Cost of capital applications and examples

In terms of breadth, this text treats cost of capital for uses in business valuation, projectassessment and capital budgeting, divisional cost of capital, reporting unit valuation and go

COST OF CAPITAL Applications and Examples

Third Edition

SHANNON P PRATT ROGER J GRABOWSKI

John Wiley & Sons, Inc

Cost of Capital

Applications and Examples

COST OF CAPITAL Applications and Examples

Third Edition

SHANNON P PRATT ROGER J GRABOWSKI

John Wiley & Sons, Inc

Copyright # 2008 by John Wiley & Sons, Inc All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by anymeans, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted underSection 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of thePublisher, 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 atwww.copyright.com Requests to the Publisher for permission should be addressed to the PermissionsDepartment, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, 201-748-6011, fax 201-748-6008,

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Library of Congress Cataloging-in-Publication Data:

Pratt, Shannon P

Cost of capital: applications and examples/Shannon P Pratt, Roger J Grabowski.—3rd ed

p cm

‘‘Published simultaneously in Canada.’’

Previous editions had subtitle: Estimation and applications

Includes bibliographical references and index

ISBN 978-0-470-17115-8 (cloth: alk paper)

To our families for their support and encouragement,

without which our careers and this book would not have been possible

Millie

Portland, OR

Meg Springfield, VA

Portland, OR

Graham Fayetteville, AR

Mary Ann

Notation System and Abbreviations Used in This Book xxix

Part 1 Cost of Capital Basics 1

1 Defining Cost of Capital 3

2 Introduction to Cost of Capital Applications: Valuation and Project Selection 9

3 Net Cash Flow: Preferred Measure of Economic Income 15

4 Discounting versus Capitalizing 23

5 Relationship between Risk and the Cost of Capital 39

Appendix 5A FASB’s Concepts Statement No 7: Cash Flows and Present

Value Discount Rates 48

6 Cost Components of a Company’s Capital Structure 51

Part 2 Estimating the Cost of Equity Capital and the Overall Cost of Capital 67

7 Build-up Method 69

8 Capital Asset Pricing Model 79

9 Equity Risk Premium 89

Appendix 9A Bias Issues in Compounding and Discounting 114

10 Beta: Differing Definitions and Estimates 117

Appendix 10A Formulas and Examples for Unlevering and Levering Equity Betas 143

Appendix 10B Examples of Computing OLS Beta, Sum Beta, and Full-Information

Beta Estimates 151

11 Criticism of CAPM and Beta versus Other Risk Measures 161

Appendix 11A Example of Computing Downside Beta Estimates 176

12 Size Effect 179

13 Criticisms of the Size Effect 209

Appendix 13A Other Data Issues Regarding the Size Effect 220

14 Company-Specific Risk 225

15 Alternative Cost of Equity Capital Models 243

16 Implied Cost of Equity Capital 255

17 Weighted Average Cost of Capital 265

Appendix 17A Iterative Process Using CAPM to Calculate the Cost of Equity

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Component of the Weighted Average Cost of Capital When Capital Structure Is Constant 283Appendix 17B Iterative Process Using CAPM to Calculate the Cost of Equity

Component of the Weighted Average Cost of Capital When Capital Structure Is Changing 297

18 Global Cost of Capital Models 309

19 Using Morningstar Cost of Capital Data 331

Part 3 Corporate Finance Officers: Using Cost of Capital Data 361

20 Capital Budgeting and Feasibility Studies 363

21 Cost of Capital for Divisions and Reporting Units 369

22 Cost of Capital in Evaluating Acquisitions and Mergers 385

23 Cost of Capital in Transfer Pricing 393

24 Central Role of Cost of Capital in Economic Value Added 409

Part 4 Cost of Capital for Closely Held Entities 417

25 Handling the Discount for Lack of Marketability for Operating Businesses 419

26 Private Company Discount 429

27 Cost of Capital of Interests in Pass-Through Entities 437

28 Cost of Capital in Private Investment Companies 449

29 Relationship between Risk and Returns in Venture Capital Investments 469

Part 5 Other Topics 479

30 Minority versus Control Implications of Cost of Capital Data 481

31 How Cost of Capital Relates to the Excess Earnings Method of Valuation 493

32 Adjusting the Discount Rate to Alternative Economic Measures 501

33 Estimating Net Cash Flows 505

Appendix 33A Estimating the Value of a Firm in Financial Distress 518

34 Common Errors in Estimation and Use of Cost of Capital 529

35 Cost of Capital in the Courts 539

Part 6 Real Estate and Ad Valorem 559

36 Cost of Capital of Real Property–Individual Assets 561

Appendix 36A Valuing Real Property 580

37 Cost of Capital of Real Estate Entities 587

Appendix 37A Valuing Real Estate Entities 615

38 Cost of Capital in Ad Valorem Taxation 623

Part 7 Advice to Practitioners 639

39 Dealing with Cost of Capital Issues 641

40 Questions to Ask Business Valuation Experts 647

Appendices 653

I Bibliography 655

II Data Resources 683

III International Glossary of Business Valuation Terms 693

IV Sample Report Submitted to U.S Tax Court by Roger J Grabowski 703

V ValuSource Valuation Software 725

VI Review of Statistical Analysis 733

Index 771

About the Authors

Dr Shannon P Pratt, CFA, FASA, MCBA, CM&AAis the chairman and CEO of Shannon PrattValuations, 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 thefounders of Willamette Management Associates, for which he was a managing director for almost 35years

He has performed valuation assignments for these purposes: transaction (acquisition, divestiture, organization, public offerings, public companies going private), taxation (federal income, gift, andestate and local ad valorem), financing (securitization, recapitalization, restructuring), litigationsupport and dispute resolution (including dissenting stockholder suits, damage cases, and corporateand marital dissolution cases), and management information and planning He has also managed avariety of fairness opinion and solvency opinion engagements

re-Dr Pratt has testified on hundreds of occasions in such litigated matters as dissenting stockholdersuits, various types of damage cases (including breach of contract, antitrust, and breach of fiduciaryduty), divorces, and estate and gift tax cases Among the cases in which he has testified are Estate ofMark S Gallo V Commissioner, Charles S Foltz, et al V U.S News & World Report et al., Estate ofMartha Watts V Commissioner, and Okerlund V United States He has also served as appointed arbitra-tor in numerous cases

PREVIOUS EXPERIENCE

Before founding Willamette Management Associates in 1969, Dr Pratt was a professor of businessadministration at Portland State University During this time, he directed a research center known asthe Investment Analysis Center, which worked closely with the University of Chicago’s Center forResearch in Security Prices

EDUCATION

Doctor of Business Administration, Finance, Indiana University

Bachelor of Arts, Business Administration, University of Washington

PROFESSIONAL AFFILIATIONS

Dr Pratt is an Accredited Senior Appraiser and Fellow (FASA), Certified in Business Valuation, of theAmerican Society of Appraisers (their highest designation) He is also a Chartered Financial Analyst(CFA) of the Institute of Chartered Financial Analysts, a Master Certified Business Appraiser(MCBA) of the Institute of Business Appraisers, a Master Certified Business Counselor (MCBC),and is Certified in Mergers and Acquisitions (CM&AA) with The Alliance of Merger andAcquisition Advisors

Dr Pratt is a life member of the American Society of Appraisers, a life member of the Business luation Committee of that organization and teaches courses for the organization He is also a lifetimemember emeritus of the Advisory Committee on Valuations of The ESOP Association He is a recipient

Va-of the magna cum laude award Va-of the National Association Va-of Certified Valuation Analysts for service to

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the business valuation profession He is also the first life member of the Institute of Business Appraisers.

He is a member and a past president of the Portland Society of Financial Analysts, the recipient of the

2002 Distinguished Achievement Award, and a member of the Association for Corporate Growth Dr.Pratt is a past trustee of The Appraisal Foundation and is currently an outside director and chair ofthe 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, 5thedition (New York: McGraw-Hill, 2007); coauthor, Valuing Small Businesses and ProfessionalPractices, 3rd edition with Robert Schweihs and Robert Reilly (New York: McGraw-Hill, 1998);coauthor, Guide to Business Valuations, 18th edition with Jay Fishman, Cliff Griffith and JimHitchner (Fort Worth, TX: Practitioners Publishing Company, 2008); coauthor, Standards of Value,with William Morrison and Jay Fishman (New York: John Wiley & Sons, 2007); coauthor, BusinessValuation and Taxes: Procedure, Law, and Perspective, with Judge David Laro (New York: JohnWiley & Sons, 2005); author, Business Valuation Discounts and Premiums (New York: John Wiley &Sons, 2001); Business Valuation Body of Knowledge: Exam Review and Professional Reference, 2ndedition (New York: John Wiley & Sons, 2003); The Market Approach to Valuing Businesses, 2ndedition (New York: John Wiley & Sons, 2005); and The Lawyer’s Business Valuation Handbook(Chicago: American Bar Association, 2000) He has also published nearly 200 articles on businessvaluation topics

Roger Grabowski, ASA,is a managing director of Duff & Phelps, LLC

Mr Grabowski has directed valuations of businesses, partial interests in businesses, intellectualproperty, intangible assets, real property, and machinery and equipment for various purposes includingtax (income and ad valorem) and financial reporting; mergers, acquisitions, formation of joint ventures,divestitures, and financing He developed methodologies and statistical programs for analyzing usefullives of tangible and intangible assets, such as customers and subscribers His experience includes work

in a wide range of industries including sports, movies, recording, broadcast and other entertainmentbusinesses; 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 manufacturingbusinesses

Mr Grabowski has testified in court as an expert witness on the value of closely held businesses andbusiness interests, matters of solvency, valuation, and amortization of intangible assets, and other valua-tion issues His testimony in U.S District Court was referenced in the U.S Supreme Court opiniondecided in his client’s favor in the landmark Newark Morning Ledger income tax case Among othercases in which he has testified was Herbert V Kohler Jr., et al., v Comm (value of stock of TheKohler Company); The Northern Trust Company, et al., v Comm (the first U.S Tax Court case that re-cognized the use of the discounted cash flow method for valuing a closely held business); OaklandRaiders v Oakland-Alameda County Coliseum Inc et al (valuation of the Oakland Raiders); 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 shareholderaction); and TMR Energy Limited v The State Property Fund of Ukraine (arbitration on behalf of world’slargest private company in Stockholm, Sweden, on cost of capital for oil refinery in Ukraine in a contractdispute)

PREVIOUS EXPERIENCE

He was formerly managing director of the Standard & Poor’s Corporate Value Consulting practice, apartner 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 ofValtec Associates, and a vice president of American Valuation Consultants

PUBLICATIONS

Mr Grabowski coauthors the annual Duff & Phelps’ Risk Premium Report He lectures and publishesregularly He is the coauthor of three chapters (on equity risk premium, valuing pass-through entities,and valuing sports teams) in Robert Reilly and Robert P Schweihs, The Handbook of BusinessValuation and Intellectual Property Analysis (New York: McGraw-Hill, 2004) He teaches coursesfor the American Society of Appraisers including Cost of Capital, a course he developed He is theeditor of the Business Valuation Review, the quarterly Journal of the Business Valuation Committee

David Feinis the CEO and president of ValuSource, which for over 20 years has been the leadingprovider of business valuation software, data, and report writers for CPAs, M&A professionals, and busi-ness owners

Mr Fein’s mission is to create state-of-the-art technology to automate and standardize complex nancial analysis and reporting tasks He has a bachelor’s degree in computer science and an MBA Youmay reach him at 1.800.825.8763 100, or at dfein@valusourcesoftware.com

fi-William H Frazieris a principal and founder of the firm of Howard Frazier Barker Elliott, Inc, andmanages its Dallas office He has 30 years of experience in business valuation and corporate finance

Mr Frazier has been an Accredited Senior Appraiser of the American Society of Appraisers (ASA)since 1987 and serves on the ASA’s Business Valuation Committee as secretary He has participated

as an appraiser and/or expert witness in numerous U.S Tax Court cases, including testimony inJelke, McCord, Dunn and Gladys Cook Mr Frazier has written numerous articles on the subject of busi-ness valuation for tax purposes, appearing in such publications as the Business Valuation Review, Valua-tion Strategies, BV E-Letter, Shannon Pratt’s Business Valuation Update, and Estate Planning He is thecoauthor of the chapter on valuing family limited partnerships in Robert Reilly and Robert P Schweihs,eds., The Handbook of Business Valuation and Intellectual Property Analysis (New York: McGraw-Hill,2004) Mr Frazier serves on the Valuation Advisory Board of Trusts & Estates Journal

James Harrington, MBA,is a product manager in Morningstar’s Individual Investor Businesssegment He leads the group that produces the widely used and cited SBBI Classic and Valuation Year-books, the Cost of Capital Yearbook, the Beta Book, and various international and domestic reports

At Morningstar since March of 2006, Mr Harrington has expanded the product offerings and creased sales for Morningstar publications and reports and is an accomplished financial writer andanalyst.

in-Immediately prior to his tenure at Morningstar, he was a product manager in the financial nications group at Ibbotson Associates Before that, he was a bond and bond portfolio analyst, worked atthe Chicago Board of Trade in the bond options pit for a filling group, managed inbound and outbounddock workers at a large trucking firm, and was even a Teamster for a year

commu-Mr Harrington holds a bachelor’s degree in marketing from Ohio State University and an MBA inboth finance and economics from the University of Illinois at Chicago, where he graduated at the top inhis class

Carl Hoemkestarted in valuation when he accepted a position as a tax appraiser for the local taxassessor’s office His education is in architecture and business After working for the tax assessor’soffice for a period of about eight years, he joined a property tax consulting firm

He later became a partner at one of the big four accounting firms, first as a tax partner then ultimately

as a corporate finance partner responsible for valuations primarily in energy and telecommunications

He is currently a managing director with Duff & Phelps, where he leads the firm’s State & Local Taxpractice while continuing to provide valuation for other purposes including financial reporting and liti-gation support

Jim MacCrate, MAI, CRE, ASA owns his own boutique real estate valuation and consultingcompany, 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 rector of the Real Estate Valuation/Advisory Services Group at Price Waterhouse LLP and Pricewater-houseCoopers LLP He received a B.S Degree from Cornell University and M.B.A from Long IslandUniversity, C W Post Center He has written numerous articles for Price Waterhouse LLP, ‘‘The Coun-selors of Real Estate,’’ and has contributed to the Appraisal Journal He initiated the Land InvestmentSurvey that has been incorporated into The PricewaterhouseCoopers Korpacz Real Estate InvestorSurvey He is on the national faculty for the Appraisal Institute and adjunct professor at New York Uni-versity

di-Harold G Martin, Jr., MBA, CPA, ABV, ASA, CFE,is the principal-in-charge of the BusinessValuation, Forensic, and Litigation Services Group for Keiter, Stephens, Hurst, Gary & Shreaves,P.C He has over 25 years of experience in financial consulting, public accounting, and financial ser-vices He has appeared as an expert witness in federal and state courts, served as a court-appointedneutral business appraiser, and also served as a federal court-appointed accountant for a receivership

Mr Martin is an adjunct faculty member of The College of William & Mary Mason Graduate School

of Business and teaches valuation and forensic accounting in the Master of Accounting program.Prior to joining Keiter Stephens, he was affiliated for 13 years with Price Waterhouse and Coopers &Lybrand He is a former member of the American Institute of Certified Public Accountants BusinessValuation Committee and is a two-time recipient of the AICPA Business Valuation Volunteer of theYear Award He is an editorial advisor for the AICPA CPA Expert, a national instructor for theAICPA’s valuation education program, an AICPA faculty member for the National Judicial College, aformer member of the Appraisal Standards Board USPAP BV Task Force, and former editor of theAICPA ABV e-Alert He is a frequent speaker and author on valuation topics and is a coauthor of Finan-cial Valuation: Applications and Models, 2nd edition, published by John Wiley & Sons Mr Martin re-ceived his A.B degree in English from The College of William and Mary and his M.B.A degree fromVirginia Commonwealth University

James Morrisis a professor of finance at the University of Colorado at Denver, where he teachescourses in business valuation and financial modeling He is an AM and holds a Ph.D in Finance

from University of California, Berkeley In addition to teaching, he provides valuation services to thebusiness community.

Mark Shirleyis a licensed certified public accountant and has earned advanced accreditations:Accreditation in Business Valuation (BV), Certified Valuation Analyst (CVA), Certified Forensic Finan-cial Analyst (CFFA), and Certified Fraud Examiner (CFE) After leaving the Internal Revenue Service

in 1984, Mr Shirley’s consulting practice has concentrated on the disciplines of business valuation,forensic/investigative accounting, and financial analysis/modeling Professional engagements have in-cluded business valuation, valuation of options/warrants, projections and forecasts, statistical sampling,commercial damage modeling, personal injury loss assessment, and the evaluation of proffered experttestimony under Daubert and the Federal Rules of Evidence

Since 1988, technical contributions have been published by Wiley Law Publications, Aspen LegalPress, and in professional periodicals, including The Valuation Examiner, BewertungsPraktiker Nr (aGerman-language business valuation journal), The Practical Accountant, CPA Litigation ServicesCounselor, The Gatekeeper Quarterly, The Journal of Forensic Accounting, and local legal society pub-lications 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 advancedcourses 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 mained active since the committee’s formation He is an adjunct faculty member at the National JudicialCollege, University of Nevada, Reno, since 1998 Mr Shirley also serves on the Advisory Panel for MdexOnline; The Daubert Tracker, an online Daubert research database; and the Ethics Oversight Board forthe NACVA

re-Since 1985, Mr Shirley has provided expert witness testimony before the U.S Tax Court, FederalDistrict Court, Louisiana district courts, Tunica-Biloxi Indian Tribal Court, and local specialtycourts Court appointments have been received in various matters adjudicated before the LouisianaNineteenth Judicial District Court

The NACVA has recognized Mr Shirley’s contributions to professional education by awarding himthe Circle of Light in 2002, Instructor of the Year in 2000/2001, and multiple recognitions as Outstand-ing Member and Award of Excellence

David M Ptashne, CFA,is a Senior Associate with the Chicago office of Duff & Phelps and hasworked directly with Roger Grabowski since 2003 Mr Ptashne has performed numerous valuationstudies of businesses and interests in businesses and intangible assets across various industries includingadvertising and communications, consumer products, technology, financial services, integrated oil andgas, retail, and healthcare Mr Ptashne enjoys researching international cost of capital issues and cur-rently manages Duff & Phelps’ international cost of capital model Mr Ptashne is a member of theCFA Institute and CFA Society of Chicago He received a Bachelor of Science degree in Financewith High Honors from the University of Illinois at Urbana-Champaign

Given the central role of the cost of capital in discounted cash flow valuation, it is surprising how littlediscussion there is on the central inputs that go into its estimation Shannon Pratt and Roger Grabowskihave not only brought together all of the issues in the cost of capital computation but have done so in away that melds timely advice for practitioners with serious debate about the best practices in the area.Both authors have decades of experience, and their expertise and knowledge show not only in how theypresent the material but also in the examples they use The book has been significantly updated from thesecond edition, both in the numbers and also in the coverage of issues

In general, books on valuation-related topics take one of two paths One is to follow the cookbookstyle and provide the reader with the ‘‘right answers’’ to questions, even though there may be debateabout what is ‘‘right.’’ The other is to present the alternatives, explain the pros and cons, and trust thereader to make the right choices at the end This book adopts the latter approach and provides compre-hensive discussions not only of the standard inputs—risk-free rates, risk premiums, and debt ratios—butalso of issues that come up infrequently in valuations but often enough that practitioners look forguidance

While this book is grounded in solid theory, it is a book for practitioners, and consequently it providespractical solutions to estimation problems In the process, though, it respects their intelligence and ca-pacity to handle ambiguity by providing a balanced discussion of the choices It belongs on the book-shelves of serious valuation practitioners, and I would expect it to be widely referenced in thecoming years

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Why did we undertake writing this book? In a recent speech, Geoff Colvin, Senior Editor at Large withFortune, stated that one of the four traits of great executives is their understanding of the fundamentalreality of wealth creation—successful organizations invest to earn a rate of return in excess of theircost of capital.1These executives ingrain that understanding in the organization, and the managers atall levels come to understand their cost of capital

Our experience tells us that practitioners need assistance in better understanding and estimating thecost of capital and in communicating their results, not from the view of portfolio management but fromthe view of business owners and managers

No other valuation text designed for the practitioner treats the cost of capital in the breadth and depththat this one does In terms of breadth, this text treats cost of capital for uses in business valuation, projectassessment and capital budgeting, divisional cost of capital, reporting unit valuation and goodwill im-pairment testing, transfer pricing, utility and other regulated industry rate setting, and ad valorem (prop-erty) taxation

Emphasis is on the cost of equity capital In addition to detailed exposition of the build-up and CapitalAsset Pricing Models for estimating the cost of capital, we present in-depth analysis of the components,including the equity risk premium, beta, and the size effect We also analyze criticism of major modelsfor developing estimates of the cost of capital in use today and present procedures for a number of alter-native models

We discuss the Duff & Phelps Risk Premium studies, which are becoming more widely used tools inestimating the cost of equity capital

WHAT’S NEW IN THIS EDITION

Throughout the book, we summarize the results and practical implications of the latest research, much ofwhich has been gleaned from unpublished academic working papers

EQUITY RISK PREMIUM 4 TO 6 PERCENT

Importantly, based on empirical research on the magnitude of the equity risk premium, we conclude that

it is in the range of 4% to 6% rather than above the 7% that many analysts have used in recent years Webelieve that readers will find this research convincing, as we did

PRIVATE COMPANY DISCOUNTS OVER 25 PERCENT

We were surprised to find academic research that concludes that the typical private company sells forabout a 25% discount compared with otherwise comparable public companies This is aftercontrolling for variables such as size and industry The authors of the studies surmise that the reasonsfor this phenomenon could relate to lack of exposure to the market and lower quality of information(e.g., lack of a track record of audited financial statements)

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

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MUCH NEW DATA AND LITERATURE

We not only describe the practical procedures that can be used to apply the various theories but alsodescribe 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 publishedand unpublished, in recent and prior years The footnotes and bibliography tell the reader who wishes toget the original studies where to find them

We also added a chapter on global cost of capital models

Much of the new research cited in this book is from working papers, to locate a working paper, searchonline using any major search engine (i.e., Google or Yahoo) by author and title An example: the firstlink provided using the Google search engine when a search for ‘‘Adrian, Tobias, and FrancescoFranzoni Learning About Beta: an Explanation of the Value Premium’’ will be ‘‘SSRN–LearningAbout Beta: An Explation of the Value Premium by .’’ This link will direct you to the SocialScience Research Network, where the article is downloadable

CORPORATE FINANCE AND ADVICE TO PRACTITIONERS

We have added a five-chapter section specifically for corporate finance officers This includes, forexample, capital budgeting and feasibility studies, cost of capital for divisions and reporting units,and valuation in mergers and acquisitions

There is also a chapter on advice from the authors about dealing with controversial cost of capitalissues, a major chapter on cost of capital in the courts, and a chapter on cross examining experts oncost of capital

AUDIENCES FOR THE BOOK

In addition to the traditional professional valuation practitioner, this book is designed to serve the needsof:

 Attorneys and judges who deal with valuation issues in mergers and acquisitions, shareholder andpartner disputes, damage cases, solvency cases, bankruptcy reorganizations, property taxes, ratesetting, transfer pricing, and financial reporting

 Investment bankers for pricing, public offerings, mergers and acquisitions, and private equityfinancing

 Corporate finance officers for pricing or evaluating mergers and acquisitions, raising private orpublic equity, property taxation, and stakeholder disputes

 Academicians and students who wish to learn anywhere from the basic theory to the latest research

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

PRACTICAL APPLICATIONS

The book is designed to enhance the insights of users of cost of capital applications as well as originators

of such applications Most formulas are accompanied by examples Several chapter appendices presentdetailed expositions of the more complex procedures

Finally, the book is comprehensively indexed to serve as a reference for specific concepts and cedures within the general topic of cost of capital.

pro-Please contact the authors with any questions, comments, or suggestions for the next edition

Shannon P Pratt, CFA, FASA, MCBA, CM&AA

Shannon Pratt Valuations, Inc

6443 S.W Beaverton Hillsdale Highway, Suite 432

Portland, Oregon 97221

(503) 459-4700

E-mail: shannon@shannonpratt.com

www.shannonpratt.com

Roger J Grabowski, ASA

Duff & Phelps, LLC

311 S Wacker Drive, Suite 4200

This book has benefited immensely from review by many people with a high level of knowledge andexperience in cost of capital and valuation These people reviewed the manuscript, and the bookreflects their invaluable efforts and legions of constructive suggestions:

Professor Thomas J Frecka

University of Notre Dame

San Francisco, CADan McConaughyGrobstein, Horwath LLPSherman Oaks, CAProfessor James MorrisUniversity of ColoradoDenver, CO

George PushnerDuff & Phelps LLCNew York, NYJeffrey TarbellHoulihan Lokey Howard & ZukinSan Francisco, CA

Richard M WiseWise, Blackman, LLPMontreal (Quebec), Canada

In addition, we thank:

 Kimberly Short of Shannon Pratt Valuations, Inc., for assistance with editing and research, includingupdating of the bibliography; updating and shepherding the manuscript among reviewers,contributors, authors, and publisher; typing; obtaining permissions; and other invaluable help

 David Turney of Duff & Phelps, LLC, for preparing numerous tables and calculations that appearthroughout the book

 Harold Martin of Keiter, Stephens, Hurst, Gary & Shreaves, P.C., for contributing Appendix 17.1 onusing the weighted average cost of capital with a constant capital structure

 Professor James Morris, University of Colorado at Boulder, for contributing Appendix 17.2 on usingthe weighted average cost of capital with a changing capital structure

 David Ptashne, CFA, of Duff & Phelps, LLC, for contributing Appendices 10.1, 10.2, and 11.1 onbeta calculations

 James Harrington with Morningstar for contributing Chapter 19 on using Morningstar Inc cost ofcapital data as well as assisting in obtaining permissions for Morningstar exhibits used herein

xxiii

 Joel M Stern, G Bennett Stewart III, and Donald H Chew Jr for contributing Chapter 24 oneconomic value added.

 William Frazier of Howard Frazier Barker Elliott, Inc., for contributing Chapter 28 on the cost ofcapital in private investment companies

 Jim MacCrate, MAI, of MacCrate Associates, LLC, for his contribution of Chapters 37, Appendix37.1, and Chapter 36 and Appendix 36.1 on the cost of capital in real estate

 Carl Hoemke of Duff & Phelps, LLC, for contributing Chapter 38 on the cost of capital in ad valoremtaxation

 David Fein of ValuSource for contributing the revised and updated Appendix E on ValuSource Pro

 Mark Shirley of V & L Consultants, LLP, for contributing Appendix F on statistical analysis

 Noah Gordon of Shannon Pratt Valuations, Inc., for assistance with researching and updating legalcases, as well as general editorial assistance

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

 Professor Edwin Burmeister, Duke University

 Business Valuation Resources, LLC

 Center for Research in Security Prices, University of Chicago

 Professors Elroy Dimson, Paul Marsh, and Mike Staunton

 Duff & Phelps, LLC

 John D Emory Sr and John D Emory Jr., Emory & Co

 FactSet Mergerstat, LLC

 FMV Opinions, Inc

 Professor Arthur Korteweg, University of Chicago

 Jim MacCrate, MacCrate Associates, LLC

 The McGraw-Hill Companies, Inc

 Morningstar, Inc

 Glen Mueller, Dividend Capital Group

 National Association of Real Estate Investment Trusts

 Pluris Valuation Advisors, LLC

 PricewaterhouseCoopers, LLP

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

 Thomson Corporation

 Valuation Advisors, LLC

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

 David King, Mesirow Financial Consulting LLC

 Professor Timothy Leuhrman, Harvard University

We thank all of the people singled out above for their assistance of course, any errors herein are ourresponsibility

Shannon Pratt

Roger Grabowski

PURPOSE AND OBJECTIVE OF THIS BOOK

The purpose of this book is to present both the theoretical development of cost of capital estimation andits practical application to valuation, capital budgeting, forecasting of expected investment returns, andrate-setting problems encountered in current practice It is intended both as a learning text for those whowant to study the subject and as a handy reference for those who are interested in background or seekdirection 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 capitalestimation

2 The reviewer who needs to make an informed evaluation of another party’s methodology and dataused to produce a cost of capital estimate

OVERVIEW

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

 The theory of what drives the cost of capital

 The models currently in use to estimate cost of capital

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

 How to use the cost of capital estimate in:

 Valuation

 Feasibility studies

 Corporate finance decisions

 Forecasting expected investment returns

 How to reflect minority/control and marketability considerations

 Explanation of terminology, with its unfortunately varied and sometimes ambiguous usage incurrent-day financial analysis

IMPORTANCE OF THE COST OF CAPITAL

The cost of capital estimate is the essential link that enables us to convert a stream of expected incomeinto an estimate of present value Doing this allows us to make informed pricing decisions for purchasesand sales and to compare one investment opportunity against another

xxv

COST OF CAPITAL ESSENTIAL IN THE MARKETPLACE

In valuation and financial decision making, the cost of capital estimate is just as important as the estimate

of the expected amounts of income to be discounted or capitalized Yet we continually see incomeestimates laboriously developed and then converted to estimated value by a cost of capital that ispractically pulled out of thin air

In the marketplace, better-informed cost of capital estimation will improve literally billions ofdollars’ worth of financial decisions every day For example, small differences in discount rates, andespecially small differences in capitalization rates, can make very large differences in concluded values

SOUND SUPPORT ESSENTIAL IN THE COURTROOM

In the courts, billions of dollars turn on experts’ disputed cost of capital estimates in many contexts:

 Gift, estate, and income tax disputes

 Dissenting stockholder suits

 Corporate and partnership dissolutions

 Marital property settlements

 Employee stock ownership plans (ESOPs)

 Ad valorem (property) taxes

 Utility rate-setting

 Damages calculations

Fortunately, courts are becoming unwilling to accept the statement ‘‘Trust me, I’m a great expert’’ inthese disputes and instead are carefully weighing the quality of supporting evidence presented by oppos-ing sides Because cost of capital is critical to the valuation of any ongoing business, the thorough under-standing, analysis, and presentation of cost of capital issues will go a long way toward carrying the day in

a battle of experts in a legal setting

ORGANIZATION OF THIS BOOK

PART I COST OF CAPITAL BASICS

Chapter 1 defines the concept of cost of capital Chapter 2 describes, in a general sense, how it is used inbusiness valuation and capital budgeting Chapter 3 defines net cash flow, explains why it is the preferredeconomic income variable for valuation and capital budgeting, and discusses issues relative tomeasuring expected net cash flow Chapter 4 explains the difference between discounting andcapitalizing Chapter 5 addresses the concept of risk and the impact of risk on the cost of capital.Chapter 6 discusses the various components of a company’s capital structure

PART II ESTIMATING THE COST OF EQUITY CAPITAL

The second part explores cost of capital estimation We begin with the build-up model (Chapter 7) andthe Capital Asset Pricing Model (CAPM) (Chapter 8), two of the most widely used models for estimatingcost of equity capital With the backgrounds of those chapters, we then discuss a major input into these

models (and in all other cost of capital models), the equity risk premium, in Chapter 9 We then discussother inputs into these models: making beta estimates for CAPM (Chapter 10) and one correction to theCAPM called the size effect (Chapter 12) In Chapter 11 we discuss the criticism of CAPM and beta assole risk measures, and present alternative risk measures, while in Chapter 13 we discuss the criticisms

of the size effect

In Chapter 14 we discuss company-specific risk In Chapter 15 we then present alternative cost ofequity capital models, such as the Fama-French 3-factor model In Chapter 16 we present methodsfor deriving implied cost of capital estimates using discounted cash flow (DCF) and residual incomemodels In Chapter 17 we discuss the overall cost of capital based on the concept of a weightedaverage of the cost of each component of the capital structure (commonly referred to as the weightedaverage cost of capital) and how changes in the capital structure affect the cost of equity capital Weinclude appendices with detailed explanations of the iterative process for cost of equity capital estima-tion for nonpublic businesses (divisions, reporting units, closely held firms) in the context of theweighted average cost of capital

The models presented thus far are based on estimating the cost of equity for companies and ments in developed economies In Chapter 18 we discuss alternative models for estimating the cost ofequity capital for companies and investments in developing economies

invest-Chapter 19 contains a discussion of the various data sources available from Ibbotson Associates, nowpart of Morningstar, for use in estimating the cost of capital

PART III CORPORATE FINANCIAL OFFICERS—USING COST OF CAPITAL DATAThe third part explores use of the cost of capital data in capital budgeting and feasibility studies inChapter 20, determining cost of capital for divisions and reporting units in Chapter 21, using the data

in evaluating mergers and acquisitions in Chapter 22, cost of capital in transfer pricing in Chapter 23,and using the data within the framework of an Economic Value Added (EVA1) financialmanagement system in Chapter 24

PART IV COST OF CAPITAL FOR CLOSELY HELD ENTITIES

Part IVaddresses commonly encountered variations in cost of capital application particularly within thecontext of closely held entities Chapter 25 covers handling discounts for lack of marketability Chapters

26 to 28 address cost of capital for pass-through entities: partnerships, limited liability corporations, Scorporations, and private investment companies Chapter 29 focuses on venture capital investments

PART V OTHER TOPICS

Chapter 30 discusses minority versus controlling interest valuations In Chapter 31 we discuss how thecost of capital relates to the excess earnings valuation method Chapter 32 discusses adjusting thediscount rate when the measure of economic income is some measure other than net cash flow Whilethis book is not a treatise on forecasting, we discuss issues in estimating net cash flows in Chapter

33 Chapter 34 covers common errors in estimating discount rates and future economic income, andChapter 35 presents court case examples of cost of capital issues

PART VI REAL ESTATE AND AD VALOREM

Given the specialized nature of real estate, we present information on developing cost of capitalestimates for real property investments, real estate investment trusts, and are within the context of advalorem taxation in Chapters 36 through 38

PART VII ADVICE TO PRACTITIONERS

Chapter 39 provides issue-by-issue advice on handing real-world cost of capital estimation issues.Chapter 40 provides questions for attorneys to use in cross-examining experts on cost of capital

SUMMARY

The book is designed to serve as both a primer and a reference source Part I covers cost of capital basics.Part II covers the methods generally used to estimate cost of equity capital Part III covers a variety oftopics commonly encountered by the corporate financial officer Part IV covers issues peculiar to closelyheld entities Part V covers a variety of topics integral to users of cost of capital data Part VI covers realproperty cost of capital issues Part VII covers real-world cost estimation issues

The appendices provide a directory for further study and data sources

Notation System and Abbreviations

Used in This Book

A source of confusion for those trying to understand financial theory and methods is that financialwriters have not adopted a standard system of notation The notation system used in this volume isadapted from the fifth edition of Valuing a Business: The Analysis and Appraisal of Closely HeldCompanies, by Shannon P Pratt (New York: McGraw-Hill, 2007)

VALUE AT A POINT IN TIME

PV ¼ Present value (seen as Pi in Chapter 19)

PVb¼ Present value of net cash flows due to business operations before cost of financing

PVeu¼ Present value of unlevered equity

PVTSD¼ Present value of tax savings on debt

PVe¼ Present value of equity

FV ¼ Future value

MVIC¼ Market value of invested capital

¼ Enterprise value

¼ Meþ Mdþ Mp

BV ¼ Book value of net assets

FVRU ¼ Fair value of reporting unit

FVNWCRU ¼ Fair value of net working capital of the reporting unit

FVICRU ¼ Fair value of invested capital of the reporting unit

FVFARU ¼ Fair value of fixed assets of the reporting unit

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

reporting unit

FVUIVRU ¼ Fair value of unidentified intangibles value (i.e., ‘‘goodwill’’) of the reporting unit

FVdRU ¼ Fair value of debt capital of the reporting unit

FVeRU ¼ Fair value of equity capital of the reporting unit

FMVBE ¼ Fair market value of the business enterprise

FMVNWC¼ Fair market value of net working capital

FMVFA¼ Fair market value of fixed assets

FMVIA¼ Fair market value on intangible assets

FMVUIV ¼ Fair market value of unidentified intangibles value (i.e., ‘‘goodwill’’)

FMVe¼ Fair market value of equity capital

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COST OF CAPITAL AND RATE OF RETURN VARIABLES

k¼ Discount rate (generalized)

kc ¼ Country cost of equity

ke ¼ Discount rate for common equity capital (cost of common equity capital)

(seen as ki in Chapter 19) Unless otherwise stated, it generally is assumed that thisdiscount rate is applicable to net cash flow available to common equity

ke500 ¼ Cost of equity for the S&P 500

keu ¼ Cost of equity capital, unlevered (cost of equity capital assuming firm financedwith all equity)

kni ¼ Discount rate for equity capital when net income rather than net cash flow isthe measure of economic income being discounted

kð ptÞ ¼ Discount rate applicable to pretax cash flows

keð ptÞ ¼ Cost of equity prior to tax affect

kp ¼ Discount rate for preferred equity capital

kd ¼ Discount rate for debt (net of tax affect, if any)

(Note: For complex capital structures, there could be more than one class ofcapital in any of the preceding categories, requiring expanded subscripts.)

¼ kdð ptÞ(1 – tax rate)

kdð ptÞ ¼ Cost of debt prior to tax effect

kTS ¼ Rate of return used to present value tax savings due to deducting interest expense

on debt capital financing

kNWCð ptÞ ¼ Rate of return for net working capital financed with debt capital

(measured pretax) and equity capital

kFAð ptÞ ¼ Rate of return for fixed assets financed with debt capital (measured pretax) and

equity capital

kdRU ¼ After-tax rate of return on debt capital of the reporting unit

¼ kdð ptÞRU(1 – tax rate)

kdð ptÞRU ¼ Rate of return on debt capital of the reporting unit without taking into account the

tax deduction on interest expense (pretax cost of debt capital)

keRU ¼ After-tax rate of return on equity capital of the reporting unit

kNWC ¼ Rate of return for net working capital

kNWCRU ¼ Rate of return for net working capital of the reporting unit financed with debt

capital (return measured after-tax) and equity capital

kNWCð ptÞRU ¼ Rate of return for net working capital of the reporting unit financed with debt

capital (measured pretax) and equity capital

kFA ¼ Rate of return for fixed assets

kFARU ¼ Rate of return for fixed assets financed with debt capital (return measured

after tax) and equity capital

kFAð ptÞRU ¼ Rate of return for fixed assets of the reporting unit financed with debt capital

(measured pretax) and equity capital

kIA ¼ Rate of return for intangible assets

kIARU ¼ Rate of return for identified and individually valued intangible assets financed

with debt capital (return measured after tax) and equity capital

kUIV ¼ Rate of return for unidentified intangibles value

kUIVRU ¼ Rate of return for unidentified intangibles value of the reporting unit financed

with debt capital (return measured after tax) and equity capital

kIAþUIV ¼ After-tax rate of return on all intangible assets, identified and individually

valued, plus the unidentified intangible value

kIAþUIVð ptÞ ¼ Pretax rate of return on all intangible assets, identified and individually

valued, plus the unidentified intangible value financed with debt capital(measured pretax) and equity capital

kIAþUIVRU ¼ After-tax rate of return on all intangible assets, identified and individually valued,

of the reporting unit plus the unidentified intangible value of the reporting unit

kIAþUIVð ptÞRU ¼ Pretax rate of return on all intangible assets, identified and individually valued,

plus the unidentified intangible value of the reporting unit financed with debtcapital (measured pretax) and equity capital

kTSRU ¼ Rate of return used to present value tax savings due to deducting interest expense

on debt capital financing of the reporting unit

c¼ Capitalization rate

ce ¼ Capitalization rate for common equity capital Unless otherwise stated, it

generally is assumed that this capitalization rate is applicable to net cash flowavailable to common equity

Cni ¼ Capitalization rate for net income

cð ptÞ¼ Capitalization rate on pretax cash flows

cp¼ Capitalization rate for preferred equity capital

cd ¼ Capitalization rate for debt

(Note: For complex capital structures, there could be more than one class ofcapital in any of the preceding categories, requiring expanded subscripts.)D=P¼ Dividend yield on stock

DRj ¼ Downside risk in the local market (U.S dollars)

DRw¼ Downside risk in global (‘‘world’’) market (U.S dollars)

Pn¼ Stock price in period n

P0¼ Stock price at valuation period

R¼ Rate of return

Rf ¼ Rate of return on a risk-free security

Rf;n¼ Risk-free rate in current month

Rf local¼ Return on the local country government’s (default-risk-free) paper

Rf u:s:¼ U.S risk-free rate

Rlocaleuro$issue ¼ Current market interest rate on debt issued by the local country government

denominated in U.S dollars (‘‘euro-dollar’’ debt), same maturity as debt issued

by the local country government denominated in U.S dollars

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 forequity securities, such as the NYSE or S&P 500)

RPs ¼ Risk premium for ‘‘small’’ stocks (usually average size of lowest quintile or decile

of NYSE as measured by market value of common equity) over and above RPm

RPmþs ¼ Risk premium for the market plus risk premium for size (Duff & Phelps Risk

premium report data for use in build-up method)

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

RPw ¼ The equity risk premium on a ‘‘world’’ diversified portfolio

RPi ¼ Risk premium for the ith security (seen in Chapter 19 as IRPi)

RPlocal ¼ Equity risk premium in local country’s stock market

RIi ¼ Risk index (full-information beta) for industry i

RIiL ¼ Full-information levered beta estimate of the subject company

EðRÞ ¼ Expected rate of return

EðRmÞ 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] orStandard & Poor’s [S&P] 500)

EðRiÞ ¼ Expected rate of return on security i

EðRdivÞ ¼ Expected rate of return on dividend

EðRcapgainsÞ ¼ Expected rate of return on capital gains

B¼ Beta (a coefficient, usually used to modify a rate of return variable)

BL ¼ Levered beta for (equity) capital

BU ¼ Unlevered beta for (equity) capital

BLS ¼ Levered segment beta

Bd ¼ Beta for debt capital

BUi ¼ Beta unlevered for industry (or guideline companies) equity capital

BLi ¼ Beta levered for industry (or guideline companies) equity capital

Be ¼ Beta (equity) expanded

Bop ¼ Operating beta (beta with effects of fixed operating expense removed)

Bi ¼ Beta of company i (F-F Beta)

Bn ¼ Estimated market coefficient based on sensitivity to excess returns on marketportfolio in current month

Bn1 ¼ Estimated lagged market coefficient based on sensitivity to excess returns on

market portfolio last month

Blocal ¼ Market risk of the subject company measured with respect to the local securities

market

Bw ¼ Market or systematic risk measured with respect to a ‘‘world’’ portfolio of stocks

bcw¼ Country covariance with world

bcr ¼ Country covariance with region

K1:::Kn¼ Risk premium associated with risk factor 1 through n for the average asset

in the market (used in conjunction with arbitrage pricing theory)

si ¼ Small-minus-big coefficient in the Fama-French regression

SMBP¼ Expected small-minus-big risk premium, estimated as the difference between the

historical average annual returns on the small-cap and large-cap portfolios

hi ¼ High-minus-low coefficient in the Fama-French regression

HMLP¼ Expected 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

Fd ¼ Face value of outstanding debt

WACCð ptÞ¼ Weighted average cost of capital (pretax)

WACCBE ¼ Overall rate of return for the business enterprise

WACCð ptÞBE ¼ Pretax WACC of the business enterprise

WACCRU ¼ Overall rate of return for the reporting unit

¼ Weighted average cost of capital for the reporting unit

WACCð ptÞRU ¼ Pretax WACC of the reporting unit

Me ¼ Market value of equity capital (stock)

Md ¼ Market value of debt capital

Mp¼ Market value of preferred equity

s2¼ Variance of returns for subject company stock

s2M ¼ Variance of the returns on the market portfolio (e.g., S&P 500)

s2e ¼ Variance of error terms

s ¼ Standard deviation

sB ¼ Standard deviation of operating cash flows of business before cost of financing

srev¼ Standard deviation of revenues of output

slocal ¼ Volatility of subject (local) stock market

su:s:¼ Volatility of U.S stock market

sstock ¼ Volatility of local country’s stock market

sbond ¼ Volatility of local country’s bond market

dr ¼ Regional risk not included in RPw

CCRc ¼ Country credit rating of country

l¼ Company’s exposure to the local country risk

t¼ Tax rate (expressed as a percentage of pretax income)

ti ¼ Federal and state income tax rate for industry (or guideline companies)

r¼ Property tax rate (expressed as a percentage of total fair market value)

C¼ Proportion of the entity that is assessed property tax

h¼ Holding period

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

F ¼ Fixed operating assets (without regard to costs of financing)

Fc ¼ Fixed operating costs of the business

NI ¼ Net income (after entity-level taxes)

CF ¼ Cash flow for a specific period

NCFe ¼ Net cash flow (free cash flow) to equity

NCFf ¼ Net cash flow (free cash flow) to the firm (to overall invested capital, or entire

capital structure, including all equity and long-term debt)NCFue ¼ Net cash flow to unlevered equity

PMT ¼ Payment (interest and principal payment on debt security)

D¼ Dividends

T ¼ Tax (in U.S dollars)

TS¼ Present value of tax savings due to deducting interest expense on debt capitalfinancing

GCF ¼ Gross cash flow (usually net income plus noncash charges)

EBT ¼ Earnings before taxes

EBIT ¼ Earnings before interest and taxes

EBITD ¼ Earnings before depreciation, interest, and taxes (‘‘Depreciation’’ in this

context usually includes amortization Some writers use EBITDA specifically toindicate that amortization is included.)

EBITDA¼ Earnings before interest, taxes, depreciation, and amortization

V ¼ Variable operating assets

i¼ ith period or ith variable in a series (may be extended to the jth variable, thekth variable, etc.)

n ¼ Number of periods or variables in a series, or the last number in a series

0 ¼ Period 0, the base period, usually the latest year immediately preceding thevaluation date

py ¼ Partial year of first year following the valuation date

¼ Fair value of equity capital/FVRU

Wp¼ Weight of preferred equity in capital structure

¼ Mp=Meþ Mdþ Mp

Wd ¼ Weight of debt in capital structure

¼ Md=Meþ Md

(Note: For purposes of computing a weighted average cost of capital [WACC],

it is assumed that preceding weightings are at market value.)

Wdi ¼ Weight of interest-bearing debt in capital structure at market for industry

(or guideline companies)

WdRU ¼ Weight of debt capital in capital structure of reporting unit

¼ Fair value of debt capital/FVRU

Wei ¼ Weight of common equity in capital structure at market for industry

(or guideline companies)

Ws ¼ Weight of segment data to total business (e.g., sales, operating income)

WNWC ¼ Weight of net working capital in FMVBE

WUIV ¼ Weight of unidentified intangibles value FMVBE

¼ FMVUIV (i.e., ‘‘goodwill’’)=FMVBE

WUIVRU ¼ Weight of unidentified intangibles value FVRU

¼ FVUIVRU (i.e., ‘‘goodwill’’)=FVRU

WIAþUIV ¼ Weight of intangible assets in FMVBE plus the weight of unidentified intangible

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

gni ¼ Rate of growth in net income

S¼ Sum of (add all the variables that follow)

\ ¼ Product of (multiply together all the variables that follow)

X ¼ Mean average (the sum of the values of the variables divided by the number

DSCR ¼ Debt service coverage ratio

EGIM ¼ Effective gross income multiplier

NOI; Ip ¼ Net operating income

OER¼ Operating expense rates

ke ¼ Equity discount or yield rate (dividend plus appreciation)

km ¼ Mortgage interest rate

kp ¼ Overall property discount rate

cp ¼ Overall property capitalization rate

ce ¼ Dividend to equity capitalization rate

cm ¼ Mortgage capitalization rate or constant

cn ¼ Terminal or residual or going-out capitalization rate

cB ¼ Building capitalization rate

cL ¼ Land capitalization rate

cLF ¼ Leased fee capitalization rate

cLH ¼ Leasehold capitalization rate

A¼ Change in income and value (adjustment factor)

P¼ Principal paid off over the holding period

1=Sn ¼ Sinking fund factor at the equity discount or yield rate (ke)

D0 ¼ Change in value over the holding period

SC% ¼ Cost of sale

PGI ¼ Potential gross income

PGIM ¼ Potential gross income multiplier

MATHEMATICAL FUNCTIONS

NOTATION FOR REAL PROPERTY VALUATION (CHAPTER 36)

GROWTH

EGI ¼ Effective gross income

Fd=PVp¼ Face value of debt (loan amount outstanding) to value ratio

½1  ðF d =PVpÞ¼ Equity to value ratio

Ip¼ Overall income to the property

IL ¼ Residual income to the land

IB ¼ Residual income to the building

Ie ¼ Equity income

Im ¼ Mortgage income

ILF ¼ Income to the leased fee

ILH ¼ Income to the leasehold

ERP¼ Equity risk premium (usually the general equity risk premium for which thebenchmark for equities is either the S&P 500 stocks or the NYSE stocks)WACC¼ Weighted average cost of capital

WARA¼ Weighted average return on assets

T-Bill¼ U.S government bill (usually 30-day, but can be up to one year)

STRIPS¼ Separate trading of registered interest and principal of securities

CRSP¼ Center for Research in Security Prices, at the University of Chicago

PIPE¼ Private Investment in Public Equity

SBBI¼ Stocks, Bonds, Bills, and Inflation, published annually by Ibbotson Associates

(now Morningstar) in both a ‘‘Classic edition’’ and a ‘‘Valuation edition’’

CAPM¼ Capital Asset Pricing Model

DCF¼ Discounted cash flow

DDM¼ Discounted dividend model

TIPS¼ Treasury Inflation-Protected Security

NCF¼ Net cash flow (also sometimes interchangeably referred to as FCF, free cash flow)

BE¼ Business enterprise or reporting unit

NWC¼ Net working capital

FA¼ Fixed assets

IA¼ Intangible assets

UIV¼ Unidentified intangible value (i.e., ‘‘goodwill’’)

NOPAT¼ Net operating profit after taxes

ABBREVIATIONS