Quantitative business valuation

Quantitative Business Valuation
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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
abc
McGraw-Hill
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
difficult circumstances.
To my great teachers, Mr. Oshima and Christopher Hunt, who
brought me to my power to make this happen. And finally, to R. K. Hiatt,
who has caught my mistakes and made significant contributions to the
thought that permeates this book.
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vii
Contents
Introduction xiii
Acknowledgments xvii
List of Figures xix
List of Tables xxi
PART I
FORECASTING CASH FLOWS
1. Cash Flow: A Mathematical Derivation 3
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 Definitions.
Conclusion.
2. Using Regression Analysis 21
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. Insufficient 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-Specific
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.
3. Annuity Discount Factors and the Gordon Model 57
Introduction. Definitions. Denoting Time. ADF with End-of-Year
Cash Flows. Behavior of the ADF with Growth. Special Case of ADF
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viii Contents
when g ϭ 0: The Ordinary Annuity. Special Case when n → ϱ and r Ͼ
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 ϭ 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 Ͻ 0, Negative
Growth, and r Ͻ g. Special Case: No Growth, g ϭ 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.
Definitions. Mathematical Derivation. Conclusions.

PART II
CALCULATING DISCOUNT RATES
4. Discount Rates as a Function of Log Size 117
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.
AppendixA: Automating Iteration Using Newton’s Method.
AppendixB: Mathematical Appendix. AppendixC: Abbreviated
Review and Use.
5. Arithmetic versus Geometric Means: Empirical Evidence and
Theoretical Issues 169
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.
6. An Iterative Valuation Approach 179
Introduction. Equity Valuation Method. Table 6-1A: The First

Iteration. Table 6-1B: Subsequent Iterations of the First Scenario. Table
Contents ix
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
7. Adjusting for Levels of Control and Marketability 195
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.
8. Sample Restricted Stock Discount Study 293
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 Qualifications.
9. Sample Appraisal Report 315
Introduction. Purpose of the Report. Valuation of Considerations.
Sources of Data. History and Description of the LLC. Significant
Terms and Legal Issues. Conclusion. Economic Outlook. Economic
Growth. Inflation. 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 Profiles 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
Qualifications. 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
10. Empirical Testing of Abrams’ Valuation Theory 357
Introduction. Steps in the Valuation Process. Applying a Valuation
Model to the Steps. Table 10-1: Log Size for 1938–1986. Table 10-2:
x Contents
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.
11. Measuring Valuation Uncertainty and Error 383
Introduction. Differences Between Uncertainty and Error. Sources of
Uncertainty and Error. Measuring Valuation Uncertainty. Table 11-1:
95% Confidence Intervals. Summary of Valuation Implications of

Statistical Uncertainty in the Discount Rate. Measuring the Effects of
Valuation Error. Defining 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
12. Valuing Startups 407
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.
13. ESOPs: Measuring and Apportioning Dilution 433
Introduction. What Can Be Skipped. Definitions of Dilution. Dilution
to the ESOP (Type 1 Dilution). Dilution to the Selling Owner (Type 2
Dilution). Defining 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? AppendixA:
Contents xi
Mathematical Appendix. Appendix B: Shorter Version of Chapter
13.
14. Buyouts of Partners and Shareholders 471
Introduction. An Example of a Buyout. The Solution. Evaluating the
Benchmarks.
Glossary 475
Index 479
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xiii
Introduction
NATURE OFTHE 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, financial 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 first
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 Profiles
database related to secondary limited partnership market trades.
In both cases we found very significant 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,
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
xiv Introduction
Chapter 4 contains the ADF with constant growth not included
in the Mercer text. ADFs crop up in many valuation contexts. I
invented several new ADFs that appear in Chapter 3 that are
useful in many valuation contexts. Chapter 11 contains the first
treatise on how much statistical uncertainty we have in our
valuations 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 comprehensive, unified approach to valuation that can
be empirically tested and whose principles work for the

valuation of billion-dollar firms and ma and pa firms alike.
While this book contains more mathematics—a worm’s eye
view, if you will—than other valuation texts, it also has more of
a bird’s eye view as well.
HOW TO READ THIS BOOK
I have tried to provide paths through this book to make it easier to follow.
Chapters 4 and 13 both contain a shortcut version of the chapter at the
end for those who want the bottom line without all the detail. In general,
I have moved most of the heaviest mathematics to appendices in order
to leave the bodies of the chapters more readable. Where that was not
optimal, I have given instructions on which material can be safely
skipped.
How to read this book depends on your quantitative skills and how
much time you 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. The first three parts to this book
follow the chronological sequence of performing a valuation: (1) forecast
cash flows, (2) discount to present value, and (3) adjust for control and
marketability. The fourth part is a bird’s eye view in order to test empir-
ically whether my methodology works. Additionally, we explore (1) con-
fidence intervals around valuation estimates and (2) what happens to the
valuation when appraisers make mistakes. Part 5, on special topics, is the
place for everything else. Each of parts of the book has an introduction
preceding it that will preview the upcoming material in greater depth
than we cover here.
Because most professionals do not have abundant time, I want to
suggest another path geared for the maximum benefit from the least in-
vestment in time. The heart of the book is Chapters 4 and 7, on log size
and on adjusting for control and marketability, respectively. I recommend
the time-pressed reader follow this order:

1. Chapter 7 (discounts for lack of control and lack of
marketability)
2. Chapter 8 (this is an application of Chapter 7—a sample
restricted stock report)
3. Chapter 9 (this is an application of Chapter 7—a sample
fractional ownership interest discount report)
Introduction xv
4. Chapter 4 (the log size model for calculating discount rates)
5. Chapter 3—the following sections: from the beginning through
the section titled ‘‘A Brief Summary’’; ‘‘Periodic Perpetuity
Factors: Perpetuities for Periodic Cash Flows’’; and
‘‘Relationship of Gordon Model Multiple to the Price/Earnings
Ratio.’’ Some readers may want to read this chapter after
Chapter 7, though it will be somewhat helpful, but definitely
not necessary, to have read Chapter 3 before 4 and 7.
6. Chapter 10 (this empirically tests Chapters 4 and 7, the heart of
the book)
7. Chapter 2 (some readers may want to start with Chapter 2 first,
as the material on using regression analysis may help reading
all of the other chapters).
After these chapters, you can read the remainder of the book in any
order, though it is best to read Chapter 14 immediately after Chapter 13.
This book has close to 125 tables, many of them being two or three
pages long. To facilitate your reading, it will help you to copy tables
whose commentary in the text is extensive and sit with the separate tables
next to you. Otherwise, you will spend an inordinate amount of time
flipping pages back and forth. Note: readers with low blood pressure may
wish to ignore that advice.
BIBLIOGRAPHY
Mercer, Z. Christopher. 1997. Quantifying Marketability Discounts: Developing and Supporting

Marketability Discounts in the Appraisal of Closely Held Business Interests. Memphis,
Tenn.: Peabody.
Pratt, Shannon P., Robert F. Reilly, and Robert P. Schweihs. 1996. Valuing a Business:
The Analysis and Appraisal of Closely Held Companies, 3d ed. New York: McGraw-Hill.
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xvii
Acknowledgments
I gratefully acknowledge help beyond the call of duty from my parents,
Leonard and Marilyn Abrams. Professionally, R. K. Hiatt has been the
ideal internal editor. Without his help, this book would have suffered
greatly. He also contributed important insights throughout the book.
Robert Reilly edited the original manuscript cover-to-cover. I thank
Robert very much for the huge time commitment for someone else’s book.
Larry Kasper gave me a surprise detailed edit of the first eight chapters.
I benefited much from his input and thank him profusely.
Chris Mercer also read much or all of the book and gave me many
corrections and very useful feedback. I thank Chris very much for his
valuable time, of which he gave me much.
Michael Bolotsky and Eric Nath were very helpful to me in editing
my summary of their work.
I thank Rob Oliver and Roy Meyers of Management Planning, Inc.
for providing me with their restricted stock data. I also thank Bob Jones
of Jones, Roach & Caringella for providing me with private fractional
interest sales of real estate.
Chaim Borevitz provided important help on Chapters 8 and 9. Mark
Shayne provided me with dozens of insightful comments. Professor Wil-
liam Megginson gave me considerable feedback on Chapter 7. I thank
him for his wisdom, patience, and good humor. His colleague, Professor
Lance Nail, also was very helpful to me.
I also appreciate the following people who gave me good feedback

on individual chapters (or their predecessor articles): Don Wisehart, Betsy
Cotter, Robert Wietzke, Abdul Walji, Jim Plummer, Mike Annin, Ed Mur-
ray, Greg Gilbert, Jared Kaplan, Esq., Robert Gross, Raymond Miles, and
Steven Stamp.
I thank the following people who provided me with useful infor-
mation that appears in the book: John Watson, Jr., Esq., David Boatwright,
Esq., Douglas Obenshain, and Gordon Gregory.
I thank the following people who reviewed this book for McGraw-
Hill: Shannon Pratt, Robert Reilly, Jay Fishman, Larry Kasper, Bob Gross-
man, Terry Isom, Herb Spiro, Don Shannon, Chris Mercer, Dave Bishop,
Jim Rigby, and Kent Osborne.
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
xviii Acknowledgments
I thank my editor at McGraw-Hill, Ela Aktay, for her encouragement
and enthusiasm. I thank my publisher, Jeff Krames, for his patience and
apologize for testing it so much due to my passion for perfection and the
huge life-changes that occurred to me while writing this book.
All the people who have helped make this book a reality have my
profound gratitude. In fact, there have been so many that it is almost
impossible to remember every single person, and I apologize to anyone
who should be in this acknowledgment section and who is not due to
my human failings.
xix
List of Figures
2-1 Z-distribution vs t-distribution 34
2-2 t-distribution of B around the Estimate b35
3-1 Timeline of the ADF and Gordon Model 65
A3-1 Timeline of Cash Flows 91
A3-2 Payment Schedule 100
4-1 1926–1998 Arithmetic Mean Returns as a Function of Standard

Deviation 125
4-2 1926–1998 Arithmetic Mean Returns as a Function of Ln(FMV) 127
4-3 Decade Standard Deviation of Returns versus Decade Average
FMV per Company on NYSE 1935–1995 128
4-4 Decade Standard Deviation of Returns versus Decade Average
FMV per Company on NYSE 1945–1995 129
4-5 Average Returns Each Decade 130
4-6 The Natural Logarithm 137
4-7 Discount Rates as a Function of FMV 137
4-8 1939–1998 Decile Standard Deviations as a Function of Ln(FMV) 138
7-1 Traditional Levels of Value Chart 197
7-2 Two-Tiered Levels of Value Chart 198
7-3 3 ϫ 2 Levels of Value Chart 230
10-1 P/E Ratio as a Function of Size (From the IBA Database) 363
12-1 Decision Tree for Venture Capital Funding 421
12-2 Decision Tree for Bootstrapping Assuming Debt Restructure and
No Venture Capital 425
12-3 Sales Forecast (Decay Rate ϭ 0.5) 430
12-3A Sales Forecast (Decay Rate ϭ 0.3) 431
13-1 Post-Transaction Value of the ESOP Vs. % Sold 442
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xxi
List of Tables
1-1 Abbreviated Balance Sheets 7
1-2 Income Statement 8
1-3 Analysis of Property, Plant, and Equipment 9
1-4 Statement of Stockholders’ Equity 12
1-5 Abbreviated Statement of Cash Flows 13
1-6 Balance Sheets 15

1-7 Statement of Cash Flows 16
2-1A Adjustments to Historical Costs and Expenses 24
2-1B Regression Analysis 1988–1997 27
2-1B Calculation of 95% Confidence Intervals for Forecast 1998 Costs 28
2-2 OLS Regression: Example of Deviation from Mean 31
2-3 Abbreviated Table of T-Statistics 36
2-4 Regression Analysis 1993–1997 40
2-5 Regression Analysis of Sales as a Function of GDP [1] 43
2-6 Regression Analysis of Guideline Companies 47
A2-1 Regression Analysis 1988–1997 54
3-1 ADF: End-of-Year Formula 66
3-2 ADF: Midyear Formula 69
3-3 ADF with Cash Flows Starting in Year 3.25: End-of-Year Formula 72
3-4 ADF with Cash Flows Starting in Year Ϫ2.00: End-of-Year
Formula 74
3-5 ADF with Cash Flows Starting in Year Ϫ2.00 with Negative
Growth: End-of-Year Formula 75
3-6 ADF with Cash Flows Starting in Year Ϫ2.00 with g Ͼ r:
End-of-Year Formula 76
3-7 Periodic Perpetuity Factor (PPF): End-of-Year Formula 80
3-8 Periodic Perpetuity Factor (PPF): Midyear Formula 80
3-9 Periodic Perpetuity Factor (PPF): End-of-Year—Cash Flows Begin
Year 6 83
3-10 PV of Loan with Market Rate Ͼ Nominal Rate: ADF, End-of-Year 86
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
xxii List of Tables
3-11 ADF Equation Numbers 90
A3-1 ADF with Fractional Year: Midyear Formula 95
A3-2 ADF with Fractional Year: Midyear Formula 96
A3-3 Amortization of Principal with Irregular Starting Point 103

A3-4 PV of Principal Amortization 107
A3-5 Present Value of a Loan at Discount Rate Different than Nominal
Rate 110
4-1 NYSE Data by Decile and Statistical Analysis: 1926–1998 121
4-2 Regressions of Returns over Standard Deviation and Log of Fair
Market Value 132
4-2A Regression Comparison [1] 133
4-3 Table of Stock Market Returns Based on FMV—60-Year Model 136
4-4A Discounted Cash Flow Analysis Using 60-Year Model—First
Iteration 141
4-4B Discounted Cash Flow Analysis Using 60-Year Model—Second
Iteration 142
4-4C Discounted Cash Flow Analysis Using 60-Year Model—Final
Valuation 143
A4-5 Gordon Model Valuation Using Newton’s Iterative Process 157
5-1 Geometric versus Arithmetic Returns 171
5-2 Geometric versus Arithmetic Returns: NYSE Data by Decile &
Statistical Analysis: 1926–1997 172
5-3 Geometric Mean versus AFMV: 60 Years 174
5-4 Comparison of Discount Rates Derived from the Log Size Model
Using 60-Year Arithmetic and Geometric Means 176
6-1A Equity Valuation Approach with Iterations Beginning with Book
Equity: Iteration #1 182
6-1B Equity Valuation Approach with Iterations Beginning with Book
Equity 184
6-1C Equity Valuation Approach with Iterations Beginning with
Arbitrary Equity 185
6-2A WACC Approach with Iterations Beginning with Book Equity 187
6-2B WACC Approach with Iterations Beginning with Arbitrary
Guess of Equity Value 189

7-1 Synergies as Measured by Acquisition Minus Going-Private
Premiums 199
7-1A Acquisition and Going-Private Transactions Premiums 200
7-2 Acquisition Premiums by SIC Code 206
7-3 Analysis of Megginson Results 215
7-3A Analysis of American VRP Results 218
7-4 Mergerstat Mean Premiums: Control versus Minority Purchases 225
7-5 Abrams Regression of Management Planning Study Data 237
7-6 Calculation of Continuously Compounded Standard Deviation
Chantal Pharmaceutical, Inc.—CHTL 242
7-7 Black–Scholes Put Option—CHTL 244
7-8 Put Model Results 245
List of Tables xxiii
7-9 Calculation of Restricted Stock Discounts for 13 Stocks Using
Regression from Table 7-5 247
7-10 Calculation of Component #1—Delay To Sale [1] 254
7-11 Estimates of Transaction Costs [1] 259
7-12 Proof of Equation (7-9) 267
7-13 Proof of Equation (7-9a) 270
7-14 Sample Calculation of DLOM 272
7-15
a
QMDM Baseline Data—30% MPI Discount 275
7-16
a
Implied Returns for Holding Period—30% Discount 275
7-17
a
Implied Returns for Holding Period—20% Discount 276
7-18 Summary of Results of Applying the QMDM in 10 Example

Appraisals 277
7-19 QMDM Comparison of Restricted Stock Discount Rate versus
Mercer Example 1 279
8-1 Abrams Valuation Group Regression of Management Planning,
Inc. Data 300
8-1A Calculation of Revenue and Earnings Stability (R
2
) 304
8-1B Calculation of Price Stability 305
8-2 Black–Scholes Call and Put Options 307
8-2A Standard Deviation of Continuously Compounded Returns 309
8-3 Final Calculation of Discount 310
9-1 Member Interests at Inception on 1/6/96 322
9-2 Balance Sheets 12/25/99 [1] 324
9-3 Income Statements [1] 325
9-4 Cash Distributions 326
9-5 Economic Components Approach: 2.80% Member Interest 328
9-5A Calculation of Component #1: Delay to Sale [1] 330
9-5B Earnings and Revenue Stability 332
9-5C Calculation of DLOM: 2.80% Member Interest 334
9-6 Regression Analysis of Partnership Profiles Database—1999 [1] 339
9-6A Correlation Matrix 341
9-6B Partnership Profiles Database: Price-to-Value Discounts—1999 342
9-7 Private Fractional Interest Sales 346
9-8 Final Calculation of Fractional Interest Discount 347
10-1 Log Size Equation for 1938–1986 NYSE Data by Decile and
Statistical Analysis: 1938–1986 360
10-2 Reconciliation to IBA Database 362
10-3 Proof of Discount Calculation 364
10-4 Calculation of Component #1—Delay to Sale—$25,000 Firm 367

10-5 Calculation of Transaction Costs for Firms of All Sizes in the
IBA Study 369
10-6 Calculation of DLOM—$25,000 Firm 370
10-4A Calculation of Component #1—Delay to Sale—$75,000 Firm 371
10-4B Calculation of Component #1—Delay to Sale—$125,000 Firm 371
xxiv List of Tables
10-4C Calculation of Component #1—Delay to Sale—$175,000 Firm 372
10-4D Calculation of Component #1—Delay to Sale—$225,000 Firm 372
10-4E Calculation of Component #1—Delay to Sale—$375,000 Firm 373
10-4F Calculation of Component #1—Delay to Sale—$750,000 Firm 373
10-4G Calculation of Component #1—Delay to Sale—$10 Million
Firm 374
10-6A Calculation of DLOM—$75,000 Firm 374
10-6B Calculation of DLOM—$125,000 Firm 375
10-6C Calculation of DLOM—$175,000 Firm 375
10-6D Calculation of DLOM—$225,000 Firm 376
10-6E Calculation of DLOM—$375,000 Firm 376
10-6F Calculation of DLOM—$750,000 Firm 377
10-6G Calculation of DLOM—$10,000,000 Firm 377
11-1 95% Confidence Intervals 389
11-2 95% Confidence Intervals—60-Year Log Size Model 391
11-3 Absolute Errors in Forecasting Growth Rates 398
11-4 Percent Valuation Error for 10% Relative Error in Growth 400
11-4A Percent Valuation Error for Ϫ10% Relative Error in Growth 401
11-4B Percent Valuation Error for 10% Relative Error in Discount Rate 402
11-5 Summary of Effects of Valuation Errors 403
12-1 First Chicago Method 412
12-2 VC Pricing Approach 414
12-3 Statistical Calculation of Fair Market Value 418
12-4 Sales Model with Exponentially Declining Growth Rate

Assumption 430
13-1 Calculation of Lifetime ESOP Costs 438
13-2 FMV Calculations: Firm, ESOP, and Dilution 441
13-3 Adjusting Dilution to Desired Levels 446
13-3A Adjusting Dilution to Desired Levels—All Dilution to Owner 447
13-3B Summary of Dilution Tradeoffs 447