. 8
( 100 .)


The de¬nition of net cash ¬‚ow available for distribution to stockholders
in equation (1-20b) can be summarized in the following way:

Activity Symbol Description

Operating NI Net income
GAIN Gains ( losses) on the sale of property, plant, and equipment
DEPR Depreciation and other noncash charges
RWC Increases ( decreases) in required working capital*
Investing CAPEXP Capital expenditures
SALESFA Selling price of property, plant, and equipment disposed of or retired
Financing LTD Increases ( decreases) in long-term debt
SALSTK Proceeds received from the sale of stock
TRSTK Payments for treasury stock
AET Additional equity transactions

*After adjusting for required cash.

This is easily compared to other de¬nitions that have been provided
in the authoritative literature. For example, one group of authors (Pratt,

16. C* C CReq

PART 1 Forecasting Cash Flows
Reilly, and Schweihs 1996) have proposed the following de¬nition of net
cash ¬‚ow available for distribution to stockholders in their Formula 9-3
(at 156“157):


Net income
Depreciation and other non-cash charges
Increases ( decreases) in required working capital
Capital expenditures
Selling price of property, plant, and equipment disposed of or retired
Increases ( decreases) in long term debt

Implicitly, this de¬nition assumes that gains and losses on the sale
of property, plant, and equipment and the selling price of property, plant,
and equipment disposed of or retired are immaterial. Likewise, this def-
inition assumes that the proceeds from the sale of stock, payments made
for treasury stock, and additional equity transactions are also immaterial.
These assumptions are quite reasonable and can safely be made in a
large number of cases.17 However, it is important for the analyst to realize
that these assumptions are being made.
It is well known that when calculating value by capitalizing a single
initial cash ¬‚ow, the consequences of making adjustments to the initial
cash ¬‚ow are magni¬ed considerably. It is important for the analyst to
understand how these hidden assumptions might in¬‚uence the amount
of initial cash ¬‚ow being capitalized. Perhaps it is even more important
for the analyst to take into account how these assumptions might impact
the future cash ¬‚ows available for distribution to stockholders.
For example, if a company were to routinely to sell its equipment
for signi¬cant sums, the analyst would be remiss if he or she overlooked
the cash ¬‚ows from these sales.

Careful consideration of mathematics in this chapter should enhance the
analyst™s understanding of important accounting relationships and the
˜˜whys™™ of the Statement of Cash Flows. It should also make the analyst
aware of the simplifying assumptions embedded in abbreviated de¬ni-
tions of cash ¬‚ow available for distribution to stockholders. Hopefully,
this awareness will result in superior valuations in those instances where
the making of these simplifying assumptions is unwarranted.

Abrams, Jay B. 1997. ˜˜Cash Flow: A Mathematical Derivation.™™ Valuation (March 1994):
Pratt, Shannon P., Robert F. Reilly, and Robert P. Schweihs. 1996. Valuing a Business: The
Analysis and Appraisal of Closely Held Companies, 3rd ed. New York: McGraw-Hill.

17. With respect to the proceeds from the sale of stock, it is unlikely that a ¬rm would sell its
stock in order to obtain cash for distribution to its stockholders. However, sometimes large
sales of stock do occur.

CHAPTER 1 Cash Flow: A Mathematical Derivation 19
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Using Regression Analysis

Adjustments to Expenses
Table 2-1A: Calculating Adjusted Costs and Expenses
Standard Error of the y Estimate
The Mean of a and b
The Variance of a and b
Precise Con¬dence Intervals
Selecting the Data Set and Regression Equation
Insuf¬cient Data
Substantial Changes in Competition or Product/Service
Spreadsheet Procedures to Perform Regression
Examining the Regression Statistics
Adding Industry-Speci¬c Independent Variables
Try All Combinations of Potential Independent Variables
Table 2-6: Regression Analysis of Guideline Companies
95% Con¬dence Intervals


Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
Regression analysis is a statistical technique that estimates the mathe-
matical relationship between causal variables, known as independent var-
iables, and a dependent variable. The most common uses of regression
analysis in business valuation are:
1. Forecasting sales in a discounted cash ¬‚ow analysis
2. Forecasting costs and expenses in a discounted cash ¬‚ow
3. Measuring the relationship between market capitalization (fair
market value) as the dependent variable and several possible
independent variables for a publicly traded guideline company
valuation approach. Typical independent variables that are
candidates to affect the fair market value are net income
(including nonlinear transformations such as its square, square
root, and logarithm), book value, the debt-to-equity ratio, and so
This chapter is written to provide the appraiser with some statistical
theory, but it is primarily focused on how to apply regression analysis to
real-life appraisal assignments using standard spreadsheet regression
tools. We have not attempted to provide a rigorous, exhaustive treatment
on statistics and have put as much of the technical background discussion
as possible in the appendix to keep the body of the chapter as simple as
possible. Those who want a comprehensive refresher should consult a
statistics text, such as Bhattacharyya and Johnson (1977) and Wonnacott
and Wonnacott (1981). We present only bits and pieces of statistics that
are necessary to facilitate our discussion of the important practical issues.
Even though you may not be familiar with using regression analysis
at all, let alone with nonlinear transformations of the data, the material
in this chapter is not that dif¬cult and can be very useful in your day-to-
day valuation practice. We will explain all the basics you need to use this
very important tool on a daily basis and will lead you step-by-step
through an example, so you can use this chapter as a guide to get ˜˜hands-
on™™ experience.
For those who are unfamiliar with the mechanical procedures to per-
form regression analysis using spreadsheets, we explain that step-by-step
in the section on using regression to forecast sales.

In performing a discounted cash ¬‚ow analysis, an analyst should forecast
sales, expenses, and changes in balance sheet accounts that affect cash
¬‚ows. Frequently analysts base their forecasts of future costs on historical
averages of, or trends in, the ratio of costs as a percentage of sales.
One signi¬cant weakness of this methodology is that it ignores ¬xed
costs, leading to undervaluation in good times and possible overvaluation
in bad times. If the analyst treats all costs as variable, in good times when
he or she forecasts rapid sales growth, the ¬xed costs should stay constant
(or possibly increase with in¬‚ation, depending on the nature of the costs),
but the analyst will forecast those ¬xed costs to rise in proportion to sales.

PART 1 Forecasting Cash Flows
That leads to forecasting expenses too high and income too low in good
times, which ultimately causes an undervaluation of the ¬rm. In bad
times, if sales are forecasted ¬‚at, then costs will be accidentally forecasted
correctly. If sales are expected to decline, then treating all costs as variable
will lead to forecasting expenses too low and net income too high, leading
to overvaluation.
Ordinary least squares (OLS) regression analysis is an excellent tool
to forecast adjusted costs and expenses (which for simplicity we will call
˜˜adjusted costs™™ or ˜˜costs™™) based on their historical relationship to sales.
OLS produces a statistical estimate of both ¬xed and variable costs, which
is useful in planning as well as in forecasting. Furthermore, the regression
statistics produce feedback used to judge the robustness of the relation-
ship between sales and costs.

Adjustments to Expenses
Prior to performing regression analysis, we should analyze historical in-
come statements to ascertain if various expenses have maintained a con-
sistent pattern or if there has been a shift in the structure of a particular
expense. When past data is not likely to be representative of future ex-
pectations, we make pro forma adjustments to historical results to model
how the Company would have looked if its operations in the past had
conformed to the way we expect them to behave in the future. The pur-
pose of these adjustments is to examine longstanding ¬nancial trends
without the interference of obsolete information from the past. For ex-
ample, if the cost of advertising was 10% of sales for the ¬rst two years
of our historical analysis, decreased to 5% for the next ¬ve years, and is
expected to remain at 5% in the future, we may add back the excess 5%
to net income in the ¬rst two years to re¬‚ect our future expectations. We
may make similar adjustments to other expenses that have changed dur-
ing the historical period or that we expect to change in the future to arrive
at adjusted net income.

Table 2-1A: Calculating Adjusted Costs and Expenses
Table 2-1A shows summary income statements for the years 1988 to 1997.
Adjustments to pretax net income appear in Rows 15“20. The ¬rst ad-
justment, which appears in Rows 15“18, converts actual salary paid”
along with bonuses and pension payments”to an arm™s length salary.
This type of adjustment is standard in all valuations of privately held
The second type of adjustment is for a one-time event that is unlikely
to repeat in the future. In our example, the Company wrote off a discon-
tinued operation in 1994. As such, we add back the write-off to income
(H19) because it is not expected to recur in the future.
The third type of adjustment is for a periodic expense. We use a
company move as an example, since we expect a move to occur about
every 10 years.1 In our example, the company moved in 1993, 4 years

1. Losses from litigation are another type of expense that often has a periodic pattern.

CHAPTER 2 Using Regression Analysis 23

T A B L E 2-1A

Adjustments to Historical Costs and Expenses


4 Summary Income Statements

6 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997

7 Sales $250,000 $500,000 $750,000 $1,000,000 $1,060,000 $1,123,600 $1,191,016 $1,262,477 $1,338,226 $1,415,000
8 Cost of sales 100,000 250,000 375,000 500,000 490,000 505,000 520,000 535,000 550,000 600,000
9 S, G & A expenses 100,000 150,000 250,000 335,000 335,000 360,000 370,000 405,000 435,000 450,000
10 Operating expenses 58,000 68,000 78,000 88,000 83,000 110,000 112,000 117,000 122,000 132,000
11 Other expense 5,000 15,000 20,000 25,000 20,000 43,000 100,000 50,000 50,000 50,000
12 Pretax income $13,000 $17,000 $27,000 $52,000 $132,000 $105,600 $89,016 $155,477 $181,226 $183,000
13 Pre-tax pro¬t margin 5.20% 3.40% 3.60% 5.20% 12.45% 9.40% 7.47% 12.32% 13.54% 12.93%
14 Adjustments:
15 Actual salary 75,000 80,000 85,000 130,000 100,000 100,000 105,000 107,000 109,000 111,000
16 Bonus 3,000 4,000 4,000 20,000 5,000 5,000 5,000 7,000 9,000 10,000
17 Pension 1,000 1,000 1,500 2,000 2,000 2,000 2,000 2,000 2,000 2,000
18 Arms length salary [1] (58,015) (60,916) (63,961) (67,159) (70,517) (74,043) (77,745) (81,633) (85,714) (90,000)
19 Discontinued operations [2] 55,000
20 Moving expense [3] 20,000
21 Adjusted pretax income $7,985 $41,084 $53,539 $136,841 $168,483 $158,557 $178,271 $189,844 $215,511 $216,000
22 Adjusted pretax pro¬t margin 3.19% 8.22% 7.14% 13.68% 15.89% 14.11% 14.97% 15.04% 16.10% 15.27%


. 8
( 100 .)