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%