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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%
Calculation of adjusted costs
24 and expenses
25 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
26 Adjusted pretax net income $7,985 $41,084 $53,539 $136,841 $168,483 $158,557 $178,271 $189,844 $215,511 $216,000
27 Adjusted costs and $242,015 $458,916 $696,461 $863,159 $891,517 $965,043 $1,012,745 $1,072,633 $1,122,714 $1,199,000

[1] Arms length salary includes bonus and pension
[2] A write-off for discontinued operations was an unusual a one-time expense already included in other expense. We reverse it our here.
[3] Moving expense is a periodic expense which occurs approximately every 10 years. For the 1993 move, we add back the $20,000 cost to pre-tax income, and use a Periodic Perpetuity Factor to calculate an adjustment to FMV, which we
apply later in the valuation process (see Chapter 3).
ago. We add back the $20,000 cost of the move in the adjustment section
(G20) and treat the cost separately as a periodic perpetuity.
In Chapter 3, we develop two periodic perpetuity factors (PPFs)2 for
periodic cash ¬‚ows occurring every j years, growing at a constant rate of
g, discounted to present value at the rate r, where the last cash ¬‚ow
occurred b years ago. Those formulas are:
PPF PPF”end-of-year (3-18a)
r) j g) j
(1 (1
1 r (1
PPF PPF”midyear (3-19a)
r) j g) j
(1 (1
We assume the move occurs at the end of the year and use equation
(3-18a), the end-of-year PPF. We also assume a discount rate of r 20%,
moves occur every j 10 years, the last move occurred b 4 years ago,
and the cost of moving grows at g 5% per year. The cost of the next
1.210 $20,000 1.62889
move, which is forecast in Year 6, is $20,000
$32,577.89. We multiply this by the PPF, which is:
PPF 0.45445
1.210 1.0510
(see Table 3-9, cell A20), which results in a present value of $14,805.14.
Assuming a 40% tax rate, the after-tax present value of moving costs
is $14,805.14 (1 40%) $8,883. Since this is an expense, we must
remember to subtract it from”not add it to”the FMV of the ¬rm before
moving expenses. For example, if we calculate a marketable minority in-
terest FMV of $1,008,883 before moving expenses, then the marketable
minority FMV would be $1 million after moving expenses.
The other possible treatment for the periodic expense, which is
slightly less accurate but avoids the complex PPF, is to allocate the peri-
odic expense over the applicable years”10 in this example. The appraiser
who chooses this method must allocate expenses from the prior move to
the years before 1993. This approach causes the regression R 2 to be arti-
¬cially high, as the appraiser has created what appears to be a perfect
¬xed cost. For example, suppose we allocated $2,000 per year moving
costs to the years 1993“1998. If we run a regression on those years only,
R 2 will be overstated, as the perfect ¬xed cost of $2,000 per year is merely
an allocation, not the real cash ¬‚ow. Other regression measures will also
be exaggerated. If the numbers being allocated are small, however, the
overstatement is also likely to be small.
Adjusted pretax income appears in Row 21. Note that as a result of
these adjustments, the adjusted pretax pro¬t margin in Row 22 is sub-
stantially higher than the unadjusted pretax margin in Row 13.

2. This is a term to describe the present value of a periodic cash ¬‚ow that runs in perpetuity. To
my knowledge, these formulas are my own invention and PPF is my own name for it. As
mentioned in Chapter 3, where we develop this, it is in essence the same as a Gordon
model, but for a periodic, noncontiguous cash ¬‚ow. As noted in Chapter 3, when sales
occur every year, j 1 and formulas (3-18a) and (3-19a) simplify to the familiar Gordon
model multiples.

CHAPTER 2 Using Regression Analysis 25
We repeat sales (Row 7) in Row 25 and adjusted pretax income (Row
21) in Row 26. Subtracting Row 26 from Row 25, we arrive at adjusted
costs and expenses in Row 27. These adjusted costs and expenses are
what is used in forecasting future costs and expenses regression analysis.

Ordinary least squares regression analysis measures the linear relation-
ship between a dependent variable and an independent variable. Its
mathematical form is y x, where:
y the dependent variable (in this case, adjusted costs).
x the independent variable (in this case, sales).
the true (and unobservable) y-intercept value, i.e., ¬xed costs.
the true (and unobservable) slope of the line, i.e., variable
Both and , the true ¬xed and variable costs of the Company, are
unobservable. In performing the regression, we are estimating and
from our historical analysis, and we will call our estimates:
a the estimated y-intercept value (estimated ¬xed costs).
the estimated slope of the line (estimated variable costs).3
OLS estimates ¬xed and variable costs (the y-intercept and slope) by
calculating the best ¬t line through the data points.4 In our case, the de-
pendent variable (y) is adjusted costs and the independent variable (x) is
sales. Sales, which is in Table 2-1A, Row 7, appears in Table 2-1B as B6
to B15. Adjusted costs and expenses, Table 2-1A, Row 27, appears in Table
2-1B as C6 to C15. Table 2-1B shows the regression analysis of these var-
iables using all 10 years of data. The resulting regression yields an inter-
cept value of $56,770 (B33) and a (rounded) slope coef¬cient of $0.80
(B34). Using these results, the equation of the line becomes:
Adjusted Costs and Expenses $56,770 ($0.80 Sales)
The y-intercept, $56,770, represents the ¬xed costs of operation, or
the cost of operating the business at a zero sales volume. The slope co-
ef¬cient, $0.80, is the variable cost per dollar of sales. This means that for
every dollar of sales, there are directly related costs and expenses of $0.80.
We show this relationship graphically at the bottom of the table. The
diamonds are actual data points, and the line passing through them is
the regression estimate. Note how close all of the data points are to the
regression line, which indicates there is a strong relationship between
sales and costs.5

3. The regression parameters a and b are often shown in statistical literature as and with a
circum¬‚ex (ˆ) over each letter.
4. The interested reader should consult a statistics text for the multivariate calculus involved in
calculating a and b. Mathematically, OLS calculates the line that minimizes the sum of the
squared deviations between the actual data points and the regression estimate.
5. We will discuss the second page of Table 2-1B later in the chapter.

PART 1 Forecasting Cash Flows
T A B L E 2-1B

Regression Analysis 1988“1997


4 Actual

5 Year Sales X [1] Adj. Costs Y [2]

6 1988 $250,000 $242,015
7 1989 $500,000 $458,916
8 1990 $750,000 $696,461
9 1991 $1,000,000 $863,159
10 1992 $1,060,000 $891,517
11 1993 $1,123,600 $965,043
12 1994 $1,191,016 $1,012,745
13 1995 $1,262,477 $1,072,633
14 1996 $1,338,226 $1,122,714
15 1997 $1,415,000 $1,199,000


19 Regression Statistics

20 Multiple R 99.88%
21 R square 99.75%
22 Adjusted R square 99.72%
23 Standard error 16,014
24 Observations 10


27 df SS MS F Signi¬cance F

28 Regression 1 8.31E 11 8.31E 11 3.24E 03 1.00E 11
29 Residual 8 2.05E 09 2.56E 08
30 Total 9 8.33E 11

32 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95%

33 Intercept [3] 56,770 14,863 3.82 5.09E-03 22,496 91,045
34 Sales [4] 0.80 0.01 56.94 1.00E-11 0.77 0.84

Regression Plot
[1] From Table 2-1A, Row 7 $1,400,000
[2] From Table 2-1A, Row 27
[3] Regression estimate of ¬xed costs
[4] Regression estimate of variable costs

y = 0.8045x + 56770
R2 = 0.9975

Adj. Costs




$0 $200,000 $400,000 $600,000 $800,000 $1,000,000 $1,200,000 $1,400,000 $1,600,000

CHAPTER 2 Using Regression Analysis 27
T A B L E 2-1B (continued)

Calculation of 95% Con¬dence Intervals for Forecast 1998 Costs


4 Actual

x2 x21998 / Sum x2
5 Year Sales X [1] Adj. Costs Y [2] x

6 1988 $250,000 $242,015 739,032 5.5E 11
7 1989 $500,000 $458,916 489,032 2.4E 11
8 1990 $750,000 $696,461 239,032 5.7E 10
9 1991 $1,000,000 $863,159 10,968 1.2E 08
10 1992 $1,060,000 $891,517 70,968 5.0E 09
11 1993 $1,123,600 $965,043 134,568 1.8E 10
12 1994 $1,191,016 $1,012,745 201,984 4.1E 10
13 1995 $1,262,477 $1,072,633 273,445 7.5E 10
14 1996 $1,338,226 $1,122,714 349,194 1.2E 11
15 1997 $1,415,000 $1,199,000 425,968 1.8E 11
16 Average/Total $989,032 $ 0 1.28E 12
17 Forecast 1998 $1,600,000 $1,343,928 610,968 3.7E 11 0.2905650

x2 x2
1 1
o o
21 Con¬dence Interval t0.025s t0.025s 1
x2 x2
n n
i i

24 Con¬dence Intervals For: Mean Speci¬c Year

25 t0.025 [t-statistic for 8 degrees of freedom] 2.306 2.306
26 s [From Table 2-1B, B23] $16,014 $16,014
27 1/n 0.1 0.1
x02 / Sum (Xi2)
28 [F17] 0.2905650 0.2905650
29 Add 0 for mean, 1 for speci¬c year™s exp. 0.0000000 1.0000000
30 Add rows 27 To 29 0.3905650 1.3905650
31 Square root of row 30 0.6249520 1.1792222
32 Con¬d interval row 25 * row 26 * row 31 $23,078 $43,547
33 Con¬d interval / forecast 1998 costs row 32 / C17 1.7% 3.2%
35 Regression Coef¬cients Coef¬cients
36 Intercept [From Table 2-1B, B33] 56,770
37 Sales [From Table 2-1B, B34] 0.80

We can use this regression equation to calculate future costs once we
generate a future sales forecast. Of course, to be useful, the regression
equation should make common sense. For example, a negative y-intercept
in this context would imply negative ¬xed costs, which makes no sense
whatsoever (although in regressions involving other variables it may well
make sense). Normally one should not use a result like that, despite oth-
erwise impressive regression statistics.
If the regression forecasts variable costs above $1.00, one should be
suspicious. If true, either the Company must anticipate a signi¬cant de-
crease in its cost structure in the near future”which would invalidate
applicability of the regression analysis to the future”or the Company
will be out of business soon. The analyst should also consider the pos-
sibility that the regression failed, perhaps because of either insuf¬cient or
incorrect data, and it may be unwise to use the results in the valuation.

PART 1 Forecasting Cash Flows
Having determined the equation of the line, we use regression statistics
to determine the strength of the relationship between the dependent and
independent variable(s). We give only a brief verbal description of re-
gression statistics below. For a more in-depth explanation, the reader
should refer to a book on statistics.
In an OLS regression, the ˜˜goodness of ¬t™™ of the line is measured
by the degree of correlation between the dependent and independent
variable, referred to as the r value. An r value of 1 indicates a perfect
direct relationship, where the independent variable explains all of the
variation of the dependent variable. A value of 1 indicates a perfect

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