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Income Statements [1] 325
Cash Distributions 326
Economic Components Approach: 2.80% Member Interest 328
Calculation of Component #1: Delay to Sale [1] 330
Earnings and Revenue Stability 332
Calculation of DLOM: 2.80% Member Interest 334
Regression Analysis of Partnership Pro¬les Database”1999 [1] 339
Correlation Matrix 341
Partnership Pro¬les Database: Price-to-Value Discounts”1999 342
Private Fractional Interest Sales 346
Final Calculation of Fractional Interest Discount 347
Log Size Equation for 1938“1986 NYSE Data by Decile and
Statistical Analysis: 1938“1986 360
Reconciliation to IBA Database 362
Proof of Discount Calculation 364
Calculation of Component #1”Delay to Sale”$25,000 Firm 367
Calculation of Transaction Costs for Firms of All Sizes in the
IBA Study 369
Calculation of DLOM”$25,000 Firm 370
Calculation of Component #1”Delay to Sale”$75,000 Firm 371
Calculation of Component #1”Delay to Sale”$125,000 Firm 371

List of Tables xxiii
Calculation of Component #1”Delay to Sale”$175,000 Firm 372
Calculation of Component #1”Delay to Sale”$225,000 Firm 372
Calculation of Component #1”Delay to Sale”$375,000 Firm 373
Calculation of Component #1”Delay to Sale”$750,000 Firm 373
Calculation of Component #1”Delay to Sale”$10 Million
Firm 374
Calculation of DLOM”$75,000 Firm 374
Calculation of DLOM”$125,000 Firm 375
Calculation of DLOM”$175,000 Firm 375
Calculation of DLOM”$225,000 Firm 376
Calculation of DLOM”$375,000 Firm 376
Calculation of DLOM”$750,000 Firm 377
Calculation of DLOM”$10,000,000 Firm 377
95% Con¬dence Intervals 389
95% Con¬dence Intervals”60-Year Log Size Model 391
Absolute Errors in Forecasting Growth Rates 398
Percent Valuation Error for 10% Relative Error in Growth 400
Percent Valuation Error for 10% Relative Error in Growth 401
Percent Valuation Error for 10% Relative Error in Discount Rate 402
Summary of Effects of Valuation Errors 403
First Chicago Method 412
VC Pricing Approach 414
Statistical Calculation of Fair Market Value 418
Sales Model with Exponentially Declining Growth Rate
Assumption 430
Calculation of Lifetime ESOP Costs 438
FMV Calculations: Firm, ESOP, and Dilution 441
Adjusting Dilution to Desired Levels 446
Adjusting Dilution to Desired Levels”All Dilution to Owner 447
Summary of Dilution Tradeoffs 447

List of Tables

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
— When and why the common practice of not using more than ¬ve
years of historical data to prevent using stale data may be
— 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


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

Cash Flow: A Mathematical

A Preliminary Explanation of Cash Flows
Analyzing Property, Plant, and Equipment Transactions
An Explanation of Cash Flows with More Detail for Equity
Considering the Components of Required Working Capital
Adjusting for Required Cash

1. This chapter was coauthored with Donald Shannon, School of Accountancy, DePaul University.
The mathematical model was published in Abrams (1997).


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

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

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-
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
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
3,150,000 1,085,000 2,065,000
Likewise, the balance sheet for the preceding year (t 1 1999) is in
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
T A B L E 1-1

Feathers R Us
For Calendar Years

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

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