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slightly over $0.5 billion to over $1.9 billion. Therefore, we might predict
from a theoretical standpoint that the standard deviation of returns
should decline over time”and it has.
As you can see, the standard deviation of returns per decade declines
exponentially from about 33% for the decade ending in 1935 to 13% in
the decade ending in 1995, for a range of 20%. If we examine the major
historical events that took place over time, the decade ending 1935 in-
cludes some of the Roaring Twenties and the Depression. It is no surprise
that it has such a high standard deviation. Figure 4-4 is identical to Figure
4-3, except that we have eliminated the decade ending 1935 in Figure
4-4. Eliminating the most volatile decade results in a ¬‚attening out of the
regression curve. The ¬tted curve in Figure 4-4 appears about half as steep
as Figure 4-3 (the standard deviation ranges from 13“22%, or a range of
9%, versus the 20% range of Figure 4-3) and much less curved.
The relationship between volatility and size when viewing the mar-
ket as a whole is somewhat loose, as the data points vary considerably
from the ¬tted curve in Figure 4-3. The R 2 52% (45% in Figure 4-4).


F I G U R E 4-4

Decade Standard Deviation of Returns versus Decade Average FMV per Company on NYSE 1945“1995

35%




30%
Decade Std Dev of Returns
Value Weighted NYSE




y = -0.0449Ln(x) + 0.1768
25% 2
R = 0.4487

1945


20%

1975
1965
1955


15%

1985

1995

10%
0.50 0.70 0.90 1.10 1.30 1.50 1.70 1.90
Decade Avg FMV per Company on NYSE (Billions of 1995 constant dollars)
(X axis derived from NYSE Fact Book, NYSE Research Library)
(Y axis derived from SBBI-1999 pp. 134-135)




7. Though 1996“1998 data are available, we choose to stop at 1995 in this graph to maintain 10
years of data in each node on the graph.




CHAPTER 4 Discount Rates as a Function of Log Size 129
For the decade ending 1945, standard deviation of returns is about one-
third lower than the previous decade (approximately 22% versus 33%),
while average ¬rm size is about the same. Standard deviation of returns
dropped again in the decade ending 1955, with only a small increase in
size. In the decade ending 1965, average ¬rm size more than doubled in
real terms, yet volatility was almost identical (we would have expected
a decrease). In the decade ending 1975, ¬rm size and volatility increased.
In the decade ending 1985, both average ¬rm size and volatility decreased
signi¬cantly, which is counterintuitive, while in the ¬nal decade ¬rm size
increased from over $1.3 billion to almost $2 billion, while volatility de-
creased slightly.
Figure 4-5 shows the relationship of average NYSE return and time,
with each data point being a decade. The relationship is a very loose one,
with R 2 0.09. The decade ending 1975 appears an outlier in this re-
gression, with average returns at half or less of the other decades (except
the one ending 1935). The regression equation is return 1.0242
(0.0006 Year). Since every decade is 10 years, this equation implies
returns increase 0.6% every 10 years. However, the relationship is not
statistically signi¬cant.
In summary, there appears to be increasing ef¬ciency of investment
over time. The market as a whole seems to deliver the same or better


F I G U R E 4-5

Average Returns Each Decade

18%



16% 1955
1995
1985


14%
1965

1945
12%
Value Weighted NYSE




10%
Return




y = 0.0006x - 1.0262
R2 = 0.0946
1935
8%



6%

1975

4%



2%



0%
1930 1940 1950 1960 1970 1980 1990 2000
Decade Ending




PART 2 Calculating Discount Rates
130
performance as measured by return experienced for risk undertaken. We
can speculate on explanations for this phenomenon: increases in the size
of the NYSE ¬rms, greater investor sophistication, professional money
management, and the proliferation of mutual funds. In any case, the risk
of investing in one portfolio (or ¬rm) relative to others still matters very
much. This may possibly be the phenomenon underlying the observations
of the nonstationarity of the data.


Which Data to Choose?
With a total of 73 years of data on the NYSE, we must decide whether
to use all of the data or some subset, and if so, which subset. In making
this choice, we will consider three sources of information:
1. Tables 4-2 and 4-2A, the statistical results of regression analyses
of the different time periods of the NYSE.
2. A study (Harrison 1998) that explores the distribution of 18th
century European stock market returns.
3. Figures 4-3 and 4-4.


Tables 4-2 and 4-2A: Regression Results for
Different Time Periods
Nonstationary data require us to consider the possibility of removing
some of the older NYSE data. In Table 4-2 we repeat regressions #1 and
#2 from Table 4-1 for the most recent 30, 40, 50, 60, and 73 years of NYSE
data. The upper table in each time period is regression #1 and the lower
table is regression #2. For example, the data for regression #1 for the last
30 years appear in Rows 7“9, 40 years in Rows 17“19, and so on. Simi-
larly, the data for regression #2 for 30 years appear in Rows 12“14, 40
years in Rows 22“24, and so on.
Table 4-2, Rows 8“14, shows regressions #1 and #2 using only the
past 30 years of data, i.e., from 1969“1998.8 Regression equation #1 for
this period is: r 14.64% (2.37% S) (B8, B9), and regression equation
#2 is r 14.14% [0.001% ln (FMV)] (B13 and B14). Note that both
the slope coef¬cient and the intercept of these equations are different from
those obtained for 73 years of data.
Rows 47“49 repeat regression #1 for the same 73 years as Table 4-1.
The y-intercept of 6.56% (B48) and the x-coef¬cient of 31.24% (B49) in
Table 4-2 are identical to those appearing in Table 4-1 (D26 and D33,
respectively). Rows 52“54 repeat regression #2 for the same period. Once
again, the y-intercept in Table 4-2 of 42.24% (B53) and the coef¬cient of
ln (FMV) of 1.284% (B54) match those found in Table 4-1 (D42 and D48,
respectively).
Table 4-2A summarizes the key regression feedback from Table 4-2.
For the ¬ve different time periods we consider, the 60-year period is sta-


8. The time sequence in Table 4-2 differs by two years from that in Figures 4-3 to 4-6. Whereas the
latter show decades ending in 19X5 (e.g., 1945, 1955, etc.), Table 4-2™s terminal year is 1998.




CHAPTER 4 Discount Rates as a Function of Log Size 131
T A B L E 4-2

Regressions of Returns over Standard Deviation and Log of Fair Market Value


A B C D E F G H I

6 30 Year

7 Coef¬cients Standard Error t Stat P-value Lower 95% Upper 95%
R square 1.35%
8 Intercept 14.64% 1.62% 9.06 0.00% 10.92% 18.37% Adjusted R square 10.98%
9 Std Dev 2.37% 7.18% 0.33 74.92% 18.92% 14.17% Standard error 0.90%
12 R square 0.00%
13 Intercept 14.14% 3.39% 4.17 0.31% 6.32% 21.95% Adjusted R square 12.50%
14 Ln(FMV) 0.001% 0.164% 0.01 99.54% 0.38% 0.38% Standard error 0.90%

16 40 Year

17 R square 67.84%
18 Intercept 10.13% 1.17% 8.66 0.00% 7.43% 12.82% Adjusted R square 63.82%
19 Std Dev 21.74% 5.29% 4.11 0.34% 9.53% 33.94% Standard error 0.75%
22 R square 78.94%
23 Intercept 27.30% 2.28% 11.95 0.00% 22.03% 32.57% Adjusted R square 76.31%
24 Ln FMV 0.605% 0.110% 5.48 0.06% 0.86% 0.35% Standard error 0.61%

26 50 Year
27 R square 77.28%
28 Intercept 11.54% 0.89% 13.00 0.00% 9.49% 13.58% Adjusted R square 74.44%
29 Std Dev 20.61% 3.95% 5.22 0.08% 11.50% 29.72% Standard error 0.54%
32 R square 89.60%
33 Intercept 27.35% 1.36% 20.08 0.00% 24.21% 30.49% Adjusted R square 88.30%
34 Ln(FMV) 0.546% 0.066% 8.30 0.00% 0.70% 0.39% Standard error 0.36%

36 60 Year

37 R square 95.84%
38 Intercept 8.90% 0.55% 16.30 0.00% 7.64% 10.16% Adjusted R square 95.31%
39 Std Dev 30.79% 2.27% 13.57 0.00% 25.56% 36.03% Standard error 0.42%
42 R square 97.29%
43 Intercept 37.50% 1.27% 29.57 0.00% 34.58% 40.43% Adjusted R square 96.95%
44 Ln(FMV) 1.039% 0.061% 16.94 0.00% 1.18% 0.90% Standard error 0.34%

46 73 Year
47 R square 98.95%
48 Intercept 6.56% 0.35% 18.94 0.00% 5.76% 7.36% Adjusted R square 98.82%
49 Std Dev 31.24% 1.14% 27.42 0.00% 28.61% 33.87% Standard error 0.27%
52 R square 90.37%
53 Intercept 42.24% 3.07% 13.78 0.00% 35.17% 49.32% Adjusted R square 89.17%
54 Ln(FMV) 1.284% 0.148% 8.66 0.00% 1.63% 0.94% Standard error 0.82%




tistically a solid winner. Regression #2 is the more important regression
for valuing privately held ¬rms, and the 60-year standard error at 0.34%
(C9) is the lowest among the ¬ve listed. The standard error of the y-
estimate using all 73 years of data (1.09%, D10) is larger than the 60-year
standard error (0.82%; C10). The next-lowest standard error is 0.90% (D8)
for 50 years of data, which is still larger than the 60-year regression. The
60-year regression also has the highest R 2 ”97% (E9)”and it has a low
standard error for regression #1, second only to the full 73 years.
The 95% con¬dence intervals for the 60 years of data are smaller than
they are for the other candidates. For regression #2 they are between

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