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