Quantitative business valuation

xxii List of TablesA3-3 Amortization of Principal with Irregular Starting Point 103 A3-5 Present Value of a Loan at Discount Rate Different than Nominal 4-1 NYSE Data by Decile and Stati

Quantitative Business Valuation

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Business Valuation

A Mathematical Approach for Today’s Professional

JAY B ABRAMS, ASA, CPA, MBA

McGRAW-HILL

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DOI: 10.1036/007138595X

abc

McGraw-Hill

To my father, Leonard Abrams, who taught me how to write To mymother, 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 onthe outside to miss the Sundays with me, but who has brought us peace.

To my parents and my brother, Mark, for their tremendous support underdifficult circumstances

To my great teachers, Mr Oshima and Christopher Hunt, whobrought me to my power to make this happen And finally, to R K Hiatt,who has caught my mistakes and made significant contributions to thethought that permeates this book

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Contents

Introduction xiii

List of Figures xix

List of Tables xxi

PART I

FORECASTING CASH FLOWS

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.

Introduction Forecasting Costs and Expenses Adjustments to

Expenses Table 2-1A: Calculating Adjusted Costs and Expenses.

Performing Regression Analysis Use of Regression Statistics to Test

the Robustness of the Relationship Standard Error of the y Estimate.

The Mean of a and b The Variance of a and b Selecting the Data Set

and Regression Equation Problems with Using Regression Analysis

for Forecasting Costs Insufficient Data Substantial Changes in

Competition or Product/Service Using Regression Analysis to

Forecast Sales Spreadsheet Procedures to Perform Regression.

Examining the Regression Statistics Adding Industry-Specific

Independent Variables Try All Combinations of Potential Independent

Variables Application of Regression Analysis to the Guideline

Company Method Table 2-5: Regression Analysis of Guideline

Companies Summary Appendix: The ANOVA table.

Introduction Definitions Denoting Time ADF with End-of-Year

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

viii Contents

when g ⫽ 0: The Ordinary Annuity Special Case when n → ⬁ and r ⬎ g: The Gordon Model Intuitively Understanding Equations (3-6) and (3-6a) Relationship between the ADF and the Gordon Model Table 3-1:

Proof of ADF Equations (3-6) through (3-6b) A Brief Summary.

Midyear Cash Flows Table 3-2: Example of Equations (3-10) through (3-10b) Special Cases for Midyear Cash Flows: No Growth, g ⫽ 0.

Gordon Model Starting Periods Other than Year 1 End-of-Year Formulas Valuation Date ⫽ 0 Table 3-3: Example of Equation (3-11).

Tables 3-4 through 3-6: Variations of Table 3-3 with S ⬍ 0, Negative Growth, and r ⬍ g Special Case: No Growth, g ⫽ 0 Generalized Gordon Model Midyear Formula Periodic Perpetuity Factors (PPFs):

Perpetuities for Periodic Cash Flows The Mathematical Formulas.

Tables 3-7 and 3-8: Examples of Equations (3-18) and (3-19) Other Starting Years New versus Used Equipment Decisions ADFs in Loan Mathematics Calculating Loan Payments Present Value of a Loan.

Relationship of the Gordon Model to the Price/Earnings Ratio.

Definitions Mathematical Derivation Conclusions.

PART II CALCULATING DISCOUNT RATES

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.

AppendixA: Automating Iteration Using Newton’s Method.

AppendixB: Mathematical Appendix AppendixC: Abbreviated Review and Use.

5 Arithmetic versus Geometric Means: Empirical Evidence and

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.

Introduction Equity Valuation Method Table 6-1A: The First Iteration Table 6-1B: Subsequent Iterations of the First Scenario Table

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

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.

Introduction Background Stock Ownership Purpose of the Appraisal.

No Economic Outlook Section Sources of Data Valuation Commentary

to Table 8-1: Regression Analysis of Management Planning Data.

Commentary to Table 8-1A: Revenue and Earnings Stability.

Commentary to Table 8-1B: Price Stability Valuation Using Options

Pricing Theory Conclusion of Discount for Lack of Marketability.

Assumptions and Limiting Conditions Appraiser’s Qualifications.

Introduction Purpose of the Report Valuation of Considerations.

Sources of Data History and Description of the LLC Significant

Terms and Legal Issues Conclusion Economic Outlook Economic

Growth Inflation Interest Rates State and Local Economics Summary.

Financial Review Commentary to Table 9-2: FMV Balance Sheets.

Commentary to Table 9-3: Income Statements Commentary to Table 9-4:

Cash Distributions Valuation Valuation Approaches Selection of

Valuation Approach Economic Components Approach Commentary to

Table 9-5: Calculations of Combined Discounts Commentary to Table

9-5A: Delay-to-Sale Commentary to Table 9-5C: Calculation of DLOM.

Commentary to Table 9-6: Partnership Profiles Approach—1999.

Commentary to Table 9-7: Private Fractional Interest Sales Commentary

to Table 9-8: Final Calculation of Fractional Interest Discounts.

Conclusion Statement of Limiting Conditions Appraiser’s

Qualifications Appendix: Tax Court’s Opinion for Discount for

Lack of Marketability Introduction The Court’s 10 Factors.

Application of the Court’s 10 Factors to the Valuation.

PART IV

PUTTING IT ALL TOGETHER

Introduction Steps in the Valuation Process Applying a Valuation

Model to the Steps Table 10-1: Log Size for 1938–1986 Table 10-2:

x Contents

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.

Introduction Differences Between Uncertainty and Error Sources of Uncertainty and Error Measuring Valuation Uncertainty Table 11-1:

95% Confidence Intervals Summary of Valuation Implications of Statistical Uncertainty in the Discount Rate Measuring the Effects of Valuation Error Defining Absolute and Relative Error The Valuation Model Dollar Effects of Absolute Errors in Forecastng Year 1 Cash Flow.

Relative Effects of Absolute Errors in Forecasting Year 1 Cash Flow.

Absolute and Relative Effects of Relative Errors in Forecasting Year 1 Cash Flow Absolute Errors in Forecasting Growth and the Discount Rate Table 11-5: Summary of Effects of Valuation Errors Summary and

Conclusions.

PART V SPECIAL TOPICS

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.

Introduction What Can Be Skipped Definitions of Dilution Dilution

to the ESOP (Type 1 Dilution) Dilution to the Selling Owner (Type 2 Dilution) Defining Terms Table 13-1: Calculation of Lifetime ESOP Costs The Direct Approach FMV Equations—All Dilution to the ESOP (Type 1 Dilution; No Type 2 Dilution) Table 13-2, Sections 1 and 2: Post-transaction FMV with All Dilution to the ESOP The 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? AppendixA:

Mathematical Appendix Appendix B: Shorter Version of Chapter

13.

Introduction An Example of a Buyout The Solution Evaluating the

Benchmarks.

Glossary 475

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Introduction

NATURE OFTHE 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, financial 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:

adjusting for control and marketability, contains the first

regression analysis of the data related to restricted stock

discounts Chapter 9, a sample fractional interest discount study,

contains a regression analysis of the Partnership Profiles

database related to secondary limited partnership market trades

In both cases we found very significant results We now know

much of what drives (a) restricted stock discounts and (b)

discounts from net asset values of the publicly registered/

privately traded limited partnerships You will also see much

empirical work in Chapter 4, ‘‘Discount Rates as a Function of

Log Size,’’ and Chapter 11, ‘‘Empirical Testing of Abrams’

Valuation Theory.’’

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,

xiv Introduction

Chapter 4 contains the ADF with constant growth not included

in the Mercer text ADFs crop up in many valuation contexts Iinvented several new ADFs that appear in Chapter 3 that areuseful in many valuation contexts Chapter 11 contains the firsttreatise on how much statistical uncertainty we have in ourvaluations and how value is affected when the appraiser makesvarious errors

present a comprehensive, unified approach to valuation that can

be empirically tested and whose principles work for thevaluation of billion-dollar firms and ma and pa firms alike.While this book contains more mathematics—a worm’s eyeview, 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 theend 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 notoptimal, I have given instructions on which material can be safelyskipped

How to read this book depends on your quantitative skills and howmuch time you have available For the reader with strong quantitativeskills and abundant time, the ideal path is to read the book in its exactorder, as there is a logical sequence The first three parts to this bookfollow the chronological sequence of performing a valuation: (1) forecastcash flows, (2) discount to present value, and (3) adjust for control andmarketability The fourth part is a bird’s eye view in order to test empir-ically whether my methodology works Additionally, we explore (1) con-fidence intervals around valuation estimates and (2) what happens to thevaluation when appraisers make mistakes Part 5, on special topics, is theplace for everything else Each of parts of the book has an introductionpreceding it that will preview the upcoming material in greater depththan we cover here

Because most professionals do not have abundant time, I want tosuggest another path geared for the maximum benefit from the least in-vestment in time The heart of the book is Chapters 4 and 7, on log sizeand on adjusting for control and marketability, respectively I recommendthe time-pressed reader follow this order:

marketability)

restricted stock report)

fractional ownership interest discount report)

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

the section titled ‘‘A Brief Summary’’; ‘‘Periodic Perpetuity

Factors: Perpetuities for Periodic Cash Flows’’; and

‘‘Relationship of Gordon Model Multiple to the Price/Earnings

Ratio.’’ Some readers may want to read this chapter after

Chapter 7, though it will be somewhat helpful, but definitely

not necessary, to have read Chapter 3 before 4 and 7

the book)

as the material on using regression analysis may help reading

all of the other chapters)

After these chapters, you can read the remainder of the book in any

order, though it is best to read Chapter 14 immediately after Chapter 13

This book has close to 125 tables, many of them being two or three

pages long To facilitate your reading, it will help you to copy tables

whose commentary in the text is extensive and sit with the separate tables

next to you Otherwise, you will spend an inordinate amount of time

flipping pages back and forth Note: readers with low blood pressure may

wish to ignore that advice

BIBLIOGRAPHY

Mercer, Z Christopher 1997 Quantifying Marketability Discounts: Developing and Supporting

Marketability Discounts in the Appraisal of Closely Held Business Interests Memphis,

Tenn.: Peabody.

Pratt, Shannon P., Robert F Reilly, and Robert P Schweihs 1996 Valuing a Business:

The Analysis and Appraisal of Closely Held Companies, 3d ed New York: McGraw-Hill.

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Acknowledgments

I gratefully acknowledge help beyond the call of duty from my parents,

Leonard and Marilyn Abrams Professionally, R K Hiatt has been the

ideal internal editor Without his help, this book would have suffered

greatly He also contributed important insights throughout the book

Robert Reilly edited the original manuscript cover-to-cover I thank

Robert very much for the huge time commitment for someone else’s book

Larry Kasper gave me a surprise detailed edit of the first eight chapters

I benefited much from his input and thank him profusely

Chris Mercer also read much or all of the book and gave me many

corrections and very useful feedback I thank Chris very much for his

valuable time, of which he gave me much

Michael Bolotsky and Eric Nath were very helpful to me in editing

my summary of their work

I thank Rob Oliver and Roy Meyers of Management Planning, Inc

for providing me with their restricted stock data I also thank Bob Jones

of Jones, Roach & Caringella for providing me with private fractional

interest sales of real estate

Chaim Borevitz provided important help on Chapters 8 and 9 Mark

Shayne provided me with dozens of insightful comments Professor

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

xviii Acknowledgments

I thank my editor at McGraw-Hill, Ela Aktay, for her encouragementand enthusiasm I thank my publisher, Jeff Krames, for his patience andapologize for testing it so much due to my passion for perfection and thehuge life-changes that occurred to me while writing this book

All the people who have helped make this book a reality have myprofound gratitude In fact, there have been so many that it is almostimpossible to remember every single person, and I apologize to anyonewho should be in this acknowledgment section and who is not due to

my human failings

List of Figures

4-1 1926–1998 Arithmetic Mean Returns as a Function of Standard

4-2 1926–1998 Arithmetic Mean Returns as a Function of Ln(FMV) 127

4-3 Decade Standard Deviation of Returns versus Decade Average

4-4 Decade Standard Deviation of Returns versus Decade Average

4-8 1939–1998 Decile Standard Deviations as a Function of Ln(FMV) 138

12-2 Decision Tree for Bootstrapping Assuming Debt Restructure and

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List of Tables

2-1B Calculation of 95% Confidence Intervals for Forecast 1998 Costs 28

2-5 Regression Analysis of Sales as a Function of GDP [1] 43

3-3 ADF with Cash Flows Starting in Year 3.25: End-of-Year Formula 72

3-4 ADF with Cash Flows Starting in Year ⫺2.00: End-of-Year

3-5 ADF with Cash Flows Starting in Year ⫺2.00 with Negative

3-6 ADF with Cash Flows Starting in Year⫺2.00 with g ⬎ r:

3-7 Periodic Perpetuity Factor (PPF): End-of-Year Formula 80

3-9 Periodic Perpetuity Factor (PPF): End-of-Year—Cash Flows Begin

3-10 PV of Loan with Market Rate⬎ Nominal Rate: ADF, End-of-Year 86

xxii List of Tables

A3-3 Amortization of Principal with Irregular Starting Point 103

A3-5 Present Value of a Loan at Discount Rate Different than Nominal

4-1 NYSE Data by Decile and Statistical Analysis: 1926–1998 121

4-2 Regressions of Returns over Standard Deviation and Log of Fair

4-3 Table of Stock Market Returns Based on FMV—60-Year Model 136

4-4A Discounted Cash Flow Analysis Using 60-Year Model—First

A4-5 Gordon Model Valuation Using Newton’s Iterative Process 157

5-2 Geometric versus Arithmetic Returns: NYSE Data by Decile &

5-4 Comparison of Discount Rates Derived from the Log Size Model

6-1A Equity Valuation Approach with Iterations Beginning with Book

6-2A WACC Approach with Iterations Beginning with Book Equity 187

6-2B WACC Approach with Iterations Beginning with Arbitrary

7-1 Synergies as Measured by Acquisition Minus Going-Private

7-1A Acquisition and Going-Private Transactions Premiums 200

7-4 Mergerstat Mean Premiums: Control versus Minority Purchases 225

7-5 Abrams Regression of Management Planning Study Data 237

7-6 Calculation of Continuously Compounded Standard Deviation

7-9 Calculation of Restricted Stock Discounts for 13 Stocks Using

7-16a Implied Returns for Holding Period—30% Discount 275

7-17a Implied Returns for Holding Period—20% Discount 276

7-18 Summary of Results of Applying the QMDM in 10 Example

8-1A Calculation of Revenue and Earnings Stability (R2 ) 304

8-2A Standard Deviation of Continuously Compounded Returns 309

9-5 Economic Components Approach: 2.80% Member Interest 328

9-6 Regression Analysis of Partnership Profiles Database—1999 [1] 339

9-6B Partnership Profiles Database: Price-to-Value Discounts—1999 342

10-1 Log Size Equation for 1938–1986 NYSE Data by Decile and

10-4 Calculation of Component #1—Delay to Sale—$25,000 Firm 367

10-5 Calculation of Transaction Costs for Firms of All Sizes in the

10-4A Calculation of Component #1—Delay to Sale—$75,000 Firm 371

10-4B Calculation of Component #1—Delay to Sale—$125,000 Firm 371

xxiv List of Tables

10-4C Calculation of Component #1—Delay to Sale—$175,000 Firm 372

10-4D Calculation of Component #1—Delay to Sale—$225,000 Firm 372

10-4E Calculation of Component #1—Delay to Sale—$375,000 Firm 373

10-4F Calculation of Component #1—Delay to Sale—$750,000 Firm 373

10-4G Calculation of Component #1—Delay to Sale—$10 Million

11-4 Percent Valuation Error for 10% Relative Error in Growth 400

11-4A Percent Valuation Error for ⫺10% Relative Error in Growth 401

11-4B Percent Valuation Error for 10% Relative Error in Discount Rate 402

12-4 Sales Model with Exponentially Declining Growth Rate

13-3A Adjusting Dilution to Desired Levels—All Dilution to Owner 447

Forecasting Cash Flows

Part 1 of this book focuses on forecasting cash flows, the initial step in

the valuation process In order to forecast cash flows, it is important to:

of cash flows

In Chapter 1, we mathematically derive the cash flow 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 flows derive from and relate to the balance

sheet and income statement It may help eliminate errors made by

ap-praisers who perform discounted cash flow analysis using shortcut or

even incorrect definitions of cash flow

In Chapter 2, we demonstrate in detail:

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

regression

years of historical data to prevent using stale data may be

wrong

traded guideline companies information While this is not related

to forecasting sales and expenses, it fits 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 benefit we derive from the variation in size of the publicly

2 PART 1 Forecasting Cash Flows

traded guideline companies, as we discuss in the chapter The final 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 regressiontools in both Microsoft Excel and Lotus 123 Excel does a better job ofpresentation and offers much more comprehensive statistical feedback.Lotus has one significant advantage: it can provide multiple regressionanalysis 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) 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:

constant growth This application has become far more importantsince the Mercer Quantitative Marketability Discount Modelrequires an ADF with growth

lawsuits, etc

with growth, e.g., what is the PV of keeping one airplane of acertain class in service perpetually

calculating the cash equivalency selling price of a business, asseller financing 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 office, I unofficially titled Chapter 3,

‘‘The Chapter That Would Not Die!!!’’ I edited and rewrote this chapterclose 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 andwhat should be relegated to the appendix In an effort to maximize read-ability, the most practical formulas appear in the body of chapter 3 andthe least useful and most mathematical work appears in the appendix

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).

4 PART 1 Forecasting Cash Flows

INTRODUCTION

In 1987, the Financial Accounting Standards Board (FASB) issued ment of Financial Accounting Standards No 95, ‘‘Statement of CashFlows.’’ This standard stipulates that a statement of cash flows is required

State-as part of a full set of financial statements for almost all business prises

enter-This chapter, which discusses the Statement of Cash Flows, is tended for readers who already have a basic knowledge of accounting.Much of what follows will involve alternating between accrual and cashreporting, which can be very challenging material Also, a parsimoniousstyle 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 flows is to provide rel-evant information about the cash receipts and cash payments of an en-terprise These receipts and payments must be classified according tothree basic types of activities: operating, investing, and financing

in-Operating activities involve those transactions that enter into the termination of net income Examples of these activities are sales of goods

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

em-ployees Net income reports these activities when they are earned or curred Cash flows from operations reports these activities only when they

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

is made even though no cash is collected Cash flows from operationswould reflect the increase only at the time the cash is collected Also, netincome is decreased when, say, insurance expense is incurred eventhough no payment is made Cash flows from operations would reflectthe decrease only at the time the payment is made

Of course, companies engage in numerous transactions involvingcash but having no impact on the income statement These transactionsare classified as investing or financing activities Investing activities in-clude the acquisition of long-lived assets as well as their disposition when

and repaying funds from debt and equity holders and providing the ers with a return on their investment

own-Either the direct or the indirect method may be used as a basis forreporting cash flows from operating activities Under the direct methodthe 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 flow from operations

Under the indirect method, net cash flow 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.

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 flow 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 flow from operations

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

infer the cash flow associated with any single reported revenue or expense.

However, most statements of cash flows require a number of such

ad-justments, which often result in confusing entanglements

Many business and real estate appraisers spend a significant part of

their careers forecasting cash flows The objective of this chapter is to

improve their understanding of the cash flow 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 flow logic and distinguish true cash flows from shortcut

approxi-mations thereof

To achieve this result, this chapter provides a mathematical

deriva-tion of the cash flow statement using the indirect method A realistic

numerical example and an intuitive explanation accompany the

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 specific 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

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

transaction and its relation to cash flow 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

benefits.

6 PART 1 Forecasting Cash Flows

Property, plant, and Equipment

equip-ment

of or retired

Stockholders’ Equity

Required Working Capital

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 tothe next are shown in the right column On the far left the symbols usedlater to refer to these accounts in mathematical expressions have beenrepeated

total assets equal $3,150,000, total liabilities equal $1,085,000, and the totalliabilities and equity also equal $3,150,000 This can be shown as:

Subtracting the beginning balance sheet from the ending balance sheet

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

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

plant, and equipment Net property, plant, and equipment (NPPE) isgross property, plant, and equipment (GPPE) less the accumulated de-preciation (AD) on these assets As shown in Table 1-3 below, the change

Feathers R Us ABBREVIATED BALANCE SHEETS For Calendar Years

Increase (Decrease)

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

subtract-ing the change in accumulated depreciation from the change in gross

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

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

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.

8 PART 1 Forecasting Cash Flows

Feathers R Us INCOME STATEMENT For Calendar Year 2000

60,000

stockholder’s equity The total change in stockholder’s equity (⌬CAP) is

equal to net income (NI) and other equity transactions (OET) (definition

is given below equation [1-6])

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

descrip-tion of these transacdescrip-tions will be provided later in the chapter (refer toTable 1-4)

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

⌬C ⫹ ⌬OCA ⫹ (⌬GPPE ⫺ ⌬AD)

⫽ 35,000 ⫹ (25,000) ⫹ 90,000 ⫹ 250,000Equation (1-7) can be rearranged to satisfy the objective of the statement

of cash flows—providing an explanation of the change in the cash ance

bal-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 (1-5): ⌬L ⫽ ⌬CL + ⌬LTD Equation (1-6): ⌬CAP ⫽ NI + ⌬OET

T A B L E 1-3

Feathers R Us ANALYSIS OF PROPERTY, PLANT,

& EQUIPMENT For Calendar Year 2000

Symbols

GPPE

Gross Prop, Plant & Equip

AD

Accumulated Depreciation

NPPE

Net Prop, Plant & Equp

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

CAPEXP Capital expenditures 175,000 175,000

Retirements

RETGBV Gross book value 105,000 105,000

RETAD Accumulated depreciation 20,000 20,000

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)

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)

10 PART 1 Forecasting Cash Flows

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

To this point, only the book value of any assets retired has beenconsidered Most often, the retirement or disposition of assets involves again or a loss (a ‘‘negative’’ gain) This gain is the difference between theselling price of the property, plant, and equipment (SALESFA) and their

sold for $115,000, producing a gain of $30,000 This is shown in equation(1-12)

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

OET

After canceling the ⫹ RETAD and ⫺ RETAD terms and rearranging,

⌬C ⫽ NI ⫺ GAIN ⫹ DEPR ⫺ ⌬OCA ⫹ ⌬CL

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 addsubtract-ing 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 flows from

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

cash flows from financing 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

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 financing 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 flows but are appended in a separate schedule.

OET

12 PART 1 Forecasting Cash Flows

Feathers R Us STATEMENT OF STOCKHOLDERS’ EQUITY

For Calendar Year 2000

Symbols

Capital Stock

Additional Paid in Capital

Retained Earnings

Treasury Stock

Total Shareholder Equity

Balance, 1999 100,000 200,000 1,425,000 0 1,725,000

Other equity transactions

Equation (1-17) can be simplified to the more familiar form:

⌬C ⫽ Cash flows from operating activities

⫹ Cash flows from investing activities

⫹ (60,000)

⫹ 225,000Equations (1-17) and (1-18) describe the conventional Statement ofCash Flows shown in Table 1-5

For the moment we will define the required change in working capital

as the change in current assets other than cash, less the change in current

This illustration is somewhat unusual Here working capital is being

reduced This reduction is a source of the cash from operating activities.

12 The definition in equation (1-19) will be modified later in the chapter.

Feathers R Us ABBREVIATED STATEMENT OF CASH FLOWS For Calendar Year 2000

Cash flows from operating activities

Adjustments to reconcile net income to net cash provided by operating activities:

Cash flows from investing activities

CAPEXP Purchase of property, plant, & equipment (175,000)

SALESFA Sale of property, plant, & equipment 115,000

Cash flows from financing activities

SALSTK Sale of stock 350,000

TRSTK Purchase of treasury stock (50,000)

(In the typical case working capital is being increased This is usually true

when sales are growing In these cases, the increase in working capital

represents a use of cash.)

14 PART 1 Forecasting Cash Flows

Operating NI ⫹ Net income

When deriving the cash flows from operating activities, we subtractthe gain (or add the loss) on the sale of property, plant, and equipment

for several reasons First, these gains and losses simply are not the result

of ‘‘operating’’ activities They are the result of ‘‘investing’’ activities.These gains and losses arise when property, plant, and equipment are

sold for more or less than their net book value Furthermore, the full

amount received for such sales (SALESFA) is included as part of the cash

flows from investing activities To show these gains or losses again as part

of cash flows from operating activities would erroneously double counttheir impact

Depreciation and other noncash expenses do reduce net income, but they do not involve any payments during the current period Therefore,

when the indirect method is used and net income is the starting point forarriving at a firm’s net cash flow, these noncash expenses must be addedback

The rationale for subtracting required increases (or adding decreases)

in working capital will be discussed at some length in the next sectionafter introducing the components of the other current assets (⌬OCA) andthe current liabilities (⌬CL)

To complete the summary of equations (1-17), (1-18), and (1-20), the

second and third lines consist of 14

Activity Symbol Description

Investing CAPEXP ⫺ Capital expenditures

SALESFA ⫹ Selling price of property, plant, and equipment disposed of or retired

Financing ⌬LTD ⫹ Increases (⫺ decreases) in long-term debt

Considering the Components of Required Working Capital

Before discussing required working capital further, it will be helpful tobreak down changes in (⌬OCA) other current assets and (⌬CL) currentliabilities into some typical component parts Table 1-6 is a restatement

of Table 1-1 with this additional detail provided in the boxed sections

14 The second line of both equations (1-17) and (1-20) is: ⫺ CAPEXP + SALESFA The third line of both equations (1-17) and (1-20) is: ⌬LTD + SALSTK ⫺ TRSTK ⫺ DIV + AET

Feathers R Us BALANCE SHEETS For Calendar Years

Increase (Decrease)

Here, other current assets consist of accounts receivable, inventory,

and additional current assets Current liabilities include accounts payable,

short-term notes payable, and accrued expenses

Accounts receivable, inventory, and additional current assets should

all be treated in the same way that other current assets was treated When

using the indirect method, increases (decreases) in these component

ac-counts should be subtracted from (added to) net income to arrive at net

cash provided by operating activities

Likewise, accounts payable, short-term notes payable, and accrued

expenses should all be treated in the same way that current liabilities was

treated When using the indirect method, increases (decreases) in these

component accounts should be added to (subtracted from) net income to

arrive at net cash provided by operating activities

Applying the procedures outlined in the two preceding paragraphs

results in the Statement of Cash Flows shown in Table 1-7 which is simply

a restatement of Table 1-5 with the boxed detail added