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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).

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
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
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
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

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
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


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.

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%

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