<<

. 7
( 18)



>>

model to calculate a size premium over the average market return and
add that to a CAPM calculation of the discount rate using Ibbotson™s sum
betas. It will take more research to determine whether than is a worth-
while improvement in methodology.


The Fama“French Cost of Equity Model18
The Fama“French cost of equity model is a multivariable regression
model that uses size (˜˜small minus big™™ premium SMB) and book to


18. The precise method of calculating beta, SMB, and HML using the three-factor model, along
with the regression equation, is more fully explained in Ibbotson Associates™ Beta Book.


CHAPTER 4 Discount Rates as a Function of Log Size 147
market equity (˜˜high minus low™™ premium HML) in addition to beta
as variables that affect market returns. Michael Annin (1997) examined
the model in detail and found that it does appear to correct for size, both
in the long term and short term, over the 30-year time period tested.
The cost of equity model, however, is neither generally accepted nor
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
Stocks.™™ Journal of Financial Economics 9: 3“18.
Fama, Eugene F., and Kenneth R. French. 1992. ˜˜The Cross-Section of Expected Stock
Returns.™™ Journal of Finance 47: 427“65.
” ”. 1993. ˜˜Common Risk Factors in the Returns on Stocks and Bonds.™™ Journal of Fi-

nancial Economics 33: 3“56.
Gilbert, Gregory A. 1990. ˜˜Discount Rates and Capitalization Rates: Where are We?™™ Busi-
ness Valuation Review (December): 108“13.
Grabowski, Roger, and David King. 1995. ˜˜The Size Effect and Equity Returns.™™ Business
Valuation Review (June): 69“74.
” ”. 1996. ˜˜New Evidence on Size Effects and Rates of Return.™™ Business Valuation Re-

view (September): 103“15.
” ”. 1999. ˜˜New Evidence on Size Effects and Rates of Return.™™ Business Valuation Re-

view (September): 112“30.
Harrison, Paul. 1998. ˜˜Similarities in the Distribution of Stock Market Price Changes be-
tween the Eighteenth and Twentieth Centuries.™™ Journal of Business 71, no. 1 (Janu-
ary).
Hayes, Richard. 1726. The Money™d Man™s Guide: or, the Purchaser™s Pocket-Companion. Lon-
don: W. Meadows.
Ibbotson & Associates. 1999. Stocks, Bonds, Bills and In¬‚ation: 1999 Yearbook. Chicago: The
Associates.
Jacobs, Bruce I., and Kenneth N. Levy. 1988. ˜˜Disentangling Equity Return Regularities:
New Insights and Investment Opportunities.™™ Financial Analysts Journal (May“June):
18“42.
Neal, L. 1990. The Rise of Financial Capitalism: International Capital Markets in the Age of
Reason. Cambridge: Cambridge University Press.
Peterson, James D., Paul D. Kaplan, and Roger G. Ibbotson. 1997. ˜˜Estimates of Small
Stock Betas Are Much Too Low.™™ Journal of Portfolio Management 23 (Summer): 104“
11.
Schwert, G. William, and Paul J. Seguin. 1990. ˜˜Heteroscedasticity in Stock Returns.™™ Jour-
nal of Finance 45: 1129“56.
Thomas, George B., Jr. 1972. Calculus and Analytic Geometry. Reading, Mass: Addison-
Wesley.




CHAPTER 4 Discount Rates as a Function of Log Size 155
APPENDIX A
Automating Iteration using Newton™s Method
This appendix is optional. It is mathematically dif¬cult and is more an-
alytically interesting than practical.
In this section we present a numerical method for automatically it-
erating to the correct log size discount rate. Isaac Newton invented an
iterative procedure using calculus to provide numerical solutions to equa-
tions with no analytic solution. Most calculus texts will have a section on
his method (for example, see Thomas 1972). His procedure involves mak-
ing an initial guess of the solution, then subtracting the equation itself
divided by its own ¬rst derivative to provide a second guess. We repeat
the process until we converge to a single answer.
The bene¬t of Newton™s method is that it will enable us to simply
enter assumptions for the cash ¬‚ow base and the perpetual growth, and
the spreadsheet will automatically calculate the value of the ¬rm without
our having to manually go through the iterations as we did in Tables
4-4A, B, and C. Remember, some iteration process is necessary when
using log size discount rates because the discount rate is not independent
of size, as it is using other discount rate models.
To use Newton™s procedure, we rewrite equation (4-13) as:
CF
Let f(V) V 0 (A4-1)
(a b ln V g)
bCF
f (V) 1 (A4-2)
g)2
V(a b ln V
Assuming our initial guess of value is V0, the formula that de¬nes our
next iteration of value, V1, is:
CF
V0
(a b ln V0 g)
V1 V0 (A4-3)
bCF
1
g)2
V0(a b ln V0
Table A4-5 shows Newton™s iterative process for the simplest valu-
ation. In B22“B26 we enter our initial guess of value of an arbitrary $2
trillion (B22), our forecast cash ¬‚ow base of $100,000 (B23), perpetual
growth g 7% (B24), and our regression coef¬cients a and b (B25 and
B26, which come from Table 4-1, E42 and E48, respectively).
In B7 we see our initial guess of $2 trillion. The iteration #2 value
of $280,530 (B8) is the result of the formula in the note immediately below
Table A4-5, which is equation (A4-3).33 B9 to B12 are simply the formula
in B8 copied to the remaining spreadsheet cells.
Once we have the formula, we can value any ¬rm with constant
growth in its cash ¬‚ows by simply changing the parameters in B23 to
B24.


33. Cell B7, our initial guess, is V0 in equation (A4-3).




PART 2 Calculating Discount Rates
156
T A B L E A4-5

Gordon Model Valuation Using Newton™s Iterative Process


A B

5 Iteration Value
6 t V(t)
0 2,000,000,000,000
8 1 280,530
9 2 612,879
10 3 599,634
11 4 599,625
12 5 599,625

14 Proof of Calculation:
16 Discount rate 23.68%
17 Gordon multiple 5.9963
18 CF FMV $599,625
30 Parameters
22 V(0) 2,000,000,000,000
23 CF 100,000
24 g 7%
25 a 37.50%
26 b 1.039%
29 Model Sensitivity

30 FMV Initial Guess V(0)
31 Explodes 3,000,000,000,000
32 599,625 2,000,000,000,000
33 599,625 27,000
34 Explodes 26,000

Formula in Cell B8:
B7 ((B7 (CF/(A B * LN(B7) G)))/(1 (B * CF)/(B7 * (A B * LN(B7) G) 2)))
Note: The above formula assumes an End-Year Gordon Model. Newton™s Method converges for the midyear Gordon Model, but too
slowly to be of practical use.




B31 to B34 show the sensitivity of the model to the initial guess. If
we guess poorly enough, the model will explode instead of converging
to the right answer. For this particular set of assumptions, an initial guess
of anywhere between $27,000 and $2 trillion will converge to the right
answer. Assumptions above $3 trillion or below $26,000 explode the
model.
Unfortunately, the midyear Gordon model, which is more accurate,
has a much more complex formula. The iterative process does converge,
but much too slowly to be of any practical use. One can use the end-of-
year Gordon model and multiply the result by the square root of (1 r).




CHAPTER 4 Discount Rates as a Function of Log Size 157
APPENDIX B
Mathematical Appendix
This appendix provides the mathematics behind the log size model, as
well as some philosophical analysis of the mathematics”speci¬cally on
the nature of exponential decay function and how that relates to phenom-
ena in physics as well as our log size model. This is intended more as
intellectual observation than as required information.
We will begin with two de¬nitions:
r return of a portfolio
S standard deviation of returns of the portfolio
Equation (B4-1) states that the return on a portfolio of securities (each
decile is a portfolio) varies positively with the risk of the portfolio, or:

r a1 b1S (B4-1)

This is a generalization of equation (4-1) in the chapter. This rela-
tionship is not directly observable for privately held ¬rms. Therefore, we
use the next equation, which is a generalization of equation (4-2) from
the chapter, to calculate expected return.
The parameter a1 is the regression estimate of the risk-free rate,34
while the parameter b1 is the regression estimate of the slope, which is
the return for each unit increase of risk undertaken, i.e., the standard
deviation of returns. Thus, b1 is the regression estimate of the price of or
the reward for taking on risk.

r a2 b 2 ln FMV, b 2 0 (B4-2)

Equation (B4-2) states that return decreases in a linear fashion with
the natural logarithm of ¬rm value. The parameter a2 is the regression
estimate of the return for a $1 ¬rm35 ”the valueless ¬rm”while the pa-
rameter b 2 is the regression estimate of the slope, which is the return for
each increase in ln FMV. Thus, b 2 is the regression estimate of the reduc-
tion in return investors accept for investing in smaller ¬rms. The terms
a1, a2, b1, and b 2 are all parameters determined in regression equations
(4-1) and (4-2).
Using all 73 years of stock market data, our regression estimate of
a1 6.56% (Table 4-1, D26), which compares well with the 73-year mean
Treasury Bond yield of 5.28%. It would initially appear that the log size
regression does a reasonable job of also providing an estimate of the risk-
free return. Unfortunately, it is not all that simple, as the log size estimate
using 60 years of data fares worse. The log size 60-year estimate of a1 is
8.90% (Table 4-1, E26), which is a long way off from the 60-year mean
treasury bond yield of 5.70% (Table 4-1, E27). Thus, eliminating the ¬rst


34. A zero risk asset would have no standard deviation of returns. Thus, S 0 and r a1.
35. A ¬rm worth $1 would have ln FMV ln $1 0. Thus in equation (B4-2), for FMV $1,
r a2.




PART 2 Calculating Discount Rates
158
13 years of data had the effect of shifting the regression line upwards and
¬‚attening it slightly.
We already knew from our analysis of Table 4-2 in the chapter that
using 60 years of data was the overall best choice because of its superi-
ority in the log size equation estimates, but it was not the best choice for
estimating equation (4-1). Its R 2 is lower and standard error is higher
than the 73-year results.
Focusing now on equation (B5-2), the log size equation, the 60-year
regression estimate of b1 1.0309% (Table 4-1, E48), which is signi¬-
cantly lower in absolute value than the 73-year estimate of 1.284%
(D48). The parameter b 2 is the reduction in return that comes about from
each unit increase in company value (in natural logarithms). The param-
eter a2 is the y-intercept. It is the return (discount rate) for a valueless
¬rm”more speci¬cally, a $1 ¬rm in value”as ln($1) 0.
Equating the right-hand sides of equation (B4-1) and (B4-2) and solv-
ing for S, we see how we are implicitly using the size of the ¬rm as a
proxy for risk.
a2 a1 b2
S ln FMV (B4-3)
b1 b1
Since a2 is the rate of return for the valueless ¬rm and a1 is the re-
gression estimate of the risk-free rate”¬‚awed as it is”the difference be-
tween them, a2 a1 is the equity premium for a $1.00 ¬rm, i.e., the val-
ueless ¬rm. Dividing by b1, the price of risk (or reward) for each
increment of standard deviation, we get (a2 a1)/b1, the standard devi-
ation of a $1 ¬rm. We then reduce our estimate of the standard deviation
by the ratio of the relative prices of risk in size divided by the price of
risk in standard deviation, and multiply that ratio by the log of the size
of the ¬rm. In other words, we start with the maximum risk, a $1 ¬rm,
and reduce the standard deviation by the appropriate price times the log
of the value of the ¬rm in order to calculate the standard deviation of the
¬rm.
Rearranging equation (B4-3), we get
(a1 a2) b1S
ln FMV (B4-4)
b2
Raising both sides of the equation as powers of e, the natural exponent,
we get:
(a1 a2) b1S (a1 a2) b1S

FMV e e e b , or (B4-5)
b2 b2 2




b1
(a1 a2)
kS
FMV Ae , where A e ,k 0 (B4-6)
b2
b2
Here we see that the value of the ¬rm or portfolio declines exponentially
with risk, i.e., the standard deviation.
Unfortunately, the standard deviation of most private ¬rms is un-
observable since there are no reliable market prices. Therefore, we must




CHAPTER 4 Discount Rates as a Function of Log Size 159
solve for the value of a private ¬rm another way. Restating equation
(B4-2),
r a2 b 2 ln(FMV) (B4-7)
Rearranging the equation, we get:
(r a2)
ln FMV (B4-8)
b2
Raising both sides by e, i.e., taking the antilog, we get:
(r a2)

FMV e (B4-9)
b2


or (B4-10)
a2
1
mr
FMV Ce , where C e and m
b2
b2
This shows the FMV of a ¬rm or portfolio declines exponentially
with the discount rate. This is reminiscent of a continuous time present
value formula; in this case, though, instead of traveling through time we
are traveling though expected rates of return. The same is true of equation
(B4-6), where we are traveling through degree of risk.


What Does the Exponential Relationship Mean?
Let™s try to get an intuitive feel for what an exponential relationship
means and why that makes intuitive sense. Equation (B4-6) shows that
the fair market value of the ¬rm is an exponentially declining function
of risk, as measured by the standard deviation of returns. Repeating equa-
tion (B4-6), FMV Ae k S, k 0. Because we ¬nd that risk itself is primarily
related to the size of the ¬rm, we come to a similar equation for size.
Cemr, m
Repeating equation (B4-10), we see that FMV 0.
In physics, radioactive minerals such as uranium decay exponen-
tially. That means that a constant proportion of uranium decays at every
moment. As the remaining portion of uranium is constantly less over time
due to the radioactive decay, the amount of decay at any moment in time
or during any ¬nite time period is always less than the previous period.
A graph of the amount of uranium remaining over time would be a
downward sloping curve, steep at ¬rst and increasing shallow over time.
Figure 4-3 shows an exponential decay curve.
It appears the same is true of the value of ¬rms. Instead of decaying
over time, their value decays over risk. Because it turns out that risk is
so closely related to size and the rate of return is so closely related to
size, the value also decays exponentially with the market rate of return,
i.e., the discount rate. The graph of exponential decay in value over risk
has the same general shape as the uranium decay curve.
Imagine the largest ship in the world sailing on a moderately stormy
ocean. You as a passenger hardly feel the effects of the storm. If instead
you sailed on a slightly smaller ship, you would feel the storm a bit more.
As we keep switching to increasingly smaller ships, the storm feels in-



PART 2 Calculating Discount Rates
160
creasingly powerful. The smallest ship on the NYSE might be akin to a
35-foot cabin cruiser, while appraisers often have to value little paddle-
boats, the passengers of which would be in danger of their lives while
the passengers of the General Electric boat would hardly feel the turbu-
lence.
That is my understanding of the principle underlying the size effect.
Size offers diversi¬cation of product and service. Size reduces transaction
costs in proportion to the entity, e.g., the proceeds of ¬‚oating a $1 million
stock issue after ¬‚otation costs are far less in percentage terms than ¬‚oat-
ing a $100 million stock issue. Large ¬rms have greater depth and breadth
of management, and greater staying power. Even the chances of beating
a bankruptcy exist for the largest businesses. Remember Chrysler? If it
were not a very big business, the government would never have jumped
in to rescue it. The same is true of the S&Ls. For these and other reasons,
the returns of big businesses ¬‚uctuate less than small businesses, which
means that the smaller the business, the greater the risk, the greater the
return.
The FMV of a ¬rm or portfolio declining exponentially with the dis-
count rate/risk is reminiscent of a continuous time present value formula,
e r t; in this case, though, instead of
where Present Value Principal
traveling through time we are traveling though expected rates of return/
risk.




CHAPTER 4 Discount Rates as a Function of Log Size 161
APPENDIX C
Abbreviated Review and Use
This abbreviated version of the chapter is intended for those who simply
wish to learn the model without the bene¬t of additional background and
explanation, or wish to use it as a quick reference for review.


INTRODUCTION
Historically, small companies have shown higher rates of return than
large ones, as evidenced by New York Stock Exchange (NYSE) data over
the past 73 years (Ibbotson Associates 1999). Further investigation into
this phenomenon has led to the discovery that return (the discount rate)
strongly correlates with the natural logarithm of the value of the ¬rm
(¬rm size), which has the following implications:
— The discount rate is a linear function negatively related to the
natural logarithm of the value of the ¬rm.
— The value of the ¬rm is an exponential decay function, decaying
with the investment rate of return (the discount rate).
Consequently, the value also decays in the same fashion with the
standard deviation of returns.
As we have already described regression analysis in Chapter 3, we
now apply these techniques to examine the statistical relationship be-
tween market returns, risk (measured by the standard deviation of re-
turns) and company size.


REGRESSION #1: RETURN VERSUS STANDARD
DEVIATION OF RETURNS
Columns A“F in Table 4-1 contain the input data from the Stocks, Bonds,
Bills and In¬‚ation 1999 Yearbook (Ibbotson Associates 1999) for all of the
regression analyses as well as the regression results. We use 73-year av-
erage returns in both regressions. For simplicity, we have collapsed 730
data points (73 years 10 deciles) into 73 data points by using averages.
Thus, the regressions are cross-sectional rather than time series. In Col-
umn A we list Ibbotson Associates™ (1999) division of the entire NYSE
into 10 different divisions”known as deciles”based on size, with the
largest ¬rms in decile 1 and the smallest in decile 10.36 Columns B through
F contain market data for each decile which is described below.
Note that the 73-year average market return in Column B rises with
each decile, as does the standard deviation of returns (Column C). Col-
umn D shows the 1998 market capitalization of each decile, which is the
price per share times the number of shares. It is also the fair market value
(FMV).
Dividing Column D (FMV) by Column F (the number of ¬rms in the
decile), we obtain Column G, the average capitalization, or the average


36. All of the underlying decile data in Ibbotson originate with the University of Chicago™s Center
for Research in Security Prices (CRSP), which also determines the composition of the deciles.


PART 2 Calculating Discount Rates
162
fair market value of the ¬rms in each decile. Column H, the last column
in the table titled ln (FMV), is the natural logarithm of the average FMV.
Regression of ln (FMV) against standard deviation of returns for the
period 1926“1998 (D26 to D36, Table 4-1), gives rise to the equation:

r 6.56% (31.24% S) (4-1)

where r return and S standard deviation of returns.
The regression statistics of adjusted R 2 of 98.82% (D30) a t-statistic
of the slope of 27.4 (D35), a p-value of less than 0.01% (D36), and the
standard error of the estimate of 0.27% (D28), all indicate a high degree
of con¬dence in the results obtained. Also, the constant of 6.56% (D26) is
the regression estimate of the long-term risk-free rate, which compares
favorably with the 73-year arithmetic mean income return from 1926“
1998 on long-term Treasury Bonds of 5.20%.37
The major problem with direct application of this relationship to the
valuation of small businesses is coming up with a reliable standard de-
viation of returns. Appraisers cannot directly measure the standard de-
viation of returns for privately held ¬rms, since there is no objective stock
price. We can measure the standard deviation of income, and we covered
that in our discussion in the chapter of Grabowski and King (1999).


REGRESSION #2: RETURN VERSUS LOG SIZE
Fortunately, there is a much more practical relationship. Notice that the
returns are negatively related to the market capitalization, i.e., the fair
market value of the ¬rm. The second regression in Table 4-1 (D42“D51)
is the more useful one for valuing privately held ¬rms. Regression #2
shows return as a function of the natural logarithm of the FMV of the
¬rm. The regression equation for the period 1926“1998 is:

r 42.24% [1.284% ln (FMV)] (4-2)

The adjusted R 2 is 92.3% (D45), the t-statistic is 10.4 (D50), and the p-
value is less than 0.01% (D51), meaning that these results are statistically
robust. The standard error for the Y-estimate is 0.82% (D43), which means
that we can be 95% con¬dent that the regression forecast is accurate
within approximately 2 0.82% 1.6.


Recalculation of the Log Size Model Based on 60 Years
NYSE data from the past 60 years are likely to be the most relevant for
use in forecasting the future (see chapter for discussion). This time frame
still contains numerous data points, but it excludes the decade of highest
volatility, attributed to nonrecurring historical events, i.e., the Roaring
Twenties and Depression years. Also, Table 4-2A shows that the 60-year
regression equation has the highest adjusted R 2 and lowest standard error


37. SBBI-1999, p. 140 uses this measure as the risk-free rate for CAPM. Arguably, the average bond
yield is a better measure of the risk-free rate, but the difference is immaterial.


CHAPTER 4 Discount Rates as a Function of Log Size 163
when compared to the other four examined. Therefore, we repeat all three
regressions for the 60-year time period from 1939“1998, as shown in Table
4-1, Column E. Regression #1 for this time period for is:
r 8.90% (30.79% S) (4-3)
where S is the standard deviation.
The adjusted R 2 in this case falls to 95.31% (E30) from the 98.82%
(D30) obtained from the 73-year equation, but is still indicative of a strong
relationship.
The corresponding log size equation (regression #2) for the 60-year
period is:
r 37.50% [1.039% ln (FMV)] (4-4)
The regression statistics indicate a good ¬t, with an adjusted R 2 of 96.95%
(E45).38 Equation (4-4) will be used for the remainder of the book to cal-
culate interest rates, as this time period is the most appropriate for cal-
culating current discount rates.

Need for Annual Updating
Table 4-1 should be updated annually, as the Ibbotson averages change,
and new regression equations should be generated. This becomes more
crucial when shorter time periods are used, because changes will have a
greater impact on the average values. Additionally, it is important to be
careful to match the regression equation to the year of the valuation. If
the valuation assignment is retroactive and the valuation date is 1994,
then don™t use the regression equation for 1939“1998. Instead, either use
the regression equation in the original article, run your own regression
on the Ibbotson data, or contact the author to provide the right equation.

Computation of Discount Rate Is an Iterative Process
In spite of the straightforwardness of these relationships, we have a prob-
lem of circular reasoning when it comes to computing of the discount
rate. We need FMV to obtain the discount rate, which is in turn used to
discount cash ¬‚ows or income to calculate the FMV! Hence, it is necessary
to make sure that our initial estimate of FMV is consistent with the ¬nal
result. If it is not, then we have to keep repeating the process until the
results are consistent. Fortunately, discount rates remain virtually con-
stant over large ranges of values, so this should not be much of a problem.


Practical Illustration of the Log Size Model: Discounted
Cash Flow Valuations
Let™s illustrate how the iterative process works with a speci¬c example.
The assumptions in Tables 4-4A, 4-4B, and 4-4C are identical, except for
the discount rate. Table 4-4A is a very simple discounted cash ¬‚ow (DCF)
analysis of a hypothetical ¬rm. The basic assumptions appear in Rows
B7 through B12. We assume the ¬rm had $100,000 cash ¬‚ow in 1998. We


38. For 1938“1997 data, adjusted R 2 was 99.54%. The ˜˜perverse™™ results of 1998 caused a
deterioration in the relationship.


PART 2 Calculating Discount Rates
164
forecast annual growth through the year 2003 in B8 through F8 and per-
petual growth at 6% thereafter in B10. In B9 we assume a 20% discount
rate.
The DCF analysis in Rows B22 through B32 is standard and requires
little explanation other than that the present value factors are midyear,
and the value in B28 is a marketable minority interest. It is this value
($943,285) that we use to compare the consistency between the assumed
discount rate (in Row 4) and calculated discount rate according to the log
size model.
We begin calculating the of discount rate using the log size model in
B34, where we compute ln (943,285) 13.7571. This is the natural log of
the marketable minority value of the ¬rm. In B35 we multiply that result
by the x-coef¬cient from the regression, or 0.01039, to come to 0.1429.
We then add that product to the regression constant of 0.3750, which
appears in B36, to obtain an implied discount rate of 23% (rounded, B37).
Comparison of the two discount rates (assumed and calculated) re-
veals that we initially assumed too high a discount rate, meaning that we
undervalued the ¬rm. B29“B31 contain the control premium and discount
for lack of marketability. Because the discount rate is not yet consistent,
ignore these numbers in this table, as they are irrelevant.
In Chapter 7, we discuss the considerable controversy over the ap-
propriate magnitude of control premiums. Nevertheless, it is merely a
parameter in the spreadsheet, and its magnitude does not affect the logic
of the analysis.

The Second Iteration: Table 4-4B
Having determined that a 20% discount rate is too low, we revise our
assumption to a 23% discount rate (B9) in Table 4-4B. In this case, we
arrive at a marketable minority FMV of $ 783,919 (B28). When we perform
the discount rate calculation with this value (B34“B37), we obtain a
matching discount rate of 23%, indicating that no further iterations are
necessary.

Consistency in Levels of Value
In calculating discount rates, it is important to be consistent in the level
of fair market value that we are using. Since the log size model is based
on returns from the NYSE, the corresponding values generated are on a
marketable minority basis. Consequently, it is this level of value that we
should use for the discount rate calculations.
Frequently, however, the marketable minority value is not the ulti-
mate level of fair market value that we are calculating. Therefore, it is
crucial to be aware of the differing levels of FMV that occur as a result
of valuation adjustments. For example, if our valuation assignment is to
calculate an illiquid control interest, we will add a control premium and
subtract a discount for lack of marketability from the marketable minority
value.39 Nevertheless, we use only the marketable minority level of FMV
in iterating to the proper discount rate.


39. Not all authorities would agree with this statement. There is considerable disagreement on the
levels of value. We cover those controversies in Chapter 7.


CHAPTER 4 Discount Rates as a Function of Log Size 165
Adding Speci¬c Company Adjustments to the DCF Analysis:
Table 4-4C
The ¬nal step in our DCF analysis is performing speci¬c company ad-
justments. Let™s suppose for illustrative purposes that there is only one
owner of this ¬rm. She is 62 years old and had a heart attack three years
ago. The success of the ¬rm depends to a great extent on her personal
relationships with customers, which may not be easily duplicated by a
new owner. Therefore, we decide to add a 2% speci¬c company adjust-
ment to the discount rate to re¬‚ect this situation.40 If there are no speci¬c
company adjustments, then we would proceed with the calculations in
B22“B32.
Prior to adding speci¬c company adjustments, it is important to
achieve internal consistency in the ex ante and ex post marketable mi-
nority values, as we did in Table 4-4B. Next, we merely add the 2% to
get a 25% discount rate, which we place in B9. The remainder of the table
is identical to its predecessors, except that we eliminate the ex post cal-
culation of the discount rate in B34“B37, since we have already achieved
consistency.
It is at this point in the valuation process that we make adjustments
for the control premium and discount for lack of marketability, which
appear in B29 and B31. Our ¬nal fair market value of $642,139 (B32) is
on an illiquid control basis.
In a valuation report, it would be unnecessary to show Table 4-4A.
One should show Tables 4-4B and 4-4C only.

Total Return versus Equity Premium
CAPM uses an equity risk premium as one component for calculating
return. The discount rate is calculated by multiplying the equity premium
by beta and adding the risk free rate. In my ¬rst article on the log size
model (Abrams 1994), I used an equity premium in the calculation of
discount rate. Similarly, Grabowski and King (1995) used an equity risk
premium in the computation of discount rate.
The equity premium term was eliminated in my second article
(Abrams 1997) in favor of total return because of the low correlation be-
tween stock returns and bond yields for the past 60 years. The actual
correlation is 6.3%”an amount small enough to ignore.

Adjustments to the Discount Rate
Privately held ¬rms are generally owned by people who are not well
diversi¬ed. The NYSE decile data were derived from portfolios of stocks


40. A different approach would be to take a discount from the ¬nal value, which would be
consistent with key person discount literature appearing in a number of articles in Business
Valuation Review (see the BVR index for cites). Another approach is to lower our estimate of
earnings to re¬‚ect our weighted average estimate of decline in earnings that would follow
from a change in ownership or the decreased capacity of the existing owner, whichever is
more appropriate, depending on the context of the valuation. In this example I have already
assumed that we have done that. There are opinions that one should lower earnings
estimates and not increase the discount rate. It is my opinion that we should de¬nitely
increase the discount rate in such a situation, and we should also decrease the earnings
estimates if that has not already been done.


PART 2 Calculating Discount Rates
166
that were diversi¬ed in every sense except for size, as size itself was the
method of sorting the deciles. In contrast, the owner of the local bar is
probably not well diversi¬ed, nor is the probable buyer. The appraiser
may want to add 2% to 5% to the discount rate to account for that. On
the other hand, a $1 million FMV ¬rm is likely to be bought by a well-
diversi¬ed buyer and may not merit increasing the discount rate.
Another common adjustment to discount rates would be for the
depth and breadth of management of the subject company compared to
other ¬rms of the same size. In general, the regression equation already
incorporates the size effect. No one expects a $100,000 FMV ¬rm to have
three Harvard MBAs running it, but there is still a difference between a
complete one-man show and a ¬rm with two talented people. In general,
this methodology of calculating discount rates will increase the impor-
tance of comparing the subject company to its peers via RMA Associates
or similar data. Differences in leverage between the subject company and
its RMA peers could well be another common adjustment.


Discounted Cash Flow or Net Income?
Since the market returns are based on the cash dividends and the market
price at which one can sell one™s stock, the discount rates obtained with
the log size model should be properly applied to cash ¬‚ow, not to net
income. We appraisers, however, sometimes work with clients who want
a ˜˜quick and dirty valuation,™™ and we often don™t want to bother esti-
mating cash ¬‚ow. I have seen suggestions in Business Valuation Review
(Gilbert 1990, for example) that we can increase the discount rate and
thereby apply it to net income, and that will often lead to reasonable
results. Nevertheless, it is better to make an adjustment from net income
based on judgment to estimate cash ¬‚ow to preserve the accuracy of the
discount rate.


SATISFYING REVENUE RULING 59-60
As discussed in more detail in the body of this chapter, a study (Jacobs
and Levy 1988) found that, in general, industry was insigni¬cant in de-
termining rates of return.41 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 results with the log size model,42 combined with
Jacobs and Levy™s general ¬nding of industry insigni¬cance, satisfy the
intent of Revenue Ruling 59-60 without the need to actually perform a
guideline publicly traded company method (GPCM).
The PE multiple43 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. Then the only new information to come


41. For the appraiser who wants to use the rationale in this section as a valid reason to eliminate
the GPCM from an appraisal, there are some possible exceptions to the ˜˜industry doesn™t
matter conclusion™™ that one should read in the body of the chapter.
42. In the context of performing a discounted cash ¬‚ow approach.
43. Included in this discussion are the variations of PE, e.g., P/CF, etc.


CHAPTER 4 Discount Rates as a Function of Log Size 167
out of a GPCM is the market™s estimate of g,44 the growth rate of the
public ¬rm. There are much easier and less expensive ways to estimate
g than doing a GPCM. When all the market research is ¬nished, the ap-
praiser 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.




44. This is under the simplest assumption that g1 g.




PART 2 Calculating Discount Rates
168
CHAPTER 5


Arithmetic versus Geometric
Means: Empirical Evidence and
Theoretical Issues




INTRODUCTION
THEORETICAL SUPERIORITY OF ARITHMETIC MEAN
Table 5-1: Comparison of Two Stock Portfolios
EMPIRICAL EVIDENCE OF THE SUPERIORITY OF THE
ARITHMETIC MEAN
Table 5-2: Regressions of Geometric and Arithmetic Returns for
1927“1997
Table 5-3: Regressions of Geometric Returns for 1938“1997
The Size Effect on the Arithmetic versus Geometric Means
Table 5-4: Log Size Comparison of Discount Rates and Gordon Model
Multiples Using AM versus GM
INDRO AND LEE ARTICLE




169




Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
This chapter compares the attributes of the arithmetic and geometric
mean returns and presents theoretical and empirical evidence why the
arithmetic mean is the proper one for use in valuation.


INTRODUCTION
There has been a ¬‚urry of articles about the relative merits of using the
arithmetic mean (AM) versus the geometric mean (GM) in valuing busi-
nesses. The SBBI Yearbook (see Ibbotson Associates 1998) for many years
has taken the position that the arithmetic mean is the correct mean to use
in valuation. Conversely, Allyn Joyce (1995) initiated arguments for the
GM as the correct mean. Previous articles have centered around Professor
Ibbotson™s famous example using a binomial distribution with 50%“50%
probabilities of a 30% and 10% return. His example is an important
theoretical reason why the AM is the correct mean. The articles critical of
Ibbotson are interesting but largely incorrect and off on a tangent. There
are both theoretical and empirical reasons why the arithmetic mean is the
correct one.


THEORETICAL SUPERIORITY OF ARITHMETIC MEAN
We begin with a quote from Ibbotson: ˜˜Since the arithmetic mean equates
the expected future value with the present value, it is the discount rate™™
(Ibbotson Associates 1998, p. 159). This is a fundamental theoretical rea-
son for the superiority of AM.
Rather than argue about Ibbotson™s much-debated above example,
let™s cite and elucidate a different quote from his book (Ibbotson Associ-
ates 1998, p. 108). ˜˜In general, the geometric mean for any time period is
less than or equal to the arithmetic mean. The two means are equal only
for a return series that is constant (i.e., the same return in every period).
For a non-constant series, the difference between the two is positively
related to the variability or standard deviation of the returns. For exam-
ple, in Table 6-7 [the SBBI table number], the difference between the ar-
ithmetic and geometric mean is much larger for risky large company
stocks than it is for nearly riskless Treasury bills.™™
The GM measures the magnitude of the returns as the investor starts
with one portfolio value and ends with another. It does not measure the
variability (volatility) of the journey, as does the AM.1 The GM is back-
ward looking, while the AM is forward looking (Ibbotson Associates
1997). As Mark Twain said, ˜˜Forecasting is dif¬cult”especially into the
future.™™


Table 5-1: Comparison of Two Stock Portfolios
Table 5-1 contains an illustration of two differing stock series. The ¬rst is
highly volatile, with a standard deviation of returns of 65% (C17), while
the second has a zero standard deviation. Although the arithmetic mean


1. Technically it is the difference of the AM and GM that measures the volatility. Put another way,
the AM consists of two components: the GM plus the volatility.


PART 2 Calculating Discount Rates
170
T A B L E 5-1

Geometric versus Arithmetic Returns


A B C D E

4 (Stock (or Portfolio) #1 Stock (or Portfolio) #2

5 Year Price Annual Return Price Annual Return

6 0 $100.00 NA $100.00 NA
7 1 $150.00 50.0000% $111.61 11.6123%
8 2 $68.00 54.6667% $124.57 11.6123%
9 3 $135.00 98.5294% $139.04 11.6123%
10 4 $192.00 42.2222% $155.18 11.6123%
11 5 $130.00 32.2917% $173.21 11.6123%
12 6 $79.00 39.2308% $193.32 11.6123%
13 7 $200.00 153.1646% $215.77 11.6123%
14 8 $180.00 10.0000% $240.82 11.6123%
15 9 $250.00 38.8889% $268.79 11.6123%
16 10 $300.00 20.0000% $300.00 11.6123%
17 Standard deviation 64.9139% 0.0000%
18 Arithmetic mean 26.6616% 11.6123%
19 Geometric mean 11.6123% 11.6123%




differs signi¬cantly for the two, both give rise to an identical geometric
mean return. It makes no sense intuitively that the GM is the correct one.
That would imply that both stocks are equally risky since they have the
same GM; yet no one would really consider stock #2 equally as risky as
#1. A risk-averse investor will always pay less for #1 than for #2.

EMPIRICAL EVIDENCE OF THE SUPERIORITY OF THE
ARITHMETIC MEAN
Much of the remainder of this chapter is focused on empirical evidence
of the superiority of the AM using the log size model. The heart of the
evidence in favor of the AM can be found in Chapter 4, Table 4-1, which
demonstrates that the arithmetic mean of stock market portfolio returns
correlate very well (98% R 2) with the standard deviation of returns, i.e.,
risk as well as the logarithm of ¬rm size, which is related to risk. We
show that the AM correlates better with risk than the GM. Also, the de-
pendent variable (AM returns) is consistent with the independent variable
(standard deviation of returns) in the regression. The latter is risk, and
the former is the fully risk-impounded rate of return. In contrast, the GM
does not fully impound risk.

Table 5-2: Regressions of Geometric and Arithmetic
Returns for 1927“1997
Table 5-2 contains both the geometric and arithmetic means for the Ib-
botson deciles for 1926“1997 data2 and regressions of those returns


2. Note that this will not match Table 4-1, because the latter contains data through 1998. While
both chapters were originally written in the same year, we chose to update all of the
regressions in Chapter 4 to include 1998 stock market data, while we did not do so in this
and other chapters.


CHAPTER 5 Arithmetic versus Geometric Means 171
T A B L E 5-2

Geometric versus Arithmetic Returns: NYSE Data by Decile & Statistical Analysis:
1926“1997


A B C D E F

5 Geometric Arithmetic Avg Cap

Mean
6 Decile Mean Return Std Dev FMV [1] Ln(FMV)

7 1 10.17% 11.89% 18.93% $28,650,613,989 24.0784
8 2 11.30% 13.68% 22.33% $5,987,835,737 22.5130
9 3 11.67% 14.29% 24.08% $3,066,356,194 21.8438
10 4 11.86% 14.99% 26.54% $1,785,917,011 21.3032
11 5 12.33% 15.75% 27.29% $1,126,473,849 20.8424
12 6 12.08% 15.82% 28.38% $796,602,581 20.4959
13 7 12.17% 16.39% 30.84% $543,164,462 20.1129
14 8 12.40% 17.46% 35.57% $339,165,962 19.6420
15 9 12.54% 18.21% 37.11% $209,737,489 19.1614
16 10 13.85% 21.83% 46.14% $68,389,789 18.0407
17 Std dev 0.94% 2.7%
18 Value wtd index 10.7% 12.6%

20 Regression #1: Return f(Std Dev. of Returns)

22 Arithmetic Geometric
23 Mean Mean

24 Constant 5.90% 8.76%
25 Std err of Y est 0.32% 0.36%
26 R squared 98.76% 86.93%
27 Adjusted R squared 98.60% 85.29%
28 No. of observations 10 10
29 Degrees of freedom 8 8
30 X coef¬cient(s) 34.19% 11.05%
31 Std err of coef. 1.35% 1.52%
32 T 25.2 7.2
33 P .01% 0.01%

35 Regression #2: Return f [Ln(FMV)]
37 Arithmetic Geometric
38 Mean Mean

39 Constant 47.62% 22.90%
40 Std err of Y est 0.76% 0.27%
41 R squared 93.16% 92.79%
42 Adjusted R squared 92.30% 91.89%
43 No. of observations 10 10
44 Degrees of freedom 8 8
45 X coef¬cient(s) 1.52% 0.52
46 Std err of coef. 0.15% 0.05%
47 T 10.4 10.1
48 P 0.01% 0.01%

[1] See Table 4-1 of Chapter 4 for speci¬c inputs and method of calcuation




PART 2 Calculating Discount Rates
172
against the standard deviation of returns and the natural logarithm of the
average market capitalization of the ¬rms in the decile. It is a repetition
of Table 4-1, with the addition of the GM data.
The arithmetic mean outperforms3 the geometric mean in regression
#1, with adjusted R 2 of 98.60% (C27) versus 85.29% (D27) and t-statistic
of 25.2 (C32) versus 7.2 (D32). In regression #2, which regresses the return
as a function of log size, the arithmetic mean slightly outperforms the
geometric mean in terms of goodness of ¬t with the data. Its adjusted
R 2 is 92.3% (C42), compared to 91.9% (D42) for the geometric mean. The
absolute value of its t-statistic is 10.4 (C47), compared to 10.1 (D47) for
the geometric mean. However, the geometric mean does have a lower
standard error of the estimate.


Table 5-3: Regressions of Geometric Returns
for 1938“1997
In Chapter 4 we discussed the relative merits of using the log size model
based on the past 60 years of NYSE return data rather than 73 years.
Table 5-3 shows the regression of ln (FMV) against the geometric mean
for the 61-year period 1937“1997.
Comparing the results in Table 5-3 to Table 4-1, the arithmetic mean
signi¬cantly outperforms the geometric mean. Looking at Regression #2,
the Adjusted R 2 in Table 4-1, cell E45 for the arithmetic mean is 99.54%,
while the geometric mean adjusted R 2 in Table 5-3, B22 is 81.69%. The t-
statistic for the AM is 44.1 (Table 4-1, E50), while it is 6.41 (D34) for
the GM. The standard error of the estimate is 0.34% (Table 4-1, E43) for
the AM versus 0.47% for the GM.4 Looking at Regression #1, in Table
4-1, E30, Adjusted R 2 for the AM is 95.31%, while it is 51.52% (B41) for
the GM. T-statistics are 13.6 for the AM (Table 4-1, E35) and 3.3 (D53)
for the GM. The standard error of the estimate is 0.42% (Table 4-1, E28)
for the AM and 0.76% (B42) for the GM. Using the past 60 years of data,
the AM signi¬cantly outperforms the GM by all measures.
GM does correlate to risk. Its R 2 value in the various regressions is
reasonable, but it is just not as good a measure of risk as the AM.
Eliminating the volatile period of 1926“1936 reduces the difference
between the geometric and arithmetic means in the calculation of dis-
count rates. We illustrate this at the bottom of Table 4-3, where discount
rates are compared for a $20 million and $300,000 FMV ¬rm using both
regression equations. For the $20 million ¬rm, the difference in discount
rate decreases from 7.9% (E57) using the 72-year equations to 4.9% (E58)
for the 60-year equations. We see a larger difference for smaller ¬rms, as
shown in Rows 59“60 for the $300,000 FMV ¬rm. In this case, the differ-
ence in discount rates falls from 12.1% (E59) to 7.5% (E60), or almost by
half.



3. In other words, the AM is more highly correlated with risk than the GM.
4. The standard error was 0.14% for the AM for the years 1938“1997.




CHAPTER 5 Arithmetic versus Geometric Means 173
T A B L E 5-3

Geometric Mean versus FMV: 60 Years


A B C D E F G

4 Year End Index Value [1]

5 Decile 1937 1997 GM 1937“1997 [2] Ln FMV Std Dev.

6 1 1.369 1064.570 11.732% 24.0784 15.687%
7 2 1.345 2232.833 13.154% 22.5130 17.612%
8 3 1.182 2834.406 13.849% 21.8438 18.758%
9 4 1.154 3193.072 14.121% 21.3032 20.704%
10 5 1.141 4324.787 14.721% 20.8424 21.829%
11 6 0.983 3686.234 14.701% 20.4959 22.750%
12 7 0.957 3906.82 14.863% 20.1129 24.909%
13 8 0.894 4509.832 15.269% 19.6420 26.859%
14 9 1.093 4958.931 15.066% 19.1614 28.415%
15 10 2.647 11398.583 14.966% 18.0407 36.081%

17 SUMMARY OUTPUT: GM vs Ln FMV, 60 years
19 Regression Statistics

20 Multiple R 91.50%
21 R square 83.73%
22 Adjusted R square 81.69%
23 Standard error 0.47%
24 Observations 10

26 ANOVA

27 df SS MS F Signi¬cance F

28 Regression 1 0.0009 0.0009 41.1611 0.0002
29 Residual 8 0.0002 0.0000
30 Total 9 0.0011


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

33 Intercept 26.20% 1.87% 14.0 0.00% 21.89% 30.51%
34 Ln (FMV) 0.57% 0.09% 6.4 0.02% 0.78% 0.37%

36 SUMMARY OUTPUT: GM vs. Std. Dev., 60 Years

38 Regression Statistics

39 Multiple R 75.44%
40 R square 56.91%
41 Adjusted R square 51.52%
42 Standard error 0.76%
43 Observations 1000.00%




The Size Effect on the Arithmetic versus Geometric Means
It is useful to note that the greater divergence between the AM and GM
as ¬rm size decreases and volatility increases means that using the GM
results in overvaluation that is inversely related to size, i.e., using the GM
on a small ¬rm will cause a greater percentage overvaluation than using
the GM on a large ¬rm.



PART 2 Calculating Discount Rates
174
T A B L E 5-3 (continued)

Geometric Mean versus AFMV: 60 Years

A B C D E F G

45 ANOVA
46 df SS MS F Signi¬cance F

47 Regression 1 0.0006 0.0006 10.5650 0.0117
48 Residual 8 0.0005 0.0001
49 Total 9 0.0011

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

52 Intercept 11.04% 1.01% 10.9 0.00% 8.70% 13.38%
53 Std dev. 13.71% 4.22% 3.3 1.17% 3.98% 23.44%

<<

. 7
( 18)



>>