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.

FORECASTING COSTS AND EXPENSES

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

22

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

companies.

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

24

T A B L E 2-1A

Adjustments to Historical Costs and Expenses

A B C D E F G H I J K

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

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

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

28

USE OF REGRESSION STATISTICS TO TEST THE

ROBUSTNESS OF THE RELATIONSHIP

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