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

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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.

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%