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