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