estimate of $16,014, which came from Table 2-1B, B23. Row 27 is 1/n

1/10 0.1, where n is the number of observations.

Row 28 is a repetition of F17, the ratio of the squared deviation of

the forecast to the sum of the squared deviations of the independent

variables from their mean.

In B29 we add zero, and in C29 we add 1, according to equations

(2-8) and (2-9), respectively. We will explain the difference in the two

formulas shortly.

In Row 30 we add Rows 27 to 29, which are the terms in the square

root sign in the equations. Obviously, C30 B30 1. In Row 31 we take

the square root of Row 30.

Finally, we are able to calculate our 95% con¬dence intervals as Row

25 Row 26 Row 31. The 95% con¬dence interval for the mean is

2.306 $16,014 0.6249520 $23,078 (B32), approximately 1.44 times

the size of the standard error of the y-estimate. The 95% con¬dence in-

PART 1 Forecasting Cash Flows

38

terval for the speci¬c year™s cost forecast is $43,547 (C32), approximately

2.72 times the size of the standard error of the y-estimate. The 95% con-

¬dence intervals are 1.7% (B33) and 3.2% (C33) of the forecast costs for

the mean and the speci¬c year™s forecast, respectively.

You can see that both the calculation of 95% con¬dence interval for

the mean and the speci¬c year™s forecast cost is roughly two times the

standard error of the y-estimate. Statisticians often loosely approximate

the 95% con¬dence intervals as two standard errors below and above the

regression estimate. Equations (2-8) and (2-9) are more precise.

Now we will discuss the difference between equations (2-8) and

(2-9). We forecast sales to be $1.6 million in 1998, which means that our

forecast of adjusted costs for that year according to the regression equa-

tion is $1,343,928. Of course, the actual expenses will not equal that num-

ber, even if actual sales by some miracle will equal forecast sales. The

95% con¬dence interval for the mean tells us that if we add and subtract

$23,078 to our forecast of $1,343,928, then we are 95% sure that the true

regression line at sales of $1.6 million should have been between

$1,320,850 and $1,367,006. If we would experience sales of $1.6 million

many times”say 1,000 times”we would be 95% sure that the average

cost would fall in our con¬dence interval.13 Equation (2-8) is the equation

describing this con¬dence interval.

That does not mean that we are 95% sure that costs would be be-

tween $1,320,850 and $1,367,006 in any particular year when sales is $1.6

million. We need a wider con¬dence interval to be 95% sure of costs in

a particular year, given a particular level of sales. Equation (2-9) describes

the con¬dence interval for a particular year.

Thus, the $23,078 con¬dence interval”meaning that we add and

subtract that number from forecast costs”appropriately quanti¬es our

long-run expectation of the con¬dence interval around forecast costs,

given the level of sales. In business valuation we are not very concerned

that every individual year conform to our forecasts. Rather, we are con-

cerned with the long-run accuracy of the regression equation. Thus, equa-

tion (2-8) is the relevant equation for 95% con¬dence intervals for valu-

ation analysts. Remember that the con¬dence interval expands the further

we move away from the mean of the historical period. Therefore, if we

forecast the costs to go with a forecast sales of, say, $5 million in the year

2005, the con¬dence interval around the cost estimate is wider than the

1.7% (B33) around 1998 forecast.

Selecting the Data Set and Regression Equation

Table 2-4 is otherwise identical to Table 2-1B, except that instead of all 10

years of data, it only contains the last 5 years. The regression equation

for the 5 years of data is (Table 2-4, B27 and B28)

Adjusted costs $71,252 ($0.79 Sales)

Examining the regression statistics, we ¬nd that the adjusted R 2 is

13. This ignores the need to recompute the regression equation with new data.

CHAPTER 2 Using Regression Analysis 39

T A B L E 2-4

Regression Analysis 1993“1997

A B C D E F G

4 Year Sales Adjusted Costs

5 1993 $1,123,600 $965,043

6 1994 $1,191,016 $1,012,745

7 1995 $1,262,477 $1,072,633

8 1996 $1,338,226 $1,122,714

9 1997 $1,415,000 $1,199,000

11 SUMMARY OUTPUT

13 Regression Statistics

14 Multiple R 99.79%

15 R square 99.58%

16 Adjusted R square 99.44%

17 Standard error 6,840

18 Observations 5

20 ANOVA

21 df SS MS F Signi¬cance F

22 Regression 1 3.35E 10 3.35E 10 716 1.15E 04

23 Residual 3 1.40E 08 4.68E 07

24 Total 4 3.36E 10

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

27 Intercept [1] 71,252 37,624 1.89 0.15 (48,485) 190,989

28 Sales [2] 0.79 0.03 26.75 0.00 0.70 0.89

Regression Plot

[1] This is the regression estimate of ¬xed costs $1,250,000

[2] This is the regression estimate of variable costs

$1,200,000 y = 0.7924x + 71252

R2 = 0.9958

$1,150,000

$1,100,000

$1,050,000

Adj. Costs

$1,000,000

$950,000

$900,000

$850,000

$800,000

$1,100,000 $1,200,000 $1,300,000 $1,400,000 $1,500,000

Sales

PART 1 Forecasting Cash Flows

40

99.44% (B16), still indicating an excellent relationship. We do see a dif-

ference in the t-statistics for the two regressions.

The t-statistic for the intercept is now 1.89 (D27), indicating it is no

longer signi¬cant at the 95% level, whereas it was 3.82 in Table 2-1B.

Another effect of fewer data is that the 95% con¬dence interval for the

intercept value is $48,485 (F27) to $190,989 (G27), a range of $239,475.

In addition, the t-statistic for the slope coef¬cient, while still signi¬cant,

has fallen from 56.94 (Table 2-1B, D34) to 26.75 (D28). The 95% con¬dence

interval for the slope now becomes $0.70 (F28) to $0.89 (G28), a range

that is 31„2 times greater than that in Table 2-1B and indicates much more

uncertainty in the variable cost than we obtain using 10 years of data.

The standard error of the Y-estimate, however, decreases from

$16,014 (Table 2-1B, B23) to $6,840. This indicates that decreasing the

number of data points improves the Y-estimate, an opposite result from

all of the preceding. Why?

Earlier, we pointed out that using only a small range for the inde-

pendent variable leads to a small denominator in the variance of b, i.e.,

2

n

x2

i

i1

which leads to larger con¬dence intervals. However, larger data sets (us-

ing more years of data) tend to lead to a larger standard error of the y-

estimate, s. As we mentioned earlier,

n

1 ˆ

Yi)2

s (Yi

n 2 i1

ˆ

where Yi are the forecast (regression ¬tted) costs, Yi are the historical

costs, and n is the number of observations.14 Thus, we often have a trade-

off in deciding how many years of data to include in the regression. More

years of data leads to better con¬dence intervals, but fewer years may

lead to smaller standard errors of the y-estimate.

Table 2-4 was constructed to demonstrate that you should evaluate

all of the regression statistics carefully to determine if the relationship is

suf¬ciently strong to merit using it and which data set is best to use.

Simply looking at the adjusted R 2 value is insuf¬cient; all the regression

statistics should be evaluated in their entirety, as an improvement in one

may be counterbalanced by a deterioration in another. Therefore, it is best

to test different data sets and compare all of the regression statistics to

select the regression equation that represents the best overall relationship

between the variables.

14. We divide by n 2 instead of n because it takes two points to determine a line. If we only had

two years of historical data, we could determine a regression line, but we would know

absolutely nothing about the variance around the line. It takes a minimum of three years of

data to be able to say anything at all about how well the regression line ¬ts the data, and

three years is usually insuf¬cient. It is much better to have at least ¬ve years of data,

though four years can often suf¬ce.

CHAPTER 2 Using Regression Analysis 41

PROBLEMS WITH USING REGRESSION ANALYSIS FOR

FORECASTING COSTS

Although regression analysis is a powerful tool, its blind application can

lead to serious errors. Various problems can be encountered, and one

should be cognizant of the limitations of this technique. Aside from the

obvious problems of poor ¬t and insuf¬cient data, structural changes in

the company can also invalidate the historical relationship of sales and

costs.

Insuf¬cient Data

Insuf¬cient data leads to increased error in the regression, which in turn

will lead to increased error in the forecast data. As mentioned previously,

to optimize the regression equation it is best to examine overlapping data

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

16 Average 41,260,912 5,123,149 17.0251 14.9651 13.0057 0.0852 8.1004

17 Standard deviation 44,558,275 5,233,919 1.1212 1.0814 4.8135 0.0252 1.9954

20 SUMMARY OUTPUT

22 Regression Statistics

23 Multiple R 0.997820486

24 R square 0.995645723

25 Adjusted R square 0.994557153

26 Standard error 0.082720079

27 Observations 11

29 ANOVA

30 df SS MS F Signi¬cance F

31 Regression 2 12.51701011 6.258505055 914.6369206 3.59471E-10

32 Residual 8 0.054740892 0.006842611

33 Total 10 12.571751

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

Error 95.0%

36 Intercept 3.430881701 0.390158993 8.79354767 2.19714E-05 2.531172869 4.330590533 2.531172869 4.330590533

37 ln NI 0.957081978 0.024655341 38.81844378 2.13125E-10 0.900226622 1.013937333 0.900226622 1.013937333

38 1/g 0.056021702 0.005538834 10.11434967 7.79687E-06 0.068794284 0.04324912 0.068794284 0.04324912

47

48

T A B L E 2-6 (continued)

Regression Analysis of Guideline Companies

A B C D E F G H I

40 Valuation

41 NI 100,000 200,000 300,000 400,000 500,000 1,000,000

42 In NI 11.5129 12.2061 12.6115 12.8992 13.1224 13.8155

43 X coef¬cient-NI 0.957081978 0.957081978 0.957081978 0.957081978 0.957081978 0.957081978

44 In NI X coef¬cient 11.01881347 11.68221215 12.07027549 12.34561082 12.55917749 13.22257672

45 g 0.05 0.055 0.06 0.065 0.07 0.075

46 1/g 20 18.18181818 16.66666667 15.38461538 14.28571429 13.33333333

47 X coef¬cient-1 / g 0.056021702 0.056021702 0.056021702 0.056021702 0.056021702 0.0560217

48 1/g X coef¬cient 1.120434033 1.018576394 0.933695028 0.861872333 0.800310024 0.746956022

49 Add intercept 3.430881701 3.430881701 3.43088176 3.430881701 3.430881701 3.430881702

50 Total ln FMV 13.329261101 14.09451745 14.56746217 14.91462019 15.18974917 15.90650185

51 FMV $614,928 $1,321,816 $2,121,136 $3,001,492 $3,952,067 $8,092,934

52 PE Ratio 6.149284138 6.609082291 7.070452024 7.50373099 7.904133036 8.09293361

54 95% Con¬dence Intervals

55 2 Standard errors 0.165440158

e2 Std Err

56 1.179912352

e-2 Std Err

57 0.847520579

First we will brie¬‚y describe the regression results for the regression

of FMV against net income. The regression yields an adjusted R 2 of 94.6%

and a t-statistic for the x-coef¬cient of 12.4, which seems to indicate a

successful regression. The regression equation obtained for the complete

data set is:

FMV $1,272,335 (8.3 Net Income)

If we were to use to value a ¬rm with net income of $100,000, the re-

gression would produce a value of $442,000. Something is wrong!

The problem is that the full regression equation is:

FMV a b Net Income ui (2-10)

where ui is an error term, assumed to be normally distributed with an

expected value of zero. Our speci¬c regression equation is:

$1,272,335 (8.3 Net Income) ui (2-11)

The problem is that this error term is additive and likely to be cor-

related to the size of the ¬rm. When that occurs, we have a problem called

˜˜heteroscedasticity.™™

There are two possible solutions to the problem. The ¬rst is to use

weighted least squares (WLS) instead of ordinary least squares regression.

In WLS, we weight the extreme values less than the more mainstream

values. This usually will not produce a usable solution for a privately

held ¬rm that is much smaller than the publicly traded guideline com-

panies.

The second possible solution is to use a log“log speci¬cation. In do-

ing so, we regress the natural logarithm of market capitalization as a

function of the natural logarithm of net income. Its form is:

ln FMVi a bi ln NI ui, i guideline company 1, 2, 3, . . . n

(2-12)

When we take antilogs, the original equation is:

A NIib vi

FMVi (2-13)

e a, vi e ui is Euler™s constant, and the expected value of

where A

vi 1.

In equation (2-13), the regression equation x-coef¬cient, bi, from equa-

tion (2-12) for net income thus becomes an exponent to net income. If

b 1, then size has no scaling effect on the FMV, and we would expect

price earnings ratios to be uncorrelated to size, all other things being

constant. If b 1, then the price earnings multiple should rise with net

income, and the opposite is true of b 1. Relating this to the log size

model in Chapter 4, we would thus expect to ¬nd b 1 because over

long periods of time large ¬rms have lower discount rates than small

¬rms, which means larger values relative to earnings.

Using equation (2-13), consider two identical errors of 20% for ¬rms

i and j, where ¬rm i has net income of $100,000 and ¬rm j has net income

of $200,000. In other words, the error terms vi and vj are both 1.2.19 For

19. This means the error terms ui and uj in equation (2-12) are equal to ln (1.2) 0.182.

CHAPTER 2 Using Regression Analysis 49

simplicity, suppose that b 1 for both ¬rms. The same statistical error in

the log of the fair market value of both ¬rms produces an error in fair

market value that is twice as large in ¬rm j as in ¬rm i. This is a desirable

property, as it corresponds to our intuition that large ¬rms will tend to

have larger absolute deviations from the regression determined values.

Thus, this form of regression is likely to be more successful than equation

(2-10) for valuing small ¬rms.

Equation (2-10) is probably ¬ne for valuing ¬rms of the same size as

the guideline companies. When we apply equation (2“10) to various lev-

els of net income, we ¬nd the forecast FMVs are $442,000, $0 (rounded),

$2.9 million, and $7.0 million for net incomes of $100,000, $154,000,

$500,000, and $1 million. Obviously equation (3-10) works poorly at the

low end. We would also have a similar, but opposite, scaling problem

forecasting value for a ¬rm with net income of $5 billion. The additive

error term restricts the applicability of equation (2-10) to subject compa-

nies of similar size to the guideline companies.

There is an important possible enhancement to the regression equa-

tion, and that is the introduction of forecast growth as an independent

variable. The emergence of the Internet makes it easier to obtain growth

forecasts, although frequently there are no such estimates for smaller pub-

licly traded ¬rms.

For a ¬rm with constant forecast growth, a midyear Gordon model

is its proper valuation equation.

1 r

FMV CFt (2-14)

1

r g

In Chapter 4, we show that New York Stock Exchange returns are nega-

tively related to the natural logarithm of market capitalization (which can

also be referred to as fair market value or size), which means that there

is a nonlinear relationship between return and size. Therefore, the dis-

count rate, r, in equation (2-14) impounds a nonlinear size effect. To the

extent that there is a nonlinear size effect in equation (2-13), we should

hopefully pick that up in the b coef¬cient.

Note that in equation (2-14) there is a growth term, g, which appears

in the denominator of the Gordon model multiple. Thus, it is reasonable

to try 1/g as an additional independent variable in equation (2-13).

Continuing our description of Table 2-6, Column C is net income and

Columns D and E are the natural logarithms of FMV and net income.

These are actual data from a real valuation. Column G shows a growth

rate, and it is not actual data (which were unavailable). Column F is the

inverse of Column G, i.e., 1/g. Thus, Column D is our dependent variable

and Columns E and F are our independent variables.20

Adjusted R 2 is 99.5% (B25), an excellent result. The standard error of

the y-estimate is 0.08272 (B26). The y-intercept is 3.43 (B36) and the x-

coef¬cients for ln NI and 1/g are 0.95708 and “0.05602 (B37, B38), re-

spectively.

20. Electronic spreadsheets require that the independent variables be in contiguous columns.

PART 1 Forecasting Cash Flows

50

On page 2 of Table 2-6, we show valuations for subject companies

with differing levels of net income and expected growth. Row 41 shows

¬rms with net incomes ranging from $100,000 to $1 million. Row 42 is

the natural log of net income.21 We multiply that by the x-coef¬cient for

net income in Row 43, which produces a subtotal in Row 44.

Row 45 contains our forecast of constant growth for the various sub-

ject companies. We are assuming growth of 5% per year for the $100,000

net income ¬rm in Column B, and we increase the growth estimate by

0.5% for each ¬rm. Row 46 is one divided by forecast growth.

In Row 47 we repeat the x-coef¬cient for 1/g from the regression,

and we multiply Row 46 Row 47 Row 48, which is another subtotal.

In Row 49 we repeat the y-intercept from the regression. In Row 50

we add Rows 44, 48, and 49, which is the natural logarithm of the forecast

FMV (at the marketable minority interest level). We must then exponen-

tiate that result, i.e., take the antilog. The Excel formula for B51 is

EXP(B50).22 Finally, we calculate the P/E ratio in Row 52 as Row 51 di-

vided by Row 41.

The P/E ratio rises because of the increase in the forecast growth rate

across the columns. If all cells in Row 45 were equal to 0.05, then the PE

ratios in Row 52 would actually decline going to the right across the

columns. The reason for this is that the x-coef¬cient for ln NI is 0.95708

(page 1, B37) 1. This is contrary to our expectations. If B38 were greater

than 1, then P/E ratios would rise with ¬rm size, holding forecast growth

constant. Does this disprove the log size model? No. While all the rest of

the data are real, these growth rates are not actual. They are made up.

Also, one small sample of one industry at one point in time does not

generalize to all ¬rms at all times.

In the absence of the made-up growth rates, the actual regression

yielded an adjusted R 2 of 93.3% and a standard error of 0.2896 (not

shown).

95% Con¬dence Intervals

We multiply the standard error in B26 by 2 0.16544 (B55). To convert

the standard error of ln FMV to the standard error of FMV, we have to

exponentiate the two standard errors. In B56 we raise e, Euler™s constant,

to the power of B55. Thus, e0.16544 1.1799, which means the high side of

our 95% con¬dence interval is 18% higher than our estimate.23 To calcu-

late the low side of our 95% con¬dence interval, we raise e to the power

of two standard errors below the regression estimate. Thus B57 e 0.16544

0.8475, which is approximately 15% below the regression estimate.

Thus our 95% con¬dence interval is the regression estimate 18% and

15%. Using only the actual data that were available at the time, the

same regression without 1/g yielded con¬dence intervals of the regres-

21. The Excel formula for cell B42, for example, is ln(B41). The Lotus 123 formula would be

@ln(B41).

22. In Lotus 123 the formula would be @exp(B50)

23. The Excel formula for cell B56 is EXP(B55) and the Lotus 123 formula is @EXP(B55).

Similarly, the Excel formula for B57 is EXP( B55), and the Lotus 123 formula is

@EXP( B55).

CHAPTER 2 Using Regression Analysis 51

sion estimate 78% and 56%. Obviously, growth can make a huge dif-

ference. Also, without growth, the x-coef¬cient for ln NI was slightly

above one, indicating increasing P/E multiples with size.

SUMMARY

Regression analysis is a powerful tool for use in forecasting future costs,

expenses, and sales and estimating fair market value. We should take care

in evaluating and selecting the input data, however, to arrive at mean-

ingful answer. Similarly, we should carefully scrutinize the regression out-

put to determine the signi¬cance of the variables and the amount of error

in the Y-estimate to determine if the overall relationship is meaningful.

BIBLIOGRAPHY

Bhattacharyya, Gouri K., and Richard A. Johnson. 1977. Statistical Concepts and Methods.

New York: John Wiley & Sons.

Pratt, Shannon P., Robert F. Reilly, and Robert P. Schweihs. 1996. Valuing a Business: The

Analysis and Appraisal of Closely Held Companies; 3d ed. New York: McGraw-Hill.

Wonnacott, Thomas H., and Ronald J. Wonnacott. 1981. Regression: A Second Course in

Statistics. New York: John Wiley & Sons.

PART 1 Forecasting Cash Flows

52

APPENDIX

The ANOVA table (Rows 28“32)

We have already discussed the importance of variance in regression anal-

ysis. The center section of Table A2-1, which is an extension of Table

2-1B, contains an analysis of variance (ANOVA) automatically generated

by the spreadsheet. We calculate the components of ANOVA in the top

portion of the table to ˜˜open up the black box™™ and show the reader

where the numbers come from.

First, we calculate the regression estimate of adjusted costs in Col-

umn D using the regression equation:

Costs $56,770 (0.80 Sales) (B35, B36)

Next, we subtract the average actual adjusted cost of $852,420 (C18) from

the calculated costs in Column D to arrive at the deviation from the mean

in Column E. Note that the sum of the deviations is zero in cell E17, as

expected.

In Column F we square each deviation term in Column E and total

them in F17. The total, 831,414,202,481, is known as the sum of squares

and measures the amount of variation explained by the regression. In the

absence of a regression, our best estimate of costs for any year during the

1988“1997 period would be Y, the mean costs. Therefore, the difference

between the historical mean and the regression estimate (Column E) is

the absolute deviation explained by the regression. The square of that

(Column F) is the variance explained by the regression. This term appears

in the ANOVA table in C30 under SS (sum of squares).

The next term to the right in the ANOVA table is the mean squared

error (MS), which measures the variance explained by the regression. In

our case, the number is identical to the SS term (D30 C30). This occurs

because we have only one independent variable, sales, and thus one de-

gree of freedom (B30) in the regression.

In Column G we calculate the difference between the each actual cost

and the calculated cost (the regression estimate) by subtracting the values

in Column D from Column C. Again, the sum of the deviations is zero.

We square the deviations and sum them to arrive at a value of

2,051,637,107 (H17). This second sum of squares, which appears in the

ANOVA table in cell C31, is the unexplained variation. We calculate the

corresponding mean square error term in Column I by dividing the val-

ues in Column H by 8, the number of degrees of freedom (B30). The sum

is 256,454,638 (I17), which appears in the ANOVA table in D31. This num-

ber represents the unexplained variance. Finally, we calculate the F-

statistic of 3,241 (E30) by dividing the explained variance (D30) by the

unexplained variance (D31).

The explained variation plus the unexplained equals the total vari-

ation. The correlation coef¬cient is

Explained Variation of Y

R2

Total Variation of Y

In our case, the explained variation (C30) divided by the total variation

(C32) is equal to 99.75%, as seen in B23.

CHAPTER 2 Using Regression Analysis 53

54

T A B L E A2-1

Regression Analysis 1988“1997

A B C D E F G H I

4 Actual Calculated Deviation of Sum of Squares Deviation of Deviation from Mean

Calc. [5] Actual Actual Square [7]

ˆ

5 Costs Y [3] from Mean from Calc. Squared [5]

[4] [6]

ˆ ˆ ˆ ˆ

Y)2 Y)2 Y)2/8

6 Year Sales X [1] Adj. Costs Y [2] Y Y (Y Y Y (Y (Y

7 1988 $250,000 $242,015 $257,889 $594,532 353,467,822,773.69 $15,874 251,983,658 31,497,957

8 1989 $500,000 $458,916 $459,007 -$393,413 154,773,949,895.09 $92 8,399 1,050

9 1990 $750,000 $696,461 $660,126 $192,295 36,977,294,181.16 $36,336 1,320,285,654 165,035,707

10 1991 $1,000,000 $863,159 $861,244 $8,824 77,855,631.91 $1,915 3,668,783 458,598

11 1992 $1,060,000 $891,517 $909,512 $57,092 3,259,496,294.19 -$17,995 323,821,415 40,477,677

12 1993 $1,123,600 $965,043 $960,677 $108,257 11,719,473,702.15 $4,366 19,064,659 2,383,082

13 1994 $1,191,016 $1,012,745 $1,014,911 $162,491 26,403,295,435.15 $2,166 4,691,209 586,401

14 1995 $1,262,477 $1,072,633 $1,072,400 $219,979 48,390,920,118.80 $233 54,240 6,780

15 1996 $1,338,226 $1,122,714 $1,133,338 $280,917 78,914,430,752.87 $10,623 112,853,095 14,106,637

16 1997 $1,415,000 $1,199,000 $1,195,101 $342,680 117,429,663,696.32 $3,899 15,205,993 1,900,749

17 Total $0 831,414,202,481 $0 2,051,637,107 256,454,638

18 $852,420 Average Actual Adjusted Costs (Y)

19 SUMMARY OUTPUT

21 Regression Statistics

22 Multiple R 0.998768455

23 R square 0.997538427

24 Adjusted R square 0.99723073

25 Standard error 16014.20115

26 Observations 10

28 ANOVA

29 df SS MS F Signi¬cance F

30 Regression 1 8.31414E 11 8.31414E 11 3241.954241 1.00493E-11

31 Residual 8 2051637107 256454638.3

32 Total 9 8.33466E 11

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

35 Intercept 56770.40117 14863.25124 3.819514334 0.005093239 22495.66018 91045.14216 22495.66018 91045.14216

36 Sales X [1] 0.804473578 0.0141289 56.93816156 1.00493E-11 0.771892255 0.8370549 0.771892255 0.8370549

[a] This sheet is an extension of Table 2-1B.

[1] from Table 2-1A, Row 7

[2] from Table 2-1A, Row 27

[3] Calculated costs using Costs 0.80 Sales $56,806 with sales ¬gures in Column B

ˆ

[4] Deviation of calculated costs from average actual costs (Column D C17) Y Y

[5] Deviations squared

[6] Deviation of actual costs from calculated costs (Column C Column D)

[7] Deviations squared / 8 (degrees of freedom)

[8] Regression estimate of ¬xed costs

[9] Regression estimate of variable costs

55

CHAPTER 3

Annuity Discount Factors and

the Gordon Model

INTRODUCTION

De¬nitions

Denoting Time

ADF WITH END-OF-YEAR CASH FLOWS

Behavior of the ADF with Growth

Special Case of ADF when g 0: The Ordinary Annuity

Special Case when n ’ and r g: The Gordon Model

Intuitively Understanding Equations (3-6) and (3-6a)

Relationship between the ADF and the Gordon Model

Table 3-1: Proof of ADF Equations (3-6) through (3-6b)

A Brief Summary

MIDYEAR CASH FLOWS

Table 3-2: Example of Equations (3-10) through (3-10b)

Special Cases for Midyear Cash Flows: No Growth, g 0

Gordon Model

STARTING PERIODS OTHER THAN YEAR 1

End-of-Year Formulas

Valuation Date 0

Table 3-3: Example of Equation (3-11)

Tables 3-4 through 3-6: Variations of Table 3-3 with S 0, Negative

Growth, and r g

Special Case: No Growth, g 0

Generalized Gordon Model

Midyear Formula

PERIODIC PERPETUITY FACTORS (PPFs): PERPETUITIES FOR

PERIODIC CASH FLOWS

The Mathematical Formulas

Tables 3-7 and 3-8: Examples of Equations (3-18) and (3-19)

Other Starting Years

New versus Used Equipment Decisions

57

Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

ADFs IN LOAN MATHEMATICS

Calculating Loan Payments

Present Value of a Loan

Table 3-10: Example of Equation (3-23)

RELATIONSHIP OF THE GORDON MODEL TO THE

PRICE/EARNINGS RATIO

De¬nitions

Mathematical Derivation

CONCLUSIONS

PART 1 Forecasting Cash Flows

58

INTRODUCTION

This chapter describes the derivation of annuity discount factors (ADFs)

and the Gordon model (Gordon and Shapiro 1956).1 The ADF is the pres-

ent value of a ¬nite stream of cash ¬‚ows (CF) with constant or zero

growth, assuming the ¬rst cash ¬‚ow $1.00. Thus, the actual ¬rst year™s

cash ¬‚ow times the ADF is the present value as of time zero of the stream

of cash ¬‚ows from years 1 to n. Growth rates in cash ¬‚ows may be pos-

itive, zero, or negative, the latter being a decline in cash ¬‚ows.

The Gordon model is identical to the ADF, except that it produces

the present value of a perpetuity for each $1.00 of initial cash ¬‚ow. The

resulting present value is known as the Gordon model multiple. When

using the Gordon model multiple, the discount rate must be larger than

the constant growth rate, which is not true of the ADF.

There are several varieties of ADFs, depending on whether the cash

¬‚ows:

— Are constant or grow/decline.

— Occur midyear or at the end of the year.

— Begin in the ¬rst year or at some other time.

— Occur every year or at regular, skipped intervals.

— Finish on a whole year or a fractional year.

This chapter begins with the derivation of the ADF and later shows

that the Gordon model, which is the present value of a perpetual annuity

with constant growth, is simply a special case of the ADF. We will dem-

onstrate that an ADF is actually the difference of two perpetuities.

There are several uses of ADFs, including:

— Calculating the present value of annuities. This application has

become far more important since the quantitative marketability

discount model (Mercer 1997) requires an ADF with growth (see

Chapter 8). While Mercer™s book has an approximation of the

ADF (at 276) that appears to be fairly accurate, this chapter

contains the exact formulas.

— Valuing periodic cash ¬‚ows such as moving expenses, losses

from lawsuits, etc. This requires a specialized ADF called a

periodic perpetuity factor (PPF), which we develop later in the

chapter. Additionally, PPFs are useful for decisions in buying

new versus used income-producing equipment (such as CAT

scans, ships, or taxicabs) and for calculating the value of used

equipment.

— Calculating loan payments.

— Calculating loan principal amortization.

— Calculating the present value of a loan. This is important in

calculating the correct selling price of a business, as seller

¬nancing typically takes place at less-than-market rates. The

present value of a loan is also important in ESOP valuation.

1. Gordon and Shapiro were preceded by Williams (1938). See also Gordon (1962).

CHAPTER 3 Annuity Discount Factors and the Gordon Model 59

At ¬rst glance this chapter appears mathematically very intensive

and daunting in its use of geometric sequences. However, because the

primary concepts appear in equations (3-1) through (3-9), once you un-

derstand those equations, the remainder are merely special cases or slight

variations on the original theme and can easily be comprehended. While

the formulas look complex, we decompose them into units that behave

as modular building blocks, each of which has an intuitive explanation.

You will bene¬t from understanding the math in the body of the chapter,

as this material is useful in several areas of business valuation. Addition-

ally, you will also gain a much better understanding of the Gordon model,

which appraisers often use in discounted future net income or discounted

cash ¬‚ow valuation.

ADFs are an area that many practitioners ¬nd dif¬cult, leading to

many mistakes. Timing errors in ADFs frequently result from the fact that

the guideline company method uses the most recent historical earnings

for calculating P/E multiples, whereas the Gordon model uses the ¬rst

future period (forecast) cash ¬‚ow as its earnings base. Many practitioners

confuse the two and use historical rather than forecast earnings as their

base in a discounted cash ¬‚ow or discounted future net income approach.

Another common error is the use of end-of-year multiples when midyear

Gordon model multiples are appropriate.

The ADF formulas given within the chapter apply only to cash ¬‚ow

streams that have a whole number of years associated with them. If the

cash ¬‚ow stream ends in a fractional year, you should use the formulas

in the appendix for ADFs with stub periods.

Unless otherwise speci¬ed, all ADF formulas are for cash ¬‚ows with

constant growth. At speci¬c points in the chapter, we make the simpli-

fying assumption that growth is zero and clearly state when that is the

case. Otherwise the reader may assume growth is constant and non-zero.

De¬nitions

Let us initially consider an ADF with constant growth in cash ¬‚ows,

where the last cash ¬‚ow occurs in period n. We will use the following

de¬nitions:

r discount rate

g annual growth rate in cash ¬‚ows

ADF annuity discount factor

PV present value

CF cash ¬‚ow

LHS left-hand side of the equation

RHS right-hand side of the equation

n terminal year of the cash ¬‚ows

t time (which can refer to a point in time or a year)

Denoting Time

Timing is frequently a source of confusion. Time t denotes the time period

under discussion. It generally refers to a speci¬c year.2 Time t refers to

2. In the context of loan amortization, periods are usually months.

PART 1 Forecasting Cash Flows

60

the entire year, except for two contexts that we discuss in the paragraph

below. Thus, time t is a span of time, not a point in time.

There are two contexts in which time t means a point in time. The

¬rst occurs with the statement t 0, which means the beginning of the

period t 1, i.e., usually the beginning of the ¬rst year of cash ¬‚ows.

For example, if t 1 represents the calendar year 2000, then t 0 means

January 1, 2000, the ¬rst day of t 1. Usually, but not always, t 0 is

the valuation date. The other context in which t means a point in time is

when we specify either the beginning, midpoint, or end of t.

In business valuation, we generally assume that cash ¬‚ows occur

approximately evenly throughout time t. In present value terms, that is

equivalent to assuming they occur at the midpoint of time t. Occasionally

it is appropriate to assume that cash ¬‚ows occur at the end of the year,

which can be the case with annuities, royalties, etc. The former is com-

monly known as the midyear assumption, while the latter is known as the

end-of-year (or end year) assumption.

Another important concept related to time that can be confusing is

the valuation date, the point in time to which we discount the cash ¬‚ows.

The valuation date is rarely the same as the ¬rst cash ¬‚ow. The most

common valuation date in this chapter is as of time zero, i.e., t 0. The

cash ¬‚ows usually, but not always, either begin during Year 1 or occur at

the end of Year 1.

ADF WITH END-OF-YEAR CASH FLOWS

The ADF is the present value of a series of cash ¬‚ows over n years with

constant growth, beginning with $1 of cash ¬‚ow in Year 1. We multiply

by the ¬rst year™s forecast cash ¬‚ow by the ADF to arrive at the PV of

the cash ¬‚ow stream. For example, if the ADF is 9.367 and the ¬rst year™s

cash ¬‚ow is $10,000, then the PV of the annuity is 9.367 $10,000

$93,670.

We begin the calculation of the ADF by de¬ning the cash ¬‚ows and

discounting them to their present value. Initially, for simplicity, we as-

sume end-of-year cash ¬‚ows. The PV of an annuity of $1, paid at the end

of the year for each of n years, is:

g)n 1

$1 (1 g) $1 (1

$1

PV (3-1)

r)1 r)2 r)n

(1 (1 (1

Factoring out the $1:

g)n 1

(1 g) (1

1

PV $1 (3-1a)

r)1 r)2 r)n

(1 (1 (1

The ADF is the PV of the constant growth cash ¬‚ows per $1 of starting

year cash ¬‚ow. Dividing both sides of equation (3-1a) by $1, the left-hand

side becomes PV/$1, which equals the ADF. Thus, equation (3-1a) sim-

pli¬es to:

g)n 1

(1 g) (1

1

ADF (3-1b)

r)1 r)2 r)n

(1 (1 (1

The numerators in equation (3-1b) are the forecast cash ¬‚ows them-

CHAPTER 3 Annuity Discount Factors and the Gordon Model 61

selves, and the denominators are the present value factors for each cash

¬‚ow. As mentioned previously, the ¬rst year™s cash ¬‚ow in an ADF cal-

culation is always de¬ned as $1. With constant growth in cash ¬‚ow, each

successive year is (1 g) times the previous year™s cash ¬‚ow, which

g)n 1. The cash ¬‚ow is not

means that the cash ¬‚ow in period n is (1

(1 g)n, because the ¬rst year™s cash ¬‚ow is $1.00, not 1 g. For example,

if g 10%, the ¬rst year™s cash ¬‚ow is, by de¬nition, $1.00. The second

year™s cash ¬‚ow is 1.1 $1.00 $1.10. The third year™s cash ¬‚ow is 1.1

2

$1.00 1.21. The fourth year™s cash ¬‚ow is 1.13 $1.00

$1.10 1.1

$1.331, etc. The denominators in equation (3-1b) discount the cash

¬‚ows in the numerator to their present value.

Next, we begin a series of algebraic manipulations which will ulti-

mately enable us to solve for the ADF and specify it in a formula. Mul-

tiplying equation (3-1b) by (1 g)/(1 r), we get:

g)n 1 g)n

(1 g) (1 g) (1 (1

ADF (3-2)

r)2 r)n 1

(1 r) (1 (1 r)n (1

Notice that most of the terms in equation (3-2) are identical to equation

(3-1b). We next subtract equation (3-2) from equation (3-1b). All of the

terms in the middle of the equation are identical and thus drop out. The

only terms that remain on the RHS after the subtraction are the ¬rst term

on the RHS of equation (3-1b) and the last term on the RHS of equation

(3-2).

g)n

1 g (1

1

ADF ADF (3-3)

r)n 1

1 r 1 r (1

Next, we wish to simplify only the left-hand side of equation (3-3):

1 g 1 g

ADF ADF ADF 1 (3-3a)

1 r 1 r

Multiplying the 1 in the square brackets on the RHS of the equation

by (1 r)/(1 r), we get:

1 g 1 g

1 r

ADF 1 ADF

1 r 1 r 1 r

(1 r) (1 g) r g

ADF ADF (3-3b)

1 r 1 r

Substituting the last expression of equation (3-3b) into the left-hand

side of equation (3-3), we get:

g)n

(r g) (1

1

ADF (3-4)

r)n 1

(1 r) (1 r) (1

Multiplying both sides of the equation by (1 r)/(r g), we obtain:

g)n

(1

(1 r) 1

ADF (3-5)

r)n 1

(r g) (1 r) (1

After canceling out the (1 r), this simpli¬es to:

PART 1 Forecasting Cash Flows

62

n

1 g

1 1

ADF (3-6)

r g 1 r r g

ADF with growth and end-of-year cash ¬‚ows

There are three alternative ways to regroup the terms in equation

(3-6) that will prove useful, which we label as equations (3-6a), (3-6b),

and (3-6c). In the ¬rst alternative expression for equation (3-6), we split

up the ¬rst term in the square brackets into two separate terms, placing

the denominator at the far right.

1 1 1

g)n

ADF (1

r)n

r g r g (1 (3-6a)

first alternative expression for (3-6)

We derive the second alternative expression by simply factoring out

the 1/(r g) from equation (3-6) and restate the equation as equation

(3-6b). It has the advantage of being more compact than equation (3-6).

n

1 g

1

ADF 1

r g 1 r (3-6b)

second alternative expression for (3-6)

After we develop some additional results, we will be able to explain

equations (3-6) through (3-6b) intuitively. In the meantime, we will make

some substitutions in equation (3-6b) that will greatly simplify its form

and eventually make the ADF much more intuitive.

Note that the ¬rst term on the right-hand-side of equation (3-6b) is

the classical Gordon model multiple, 1/(r g). Let™s denote it GM. The

next substitution that will simplify the expression is to let x (1 g)/

(1 r). Then we can restate equation (3-6b) as:

xn)

ADF GM (1 third alternative expression for (3-6) (3-6c)

Behavior of the ADF with Growth

The ADF is inversely related to r and directly related to g, i.e., an increase

in the discount rate decreases the ADF and vice-versa, while an increase

in the growth rate causes an increase in the ADF, and vice-versa.

Special Case of ADF when g 0: The Ordinary Annuity

When g 0, there is no growth in cash ¬‚ows, and equation (3-6) sim-

pli¬es to equation (3-6d), the formula for an ordinary annuity.

1

1

r)n

1 1 1 (1

ADF , or ADF (3-6d)

r)n r

r (1 r

1/r is the PV of a perpetuity that is constant in nominal dollars, or a

Gordon model with g 0.

CHAPTER 3 Annuity Discount Factors and the Gordon Model 63

Special Case when n ’ and r g: The Gordon Model

The Gordon model is a ¬nancial formula that every business appraiser

knows”at least in the end-of-year form. It is the formula necessary to

calculate the present value of the perpetuity with constant growth in cash

¬‚ows in the terminal period (also known as the residual or reversion

period), i.e., from years n 1 to in¬nity (after discounting the ¬rst n

years of cash ¬‚ows or net income). To be valid, the growth rate must be

less than the discount rate.

What few practitioners know, however, is that the Gordon model is

merely a special case of the ADF. The Gordon model contains two ad-

ditional assumptions that the ADF in equation (3-6) does not have.

— The time horizon is in¬nite, which means that we assume cash

¬‚ows will grow at the constant rate of g forever. This means that

n, the terminal year of the cash ¬‚ows, equals in¬nity.

— The discount rate is greater than the growth rate, i.e., r g.

Since r g,

n

1 g

1 r

goes to zero as n goes to in¬nity. Therefore, the entire term in square

brackets in equation (3-6) goes to zero, which simpli¬es to:

1

ADF Gordon model multiple, end-of-year cash flows (3-7)

r g

Equation (3-7) is the end-of-year Gordon model multiple. In other

words, the Gordon model multiple is just a special case of the ADF when

n equals in¬nity. Using this multiple, we obtain the Gordon model, with

end-of-year cash ¬‚ows:

CF

PV (3-8)

(r g)

Another way of expressing equation (3-8) is rewriting it as:

1

PV CF (3-9)

(r g)

Thus, the present value of a perpetuity with growth contains two terms

conceptually:

— CF, the starting year™s forecast cash ¬‚ow.3

— 1/(r g), the Gordon model multiple, which when multiplied

by the ¬rst year™s forecast cash ¬‚ow gives us the present value of

the perpetuity.

3. Note that you do not use historical cash ¬‚ow (or earnings).

PART 1 Forecasting Cash Flows

64

Intuitively Understanding Equations (3-6) and (3-6a)

Now that we understand the Gordon model, we can gain deeper insight

into equation (3-6). The ADF is the difference of two perpetuities. The

¬rst term, 1/(r g), is the PV as of t 0 of a perpetuity with cash ¬‚ows

going from t 1 to in¬nity. The second term is the PV as of t 0 of a

perpetuity going from t n 1 to in¬nity, which is explained in the

next paragraph. The difference of the two is the PV as of t 0 of the

annuity from t 1 to n.

g)n

Let™s give an intuitive explanation of equation (3-6a). The (1

is the forecast cash ¬‚ow4 for Year (n 1), which we then multiply by

1/(r g), our familiar Gordon model multiple. The result is the PV as

of t n of the forecast cash ¬‚ows from n 1 to in¬nity. Dividing by

n

(1 r) transforms the PV as of t n to the PV as of t 0.

Relationship between the ADF and the Gordon Model

The relationship between the ADF and Gordon model is so intimate that

we can derive the Gordon model from the ADF and vice-versa. The ADF

is the difference of two Gordon models, as illustrated graphically below

in Figure 3-1.

In graphical terms, the top line represents the Gordon model with

cash ¬‚ows from t 1 to in¬nity (our valuation date is actually time zero,

which is not shown on the graph). The cash ¬‚ows in the second Gordon

model begin at t n 1 and continue to in¬nity. The difference between

these two Gordon models is simply the ADF from t 1 to n.

F I G U R E 3-1

Timeline of the ADF and Gordon Model

Gordon 1’∞

Minus

Gordon n+1’∞

Equals

ADF 1’n

∞

1 n n+1

Table 3-1: Proof of ADF Equations (3-6) through (3-6b)

Table 3-1 is the valuation of a 10-year annuity, with a discount rate of

15% and an annual growth rate of 5.1%. All assumptions appear in cells

4. The ¬rst year™s cash ¬‚ow is 1, or (1 + g)0. The second year™s cash ¬‚ow is (1 + g)1. In general,

cash ¬‚ow in Year t (1 + g)t 1.

CHAPTER 3 Annuity Discount Factors and the Gordon Model 65

T A B L E 3-1

ADF: End-of-Year Formula

A B C D E F

g)t 1

4 t (Yrs) Cash Flow (CF) Growth in CF (1 PV Factor NPV

5 1 1.00000 0.00000 1.00000 0.86957 0.86957

6 2 1.05100 0.05100 1.05100 0.75614 0.79471

7 3 1.10460 0.05360 1.10460 0.65752 0.72629

8 4 1.16094 0.05633 1.16094 0.57175 0.66377

9 5 1.22014 0.05921 1.22014 0.49718 0.60663

10 6 1.28237 0.06223 1.28237 0.43233 0.55440

11 7 1.34777 0.06540 1.34777 0.37594 0.50668

12 8 1.41651 0.06874 1.41651 0.32690 0.46306

13 9 1.48875 0.07224 1.48875 0.28426 0.42320

14 10 1.56468 0.07593 1.56468 0.24718 0.38676

15 Totals 5.99506

17 Calculation of NPV by formulas:

18 Grand

19 Time 1 to In¬nity (n 1) to In¬nity 1 to n Total

20 NPV 10.10101 4.10595 5.99506 5.99506

22 Assumptions:

24 n Number of years of cash ¬‚ows 10

24 r Discount rate 15.0%

26 g Growth rate in net inc/cash ¬‚ow 5.1%

27 x (1 g)/(1 r) 0.9139

28 Gordon model multiple GM 1/(r g) 10.101010

30 Spreadsheet formulas:

32 B20: GM 1/(r g)

33 C20: GM*x n

34 D20 B20 C20

35 E20 GM * (1 x n) This is equation (3-6c)

0.9139 (F27).5 If

F24 to F28. Recall that we de¬ne x (1 g)/(1 r)

this were a perpetuity, the Gordon model multiple would be 10.101010

(F28).

We begin with a cash ¬‚ow of $1.00 at the end of Year 1 (B5). Column

C shows the annual growth in cash ¬‚ows at 5.1%.6 The cash ¬‚ow in

Column B is always equal to the previous cash ¬‚ow plus the growth in

the current period, where Cash Flowt Cash Flowt 1 Growtht. Column

D replicates the cash ¬‚ow in Column C using the formula Cash Flow

(1 g)t 1, which thus provides us with a general formula for the cash

¬‚ows. We multiply the cash ¬‚ows in Column C by the end-of-year present

value factor in Column E to arrive at the present value of the cash ¬‚ows

5. As mentioned in a previous footnote, we use i synonymously with r.

6. We can use the same formulas for other time periods, e.g., months instead of years. Then we

must use the monthly growth rate of 5.1%/12 0.4267% instead of the annual.

PART 1 Forecasting Cash Flows

66

in Column F. The sum of the present values of the 10 years of cash ¬‚ows

is 5.99506 in F15. This is the ˜˜brute force™™ method of calculating the an-

nuity.

As we will demonstrate, equation (3-6) is a more compact and ele-

gant solution. Cell B20 contains the end-of-year Gordon multiple results

of the ¬rst term in equation (3-6), which equals F28. This is the present

value of the perpetuity of $1.00 growing at a constant 5.1% from Year 1

to in¬nity. In C20 we subtract the present value of the perpetuity from

Year n 1 to in¬nity, which equals 4.10595 and is the term in equation

(3-6) in square brackets. The difference of the two perpetuities is 5.99506,

which equals F15, our brute force solution. Finally, E20 is the formula for

the entire equation, which equals the same 5.99506 calculated in D20 and

F15, proving the validity of equation (3-6), including its components. We

show the formulas for Row 20 at the bottom of Table 3-1. Note that the

formula in E20 is equation (3-6c).

A Brief Summary

To help you decide if you should read on, let™s take a look at what we

have covered so far, what we will cover in the remainder of the chapter,

and how dif¬cult the material will be. We have thus far derived the end-

of-year ADF, examined its special cases (the Gordon model and the no-

growth formula), explained the intimate relationship of the ADF and the

Gordon model, explained the intuition behind the components of the

ADF model, and proved the model with an example.

The reader now should understand the principles of ADFs and Gor-

don models. If you are having dif¬culty with the mathematics, you may

wish to skip to the sections on Periodic Perpetuity Factors (PPFs) and

Relationship of the Gordon Model to the Price/Earnings Ratio, which are

of practical signi¬cance to most readers. However, you now should un-

derstand almost everything you will need to easily comprehend the rest

of the chapter. The rest of the chapter is primarily simple variations of

the derivations we have done thus far.

In the remainder of the chapter, we will cover:

— The midyear version of the ADF (with the same special cases of

the Gordon model and g 0).

— Starting periods for the cash ¬‚ows that are different than Year 1,

which is of practical signi¬cance in discounted cash ¬‚ow analysis

in the calculation of the PV of the reversion.

— Calculating periodic perpetuity factors (PPFs), which are a

variation of the Gordon model for periodic expenses such as

moving expense and losses from lawsuits. Additionally, PPFs are

useful for decisions in buying new versus used income-

producing equipment (such as CAT scans, ships, or taxicabs) and

for calculating the value of used equipment.

— Calculating loan payments.

— Calculating the present value of loans.

CHAPTER 3 Annuity Discount Factors and the Gordon Model 67

— The relationship of the Gordon model to the PE multiple, the

misunderstanding of which may well be the single most

common source of technical error in business valuation.

MIDYEAR CASH FLOWS

Most businesses have cash ¬‚ows that more or less occur evenly through-

out the year. In a present value sense, this is approximately equivalent to

having all cash ¬‚ows occur midway through the year. Thus, in valuing

most businesses, it is appropriate to use midyear cash ¬‚ows rather than

end-of-year cash ¬‚ows.

Midyear cash ¬‚ows occur six months (one half-year) earlier than end-

of-year cash ¬‚ows. We derive this formula in exactly the same fashion as

equation (3-6). We start with equation (3-1b); however, the denominators,

which are the time periods by which we discount the cash ¬‚ows, are one

half-year less than those in equation (3-1b). We adjust for this difference

by multiplying every numerator by 1 r, which has the same effect

as reducing the denominators by 0.5 years. We then factor the 1 r

out of the sequence, resulting in a the midyear ADF that equals 1 r

times the end-of-year ADF.

n

1 r 1 r

1 g

ADF midyear ADF (3-10)

r g 1 r r g

We interpret equation (3-10) in exactly the same fashion as equation

(3-6). We can factor out the Gordon model multiple as before and restate

equation (3-10) as equations (3-10a) and (3-10b) below. Note that equa-

tions (3-10a) and (3-10b) are identical to equations (3-6b) and (3-6c), re-

spectively, except that the Gordon model multiple is midyear instead of

end-of-year.

n

1 r 1 g

ADF 1 alternative expression for (3-10)

r g 1 r

(3-10a)

n

ADF GM (1 x ) second alternative expression for (3-10) (3-10b)