easy to use (Annin 1997), and using it to determine discount rates for

privately held ¬rms is particularly problematic. Market returns are not

available for these ¬rms, rendering direct use of the model impossible.19

Discount rates based on using the three-factor model are published by

Ibbotson Associates in the Cost of Capital Quarterly by industry SIC code,

with companies in each industry sorted from highest to lowest. Deter-

mining the appropriate percentile grouping for a privately held ¬rm is a

major obstacle, however. The Fama“French model is a superior model for

calculating discount rates of publicly held ¬rms. It is not practical for

privately held ¬rms.

Log Size Models

The log size model is a superior approach because it better correlates with

historical equity returns. Therefore, it enables business appraisers to dis-

pense with CAPM altogether and use ¬rm size as the basis for deriving

a discount rate before adjustments for qualitative factors different from

the norm for similarly sized companies.

In another study on stock market returns, analysts at an investment

banking ¬rm regressed P/E ratios against long-term growth rate and mar-

ket capitalization. The R 2 values produced by the regressions were 89%

for the December 1989 data and 73% for the November 1990 data. Sub-

stituting the natural logarithm of market capitalization in place of market

capitalization, the same data yields an R 2 value of 91% for each data set,

a marginal increase in explanatory power for the ¬rst regression but a

signi¬cant increase in explanatory power for the second regression.

From Chapter 3, equation (3-28), the PE multiple is equal to

1 r

PE (1 b)(1 g1)

r g

Using a log size model to determine r, the PE multiple is equal to:

1 a b ln (FMV)

PE (1 RR)(1 g1) (4-18)

a b ln(FMV) g

where g1 is expected growth in the ¬rst forecast year, RR is the retention

ratio,20 a and b are the log size regression coef¬cients, and g is the long-

term growth rate. Looking at equation (4-18), it is clear why using the

log of market capitalization improved the R 2 of the above regression.

Grabowski and King (1995) applied a ¬ner breakdown of portfolio

returns than was previously used to relate size to equity premiums. When

19. Based on a conversation with Michael Annin.

20. Equation (3-28) uses the more conventional term b instead of RR to denote the retention ratio.

Here we have changed the notation in order to eliminate confusion, as we use the term b

for the regression x-coef¬cient.

PART 2 Calculating Discount Rates

148

they performed regressions with 31-year data for 25 and 100 portfolios

(as compared to our 10), they found results similar to the equity premium

form of log size model, i.e., the equity premium is a function of the neg-

ative of the log of the average market value of equity, further supporting

this relationship.21

Grabowski and King (1996) in an update article also used other prox-

ies for ¬rm size in their log size discount rate model, including sales, ¬ve-

year average net income, and EBITDA. Following is a summary of their

regression results sorted ¬rst by R 2 in descending order, then by the stan-

dard error of the y-estimate in ascending order. Overall, we are attempt-

ing to present their best results ¬rst.

R2

Measure of Size Standard Error of Y-Estimate

1. Mkt cap”common equity 93% 0.862%

2. Five-year average net income 90% 0.868%

3. Market value of invested capital 90% 1.000%

4. Five-year average EBITDA 87% 0.928%

5. Book value”invested capital 87% 0.989%

6. Book value”equity 87% 0.954%

7. Number of employees 83% 0.726%

8. Sales 73% 1.166%

Note that the market value of common equity, i.e., market capitali-

zation of common equity, has the highest R 2 of all the measures. This is

the measure that we have used in our log size model. The ¬ve-year av-

erage net income, with an R 2 of 90%, is the next-best independent vari-

able, superior to the market value of invested capital by virtue of its lower

standard error.

This is a very important result. It tells us that the majority of the

information conveyed in the market price of the stock is contained in net

income. When we use a log size model based on equity in valuing a

privately held ¬rm, we do not have the bene¬t of using a market-

determined equity. The value will be determined primarily by the mag-

nitude and timing of the forecast cash ¬‚ows, the primary component of

which is forecast net income. If we did not know that the log of net

income was the primary causative variable of the log size effect, it is

possible that other variables such as leverage, sales, book value, etc. sig-

ni¬cantly impact the log size effect. If we failed to take those variables

into account and our subject company™s leverage varied materially from

the average of the market (in each decile) as it is impounded into the log

size equation, our model would be inaccurate. Grabowski and King™s

research eliminates this problem. Thus, we can be reasonably con¬dent

that the log size model as presented is accurate and is not missing any

signi¬cant variable.

Of Grabowski and King™s eight different measures of size, only mar-

ket capitalization (#1) and the market value of invested capital (#3) have

21. Grabowski and King actually used base 10 logarithms.

CHAPTER 4 Discount Rates as a Function of Log Size 149

the circular reasoning problem of our log size model. The other measures

of size have the advantage in a log size model of eliminating the need

for iteration since the discount rate equation does not depend on the

market value of equity, the determination of which is the ultimate pur-

pose of the discount rate calculation. For example, if we were to use #2,

net income, we would simply insert the subject company™s ¬ve-year av-

erage net income into Grabowski and King™s regression equation and it

would determine the discount rate. This is problematic, however, for de-

termining discount rates for high-growth ¬rms, due to the inability to

adequately capture signi¬cant future growth in sales, net income, and so

on. Start-up ¬rms in high technology industries frequently have negative

net income for the ¬rst several years due to their investment in research

and development. Sales may subsequently rise dramatically once prod-

ucts reach the market. Therefore, ¬ve-year averages are not suitable in

this situation.

Another problem with Grabowski and King™s results is that their data

only encompass 1963“1994, 31 years”the years for which Compustat

data were available for all companies. Thus, their equations suffer from

the same wide con¬dence intervals that our 30-year regressions have.

Their standard error of the y-estimate is 0.862% (Exhibit A, p. 106), which

is six times larger than our 1938“1997 con¬dence intervals.22 Thus, their

95% con¬dence intervals will also be approximately six times wider

around the regression estimate.

As mentioned in the introduction, in their latest article (Grabowski

and King 1999) they demonstrate a negative logarithmic relationship be-

tween returns and operating margin and a positive logarithmic relation-

ship between returns and the coef¬cient of variation of operating margin

and accounting return on equity.

This is their most important result so far because it relates returns to

fundamental measures of risk. Actually, it appears to me that operating

margin in itself works because of its strong correlation of 0.97 to market

capitalization, i.e., value. However, the coef¬cient of variations (CV) of

operating margin and return on equity seem to be more fundamental

measures of risk than size itself. In other words, it appears that size itself

is a proxy for the volatility of operating margin, return on equity, and

possibly other measures. Thus, we must pay serious attention to their

results.

Below is a summary of their statistical results.

R2

Measure of Risk Standard Error of Y-Estimate

1. Log of ¬ve-year operating margin 76% 1.185%

2. Log CV(operating margin) 54% 0.957%

3. Log CV (return on equity) 54% 0.957%

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

150

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

size:

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.

Heteroscedasticity

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“

1962.

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

least.

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.

INDUSTRY EFFECTS

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-

turns.27

SATISFYING REVENUE RULING 59-60 WITHOUT A

GUIDELINE PUBLIC COMPANY METHOD

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

insigni¬cance.

PART 2 Calculating Discount Rates

152

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

(3-28)

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

model.

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

SUMMARY AND CONCLUSIONS

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

¬rms.

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

154

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

calculations.

BIBLIOGRAPHY

Abrams, Jay B. 1994. ˜˜A Breakthrough in Calculating Reliable Discount Rates.™™ Business

Valuation Review (August): 8“24.

Abrams, Jay B. 1997. ˜˜Discount Rates as a Function of Log Size and Valuation Error

Measurement.™™ The Valuation Examiner (Feb./March): 19“21.

Annin, Michael. 1997. ˜˜Fama-French and Small Company Cost of Equity Calculations.™™

Business Valuation Review (March 1997): 3“12.

Banz, Rolf W. 1981. ˜˜The Relationship Between Returns and Market Value of Common