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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
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
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
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.

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

Arithmetic versus Geometric
Means: Empirical Evidence and
Theoretical Issues

Table 5-1: Comparison of Two Stock Portfolios
Table 5-2: Regressions of Geometric and Arithmetic Returns for
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


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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.

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.

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

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
T A B L E 5-1

Geometric versus Arithmetic Returns


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%


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( 100 .)