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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
expenses

[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:
r)b
(1
PPF PPF”end-of-year (3-18a)
r) j g) j
(1 (1
r)b
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:
1.24
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.


PERFORMING 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
costs.
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
b
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
26
T A B L E 2-1B

Regression Analysis 1988“1997


A B C D E F G

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

17 SUMMARY OUTPUT

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

26 ANOVA

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
$1,200,000




$1,000,000
y = 0.8045x + 56770
R2 = 0.9975



$800,000
Adj. Costs




$600,000




$400,000




$200,000




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




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

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


A B C D E F

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

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( 100 .)



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