quantitative second edition quantitative business valuation · own, and his conclusions. it is a...

• Lost inventory and lost profits dam-age formulas in litigation

Organized into seven sections, the fi rst three parts of this book follow the chron-ological sequence of performing a dis-counted cash fl ow. The fourth part puts it all together, covering empirical testing of Abrams’ valuation theory and mea-suring valuation uncertainty and error. Parts fi ve to seven round it all out with discussion of litigation, valuing ESOPs and partnership buyouts, and probabilistic methods including valuing start-ups.

The resulting work, solidly grounded in economic theory and including all nec-essary mathematics, integrates existing science into the valuation profession—and develops valuation formulas and models that you will find useful on a daily basis.

JAY B. ABRAMS is a leading valuation and litigation econ-omist. He is the principal in Abrams Valuation Group, Inc. and is credited with numer-

ous inventions, including the Abrams Log Size Model for calculating discount rates. He is also a popular finance lecturer and the author of numerous journal articles.

(continued from front flap)

“Jay Abrams’ book is close to the equivalent of several graduate dissertations rolled into one book. For each topic (covered), he presents a scholarly summary of past research, new empirical research of his own, and his conclusions. It is a well-documented contribution to in-depth understanding of important business valuation issues, and should not be overlooked by the serious practitioner.”

— Shannon Pratt, DBAManaging Director, Willamette Management AssociatesCoauthor, Valuing a Business

“A must-read for the serious business appraiser.”

— Jay E. Fishman, ASA, CBAPresident, Financial Research, Inc.

“The problem of simplified valuation procedures and coherent theory still remains complex and is ever evolving. Jay Abrams deals very effectively with this complexity through the use of mathematical formu-las. Input to his models is explained with clarity and effectiveness, which adds to the overall value of this advanced text on business valuation.”

—Terry A. Isom, Chairman, National Association of Certified Valuation Analysts

“Jay Abrams’ book strives to provide mathematical modeling for what practitioners often do by reason-ing alone. This book is a must-read for practitioners who are searching for additional techniques for dealing with some of business valuation’s imponderables.”

— David M. Bishop, FIBA, BVAL, ASA, MCBAPresident, American Business Appraisers, Inc.

“Jay Abrams’ book will not only challenge the top theoreticians in the field; his step-by-step explana-tions will make advanced quantitative techniques available to the many appraisers who are not capable of independently creating the underlying mathematical analysis.”

— Kent Osborne, ASAChairman, Editorial Review Board of the American Society of Appraisers

“While a proliferation of business valuation treatises and guides exists in the market, most are very general in nature and do nothing more than rehash fundamental concepts. I am unaware of any author who has stepped into the unknown as Jay Abrams has and compiled and developed a treatise of extremely useful analytical tools for the serious valuator.”

— Robert J. Grossman, CPA/ABV, ASA, CVAPartner, Grossman Yanak & Ford

“Jay Abrams develops unusual approaches which merit consideration when ‘cookie cutter’ methodolo-gies are inadequate. This manuscript contributes to the dialogue among practitioners and strengthens the theoretical foundations of business valuation.”

— Herbert T. SpiroPresident, American Valuation Group, Inc.

“There is no question about it, the use of rigorous quantitative methods is the cure for subjective valu-ation analysis. This book not only satisfies this need—which has grown considerably in recent years—but is chock-full of new tools that have been carefully developed.”

— Edward MurrayValuation Partner, Arthur Andersen, LLP

$150.00 USA/$180.00 CAN

Essential reading for the serious business appraiser, Quantitative Business Valuation, Second Edition

is the definitive guide to quantitative measurements in the valuation pro-cess. No other book written on business valuation is as well researched, innova-tive, and bottom-line beneficial to you as a practitioner.

Written by leading valuation and litiga-tion economist Jay B. Abrams, this text is a rigorous and eye-opening treatment filled with applications for a wide variety of scenarios in the valuation of your pri-vately held business.

Substantially revised for greater clar-ity and logical flow, the Second Edition includes new coverage of:

• Converting forecast net income to forecast cash flow

• Damages in manufacturing firms

• Regressing scaled y-variables as a way to control for heteroscedasticity

• Mathematical derivation of the Price-to-Sales (PS) ratio

• Monte Carlo Simulation (MCS) and Real Options (RO) Analysis

• Venture capital and angel investor rates of return

(continued on back flap)

QuantitativeBusiness Valuation

A Mathematical Approachfor Today’s Professionals

SECOND EDITION

Praise for the First Edition of

Quantitative Business ValuationA Mathematical Approach for Today’s Professionals

SECONDEDITION

A Mathematical Approach for Today’s Professionals

J a y B . A b r a m s

QuantitativeBusinessValuation

PMS 143 BLACK GLOSSY

Quantitative Business ValuationA

Math

ematical A

pproach

for Today

’s Profession

als

Abrams

SECONDEDITION

P1: OTA/XYZ P2: ABCFM JWBT212-Abrams January 22, 2010 11:6 Printer Name: Yet to Come

ii


Quantitative Business Valuation

i


ii


Quantitative Business Valuation

A Mathematical Approach forToday’s Professionals

Second Edition

JAY B. ABRAMS

John Wiley & Sons, Inc.

iii


Copyright C© 2010 by John Wiley & Sons, Inc. All rights reserved.

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

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in anyform or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise,except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, withouteither the prior written permission of the Publisher, or authorization through payment of theappropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers,MA 01923, (978) 750-8400, fax (978) 750-4470, or on the Web at www.copyright.com. Requests tothe Publisher for permission should be addressed to the Permissions Department, John Wiley &Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online atwww.wiley.com/go/permissions.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their bestefforts in preparing this book, they make no representations or warranties with respect to theaccuracy or completeness of the contents of this book and specifically disclaim any impliedwarranties of merchantability or fitness for a particular purpose. No warranty may be created orextended by sales representatives or written sales materials. The advice and strategies containedherein may not be suitable for your situation. You should consult with a professional whereappropriate. Neither the publisher nor author shall be liable for any loss of profit or any othercommercial damages, including but not limited to special, incidental, consequential, or otherdamages.

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Library of Congress Cataloging-in-Publication Data:

Abrams, Jay B.Quantitative business valuation : a mathematical approach for today’s professionals /

Jay B. Abrams. — 2nd ed.p. cm.

Includes index.ISBN 978-0-470-39016-0 (cloth)

1. Business enterprises—Valuation—Mathematical models. I. Title.HF5681.V3A28 2010657′.73—dc22

2009042697

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

iv

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www.wiley.com/go/permissions

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To my father, Leonard Abrams, who taught me how to write. To my mother,Marilyn Abrams, who taught me mathematics. To my wife, Cindy, who believes

in me. To my children, Yonatan, Binyamin, Miriam, Nechamah Leah, andRivkah, who gave up countless Sundays with Abba (Dad) for this book.

To my great teachers, Mr. Ohshima and Christopher Hunt, who brought me tomy power to make this happen. And finally, to R. K. Hiatt and Scott Deifik, who

caught my mistakes and made significant contributions to the thought thatpermeates this book.

v


vi


Contents

List of Tables and Figures xiii

Introduction xxi

Acknowledgments xxvii

PART I FORECASTING CASH FLOW 1

CHAPTER 1 Cash Flow: A Mathematical Derivation 5

Introduction 7

The Mathematical Model 11

Analysis of the Mathematical Model 25

Summary 27

References 27

CHAPTER 2 Forecasting Cash Flow: Mathematics of the Payout Ratio 29

Introduction 31

The Mathematics 32

Forecasting Gross Cash Flow Is Incorrect 43

Conclusion 44

References 44

CHAPTER 3 Using Regression Analysis 45

Introduction 47

Forecasting Costs and Expenses 48

Performing Regression Analysis 51

Use of Regression Statistics to Test the Robustness ofthe Relationship 52

Problems with Regression Analysis for ForecastingCosts 63

Using Regression Analysis to Forecast Sales 64

Autocorrelation in Time Series Analysis 69

vii


viii Contents

Application of Regression Analysis to the Guideline Company(GC) Methods 69

Summary 73

References 74

APPENDIX 3A The ANOVA Table (Table A3.1, Rows 28–32) 75

CHAPTER 4 Annuity Discount Factors and the Gordon Model 79

Introduction 81

ADF with End-of-Year Cash Flows 83

Midyear Cash Flows 91

Starting Periods Other Than Year 1 93

Periodic Perpetuity Factors (PPFs): Perpetuities for PeriodicCash Flows 101

ADFs in Loan Mathematics 107

Relationship of the Gordon Model to the Price/Earnings andPrice/Sales Ratios 110

The Bias in Annual (versus Monthly) DiscountingIs Immaterial 113

Conclusions 119

References 121

APPENDIX 4A Mathematical Appendix 123

APPENDIX 4B Mathematical Appendix: Monthly ADFs 141

PART II CALCULATING DISCOUNT RATES 145

CHAPTER 5 Discount Rates as a Function of Log Size 149

Research Included in the First Edition 151

Table 5.1: Analysis of Historical Stock Returns 152

Application of the Log Size Model 167

Discussion of Models and Size Effects 181

Industry Effects 191

The Wedge between Public and Private FirmValuations 192

Satisfying Revenue Ruling 59-60 196

Summary and Conclusions 198

References 199

APPENDIX 5A Automating Iteration Using Newton’s Method 203


Contents ix

APPENDIX 5B Mathematical Appendix 207

APPENDIX 5C Abbreviated Review and Use 211

CHAPTER 6 Arithmetic versus Geometric Means: Empirical Evidenceand Theoretical Issues 223

Introduction 225

Theoretical Superiority of the Arithmetic Mean 226

Empirical Evidence of the Superiority of theArithmetic Mean 227

Indro and Lee Article 232

References 233

CHAPTER 7 An Iterative Valuation Approach 235

Introduction 237

Equity Valuation Method 237

Invested Capital Approach 243

Log Size 245

Summary 245

References 247

PART III ADJUSTING FOR CONTROL AND MARKETABILITY 249

CHAPTER 8 Adjusting for Levels of Control and Marketability 253

Introduction 257

The Value of Control and Adjusting for Levelof Control 257

Discount for Lack of Marketability (DLOM) 301

Conclusion 358

References 359


PART IV PUTTING IT ALL TOGETHER 375

CHAPTER 9 Empirical Testing of Abrams’s Valuation Theory 377

Introduction 379

Table 9.1: Log Size for 1938–1986 380

Table 9.2: Reconciliation to the IBA Database 382

Calculation of DLOM 387


x Contents

Interpretation of the Error 400

Conclusion 401

References 401

CHAPTER 10 Measuring Valuation Uncertainty and Error 403

Introduction 405

Measuring Valuation Uncertainty 406

Measuring the Effects of Valuation Error 410

Summary and Conclusions 422

Reference 423

PART V LITIGATION 425

CHAPTER 11 Demonstrating Expert Bias 427

Introduction 429

Market Methods 429

A Balanced DCF Valuation 432

Summary 434

CHAPTER 12 Lost Inventory and Lost Profits Damage Formulas in Litigation 435

Introduction 437

Commentary to Table 12.1: Sample Damage Calculations withVM = $95 438

Table 12.1B: Lost Profits Formulas Based on EBITDA for LostSales on Inventory Never Produced 445

When Reality May Vary with Our Assumptions 446

Modification of Formulas for Wholesaleand Retail Businesses 447

Legal Treatment 447

Summary 448

Reference 448

PART VI VALUING ESOPs AND BUYOUTS OF PARTNERS ANDSHAREHOLDERS 449

CHAPTER 13 ESOPs: Measuring and Apportioning Dilution 451

Introduction 453

Definitions of Dilution 454

Table 13.1: Calculation of Lifetime ESOP Costs 456

The Direct Approach 457


Contents xi

The Iterative Approach 466

Summary 469

References 474


CHAPTER 14 The Trade-off in Selling to an ESOP versus an Outside Buyer 477

Section 1: Introduction 479

Section 2: Advantages and Disadvantages of Selling to anESOP versus a Third Party 480

Section 3: The Mathematics 481

Section 4: Sample Calculations in the Tables 486

Section 5: Conclusion 494

References 494

CHAPTER 15 Buyouts of Partners and Shareholders 497

Introduction 499

Table 15.1: Pre- and Post-Transaction Valuations 499

Table 15.2: Dilution in FMV as a Result of the Partner Buyout 501

Sharing the Dilution 503

Conclusion 506

PART VII PROBABILISTIC METHODS 507

CHAPTER 16 Valuing Start-Ups 511

Issues Unique to Start-Ups 513

Organization of the Chapter 513

Part 1: First Chicago Approach 514

Venture Capital Valuation Approach 520

Part 2: Debt Restructuring Study 521

Part 3: Exponentially Declining Sales Growth Model 534

References 536

CHAPTER 17 Monte Carlo Risk Simulation, by Dr. Johnathan Mun 539

What Is Monte Carlo Risk Simulation? 541

Comparing Simulation with Traditional Analyses 543

Running a Monte Carlo Simulation Using Risk Simulator 543

Using Forecast Charts and Confidence Intervals 554

Tornado and Sensitivity Tools in Simulation 556

Sensitivity Analysis 563

Distributional Fitting: Single Variable and Multiple Variables 567

Getting the Risk Simulator Software 571


xii Contents

CHAPTER 18 Real Options, by Dr. Johnathan Mun 573

Part 1: Introduction to Real Options 575

Part 2: Traditional Valuation Approaches 585

Part 3: Application: Real Options SLS Software 597

Glossary 617

About the Author 621

Index 623


List of Tables and Figures

Chapter 1: Cash Flow: A Mathematical Derivation 5

Table 1.1 Feathers R Us Abbreviated Balance Sheets for Calendar Years 13Table 1.2 Feathers R Us Income Statement for Calendar Year 2009 15Table 1.3 Feathers R Us Analysis of Property, Plant, and Equipment for

Calendar Year 2009 16Table 1.4 Feathers R Us Statement of Stockholders’ Equity for Calendar

Year 2009 19Table 1.5 Feathers R Us Abbreviated Statement of Cash Flows for

Calendar Year 2009 20Table 1.6 Feathers R Us Balance Sheets for Calendar Years 23Table 1.7 Feathers R Us Statement of Cash Flows—Detailed for Calendar

Year 2009 24

Chapter 2: Forecasting Cash Flow: Mathematics of the Payout Ratio 29

Table 2.1 Analysis of Depreciation and Capital Expenditures 37Table 2.2 How Capital Expenditures Exceeds Depreciation 39Table 2.3 Analysis of Depreciation and Capital Expenditures 41

Chapter 3: Using Regression Analysis 45

Table 3.1A Adjustments to Historical Costs and Expenses: SummaryIncome Statements 49

Table 3.1B Regression Analysis, 1998–2007 53Figure 3.1 Adjusted Costs and Expenses as a Function of Sales 53Table 3.2 OLS Regression: Example of Deviation from Mean 56Figure 3.2 Z-Distribution versus t-Distribution 58Figure 3.3 T -Distribution of 95% Confidence Interval of β around the

Estimate b 59Table 3.3 Abbreviated Table of t-Statistics 60Table 3.4 Regression Analysis, 2003–2007 64Figure 3.4 Adjusted Costs and Expenses as a Function of Sales 64Table 3.5 Regression Analysis of Sales as a Function of GDP 65Table 3.6 Regression Analysis of Guideline Companies 70Table A3.1 Regression Analysis, 1998–2007 76

xiii


xiv List of Tables and Figures

Chapter 4: Annuity Discount Factors and the Gordon Model 79

Table 4.1 ADFs for n = 3 Years 86Table 4.1A ADFs for n = 20 Years 87Figure 4.1 Timeline of the ADF and the Gordon Model 89Table 4.2 ADF: End-of-Year Formula 90Table 4.3 ADF: Midyear Formula 93Table 4.4 ADF with Cash Flows Starting in Year 3.25 End-of-Year Formula 96Table 4.5 ADF with Cash Flows Starting in Year −2.00 End-of-Year

Formula 97Table 4.6 ADF with Cash Flows Starting in Year −2.00 with Negative

Growth End-of-Year Formula 98Table 4.7 ADF with Cash Flows Starting in Year −2.00 with g > r

End-of-Year Formula 99Table 4.8 Periodic Perpetuity Factor (PPF)—End-of-Year Formula 103Table 4.9 Periodic Perpetuity Factor (PPF)—Midyear Formula 103Table 4.10 Periodic Perpetuity Factor (PPF)—End-of-Year—Cash Flows

Begin Year 6 106Table 4.11 PV of Loan with Market Rate > Nominal Rate: ADF,

End-of-Year 109Table 4.12 Present Values of Cash Flows: r = 12%, g = 6% 116Table 4.13 Present Values of Daily Cash Flows 120Table 4.14 Table of ADF Equation Numbers 121Figure A4.1 Timeline of Cash Flows 124Table A4.1 ADF with Fractional Year Midyear Formula 127Table A4.2 ADF with Fractional Year End-of-Year Formula 128Figure A4.2 Payment Schedule 131Table A4.3 Amortization of Principal with Irregular Starting Point 134Table A4.4 PV of Principal Amortization 136Table A4.5 Present Value of a Loan at Discount Rate Different than

Nominal Rate 140

Chapter 5: Discount Rates as a Function of Log Size 149

Table 5.1 NYSE/AMEX/NASDAQ Data by Decile and Statistical Analysis:1926–2007 153

Figure 5.1 1926–2007 Arithmetic Mean Returns as a Function of StandardDeviation 155

Table 5.1A Regression of Decile Portfolios, 1926–2008 157Figure 5.2 1926–2007 Arithmetic Mean Returns as a Function of ln(FMV) 158Figure 5.3 Decade Standard Deviation of Returns vs. Avg. FMV per NYSE

Company 1935–1995 159Figure 5.3A Decade Standard Deviation of Returns vs. Avg. FMV per NYSE

Company 1935–2005 160Figure 5.4 Decade Standard Deviation of Returns vs. Avg. FMV per NYSE

Company 1945–1995 160Figure 5.4A Decade Standard Deviation of Returns vs. Avg. FMV per NYSE

Company 1945–2005 161


List of Tables and Figures xv

Figure 5.5 Average Returns Each Decade 162Table 5.2 Regressions of Returns over Standard Deviation and Log of Fair

Market Value 163Table 5.2A Regression Comparison 164Table 5.3 Table of Discount Rates Based on FMV: 1926–2007 SBBI Data 168Figure 5.6 Natural Logarithm 170Figure 5.7 Discount Rates as a Function of FMV 170Table 5.4A DCF Analysis Using 1926–2007 Regression Data—1st Iteration 173Table 5.4B DCF Analysis Using 1926–2007 Regression Data—2nd Iteration 174Table 5.4C DCF Analysis Using 1926–2007 Regression Data—3rd Iteration 175Table 5.5 Correlation of Large Stock Returns and Bond Yields 178Table A5.1 Gordon Model Valuation Using Newton’s Iterative Process 204

Chapter 6: Arithmetic versus Geometric Means: Empirical Evidence andTheoretical Issues 223

Table 6.1 Geometric versus Arithmetic Returns 228Table 6.2 Geometric versus Arithmetic Returns NYSE/AMEX/NASDAQ

Data by Decile and Statistical Analysis: 1926–2007 229Table 6.3 The Size Effect on Discount Rates Based on the Arithmetic

versus Geometric Means 230Table 6.4 Comparison of Discount Rates Derived from the Log Size

Model Using Arithmetic and Geometric Means 231Table 6.4A Comparison of Discount Rates Derived from the Log Size

Model Using Arithmetic and Geometric Means (g = 6% for AM,5% for GM) 232

Chapter 7: An Iterative Valuation Approach 235

Table 7.1A Equity Valuation Approach with Iterations Beginning withBook Equity Iteration #1 239

Table 7.1B Equity Valuation Approach with Iterations Beginning withBook Equity 241

Table 7.1C Equity Valuation Approach with Iterations Beginning withArbitrary Equity 242

Table 7.2A WACC Approach with Iterations Beginning with Book Equity 244Table 7.2B WACC Approach with Iterations Beginning with Arbitrary

Guess of Equity Value 246

Chapter 8: Adjusting for Levels of Control and Marketability 253

Figure 8.1 Chart of Traditional Levels of Value and Mercer’s (1998) Chartof Modified Traditional Levels of Value 258

Figure 8.2 Chart of Two-Tiered Levels of Value 259Table 8.1 Synergies as Measured by Acquisition Minus Going-Private

Premiums 260Table 8.1A Summary Statistics from Mergerstat Database for Different

Populations 262


xvi List of Tables and Figures

Table 8.2 Acquisition Premiums by SIC Code 271Table 8.3 Analysis of Megginson Results 282Table 8.3A Analysis of American VRP Results—Lease, McConnell, and

Mikkelson Results 283Figure 8.3 Chart of 4 × 2 Levels of Value 295Table 8.4 Mergerstat Mean Premiums—Control versus Minority Purchases 299Table 8.5 Abrams Regression of Management Planning Study Data 307Table 8.6 Calculation of Continuously Compounded Standard Deviation

Chantal Pharmaceutical Inc.—CHTL 310Table 8.7 Black-Scholes Put Option—CHTL 311Table 8.8 Put Model Results 313Table 8.9 Calculation of Restricted Stock Discounts for 13 Stocks Using

Regression from Table 8.5 314Table 8.10 Calculation of Component #1—Delay to Sale 322Table 8.11 Estimates of Transaction Costs 326Table 8.12 Proof of Equation (8.9) 332Table 8.13 Proof of Equation (8.9a) 334Table 8.14 Sample Calculation of DLOM 337Table 8.15 Using the MPI Study 30% Average Discount 340Table 8.16 Using the MPI Study 30% Average Discount 341Table 8.17 Using the Johnson Study 20% Average Discount 341Table 8.18 Summary of Results of Applying the QMDM in 10 Example

Appraisals 342Table 8.19 QMDM Comparison of Restricted Stock Discount Rate versus

Mercer Example 1 344Table 8.20 Predictive Power of QMDM versus ECM 348Table 8.21 QMDM DLOM Calculations 351Table 8.22 MPI’s 2008 Study Discount by Time Period Unregistered

Restricted Stock Only 353Table 8.23 Predictive Power of QMDM versus ECM 354Table 8.24 Predictive Power of QMDM vs. ECM—21% Discount Rate 355Table 8.25 FMV Opinions’ Restricted Stock Discounts 356Table 8.26 FMV Opinions’ 2008 Study Discount by Time Period 357Table 8.27 FMV Opinions’ 2008 Study Discount by Time Period Excludes

Transactions with Registration Rights 357

Chapter 9: Empirical Testing of Abrams’s Valuation Theory 377

Table 9.1 Log Size Equation for 1938–1986 NYSE Data by Decile andStatistical Analysis: 1938–1986 381

Table 9.2 Reconciliation to IBA Database 382Figure 9.1 PE Ratio as a Function of Size from the IBA Database 383Table 9.3 Proof of Discount Calculation 385Table 9.4 Calculation of Component #1—Delay to Sale—$25,000 Firm 388Table 9.5 Calculation of Transaction Costs for Firms of All Sizes in the

IBA Study 390Table 9.6 Calculation of DLOM 391


List of Tables and Figures xvii

Table 9.4A Calculation of Component #1—Delay to Sale—$75,000 Firm 393Table 9.6A Calculation of DLOM 393Table 9.4B Calculation of Component #1—Delay to Sale—$125,000 Firm 394Table 9.6B Calculation of DLOM 394Table 9.4C Calculation of Component #1—Delay to Sale—$175,000 Firm 395Table 9.6C Calculation of DLOM 395Table 9.4D Calculation of Component #1—Delay to Sale—$225,000 Firm 396Table 9.6D Calculation of DLOM 396Table 9.4E Calculation of Component #1—Delay to Sale—$375,000 Firm 397Table 9.6E Calculation of DLOM 397Table 9.4F Calculation of Component #1—Delay to Sale—$750,000 Firm 398Table 9.6 F Calculation of DLOM 398Table 9.4G Calculation of Component #1—Delay to Sale—$10 Million

Firm 399Table 9.6G Calculation of DLOM 399

Chapter 10: Measuring Valuation Uncertainty and Error 403

Table 10.1 Ninety-Five Percent Confidence Intervals 407Table 10.2 Absolute Errors in Forecasting Growth Rates 417Table 10.3 Percent Valuation Error for 10% Relative Error in Growth 419Table 10.3A Percent Valuation Error for -10% Relative Error in Growth 420Table 10.3B Percent Valuation Error for 10% Relative Error in Discount Rate 421Table 10.4 Summary of Effects of Valuation Errors 422

Chapter 11: Demonstrating Expert Bias 427

Table 11.1 Binomial Distribution p = Probability of Randomly TooAggressive = 25% 433

Table 11.2 Binomial Distribution p = Probability of Randomly TooAggressive = 12.5% 433

Chapter 12: Lost Inventory and Lost Profits Damage Formulas in Litigation 435

Table 12.1 Sample Damage Calculations with VM = $95 438Table 12.1A Sample Damage Calculations with VM = $65 439Table 12.1B Lost Profits Formulas and Calculations Based on EBITDA 440Table 12.1C Lost Profits Formulas and Calculations When Company Paid

Employees for the Day Off 441

Chapter 13: ESOPs: Measuring and Apportioning Dilution 451

Table 13.1 Calculation of Lifetime ESOP Costs 457Table 13.2 FMV Calculations: Firm, ESOP, and Dilution 459Figure 13.1 Post-Transaction FMV-ESOP versus Percent Sold 460Table 13.3 Adjusting Dilution to Desired Levels 464Table 13.3A Adjusting Dilution to Desired Levels—All Dilution to Owner 465Table 13.3B Summary of Dilution Trade-offs 466


xviii List of Tables and Figures

Chapter 14: The Trade-off in Selling to an ESOP versus an Outside Buyer 477

Table 14.1 ESOP Valuation Advantage 487Table 14.2A Shareholder Wealth Calculations—Arbitrary Percentage

Sold (p) 488Table 14.2B Shareholder Wealth Calculations—p* Calculated 489

Chapter 15: Buyouts of Partners and Shareholders 497

Table 15.1 Pre- and Post-Transaction Valuations 500Table 15.2 Dilution in FMV as a Result of the Partner Buyout 500Table 15.3 Sharing the Dilution in FMV per Share 501

Chapter 16: Valuing Start-Ups 511

Table 16.1 First Chicago Method 518Table 16.2 VC Pricing Approach 520Table 16.3 Statistical Calculation of Fair Market Value 523Figure 16.1 Decision Tree for Venture Capital Funding 525Figure 16.2 Decision Tree for Bootstrapping Assuming Debt Restructure

and No Venture Capital 526Table 16.4 Sales Model with Exponentially Declining Growth Rate

Assumption 535Figure 16.3 Sales Forecast (Decay Rate = 0.5) 536Figure 16.4A Sales Forecast (Decay Rate = 0.3) 536

Chapter 17: Monte Carlo Risk Simulation 539

Figure 17.1 Point Estimates, Sensitivity Analysis, Scenario Analysis,Probabilistic Scenarios, and Simulations 544

Figure 17.1A Conceptualizing the Lognormal Distribution 545Figure 17.2 Risk Simulator Icons in Excel 546Figure 17.3 New Simulation Profile 546Figure 17.4 Change Active Simulation 548Figure 17.5 Setting an Input Assumption 548Figure 17.6 Assumption Properties 549Figure 17.7 Set Output Forecast 551Figure 17.8 Forecast Chart 552Figure 17.9 Forecast Statistics 553Figure 17.10 Forecast Chart Preferences 553Figure 17.11 Forecast Chart Options 554Figure 17.12 Forecast Chart Two-Tail Confidence Interval 554Figure 17.13 Forecast Chart One-Tail Confidence Interval 555Figure 17.14 Forecast Chart Probability Evaluation: Left-Tail 556Figure 17.15 Forecast Chart Probability Evaluation: Right-Tail 557Figure 17.16 Sample Discounted Cash Flow Model 558Figure 17.17 Running Tornado Analysis 559Figure 17.18 Tornado Analysis Report 560Figure 17.19 Sensitivity Table 560Figure 17.20 Spider Chart 561


List of Tables and Figures xix

Figure 17.21 Tornado Chart 562Figure 17.22 Nonlinear Spider Chart 563Figure 17.23 Sensitivity Chart without Correlations 564Figure 17.24 Sensitivity Chart with Correlations 564Figure 17.25 Running Sensitivity Analysis 565Figure 17.26 Rank Correlation Chart 566Figure 17.27 Contribution to Variance Chart 567Figure 17.28 Single Variable Distributional Fitting 568Figure 17.29 Distribution Fitting Result 569Figure 17.30 Single-Variable Distributional Fitting Report 570

Chapter 18: Real Options 573

Table 18.1 Disadvantages of DCF: Assumptions versus Realities 588Figure 18.1 Applying Discounted Cash Flow Analysis 589Figure 18.2 Shortcomings of Discounted Cash Flow Analysis 590Figure 18.3 Using the Appropriate Analysis 594Figure 18.4 An Analytical Perspective 595Figure 18.5 Financial Options versus Real Options 595Figure 18.6 Option Payoff Charts 596Figure 18.7 Single Super Lattice Solver (SLS) 600Figure 18.8 SLS Results of a Simple European and American Call Option 601Figure 18.9 SLS Comparing Results with Benchmarks 602Figure 18.10 Custom Equation Inputs 603Figure 18.11 SLS Generated Audit Worksheet 604Figure 18.12 SLS Results with a 10-Step Lattice 605Figure 18.13 Multiple Super Lattice Solver 607Figure 18.14 MSLS Solution to a Simple Two-Phased Sequential

Compound Option 608Figure 18.15 Strategy Tree for Two-Phased Sequential Compound Option 608Figure 18.16 Multinomial Lattice Solver 609Figure 18.17 A Simple Call and Put Using Trinomial Lattices 609Figure 18.18 Customized Abandonment Option Using SLS 610Table 18.2 Binomial versus Trinomial Lattices 610Figure 18.19 Customized Abandonment Option Using SLS Excel Solution 611Figure 18.20 Complex Sequential Compound Option Using SLS Excel

Solver 612Figure 18.21 Changing Volatility and Risk-Free Rate Option 613Figure 18.22 Excel’s Equation Wizard 614Figure 18.23 Using SLS Functions in Excel 614Figure 18.24 Lattice Maker 615


xx


Introduction

Nature of the Book

This is an advanced book in the science and art of valuing privately held businesses.In order to read this book, you must already have read at least one introductorybook such as Valuing a Business (Pratt, Reilly, and Schweihs, 1996 and subsequent).Without such a background, you will be lost.

I have written this book with the professional business appraiser as my primaryintended audience, though I think this book is also appropriate for attorneys whoare very experienced in valuation matters, investment bankers, venture capitalists,financial analysts, and MBA students.

Throughout this book, I generally write to you, the reader, as if you are sittingnext to me and we are conversing. I am writing to you as my colleague with whomI share my thinking process. I prefer a conversational tone to a more formal one.

Uniqueness of This Book

This is a rigorous book, and it is not easy reading. However, the following uniqueattributes of this book make reading it worth the effort:

1. It emphasizes regression analysis of empirical data. Chapter 8, “Adjusting forLevels of Control and Marketability,”1 contains the first regression analysis ofthe data related to restricted stock discounts. Chapter 9 from the first editionwas a sample fractional interest discount study containing a regression analysisof the Partnership Profiles database related to secondary limited partnershipmarket trades. In both cases, we found very significant results. We now knowmuch of what drives (a) restricted stock discounts and (b) discounts from netasset values of the publicly registered/privately traded limited partnerships. Wemoved the old Chapter 9 out of this book. It is our intention eventually topublish a workbook to accompany this book—probably when we produce thethird edition. In the meantime, we intend to provide the old Chapter 9 onour website, www.abramsvaluation.com, under “Books,” “Quantitative BusinessValuation.” You will also see much empirical work in Chapter 5, “DiscountRates as a Function of Log Size,” and Chapter 9, “Empirical Testing of Abrams’Valuation Theory.”

1Chapter 7 in the first edition of the book.

xxi


xxii Introduction

2. It emphasizes quantitative skills. Chapter 3 focuses on using regression analysisin business valuation. Chapter 4, the official title of which is “Annuity DiscountFactors and the Gordon Model” (and the unofficial title of which is “The Chapterthat Would Not Die!”) is the most comprehensive treatment of ADFs in print.For anyone wishing to use the Mercer quantitative marketability discount model,Chapter 4 contains the ADF with constant growth not included in Mercer (1997).2

ADFs crop up in many valuation contexts. I invented several new ADFs thatappear in Chapter 4 that are useful in many valuation contexts. Chapter 10contains the first treatise on how much statistical uncertainty we have in ourvaluations and how value is affected when the appraiser makes various errors.

3. It emphasizes putting all the pieces of the puzzle together to present a com-prehensive, unified approach to valuation that can be empirically tested andwhose principles work for the valuation of billion-dollar firms and ma-and-pafirms alike. While this book contains more mathematics—a worm’s-eye view, ifyou will—than other valuation texts, we also refocus to the bird’s-eye view inthis section.

Organization

There are seven parts to this book:

1. Forecasting Cash Flows (Chapters 1 through 4)2. Calculating Discount Rates (Chapter 5 through 7)3. Adjusting for Control and Marketability (i.e., valuation premiums and discounts)

(Chapter 8)4. Putting It All Together (Chapters 9 and 10)5. Litigation (Chapters 11 and 12)6. Valuing ESOPs and Buyouts of Partners and Shareholders (Chapters 13

through 15)7. Probabilistic Valuation Methods (Chapters 16 through 18)

The first three parts of this book follow the chronological sequence of perform-ing a discounted cash flow, although the regression analysis material in Chapter 3applies to market methods as well.

The fourth part is empirically testing whether my methodology in the first threeparts works (i.e., yields reasonable results). Additionally, we explore (1) confidenceintervals around valuation estimates and (2) what happens to the valuation whenappraisers make mistakes.

The reason for moving partnership and shareholder buyouts into Part VI, theESOP section, is they share the common intellectual problem of post-transactiondilution. While the specific topic applications differ, the intellectual problem andprocess to solve it are similar.

The appraisal profession is still in the relatively early stages of using probabilisticvaluation methods. However, it is a topic that is rapidly growing in importance.

2It is possible that he included this in a later edition, but I have not verified that.


Introduction xxiii

Hence we have added Chapters 17 and 18, Monte Carlo Simulation (MCS) and RealOptions (RO) Analysis, to the book. Because valuing start-ups, which was Chapter12 in the first edition, makes use of probabilistic valuation methods, it logically fitstogether with Chapters 17 and 18, which is why I moved it to Chapter 16 in thesecond edition.

I invited Dr. Johnathan Mun, author of Wiley books Modeling Risk and RealOptions Analysis, in addition to many other books, to write Chapters 17 and 18.They are introductions to these two topics and to Dr. Mun’s software. We intendto cover practical examples of using MCS and RO in the workbook. Since that islikely to wait to accompany the third edition of this book, in the meantime look forit on our website somewhere between June 2010 to June 2011. I encourage readerswho want to develop a deep understanding of each topic to buy Dr. Mun’s booksand software, and watch for the workbook and updates on our website. It is simplyimpossible to cover these complex topics in one chapter each.

Differences in the Chapter Numbering

I added a new chapter as Chapter 2 in the second edition. That means that Chapters2 through 7 in the first edition are now 3 through 8, respectively. I moved Chapters8 and 9 from the first edition to our website—eventually to appear in the workbook.Thus, Chapters 10 and 11 from the first edition are now 9 and 10 in the secondedition.

Part V, the “Litigation” section, which consists of Chapters 11 and 12 in thesecond edition, is entirely new. “Valuing Start-Ups” moved from Chapter 12 inthe first edition to Chapter 16 in this edition, as it now fits in a new section ofthe book, “Probabilistic Valuation Methods.”

Chapter 13 has kept the same number in the second edition. Chapter 14 is newin the second edition. Chapter 14 in the first edition is now Chapter 15 in the secondedition. Finally, Chapters 17 and 18 are new in the second edition.

The following two tables should help you reference between chapter numbersin the two editions. The first one is in chapter order number of the first edition,whereas the second one is in chapter order number of the second edition.

First Edition Second Edition1 12 33 44 55 66 77 88 —9 —

10 911 1012 1613 1314 15


xxiv Introduction

The missing chapters in the second edition sequence are new to the secondedition: Chapters 2, 11, 12, 14, 17, and 18.

First Edition Second Edition1 1

NA 22 33 44 55 66 77 88 —9 —

10 911 10NA 11NA 1213 13NA 1414 1512 16NA 17NA 18

Similarities and Differences in the First and Second Editions

While the intellectual content of Chapter 1, “Cash Flow: A Mathematical Derivation,”is largely the same, I nevertheless made a substantial rewrite for better clarity andlogical flow. In general, all chapters that were in the first edition have undergoneintensive editing, even if there is no or little new material. Chapter 2, “ForecastingCash Flow: Mathematics of the Payout Ratio,” is a new chapter that did not existin the first edition. It should help the reader in converting forecast net income toforecast cash flow.

Chapter 3 (Chapter 2 in the first edition), “Using Regression Analysis,” islargely the same as in the first edition, with the important addition of regressingscaled y-variables (Price-to-Sales and Price-Earnings ratios) as a way to control forheteroscedasticity.

Chapter 4 (3 in the first edition), “Annuity Discount Factors and the GordonModel,” is largely the same. However, there are two new sections added: (a) Math-ematical Derivation of the PS Multiple;3 (b) The Bias in Annual (versus Monthly)Discounting Is Immaterial.

Chapter 5 (4 in the first edition) has the following new material: (a) Keeping inthe Roaring Twenties and the Great Depression; (b) Ibbotson’s Opinion of Outliersand the Financial Crisis of 2008; (c) Is the Equity Premium Declining?; (d) Growth

3The first edition had a mathematical derivation of the Price-to-Earnings (PE) ratio. Now thesetwo topics are combined in one section.


Introduction xxv

versus Value Stocks; and (e) The Wedge between Public and Private Firm Valuations[This section is extremely important, being a reconciliation between the Ibbotsontotal returns equation r = d (dividend yield) + g (growth, i.e., capital gains) and theGordon model.]; (f) Satisfying Revenue Ruling 59-60 is substantially different.

Chapters 6 and 7 (5 and 6, respectively, in the first edition), “Arithmetic versusGeometric Means” and “An Iterative Valuation Approach,” are largely the same.

Chapter 8 (7 in the first edition), “Adjusting for Levels of Control and Mar-ketability,” is the largest chapter in the book and requires some explanation. Unlikeother chapters, time pressure with the publishing schedule necessitated finishingthe chapter before I would have preferred. This chapter could use another 3 to 6months’ more research. Of course, by that time, it may well be large enough tobecome a book by itself. When I write the third edition of this book, it is likelyeither that Chapter 8 will become a book by itself, or that I will split it into two ormore chapters.

Table 8.1A contains new data on the Mergerstat database. Chris Mercer extendedthe debate that we had in the first edition into a Business Valuation Review article,and I responded in kind. I have added my response to this chapter, which iscovered in Tables 8.20, 8.21, 8.23, and 8.24. Table 8.22 shows summary statisticsof Management Planning Inc.’s 2008 restricted stock study, and Tables 8.25 through8.27 do the same with FMV Opinions’ 2008 restricted stock study.

In general, I cite and summarize new academic and professional articles andinclude those into our analysis. The analysis is more complex, the data conflict more,and conclusions are murkier in the second edition.

Chapters 9 and 10 (10 and 11 in the first edition), “Empirical Testing of Abrams’Valuation Theory” and “Measuring Valuation Uncertainty and Error,” are largely thesame. Chapters 11 and 12, “Demonstrating Expert Bias” and “Lost Inventory and LostProfits Damage Formulas in Litigation,” are new to the second edition and comprisethe litigation section.

The next three chapters comprise Part 6. Chapter 13, “ESOPs: Measuring andApportioning Dilution,” is largely the same, while Chapter 14, “The Tradeoff inSelling to an ESOP versus an Outside Buyer,” is new. Chapter 15 (14 in the firstedition), “Buyouts of Partners and Shareholders,” while covering the same topic, iscompletely different in the second edition. I use a different model for the effects ofpost-transaction dilution.

Chapter 16 (12 in the first edition), “Valuing Start-Ups,” has little change to thequantitative sections. However, there is some important new research on venturecapital and angel investor rates of return. Chapters 17 and 18, “Monte Carlo RiskSimulation” and “Real Options,” are new.

How to Read This Book

Because this book is more difficult than most, I have done my best to try to providemore paths through it. Chapter 5 contains a shortcut version of the chapter at theend for those who want the bottom line without all the detail. In general, I haveattempted to move most of the heaviest mathematics to appendices in order toleave the bodies of the chapters more readable. Where that was not optimal, I havegiven instructions on which material can be safely skipped.


xxvi Introduction

How you read this book depends on your quantitative skills and how much timeyou have available. For the reader with strong quantitative skills and abundant time,the ideal path is to read the book in its exact order, as there is a logical sequence.

Because most professionals do not have abundant time, I want to suggest an-other path geared for the maximum benefit from the least investment in time. Theheart of the book is “Discount Rates as a Function of Log Size” and “Adjusting forLevels of Control and Marketability,” Chapters 5 and 8, respectively. I recommendthe time-pressed reader follow this order:

1. Chapter 4—the following sections: from the beginning through the section titled“A Brief Summary”; “Periodic Perpetuity Factors: Perpetuities for Periodic CashFlows”; and “Relationship of the Gordon Model Multiple to the Price/Earningsand Price/Sales Ratios.”

2. Chapter 5 (the log size model for calculating discount rates)3. Chapter 8 (“Adjusting for Levels of Control and Marketability”)4. Chapter 9 (this empirically tests Chapters 5 and 9, the heart of the book)

After these chapters, you can read the remainder of the book in any order,though it is best to read each part of the book in order and, better yet, to read theentire book in order.

This book has well over 100 tables, many of them being two or threepages long. To facilitate your reading, you can go to my company’s Web site,www.abramsvaluation.com, click under “Publications” (on the left), then “Books,”then “Quantitative Business Valuation,” and then look for the file download for theQBV tables in PDF format. Then print out the tables and have them handy as youread the book. Otherwise, you will spend an inordinate amount of time flippingpages back and forth.

My Thanks to You

I thank you for investing your valuable time and money to understand my work. Isincerely hope you will greatly benefit from it.

References

Mercer, Z. Christopher. 1997. Quantifying Marketability Discounts: Developing andSupporting Marketability Discounts in the Appraisal of Closely Held BusinessInterests. Memphis, TN: Peabody.

Pratt, Shannon P., Robert F. Reilly, and Robert P. Schweihs. 1996. Valuing a Busi-ness: The Analysis and Appraisal of Closely Held Companies, 3rd ed. New York:McGraw-Hill.


Acknowledgments

I gratefully acknowledge help beyond the call of duty from my parents, Leonardand Marilyn Abrams. Professionally, R. K. Hiatt was my internal editor for the first

edition of this book, and Scott Deifik is for the second edition. Without their help,this book would have suffered greatly. They also contributed important insightsthroughout the book.

Robert Reilly edited the first edition manuscript cover-to-cover. I thank Robertvery much for the huge time commitment for someone else’s book. Larry Kaspergave me a surprise detailed edit of the first eight chapters. I benefited much fromhis input and thank him profusely.

Chris Mercer also read much of the book and gave me many corrections andvery useful feedback. I thank Chris very much for his valuable time, of which hegave me much.

Michael Bolotsky and Eric Nath were very helpful to me in editing my summaryof their work.

I thank Hal Curtiss, Roy Meyers, and Ezra Angrist of Management Planning, Inc.for providing me with their restricted stock data. I also thank Kyle Vataha and LanceHall of FMV Opinions, Inc. for providing me with theirs.

I also thank Bob Jones of Jones, retired from Roach & Caringella for providingme with private fractional interest sales of real estate.

Chaim Borevitz, our vice president, provided important help. Mark Shayne pro-vided me with dozens of insightful comments. Professor William Megginson gaveme considerable feedback on Chapter 8. I thank him for his wisdom, patience, andgood humor. His colleague, Professor Lance Nail, also was very helpful to me.

I also appreciate the following people who gave me good feedback on individualchapters (or their predecessor articles): Don Wisehart, Betsy Cotter, Robert Wietzke,Abdul Walji, Jim Plummer, Mike Annin, Ed Murray, Greg Gilbert, Jared Kaplan, Esq.,John Kober, Esq., Robert Gross, Raymond Miles, and Steven Stamp.

I thank the following people who provided me with useful information thatappears in the book: John Watson, Jr., Esq., David Boatwright, Esq., DouglasObenshain, and Gordon Gregory.

I thank the following people, who reviewed the first edition of this book:Shannon Pratt, Robert Reilly, Jay Fishman, Larry Kasper, Bob Grossman, Terry Isom,Herb Spiro, Don Shannon, Chris Mercer, Dave Bishop, Jim Rigby, and Kent Osborne.

Many people have helped make this book a reality. You all have my profoundgratitude. In fact, there have been so many that it is almost impossible to remember

The majority of these acknowledgments apply to the first edition.

xxvii


xxviii Acknowledgments

every single person, and I apologize to anyone who should be in this acknowledg-ment section whom I forgot. Please forgive me.

Thanks go to John De Remegis at John Wiley & Sons for his commitment to thisbook, and my editor, Judy Howarth, for her pleasant helpfulness.

I thank Dr. Johnathan Mun for graciously providing Chapters 17 and 18 of thisbook. He is a great scholar in our profession.

Finally, I thank my wife, Cindy, for holding down the fort on endless Sundayswhile I was working on this book.

quantitative second edition quantitative business valuation · own, and his conclusions. it is a...

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