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Quantitative

Business Valuation

A Mathematical Approach

for Todayā™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 wasnā™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

difļ¬cult circumstances.

To my great teachers, Mr. Oshima and Christopher Hunt, who

brought me to my power to make this happen. And ļ¬nally, to R. K. Hiatt,

who has caught my mistakes and made signiļ¬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 Deļ¬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. Insufļ¬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-Speciļ¬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. Deļ¬nitions. Denoting Time. ADF with End-of-Year

Cash Flows. Behavior of the ADF with Growth. Special Case of ADF

vii

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0: The Ordinary Annuity. Special Case when n ā’ 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.

Deļ¬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

Famaā“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 Newtonā™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 1927ā“1997. Table 5-3: Regressions

of Geometric Returns for 1938ā“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

Contents

viii

6-1C: Initial Choice of Equity Doesnā™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 Doesnā™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 Researchā”Qualitative Professional. Prior Researchā”

Academic. My Synthesis and Analysis. Discount for Lack of

Marketability (DLOM). Mercerā™s Quantitative Marketability Discount

Model. Kasperā™s BAS Model. Restricted Stock Discounts. Abramsā™

Economic Components Model. Mercerā™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. Appraiserā™s Qualiļ¬cations.

315

9. Sample Appraisal Report

Introduction. Purpose of the Report. Valuation of Considerations.

Sources of Data. History and Description of the LLC. Signiļ¬cant

Terms and Legal Issues. Conclusion. Economic Outlook. Economic

Growth. Inļ¬‚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 Proļ¬les Approachā”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. Appraiserā™s

Qualiļ¬cations. Appendix: Tax Courtā™s Opinion for Discount for

Lack of Marketability. Introduction. The Courtā™s 10 Factors.

Application of the Courtā™s 10 Factors to the Valuation.

PART IV

PUTTING IT ALL TOGETHER

357

10. Empirical Testing of Abramsā™ Valuation Theory

Introduction. Steps in the Valuation Process. Applying a Valuation

Model to the Steps. Table 10-1: Log Size for 1938ā“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 Componentā“$25,000 Firm.

Table 10-5: Calculation of Transactions Costs. Table 10-6: Calculation of

DLOM. Table 10-6Aā“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% Conļ¬dence Intervals. Summary of Valuation Implications of

Statistical Uncertainty in the Discount Rate. Measuring the Effects of

Valuation Error. Deļ¬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. Deļ¬nitions of Dilution. Dilution

to the ESOP (Type 1 Dilution). Dilution to the Selling Owner (Type 2

Dilution). Deļ¬ning Terms. Table 13-1: Calculation of Lifetime ESOP

Costs. The Direct Approach. FMV Equationsā”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 Equationsā”All Dilution to the

Owner (Type 2 Dilution). Table 13-2, Section 3: FMV Calculationsā”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

x

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, ļ¬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 ļ¬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 Proļ¬les

database related to secondary limited partnership market trades.

In both cases we found very signiļ¬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, ā˜ā˜Discount Rates as a Function of

Log Size,ā™ā™ and Chapter 11, ā˜ā˜Empirical Testing of Abramsā™

Valuation Theory.ā™ā™

2. It emphasizes quantitative skills. Chapter 2 focuses on using

regression analysis in business valuation. Chapter 3, ā˜ā˜Annuity

Discount Factors and the Gordon Model,ā™ā™ is the most

comprehensive treatment of ADFs in print. For anyone wishing

to use the Mercer quantitative marketability discount model,

xiii

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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 ļ¬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, uniļ¬ed approach to valuation that can

be empirically tested and whose principles work for the

valuation of billion-dollar ļ¬rms and ma and pa ļ¬rms alike.

While this book contains more mathematicsā”a wormā™s eye

view, if you willā”than other valuation texts, it also has more of

a birdā™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 ļ¬rst three parts to this book

follow the chronological sequence of performing a valuation: (1) forecast

cash ļ¬‚ows, (2) discount to present value, and (3) adjust for control and

marketability. The fourth part is a birdā™s eye view in order to test empir-

ically whether my methodology works. Additionally, we explore (1) con-

ļ¬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 beneļ¬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 7ā”a sample

restricted stock report)

3. Chapter 9 (this is an application of Chapter 7ā”a sample

fractional ownership interest discount report)

Introduction

xiv

4. Chapter 4 (the log size model for calculating discount rates)

5. Chapter 3ā”the following sections: from the beginning through

the section titled ā˜ā˜A Brief Summaryā™ā™; ā˜ā˜Periodic Perpetuity

Factors: Perpetuities for Periodic Cash Flowsā™ā™; and

ā˜ā˜Relationship of Gordon Model Multiple to the Price/Earnings

Ratio.ā™ā™ Some readers may want to read this chapter after

Chapter 7, though it will be somewhat helpful, but deļ¬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 ļ¬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

ļ¬‚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 elseā™s book.

Larry Kasper gave me a surprise detailed edit of the ļ¬rst eight chapters.

I beneļ¬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

1926ā“1998 Arithmetic Mean Returns as a Function of Standard

4-1

Deviation 125

1926ā“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 1935ā“1995 128

Decade Standard Deviation of Returns versus Decade Average

4-4

FMV per Company on NYSE 1945ā“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

1939ā“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

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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 Stockholdersā™ 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 1988ā“1997 27

2-1B

Calculation of 95% Conļ¬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 1993ā“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 1988ā“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-Yearā”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: 1926ā“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 FMVā”60-Year Model 136

4-3

Discounted Cash Flow Analysis Using 60-Year Modelā”First

4-4A

Iteration 141

Discounted Cash Flow Analysis Using 60-Year Modelā”Second

4-4B

Iteration 142

Discounted Cash Flow Analysis Using 60-Year Modelā”Final

4-4C

Valuation 143

Gordon Model Valuation Using Newtonā™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: 1926ā“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.ā”CHTL 242

Blackā“Scholes Put Optionā”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 #1ā”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 Dataā”30% MPI Discount 275

a

Implied Returns for Holding Periodā”30% Discount 275

7-16

7-17a Implied Returns for Holding Periodā”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

Blackā“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

Income Statements [1] 325

9-3

Cash Distributions 326

9-4

Economic Components Approach: 2.80% Member Interest 328

9-5

Calculation of Component #1: Delay to Sale [1] 330

9-5A

Earnings and Revenue Stability 332

9-5B

Calculation of DLOM: 2.80% Member Interest 334

9-5C

Regression Analysis of Partnership Proļ¬les Databaseā”1999 [1] 339

9-6

Correlation Matrix 341

9-6A

Partnership Proļ¬les Database: Price-to-Value Discountsā”1999 342

9-6B

Private Fractional Interest Sales 346

9-7

Final Calculation of Fractional Interest Discount 347

9-8

Log Size Equation for 1938ā“1986 NYSE Data by Decile and

10-1

Statistical Analysis: 1938ā“1986 360

Reconciliation to IBA Database 362

10-2

Proof of Discount Calculation 364

10-3

Calculation of Component #1ā”Delay to Saleā”$25,000 Firm 367

10-4

Calculation of Transaction Costs for Firms of All Sizes in the

10-5

IBA Study 369

Calculation of DLOMā”$25,000 Firm 370

10-6

Calculation of Component #1ā”Delay to Saleā”$75,000 Firm 371

10-4A

Calculation of Component #1ā”Delay to Saleā”$125,000 Firm 371

10-4B

List of Tables xxiii

Calculation of Component #1ā”Delay to Saleā”$175,000 Firm 372

10-4C

Calculation of Component #1ā”Delay to Saleā”$225,000 Firm 372

10-4D

Calculation of Component #1ā”Delay to Saleā”$375,000 Firm 373

10-4E

Calculation of Component #1ā”Delay to Saleā”$750,000 Firm 373

10-4F

Calculation of Component #1ā”Delay to Saleā”$10 Million

10-4G

Firm 374

Calculation of DLOMā”$75,000 Firm 374

10-6A

Calculation of DLOMā”$125,000 Firm 375

10-6B

Calculation of DLOMā”$175,000 Firm 375

10-6C

Calculation of DLOMā”$225,000 Firm 376

10-6D

Calculation of DLOMā”$375,000 Firm 376

10-6E

Calculation of DLOMā”$750,000 Firm 377

10-6F

Calculation of DLOMā”$10,000,000 Firm 377

10-6G

95% Conļ¬dence Intervals 389

11-1

95% Conļ¬dence Intervalsā”60-Year Log Size Model 391

11-2

Absolute Errors in Forecasting Growth Rates 398

11-3

Percent Valuation Error for 10% Relative Error in Growth 400

11-4

Percent Valuation Error for 10% Relative Error in Growth 401

11-4A

Percent Valuation Error for 10% Relative Error in Discount Rate 402

11-4B

Summary of Effects of Valuation Errors 403

11-5

First Chicago Method 412

12-1

VC Pricing Approach 414

12-2

Statistical Calculation of Fair Market Value 418

12-3

Sales Model with Exponentially Declining Growth Rate

12-4

Assumption 430

Calculation of Lifetime ESOP Costs 438

13-1

FMV Calculations: Firm, ESOP, and Dilution 441

13-2

Adjusting Dilution to Desired Levels 446

13-3

Adjusting Dilution to Desired Levelsā”All Dilution to Owner 447

13-3A

Summary of Dilution Tradeoffs 447

13-3B

List of Tables

xxiv

PART ONE

Forecasting Cash Flows

Part 1 of this book focuses on forecasting cash ļ¬‚ows, the initial step in

the valuation process. In order to forecast cash ļ¬‚ows, it is important to:

ā— Precisely deļ¬ne the components of cash ļ¬‚ow.

ā— Develop statistical tools to aid in forecasting cash ļ¬‚ows.

ā— Analyze different types of annuities, which are structured series

of cash ļ¬‚ows.

In Chapter 1, we mathematically derive the cash ļ¬‚ow statement as

the result of creating and manipulating a series of accounting equations

and identities. This should give the appraiser a much greater depth of

understanding of how cash ļ¬‚ows derive from and relate to the balance

sheet and income statement. It may help eliminate errors made by ap-

praisers who perform discounted cash ļ¬‚ow analysis using shortcut or

even incorrect deļ¬nitions of cash ļ¬‚ow.

In Chapter 2, we demonstrate in detail:

ā— How appraisers can use regression analysis to forecast sales and

expenses, the latter by far being the more important use of

regression.

ā— When and why the common practice of not using more than ļ¬ve

years of historical data to prevent using stale data may be

wrong.

ā— How to use regression analysis in valuation using publicly

traded guideline companies information. While this is not related

to forecasting sales and expenses, it ļ¬ts in with our other

discussions about using regression analysis.

When using publicly traded guideline companies of widely varying

sizes, ordinary least squares (OLS) regression will usually fail, as statis-

tical error is generally proportional to the market value (size) of the

guideline company. However, there are simple transformations the ap-

praiser can make to the data that will (1) enable him or her to minimize

the negative impact of differences in size and (2) still preserve the very

important beneļ¬t we derive from the variation in size of the publicly

1

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traded guideline companies, as we discuss in the chapter. The ļ¬nal result

is valuations that are more reliable, realistic, and objective.

Most electronic spreadsheets provide a least squares regression that

is adequate for most appraisal needs. I am familiar with the regression

tools in both Microsoft Excel and Lotus 123. Excel does a better job of

presentation and offers much more comprehensive statistical feedback.

Lotus has one signiļ¬cant advantage: it can provide multiple regression

analysis for a virtually unlimited number of variables, while Excel is lim-

ited to 16 independent variables.

In Chapter 3, we discuss annuity discount factors (ADFs). Histori-

cally, ADFs have not been used much in business valuation and thus,

have had relatively little importance. Their importance is growing, how-

ever, for several reasons. They can be used in:

ā— Calculating the present value of annuities, including those with

constant growth. This application has become far more important

since the Mercer Quantitative Marketability Discount Model

requires an ADF with growth.

ā— Valuing periodic expenses such as moving expenses, losses from

lawsuits, etc.

ā— Calculating the present value of periodic capital expenditures

with growth, e.g., what is the PV of keeping one airplane of a

certain class in service perpetually.

ā— Calculating loan payments.

ā— Calculating loan principal amortization.

ā— Calculating the present value of a loan. This is important in

calculating the cash equivalency selling price of a business, as

seller ļ¬nancing typically takes place at less-than-market rates.

The present value of a loan is also important in ESOP valuations.

Among my colleagues in the ofļ¬ce, I unofļ¬cially titled Chapter 3,

ā˜ā˜The Chapter That Would Not Die!!!ā™ā™ I edited and rewrote this chapter

close to 40 times striving for perfection, the elusive and unattainable goal.

It was quite a task to decide what belongs in the body of the chapter and

what should be relegated to the appendix. In an effort to maximize read-

ability, the most practical formulas appear in the body of chapter 3 and

the least useful and most mathematical work appears in the appendix.

PART 1 Forecasting Cash Flows

2

CHAPTER 1

Cash Flow: A Mathematical

Derivation1

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

1. This chapter was coauthored with Donald Shannon, School of Accountancy, DePaul University.

The mathematical model was published in Abrams (1997).

3

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

INTRODUCTION

In 1987, the Financial Accounting Standards Board (FASB) issued State-

ment of Financial Accounting Standards No. 95, ā˜ā˜Statement of Cash

Flows.ā™ā™ This standard stipulates that a statement of cash ļ¬‚ows is required

as part of a full set of ļ¬nancial statements for almost all business enter-

prises.

This chapter, which discusses the Statement of Cash Flows, is in-

tended for readers who already have a basic knowledge of accounting.

Much of what follows will involve alternating between accrual and cash

reporting, which can be very challenging material. Also, a parsimonious

style has been used to keep the chapter to a reasonable length. Accord-

ingly, certain sections and derivations may require more than one reading.

The primary purpose of a statement of cash ļ¬‚ows is to provide rel-

evant information about the cash receipts and cash payments of an en-

terprise. These receipts and payments must be classiļ¬ed according to

three basic types of activities: operating, investing, and ļ¬nancing.

Operating activities involve those transactions that enter into the de-

termination of net income. Examples of these activities are sales of goods

or services, purchases of component materials, and compensation of em-

ployees. Net income reports these activities when they are earned or in-

curred. Cash ļ¬‚ows from operations reports these activities only when they

are collected or paid. For example, net income is increased when a sale

is made even though no cash is collected. Cash ļ¬‚ows from operations

would reļ¬‚ect the increase only at the time the cash is collected. Also, net

income is decreased when, say, insurance expense is incurred even

though no payment is made. Cash ļ¬‚ows from operations would reļ¬‚ect

the decrease only at the time the payment is made.

Of course, companies engage in numerous transactions involving

cash but having no impact on the income statement. These transactions

are classiļ¬ed as investing or ļ¬nancing activities. Investing activities in-

clude the acquisition of long-lived assets as well as their disposition when

no gains or losses are involved.2 Financing activities include obtaining

and repaying funds from debt and equity holders and providing the own-

ers with a return on their investment.

Either the direct or the indirect method may be used as a basis for

reporting cash ļ¬‚ows from operating activities. Under the direct method

the enterprise lists its major categories of cash receipts from operations

(such as receipts from product sales and receipts from consulting services)

and cash disbursements for operations (such as payments for inventory,

wages, interest, and taxes). The difference between these receipts and dis-

bursements is the net cash ļ¬‚ow from operations.

Under the indirect method, net cash ļ¬‚ow from operations is found

by adjusting net income for changes in related asset and liability accounts.

For example, an increase in accounts receivable indicates that cash receipts

from sales are less than reported revenues. Receivables increase as a result

2. This introductory comment presumes the long-lived assets are sold for their net book values. Of

course, when gains or losses on disposition are involved they do appear in the income

statement. The treatment of these gains and losses is addressed later in the chapter.

PART 1 Forecasting Cash Flows

4

of failing to collect all revenues reported. Therefore, the amount of the

increase in accounts receivable would have to be subtracted from net

income to arrive at net cash ļ¬‚ow from operations. Likewise, a decrease

in wages payable would indicate that cash payments for wages were

greater than the expenses shown in the income statement. Payables de-

crease when payments exceed the amount of expenses reported. There-

fore, the amount of the decrease in wages payable also would have to be

subtracted from net income to arrive at net cash ļ¬‚ow from operations.

Usually it is easy to follow the logic of the adjustment required to

infer the cash ļ¬‚ow associated with any single reported revenue or expense.

However, most statements of cash ļ¬‚ows require a number of such ad-

justments, which often result in confusing entanglements.

Many business and real estate appraisers spend a signiļ¬cant part of

their careers forecasting cash ļ¬‚ows. The objective of this chapter is to

improve their understanding of the cash ļ¬‚ow statement and its interre-

lationship with the balance sheet and the income statement. Appraisers

who read this chapter will, we hope, be able to understand better the

cash ļ¬‚ow logic and distinguish true cash ļ¬‚ows from shortcut approxi-

mations thereof.

To achieve this result, this chapter provides a mathematical deriva-

tion of the cash ļ¬‚ow statement using the indirect method. A realistic

numerical example and an intuitive explanation accompany the mathe-

matical derivation.3

THE MATHEMATICAL MODEL

In what follows, be careful to distinguish between equations and tables,

as they both have the same numbering system to describe them. Equa-

tions always have some algebraic expression at the top, even if there are

numbers below that serve as speciļ¬c examples of the equations.

A Preliminary Explanation of Cash Flows

The following is a list of the symbols that will be used in this chapter.

Balance Sheet

C cash

OCA other current assets

GPPE gross property, plant, and equipment

AD accumulated depreciation

NPPE net property, plant, and equipment

A total assets

CL current liabilities

LTD long-term debt

3. Surely it would be possible to examine in detail every conceivable type of accounting

transaction and its relation to cash ļ¬‚ow. Here, certain transactions such as recapitalizations,

the effects of accounting changes, and inventory write-downs have not been considered. The

authors feel the additional complication of their inclusion would more than offset any

beneļ¬ts.

CHAPTER 1 Cash Flow: A Mathematical Derivation 5

L total liabilities

CAP total stockholderā™s equity

Property, plant, and Equipment

CAPEX capital expenditures

DEPR depreciation expense

RETGBV gross book value of retired property, plant and equip-

ment

RETAD accumulated depreciation on retired assets

SALESFA selling price of property, plant and equipment disposed

of or retired

Stockholdersā™ Equity

NI net income

DIV dividends paid

SALSTK sale of stock

TRSTK purchase of stock

OET other equity transactions

AET additional equity transactions

Required Working Capital

RWC required working capital

CReq required cash

The balance sheets for Feathers R Us for 1999 and 2000 are presented

in Table 1-1. The changes in the balance sheet accounts from one year to

the next are shown in the right column. On the far left the symbols used

later to refer to these accounts in mathematical expressions have been

repeated.

The balance sheet for the current year (t 2000) is in balance. The

total assets equal $3,150,000, total liabilities equal $1,085,000, and the total

liabilities and equity also equal $3,150,000. This can be shown as:

At Lt CAPt

(1-1)

3,150,000 1,085,000 2,065,000

Likewise, the balance sheet for the preceding year (t 1 1999) is in

balance.

At Lt CAPt

1 1 1 (1-2)

2,800,000 1,075,000 1,725,000

Subtracting the beginning balance sheet from the ending balance sheet

shows that the changes from one year to the next are also in balance.

A L CAP (1-3)

350,000 10,000 340,000

Greater detail can be shown for each of the terms in equation (1-3).

The change in total assets ( A) consists of the change in cash ( C), the

change in other current assets ( OCA), and the change in net property,

plant, and equipment. Net property, plant, and equipment (NPPE) is

gross property, plant, and equipment (GPPE) less the accumulated de-

preciation (AD) on these assets. As shown in Table 1-3 below, the change

PART 1 Forecasting Cash Flows

6

T A B L E 1-1

Feathers R Us

ABBREVIATED BALANCE SHEETS

For Calendar Years

Increase

Symbols ASSETS: 1999 2000 (Decrease)

C Cash 1,125,000 1,500,000 375,000

OCA Other current assets 875,000 790,000 (85,000)

Total current assets 2,000,000 2,290,000 290,000

GPPE Gross property, plant, & equipment 830,000 900,000 70,000

AD Accumulated depreciation 30,000 40,000 10,000

NPPE Net property, plant, & equipment 800,000 860,000 60,000

A Total assets 2,800,000 3,150,000 350,000

LIABILITIES

Current liabilities 325,000 360,000 35,000

LTD Long-term debt 750,000 725,000 (25,000)

L Total liabilities 1,075,000 1,085,000 10,000

STOCKHOLDERSā™ EQUITY

Capital stock 100,000 150,000 50,000

Additional paid in capital 200,000 500,000 300,000

Retained earnings 1,425,000 1,465,000 40,000

Treasury stock 0 50,000 50,000

CAP Total stockholdersā™ equity 1,725,000 2,065,000 340,000

Total liabilities & equity 2,800,000 3,150,000 350,000

in net property, plant, and equipment ( NPPE) can be found by subtract-

ing the change in accumulated depreciation from the change in gross

AD).4

property, plant, and equipment ( GPPE

A C OCA ( GPPE AD) (1-4)

350,000 375,000 (85,000) (70,000 10,000)

The change in total liabilities ( L) consists of the change in current lia-

bilities ( CL) and the change in long-term debt ( LTD).

L CL LTD (1-5)

10,000 35,000 (25,000)

To explain the change in stockholderā™s equity, the analyst would have

to know the companyā™s net income, provided in Table 1-2.

This income statement shows that Feathers R Us had net income after

tax (NI) of $90,000. This explains only a portion of the change in the

4. Other long-lived assets, such as intangibles and certain investments, are treated the same as

property, plant, and equipment.

CHAPTER 1 Cash Flow: A Mathematical Derivation 7

T A B L E 1-2

Feathers R Us

INCOME STATEMENT

For Calendar Year

2000

Sales 1,000,000

Cost of sales 600,000

Gross proļ¬t 400,000

Sales expense 100,000

General & administrative expense 150,000

Depreciation 30,000

Total expense 280,000

Operating income 120,000

Gain on sale of assets 30,000

Net income before taxes 150,000

60,000

Net Income 90,000

stockholderā™s equity. The total change in stockholderā™s equity ( CAP) is

equal to net income (NI) and other equity transactions (OET) (deļ¬nition

is given below equation [1-6]).

CAP NI OET (1-6)

340,000 90,000 250,000

The other equity transactions consist of the purchase and sale of the

companyā™s stock and the payment of cash dividends.5 A detailed descrip-

tion of these transactions will be provided later in the chapter (refer to

Table 1-4).

Substituting equations (1-4), (1-5), and (1-6) into equation (1-3) results

6

in:

C OCA ( GPPE AD)

CL LTD NI OET (1-7)

375,000 (85,000) (70,000 10,000)

35,000 (25,000) 90,000 250,000

Equation (1-7) can be rearranged to satisfy the objective of the statement

of cash ļ¬‚owsā”providing an explanation of the change in the cash bal-

ance.

5. Here it is assumed that all dividends declared are paid.

6. For the readerā™s convenience certain equations are repeated in the footnotes.

Equation (1-3): A L + CAP

Equation (1-4): A C + OCA + ( GPPE AD)

Equation (1-5): L CL + LTD

Equation (1-6): CAP NI + OET

PART 1 Forecasting Cash Flows

8

C NI OCA CL

( GPPE AD)

LTD OET (1-8)

375,000 90,000 (85,000) 35,000

(70,000 10,000)

(25,000) 250,000

Equation (1-8) does provide an explanation of the $375,000 increase

in the cash balance from 1999 to 2000, but it is still somewhat preliminary.

Discussion of this explanation is best deferred until more details have

been incorporated into the model.

Analyzing Property, Plant, and Equipment Transactions

The balance sheets in Table 1-1 show that the net property, plant, and

equipment increased by $60,000. The analyst will want to obtain a more

detailed understanding of this change. This can be accomplished by re-

viewing an analysis of property, plant, and equipment such as the one

shown in Table 1-3.

This analysis shows that gross property, plant, and equipment are

increased by capital expenditures (CAPEXP) and decreased by original

book value of any assets retired (RETGBV). This relationship is restated

as equation (1-9).

GPPE CAPEXP RETGBV (1-9)

70,000 175,000 105,000

Likewise, accumulated depreciation is increased by depreciation ex-

pense and decreased by the accumulated depreciation on any assets re-

tired. This relationship is restated as equation (1-10).

T A B L E 1-3

Feathers R Us

ANALYSIS OF PROPERTY, PLANT,

& EQUIPMENT

For Calendar Year 2000

GPPE AD NPPE

Gross Prop, Accumulated Net Prop,

Symbols Plant & Equip Depreciation Plant & Equp

Balance, 1999 830,000 30,000 800,000

CAPEXP Capital expenditures 175,000 175,000

DEPR Depreciation expense 30,000 30,000

Retirements

RETGBV Gross book value 105,000 105,000

RETAD Accumulated depreciation 20,000 20,000

Balance, 2000 900,000 40,000 860,000

Change in the balance 70,000 10,000 60,000

CHAPTER 1 Cash Flow: A Mathematical Derivation 9

AD DEPR RETAD (1-10)

10,000 30,000 20,000

Substituting equations (1-9) and (1-10) into equation (1-8) and rearranging

the terms results in equation (1-11):7

C NI OCA CL

DEPR

CAPEXP RETGBV RETAD

LTD OET (1-11)

375,000 90,000 (85,000) 35,000

30,000

175,000 105,000 20,000

(25,000) 250,000

The bold symbols in equation (1-11) are the symbols that have been

changed by the substitutions described. For example DEPR, CAPEXP,

RETGBV, and RETAD in equation (1-11) did not appear in equation

(1-8).

To this point, only the book value of any assets retired has been

considered. Most often, the retirement or disposition of assets involves a

gain or a loss (a ā˜ā˜negativeā™ā™ gain). This gain is the difference between the

selling price of the property, plant, and equipment (SALESFA) and their

net book values (RETGBV RETAD). The assets in this illustration were

sold for $115,000, producing a gain of $30,000. This is shown in equation

(1-12).

GAIN SALESFA (RETGBV RETAD) (1-12)

30,000 115,000 (105,000 20,000)

Equation (1-13) below is simply a rearrangement of equation (1-12).

RETGBV SALESFA GAIN RETAD (1-13)

105,000 115,000 30,000 20,000

Substituting equation (1-13) into equation (1-11) results in:8

C NI DEPR OCA CL

CAPEXP RETAD

SALESFA GAIN RETAD

LTD OET

375,000 90,000 30,000 (85,000) 35,000

175,000 20,000

115,000 30,000 20,000

(25,000) 250,000

(1-14)

7. Equation (1-8): C NI OCA + CL ( GPPE AD) + LTD + OET

Equation (1-9): GPPE CAPEXP RETGBV

8. Equation (1-11): C NI + DEPR OCA + CL CAPEXP + RETGBV RETAD + LTD +

OET

PART 1 Forecasting Cash Flows

10

After canceling the RETAD and RETAD terms and rearranging,

equation (1-14) simpliļ¬es to Equation (1-15):9

C NI GAIN DEPR OCA CL

CAPEXP SALESFA

LTD OET (1-15)

375,000 90,000 30,000 30,000 (85,000) 35,000

175,000 115,000

(25,000) 250,000

The ļ¬rst line of equation (1-15) represents cash ļ¬‚ows from operating

activities, which is found by making certain adjustments to net income

such as adding back depreciation and other noncash expenses, subtract-

ing changes in other current assets, and adding changes in current lia-

bilities. These adjustments will be explained in more detail later in the

chapter. The second line in the equation represents cash ļ¬‚ows from in-

vesting activities, and the third line represents a preliminary version of

cash ļ¬‚ows from ļ¬nancing activities.

An Explanation of Cash Flows with More Detail for Equity

Transactions

Frequently the details of the other equity transactions (OET), referred to

in equation (1-6), are also important. In this example the statement of

stockholder equity included three common types of equity transactions:

(DIV) issuing cash dividends, (SALSTK) selling stock, and (TRSTK) pur-

chasing treasury stock. These are shown in Table 1-4 below.

During the year the company paid cash dividends of $50,000, sold

some additional shares of stock for $350,000, and bought back some stock

for $50,000. The net effect of these three transactions (OET) is a $250,000

increase in stockholderā™s equity. This is summarized in equation (1-16).

The term AET has been added to equation (1-16) to represent additional

equity transactions.10

OET SALSTK TRSTK DIV AET (1-16)

250,000 350,000 50,000 50,000 0

Substituting this last expression into equation (1-15) results in equation

(1-17) below.11

9. Equation (1-14): C NI DEPR OCA CL CAPEXP SALESFA GAIN

RETAD RETAD LTD OET

10. The term additional equity transactions was used to describe equity transactions other than the

sale or purchase of the companyā™s stock and the payment of dividends. One example of an

additional equity transaction would be the contribution of property to the company in

exchange for an equity interest. For analytical purposes, the increase in equity could be

treated as a source of cash from ļ¬nancing activities. The corresponding increase in assets

could be treated as a use of cash from investing activities. The net result would be overall

zero effect on cash. Normally, noncash transactions of this nature are not incorporated in

formal statements of cash ļ¬‚ows but are appended in a separate schedule.

11. Equation (1-15): C NI GAIN + DEPR OCA + CL CAPEXP + SALESFA + LTD +

OET

Equation (1-16): OET SALESTK TRSTK DIV + AET

ńņš. 1 |