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Quantitative
Business Valuation
A Mathematical Approach
for Today™s Professional




JAY B. ABRAMS, ASA, CPA, MBA




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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
dif¬cult circumstances.
To my great teachers, Mr. Oshima and Christopher Hunt, who
brought me to my power to make this happen. And ¬nally, to R. K. Hiatt,
who has caught my mistakes and made signi¬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 De¬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. Insuf¬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-Speci¬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. De¬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 ’ 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.
De¬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
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
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 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.

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 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

195
7. Adjusting for Levels of Control and Marketability
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.

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. Appraiser™s Quali¬cations.

315
9. Sample Appraisal Report
Introduction. Purpose of the Report. Valuation of Considerations.
Sources of Data. History and Description of the LLC. Signi¬cant
Terms and Legal Issues. Conclusion. Economic Outlook. Economic
Growth. In¬‚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 Pro¬les 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
Quali¬cations. 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

357
10. Empirical Testing of Abrams™ Valuation Theory
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.

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% Con¬dence Intervals. Summary of Valuation Implications of
Statistical Uncertainty in the Discount Rate. Measuring the Effects of
Valuation Error. De¬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. De¬nitions of Dilution. Dilution
to the ESOP (Type 1 Dilution). Dilution to the Selling Owner (Type 2
Dilution). De¬ning Terms. Table 13-1: Calculation of Lifetime ESOP
Costs. The Direct Approach. FMV Equations”All Dilution to the
ESOP (Type 1 Dilution; No Type 2 Dilution). Table 13-2, Sections 1 and
2: Post-transaction FMV with All Dilution to the ESOP. The Post-
transaction Value Is a Parabola. FMV Equations”All Dilution to the
Owner (Type 2 Dilution). Table 13-2, Section 3: FMV Calculations”All
Dilution to the Seller. Sharing the Dilution. Equation to Calculate Type 2
Dilution. Tables 13-3 and 13-3A: Adjusting Dilution to Desired Levels.
Table 13-3B: Summary of Dilution Tradeoffs. The Iterative Approach.
Iteration #1. Iteration #2. Iteration #3. Iteration #n. Summary.
Advantages of Results. Function of ESOP Loan. Common Sense Is
Required. To Whom Should the Dilution Belong? 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, ¬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 ¬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 Pro¬les
database related to secondary limited partnership market trades.
In both cases we found very signi¬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, ˜˜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 ¬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




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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
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 Pro¬les Database”1999 [1] 339
9-6
Correlation Matrix 341
9-6A
Partnership Pro¬les Database: Price-to-Value Discounts”1999 342
9-6B
Private Fractional Interest Sales 346
9-7
Final Calculation of Fractional Interest Discount 347
9-8
Log Size Equation for 1938“1986 NYSE Data by Decile and
10-1
Statistical Analysis: 1938“1986 360
Reconciliation to IBA Database 362
10-2
Proof of Discount Calculation 364
10-3
Calculation of Component #1”Delay to Sale”$25,000 Firm 367
10-4
Calculation of Transaction Costs for Firms of All Sizes in the
10-5
IBA Study 369
Calculation of DLOM”$25,000 Firm 370
10-6
Calculation of Component #1”Delay to Sale”$75,000 Firm 371
10-4A
Calculation of Component #1”Delay to Sale”$125,000 Firm 371
10-4B



List of Tables xxiii
Calculation of Component #1”Delay to Sale”$175,000 Firm 372
10-4C
Calculation of Component #1”Delay to Sale”$225,000 Firm 372
10-4D
Calculation of Component #1”Delay to Sale”$375,000 Firm 373
10-4E
Calculation of Component #1”Delay to Sale”$750,000 Firm 373
10-4F
Calculation of Component #1”Delay to Sale”$10 Million
10-4G
Firm 374
Calculation of DLOM”$75,000 Firm 374
10-6A
Calculation of DLOM”$125,000 Firm 375
10-6B
Calculation of DLOM”$175,000 Firm 375
10-6C
Calculation of DLOM”$225,000 Firm 376
10-6D
Calculation of DLOM”$375,000 Firm 376
10-6E
Calculation of DLOM”$750,000 Firm 377
10-6F
Calculation of DLOM”$10,000,000 Firm 377
10-6G
95% Con¬dence Intervals 389
11-1
95% Con¬dence Intervals”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 Levels”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 ¬‚ows, the initial step in
the valuation process. In order to forecast cash ¬‚ows, it is important to:
— Precisely de¬ne the components of cash ¬‚ow.
— Develop statistical tools to aid in forecasting cash ¬‚ows.
— Analyze different types of annuities, which are structured series
of cash ¬‚ows.
In Chapter 1, we mathematically derive the cash ¬‚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 ¬‚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 ¬‚ow analysis using shortcut or
even incorrect de¬nitions of cash ¬‚ow.
In Chapter 2, we demonstrate in detail:
— How appraisers can use regression analysis to forecast sales and
expenses, the latter by far being the more important use of
regression.
— When and why the common practice of not using more than ¬ve
years of historical data to prevent using stale data may be
wrong.
— How to use regression analysis in valuation using publicly
traded guideline companies information. While this is not related
to forecasting sales and expenses, it ¬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 bene¬t we derive from the variation in size of the publicly



1




Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
traded guideline companies, as we discuss in the chapter. The ¬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 signi¬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:
— 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.
— Valuing periodic expenses such as moving expenses, losses from
lawsuits, etc.
— 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.
— Calculating loan payments.
— Calculating loan principal amortization.
— Calculating the present value of a loan. This is important in
calculating the cash equivalency selling price of a business, as
seller ¬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 of¬ce, I unof¬cially titled Chapter 3,
˜˜The Chapter That Would Not Die!!!™™ 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, ˜˜Statement of Cash
Flows.™™ This standard stipulates that a statement of cash ¬‚ows is required
as part of a full set of ¬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 ¬‚ows is to provide rel-
evant information about the cash receipts and cash payments of an en-
terprise. These receipts and payments must be classi¬ed according to
three basic types of activities: operating, investing, and ¬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 ¬‚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 ¬‚ows from operations
would re¬‚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 ¬‚ows from operations would re¬‚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 classi¬ed as investing or ¬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 ¬‚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 ¬‚ow from operations.
Under the indirect method, net cash ¬‚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 ¬‚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 ¬‚ow from operations.
Usually it is easy to follow the logic of the adjustment required to
infer the cash ¬‚ow associated with any single reported revenue or expense.
However, most statements of cash ¬‚ows require a number of such ad-
justments, which often result in confusing entanglements.
Many business and real estate appraisers spend a signi¬cant part of
their careers forecasting cash ¬‚ows. The objective of this chapter is to
improve their understanding of the cash ¬‚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 ¬‚ow logic and distinguish true cash ¬‚ows from shortcut approxi-
mations thereof.
To achieve this result, this chapter provides a mathematical deriva-
tion of the cash ¬‚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 speci¬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 ¬‚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
bene¬ts.




CHAPTER 1 Cash Flow: A Mathematical Derivation 5
L total liabilities
CAP total stockholder™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
Stockholders™ 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

STOCKHOLDERS™ 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 stockholders™ 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 stockholder™s equity, the analyst would have
to know the company™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 pro¬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




stockholder™s equity. The total change in stockholder™s equity ( CAP) is
equal to net income (NI) and other equity transactions (OET) (de¬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
company™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 ¬‚ows”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 reader™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 ˜˜negative™™ 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) simpli¬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 ¬rst line of equation (1-15) represents cash ¬‚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 ¬‚ows from in-
vesting activities, and the third line represents a preliminary version of
cash ¬‚ows from ¬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 stockholder™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 company™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 ¬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 ¬‚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



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