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

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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.

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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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

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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