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%

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

the AM for such a small ¬rm (this is assuming a ¬rm with 6% constant

growth). Notice that the ratio declines continuously as we move down

Column F. The overvaluation of a $10 billion ¬rm using the GM is

12.57%”far less than the overvaluation of the $250,000 ¬rm. The differ-

ences are signi¬cantly greater when using the 72-year log size models, as

including the most volatile years in the regression makes for a greater

difference in the AM versus GM Gordon model multiples. These numer-

ical examples underscore the importance of using the arithmetic mean

when valuing expected future earnings or cash ¬‚ow.

INDRO AND LEE ARTICLE

This article (Indro and Lee 1997) is extremely mathematical, exceedingly

dif¬cult reading. The authors begin by citing (Brealey and Myers 1991),

CHAPTER 5 Arithmetic versus Geometric Means 175

T A B L E 5-4

Comparison of Discount Rates Derived from the Log Size Model Using 60-Year

Arithmetic and Geometric Means

A B C D E F

5 Gordon Model Ratio

Multiples Using

6 Firm Size AM [1] GM [2] AM [3] GM [3] GG / AG [4]

7 $250,000 26.76% 19.12% 5.42 8.32 153.41%

8 $1,000,000 25.09% 18.33% 5.86 8.83 150.61%

9 $25,000,000 21.21% 16.49% 7.24 10.29 142.14%

10 $50,000,000 20.38% 16.10% 7.63 10.67 139.85%

11 $100,000,000 19.54% 15.70% 8.07 11.09 137.34%

12 $500,000,000 17.60% 14.78% 9.35 12.20 130.52%

13 $10,000,000,000 14.00% 13.08% 13.35 15.03 112.57%

Conclusion: The ratio of Gordon Model Multiples decreases with ¬rm size (Column F)

Notes:

[1] Arithmetic Mean (AM) Regression Equation, 60 year model r 41.72% 0.01204 Ln (FMV)

[2] Geometric Mean (GM) Regression Equation, 60 year model. r 26.2% 0.0057 Ln (FMV)

[3] Gordon Model Multiple calculated assuming 6% growth in earnings-midyear assumption. Discount rates are not rounded in these

calculations.

[4] Geometric Gordon Model Multiple / Arithmetic Gordon Model Multiple

who say that if monthly returns are identically and independently dis-

tributed, then the arithmetic average of monthly returns should be used

to estimate the long-run expected return. They then cite empirical evi-

dence that there is signi¬cant negative autocorrelation in long-term equity

returns and that historical monthly returns are not independent draws

from a stationery distribution. This means that high returns in one time

period will tend to mean that on average there will be low returns in the

next period, and vice-versa. Based on this, Copeland, Koller, and Murrin

(1994) argue that the geometric average is a better estimate of the long-

run expected returns.

Indro and Lee show that the arithmetic and geometric means have

upward and downward biases, respectively, and that a horizon-weighted

average of the two is the least biased and most ef¬cient estimator.

If the authors are correct, it would mean that there would no longer

be a single discount rate. Every year would have its own unique

weighted-average discount rate. That would also add complexity to the

use of the Gordon model to calculate a residual value.

Because of the extremely dif¬cult mathematics in the article, it was

necessary to speak to academic sources to evaluate it. Professor Myers,

cited above, did agree that long-term (¬ve-year) returns are negatively

autocorrelated but that there are ˜˜very few data points.™™ He had not fully

read the article, is not sure of its signi¬cance, and did not have an opinion

of it. Ibbotson Associates does not feel the evidence for mean reversion

is that strong, and on that basis is not moved to change its opinion that

the AM is the correct mean. It seems that it will take some time before

this article gets enough academic attention to cause the valuation profes-

sion to make any changes in the way it operates.

PART 2 Calculating Discount Rates

176

BIBLIOGRAPHY

Brealey, R. A., and Stewart C. Myers. 1991. Principles of Corporate Finance. New York:

McGraw-Hill.

Copeland, Tom, Tim Koller, and Jack Murrin. 1994. Valuation: Measuring and Managing

the Value of Companies. John Wiley & Sons, Inc. New York, NY.

Ibbotson Associates. 1998. Stocks, Bills, Bonds and In¬‚ation: 1998 Yearbook. Chicago: The

Associates. 107“08; 153“155.

Indro, Daniel C., and Wayne Y. Lee. 1997. ˜˜Biases in Arithmetic and Geometric Averages

as Estimates of Long-Run Expected Returns and Risk Premia.™™ Financial Management

26, no. 4 (Winter): 81“90.

Joyce, Allyn A. 1995. ˜˜Arithmetic Mean vs. Geometric Mean: The Issue in Rate of Return.™™

Business Valuation Review ( June): 62“68.

CHAPTER 5 Arithmetic versus Geometric Means 177

CHAPTER 6

An Iterative Valuation Approach

INTRODUCTION

EQUITY VALUATION METHOD

Table 6-1A: The First Iteration

Table 6-1B: Subsequent Iterations of the First Scenario

Table 6-1C: Initial Choice of Equity Doesn™t Matter

Convergence of the Equity Valuation Method

INVESTED CAPITAL APPROACH

Table 6-2A: Iterations Beginning with Book Equity

Table 6-2B: Initial Choice of Equity Doesn™t Matter

Convergence of the Invested Capital Approach

LOG SIZE

SUMMARY

BIBLIOGRAPHY

179

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your best guess of the FMV of equity or you can use the net