) will result in
the largest increase in levered betas as debt increases, but the relationship between
leverage and the levered beta is linear. In fact, the correct relationship is probably
nonlinear. An example of the relationship between beta (for equity and debt capital)
as debt increases and the costs of financial distress increase is shown in Exhibit 16.8.
This figure depicts the relation between leverage and the beta of a firm’s debt,
equity, and the weighted average beta with tax benefits and costs of financial distress. Leverage is defined as the market value of debt divided by the total market
value of the firm; Bd is the beta of the company’s debt, and BL is the beta of the
firm’s levered equity. The unlevered asset beta is assumed equal to 1.
In Chapter 10, we presented data on the beta of debt. Research has shown that
returns on lower-rated high-yield bonds (e.g., B and Caa) are only minimally
30
Stuart C. Gilson, Edith S. Hotchkiss, and Richard S. Ruback, ‘‘Valuation of Bankrupt
Firms,’’ Review of Financial Studies (Spring 2000): 56. The median is 3%; 4% is the mean,
and the standard deviation is 3.3%.
31
Edward I. Altman, Tushar Kent, and Thongchai Rattanaruengyot, ‘‘Post-Chapter 11 Bankruptcy Performance: Avoiding Chapter 22,’’ Journal of Applied Corporate Finance 21(3)
(Summer 2009): 53–64.
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Distressed Businesses
3
2.5
2
Beta 1.5
1
0.5
0
0
0.2
0.4
Leverage
0.6
Weighted average beta of equity and debt
Bd
BL
EXHIBIT 16.8 Beta as a Function of Leverage
Source: Arthur G. Korteweg, ‘‘The Costs of Financial Distress across Industries,’’ Working
paper, Stanford University, January 15, 2007, 65. Used with permission. All rights reserved.
affected by changes in interest rates on U.S. government bonds but are highly correlated with the returns on common stocks.32 We briefly discussed risky debt in
Chapter 6. The estimation of default and recovery probabilities of debt is important
in estimating the market value of the debt. Whereas equity can, and many times
does, decrease in market value to a point that the equity trades like an option, debt
capital has claims on the business that stand ahead of equity and create a floor (of
varying degrees, depending on the priority of claims of the various debt classes) on
the market values.33
COST OF DISTRESS
Distress reduces the value of the enterprise. Studies have been conducted to understand the magnitude of the decrease in enterprise value that companies that have
fallen into distress have experienced.
In one study, the author quantified the net costs of financial distress in various
industries as the level of debt financing increases.34 He studied both direct and
indirect costs of financial distress because the costs can be substantial even if a firm
never actually files for bankruptcy. The study defined the net cost of financial
32
Frank K. Reilly, David Wright, and James Gentry, ‘‘Historic Changes in the High Yield
Bond Market,’’ Journal of Applied Corporate Finance 21(3) (Summer 2009): 65–79.
33
Edward I. Altman, Brooks Brady, Andrea Resti, and Andrea Sironi, ‘‘The Link between
Default and Recovery Rates,’’ Working paper, September 2003; Edward I. Altman, Andrea
Resti, and Andrea Sironi, ‘‘Default Recovery Rates in Credit Risk Modeling,’’ Working
paper, December 2003; Edward I. Altman, Brenda Karlin, and Louis Kay, ‘‘The Investment
Performance and Market Size of Defaulted Bonds and Bank Loans: 2007 Review and 2008
Outlook,’’ Working paper, February 21, 2008. Available at http://pages.stern.nyu.edu/
~ealtman/2007%20InvestPerf.pdf.
34
Arthur Korteweg, ‘‘The Cost of Financial Distress across Industries,’’ Working paper,
January 15, 2007. Available at http://ssrn.com/abstract=945425.
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distress as the expected present value of lost future cash flows due to financing decisions minus the present value of the interest tax shield.
That author used a sampling of firms for the period 1994 to 2004 and found the
average net cost of financial distress to be 4% of enterprise firm value (across industries). At levels of leverage at the time of default on debt, the net cost of leverage
averaged 13% to 26% of enterprise value at the time of bankruptcy.
In another study, the author estimated that firms are expected to lose, on the
average, 16.3% of their value (value of the business enterprise) in bankruptcy, ranging from 9.9% to 24.7%.35
Two authors used bond ratings to impute the market’s pricing of the costs of
distress. Using the current credit rating and the yield spread among the ratings
allows one to assess the market’s current assessment of the probability of default
and pricing of the costs of distress, rather than relying on historic data. They estimated the net present value of distress costs for different bond ratings (for data for
1985 to 1995) and provide a methodology to quantify the costs of distress today.36
These data can be useful either (1) directly by subtracting a discount in value
for distress from the implied value of the subject firm as if not distressed or (2) by
adding an increment to the cost of capital that reduces the implied value by an
equivalent amount.
SUMMARY
Today’s capital markets environment is making cost of capital estimation for distressed companies particularly challenging. How can one check for the reasonableness of their cost of capital estimates?
One check one can make on cost of equity capital estimates is to fall back on the
classic text, Graham and Dodd.37 Their methodology was based on the yield of the
bonds of the corporation (reflecting the leverage and the company-specific risks
embedded in the credit ratings) plus an average equity premium. More recent
research indicates that this spread goes up as the debt rating decreases. See the discussion of estimating the cost of equity capital based on yield spreads in Chapter 12.
The cost of equity capital should logically exceed the yield investors are expecting on the company’s debt capital (without reducing the yield by any income tax
deductions that might be realized by the subject company). Equity capital is riskier
than debt capital, and the market will price each component based on its relative
risk. In normal times, one would examine the spreads over long-term U.S. government bonds. In this environment, with the yields on U.S. government bonds possibly
artificially low because of the continued flight to quality (see discussion in Chapters
7 and 9), observed spreads are not as meaningful. Rather, one should look at the
35
Craig M. Lewis, ‘‘Firm-Specific Estimates of the Ex Ante Bankruptcy Discount,’’ Working
paper, April 2, 2009. Available at http://ssrn.com/abstract=1372284.
36
Heitor Almeida and Thomas Philippon, ‘‘Estimating Risk-Adjusted Costs of Financial
Distress,’’ Journal of Finance 62(6) (December 2007): 2557–2586; Journal of Applied Corporate Finance 20(4) (Fall 2008): 105–109.
37
Sidney Cottle, Roger F. Murray, and Frank E. Block, Graham & Dodd’s Security Analysis,
5th ed. (New York: McGraw-Hill, 1988).
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Distressed Businesses
345
absolute level of market yield on the company’s debt (market yield for the debt rating on the subject company’s debt level, either actual or target, based on the actual
or synthetic debt rating of the subject company), and the cost of equity capital
should exceed that yield on debt.
Has the cost of equity capital for most companies increased? These authors
believe that the market is highly divided between companies with no or limited
amounts of debt capital in the capital structure and companies with high levels of
debt. If one looks at the absolute yields on highly rated companies, one can conclude
that there probably has been only a small increase in the cost of equity capital and
the WACC for companies with no debt or highly rated debt. However, that is not
true for companies with lower-rated debt. As we work on this chapter, the following
warning that higher than average levels of distress will continue was just published:
Some companies have merely pushed out the maturities on their debt or received covenant amendments instead of restructuring and rightsizing their
balance sheets, say restructuring experts. Therefore, more so than in prior
recessions, the sustainability and strength of the economic recovery will be
a critical determinant of whether speculative-grade companies can survive
in 2010 and beyond.
Leverage overall remains high, Fitch reports, because many defaults in
the past year have been in the form of out-of-court debt exchanges—deals
that offered some debt relief but didn’t reduce debt to the extent that a formal bankruptcy would. Prior Fitch studies show that companies that
declared bankruptcy from 2000 to 2006 emerged with just one-third of
their pre-bankruptcy debt. But after undergoing debt exchanges this year,
many ailing companies are still carrying plenty of debt, ‘‘evidenced by the
fact that most remained rated ‘CCC’ or lower following the exchange,’’
says Fitch.
At the end of November, CCC-rated bonds, which carry substantial
credit risk, still represented 30% of the U.S. high-yield market. More than a
third of that $230 billion in outstanding bonds is associated with companies
that have already done some kind of debt exchange, says Fitch. A reopening
of the bond market for new issuances also rescued many noninvestmentgrade credits that were candidates for bankruptcy. Noninvestment-grade
firms were able to tap the bond market for $186 billion in new issuance as
of the end of November. More than 80% of those dollars went to refinance
existing debt, including some in bank-loan or revolver form.
‘‘A concern going into 2010 is not only the risk of new defaults but also
a heightened risk of serial defaults,’’ says Mariarosa Verde, a managing director at Fitch Credit Market Research. ‘‘If growth proves weak, some of
the debt-restructuring measures adopted over the past year may have only
been successful in helping companies defer rather than avoid bankruptcy.’’
The market seems to be pricing in expectations of further failures. After
falling for most of 2009, credit-default swap (CDS) spreads for noninvestment-grade firms have begun to plateau, as evident in the performance of
the Baird CDS Index, a proprietary index of 36 CDS contracts for the noninvestment-grade debt of nonfinancial companies. While the index dropped
4% in November, it is still six times its base level of January 31, 2006. ‘‘The
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
index is signaling that we’re not out of the woods,’’ says William
Welnhofer, a managing director of investment banking at Robert W. Baird
& Co. ‘‘There just hasn’t been the evidence of an operational turnaround.
There is a feeling that leverage is still pretty high in relation to operating
income. Fundamentally, companies that were overlevered six months ago
are still overlevered.’’38
ADDITIONAL READING
Gartland, Jessica K., and Howard Fielstein. ‘‘Valuation of Distressed Companies.’’ Valuation
Strategies (November–December 2002): 32–37.
Grabowski, Roger J. ‘‘Cost of Capital Estimation in the Current Distressed Environment.’’
Journal of Applied Research in Accounting and Finance 4(1) (2009): 31–40.
Mansi, Sattar A., William F. Maxwell, and Andrew Zhang. ‘‘Bankruptcy Prediction Models
and the Cost of Debt,’’ Working paper, June 8, 2010. Available at http://ssrn.com/
abstract=1622407.
Seago, Eugene W., and Edward J. Schnee. ‘‘Valuing a Bankrupt Corporation’s Net Operating
Loss.’’ Valuation Strategies (September–October 2009): 21–30.
TECHNICAL SUPPLEMENT CHAPTER 7: COST OF CAPITAL
AND THE VALUATION OF WORTHLESS STOCK
In Chapter 7 of the Cost of Capital: Applications and Examples 4th ed. Workbook
and Technical Supplement we present an example of valuing worthless stock.
It includes an example of using the Black-Scholes model in valuing a distressed
business.
38
Vincent Ryan, ‘‘Default Risk to Linger in 2010,’’ CFO.com (December 10, 2009).
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CHAPTER
17
Other Methods of Estimating
the Cost of Equity Capital
Introduction
Fama-French Three-Factor Model
Arbitrage Pricing Theory
Explanation of the APT Model
APT Model Formula
Comparing Models
Market-derived Capital Pricing Model
Yield Spread Model
Implied Cost of Equity Capital
The DCF Method
Residual Income Method
Using Analyst Forecasts
Sources of Information
Summary
Additional Reading
Technical Supplement Appendix I
INTRODUCTION
The pure capital asset pricing model (CAPM) (like the Markowitz’s portfolio model,
from which it was built) provides fundamental insights about risk and return. However, while providing an introduction to fundamental concepts of asset pricing
and portfolio theory, CAPM’s empirical problems probably invalidate pure CAPM’s
use in applications. While simple to understand and easy to apply, pure
CAPM’s empirical record is poor.1 For these reasons, a number of alternative models have been developed to assist practitioners in more accurately estimating the cost
of equity capital. Many of these models are multifactor models instead of the singlefactor pure CAPM.
1
Eugene Fama and Kenneth French, ‘‘The CAPM: Theory and Evidence,’’ Journal of
Economic Perspectives (January 2004): 25–46.
347
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Whether pure CAPM or a multifactor model, all of the models share one common component: They begin with a risk-free rate or return and add one or more
factors, based on the risks of the investment.
As an alternative to using a model, you can also estimate the expected rate of
return implied by the existing price for publicly traded securities. You can use either
a discounted cash flow (DCF) method or a residual income (RI) method to reverseengineer the company’s implied cost of equity capital.
FAMA-FRENCH THREE-FACTOR MODEL
Because of the poor empirical record of pure CAPM, Eugene Fama and Kenneth
French (FF) conducted an empirical study confirming that firm size (as measured by
market capitalization), and the book-value-to-market-value of equity ratio add to
the explanation of realized returns provided by market beta. They found that the
CAPM cost of equity estimates for high-beta stocks were too high and estimates for
low-beta stocks were too low (relative to realized returns). The CAPM cost of equity
estimates for high book-value-to-market-value stocks (so-called value stocks) were
too low, and estimates for low book-value-to-market-value stocks (so-called growth
stocks) were too high (relative to realized returns). The implication of their research
is that if market betas do not suffice to explain expected returns, then the market
portfolio, M, is not efficient, and pure CAPM has potentially fatal problems. As a
result, they introduced an empirically driven model to estimate cost of equity capital
that is not dependent on beta alone.2
Fama and French developed a three-factor model that is empirically driven, not
theoretically based. They tested many factors until they found several that produced
meaningful results. As such, FF considered that investors are not constrained to
behave rationally, a tenet of pure CAPM. The opportunity cost of equity capital
depends on premiums investors require to hold stocks, whether the required market
premiums are based on rational or irrational behavior. The FF three-factor model is
summarized in Formula 17.1.
(Formula 17.1)
EðRi Þ ¼ Rf þ ðBi ERPÞ þ ðsi SMBPÞ þ ðhi HMLPÞ
where:
2
E(Ri) ¼ Expected rate of return on subject security i
Rf ¼ Rate of return on a risk-free security
Bi ¼ Beta of company i
ERP ¼ Equity risk premium
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 smallcap and large-cap portfolios
hi ¼ High-minus-low coefficient in the Fama-French regression
Eugene Fama and Kenneth French, ‘‘The Cross-Section of Expected Stock Returns,’’ Journal
of Finance (June 1992): 427–486.
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Other Methods of Estimating the Cost of Equity Capital
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
Fama and French formed six return series (realized returns for companies in one
of six categories):
Small
Big
Low-Cap
Mid-Cap
High-Cap
where small companies are those with market capitalizations below the median New
York Stock Exchange (NYSE) company, and big companies are those with market
capitalizations above the median NYSE company. Low-cap companies have bookvalue-to-market-value ratios in the bottom 30% of the NYSE companies, mid-cap
companies have book-value-to-market-value ratios in the middle 40% of the NYSE
companies, and high-cap companies have book-value-to-market-value ratios in the
top 30% of the NYSE companies. Because the universe of stocks includes
the NYSE, American Stock Exchange (AMEX), and NASDAQ, the result is that
there are more small-capitalization companies by count than big-capitalization
companies.
Fama and French then calculated average returns for the three portfolios of
small-cap companies and for the three portfolios of big-cap companies. They then
subtracted the average return for big from small to get the small-cap minus big-cap
risk premium.
Then FF calculated average returns for the two portfolios of high-cap
book-value-to-market-value ratio companies and for the two portfolios of lowcap book-value-to-market-value ratio companies. They then subtracted the average
return for high-cap from low-cap to get the high-cap minus low-cap risk premium.
Some people consider this factor to be a measure of financial distress.
Fama and French then ran regressions of historical security returns against the
three time series and calculated the Bi, si SMBP (SMBP, or size risk premium),
and hi HMLP (HMLP risk premium). The Bi is not equivalent to the single-factor
CAPM beta because this is a multiple regression; si is the sensitivity of the subject
stock’s returns to the size, and hi is the sensitivity of the subject stock’s returns to
book-value-to-market-value ratio.
Different interpretations have been given to the book-value-to-market-value ratio. High book-value-to-market-value ratio companies have been termed value
stocks or considered distressed stocks. Low book-value-to-market-value ratio stocks
have been termed growth stocks or considered nondistressed stocks.
One source of the FF factors is the Morningstar Beta Book. Another source of
the three factors is Kenneth French’s web site: http://mba.tuck.dartmouth.edu/
pages/faculty/ken.french/data_library.html.
Generally, using the FF three-factor model results in a large number of companies with high cost of equity compared to the resulting cost of equity from using
pure CAPM. This conclusion leads users to ask whether the FF three-factor model
is overcorrecting for size and/or financial distress or whether pure CAPM is
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systematically underestimating the cost of equity capital. Also be aware that the FF
factors tend to change regularly, leading to regular changes in cost of equity capital
estimates.
ARBITRAGE PRICING THEORY
The concept of the arbitrage pricing theory (APT) was introduced by academicians
in 1976.3 However, it was not until 1988 that data became available in a commercially usable form to permit application of the theory to the estimation of required
rates of return in day-to-day practice. Interestingly, despite the theory’s longevity, it
still is not widely used by practitioners today.
Explanation of the APT Model
As noted in Chapter 8, the pure CAPM is a univariate model; that is, pure CAPM
recognizes only one risk factor: systematic risk relative to a market index. In a sense,
APT is a multivariate extension of the pure CAPM. It recognizes a variety of risk
factors that may bear pervasively on an investment’s required rate of return, one of
which may be a CAPM-type market or market timing risk. It may be argued that the
pure CAPM and APT are not mutually exclusive, nor is one of greater or lesser
scope than the other. It also may be argued that the pure CAPM beta implicitly
reflects the information included separately in each of the APT factors.
However, in spite of its more limited use, most academicians consider the arbitrage pricing theory model richer in its information content and explanatory and
predictive power.4
While pure CAPM is a single regression model, APT is a multiple regression
model. In the APT model, the cost of equity capital for an investment varies according to that investment’s sensitivity to each of several different risk factors.
The theoretical model itself does not specify what the risk factors are. Most formulations of the APT theory consider only risk factors of a pervasive macroeconomic nature, such as:
&
&
3
Yield spread. The differential between risky and less risky bonds as a measure of
investors’ consensus confidence in economic prosperity
Interest rate risk. Measured by the difference between long-term and short-term
U.S. government bond yields
Stephen A. Ross, ‘‘The Arbitrage Theory of Capital Asset Pricing,’’ Journal of Economic
Theory (December 1976): 241–260; and Stephen A. Ross, ‘‘Return, Risk, and Arbitrage,’’ in
Risk and Return in Finance, Irwin I. Friend and I. Bisksler, eds., (Cambridge, MA: Ballinger,
1977), 189–218. See also Stephen A. Ross, Randolph W. Westerfield, and Jeffrey F. Jaffe,
Corporate Finance, 8th ed. (Burr Ridge, IL: McGraw-Hill, 2006).
4
See, for example, Richard Roll and Stephen A. Ross, ‘‘An Empirical Investigation of Arbitrage Pricing Theory,’’ Journal of Finance (December 1980): 1073–1103; Nai-fu Chen,
‘‘Some Empirical Tests of Arbitrage Pricing,’’ Journal of Finance (December 1983): 1393–
1414; Nai-fu Chen, Richard Roll, and Stephen A. Ross, ‘‘Economic Forces and the Stock
Market: Testing the APT and Alternative Pricing Theories,’’ Journal of Business 59 (1986):
383–403.
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Other Methods of Estimating the Cost of Equity Capital
&
&
351
Business outlook risk. Measured by changes in forecasts for economic variables
such as gross national product (GNP)
Inflation risk. Measured by changes in inflation forecasts
The beta measuring market risk may or may not be one of the risk factors included in any particular practitioner’s version of the APT. In some versions, more
industry-specific factors may be included, such as changes in oil prices. Exhibit 17.1
explains one version of APT risk factors.
EXHIBIT 17.1 Explanation of APT Risk Factors
CONFIDENCE RISK
Confidence risk is the unanticipated changes in investors’ willingness to undertake relatively
risky investments. It is measured as the difference between the rate of return on relatively risky
corporate bonds and the rate of return on government bonds, both with 20-year maturities,
adjusted so that the mean of the difference is zero over a long historical sample period. In any
month when the return on corporate bonds exceeds the return on government bonds by more
than the long-run average, this measure of confidence risk is positive. The intuition is that a
positive return difference reflects increased investor confidence because the required yield on
risky corporate bonds has fallen relative to safe government bonds. Stocks that are positively
exposed to the risk then will rise in price. (Most equities do have a positive exposure to
confidence risk, and small stocks generally have greater exposure than large stocks.)
TIME HORIZON RISK
Time horizon risk is the unanticipated changes in investors’ desired time to payouts. It is
measured as the difference between the return on 20-year government bonds and 30-day
Treasury bills, again adjusted to be mean zero over a long historical sample period. A positive
realization of time horizon risk means that the price of long-term bonds has risen relative to
the 30-day Treasury bill price. This is a signal that investors require a lower compensation for
holding investments with relatively longer times to payouts. The price of stocks that are
positively exposed to time horizon risk will rise to appropriately decrease their yields.
(Growth stocks benefit more than income stocks when this occurs.)
INFLATION RISK
Inflation risk is a combination of the unexpected components of short- and long-run inflation
rates. Expected future inflation rates are computed at the beginning of each period from
available information: historical inflation rates, interest rates, and other economic variables
that influence inflation. For any month, inflation risk is the unexpected surprise that is
computed at the end of the month (i.e., it is the difference between the actual inflation for that
month and what had been expected at the beginning of the month). Since most stocks have
negative exposures to inflation risk, a positive inflation surprise causes a negative contribution
to return, whereas a negative inflation surprise (a deflation shock) contributes positively
toward return.
Industries whose products tend to be ‘‘luxuries’’ are most sensitive to inflation risk.
Consumer demand for luxuries plummets when real income is eroded through inflation, thus
depressing profits for industries such as retailers, services, eating places, hotels and motels,
and toys. In contrast, industries least sensitive to inflation risk tend to sell ‘‘necessities,’’ the
demands for which are relatively insensitive to declines in real income. Examples include
foods, cosmetics, tire and rubber goods, and shoes. Also companies that have large asset
holdings such as real estate or oil reserves may benefit from increased inflation.
(continued )
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EXHIBIT 17.1 (Continued )
BUSINESS CYCLE RISK
Business cycle risk represents unanticipated changes in the level of real business activity. The
expected values of a business activity index are computed both at the beginning and end of the
month, using only information available at those times. Then business cycle risk is calculated
as the difference between the end-of-month value and the beginning-of-month value. A
positive realization of business cycle risk indicates that the expected growth rate of the
economy, measured in constant dollars, has increased. Under such circumstances firms that
are more positively exposed to business cycle risk—for example, firms such as retail stores
that do well when business activity increases as the economy recovers from a recession—will
outperform those such as utility companies that do not respond much to increased levels in
business activity.
MARKET TIMING RISK
Market timing risk is computed as that part of the S&P 500 total return that is not explained
by the first four macroeconomic risks and an intercept term. Many people find it useful to
think of the APT as a generalization of the CAPM, and by including this Market Timing
factor, the CAPM becomes a special case: If the risk exposures to all of the first four
macroeconomic factors were exactly zero, then market timing risk would be proportional to
the S&P 500 total return. Under these extremely unlikely conditions, a stock’s exposure to
market timing risk would be equal to its CAPM beta. Almost all stocks have a positive
exposure to market timing risk, and hence positive market timing surprises increase returns,
and vice versa.
A natural question, then, is: Do confidence risk, time horizon risk, inflation risk, and
business cycle risk help to explain stock returns better than I could do with just the S&P 500?
This question has been answered using rigorous statistical tests, and the answer is very clearly
that they do.
Source: Presented in a talk based on a paper, ‘‘A Practitioner’s Guide to Arbitrage Pricing
Theory,’’ by Edwin Burmeister, Richard Roll, and Stephen A. Ross, written for the Research
Foundation of the Institute of Chartered Financial Analysts, 1994. The exhibit is drawn from
notes for ‘‘Controlling Risks Using Arbitrage Pricing Techniques,’’ by Edwin Burmeister.
Reprinted with permission.
APT Model Formula
The econometric estimation of the APT model with multiple risk factors yields this
formula:
(Formula 17.2)
EðRi Þ ¼ Rf þ ðBi1 RP1 Þ þ ðBi2 RP2 Þ þ . . . þ ðBin RPn Þ
where:
E(Ri) ¼ Expected rate of return on the subject security
Rf ¼ Rate of return on a risk-free security
RP1 . . . RPn ¼ Risk premium associated with risk factor 1 through n for the
average asset in the market
Bi l . . . Bin ¼ Sensitivity of security i to each risk factor relative to the market average sensitivity to that factor
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353
Roger Ibbotson and Gary Brinson make these observations regarding APT:
In theory, a specific asset has some number of units of each risk; those units
are each multiplied by the appropriate risk premium. Thus, APT shows that
the equilibrium expected return is the risk-free rate plus the sum of a series
of risk premiums. APT is more realistic than CAPM because investors can
consider other characteristics besides the beta of assets as they select their
investment portfolios.5
Edwin Burmeister says this about APT:
The APT takes the view that there need not be any single way to measure
systematic risk. While the APT is completely general and does not specify
exactly what the systematic risks are, or even how many such risks exist,
academic and commercial research suggests that there are several primary
sources of risk which consistently impact stock returns. These risks arise
from unanticipated changes in the following fundamental economic
variables:
&
&
&
&
&
Investor confidence
Interest rates
Inflation
Real business activity
A market index
Every stock and portfolio has sensitivity (or betas) with respect to each
of these systematic risks. The pattern of economic betas for a stock or portfolio is called its risk exposure profile. Risk exposures are rewarded in the
market with additional expected return, and thus the risk exposure profile
determines the volatility and performance of a well-diversified portfolio.
The profile also indicates how a stock or portfolio will perform under different economic conditions. For example, if real business activity is greater
than anticipated, stocks with a high exposure to business activity, such as
retail stores, will do relatively better than those with low exposures to business activity, such as utility companies.6
Research has shown that the cost of equity capital as estimated by the APT tends
to be higher for some industries and lower for others than the cost of equity capital
using the CAPM. Early research also suggested that the multivariate APT model
explains expected rates of return better than does the univariate CAPM.7
5
Roger G. Ibbotson and Gary P. Brinson, Investment Markets (New York: McGraw-Hill,
1987), 32. For a more extensive discussion of APT, see Frank K. Reilly, Investment Analysis
and Portfolio Management, 8th ed. (Fort Worth, TX: Dryden Press, 2005).
6
Edwin Burmeister, ‘‘Using Macroeconomic Factors to Control Portfolio Risk,’’ Working
paper, Duke University, March 9, 2003, 3. Available at http://web.econ.unito.it/nicodano/
roll_ross_apt_portfolio_management.pdf.
7
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), 317.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
So, if the APT is more powerful than the pure CAPM, why is the APT not used
more? For one thing, the variables are not specified. Also, there is no universal consensus about which variables are likely to have the greatest efficacy. Furthermore,
implementing a model based on APT is complicated in that coefficients for several
factors, rather than just one factor, must be worked out for each company for each
specific time it is going to be applied.
BIRR Portfolio Analysis, Inc. is a source for inputs to their version of the
APT model. The BIRR Risk Index is a single overall risk measure for a stock or
portfolio; one can think of this risk index as a multifactor equivalent to the
single-factor CAPM beta. Exhibit 17.2 shows an example of comparison of the
exposures of the Standard & Poor’s (S&P) 500 and Nike to the risk factors presented in Exhibit 17.1. Contact information for sources of information is given in
Appendix II.
COMPARING MODELS
Researchers have studied the relative predictive power of the various models. In one
such study, the author investigated the ability of models to capture time-varying predictability of returns. He studied the pure CAPM, FF three-factor model, and a fivefactor economic model (e.g., an APT-type model). At the industry level, he found
that the pure CAPM is best in capturing time-variation of industry expected returns
while the FF three-factor model was the worst. His five-factor economic model best
captures effect of size (as measured by market capitalization) and book-value-tomarket-value ratio on the returns of portfolios.8
In another study, the authors found that the degree of financial leverage explains
the book-to-market effect observed by FF.9
Two authors in another study expanded the FF three-factor model to include
measures of company asset liquidity and proxies for business risk (volatility measures of assets’ profitability). They found that the expanded FF three-factor model
can better explain stock returns.10
In another study, the authors tested various forms of the FF three-factor model,
including adding yield spreads on bonds as an added variable explaining realized
stock returns. For the period studied, 1973 to 1998, they found that the market,
size, and book-value-to-market-value factors of the FF model were priced by the
market and that adding yield spreads increased the ability of the model to explain
differences in returns among stocks.11
8
Alex P. Taylor, ‘‘Conditional Factor Models and Return Predictability,’’ AFA 2006 Boston
Meetings Paper, February 2005.
9
Lorenzo Garlappi and Hong Yan, ‘‘Financial Distress and the Cross Section of Equity
Returns,’’ AFA 2008 New Orleans Meetings Paper, September 29, 2008. Available at http://
ssrn.com/abstract=970644.
10
Antonio Camara and Ali Nejadmalayeri, ‘‘Asset Liquidity, Business Risk and Beta,’’ Working paper, June 14, 2009. Available at http://ssrn.com/abstract=1360049.
11
Murillo Camello, Long Chen, and Lu Zhang, ‘‘Expected Returns, Yield Spreads and Asset
Pricing Tests,’’ Review of Financial Studies (21) 2007: 1297–1338.
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Other Methods of Estimating the Cost of Equity Capital
EXHIBIT 17.2 Example of BIRR Risk Index for S&P 500 and Nike as of September 2009
The risk exposure profile for the S&P 500 and the corresponding prices of risk (the risk premiums) are:
Risk Factor
Exposure for S&P 500
Price of Risk (%/yr)
0.9208
0.4516
0.1340
2.2557
1.00
0.5982
3.9849
0.2389
0.2759
3.0312
Confidence Risk
Time Horizon Risk
Inflation Risk
Business Cycle Risk
Market Timing Risk
For each risk factor, the contribution to expected return is the product of the risk exposure
and the corresponding price of risk, and the sum of the products is equal to the expected
return in excess of the 30-day Treasury bill rate:
Exposure
for S&P 500
Risk Factor
Confidence Risk
Time Horizon Risk
Inflation Risk
Business Cycle Risk
Market Timing Risk
0.9208
0.4516
0.1340
2.2557
1.00
Price of Risk
(%/yr)
0.5982
3.9849
0.2389
0.2759
3.0312
Contribution of Risk
Factor to Expected Return
¼
¼
¼
¼
¼
0.5509
1.7995
0.032
0.6224
3.0312
Sum ¼ Expected Excess Return for the S&P 500 ¼ 6.04%
The price of each risk factor tells you how much expected return will change due to an increase or decrease in your portfolio’s exposure to that type of risk. The risk exposure profile
for Nike is:
Risk Factor
Confidence Risk
Time Horizon Risk
Inflation Risk
Business Cycle Risk
Market Timing Risk
Exposure for Nike
Exposure for S&P 500
0.2635
0.8371
0.1325
0.4241
1.2108
0.9208
0.4516
0.1340
2.2557
1.00
These exposures give rise to an expected excess rate of return for Nike equal to 7.25%/yr
compared with the 6.04%/yr that is computed for the S&P 500.
Source: Edwin Burmeister, ‘‘Using Macroeconomic Factors to Control Portfolio Risk,’’
Working paper, Duke University, March 9, 2003. (This paper is based on an early version
of ‘‘A Practitioner’s Guide to Arbitrage Pricing Theory,’’ in A Practitioner’s Guide to
Factor Models (Research Foundation of the Institute of Chartered Financial Analysts,
1994)). Source for BIRR update: BIRR Risk and Returns Analyzer, July 2006 release
updated to September 2009.
Are the FF model factors simply proxies for more commonly used measures
of risk? One study found that a company’s sensitivity to the book-valueto-market-value factor is related to the degree of operating leverage. They
also reported that a company’s sensitivity to the size factor is related to its
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
degree of financial leverage (though not as strongly as the operating leverage
relationship).12
Other authors proposed alternative models to the FF three-factor model. For
example, in one paper, the authors proposed that the model consist of market risk,
amount of investment to assets, and the return on assets. They compared the results
of employing their model to the FF three-factor model and found that it better
explains realized returns.13
Exhibit 17.3 displays a comparison of the cost of equity capital obtained by
using the pure single-factor CAPM, the FF three-factor model, and the BIRR version
of APT.
MARKET-DERIVED CAPITAL PRICING MODEL
In cases when a company has troubled operations and is declining (negative returns)
while the market returns are stable or rising, any of the risk measures that rely on
comparing the historical realized returns of the subject company to the market
returns will probably give measures that are unrepresentative of the company’s true
risk. For example, one could get a very low beta indicating low risk when in fact the
subject company’s cost of equity capital should reflect high risk. What does one do?
One can certainly turn to the fundamental risk studies, such as the Duff &
Phelps Risk Study, and use the build-up method to estimate the cost of equity capital.
Alternatively, if the subject company is a publicly traded company with publicly
traded options, one can extract implied variance statistics from publicly traded
options and develop a risk measure that is current and forward-looking rather than
based on historical returns. One such model is the market-derived capital pricing
model.14
The technique can be summarized in five steps for a public company with traded
options.
Step 1: Calculate the forward break-even price, Pn. This price represents the
minimum amount equity investors must be compensated, knowing that
the return on stock must be greater than return on bonds of the subject company and the current bond yield reflects the specific-company risk of the
company. The expected return of the subject company stock equals
the expected return due to dividends plus the expected return resulting from
capital gains:
EðRi Þ ¼ EðRdiv Þ þ E Rcap gains
12
Luis Garcia-Feijoo and Randy Jorgensen, ‘‘Can Operating Leverage Be the Cause of the
Value Premium?’’ Working paper, December 2007. Available at http://ssrn.com/
abstract=1077739.
13
Long Chen and Lu Shang, ‘‘A Better Three-Factor Model That Explains More Anomalies,’’
Working paper, June 2009, forthcoming in the Journal of Finance.
14
James McNulty, Tony Yeh, William Schulze, and Michael Labatkin, ‘‘What’s Your Real
Cost of Capital?’’ Harvard Business Review (October 2002): 114.
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Other Methods of Estimating the Cost of Equity Capital
EXHIBIT 17.3 Comparative Cost of Equity Capital Models: CAPM versus FF versus APT
Comparative Equity Return Models as of September 30, 2009
CAPM
F-F 3BIRR
Company
þSP1 CAPM Factor
APT
CAPM F-F
Beta Beta
F-F
SMB
F-F
HML
Systems Software
Microsoft (MSFT)
Novell (NOVL)
Symantec (SYMC)
8.95%
10.72%
9.82%
6.12%
4.98%
6.67%
0.96
0.92
0.94
1.07 0.80% 1.33%
1.32 2.14% 1.91%
0.62 3.52% 0.50%
Application Software
Adobe Systems (ADBE)
Autodesk (ADSK)
JDA Software (JDAS)
14.72% 14.10% 15.31% 6.40%
17.20% 16.46% 16.39% 12.47%
14.91% 12.56% 13.34% 12.21%
1.83
2.26
1.55
1.76 2.48% 0.87%
2.23 0.34% 0.43%
1.58 2.21% 1.59%
9.31% 9.39%
9.09% 7.24%
9.20% 11.46%
Healthcare Equipment
CR Bard (BCR)
5.81%
Zimmer Holdings (ZMH) 10.54%
Stryker (SYK)
9.88%
Varian Medical (VAR)
9.01%
5.19% 4.40% 4.37%
9.92% 11.50% 10.64%
9.26% 10.31% 5.58%
8.27% 9.89% 4.12%
0.21
1.07
0.95
0.77
0.3 1.26% 0.02%
0.65 1.39% 2.50%
0.84 2.07% 0.41
0.58 2.68% 0.01
Integrated Petroleum
Chevron
Exxon Mobil
Occidental Petroleum
7.62%
6.56%
9.75%
Apparel Retail
Ross Stores (ROST)
Abercrombie & Fitch
(ANF)
Gap (GPS)
6.32%
5.39%
9.03%
8.50%
6.54%
5.03%
0.65
0.46
1.04
0.97 1.18% 1.87%
0.73 1.27% 1.39%
1.34 0.21% 2.58%
9.01% 8.27% 9.68%
13.03% 12.06% 12.21%
7.12%
9.01%
0.77
1.46
0.53
1.44
1.96%
0.20%
0.77%
0.06%
1.2
0.98
0.28%
1.61%
0.66
1.19
1.48
0.39 1.42% 1.33%
0.98 1.10% 1.02%
1.53 0.48% 0.10%
7.25%
6.20%
9.39%
11.25% 10.63% 11.31% 14.19%
Aerospace and Defense
Raytheon (RTN)
7.30% 7.66% 8.93% 10.32%
General Dynamics (GD)
10.22% 10.58% 11.54% 9.89%
Precision Castparts (PCP) 12.79% 12.17% 11.87% 15.91%
1
Size Premiums from SBBI 2009 Valuation Yearbook, Table 7.5, p. 94.
Source: Source for CAPM betas and three-factor premiums: Ibbotson Beta Book, First 2009
Edition. Copyright # 2009 Morningstar Inc. Source for BIRR estimate: BIRR Risk and
Returns Analyzer, July 2006 release. Updated to September 2009. Source for Small Stock
Premium (SSP): 2009 Stocks, Bonds, Bills, and Inflation1 Valuation Yearbook. Copyright #
2009 Morningstar, Inc. Assumptions used in CAPM and three-factor calculations: Rf ¼
4.04% (20-year Treasury security yield on September 30, 2009), ERP ¼ 5.5%. Calculations
by Duff & Phelps LLC. Used with permission. All rights reserved.
The minimum return resulting from capital gains must be the yield on the subject company debt minus the expected return on the stock due to dividends:
Minimum E Rcap gains ¼ kd EðRdiv Þ ¼ company bond yield ðD=PÞ
The price the subject company stock must reach at the end of the period n to
earn the minimum rate of return is represented by Pn. That is, the stock
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
price must at a minimum increase from P0 to Pn to earn a compound rate of
return equal to (kd D/P) in each of n periods:
Pn ¼ P0 ð1 þ kd D=P0 Þn
E(Ri) ¼ Expected rate of return on equity i
E(Rdiv) ¼ Expected equity rate of return on dividend
E(Rcapgains) ¼ Expected equity rate of return on capital gains
kd ¼ Discount rate for debt (net of tax affect, if any)
D/P0 ¼ Dividend yield on stock
Pn ¼ Stock price in period n
P0 ¼ Stock price at valuation period
where:
Step 2: Estimate the stock’s future volatility (s) using an option pricing model
like the Black-Scholes option pricing model to solve for the implied
volatility.
Step 3: Calculate the cost of downside insurance using Black-Scholes option
pricing model, and calculate the value of a theoretical put option with the
strike price equal to the forward break-even price Pn (from step 1) and implied volatility s (from step 2) and the cost of funds (corporate interest
rate).
Step 4: Derive the annualized excess equity return; that is, divide the theoretical
put option price (step 3) by P0 and convert it to an annual percentage rate
following an annuity formula. This gives the rate of excess return required
on the company’s shares.
Step 5: Add the excess return to the corporate bond yield to derive E(Ri).
The market-derived capital pricing model is responsive to changes in the stock
market’s pricing of the subject company, as news is reflected in the stock prices and
implied volatility of the option pricing.
YIELD SPREAD MODEL
The authors of one paper estimated the expected return on debt and equity
based on yield spreads. To do this, one looks at the differences in market yields
on bonds of different ratings. Using historical default rates on bonds, one can
estimate expected default rates on bonds and, from that, a firm’s current cost of
debt for use in its cost of capital. One can estimate a market consensus equity
risk premium using these debt ratings (i.e., the equity risk premium for specific
ratings classes based on differences in leverage), which can be used to estimate a
firm-specific cost of equity, given the subject company’s debt rating. The data
on which the authors built their analyses were drawn from 1994 to 1999. The
authors estimated equity risk premiums ranging from 3.1% for AA-rated firms
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359
to 8.5% for B-rated companies over U.S. government bonds of comparable
duration.15
As we discussed in Chapter 12, these results can be used as a reasonableness
check for the cost of equity capital.
IMPLIED COST OF EQUITY CAPITAL
There are at least two methods of estimating the cost of equity capital implied using
a company’s current stock price: the discounted cash flow (DCF) method and the
residual income (RI) method. These methods of estimating the cost of equity capital,
like the market-derived capital pricing model, are responsive to changes in the stock
market’s pricing of the subject company as news is reflected in the stock prices. They
provide direct evidence of the returns that actual investors require for a public company stock on a given date.16
The DCF Method
Implying the cost of equity capital using the DCF method is equivalent to applying the
DCF method in reverse. This implies that the current stock price is equal to the expected
future returns discounted to a present value at a discount rate that represents the equity
cost of capital for the company. Since the present value (i.e., the current stock price) is
known, the calculations are configured to solve for ke, the cost of equity capital.
Two main types of models are used to implement the DCF method as it is
applied to estimating cost of equity capital. The first, and most popular, is the
single-stage model. The second, and most accurate (in most instances), is the multistage model. Although these models can be used to imply the weighted average cost
of capital, they typically are used to estimate the cost of equity capital. The discussion that follows is based on equity rates of return only.
Single-Stage DCF Model The single-stage DCF model is based on a rewrite (an algebraic manipulation) of a constant growth model, such as the Gordon Growth
Model, presented earlier as Formula 4.6 and repeated here:
(Formula 17.3)
PV ¼
15
NCF0 ð1 þ gÞ
ke g
Ian Cooper and Sergei Davydenko, ‘‘Using Yield Spreads to Estimate Expected Returns on
Debt and Equity,’’ Working paper, February 2003. Available at http://ssrn.com/
abstract=387380. In another paper, Harjoat Bhamra, Lars-Alexander Kuehn, and llya Strebuaev, ‘‘The Levered Equity Risk Premium and Credit Spreads: A Unified Framework,’’
Working paper, July 18, 2007, study the substantial empirical evidence that stock returns
can be predicted by credit spreads, and movement in stock-return volatility can explain
movements in credit spreads and explore the joint pricing of corporate bonds and stocks.
Available at http://ssrn.com/abstract=1016891.
16
Perry Ukren, ‘‘Estimating Discount Rates: An Alternative to the CAPM,’’ Valuation Strategies (March–April 2005): 10–15.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
where:
PV ¼ Present value
NCF0 ¼ Net cash flow in period 0, the period immediately preceding
the valuation date
ke ¼ Cost of equity capital (discount rate)
g ¼ Expected long-term sustainable growth rate in net cash
flow to investor
When the present value (i.e., the market price) is known, but the discount rate
(i.e., the cost of capital) is unknown, Formula 17.3 can be rearranged to solve for the
cost of capital:
(Formula 17.4)
NCF0 ð1 þ gÞ
þg
PV
where the variables have the same definitions as in Formula 17.3.
In publicly traded companies, the net cash flow that the investor actually
receives is the dividend. We can substitute some numbers into Formula 17.3 and
thus illustrate estimating the cost of equity capital for Alpha Utilities, Inc. (AUI), an
electric, gas, and water utility conglomerate, by making these three assumptions:
ke ¼
1. Dividend. AUI’s dividend for the latest 12 months was $3.00 per share.
2. Growth. Analysts’ consensus estimate is that the long-term growth in AUI’s dividend will be 5%.
3. Present value. AUI’s current stock price is $36.00 per share.
Substituting this information into Formula 17.4, we have:
(Formula 17.5)
ke ¼
$3:00ð1 þ 0:05Þ
þ 0:05
$36:00
$3:15
þ 0:05
$36:00
¼ 0:088 þ 0:05
¼
¼ 13:8%
Thus, according to this computation, AUI’s cost of equity capital is estimated to
be 13.8% (8.8% dividend yield plus 5.0% expected stock price increase).
The preceding is the formulation used in the Morningstar Cost of Capital Yearbook, ‘‘Analysts Single-Stage Discounted Cash Flow’’ cost of equity capital estimate.
The source of the Morningstar growth estimates is the I/B/E/S database (now
Thomson Financial) of long-term growth rate estimates. A number of other sources
of growth estimates are included in Appendix II. This single-stage DCF model often
is used in utility rate hearings to estimate a utility’s cost of equity capital.17
17
For a concise discussion of the use of this model for utility rate-setting, see Richard A.
Brealey, Stewart C. Myers, and Franklin Allen, Principles of Corporate Finance, 8th ed.
(Boston: Irwin McGraw-Hill, 2006), 67–68.
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361
Like the capitalization shortcut version of the discounting model used for valuation, the single-stage DCF model for estimating cost of capital is deceptively simple.
In utility settings, the dividend yield is assumed to be an appropriate estimate of
the first input, cash flow yield. This is reasonable, because publicly traded utilities
typically pay dividends, and these dividends represent a high percentage of available
cash flows. In cases where the utility’s dividend yield is abnormally high or low, a
normal dividend yield is used. It is difficult, however, to use dividend yields with all
publicly traded companies.
For many companies, dividend payments may be unrelated to the level of or
growth in earnings or cash flows. A large number of companies do not pay dividends
or pay only a token amount. In these cases, theoretically, the growth component, g,
will be larger than that of an otherwise similar company that pays higher dividends.
In practice, properly adjusting for this lack of dividends is extremely difficult.
One way to avoid the dividend issue is to define cash flows more broadly.
Instead of considering only the cash flows investors actually receive (dividends), the
analyst might define net cash flows as those amounts that could be paid to equity
investors without impeding a company’s future growth. As noted in Chapter 3, net
cash flow to equity, NCFe, is usually defined as in Formula 3.1.
Of course, these cash flows are not those paid to investors, but, presumably, investors ultimately realize the benefit of these amounts through higher future dividends, a special dividend, or, more likely, stock price appreciation. Some analysts
assume that over the very long run, net (after-tax) income should be quite close to
cash flows. Therefore, they assume that net income can be used as a proxy for net
cash flow. This assumption should be questioned on a case-by-case basis. For a growing company, capital expenditure and working capital requirements may make the
assumed equivalence of net income and net cash flow so remote as to be irrelevant.
The other, and perhaps more problematic, input is the expected growth rate.
An important characteristic of the growth rate is that it is the perpetual annual
growth rate. Future growth rates do not have to be the same for every year; however, the average rate should be equal to this perpetual rate. For example, if a company is expected to grow at 10% per year for the next four years and 3% per year
thereafter, then the average growth rate into perpetuity could be estimated as about
5%. If the company is expected to grow by 10% per year for the next 20 years and
3% per year thereafter, the average growth rate is probably closer to 9%. However,
this would be an extreme case. It is theoretically impossible for the sustainable perpetual growth rate for a company to significantly exceed the growth rate in the
economy. Any rate over a 6% to 7% perpetual growth rate should be questioned
carefully.
Multistage DCF Models Multistage models come closer to reversing the discounting
process than do single-stage models that simply reverse the capitalization process.
However, they involve additional variables that require estimation. Multistage models do not incorporate specific expected return amounts for specific years, but they
do incorporate different growth rates for different expected growth stages, most often three stages.
Multistage models have one main advantage over single-stage models: Using
more than one growth rate reduces reliance on a single such rate. Furthermore, it is
unnecessary to compute a blended growth rate.
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The main disadvantage of a multistage model is its computational complexity
relative to the single-stage model. Unlike the single-stage model, the multistage
model must be solved iteratively.
It also differs from the single-stage model in that there is no single form of the
multistage model. Two main factors determine the form of the model:
1. Number of growth stages. Usually there are either two or three growth
stages.
2. Length of each stage. Usually each stage is between three and five years long.
In a three-stage model, the discounting formula that must be reversed to solve
for k, the cost of capital, looks like this:
(Formula 17.6)
PV ¼
h
i
5
NCF0 ð1 þ g1 Þn
X
n¼1
where:
ð1 þ ke Þn
þ
h
i
n5
10 NCF 5 ð1 þ g Þ
X
2
n¼6
ð1 þ ke Þn
þ
NCF10 ð1 þ g3 Þ
ke g3
ð1 þ ke Þ10
NCF0 ¼ Net cash flow (or dividend) in the immediately preceding
year
NCF5 ¼ Expected net cash flow (or dividend) in the fifth year
NCF10 ¼ Expected net cash flow (or dividend) in the tenth year
g1, g2, and g3 ¼ Expected growth rates in NCF (or dividends) through each
of stages 1, 2, and 3, respectively
ke ¼ Cost of equity capital (discount rate)
These stages can be formed in three-year increments, five-year increments, or
increments of any number of years. Also, the length of the second stage can differ
from the length of the first stage.
As noted earlier, this equation must be solved iteratively for k. Fortunately,
many spreadsheet software packages, such as Microsoft Excel, can perform this calculation. Morningstar, for example, in its Cost of Capital Yearbook, uses two 5-year
stages and then a growth rate applicable to earnings over all future years after the
first 10 years. In the first and second stages, Morningstar uses estimated cash flows
instead of dividends. It defines cash flows for this purpose as net income plus noncash charges less capital expenditures. This definition comes close to our definition
of net cash flow to equity, except that it does not subtract additions to working capital or adjust for changes in outstanding debt principal. Morningstar’s third-stage
(long-term) growth rate is the expected long-term inflation forecast plus the historical real gross domestic product (GDP) growth rate.
Residual Income Method
Like the DCF method, the RI method (or the related abnormal earnings growth
[AEG] method) is considered by some to be more direct and simpler than the buildup model or the CAPM for a public company.
The single-stage residual income model is based on a rewrite of a constant
growth model presented in Formula 4.20.
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363
When the present value (i.e., the market price) is known, but the discount rate
(i.e., the cost of capital) is unknown, Formula 4.20 can be rearranged to solve for the
cost of capital, as shown:
(Formula 17.7)
ke ¼ g þ ½ðPV 0 BV 0 Þ=PV 0 þ RIe;1 =PV 0
PV 0 BV 0
RIe;1
þ
ke ¼ g þ
PV 0
PV 0
where: PV0 ¼ Present value
BV0 ¼ Book value (net asset value) for period 0, the period immediately preceding the valuation date
RIe,1 ¼ Residual income to common equity capital for period 1
ke ¼ Cost of equity capital
g ¼ Expected long-term sustainable growth rate in net cash flow to common equity investors
Similarly the AEG model is a rewrite of a constant growth model presented as
Formula 4.22 and can be similarly rearranged to solve for the cost of capital.
Peter Easton has used the concepts of the AEG model to improve on the popular
PEG ratio.18 The PEG ratio (the price-to-earnings ratio divided by the short-term
earnings growth rate) has become a fairly widely used means of combining prices
and forecasts of earnings and earnings growth into a ratio. Advocates of the PEG
ratio hold that it takes into account differences in short-run earnings growth. The
author developed a methodology that simultaneously estimates the expected rate of
return and the rate of change in abnormal earnings growth in earnings beyond the
short forecast horizon. He then demonstrated the use of the methodology with prices
and analysts’ short-term earnings forecasts for years 1981 to 1995 and estimated the
implied cost of capital and the long-run change in abnormal earnings growth.
USING ANALYST FORECASTS
A common approach to deriving a perpetual growth rate is to obtain stock analysts’
estimates of earnings growth rates. The advantage of using these growth estimates is
that they are prepared by people who follow these companies on an ongoing basis.
These professional stock analysts develop a great deal more insight into these companies than a casual investor or a valuation analyst not specializing in the industry is
likely to achieve.
There are, however, four caveats when using this information:
1. These earnings growth estimates typically are for only the next two to five years;
they are not perpetual. Therefore, any use of these forecasts in a single-stage
DCF model must be tempered with a longer-term forecast.
18
Peter Easton, ‘‘PE Ratios, PEG Ratios, and Estimating the Implied Expected Rate of Return
on Equity Capital,’’ Accounting Review 79 (2004): 73–96.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
2. Most published analysts’ estimates come from sell-side stock analysts who work
for firms that are in business to sell publicly traded equities. Thus, although their
earnings forecasts fall within the range of reasonable possibilities, they may be
at the high end of the range. Furthermore, they rarely publicize negative earnings estimates.
3. Usually these estimates are obtained from firms that provide consensus earnings
forecasts; that is, they aggregate forecasts from a number of analysts and report
certain summary statistics (mean, median, etc.) on these forecasts. For a small
publicly traded company, there may be only one or even no analyst following
the company. As is discussed in Chapters 14 and 15, the potential for forecasting errors is greater when the forecasts are obtained from a very small number of
analysts. These services typically report the number of analysts who have provided earnings estimates, which should be considered in determining how much
reliance to place on forecasts of this type.
4. The analysts’ estimates are typically denominated as net income. Any significant
variations between net income and net cash flow require adjustment.
The issues surrounding using analyst forecasts just described are equally applicable if you use the DCF method or the RI method to estimate the cost of equity
capital.
Several authors have published studies on the usefulness and bias of analysts’ forecasts. For example, one study ‘‘examines the ability of na€ve investor expectations models to explain the higher returns to contrarian investment strategies.’’ The authors:
find no systematic evidence that stock prices reflect naive extrapolation of
past trends in earnings or sales growth. . . . [H]owever, we find that stock
prices appear to naively reflect analysts’ biased forecasts of future earnings
growth. Further, we find that naı̈ve reliance on analysts’ forecast of future
earnings growth can explain over half of the higher returns to contrarian
investment strategies.
[T]he evidence suggests that stock prices naively incorporate analysts’
forecasts of long-term earnings growth. In particular, our results indicate that
earnings tend to grow at less than half the rate predicted by analysts, but that
stock prices initially reflect substantially all of the forecast earnings growth.19
In another study, the authors tested the relationship of accounting information and
firm value based on a residual income model (book value plus present value of future
residual income which is, income in excess of the cost of capital) using analyst earnings
forecasts (I/B/E/S consensus earnings forecasts to proxy for market expectations of
future earnings). That study provided evidence on the reliability of I/B/E/S consensus
forecasts for valuation (and a method for correcting predictable forecast errors).20
19
Patricia Dechow and Richard Sloan, ‘‘Returns to Contrarian Investment Strategies: Tests of
Nai€ve Expectations Hypotheses,’’ Journal of Financial Economics (January 1997): 3–27.
Quotes are from 3, 4.
20
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.
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365
Are one set of analyst projections more accurate than another? In one study, the
authors compared I/B/E/S quarterly forecasts to Value Line’s and found for
the study period that the I/B/E/S forecasts outperformed Value Line forecasts in
terms of accuracy and proxies for market expectations. The I/B/E/S long-term forecasts are less biased and more accurate:
Using more recent data . . . we reach different conclusions [than earlier
studies]. . . . We find that . . . I/B/E/S quarterly earnings forecasts significantly outperform Value Line in terms of accuracy and as proxies for
market expectations. . . . We also evaluate long-term forecasts and find
that I/B/E/S forecasts are less biased and more accurate.21
Those authors reported the results of projected earnings amounts rather than
growth rates. (They used the I/B/E/S long-term growth rate to project the earnings
per share four years into the future and compared this with the actual earnings per
share four years out.) The results indicated that I/B/E/S mean forecast error in Year
4 can be translated into a typical growth rate adjustment for, say, 15% growth,
implying a ratio of actual to forecast of 0.89.22
Analyst cash flow forecasts seem to be even less accurate than their earnings
forecasts.23
Peter Easton highlights the errors that will be introduced if invalid assumptions
are made about growth beyond the short horizon for which analysts’ forecasts of
earnings are available.24 Easton concludes that
in light of the analysts’ tendency to be optimistic, the estimate of the expected
rate of returns are generally likely to be higher than the cost of capital.25
Other studies estimated the bias resulting from using analysts’ forecasts in estimating the cost of equity capital from the residual income method. One study
reported that cost of capital estimates were on the average 2.8% too high because of
analyst forecast bias (an average of 9.4% based on analysts’ estimates versus 6.6%
after removing the bias in the analysts’ estimates).26 The authors found that the bias
21
Sundaresh Ramnath, Steve Rock, and Philip Shane, ‘‘Value Line and I/B/E/S Earnings Forecast,’’ International Journal of Forecasting (January 2005): 185–198.
22
Sundaresh Ramnath, Steve Rock, and Philip Shane, ‘‘Value Line and I/B/E/S Earnings Forecast,’’ International Journal of Forecasting (January 2005): 185–198, Table 6, panel A. I/B/
E/S mean forecast error in Year 4 can be translated into a typical growth rate adjustment
for 15% growth in this way: ((1.15^4)(1 .0545))^.25 1 ¼ 13.4%, implying a ratio of
actual to forecast of 13.4/15 = 0.89.
23
Dan Givoly, Carla Hayn, and Reuven Lehavy, ‘‘The Quality of Analysts’ Cash Flow Forecasts,’’ Working paper, February 16, 2009, forthcoming, Accounting Review. Available at
http://ssrn.com/abstract=1423137.
24
Peter D. Easton, ‘‘Use of Forecasts of Earnings to Estimate and Compare Cost of Capital
across Regimes,’’ Journal of Business Finance & Accounting (April–May 2006): 374–394.
25
Peter D. Easton, ‘‘Use of Forecasts of Earnings to Estimate and Compare Cost of Capital
across Regimes,’’ Journal of Business Finance & Accounting (April–May 2006): 376.
26
Peter D. Easton and Gregory A. Sommers, ‘‘Effect of Analysts’ Optimism on Estimates of
the Expected Rate of Return Implied by Earnings Forecasts,’’ Journal of Accounting
Research 45(5) (2007): 983–1015.
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
in analysts’ estimates decreased as the size of the company increased. In a later
study, the author reported that the cost of capital estimates were on the average
2.4% too high because of analyst forecast bias (an average of 11.0% based on analysts’ estimates versus 8.6% after removing the bias in the analysts’ estimates).27,28
What causes the analysts to produce such optimistic forecasts? One study found
that analysts do not adequately take into account the impact of earnings volatility
on earnings predictability.29
Another study examined the information content in changes in analyst forecasts. When an analyst changes his forecast, it indicates to the market that his valuation differs from that of the market. That difference can come from changes in his
earnings forecast, discount rate, or growth forecast. Are any of these more accurate
changes? The authors found that earnings-based forecast changes are characterized
by hard information, greater verifiability, and shorter forecast horizons. They concluded that earnings-based forecast changes are less subject to bias than changes in
forecasts based on changes in discount rates and growth rates. Their finding is consistent with other studies that find that the longer the forecast horizon, the more
optimistic typically are the forecasts.30
The market appears to recognize the better reliability in earnings-based, shortterm forecast changes that are more informative.31 Still another study found that
analysts’ forecasts are no better than a simple random-walk time-series model forecast over longer forecast periods (36 months from the forecast date). Analysts’ forecasts are particularly poor for smaller and younger firms.32 The authors do point out
that their results do not refute studies that use analysts’ forecasts to proxy for market expectations.
Many of the problems inherent in using the single-stage model to estimate cost
of capital are addressed by using a multistage model.
27
Stephannie Larocque, ‘‘Disclosure, Analyst Forecast Bias, and the Cost of Equity Capital,’’
Working paper, December 2008: 18, Available at http://ssrn.com/abstract=1282170.
28
If one aggregates cost of capital estimates based on analysts’ forecasts with the estimated
bias removed into value-weighted averages (similar to the S&P 500), the Easton and
Sommers study (footnote 28) results in an implied ERP estimate of 4.43% as of 2004 and
the Larocque study (footnote 29) results in an implied ERP estimate of 3.6% as of 2006.
29
Ilia D. Dichev and Vicki Wei Tang, ‘‘Earnings Volatility and Earnings Predictability,’’
Working paper, September 2008, Available at http://ssrn.com/abstract=927305, forthcoming in Journal of Accounting and Economics.
30
Ambrus Kecskes, Roni Michaely, and Kent L. Womack, ‘‘What Drives the Value of Analysts’ Recommendations: Earnings Estimates or Discount Rate Estimates?’’ Working
paper, February 21, 2010, Available at http://papers.ssrn.com/sol3/papers.cfm?
abstract_id=1478451.
31
Ambrus Kecskes, Roni Michaely, and Kent Womack, ‘‘What Drives the Value of Analysts’
Recommendations: Earnings Estimates or Discount Rate Estimates?’’ Working paper,
September 2009. Available at http://ssrn.com/abstract=1478451. see also Randolph B.
Cohen, Christopher Polk, and Tuomo Vuolteenaho, ‘‘The Value Spread,’’ Journal of Finance 58 (2003): 609–641, who conclude that changes in cash flows typically explain
roughly 75% of the variation in stock prices and stock returns.
32
Mark Bradshaw, Michael Drake, James Myers, and Linda Myers, ‘‘A Re-Examination of
Analysts’ Superiority over Time-Series Forecasts,’’ Working paper, December 2009. Available at http://ssrn.com/abstract=1528987.
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367
SOURCES OF INFORMATION
To perform an implied cost of capital analysis rather than use data compiled by one
of the services, a variety of inputs are necessary, including company-specific data,
industry outlook data, and long-term macroeconomic forecasts. Company data can
be obtained from Securities and Exchange Commission (SEC) filings or services such
as Standard & Poor’s (a division of McGraw-Hill), Moody’s (published by Mergent,
Inc.), or Value Line Publishing, Inc. Analysts’ estimates can be compiled from individual analysts’ reports or from one of the three earnings consensus reporting services: Thomson Financial (formerly First Call and I/B/E/S), Multex-Ace, and Zack’s
Investment Research, Inc. There are a great number of different industry forecasts.
For some industries, excellent material is available from industry trade associations,
although they tend to focus primarily on revenues rather than on cash flows. There is
also a wide variety of macroeconomic forecast information. Appendix II lists details
on many sources providing data in all these categories. A more comprehensive compilation of the industry forecasts is the Business Valuation Data, Publications &
Internet Directory, published annually by Business Valuation Resources, LLC
(www.BVResources.com).
SUMMARY
Various models have been developed and are in active use by academics and practitioners because the single-factor pure CAPM has generally proven to provide
unreliable estimates of the cost of equity capital.
The FF three-factor model is an empirically derived model that has gained wide
acceptance but not wide use by practitioners, though that use is increasing particularly in the utility rate making area. The three risk factors can be obtained from
Morningstar’s Beta Book or from Kenneth French’s web site, making implementation of the model relatively easy. However, the FF three-factor model has not proven
to provide a consistently reliable estimate of the cost of equity capital.
The FF three-factor model is widely used by academic researchers and is often
used instead of the pure CAPM in such research. For example, in Chapter 15, we
discussed the study of unsystematic risk as the study of residuals resulting from
fitting models to realized stock returns; most of those researchers use the FF threefactor model or variations of the model (adding more factors).
The APT is a multivariate model for estimating the cost of equity capital. The
risk factor variables are not specified, but most formulations use macroeconomic
factors that may affect the rates of return of different companies to different degrees.
The beta in the CAPM may or may not be one of the factors. Partly because of lack
of consensus on the specific factors and the complexity of the model, it has not
enjoyed wide usage. Moreover, the macroeconomic factors used in current applications of APT may have a considerably less significant systematic impact on the cost
of capital for smaller companies or on individual divisional or project decisions than
for large national companies.
The market-derived capital pricing model and the yield spread model use current bond yields on the subject company bonds as a base risk measure. The bond
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
yields incorporate current market conditions and the entire risk profile, including the
company-specific risk of the subject company.
The implied method of cost of capital estimation uses current public stock price
information to estimate implied costs of equity capital. You can use either a DCF
method or an RI method. The single-stage DCF method uses a Gordon Growth
Model type of formula, with the present value (i.e., the stock price) known, solving
for k, the cost of equity capital. The multistage DCF method uses two or more
growth estimates for different future periods. As with the pure CAPM, applying the
method to closely held companies involves using publicly traded companies as proxies in a similar industry group to develop a starting point, with modifications for
differences in the characteristics between the public guideline companies and the
subject company. Analysts can obtain DCF-based cost of capital estimates for public
companies and industries from several services that compile them or can build their
own estimates.
Research continues to improve the methods for estimating the cost of equity
capital. Alternative models are being explored.33 Recent research has been reported
on using option prices on traded options of public stocks to find the implied cost of
equity capital. Options have been used to estimate expected volatility, and these
authors expanded the option pricing models to estimate the cost of equity capital
that must be implied by the price and volatility of call and put options. Their results
of applying the methodology to the stocks comprising the S&P 100 firms are promising in that their implied cost of equity estimates are reasonable and consistent with
estimates obtained using the FF three-factor model.34
ADDITIONAL READING
Palkar, Darshana D., and Stephen E. Wilcox. ‘‘Adjusted Earnings Yields and Real
Rates of Return.’’ Financial Analysts Journal (September–October 2009): 66–79.
TECHNICAL SUPPLEMENT APPENDIX I
We include an excerpt from the report submitted by Roger Grabowski in the case
Herbert V. Kohler, Jr. et al. v. Commissioner of Internal Revenue in the Cost of
Capital: Applications and Examples 4th ed. Workbook and Technical Supplement,
Appendix I, which appears on the companion John Wiley & Sons web site. In that
case, Grabowski used the CAPM, the Duff & Phelps Risk Premium Report, and the
FF three-factor model in estimating the cost of capital. He submitted a report using
multiple methods in the belief that no one method is likely to produce the true cost
of capital. The use of multiple methods provides a range of market indications upon
which the analyst then applies judgment to reach a final conclusion.
33
Long Chen and Lu Zhang, ‘‘A Better Three-Factor Model That Explains More Anomalies,’’
Working paper, June 2009. Forthcoming, Journal of Finance. Available at http://ssrn.com/
abstract=1418117.
34
Antonio Camara, San-Lin Chung, and Yaw-Huei Wang, ‘‘The Cost of Equity Capital Implied by Option Market Prices,’’ Working paper, June 19, 2007.
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CHAPTER
18
Weighted Average Cost of Capital
Introduction
Where Does WACC Come From?
When to Use WACC
Valuing the Levered Business Enterprise
After-Tax WACC
Debt Capacity and Optimal Capital Structure
Computing WACC for a Public Company
Income Tax Rates Impact WACC
Market Value of Debt
Computing WACC for a Nonpublic Company
Should an Actual or a Hypothetical Capital Structure Be Used?
Should a Constant or Variable Capital Structure Be Used?
Fixed Book-Value Leverage Ratio
Pre-Interest-Tax-Shield WACC
Capital Cash Flows
Equivalence of Valuation Methodologies
Other Tax Shields
Summary
Additional Reading
Technical Supplement Chapters 5 and 6
INTRODUCTION
In Chapter 6, we identified components of a company’s capital structure. To estimate the weighted cost for all of the company’s overall capital, we blend their costs
together to derive the company’s weighted average cost of capital (WACC), often
called the overall cost of capital. In other words, we want to estimate the weighted
cost for all of the company’s invested capital. This requires a discussion of the
appropriate amount of debt and equity in the capital structure and how much value,
if any, the debt adds to the value of the providers of equity capital because of the
interest tax shield we discuss later in this chapter.
The authors wish to thank Nick Arens and William Susott for help with compiling data.
369
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The most common assumptions made by analysts are that (1) the market value
of debt capital equals its face value or carrying value on the balance sheet (their reasoning is often that data services include the balance sheet debt in their reporting of
company debt) and (2) the ratio of debt capital (measured at book value) to equity
capital (at market) in recent years is representative of the appropriate current capital
structure. The events during the financial crisis of 2008–2010 proved that both of
these commonly made assumptions can be wrong. Today, one of the most difficult
analyses to perform is estimating the capital structure that matches the debt capacity
of the company and the appetite for debt in the debt market.
WHERE DOES WACC COME FROM?
Consider the stylized balance sheet in Exhibit 18.1, where we have one class of interestbearing debt capital and common equity capital, stated at market values, and assets
stated at market values. It is useful to begin with the definition of a business enterprise
(enterprise value):
BE ¼ NWC þ FA þ IA þ UIV
where:
BE ¼ Business enterprise value
NWC ¼ Net working capital value
FA ¼ Fixed assets value
IA ¼ Intangible assets value
UIV ¼ Unidentified intangible value (i.e., goodwill and other unidentified
assets)
All risks inherent in the assets of the business are borne by the investors who
provided debt and equity capital. Alternatively, we can express these concepts in a
generalized formula:
(Formula 18.1)
k ¼ Rf þ Business risk premium
Market Value Balance Sheet
Assets
Debt
Equity
Cash is
generated
here.
Risk
originates
here.
Cash is
distributed
here.
All risk of the
assets must be
borne by investors
EXHIBIT 18.1 Business Enterprise
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371
Weighted Average Cost of Capital
WACC
risk
premium
Expected return
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Debt risk
premium
Equity risk
premiums (total)
Time value (risk-free rate)
0
0
Riskiness of Cash Flows
EXHIBIT 18.2 Revisiting the Risk/Return Trade-off: Expected Return Is an Increasing
Function of Risk
where:
k ¼ Overall discount rate given the business risk
Rf ¼ Risk-free rate of return
Business risk premium ¼ Rate of return in excess of the risk-free rate appropriate for business risk inherent in the assets
Since we cannot generally observe the business risk premium, we must impute
the overall cost of capital from the cost of capital for the debt capital and equity
capital as is shown in Formula 18.2, which describes the overall risk premium:
(Formula 18.2)
Assets’ Value Business risk premium ¼ ðW d Risk premiums on debt capitalÞ
þ ðW e Risk premiums on equity capitalÞ
where: Wd ¼ Percentage of debt capital in the capital structure, at market value
We ¼ Percentage of equity capital in the capital structure, at market value
Graphically, one can see the relationship in Exhibit 18.2. The total equity risk
premium depicted in Exhibit 18.2 includes the market risk premium, the size risk
premium (if applicable), and the company-specific risk premium (if applicable).
We can make the two following observations:
1. The risk premium embedded in WACC is a weighted average of a (higher) total of
the risk premiums on equity capital and a (lower) risk premium on debt capital.
2. This risk premium embedded in WACC often approximates the risk premium
for the enterprise, or unlevered, risk premium, that is, the premium implied by
the unlevered beta, except for highly-levered companies.
WACC generally works as a substitute for the enterprise-cash-flow discount rate
(k) because the risk premium on the left-hand side of the balance sheet must equal
the risk premiums on the right-hand side (Exhibit 18.1).
WHEN TO USE WACC
WACC can be applied in a single year capitalization of net cash flows or multiyear
discounted net cash flows valuation. It can also be used in valuing a control or
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
minority interest position. The most obvious instance in which to use weighted
average cost of capital WACC is when the objective is to value the overall business
enterprise.
An example would be when, in considering an acquisition, the buyer expects to
pay off all existing equity and debt investors and refinance the entire business in a
different manner that better suits the buyer. Such an analysis may result in investment value instead of fair market value if the financing plan was significantly different from the capital structure that a typical buyer would employ.
Alternatively, WACC can be used even when the objective is ultimately to value
only the equity capital. One would value the overall business enterprise and then
subtract the market value of the debt to estimate the value of the equity capital.
Valuing the overall firm is frequently done in highly leveraged situations to understand the value of the operations separately from currently debt-burdened equity.
However, WACC can be used in all valuation situations—it is not limited to valuing
highly leveraged companies.
It is especially appropriate for project selection in capital budgeting. The proportions of debt capital and equity capital that could be available to finance various
projects might differ according to the project (e.g., asset-intensive projects may be
financed with more debt than the company’s overall capital structure), and the cost
of capital generally should be based on the debt capacity of the specific investment.
The idea of differing proportions of debt and equity for financing various projects introduces the idea that we have to compute or estimate the weight (percentage
of the total) for each component of the capital structure. The critical point is that the
relative weightings of debt and equity or other capital components are based on the
market values of each component, not on the book values.
In Chapter 11, we discussed the various formulas for adjusting the cost of equity
capital for the amount of leverage. Even the Fernandez formulas (11.9 and 11.10),
which are based on debt capital (at market value) increasing or decreasing in proportion to the book value of equity capital, use the market value weights of both
debt capital and equity capital.1
VALUING THE LEVERED BUSINESS ENTERPRISE
There are two equivalent formulations in the literature for valuing a levered business
enterprise as depicted in Exhibit 11.1, reproduced here in part as Exhibit 18.3. The
values of debt and equity capital are market values.
The tax shield is the reduction of the cost of debt capital due to the tax deductibility of interest expense on debt capital. In the first formulation, cost of debt capital is measured after the tax affect (kd), as the value of the tax deduction on interest
payment reduces the effective cost of debt capital. This formulation uses as the discount rate the WACC. It is applied to net after-tax (but before interest) net cash
flows of the business enterprise.
1
The formulas are based on the relationship between the growth in book value of equity and
market value of equity. This is also true for the Bodt-Levasseur formulas we reference in
Chapter 11.
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Weighted Average Cost of Capital
373
EXHIBIT 18.3 Value of a Levered Business Enterprise (BE)
Formulation 1
Value of Levered BE ¼ Value of Levered Assets
Formulation 2
Value of the Levered BE ¼ Value of the Unlevered Assets þ Present Value of Tax Shield
In the second formulation, the cost of debt capital is measured prior to the tax
effect (kd(pt)), as the present value of the tax deduction on the interest payments
equals the value of the tax shield.
In the first formulation, you attach value to the assets of the business based on
their being partially financed with debt capital. In the second formulation, you
attach value to the assets of the business as if they were financed with all equity capital, and then the tax shield is valued separately.
In the second formulation, the tax savings due to interest expense deductions are
directly valued as a cash flow. Therefore, the discount rate is the weighted pretax
weighted cost of debt capital and the cost of equity capital components (pre-interesttax-shield weighted average cost of capital). It is applied to the net after-tax (but
before interest) net cash flows of the firm and the cash flows due to the tax shield.
AFTER-TAX WACC
As noted in the discussion of debt in Chapter 6, the so-called after-tax WACC is
based on the cost of each capital structure component net of any corporate-level tax
effect of that component. Interest expense is a tax-deductible expense to a corporate
taxpayer. Whatever taxes are paid are an actual cash expense to the company,
and the returns available to equity holders are after the payment of corporate-level
income taxes.
Because we are interested in cash flows after entity-level taxes, literature and
practitioners typically refer to this formulation of the WACC as an after-tax
WACC. The basic formula for computing the after-tax WACC for an entity with
three capital structure components is:
(Formula 18.3)
WACC ¼ ðke W e Þ þ kp W p þ kdðptÞ ½1 t W d
where: WACC ¼ Weighted average cost of capital (after-tax)
ke ¼ Cost of common equity capital
We ¼ Percentage of common equity capital in the capital structure, at
market value
kp ¼ Cost of preferred equity capital
Wp ¼ Percentage of preferred equity capital in the capital structure, at
market value
kd(pt) ¼ Cost of debt capital (pretax)
t ¼ Income tax rate
Wd ¼ Percentage of debt capital in the capital structure, at market value
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Market Value Balance Sheet
EBIT(1 – t)
+ Depreciation
– Capital expenditures
– Change in NWC
Debt
Capital
+
Equity
Capital
Assets
= Enterprise Cash Flow
Asset
Value
NCFf discounted
at WACC to
obtain Enterprise
Value
=
Capital
Value
These NCFe are
discounted at the
levered cost of equity
to get Equity Value
Interest
+ Net principal payment
= Debt Cash Flow
These NCFd are
EBIT
discounted at the cost
– Interest
of debt to Debt Value
= EBT
– Inc. Taxes
= Net income
+ Depreciation
– Capital expenditures
– Change in NWC
= Cash flow available
for debt repayment
– Net debt principal
payment
= Equity Cash Flow
EXHIBIT 18.4 Comparing Different Net Cash Flows
The adjustment to the cost of debt capital, (1 t), is the interest tax shield
due to the deductibility of interest and the resulting reduction in income tax payments. It is a correction for the fact that the asset cash flows typically overstate
taxes because they omit the interest tax shield. The asset cash flows are equal to
the cash flows to invested capital (NCF f) which was defined in Formula 3.2.
The NCFf equals the net cash flows to the business enterprise and are before the
additional net cash flow due to the interest tax shield. Simplifying Formula 3.2,
the business enterprise cash flows can generally be summarized as shown in
Exhibit 18.4
Because the interest tax shield is missing in the NCFf formula, it is not true that:
Enterprise Cash Flows ¼ Debt Cash Flows þ Equity Cash Flows
One needs to directly value the interest tax shield for the equality to hold as
follows:
Enterprise Cash Flows þ Interest expense ðtÞ ¼
Debt Cash Flows þ Equity Cash Flows
Multiplying (1 t) by the cost of debt capital in the WACC is designed to give
us the right value even when applied to the ‘‘wrong’’ cash flows—to NCFf (i.e., without the interest tax shield).
Formula 18.3 is a simplification. Implicit in this formula is the assumption that
the interest tax shield equals the cost of debt capital times the market value of debt
and that the interest deductions reduce income taxes in the period in which the interest is paid. More likely, the interest deduction equals the face amount of debt times a
coupon rate, and, for many companies, there is some risk of realizing the interest tax
shield.
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Weighted Average Cost of Capital
The assumptions implicit in Formula 18.3 are violated:
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Any time the market value of debt differs from the book value of debt
When the coupon on the debt does not equal the expected return on the market
value of debt capital
When the income tax deduction does not equal the coupon multiplied by the
face value of the debt
When the interest tax shields do not reduce cash income taxes in the period the
interest is paid
DEBT CAPACITY AND OPTIMAL CAPITAL STRUCTURE
The traditional view of the optimal capital structure is that a company should
increase debt until its weighted average cost of capital minimizes its WACC. Or
the amount of debt should be increased until the after-tax cost of debt exceeds
the increase in the risk of financial distress. This relationship is depicted in
Exhibit 18.5.
Commonly made computations of WACC ignore costs of financial distress,
thereby systematically underestimating WACC for highly levered companies. As
the proportion of debt is increased in the capital structure, the formulas commonly used for levering equity betas (and increasing the cost of equity as debt
increases) are linear and likely to understate the cost of equity capital at high
amounts of leverage. Similarly, the traditional depiction of the cost of debt capital fails to consider the gradually increasing cost of debt as debt is added to the
capital structure.
Many practitioners’ WACC templates are especially prone to error when leverage is high. WACC templates often give results that do not increase with increases in
leverage, which cannot be true when debt is greater than a certain amount.
Weighted Average Cost of Capital
Cost of Equity
WACC
Cost of Capital
E1C18
Cost of Debt
Debt
Total Capital
EXHIBIT 18.5 Traditional View of the Optimal Capital Structure
E1C18
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ESTIMATING THE COST OF EQUITY CAPITAL AND THE OVERALL COST OF CAPITAL
3
2.5
2
Beta 1.5
1
0.5
0
0
0.2
0.4
Leverage
0.6
Weighted average beta of equity and debt
Bd
BL
EXHIBIT 18.6 Beta as a Function of Leverage
Source: Arthur G. Korteweg, ‘‘The Costs of Financial Distress across Industries,’’ Working
paper, Stanford University, January 15, 2007, 65. Used with permission. All rights reserved.
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Many templates treat the cost of debt as unchanged with respect to increased
leverage