Cost of C

apital WORKBOOK AND TECHNICAL SUPPLEMENT

FOURTHEDITION

Praise for Fourth Edition of Cost of Capital WORKBOOK AND TECHNICAL SUPPLEMENT“Pratt and Grabowski went the extra mile to supplement their magnum opus by providing this Workbook and Technical Supplement. As a finance professor for many years, I know from experience that students and teachers really value supplements to textbooks. It allows the teacher to help the student to review and apply what was presented in the text, and the PowerPoints are a great service to teachers in course preparation. The website provides various worksheets that show the inner workings of the models. I enthusiastically recommend the Workbook and Technical Supplement to finance professors and teachers and their students. —Daniel L. McConaughy, PhD, ASA, Professor of Finance, California State University, Northridge,

Valuation Services, Crowe Horwath LLP

“The Workbook and Technical Supplement provides a detailed tutorial on understanding and executing the theoretical concepts explained in the Fourth Edition. This supplement is three books in one. Part One is a step-by-step tutorial on estimating certain key components of the cost of equity capital. Part Two provides a bridge between the theory and some practical applications, such as estimating the cost of capital for real property. Parts Three and Four allow the readers to test their comprehension of the concepts and identify areas for a review. It is almost as good as having Professors Pratt and Grabowski looking over your shoulder to ensure that one is both comprehending and correctly implementing the complex concepts..”

—Ashok Abbott, PhD, Associate Professor of Finance, College of Business & Economics, West Virginia University

“This text provides the most comprehensive coverage of cost of capital issues that I have seen to date. Messrs. Pratt and Grabowski have created a very accessible and lucid treatment of what most would consider an opaque subject. The Fourth Edition is especially important for its new topics as well as expanded coverage of concepts from earlier editions. Of particular interest is the review of the extreme market conditions during the 2008–2009 crisis and the effect that the unprecedented volatility had on traditional cost of capital models. For years, Pratt and Grabowski’s research has informed the business valuation curriculum of the American Society of Appraisers. This book will be added to our reading list, and thousands of students worldwide will benefit from the state-of-the-art content of the Fourth Edition and the companion Workbook and Technical Supplement. Furthermore, Cost of Capital, Fourth Edition should be a mandatory part of every valuation practitioner’s library. If you buy this book, you can expect it to become well worn and remain on your desk within arm’s length until the publication of the Fifth Edition.” —John Barton, ASA, CPA, Chairman, Business Valuation Committee, ASA

“Cost of capital is so much more complex than it used to be. With so many additional considerations regarding each variable of the cost of capital formula, this book is a must for anyone that needs to understand or develop a discount rate. Even the most experienced practitioner will benefit from the outstanding work of Pratt and Grabowski. This book has to become part of your library.”

—Gary R. Trugman, CPA/ABV, MCBA, ASA, MVS, President, Trugman Valuation Associates, Inc.

SHANNON P. PRATT, CFA, ARM, ABAR, FASA, 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 Portland, Oregon, 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.

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.

colors=PMS 319, PMS 280

WORKBOOK ANDTECHNICAL SUPPLEMENT

FOURTH EDITION

Cost of Capital

Shannon P. PrattRoger J. Grabowski

with WEBSITEwith WEBSITE

Pratt

Grab

owski

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CostofCapital FourthEdition Workbook

and TechnicalSupplement

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SHANNON P. PRATTROGER J. GRABOWSKI

John Wiley & Sons, Inc.

CostofCapital FourthEdition Workbook

and TechnicalSupplement

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Copyright# 2011 by JohnWiley & Sons, Inc. All rights reserved.

Published by JohnWiley & 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 aspermitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior

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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 salesrepresentatives 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 tospecial, 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 (317)572-3993 or fax (317) 572-4002.

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 atwww.wiley.com.

Library of Congress Cataloging-in-Publication Data:

ISBN 978-0-470-47606-2; ISBN 978-0-470-94492-9 (ebk); ISBN 978-0-470-94934-4 (ebk);

ISBN 978-0-470-94935-1 (ebk)

Printed in the United States of America

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Contents

Preface ix

About the Author xi

Acknowledgments xix

Notation System and Abbreviations Using in This Book xxi

PART ONE

Technical Supplement—Supplements to Chapters ofCost of Capital: Applications and Examples, 4th ed.

CHAPTER 1Alternative Net Cash Flow Definitions—Supplement to Chapter 3 3

Introduction 3Equity Cash FlowMethod 3Invested Capital Method 4Capital Cash FlowMethod 5Adjusted Present Value Method 7Residual IncomeMethod 8

CHAPTER 2Examples of Computing OLS Beta, Sum Beta, and Full InformationBeta Estimates—Supplement to Chapter 10 10

Introduction 10Computing OLS and Sum Beta Estimates—An Example 11Computing Full-Information Beta Estimate—An Example 15

CHAPTER 3Estimating Beta: Interpreting RegressionStatistics—Supplement to Chapter 10 22

Introduction 22Evaluating Beta Estimation Output 25Evaluating Regression Output 26

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CHAPTER 4Example of Computing Downside Beta Estimates—Supplement to Chapter 12 31

Introduction 31Computing Downside Beta Estimates 31

CHAPTER 5Iterative Process Using CAPM to Calculate the Cost of EquityComponent of the Weighted Average Cost of Capital WhenCapital Structure Is Constant—Supplement to Chapter 18 34

Introduction 35Capital Asset Pricing Model and Beta 35Solution: The Iterative Process 36Iterative Process Using a Financial Spreadsheet Model 39Summary 48Additional Reading 48

CHAPTER 6Iterative Process Using CAPM to Calculate the Cost of EquityComponent of the Weighted Average Cost of Capital WhenCapital Structure Is Changing—Supplement to Chapter 18 49

Introduction 49Assumptions Inherent in Weighted Average Cost of Capital 50Solution: Iterative Process with Changing Capital Structure 50Iterative Process Using Financial Spreadsheet Model 52Equity Value 60Summary 62Additional Reading 62

CHAPTER 7Cost of Capital and the Valuation of Worthless Stock—Supplementto Chapter 16 63

Introduction 64Liquidating Value 64Potential Value 66Example 67Possibility That Value of the Business Enterprise Exceeds

the Face Value of Debt—Pricing Equity as a Call Option 67Potential Future Value: Probability That Business

Enterprise Exceeds Face Value of Debt 73Additional Considerations 77Summary 79

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PART TWO

Technical Supplement—Specific Applications of Cost of Capital

CHAPTER 8Cost of Capital of Private Investment Company Interests 83

Introduction 83The Private Investment Company 84Relationships between Time to a Liquidity Event and Value 85Lack of Control 89Lack of Marketability or Illiquidity 93Example Valuation of PIC Interest 99Summary 105

CHAPTER 9Cost of Capital of Real Property—Individual Assets 107

Introduction 107Typical Structure of a Real Estate Transaction 108Real Property Competes with Other Asset Classes 109Direct Capitalization Method 111Discounted Cash FlowMethod 125Estimating the Property Discount Rate 126Summary 130

Appendix 9A: Valuing Real Property 131

CHAPTER 10Cost of Capital of Real Estate Entities 139

Introduction 139Definition of a Real Estate Entity 140Measuring Net Cash Flow for Real Estate Entities 153Valuation of Real Estate Entities 154Summary 175Additional Reading 176

Appendix 10A: Valuing Real Estate Entities 177

PART THREE

Learning Objectives, Questions, and Problems 185

PART FOUR

Answers and Solutions 227

Index 247

Contents vii

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Preface

Why did we add the Workbook and Technical Supplement to the Cost of Cap-ital: Applications and Examples, 4th ed.? We wanted to further assist practi-

tioners in better understanding how to estimate the cost of capital. This text addsmore detailed examples to the Cost of Capital: Applications and Examples,4th ed. It also contains questions and problems covering the material containedin the Cost of Capital: Applications and Examples, 4th ed. designed to help thereader better grasp that material.

This book uses the identical notation and abbreviations as those used in the text.Those can be referenced either in Cost of Capital: Applications and Examples,4th ed. or on the companion web site (see later).

Part One contains technical supplements to several chapters. These will helpthe reader to be better able to implement the methods of analyses discussed in themain book.

Part Two contains an example of specific applications of applying the theory tothe cost of capital for private investment companies, including one approach to in-corporating the discount for lack of control and lack of marketability into the cost ofcapital. It also contains chapters extending the general concepts of developing costof capital to real estate properties and real estate entities. These investments havetheir own set of terminologies unique to the industry, and we cover the terminologyand methods of analysis commonly used in the industry in detail.

Part Three contains learning objectives, questions, and problems to help thepractitioner better understand the content of the first 34 chapters of the Cost ofCapital: Applications and Examples, 4th ed.

Part Four contains the answers to the questions and solutions to the problemspresented in Part Three.

Finally, this book includes a companion web site, which can be found at www.wiley.com/go/coc4e. The web site includes the following:

1. The notation system and abbreviations used in this book.2. The worksheets that are presented as exhibits in Chapters 5, Iterative Process

Using CAPM to Calculate the Cost of Equity Component of the Weighted Aver-age Cost of Capital When Capital Structure Is Constant; in Chapter 6, IterativeProcess Using CAPM to Calculate the Cost of Equity Component of theWeighted Average Cost of Capital When Capital Structure Is Changing; and inChapter 8, Cost of Capital of Private Investment Company Interests. Theseexhibits are provided for your reference so that you can track the methodologiesdiscussed in the book and see how they are implemented through the MicrosoftExcel worksheets. This will assist you in building models of your own using theworksheets as templates. Note that each file, however, typically contains addi-tional information in different worksheets within that file.

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3. Three appendices:& Appendix I: Sample Report Submitted to U.S. Tax Court (Supplement to

Chapters 7 and 17) is an example of a report submitted to the U.S. Tax Courtto help readers communicate the cost of capital methods in a straightforwardway to the nontechnical reader.

& Appendix II discusses the ValuSource Valuation Software, which is a helpfultool for the practitioner.

& Appendix III contains a comprehensive review of the statistics discussed inthe Cost of Capital: Applications and Examples, 4th ed. and used in develop-ing the cost of capital. We included this appendix so practitioners who mayneed a refresher in basic statistics do not need to try to locate their statisticsbooks from college. It covers many topics including probability theory(important for understanding and measuring risk), the statistics (e.g., mean,mode, standard deviation, beta, etc.) that are used to summarize return andrisk data, and basic concepts of risk neutral payoffs and probabilities. It alsoincludes formulas, terminology, and the statistical tools of the MicrosoftExcel Analysis Toolpak.

4. PowerPoints that accompany the chapters of Cost of Capital: Applications andExamples, 4th ed. to assist those that want to use the book in seminars.

x PREFACE

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About the Authors

Dr. Shannon P. Pratt, CFA, FASA, ARM,MCBA, ABAR, CM&AA, is the chairmanand CEO of Shannon Pratt Valuations, Inc., a nationally recognized business valua-tion firm headquartered in Portland, Oregon. He is also the founder and editor emer-itus of Business Valuation Resources, LLC, and one of the founders of WillametteManagement Associates, for which he was a managing director for almost 35 years.

He has performed valuation assignments for these purposes: transaction (acquis-ition, divestiture, reorganization, public offerings, public companies going private),taxation (federal income, gift, and estate and local ad valorem), financing (securitiza-tion, recapitalization, restructuring), litigation support and dispute resolution(including dissenting stockholder suits, damage cases, and corporate and marital dis-solution cases), and management information and planning. He has also managed avariety of fairness opinion and solvency opinion engagements. He regularly reviewsbusiness valuation reports for attorneys in litigation matters.

Dr. Pratt has testified on hundreds of occasions in such litigated matters as dis-senting stockholder suits, various types of damage cases (including breach of con-tract, 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. Commis-sioner, Charles S. Foltz, et al. v. U.S. News & World Report et al., Estate of MarthaWatts v. Commissioner, and Okerlund v. United States. He has also served asappointed arbitrator in numerous cases.

Prev i o us Exper i e nce

Before foundingWillamette Management Associates in 1969, Dr. Pratt was a profes-sor of business administration at Portland State University. During this time, hedirected a research center known as the Investment Analysis Center, which workedclosely with the University of Chicago’s Center for Research in Security Prices.

E duca t i o n

Doctor of Business Administration, Finance, Indiana University.Bachelor of Arts, Business Administration, University of Washington.

Pro f ess i ona l A f fi l i a t i o ns

Dr. Pratt is an Accredited Senior Appraiser and Fellow (FASA), Certified in Busi-ness Valuation, of the American Society of Appraisers (their highest designation)and is also accredited in Appraisal Review and Management (ARM). He is aChartered Financial Analyst (CFA), a Master Certified Business Appraiser

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(MCBA) and Accredited in Business Appraisal Review (ABAR) by the Instituteof Business Appraisers, a Master Certified Business Counselor (MCBC), and isCertified in Mergers and Acquisitions (CM&AA) with the Alliance of Mergerand Acquisition Advisors.

Dr. Pratt is a life member of the American Society of Appraisers, a life mem-ber of the Business Valuation Committee of that organization, and a teacher ofcourses for the organization. He is also a lifetime member emeritus of the Advi-sory Committee on Valuations of the ESOP Association. He is a recipient of themagna cum laude award of the National Association of Certified Valuation Ana-lysts for service to the business valuation profession. He is also the first life mem-ber of the Institute of Business Appraisers. He is a member and a past presidentof the Portland Society of Financial Analysts, the recipient of the 2002 Distin-guished Achievement Award, and a member of the Association for CorporateGrowth. Dr. Pratt is a past trustee of the Appraisal Foundation and is currentlyan outside director and chair of the audit committee of Paulson Capital Corp., aNASDAQ-listed investment banking firm specializing in small initial public offer-ings (usually under $50 million).

Pub l i c a t i ons

Dr. Pratt is the author of Valuing a Business: The Analysis and Appraisal ofClosely Held Companies, 5th ed. (New York: McGraw-Hill, 2008); co-author,Valuing Small Businesses and Professional Practices, 3rd ed., with RobertSchweihs and Robert Reilly (New York: McGraw-Hill, 1998); co-author, Guideto Business Valuations, 20th ed., with Jay Fishman, Cliff Griffith, and JimHitchner (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: Proce-dure, Law, and Perspective, 2nd ed., with Judge David Laro (Hoboken, NJ:John Wiley & Sons, 2010); and author, Business Valuation Discounts and Pre-miums, 2nd ed. (Hoboken, NJ: John Wiley & Sons, 2009); Business ValuationBody of Knowledge: Exam Review and Professional Reference, 2nd ed. (Hoboken,NJ: John Wiley & Sons, 2003); The Market Approach to Valuing Businesses, 2nded. (Hoboken, NJ: John Wiley & Sons, 2005); and The Lawyer’s Business Valu-ation Handbook, 2nd ed. (Chicago: American Bar Association, 2010). He hasalso 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 machineryand equipment for various purposes, including tax (income and ad valorem) andfinancial reporting; mergers, acquisitions, formation of joint ventures, divesti-tures, and financing. He developed methodologies and statistical programs foranalyzing useful lives of tangible and intangible assets, such as customersand subscribers. His experience includes work in a wide range of industries, in-cluding 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

xii ABOUT THE AUTHORS

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other transportation companies; mining ventures; software and electronic com-ponent businesses; and a variety of manufacturing businesses.

Mr. Grabowski has testified in court as an expert witness on the value ofclosely held businesses and business interests; matters of solvency, valuation, andamortization of intangible assets; and other valuation issues. His testimony inU.S. District Court was referenced in the U.S. Supreme Court opinion decided inhis 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 Com-pany, et al. v. Comm. (the first U.S. Tax Court case that recognized the use ofthe discounted cash flow method for valuing a closely held business); OaklandRaiders v. Oakland–Alameda County Coliseum Inc. et al. (valuation of theOakland Raiders); In re: Louisiana Riverboat Gaming Partnership, et al. Debtors(valuation of business enterprise owning two riverboat casinos and feasibility ofplan of reorganization); ABC-NACO, Inc. et al., Debtors, and The Official Com-mittee of Unsecured Creditors of ABC-NACO v. Bank of America, N.A. (valua-tion of collateral); Wisniewski and Walsh v. Walsh (oppressed shareholderaction); and TMR Energy Limited v. The State Property Fund of Ukraine (arbi-tration on behalf of world’s largest private company in Stockholm, Sweden, oncost of capital for oil refinery in Ukraine in a contract dispute).

Prev i o us Exper i e nce

Mr. Grabowski was formerly managing director of the Standard & Poor’s Cor-porate 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). Priorto Price Waterhouse, he was a finance instructor at Loyola University of Chica-go, a cofounder of Valtec Associates, and a vice president of American ValuationConsultants.

E duca t i o n

Mr. Grabowski received his BBA–Finance from Loyola University of Chicago andcompleted all coursework in the doctoral program, Finance, at Northwestern Uni-versity, Chicago.

Pro f ess i ona l A f fi l i a t i o ns

He serves on the Loyola University School of Business Administration Dean’s Boardof Advisors. Mr. Grabowski is an Accredited Senior Appraiser of the American Soci-ety of Appraisers (ASA) certified in business valuation. He serves as Editor of theBusiness Valuation Review, the quarterly journal of the Business Valuation Commit-tee of the American Society of Appraisers.

Pub l i c a t i o ns

Mr. Grabowski authors the annual Duff & Phelps Risk Premium Report. He lec-tures and publishes regularly. Recent articles include ‘‘The Cost of Capital,’’ Journal

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of Business Valuation, the Canadian Institute of Chartered Business Valuators,August 2009; ‘‘Problemas relacionados con el calculo del coste de capital en el entor-no actual: actualizacion,’’ co-authored with Mathias Schumacher, Analisis Finan-ciero Internactional, Sumario No 137 Tercer trimestre 2009; ‘‘Cost of CapitalEstimation in the Current Distressed Environment,’’ The Journal of AppliedResearch in Accounting and Finance, July 2009; ‘‘Cost of Capital in Valuation ofStock by the Income Approach: Updated for an Economy in Crisis,’’ with ShannonP. Pratt, Jahreskonferenz der NACVA, Bewertungs Praktiker, January 2009; ‘‘Prob-lems with Cost of Capital Estimation in the Current Environment—2008 Update,’’Business Valuation Review, Winter 2008 and Business Valuation E-Letter, February2009; and ‘‘Cost of Capital in Valuation of Stock by the Income Approach: Updatedfor Economy in Crisis,’’ The Value Examiner, January–February 2009.

He is the co-author Cost of Capital: Applications and Examples, 3rd ed., withShannon P. Pratt (Hoboken, NJ: John Wiley & Sons, 2008) and co-author of threechapters (on equity risk premium, valuing pass-through entities, and valuing sportsteams) in Robert Reilly and Robert P. Schweihs, The Handbook of Business Valua-tion and Intellectual Property Analysis (New York: McGraw-Hill, 2004).

He teaches courses for the American Society of Appraisers including Cost ofCapital, a course he developed.

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 BusinessAdministration, both from the University of Michigan, Ross School of Business.

Ms. Fong co-authored Chapter 7 of the Cost of Capital: Applications and Exam-ples, 4th ed.Workbook and Technical Supplement.

William H. Frazier, ASA, is a principal and founder of the firm of HowardFrazier Barker Elliott, Inc, and manages its Dallas office. He has 30 years of experi-ence in business valuation and corporate finance. Mr. Frazier has been an AccreditedSenior Appraiser of the American Society of Appraisers (ASA) since 1987 and serveson the ASA’s Government Relations Committee. He has participated as an appraiserand/or expert witness in numerous U.S. Tax Court cases, including testimony inJelke, McCord, Dunn, and Gladys Cook. Mr. Frazier has written numerous articleson the subject of business valuation for tax purposes, appearing in such publicationsas the Business Valuation Review, Valuation Strategies, BV E-Letter, ShannonPratt’s Business Valuation Update, and Estate Planning. He is the co-author of thechapter on valuing family limited partnerships in Robert Reilly and Robert P.Schweihs, eds., The Handbook of Business Valuation and Intellectual PropertyAnalysis (New York: McGraw-Hill, 2004). Mr. Frazier serves on the IRS AdvisoryCouncil (IRSAC) and the Valuation Advisory Board of Trusts & Estates Journal.

Mr. Frazier contributed Chapter 8 of the Cost of Capital: Applications and Ex-amples, 4th ed. Workbook and Technical Supplement and the companion Excelworksheets that appear on the John Wiley & Sons web site.

Terry V. Grissom, PhD, CRE, MAI, serves on the faculty at the University of Wash-ington. He just completed a faculty assignment at the University of Ulster, Built En-vironment Research Institute. He received his PhD in Business from the University ofWisconsin, Madison, majoring in Real Estate and Urban Land Economics, with mi-nors in Finance/Risk Management and Civil-Environmental Engineering. He

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received an MS in Real Estate Appraisal and Investment Analysis, also from theUniversity of Wisconsin, and an MBA in Finance, Real Estate, and Urban Affairsfrom Georgia State University. He did postdoctoral work at Texas A&M Universityin Econometrics and Statistics.

Dr. Grissom was formerly Professor of Real Estate and Urban Land Economicsat Georgia State University, Atlanta, in the Robinson College of Business. Prior tohis tenure at GSU, he was Vice-President of Investment Research for Equitable RealEstate Investment Management, an institutional investment advisory for pensionfunds, insurance companies, and other financial institutions. From 1992 throughOctober 1994, he was the National Research Director for Price Waterhouse’s Finan-cial 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 investmentanalysis, market analysis, and real estate development and land economics. He hasalso authored chapters in books on real estate development, investment analysis,business and property valuation techniques, and education theory and practice forboth academics and practitioners and for both domestic and internationalaudiences.

Dr. Grissom co-authored Chapters 9 and 10 of the Cost of Capital: Applicationsand Examples, 4th ed.Workbook and Technical Supplement.

Jim MacCrate, MAI, CRE, ASA, owns his own boutique real estate valuation andconsulting company, MacCrate Associates, LLC, located in the New York City met-ropolitan area, concentrating on complex real estate valuation issues. Formerly, hewas the Northeast regional practice leader and director of the Real Estate Valuation/Advisory Services Group at Price Waterhouse LLP and Pricewaterhouse CoopersLLP. He received a BS degree from Cornell University and anMBA from Long IslandUniversity, C. W. Post Center.

Mr. MacCrate has written numerous articles for Price Waterhouse LLP, ‘‘TheCounselors of Real Estate,’’ and has contributed to the Appraisal Journal. He ini-tiated the Land Investment Survey that has been incorporated into the Pricewater-houseCoopers Korpacz Real Estate Investor Survey. He is on the national facultyfor the Appraisal Institute and adjunct professor at New York University.

Mr. MacCrate co-authored Chapters 9 and 10 in the Cost of Capital: Applica-tions and Examples, 4th ed.Workbook and Technical Supplement.

Harold G. Martin Jr., CPA/ABV/CFF, ASA, CFE, is the Principal-in-Charge of theBusiness Valuation, Forensic, and Litigation Services Group for Keiter, Stephens,Hurst, Gary & Shreaves, P.C., in Richmond and Charlottesville, Virginia. He hasmore than 25 years of experience in financial consulting, public accounting, and fi-nancial 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 federalcourt–appointed accountant for receiverships. He is an adjunct faculty member ofthe College of William and Mary Mason Graduate School of Business and teachesforensic accounting and valuation in the Master of Accounting program. He is alsoa guest lecturer on valuation in the MBA program.

Prior to joining Keiter Stephens, he served as a Senior Manager in ManagementConsulting Services for Price Waterhouse and as a Director in Financial Advisory

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Services for Coopers & Lybrand. He currently serves as an instructor for theAmerican Institute of Certified Public Accountants National Business ValuationSchool and ABV Exam Review Course and also as an editorial advisor and contribu-ting author for the AICPA CPA Expert. He is a former member of the AICPA Busi-ness Valuation Committee, former editor of the AICPA ABV e-Alert, and a two-timerecipient of the AICPA Business Valuation Volunteer of the Year Award. He is afrequent speaker and author on valuation topics and is a co-author of Financial Val-uation: Applications and Models, 2nd ed. (Hoboken, NJ: JohnWiley & Sons, 2006).

Mr. Martin received his AB degree in English in 1979 from the College of WilliamandMary and hisMBA degree in 1991 fromVirginia Commonwealth University.

Mr. Martin contributed Chapter 10 of the companion Cost of Capital: Applica-tions and Examples, 4th ed. Workbook and Technical Supplement and the compan-ion Excelworksheets that appear on the JohnWiley & Sons web site.

James Morris, PhD, AM, received his PhD in Finance from University ofCalifornia, Berkeley. He is a professor of finance at the University of Colorado atDenver, where he teaches courses in business valuation, financial modeling, andfinancial management, and he has also served on the finance faculties at the WhartonSchool of University of Pennsylvania and at the University of Houston and taughtfinance courses at business schools in England, France, and Germany.

Dr. Morris’s recent publications include Introduction to Financial Models forManagement and Planning with J. Daley (CRC Press, 2009); ‘‘Life and Death ofBusinesses: A Review of Research on Firm Mortality,’’ Journal of Business Valua-tion and Economic Analysis (2009); ‘‘Firm Mortality and Business Valuation,’’ Val-uation Strategies (September–October 2009); ‘‘The Iterative Process Using CAPM toCalculate the Cost of Equity Component of the Weighted Average Cost of CapitalWhen Capital Structure is Changing,’’ Appendix 7.2 in Pratt and Grabowski, Costof 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 Capi-tal,’’ Business Valuation Review (Fall 2005); and ‘‘Reconciling the Equity andInvested Capital Methods of Valuation When the Capital Structure is Changing,’’Business Valuation Review (March 2004). In addition, his research articles havebeen published in the Journal of Finance, Journal of Financial & Quantitative Anal-ysis, Journal of Applied Psychology, Academy of Management Journal, and Man-agement Science, among others. In addition to teaching, he provides valuationservices to the business community.

Dr. Morris contributed Chapter 6 of the Cost of Capital: Applications andExamples, 4th ed. Workbook and Technical Supplement and the companion Excelworksheets that appear on the John Wiley & Sons web site.

David M. Ptashne, CFA, is an Associate Director with Ceteris, a global economicconsulting firm that provides transfer pricing and business valuation services. Mr.Ptashne has performed numerous valuation studies of businesses, interests in busi-nesses, and intangible assets across various industries, including advertising andcommunications, consumer products, technology, financial services, integrated oiland gas, retail, and health care. He received a Bachelor of Science degree in Financewith High Honors from the University of Illinois at Urbana-Champaign.

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Mr. Ptashne contributed Chapters 2 and 4 of the Cost of Capital: Applicationsand Examples, 4th ed.Workbook and Technical Supplement.

Mark Shirley, CPA/ABV/CFE, has earned advanced accreditations: Certified Valua-tion Analyst and Certified Forensic Financial Analyst. After leaving the Internal Rev-enue Service in 1984, Mr. Shirley’s consulting practice has concentrated on thedisciplines of business valuation, forensic/investigative accounting, and financialanalysis/modeling. Professional engagements have included business valuation, valu-ation of options/warrants, projections and forecasts, statistical sampling, commer-cial damage modeling, personal injury loss assessment, and the evaluation ofproffered expert testimony underDaubert and the Federal Rules of Evidence.

Since 1988, his technical contributions have been published by Wiley LawPublications, Aspen Legal Press, and in professional periodicals, including Valu-ation Examiner, BewertungsPraktiker Nr. (a German-language business valua-tion journal), Practical Accountant, CPA Litigation Services Counselor,Gatekeeper Quarterly, Journal of Forensic Accounting, and local legal societypublications. Since 1997, Mr. Shirley has authored courses for NACVA’s Fun-damentals, Techniques & Theory; Forensic Institute, and Consultant’s TrainingInstitute. He also has developed several advanced courses for the NACVA inapplied 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 anadjunct 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 Over-sight 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 IndianTribal Court, and local specialty courts. Court appointments have been received invarious matters adjudicated before the Louisiana Nineteenth Judicial District Court.

The NACVA has recognized Mr. Shirley’s contributions to professional educa-tion 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 andExamples, 4th ed.Workbook and Technical Supplement andAppendix III of theWork-book and Technical Supplementwhich appears on the JohnWiley& Sons web site.

About the Authors xvii

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Acknowledgments

This book has benefited immensely from review by many people with a high levelof knowledge and experience in cost of capital and valuation. These people

reviewed the manuscript, and the book reflects their invaluable efforts and legions ofconstructive suggestions:

Bruce Bingham Mark LeeCapstone Advisory Group LLC Eisner LLPNew York, NY New York, NY

Stephen J. Bravo DanMcConaughyApogee Business Valuation Grobstein, Horwath LLPFramingham, MA Sherman Oaks, CA

James Budyak George PushnerValuation Research Corp. Duff & Phelps LLCMilwaukee, WI New York, NY

David Clarke Raymond RathThe Griffing Group PricewaterhouseCoopers LLCOak Park, IL Los Angeles, CA

Stan Deakin Jeffrey TarbellMosaic Capital LLC Houlihan LokeyLos Angeles, CA San Francisco, CA

Donald A. Erickson Terence TchenErickson Partners, LLC Houlihan LokeyDallas, TX Los Angeles, CA

Aaron A. Gilcreast Marianna TodorovaPricewaterhouseCoopers LLC Duff & Phelps LLCAtlanta, GA New York, NY

Professor Joao Gomes Richard M. WiseThe Wharton School of theUniversity of Pennsylvania

Wise, Blackman, LLPMontreal (Quebec), Canada

Philadelphia, PA

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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manuscript among reviewers, contributors, authors, and publisher; typing; ob-taining permissions; and other invaluable help.

& David Fein of ValuSource for contributing Appendix II of the Workbook andTechnical Supplement on ValuSource Pro.

& Noah Gordon of Shannon Pratt Valuations, Inc., for general editorial assistance.

We thank all of the people singled out for their assistance. Of course, any errorsare our responsibility.1

Shannon PrattRoger 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 isoffered for educational purposes with the understanding that neither the authors nor Duff &Phelps, LLC, are engaged in rendering legal, accounting, or any other professional servicethrough presentation of this material. The information presented in this book has beenobtained with the greatest of care from sources believed to be reliable, but is not guaranteedto be complete, accurate, or timely. The authors and Duff & Phelps LLC expressly disclaimany liability, including incidental or consequential damages, arising from the use of thismaterial or any errors or omissions that may be contained in it.

xx ACKNOWLEDGMENTS

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Notation System and AbbreviationsUsed in This Book

A source of confusion for those trying to understand financial theory and methodsis that financial writers have not adopted a standard system of notation. The no-

tation system used in this volume is adapted from the fifth edition of Valuing a Busi-ness: The Analysis and Appraisal of Closely Held Companies, by Shannon P. Pratt(New York: McGraw-Hill, 2008).

VALUE AT A PO INT IN T IME

Pn ¼ Stock price in period nP0 ¼ Stock price at valuation periodPi ¼ Price per share for company i (seen elsewhere as PV)PV ¼ Present valuePVb ¼ Present value of net cash flows due to business operations before cost of

financingPVkeu ¼ Present value of net cash flows using unlevered cost of equity capital, keu, as

the discount ratePVts ¼ Present value of tax shield due to interest expense on debt capitalPVdc ¼ Present value of net distress-related costsPVTSn ¼ Present value of the tax shield as of time¼ nPVf ¼ Present value of invested capitalTVn ¼Terminal value at time nMe ¼Market value of equity capital (stock)Md ¼Market value of debt capitalMp ¼Market value of preferred equityMVIC ¼Market value of invested capital

¼Enterprise value¼Me þMd þMp

BV ¼Book value of net assetsBVn ¼Book value of equity at time¼ nBVi ¼Measure of book value (typically book value to market value) of stock of

company iFd ¼ Fair value of debtFVRU ¼ Fair value of reporting unitFVNWCRU ¼ Fair value of net working capital of the reporting unitFVICRU ¼ Fair value of invested capital of the reporting unitFVFARU ¼ Fair value of fixed assets of the reporting unit

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FVIARU ¼ Fair value on intangible assets, identified and individually valued, of thereporting unit

FVUIVRU ¼ Fair value of unidentified intangibles value (i.e., goodwill) of the reportingunit

FVdRU ¼ Fair value of debt capital of the reporting unitFVeRU ¼ Fair value of equity capital of the reporting unitFMVBE ¼ Fair market value of the business enterpriseFMVNWC ¼ Fair market value of net working capitalFMVFA ¼ Fair market value of fixed assetsFMVIA ¼ Fair market value on intangible assetsFMVUIV ¼ Fair market value of unidentified intangibles value (i.e., goodwill)FMVe ¼ Fair market value of equity capitalFMVe;n;up ¼ Fair market value of equity at time¼ n assuming ‘‘up’’ scenario (value of BE

increases)FMVBE;n;down ¼ Fair market value of business enterprise at time¼ n assuming ‘‘down’’

scenario (value of BE decreases)FMVe;n;down ¼ Fair market value of equity at time¼ n assuming ‘‘down’’ scenario (value of

BE decreases)

COST OF CAP I TA L AND RATE OF RETURN VAR IABL ES

k ¼Discount rate (generalized)kc ¼Country cost of equityke ¼Discount rate for common equity capital (cost of common equity capital).

Unless otherwise stated, it generally is assumed that this discount rate isapplicable to net cash flow available to common equity.

ke,local ¼Discount rate for equity capital in local country for discounting expectedcash flows in local currency

ke,u.s. ¼Discount rate for equity capital in the United StateskBV ¼Rate of return on book value, retained portion of net income, usually esti-

mated as¼NInþ1/BVn

keu ¼Cost of equity capital, unlevered (cost of equity capital assuming firm fi-nanced with all equity)

klocal ¼Cost of equity capital in local countryki ¼Discount rate for company ikni ¼Discount rate for equity capital when net income rather than net cash flow is

the measure of economic income being discountedk ptð Þ ¼Discount rate applicable to pretax cash flowskp ¼Discount rate for preferred equity capitalkd ¼Discount 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 preced-ing categories, requiring expanded subscripts.)

¼ kd ptð Þ � 1� tax rateð Þkd ptð Þ ¼Cost of debt prior to tax effectkA ¼Discount rate for the firm’s assetskTS ¼Rate of return used to present value tax savings due to deducting interest

expense on debt capital financingkeRU ¼After tax rate of return on equity capital of reporting unit

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kNWC ptð Þ ¼Rate of return for net working capital financed with debt capital (measuredbefore interest tax shield) and equity capital

kFA ptð Þ ¼Rate of return for fixed assets financed with debt capital (measured beforeinterest tax shield) and equity capital

kdRU ¼Rate of return on debt capital of the reporting unit net of tax effect¼ kd ptð ÞRU � 1� tax rateð Þ

kNWCRU ¼Rate of return for net working capital of the reporting unit financed withdebt capital (return measured net of the tax effect on debt financing, if any)and equity capital

kFARU ¼Rate of return for fixed assets financed with debt capital (return measured netof the tax effect on debt financing, if any) and equity capital

kIARU ¼Rate of return for identified and individually valued intangible assets fi-nanced with debt capital (return measured net of the tax effect on debtfinancing, if any) and equity capital

kUIVRU ¼Rate of return for unidentified intangibles value of the reporting unit financedwith debt capital (return measured net of the tax effect on debt financing, ifany) and equity capital

kIAþUIV ptð Þ ¼ Pretax rate of return on all intangible assets, identified and individually val-ued, plus the unidentified intangible value financed with debt capital (meas-ured before interest tax shield) and equity capital

c ¼Capitalization ratec ptð Þ ¼Capitalization rate on pretax cash flows (Note: For complex capital struc-

tures, there could be more than one class of capital in any of the precedingcategories, requiring expanded subscripts.)

D/P0 ¼Dividend yield on stockDRj ¼Downside risk in the local market (U.S. dollars)DRw ¼Downside risk in global (‘‘world’’) market (U.S. dollars)R ¼Rate of returnRi ¼Return on stock iRd ¼Rate of return on subject debt (e.g., bond) capitalRm,n ¼Return on market portfolio in current month nRf ¼Rate of return on a risk-free securityRf;n ¼Risk-free rate in current month nRf; local ¼Return on the local country government’s (default-risk-free) paperRf;u:s: ¼U.S. risk-free rateRlocal euro $issue ¼Current market interest rate on debt issued by the local country government

denominated in U.S. dollars (‘‘euro-dollar’’ debt), same maturity as debtissued by the local country government denominated in U.S. dollars

(Rlocal euro $issue

�Rf;u:s:) ¼Yield spread between government bonds issued by the local country versusU.S. government bonds

Rn ¼Return on individual security subject stock in current monthRm ¼Historical rate of return on the ‘‘market’’RP ¼Risk premiumRPm ¼Risk premium for the ‘‘market’’ (usually used in the context of a market for

equity securities, such as the NYSE or S&P 500)RPs ¼Risk premium for ‘‘small’’ stocks (usually average size of lowest quintile or

decile of NYSE as measured by market value of common equity) over andabove RPm

RPmþs ¼Risk premium for the market plus risk premium for size (Duff & Phelps RiskPremium Report data for use in build-up method)

Notation System and Abbreviations Used in This Book xxiii

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RPsþu ¼Risk premium for small size plus risk premium attributable to the specificdistressed company

RPmþsþu ¼Risk premium for the ‘‘market’’ plus risk premium for size plus risk attribut-able to the specific company

RPu ¼Risk premium for company-specific or unsystematic risk attributable to thespecific company

RPw ¼The equity risk premium on a ‘‘world’’ diversified portfolioRPi ¼Risk premium for the ith securityRPi,s ¼Bi,s � Si¼Risk premium for size of company iRPi,BV ¼Bi,BV � BVi¼Risk premium for book value of company iRPi,u ¼Bi,u � Ui¼Risk premium for unique or unsystematic risk of company iRPlocal ¼Equity risk premium in local country’s stock marketRIiL ¼ Full-information levered beta estimate of the subject companyE Rð Þ ¼Expected rate of returnE Rmð Þ ¼Expected 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)

E Rið Þ ¼Expected rate of return on security iE Rdivð Þ ¼Expected rate of return on dividendE Rcap�gains

� � ¼Expected rate of return on capital gainsE(Ri,j) ¼Expected rate of return on security i for undiversified investor jB ¼Beta (a coefficient, usually used to modify a rate of return variable)Bi ¼Expected beta of the stock of company iBL ¼Levered beta for (equity) capitalBU ¼Unlevered beta for (equity) capitalBLS ¼Levered segment betaBd ¼Beta for debt capitalBp ¼Beta of preferred capitalBe ¼Beta (equity) expandedBop ¼Operating beta (beta with effects of fixed operating expense removed)Bi ¼Beta of company i (F-F beta)Bi,m ¼ Sensitivity of return of stock of company i to the market risk premium

or ERPBi,s ¼ Sensitivity of return of stock of company i to a measure of size, S, of

company iBi,BV ¼ Sensitivity of return of stock of company i to a measure of book value (typi-

cally measure of book-value-to-market-value) of stock of company iBi,u ¼ Sensitivity of return of stock of company i to a measure of unique or un-

systematic risk of company iBn ¼Estimated market coefficient based on sensitivity to excess returns on market

portfolio in current monthBlocal ¼Market risk of the subject company measured with respect to the local securi-

ties marketBw ¼Market or systematic risk measured with respect to a ‘‘world’’ portfolio of

stocksBi1 . . . Bin ¼ Sensitivity of security i to each risk factor relative to the market average sen-

sitivity to that factorB0i ¼True beta estimate for stock of company i based on relationship to excess

returns on market portfolio of equity plus debt,ME þMD

Bu.s.� RPu.s. ¼Risk premium appropriate for a U.S. company in similar industry as the sub-ject company in local country, expressed in U.S. dollar-denominated returns

xxiv NOTATION SYSTEM AND ABBREVIATIONS USED IN THIS BOOK

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FI-Beta ¼ Full-information beta for industryTBi ¼Total beta for security ibcr ¼Country covariance with regionbcw ¼Country covariance with worldSi ¼Measure of size of company iUi ¼Measure of unique or unsystematic risk of company il ¼A measure of individual stock’s liquidityRP1 . . .RPn ¼Risk premium associated with risk factor 1 through n for the average asset in

the market (used in conjunction with arbitrage pricing theory)si ¼ Small-minus-big coefficient in the Fama-French regressionSMBP ¼Expected small-minus-big risk premium, estimated as the difference between

the historical average annual returns on the small-cap and large-cap portfo-lios (also shown as SMB)

hi ¼High-minus-low coefficient in the Fama-French regressionHMLP ¼Expected high-minus-low risk premium, estimated as the difference between

the historical average annual returns on the high book-to- market and lowbook-to-market portfolios (also shown as HML)

Fd ¼ Face value of outstanding debtb ¼ 1� Payout ratio¼ retention ratioWACC ptð Þ ¼Weighted average cost of capital (before interest tax shield)WACCRU ¼Overall rate of return for the reporting unit

¼Weighted average cost of capital for the reporting unitWACC ptð ÞRU ¼Before interest tax shield WACC of the reporting units2i ¼Variance of returns for security i

s2m ¼Variance of the returns on the market portfolio (e.g., S&P 500)

s2e ¼Variance of error terms

s ¼ Standard deviationse ¼ Standard deviation of returns on firm’s common equitysA ¼ Standard deviation of returns on firm’s assetssB ¼ Standard deviation of operating cash flows of business before cost of

financingsrev ¼ Standard deviation of revenuessBE ¼ Standard deviation of value of business enterpiseslocal ¼Volatility of subject (local) stock marketsu:s: ¼Volatility of U.S. stock marketsstock ¼Volatility of local country’s stock marketsbond ¼Volatility of local country’s bond marketsi ¼ Standard deviation of returns for security ism ¼ Standard deviation of returns for the market portfolio (e.g., S&P 500)si,m ¼Variance of returns on the security, i, and the market,msD

2 ¼Variance in excess returns on market of debtsMEþMD

2 ¼Variance in excess returns on market portfolio of equity plus debt,ME þMD

r ¼Correlation coefficient between the returns on the security, i, and themarket, m

dr ¼Regional risk not included in RPw

CCRlocal ¼Country credit rating of local countryl ¼Company’s exposure to the local country riskt ¼Tax rate (expressed as a percentage of pretax income)h ¼Holding periodInflationlocal ¼Expected rate of inflation in local countryInflationu:s: ¼Expected rate of inflation in U.S.

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I NCOME VAR IAB L ES

E ¼Expected economic income (in a generalized sense; i.e., could be dividends,any of several possible definitions of cash flows, net income, etc.)

F ¼ Fixed operating assets (without regard to costs of financing)Fc ¼ Fixed operating costs of the businessNI ¼Net income (after entity-level taxes)NCIe,n ¼Net comprehensive income to common equity in period n, which includes

income terms reported directly in the equity account rather than in the in-come statement

NCIf,n ¼Net comprehensive income to the firm in period n, which includes incometerms reported directly in the equity account rather than in the incomestatement

CF ¼Cash flow for a specific periodNCFe ¼Net cash flow (free cash flow) to equityNCFf ¼Net cash flow (free cash flow) to the firm (to overall invested capital, or entire

capital structure, including all equity and long-term debt)NCFue ¼Net cash flow to unlevered equityD ¼DividendsDe,n ¼Distributions to common equity, net of new issues of common equity in pe-

riod nDf,n ¼Distributions to total capital, net of new issues of debt or equity capital in

period nRIe,n ¼Residual income for common equity capitalTS ¼ Present value of tax savings due to deducting interest expense on debt capital

financingEBT ¼Earnings before taxesEBIT ¼Earnings before interest and taxesEBITDA ¼Earnings before interest, taxes, depreciation, and amortizationV ¼Variable operating costsAEG ¼Abnormal earnings growth

PER I ODS OR VAR IABL ES IN A SER I ES

i ¼ ith period or ith variable in a series (may be extended to the jth variable, thekth variable, etc.)

n ¼Number of periods or variables in a series, or the last number in a series0 ¼ Period 0, the base period, usually the latest year immediately preceding the

valuation datepy ¼ Partial year of first year following the valuation date

WE IGHT INGS

W ¼WeightWe ¼Weight of common equity in capital structure

¼Me/(Me þMd þMp)Wp ¼Weight of preferred equity in capital structure

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¼Mp=ðMe þMd þMpÞWd ¼Weight of debt in capital structure

¼Md/(Me þ Md þ Mp) (Note: For purposes of computing a weighted averagecost of capital [WACC], it is assumed that preceding weightings are at mar-ket value.)

WdRU ¼Weight of debt capital in capital structure of reporting unit¼ Fair value of debt capital/FVRU

Ws ¼Weight of segment data to total business (e.g., sales, operating income)WNWC ¼Weight of net working capital in FMVBE

¼ FMVNWC=FMVBE

WNWCRU ¼Weight of net working capital in FVRU

¼ FVNWCRU=FVRU

WFA ¼Weight of fixed assets in FMVBE

¼ FMVFA=FMVBE

WFARU ¼Weight of fixed assets in FVRU

¼ FVFARU=FVRU

WIARU ¼Weight of intangible assets in FVRU

¼ FVIARU=FVRU

WUIVRU ¼Weight of unidentified intangibles value FVRU

¼ FVUIVRU (i.e., goodwill)=FVRU

WTS ¼Weight of TS in FMVBE

¼TS=FMVBE

GROWTH

g ¼Rate of growth in a variable (e.g., net cash flow)gi ¼Dividend growth rate for company igni ¼Rate of growth in net income

MATHEMAT ICA L FUNCT I ONS

S ¼ Sum of (add all the variables that follow)\ ¼ Product of (multiply together all the variables that follow)�X ¼Mean average (the sum of the values of the variables divided by the number

of variables)G ¼Geometric mean (product of the values of the variables taken to the root of

the number of variables)a ¼Regression constante ¼Regression error termei ¼Error term, difference between predicted return and realized return, Ri

1 ¼ InfinityN (*) ¼Cumulative normal density function (the area under the normal probability

distribution)D ¼Change in . . . (whatever follows)

Notation System and Abbreviations Used in This Book xxvii

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NOTAT I ON FOR REAL PROPERTY VALUAT I ON (CHAPTER9 OF WORKBOOK AND TECHN ICAL SUPPL EMENT )

DSCR ¼Debt service coverage ratioEGIM ¼Effective gross income multiplierNOI; Ip ¼Net operating incomeOER ¼Operating expense ratesPVp ¼Overall value or present value of the propertyke ¼Equity discount or yield rate (dividend plus appreciation)km ¼Mortgage interest ratekp ¼ Property yield discount ratecp ¼Overall property capitalization ratece ¼Dividend to equity capitalization ratecm ¼Mortgage capitalization rate or constantcn ¼Terminal or residual or going-out capitalization ratecB ¼Building capitalization ratecL ¼Land capitalization ratecLF ¼Leased fee capitalization ratecLH ¼Leasehold capitalization rateA ¼Change in income and value (adjustment factor)P ¼ Principal paid off over the holding period1=Sn ¼ Sinking fund factor at the equity discount or yield rate keð ÞDp ¼Change in value over the holding periodSC% ¼Cost of salePGI ¼ Potential gross incomePGIM ¼ Potential gross income multiplierEGI ¼Effective gross incomeNIM ¼Net income multiplierFd=PVp ¼ Face value of debt (loan amount outstanding) to value ratio[1�(Fd/PVp)] ¼Equity to value ratioMB ¼Building valueMm ¼Mortgage valueML ¼Land valueMLF ¼Leased fee valueMLH ¼Leasehold valueIp ¼Overall income to the propertyIL ¼Residual income to the landIB ¼Residual income to the buildingIe ¼Equity incomeIm ¼Mortgage incomeILF ¼ Income to the leased feeILH ¼ Income to the leasehold

ABBREV I AT I ONS

ERP ¼Equity risk premium (usually the general equity risk premium for which thebenchmark for equities is either the S&P 500 stocks or the NYSE stocks)

WACC ¼Weighted average cost of capitalWARA ¼Weighted average return on assets

xxviii NOTATION SYSTEM AND ABBREVIATIONS USED IN THIS BOOK