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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




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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




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

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