<< . .

. 20
( : 66)



. . >>

30%
Decade Std Dev of Returns
Value Weighted NYSE




y = -0.0878Ln(x) + 0.1967
25%
2
R = 0.5241

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)




PART 2 Calculating Discount Rates
128
NYSE and the x-axis being the average FMV per NYSE company in 1995
constant dollars in successive decades.7 The year adjacent to each data
point is the ¬nal year of the decade, e.g., 1935 encompasses 1926 to 1935.
The decade average FMV (in 1995 constant dollars) has increased from
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

PART 2 Calculating Discount Rates
132
T A B L E 4-2A (continued)

Regression Comparison [1]


A B C D E

4 Standard Errors

Adj R2 (Regr #2) [4]
5 Years Regr #1 [2] Regr #2 [3] Total

6 30 0.90% 0.90% 1.80% 12.50%
7 40 0.75% 0.61% 1.36% 76.31%
8 50 0.54% 0.36% 0.90% 88.30%
9 60 0.42% 0.34% 0.76% 96.95%
10 73 0.27% 0.82% 1.09% 89.17%

[1] Summary Regression Statistics from Table 4-2
[2] Table 4-2: I9, I19, ...
[3] Table 4-2: I14, I24, ...
[4] Table 4-2: I13, I23, ...




34.58% and 40.43% (Table 4-2, F43, G43) for the y-intercept”a range of
5.8%”and 1.18% to 0.90% (F44, G44) for the slope”a range of 0.28%.
For 73 years of data, the range is 14% for the y-intercept (G53“F53) and
0.69% (G54“F54) for the slope, which is 21„2 times larger than the 60-year
data. Thus, the past 60 years data are a more ef¬cient estimator of stock
market returns than other time periods, as measured by the size of con-
¬dence intervals around the regression estimates for the log size ap-
proach.

18th Century Stock Market Returns
Paul Harrison™s article (Harrison 1998) is a fascinating econometric study
which is very advanced and extremely mathematical. The data for this
study came primarily from biweekly Amsterdam stock prices published
from July 1723 to December 1794 for the Dutch East India Company and
a select group of English stocks that were traded in Amsterdam: the Bank
of England, the English East India Company, and the South Sea Company.
Harrison also examined stock prices from London spanning the 18th cen-
tury.
Harrison found the shape of the distribution of stock price returns
in the 18th and 20th centuries to be very similar, although their means
and standard deviations are different. The 18th century returns were
lower”but less volatile”than 20th century returns. He found the distri-
butions to be symmetric, like a normal curve, but leptokurtic (fat tailed),
which means there are more extreme events occurring than would be
predicted by a normal curve. The same fundamental pattern exists in both
1725 and 1995.
Harrison remarks that clearly much has changed over the last 300
years, but, interestingly, such changes do not seem to matter in his anal-
ysis. He comments that the distribution of prices is not driven by infor-
mation technology, regulatory oversight, or by the specialist”none of
these existed in the 18th century markets. However, what did exist in the
18th century bears resemblance to what exists today.
Harrison describes the following as some of the evidence for simi-
larities in the market:

CHAPTER 4 Discount Rates as a Function of Log Size 133
—Stock traders in the 18th century reacted to and affected market
prices like traders today. They competed vigorously for
information,9 and the 18th century markets followed a near
random walk”so much so that an entire pamphlet literature
sprang up in the early 18th century lamenting the
unpredictability of the market. Harrison says that
unpredictability is a theoretical result of competition in the
market.
— Eighteenth century stock markets were informationally ef¬cient,
as shown econometrically by Neal (1990).
— The practices of 18th century brokers were sophisticated.
Investors early in the 18th century valued stocks according to
their discounted stream of future dividends. Tables were
published (such as Hayes 1726) showing the appropriate
discount for different interest rates and time horizons. Traders
engaged in cash contracts, futures contracts, and options; they
sold short, issued credit, and used ˜˜modern™™ investment
strategies, such as forming portfolios, diversi¬cation, and
hedging.
To all of the foregoing, I would add an observation by King
Solomon, who said, ˜˜There is nothing new under the sun.™™ (Ecclesiastes
1:9) Also in keeping with the theme in our chapter, King Solomon
became the inventor of portfolio theory when he wrote, ˜˜Divide your
wealth into seven, even eight parts, for you cannot know what
misfortune may occur on earth™™ (Ecclesiastes, 11:2).

Conclusion on Data Set
To return to the 20th century, Ibbotson (Ibbotson Associates 1998, p. 27)
enunciated the principle that over the very long run there are very few
events that are truly outliers. Paul Harrison™s research seems to corrob-
orate this. It is in the nature of the stock market for there to be periodic
booms and crashes, indicating that we should use all 73 years of the
NYSE data. On the other hand, the statistical feedback in Table 4-2A
shows that eliminating the 1926“1938 data provides the most statistically
reliable log size relationship. Similarly, Figure 4-4 shows a ¬‚attening of
the regression curve when the decade ending 1935 is eliminated. Paul

<< . .

. 20
( : 66)



. . >>