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22. Our standard error increased after incorporating the 1998 stock market results because it was
such a perverse year, with decile #1 performing fabulously and decile #10 losing. Thus, both
our results and Grabowski and King™s would be worse with 1998 included, and the relative
difference between the two would be less.

PART 2 Calculating Discount Rates
In conclusion, Grabowski and King™s (1996) work is very important
in that it demonstrates that other measures of size can serve as effective
proxies for our regression equation. It is noteworthy that the ¬ner break-
down into 25 portfolios versus Ibbotson™s 10 has a signi¬cant impact on
the reliability of the regression equation. Our 30-year results show a neg-
ative R 2 (Table 4-2, I13), while their R 2 was 93%.23 It did not seem to
improve the standard error of the y-estimate. Overall, our log size results
using 60-year data are superior to Grabowski and King™s results because
of the signi¬cantly smaller standard error of the y-estimate, which means
the 95% con¬dence intervals around the estimate are correspondingly
smaller using the 60 years of data.24
Grabowski and King™s (1999) work is even more important. It is the
¬rst ¬nding of the underlying variables for which size is a proxy. If Com-
pustat data went back to 1926, as do the CRSP data, then I would rec-
ommend abandoning log size entirely in favor of their variables. How-
ever, there are several reasons why I do not recommend abandoning log
1. Because the Compustat database begins in 1963, it misses 1926“
1962 data.25 Because of this, their R 2™s are lower and their
standard error of y-estimates are signi¬cantly higher than ours,
leading to larger con¬dence intervals.
2. Their sample universe consists of publicly traded ¬rms that are
all subject to Securities Exchange Commission scrutiny. There is
much greater uniformity of accounting treatment in the public
¬rms than in the private ¬rms to which professional appraisers
will be applying their results. This would greatly increase
con¬dence intervals around the valuation estimates.
3. The lower R 2™s of Grabowski and King™s results may mean that
size still proxies for other currently unknown variables or that
size itself has a pure effect on returns that must be accounted
for in an asset pricing model. Thus, log size is still important,
and Grabowski and King themselves said that was still the case.

Schwert and Seguin (1990) also found that stock market returns for small
¬rms are higher than predicted by CAPM by using a weighted least
squares estimation procedure. They suggest that the inability of beta to
correctly predict market returns for small stocks is partially due to het-
eroscedasticity in stock returns.
Heteroscedasticity is the term used to describe the statistical condi-
tion that the variance of the error term is not constant. The standard
assumption in an ordinary least squares (OLS) regression is that the errors

23. Again, the 1998 anomalous stock market results had a large impact on this measure. For the 30
years ending 1997, the R 2 was 53%.
24. Again, the difference would be less after including 1998 results.
25. While we have eliminated the ¬rst 12 or 13 years of stock market data”a choice that is
reasonable, but arguable”that still means the Grabowski and King results eliminate 1938“

CHAPTER 4 Discount Rates as a Function of Log Size 151
are normally distributed, have constant variance, and are independent of
the x-variable(s). When that is not true, it can bias the results. In the
simplest case of heteroscedasticity, the variance of the error term is line-
arly related to the independent variable. This means that observations
with the largest x-values are generating the largest errors and causing
bias to the results. Using weighted least squares (WLS) instead of OLS
will correct for that problem by weighting the largest observations the
In the case of CAPM, the regression is usually done in the form of
excess returns to the ¬rm as a function of excess returns in the market,
ˆ (Rm
or: (ri rF ) ˆ RF ). Here we are using the historical market
returns as our estimate of future returns. If everything works properly,
ˆ should be equal to zero. If there is heteroscedasticity, then when excess
market returns are high, the errors will tend to be high. That is what
Schwert and Seguin found.
Schwert and Seguin also discovered that after taking heteroscedas-
ticity into account, the relationship between ¬rm size and risk-adjusted
returns is stronger than previously reported. They also found that the
spread between the risk of small and large stocks was greater during
periods of heavier market volatility, e.g., 1929“1933.

Jacobs and Levy (1988) examined rates of return in 38 different industries
by including industry as a dummy variable in their regression analyses.
Only one industry (media) showed (excess) returns different from zero
1% level,26 which the authors speculate
that were signi¬cant at the p
was possibly related to the then recent wave of takeovers. The higher
returns to media would only be relevant to a subject company if it was
a serious candidate for a takeover.
There were seven industries where (excess) returns were different
from zero at the p 10% level, but this is not persuasive, as the usual
level for rejecting the null hypothesis that industry does not matter in
investor returns is p 5% or less. Thus, Jacobs and Levy™s results lead
to the general conclusion that industry does not matter in investor re-

Revenue Ruling 59-60 requires that we look at publicly traded stocks in
the same industry as the subject company. I claim that our excellent re-

26. This means that, given the data, there is only a 1% probability that the media industry returns
were the same as all other industries.
27. Jacobs and Levy also found an interest rate-sensitive ¬nancial sector. They also found that
macroeconomic events appear to explain some industry returns. Their example was that
precious metals was the most volatile industry and its returns were closely related to gold
prices. Thus, there may be some”but not many”exceptions to the general rule of industry

PART 2 Calculating Discount Rates
sults with the log size model28 combined with Jacobs and Levy™s general
¬nding of industry insigni¬cance satis¬es the intent of Revenue Ruling
59-60 for small and medium ¬rms without the need actually to perform
a publicly traded guideline company method. Some in our profession
may view this as heresy, but I stick to my guns on this point.
We repeat equation (3-28) from Chapter 3 to show the relationship
of the PE multiple to the Gordon model.

1 r
PE (1 g1)(1 b)
r g
relationship of the PE multiple to the Gordon model multiple


The PE multiple29 of a publicly traded ¬rm gives us information on
the one-year and long-run expected growth rates and the discount rate
of that ¬rm”and nothing else. The PE multiple only gives us a combined
relationship of r and g. In order to derive either r or g, we would have
to assume a value for the other variable or calculate it according to a
For example, suppose we use the log size model (or any other model)
to determine r. Then the only new information to come out of a guideline
public company method (GPCM) is the market™s estimate of g,30 the
growth rate of the public ¬rm. There are much easier and less expensive
ways to estimate g than to do a GPCM. When all the market research is
¬nished, the appraiser still must modify g to be appropriate for the subject
company, and its g is often quite different than the public companies™. So
the GPCM wastes much time and accomplishes little.
Because discount rates appropriate for the publicly traded ¬rms are
much lower than are appropriate for smaller, privately held ¬rms, using
public PE multiples will lead to gross overvaluations of small and me-
dium privately held ¬rms. This is true even after applying a discount,
which many appraisers do, typically in the 20“40% range”and rarely
with any empirical justi¬cation.
If the appraiser is set on using a GPCM, then he or she should use
regression analysis and include the logarithm of market capitalization as
an independent variable. This will control for size. In the absence of that,
it is critical to only use public guideline companies that are approximately
the same size as the subject company, which is rarely possible.
This does not mean that we should ignore privately held guideline
company transactions, as those are far more likely to be truly comparable.
Also, when valuing a very large privately held company, where the size
effect will not confound the results, it is more likely to be worthwhile to
do a guideline public company method, though there is a potential prob-
lem with statistical error from looking at only one industry.

28. In the context of performing a discounted cash ¬‚ow method.
29. Included in this discussion are the variations of PE, e.g., P/CF, etc.
30. This is under the simplest assumption that g1 g.

CHAPTER 4 Discount Rates as a Function of Log Size 153
The log size model is not only far more accurate than CAPM for valuing
privately held businesses, but it is much faster and easier to use. It re-
quires no research,31 whereas CAPM often requires considerable research
of the appropriate comparables (guideline companies).
Moreover, it is very inaccurate to apply the betas for IBM, Compaq,
Apple Computer, etc. to a small startup computer ¬rm with $2 million in
sales. The size effect drowns out any real information contained in betas,
especially applying betas of large ¬rms to small ¬rms. The almost six-
fold improvement that we found in the 0.34% standard error in the 60-
year log size equation versus the 1.89% standard error from the 73-year
CAPM applies only to ¬rms of the same magnitude. When applied to
small ¬rms, CAPM yields even more erroneous results, unless the ap-
praiser compensates by blindly adding another 5“10% beyond the typical
Ibbotson ˜˜small ¬rm premium™™ and calling that a speci¬c company ad-
justment (SCA). I suspect this practice is common, but then it is not really
an SCA; rather, it is an outright attempt to compensate for a model that
has no place being used to value small and medium ¬rms.
Several years ago, in the process of valuing a midsize ¬rm with $25
million in sales, $2 million in net income after taxes, and very fast growth,
I used a guideline public company method”among others. I found 16
guideline companies with positive earnings in the same SIC Code. I re-
gressed the value of the ¬rm against net income, with ˜˜great™™ results”
99.5% R 2 and high t-statistics. When I applied the regression equation to
the subject company, the value came to $91 million!32 I suspect that
much of this scaling problem goes on with CAPM as well, i.e., many
appraisers seriously overvalue small companies using discount rates ap-
propriate for large ¬rms only.
When using the log size model, we extrapolate the discount rate to
the appropriate level for each ¬rm that we value. There is no further need
for a size adjustment. We merely need to compare our subject company
to other companies of its size, not to IBM. Using Robert Morris Associates
data to compare the subject company to other ¬rms of its size is appro-
priate, as those companies are often far more comparable than NYSE
Since we have already extrapolated the rate of return through the
regression equation in a manner that appropriately considers the average
risk of being any particular size, the relevant comparison when consid-
ering speci¬c company adjustments is to other companies of the same
size. There is a difference between two ¬rms that each do $2 million in
sales volume when one is a one-man show and the other has two Harvard
MBAs running it. If the former is closer to average management, you
should probably subtract 1% or 2% from the discount rate for the latter;

31. One needs only a single regression equation for all valuations performed within a single year.
32. The magnitude problem was solved by regressing the natural log of value against the natural
log of net income. That eliminated the scaling problem and led to reasonable results. That
particular technique is not always the best solution, but it sometimes works beautifully. We
cover this topic in more detail near the end of Chapter 2.

PART 2 Calculating Discount Rates
if the latter is the norm, it is appropriate to add that much to the discount
rate of the former. Although speci¬c company adjustments are subjective,
they serve to further re¬ne the discount rate obtained from discount rate

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