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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

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