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invented several new ADFs that appear in Chapter 3 that are
useful in many valuation contexts. Chapter 11 contains the ¬rst
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, uni¬ed approach to valuation that can
be empirically tested and whose principles work for the
valuation of billion-dollar ¬rms and ma and pa ¬rms 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 ¬rst three parts to this book
follow the chronological sequence of performing a valuation: (1) forecast
cash ¬‚ows, (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-
¬dence 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 bene¬t 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
xiv
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 de¬nitely
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 ¬rst,
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
¬‚ipping 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 ¬rst eight chapters.
I bene¬ted 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.



xvii




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




Acknowledgments
xviii
List of Figures




Z-distribution vs t-distribution 34
2-1
t-distribution of B around the Estimate b 35
2-2
Timeline of the ADF and Gordon Model 65
3-1
Timeline of Cash Flows 91
A3-1
Payment Schedule 100
A3-2
1926“1998 Arithmetic Mean Returns as a Function of Standard
4-1
Deviation 125
1926“1998 Arithmetic Mean Returns as a Function of Ln(FMV) 127
4-2
Decade Standard Deviation of Returns versus Decade Average
4-3
FMV per Company on NYSE 1935“1995 128
Decade Standard Deviation of Returns versus Decade Average
4-4
FMV per Company on NYSE 1945“1995 129
Average Returns Each Decade 130
4-5
The Natural Logarithm 137
4-6
Discount Rates as a Function of FMV 137
4-7
1939“1998 Decile Standard Deviations as a Function of Ln(FMV) 138
4-8
Traditional Levels of Value Chart 197
7-1
Two-Tiered Levels of Value Chart 198
7-2
3 2 Levels of Value Chart 230
7-3
P/E Ratio as a Function of Size (From the IBA Database) 363
10-1
Decision Tree for Venture Capital Funding 421
12-1
Decision Tree for Bootstrapping Assuming Debt Restructure and
12-2
No Venture Capital 425
Sales Forecast (Decay Rate 0.5) 430
12-3
Sales Forecast (Decay Rate 0.3) 431
12-3A
Post-Transaction Value of the ESOP Vs. % Sold 442
13-1




xix




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This page intentionally left blank.
List of Tables




Abbreviated Balance Sheets 7
1-1
Income Statement 8
1-2
Analysis of Property, Plant, and Equipment 9
1-3
Statement of Stockholders™ Equity 12
1-4
Abbreviated Statement of Cash Flows 13
1-5
Balance Sheets 15
1-6
Statement of Cash Flows 16
1-7
Adjustments to Historical Costs and Expenses 24
2-1A
Regression Analysis 1988“1997 27
2-1B
Calculation of 95% Con¬dence Intervals for Forecast 1998 Costs 28
2-1B
OLS Regression: Example of Deviation from Mean 31
2-2
Abbreviated Table of T-Statistics 36
2-3
Regression Analysis 1993“1997 40
2-4
Regression Analysis of Sales as a Function of GDP [1] 43
2-5
Regression Analysis of Guideline Companies 47
2-6
Regression Analysis 1988“1997 54
A2-1
ADF: End-of-Year Formula 66
3-1
ADF: Midyear Formula 69
3-2
ADF with Cash Flows Starting in Year 3.25: End-of-Year Formula 72
3-3
ADF with Cash Flows Starting in Year 2.00: End-of-Year
3-4
Formula 74
ADF with Cash Flows Starting in Year 2.00 with Negative
3-5
Growth: End-of-Year Formula 75
ADF with Cash Flows Starting in Year 2.00 with g r:
3-6
End-of-Year Formula 76
Periodic Perpetuity Factor (PPF): End-of-Year Formula 80
3-7
Periodic Perpetuity Factor (PPF): Midyear Formula 80
3-8
Periodic Perpetuity Factor (PPF): End-of-Year”Cash Flows Begin
3-9
Year 6 83
PV of Loan with Market Rate Nominal Rate: ADF, End-of-Year 86
3-10



xxi




Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
ADF Equation Numbers 90
3-11
ADF with Fractional Year: Midyear Formula 95
A3-1
ADF with Fractional Year: Midyear Formula 96
A3-2
Amortization of Principal with Irregular Starting Point 103
A3-3
PV of Principal Amortization 107
A3-4
Present Value of a Loan at Discount Rate Different than Nominal
A3-5
Rate 110
NYSE Data by Decile and Statistical Analysis: 1926“1998 121
4-1
Regressions of Returns over Standard Deviation and Log of Fair
4-2
Market Value 132
Regression Comparison [1] 133
4-2A
Table of Stock Market Returns Based on FMV”60-Year Model 136
4-3
Discounted Cash Flow Analysis Using 60-Year Model”First
4-4A
Iteration 141
Discounted Cash Flow Analysis Using 60-Year Model”Second
4-4B
Iteration 142
Discounted Cash Flow Analysis Using 60-Year Model”Final
4-4C
Valuation 143
Gordon Model Valuation Using Newton™s Iterative Process 157
A4-5
Geometric versus Arithmetic Returns 171
5-1
Geometric versus Arithmetic Returns: NYSE Data by Decile &
5-2
Statistical Analysis: 1926“1997 172
Geometric Mean versus AFMV: 60 Years 174
5-3
Comparison of Discount Rates Derived from the Log Size Model
5-4
Using 60-Year Arithmetic and Geometric Means 176
Equity Valuation Approach with Iterations Beginning with Book
6-1A
Equity: Iteration #1 182
Equity Valuation Approach with Iterations Beginning with Book
6-1B
Equity 184
Equity Valuation Approach with Iterations Beginning with
6-1C
Arbitrary Equity 185
WACC Approach with Iterations Beginning with Book Equity 187
6-2A
WACC Approach with Iterations Beginning with Arbitrary
6-2B
Guess of Equity Value 189
Synergies as Measured by Acquisition Minus Going-Private
7-1
Premiums 199
Acquisition and Going-Private Transactions Premiums 200
7-1A
Acquisition Premiums by SIC Code 206
7-2
Analysis of Megginson Results 215
7-3
Analysis of American VRP Results 218
7-3A
Mergerstat Mean Premiums: Control versus Minority Purchases 225
7-4
Abrams Regression of Management Planning Study Data 237
7-5
Calculation of Continuously Compounded Standard Deviation
7-6
Chantal Pharmaceutical, Inc.”CHTL 242
Black“Scholes Put Option”CHTL 244
7-7
Put Model Results 245
7-8



List of Tables
xxii
Calculation of Restricted Stock Discounts for 13 Stocks Using
7-9
Regression from Table 7-5 247
Calculation of Component #1”Delay To Sale [1] 254
7-10
Estimates of Transaction Costs [1] 259
7-11
Proof of Equation (7-9) 267
7-12
Proof of Equation (7-9a) 270
7-13
Sample Calculation of DLOM 272
7-14
7-15a QMDM Baseline Data”30% MPI Discount 275
a
Implied Returns for Holding Period”30% Discount 275
7-16
7-17a Implied Returns for Holding Period”20% Discount 276
Summary of Results of Applying the QMDM in 10 Example
7-18
Appraisals 277
QMDM Comparison of Restricted Stock Discount Rate versus
7-19
Mercer Example 1 279
Abrams Valuation Group Regression of Management Planning,
8-1
Inc. Data 300
2
Calculation of Revenue and Earnings Stability (R ) 304
8-1A
Calculation of Price Stability 305
8-1B
Black“Scholes Call and Put Options 307
8-2
Standard Deviation of Continuously Compounded Returns 309
8-2A
Final Calculation of Discount 310
8-3
Member Interests at Inception on 1/6/96 322
9-1
Balance Sheets 12/25/99 [1] 324
9-2

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