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Quantitative Business Valuation
Other Titles in the Irwin Library of Investment and Finance
Convertible Securities
by John P. Calamos
Pricing and Managing Exotic and Hybrid Options
by Vineer Bhansali
Risk Management and Financial Derivatives
by Satyajit Das
Valuing Intangible Assets
by Robert F. Reilly and Robert P. Schweihs
Managing Financial Risk
by Charles W. Smithson
High-Yield Bonds
by Theodore Barnhill, William Maxwell, and Mark Shenkman
Valuing Small Business and Professional Practices, 3rd edition
by Shannon Pratt, Robert F. Reilly, and Robert P. Schweihs
Implementing Credit Derivatives
by Israel Nelken
The Handbook of Credit Derivatives
by Jack Clark Francis, Joyce Frost, and J. Gregg Whittaker
The Handbook of Advanced Business Valuation
by Robert F. Reilly and Robert P. Schweihs
Global Investment Risk Management
by Ezra Zask
Active Portfolio Management 2nd edition
by Richard Grinold and Ronald Kahn
The Hedge Fund Handbook
by Stefano Lavinio
Pricing, Hedging, and Trading Exotic Options
by Israel Nelken
Equity Management
by Bruce Jacobs and Kenneth Levy
Asset Allocation, 3rd edition
by Roger Gibson
Valuing a Business, 4th edition
by Shannon P. Pratt, Robert F. Reilly, and Robert Schweihs
The Relative Strength Index Advantage
by Andrew Cardwell and John Hayden
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
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.
This page intentionally left blank.
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
vii
Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
when g ⫽ 0: The Ordinary Annuity. Special Case when n → ⬁ and r ⬎
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.
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.
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
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
Introduction. Equity Valuation Method. Table 6-1A: The First
Iteration. Table 6-1B: Subsequent Iterations of the First Scenario. Table
viii
Contents
179
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:
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.
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
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 Posttransaction 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:
x
Contents
433
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
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 appropriate for attorneys who are very experienced in valuation matters, investment bankers, venture capitalists, ﬁnancial analysts, and MBA students.
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,
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
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 empirically whether my methodology works. Additionally, we explore (1) conﬁdence 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 investment 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)
xiv
Introduction
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.
Introduction
xv
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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 William 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 Murray, Greg Gilbert, Jared Kaplan, Esq., Robert Gross, Raymond Miles, and
Steven Stamp.
I thank the following people who provided me with useful information 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 McGrawHill: 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.
xvii
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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.
xviii
Acknowledgments
List of Figures
2-1
2-2
3-1
A3-1
A3-2
4-1
4-2
4-3
4-4
4-5
4-6
4-7
4-8
7-1
7-2
7-3
10-1
12-1
12-2
12-3
12-3A
13-1
Z-distribution vs t-distribution
t-distribution of B around the Estimate b
Timeline of the ADF and Gordon Model
Timeline of Cash Flows
Payment Schedule
1926–1998 Arithmetic Mean Returns as a Function of Standard
Deviation
1926–1998 Arithmetic Mean Returns as a Function of Ln(FMV)
Decade Standard Deviation of Returns versus Decade Average
FMV per Company on NYSE 1935–1995
Decade Standard Deviation of Returns versus Decade Average
FMV per Company on NYSE 1945–1995
Average Returns Each Decade
The Natural Logarithm
Discount Rates as a Function of FMV
1939–1998 Decile Standard Deviations as a Function of Ln(FMV)
Traditional Levels of Value Chart
Two-Tiered Levels of Value Chart
3 ⫻ 2 Levels of Value Chart
P/E Ratio as a Function of Size (From the IBA Database)
Decision Tree for Venture Capital Funding
Decision Tree for Bootstrapping Assuming Debt Restructure and
No Venture Capital
Sales Forecast (Decay Rate ⫽ 0.5)
Sales Forecast (Decay Rate ⫽ 0.3)
Post-Transaction Value of the ESOP Vs. % Sold
34
35
65
91
100
125
127
128
129
130
137
137
138
197
198
230
363
421
425
430
431
442
xix
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