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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%




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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