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

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