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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