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

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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

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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