sets to determine which gives the best results.

Substantial Changes in Competition or Product/Service

Although regression analysis is applicable in most situations, substantial

structural changes in a business may render it inappropriate. As men-

tioned previously, the appraiser can often compensate for changes in the

competitive environment by making pro forma adjustments to historical

sales, keeping costs the same. However, when a company changes its

business, the past is less likely to be a good indicator of what may occur

in the future, depending on the signi¬cance of the change.

USING REGRESSION ANALYSIS TO FORECAST SALES

Table 2-5 is an example of using regression techniques to forecast sales.

In order to do this, it must be reasonable to assume that past performance

is a reasonable indicator of future expectations. If there are fundamental

changes in the industry that render the past a poor indicator of the future,

then regression may useless and even quite misleading. As cautioned by

Pratt, Reilly, and, Schweihs (1996), blind application of regression, where

past performance is the sole indicator of future sales, can be misleading

and incorrect. Instead, careful analysis is required to determine whether

past income generating forces will be duplicated in the future. Neverthe-

less, regression analysis is often useful as a benchmark in forecasting.

In our example in Table 2-5, the primary independent variable is

gross domestic product (GDP), which we show for the years 1988“1998

in billions of dollars in cells B5:B15 (the cell references separated by a

colon will be our way to indicate contiguous spreadsheet ranges). In C5:

C15, we show the square of GDP in billions of dollars, which is our

second potential independent variable.15 Our dependent variable is sales,

which appears in D5:D15.

15. Another variation of this procedure is to substitute the square root of GDP for its square.

PART 1 Forecasting Cash Flows

42

T A B L E 2-5

Regression Analysis of Sales as a Function of GDP [1]

A B C D E F G H I

GDP2

4 Year GDP Sales

5 1988 5,049.6 25,498,460.2 $1,000,000

6 1989 5,438.7 29,579,457.7 $1,090,000

7 1990 5,743.0 32,982,049.0 $1,177,200

8 1991 5,916.7 35,007,338.9 $1,259,604

9 1992 6,244.4 38,992,531.4 $1,341,478

10 1993 6,558.1 43,008,675.6 $1,442,089

11 1994 6,947.0 48,260,809.0 $1,528,614

12 1995 7,269.6 52,847,084.2 $1,617,274

13 1996 7,661.6 58,700,114.6 $1,706,224

14 1997 8,110.9 65,786,698.8 $1,812,010

15 1998 8,510.7 72,432,014.5 $1,929,791

17 SUMMARY OUTPUT

19 Regression Statistics

20 Multiple R 0.999156207

21 R square 0.998313125

22 Adjusted R square 0.997891407

23 Standard error 13893.80997

24 Observations 11

26 ANOVA

27 df SS MS F Signi¬cance F

28 Regression 2 9.13938E 11 4.5697E 11 2367.24925 8.0971E 12

29 Residual 8 1544303643 193037955.4

30 Total 10 9.15482E 11

32 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%

33 Intercept 824833.1304 182213.8131 4.526732175 0.001932674 1245019.209 404647.0522 1245019.209 404647.0522

34 GDP 412.8368996 54.65310215 7.553768832 6.5848E-05 286.8065386 538.8672622 286.8065386 538.8672607

GDP2

35 0.010625314 0.004016833 2.64519663 0.029474667 0.019888154 0.001362473 0.019888154 0.001362473

[1] GDP, Gross Domestic Product, is in billions of dollars. GDP is a proxy for the overall economy.

43

Spreadsheet Procedures to Perform Regression

It is mandatory to put the variables in columns and the time periods in

rows. Electronic spreadsheets will not permit you to perform regression

analysis with time in columns and the variables in rows. In other words,

we cannot transpose the data in Table 2-5, cells A4:D15 and still perform

a regression analysis.

Another requirement is that all cells must contain numeric data. You

cannot perform regression with blank cells or cells with alphanumeric

data in them. Also, you will receive an error message if one of your

independent variables is a multiple of another. For example, if each cell

in C5:C15 is three times the corresponding cell in B5:B15, then the x var-

iables are perfectly collinear and the regression produce an error message.

We will explain regression procedures in Microsoft Excel ¬rst, then

in Lotus 123.

In Excel, the procedure to perform the regression analysis is as fol-

lows:

1. Select Tools Data Analysis Regression. This will bring up a

dialog box and automatically places the cursor in Input Y

Range.16

2. For the Y range (which is the dependent variable, sales in our

example), click on the range icon with the red arrow

immediately to the right. Doing so minimizes the dialog box

and enables you to highlight the cell range D4:D15 with your

mouse.17 Note that we have included the label Sales in D4 in

this range. Click again on the range icon again to return to the

dialog box.

3. For the X range, which are the independent variables GDP and

GDP2 in our case, repeat the procedure in (2) and highlight the

range B4:C15.

4. Click on the box Labels, which will put a check mark in the

box.

5. Click on Output Range. Click on the box to the right, click on

the range icon with the red arrow, and then click on cell A17.

This tells the spreadsheet to begin the regression output at that

cell.

6. Click OK.

Excel now calculates the regression and outputs the data as shown

in the bottom half of Table 2-5.

The instructions for Lotus 123 are almost identical. The only differ-

ences are:

1. The command is Range Analyze Regression.

2. The ranges for the dependent and independent variables should

not include the label in Row 4. Thus they are D5:D15 and B5:

C15, respectively.

16. If Data Analysis is not yet enabled in Excel, you must select add-ins and then select

Analysis ToolPak.

17. Excel actually shows the range with dollar signs, e.g., $D$4:$D$15

PART 1 Forecasting Cash Flows

44

3. Lotus 123 does not compute t-statistics for you.18 You will have

to do that manually by creating a formula. Divide the regression

coef¬cient by its standard error. Unfortunately, Lotus 123 does

not calculate the p-values either. You will have to look up your

results in a standard table of t-statistics. We will cover that later.

Examining the Regression Statistics

Once again, we look at the statistical measures resulting from the regres-

sion to determine how strong is the relationship between sales and time.

Adjusted R 2 is 99.8% (B22), a near-perfect relationship. The t-statistics for

the independent variables, GDP and GDP2, are 7.55 (D34) and “2.65

(D35), both statistically signi¬cant. The easiest way to determine the level

of statistical signi¬cance is through the p-value. One minus the p-value

is the level of statistical signi¬cance. For GDP, the p-value is 6.5848 10 5

(E34), which is much less than 0.1%. Thus GNP is statistically signi¬cant

at a level greater than 100% 0.1% 99.9%. The square of GDP has a

p-value of 0.029 (E35), which indicates statistical signi¬cance at the 97.1%

level. We normally accept any regressor with signi¬cance greater than or

equal to 95%, and we may consider accepting a regressor that is signi¬-

cant at the 90% to 95% level.

The standard error of the y-estimate, i.e., sales, is $13,894 (B23). Our

approximate 95% con¬dence interval is two standard errors

$27,788, which is less than 2% of the mean of sales.

In actual practice, adjusted R 2 for a regression of sales of mature

¬rms is often above 90% and frequently around 98%.

Adding Industry-Speci¬c Independent Variables

One should also consider adding industry-speci¬c independent variables.

For example, when valuing a jeweler, we should try adding the price of

gold and silver (and the nonlinear transformations, i.e., squares, square

roots, and logarithms) as independent variables. When valuing a ¬rm in

the oil industry, we should try using the price of a barrel of oil (and its

nonlinear transformations).

When valuing a coffee producer, we would want to have not only

the average price of coffee as an independent variable, but also the price

of tea and perhaps even sugar. The analyst should look to the prices of

the product itself, complements, and substitutes.

Once again, it is important to examine the statistical validity of the

relationship and use professional judgment to determine the usefulness

of the equation. Sales forecasts obtained from regression analysis can

serve as a benchmark from which adjustments can be made based on

qualitative factors that may in¬‚uence future sales.

One should also keep in mind that just because a less quantitative

method of forecasting sales does not have an embarrassingly low R 2 star-

ing the analyst in the face does not mean that it is superior to the re-

18. That is true of version 5, which is already at least four years old. If Lotus has added that

feature in a later version, I would not be aware of that.

CHAPTER 2 Using Regression Analysis 45

gression. It means we have no clue as to the reliability of the forecast. We

should always be uncomfortable with our ignorance.

Try All Combinations of Potential Independent Variables

It is important to try all combinations of independent variables. With a

statistics package, this is done automatically in using automated forward

or backward regression. However, statistics packages have their draw-

backs. They are not very user friendly in communicating with spread-

sheet programs, which most appraisers use in valuation analysis. Most

appraisers will ¬nd the spreadsheet regression capabilities more than ad-

equate.

Therefore, it is important to try all combinations of potential inde-

pendent variables in the regression process. For example, in regressing

sales against both GDP and GDP2, it is not at all unusual to ¬nd both

independent variables statistically insigni¬cant when regressed together,

i.e., p-values greater than 0.05. However, they still may be statistically

signi¬cant when regressed individually. So it is important to regress sales

against GDP and perform a second regression against GDP2. This process

becomes more complicated with additional candidates for independent

variables.

APPLICATION OF REGRESSION ANALYSIS TO THE

GUIDELINE COMPANY METHOD

Valuation using the guideline company method involves the use of ratios

of stock price to: earnings (P/E multiples), cash ¬‚ow (P/CF or P/EBIT

multiples), book value (P/BV multiples), sales (P/Sales), or other mea-

sures of income, cash ¬‚ow, or value. The stock prices typically are those

of public companies in the same or similar business as the company.

Consideration is therefore given to the opinion of the informed investor

and what he or she is willing to pay for the stock of comparative public

companies adjusted for the speci¬c circumstances of the company being

valued. While the use of ratios is common in valuation, regression anal-

ysis is more sophisticated and informative because it provides us with

statistical feedback on the strength of the relationship. Pratt, Reilly, and

Schweihs (1996) present a comprehensive chapter on use of the guideline

company method, so we will only discuss it within the context of regres-

sion analysis.

Table 2-6: Regression Analysis of Guideline Companies

Table 2-6 shows data from an actual guideline company analysis, with

the company names disguised in Column A. Column B contains the fair

market values (FMVs) (market capitalization) for 11 companies, ranging

from slightly over $3 million (B5) to over $150 million (B15). The average

FMV is $41.3 million (B16), with a standard deviation of $44.6 million

(B17). Net income (Column C) averages about $5.1 million (C16), with a

range of $600,000 to $16.9 million. We had to exclude companies A and

B, which were outliers with price earnings (PE) ratios over 60.

PART 1 Forecasting Cash Flows

46

T A B L E 2-6

Regression Analysis of Guideline Companies

A B C D E F G H I

4 Company FMV Net Income ln FMV ln NI 1/g g PE Ratio

5 C 3,165,958 602,465 14.9680 13.3088 20.0000 0.0500 5.2550

6 D 6,250,000 659,931 15.6481 13.3999 10.0000 0.1000 9.4707

7 E 12,698,131 1,375,000 16.3570 14.1340 10.5263 0.0950 9.2350

8 F 24,062,948 2,325,000 16.9962 14.6592 9.0909 0.1100 10.3497

9 G 23,210,578 2,673,415 16.9601 14.7989 12.1951 0.0820 8.6820

10 H 16,683,567 2,982,582 16.6299 14.9083 20.0000 0.0500 5.5937

11 I 37,545,523 4,369,808 17.4411 15.2902 12.5000 0.0800 8.5920

12 J 46,314,262 4,438,000 17.6510 15.3057 9.3023 0.1075 10.4358

13 K 36,068,550 7,384,000 17.4009 15.8148 20.8333 0.0480 4.8847

14 L 97,482,000 12,679,000 18.3952 16.3555 9.5238 0.1050 7.6885

15 M 150,388,518 16,865,443 18.8287 16.6408 9.0909 0.1100 8.9170