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%

55 Comparison of Discount Rates Using 60 and 72 Year Models

56 FMV Regression Model Geometric Mean Arithmetic Mean Difference

57 $20,000,000 72 year 14.2% 22.1% 7.9%

58 60 year 16.6% 21.5% 4.9%

59 $300,000 72 year 16.3% 28.5% 12.1%

60 60 year 19.0% 26.5% 7.5%

[1] Values from Ibbotson™s 1998 SBBI Yearbook, Table 7-3

[vn / vo]1 / n 1

[2] Geometric mean for 1937-1997 was calculated using Year End Index Values for 1937 (for year starting 1938) and 1997 according to the formula rg

[3] From Table 4-1, Chapter 4

Table 5-4: Log Size Comparison of Discount Rates and

Gordon Model Multiples Using AM versus GM

Table 5-4 illustrates this, where discount rates are calculated using the log

size model, with both the arithmetic and geometric mean regression equa-

tions derived from Tables 4-1 and 5-3, respectively. There is a dramatic

difference in discount rates, especially with small ¬rms. The log size dis-

count rate for a $250,000 ¬rm is 26.76% using the AM (B7) and 19.12%

using the GM (C7). The resulting midyear Gordon model multiples are

5.42 (D7) using the AM and 8.32 (E7) using the GM.

Column F is the ratio of the Gordon model multiples using the ge-

ometric mean to the Gordon model multiples using the arithmetic mean.

Dividing the 8.32 GM multiple by the 5.42 AM multiple gives us a ratio

of 153.41%, i.e., the GM leads to a valuation that is 53.41% higher than