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Quantitative

Business Valuation

A Mathematical Approach

for Today™s Professional

JAY B. ABRAMS, ASA, CPA, MBA

McGRAW-HILL

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DOI: 10.1036/007138595X

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,

and Nechamah Leah, who gave up countless Sundays with Abba (Dad)

for this book. To my youngest child, Rivkah Sarah, who wasn™t yet on

the outside to miss the Sundays with me, but who has brought us peace.

To my parents and my brother, Mark, for their tremendous support under

dif¬cult circumstances.

To my great teachers, Mr. Oshima and Christopher Hunt, who

brought me to my power to make this happen. And ¬nally, to R. K. Hiatt,

who has caught my mistakes and made signi¬cant contributions to the

thought that permeates this book.

This page intentionally left blank.

Contents

Introduction xiii

Acknowledgments xvii

List of Figures xix

List of Tables xxi

PART I

FORECASTING CASH FLOWS

3

1. Cash Flow: A Mathematical Derivation

Introduction. The Mathematical Model. A Preliminary Explanation of

Cash Flows. Analyzing Property, Plant, and Equipment Transactions. An

Explanation of Cash Flows with More Detail for Equity Transactions.

Considering the Components of Required Working Capital. Adjusting for

Required Cash. Comparison to Other Cash Flow De¬nitions.

Conclusion.

21

2. Using Regression Analysis

Introduction. Forecasting Costs and Expenses. Adjustments to

Expenses. Table 2-1A: Calculating Adjusted Costs and Expenses.

Performing Regression Analysis. Use of Regression Statistics to Test

the Robustness of the Relationship. Standard Error of the y Estimate.

The Mean of a and b. The Variance of a and b. Selecting the Data Set

and Regression Equation. Problems with Using Regression Analysis

for Forecasting Costs. Insuf¬cient Data. Substantial Changes in

Competition or Product/Service. Using Regression Analysis to

Forecast Sales. Spreadsheet Procedures to Perform Regression.

Examining the Regression Statistics. Adding Industry-Speci¬c

Independent Variables. Try All Combinations of Potential Independent

Variables. Application of Regression Analysis to the Guideline

Company Method. Table 2-5: Regression Analysis of Guideline

Companies. Summary. Appendix: The ANOVA table.

57

3. Annuity Discount Factors and the Gordon Model

Introduction. De¬nitions. Denoting Time. ADF with End-of-Year

Cash Flows. Behavior of the ADF with Growth. Special Case of ADF

vii

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0: The Ordinary Annuity. Special Case when n ’ and r

when g

g: The Gordon Model. Intuitively Understanding Equations (3-6) and

(3-6a). Relationship between the ADF and the Gordon Model. Table 3-1:

Proof of ADF Equations (3-6) through (3-6b). A Brief Summary.

Midyear Cash Flows. Table 3-2: Example of Equations (3-10) through

(3-10b). Special Cases for Midyear Cash Flows: No Growth, g 0.

Gordon Model. Starting Periods Other than Year 1. End-of-Year

Formulas. Valuation Date 0. Table 3-3: Example of Equation (3-11).

Tables 3-4 through 3-6: Variations of Table 3-3 with S 0, Negative

Growth, and r g. Special Case: No Growth, g 0. Generalized

Gordon Model. Midyear Formula. Periodic Perpetuity Factors (PPFs):

Perpetuities for Periodic Cash Flows. The Mathematical Formulas.

Tables 3-7 and 3-8: Examples of Equations (3-18) and (3-19). Other

Starting Years. New versus Used Equipment Decisions. ADFs in Loan

Mathematics. Calculating Loan Payments. Present Value of a Loan.

Relationship of the Gordon Model to the Price/Earnings Ratio.

De¬nitions. Mathematical Derivation. Conclusions.

PART II

CALCULATING DISCOUNT RATES

117

4. Discount Rates as a Function of Log Size

Prior Research. Table 4-1: Analysis of Historical Stock Returns.

Regression #1: Return versus Standard Deviation of Returns. Regression

#2: Return versus Log Size. Regression #3: Return versus Beta. Market

Performance. Which Data to Choose? Recalculation of the Log Size Model

Based on 60 Years. Application of the Log Size Model. Discount Rates

Based on the Log Size Model. Practical Illustration of the Log Size

Model: Discounted Cash Flow Valuations. Total Return versus Equity

Premium. Adjustments to the Discount Rate. Discounted Cash Flow or

Net Income? Discussion of Models and Size Effects. CAPM. The

Fama“French Cost of Equity Model. Log Size Models. Heteroscedasticity.

Industry Effects. Satisfying Revenue Ruling 59-60 without a

Guideline Public Company Method. Summary and Conclusions.

Appendix A: Automating Iteration Using Newton™s Method.

Appendix B: Mathematical Appendix. Appendix C: Abbreviated

Review and Use.

5. Arithmetic versus Geometric Means: Empirical Evidence and

169

Theoretical Issues

Introduction. Theoretical Superiority of Arithmetic Mean. Table 5-1:

Comparison of Two Stock Portfolios. Empirical Evidence of the

Superiority of the Arithmetic Mean. Table 5-2: Regressions of

Geometric and Arthmetic Returns for 1927“1997. Table 5-3: Regressions

of Geometric Returns for 1938“1997. The Size Effect on the Arithmetic

versus Geometric Means. Table 5-4: Log Size Comparison of Discount

Rates and Gordon Model Multiples Using AM versus GM. Indro and

Lee Article.

179

6. An Iterative Valuation Approach

Introduction. Equity Valuation Method. Table 6-1A: The First

Iteration. Table 6-1B: Subsequent Iterations of the First Scenario. Table

Contents

viii

6-1C: Initial Choice of Equity Doesn™t Matter. Convergence of the Equity

Valuation Method. Invested Capital Approach. Table 6-2A: Iterations

Beginning with Book Equity. Table 6-2B: Initial Choice of Equity Doesn™t

Matter. Convergence of the Invested Capital Approach. Log Size.

Summary. Bibliography.

PART III

ADJUSTING FOR CONTROL AND MARKETABILITY

195

7. Adjusting for Levels of Control and Marketability

Introduction. The Value of Control and Adjusting for Level of

Control. Prior Research”Qualitative Professional. Prior Research”

Academic. My Synthesis and Analysis. Discount for Lack of

Marketability (DLOM). Mercer™s Quantitative Marketability Discount

Model. Kasper™s BAS Model. Restricted Stock Discounts. Abrams™

Economic Components Model. Mercer™s Rebuttal. Conclusion.

Mathematical Appendix.

293

8. Sample Restricted Stock Discount Study

Introduction. Background. Stock Ownership. Purpose of the Appraisal.

No Economic Outlook Section. Sources of Data. Valuation. Commentary

to Table 8-1: Regression Analysis of Management Planning Data.

Commentary to Table 8-1A: Revenue and Earnings Stability.

Commentary to Table 8-1B: Price Stability. Valuation Using Options

Pricing Theory. Conclusion of Discount for Lack of Marketability.

Assumptions and Limiting Conditions. Appraiser™s Quali¬cations.

315

9. Sample Appraisal Report

Introduction. Purpose of the Report. Valuation of Considerations.

Sources of Data. History and Description of the LLC. Signi¬cant

Terms and Legal Issues. Conclusion. Economic Outlook. Economic

Growth. In¬‚ation. Interest Rates. State and Local Economics. Summary.

Financial Review. Commentary to Table 9-2: FMV Balance Sheets.

Commentary to Table 9-3: Income Statements. Commentary to Table 9-4:

Cash Distributions. Valuation. Valuation Approaches. Selection of

Valuation Approach. Economic Components Approach. Commentary to

Table 9-5: Calculations of Combined Discounts. Commentary to Table

9-5A: Delay-to-Sale. Commentary to Table 9-5C: Calculation of DLOM.

Commentary to Table 9-6: Partnership Pro¬les Approach”1999.

Commentary to Table 9-7: Private Fractional Interest Sales. Commentary

to Table 9-8: Final Calculation of Fractional Interest Discounts.

Conclusion. Statement of Limiting Conditions. Appraiser™s

Quali¬cations. Appendix: Tax Court™s Opinion for Discount for

Lack of Marketability. Introduction. The Court™s 10 Factors.

Application of the Court™s 10 Factors to the Valuation.

PART IV

PUTTING IT ALL TOGETHER

357

10. Empirical Testing of Abrams™ Valuation Theory

Introduction. Steps in the Valuation Process. Applying a Valuation

Model to the Steps. Table 10-1: Log Size for 1938“1986. Table 10-2:

Contents ix

Reconciliation to the IBA Database. Part 1: IBA P/CF Multiples.

Part 2: Log Size P/CF Multiples. Conclusion. Calculation of DLOM.

Table 10-4: Computation of the Delay-to-Sale Component“$25,000 Firm.

Table 10-5: Calculation of Transactions Costs. Table 10-6: Calculation of

DLOM. Table 10-6A“10-6F: Calculations of DLOM for Larger Firms.

Calculation of DLOM for Large Firms. Interpretation of the Error.

Conclusion.

383

11. Measuring Valuation Uncertainty and Error

Introduction. Differences Between Uncertainty and Error. Sources of

Uncertainty and Error. Measuring Valuation Uncertainty. Table 11-1:

95% Con¬dence Intervals. Summary of Valuation Implications of

Statistical Uncertainty in the Discount Rate. Measuring the Effects of

Valuation Error. De¬ning Absolute and Relative Error. The Valuation

Model. Dollar Effects of Absolute Errors in Forecastng Year 1 Cash Flow.

Relative Effects of Absolute Errors in Forecasting Year 1 Cash Flow.

Absolute and Relative Effects of Relative Errors in Forecasting Year 1

Cash Flow. Absolute Errors in Forecasting Growth and the Discount

Rate. Table 11-5: Summary of Effects of Valuation Errors. Summary and

Conclusions.

PART V

SPECIAL TOPICS

407

12. Valuing Startups

Issues Unique to Startups. Organization of the Chapter. First

Chicago Approach. Discounting Cash Flow Is Preferable to Net Income.

Capital Structure Changes. Venture Capital Rates of Return. Table 12-1:

Example of the First Chicago Approach. Advantages of the First Chicago

Approach. Discounts for Lack of Marketability and Control. Venture

Capital Valuation Approach. Venture Capital Rates of Return.

Summary of the VC Approach. Debt Restructuring Study. Backgound.

Key Events. Decision Trees and Spreadsheet Calculations. Table 12-3:

Statistical Calculation of FMV. Conclusion. Exponentially Declining

Sales Growth Model.

433

13. ESOPs: Measuring and Apportioning Dilution

Introduction. What Can Be Skipped. De¬nitions of Dilution. Dilution

to the ESOP (Type 1 Dilution). Dilution to the Selling Owner (Type 2

Dilution). De¬ning Terms. Table 13-1: Calculation of Lifetime ESOP

Costs. The Direct Approach. FMV Equations”All Dilution to the

ESOP (Type 1 Dilution; No Type 2 Dilution). Table 13-2, Sections 1 and

2: Post-transaction FMV with All Dilution to the ESOP. The Post-

transaction Value Is a Parabola. FMV Equations”All Dilution to the

Owner (Type 2 Dilution). Table 13-2, Section 3: FMV Calculations”All

Dilution to the Seller. Sharing the Dilution. Equation to Calculate Type 2

Dilution. Tables 13-3 and 13-3A: Adjusting Dilution to Desired Levels.

Table 13-3B: Summary of Dilution Tradeoffs. The Iterative Approach.

Iteration #1. Iteration #2. Iteration #3. Iteration #n. Summary.

Advantages of Results. Function of ESOP Loan. Common Sense Is

Required. To Whom Should the Dilution Belong? Appendix A:

Contents

x

Mathematical Appendix. Appendix B: Shorter Version of Chapter

13.

471

14. Buyouts of Partners and Shareholders

Introduction. An Example of a Buyout. The Solution. Evaluating the

Benchmarks.

Glossary 475

Index 479

Contents xi

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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 introductory book such as Valuing A Business (Pratt, Reilly, and

Schweihs 1996). Without such a background, you will be lost.

I have written this book with the professional business appraiser as

my primary intended audience, though I think this book is also appro-

priate for attorneys who are very experienced in valuation matters, in-

vestment bankers, venture capitalists, ¬nancial analysts, and MBA stu-

dents.

Uniqueness of This Book

This is a rigorous book, and it is not easy reading. However, the following

unique attributes of this book make reading it worth the effort:

1. It emphasizes regression analysis of empirical data. Chapter 7,

adjusting for control and marketability, contains the ¬rst

regression analysis of the data related to restricted stock

discounts. Chapter 9, a sample fractional interest discount study,

contains a regression analysis of the Partnership Pro¬les

database related to secondary limited partnership market trades.

In both cases we found very signi¬cant results. We now know

much of what drives (a) restricted stock discounts and (b)

discounts from net asset values of the publicly registered/

privately traded limited partnerships. You will also see much

empirical work in Chapter 4, ˜˜Discount Rates as a Function of

Log Size,™™ and Chapter 11, ˜˜Empirical Testing of Abrams™

Valuation Theory.™™

2. It emphasizes quantitative skills. Chapter 2 focuses on using

regression analysis in business valuation. Chapter 3, ˜˜Annuity

Discount Factors and the Gordon Model,™™ is the most

comprehensive treatment of ADFs in print. For anyone wishing

to use the Mercer quantitative marketability discount model,

xiii

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Chapter 4 contains the ADF with constant growth not included

in the Mercer text. ADFs crop up in many valuation contexts. I