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Quantitative Business Valuation
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Quantitative
Business Valuation
A Mathematical Approach
for Today™s Professional




JAY B. ABRAMS, ASA, CPA, MBA




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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




Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
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



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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
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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




Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
Chapter 4 contains the ADF with constant growth not included
in the Mercer text. ADFs crop up in many valuation contexts. I

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