5

4 7

15% 3

2

1

10%

5%

0%

0% 4% 8% 12% 16% 20% 24% 28% 32% 36% 40% 44%

Standard Deviation of Returns

These are arithmetic mean returns for the CRSP deciles. Data labels are decile numbers.

Y intercept is regression data, not actual

Regression #1: r = 6.56% + (31.24% x Std Dev of Decile)

(or portfolio) with higher risk than another unless the expected return is

also higher. It is still a relatively new observation that we can see this

relationship in the size of the ¬rms. Figure 4-1 shows this relationship

graphically, and the regressions in Table 4-1 that follow demonstrate that

relationship mathematically.

Regression #1 in Table 4-1 (Rows 23“36) is a statistical measurement

of return as a function of standard deviation of returns. The results for

the period 1926“1998 (D26“D36) con¬rm that a very strong relationship

exists between historical returns and standard deviation. The regression

equation is:

r 6.56% (31.24% S) (4-1)

where r return and S standard deviation of returns.

2

The adjusted R for equation (4-1) is 98.82% (D30), and the t-statistic

of the slope is 27.4 (D35). The p-value is less than 0.01% (D36), which

means the slope coef¬cient is statistically signi¬cant at the 99.9% level.

The standard error of the estimate is 0.27% (D28), also indicating a high

degree of con¬dence in the results obtained. Another important result is

that the constant of 6.56% (D26) is the regression estimate of the long-

term risk-free rate, i.e., the rate of return for a no-risk (zero standard

deviation) asset. The 73-year arithmetic mean income return from 1926“

CHAPTER 4 Discount Rates as a Function of Log Size 125

1998 on long-term Treasury Bonds is 5.20%.4 Therefore, in addition to the

other robust results, the regression equation does a reasonable job of es-

timating the risk-free rate. In prior years the regression estimate was

much closer to the historical average risk-free rate, but very strong per-

formance of large cap stocks in 1995“1998 has weakened this relationship.

We will temporarily ignore the 1938“1998 data in Column E and address

that later on in the chapter.

The major problem with direct application of this relationship to the

valuation of privately held businesses is coming up with a reliable stan-

dard deviation of returns. Appraisers cannot directly measure the stan-

dard deviation of returns for privately held ¬rms, since there is no objec-

tive stock price. We can measure the standard deviation of income, and

we cover that later in the chapter in our discussion of Grabowski and

King (1999).

Regression #2: Return versus Log Size

Fortunately, there is a much more practical relationship. Notice that the

returns are negatively correlated with the market capitalization, that is,

the fair market value of the ¬rm. The second regression in Table 4-1 (D42“

D51) is the more useful one for valuing privately held ¬rms. Regression

#2 shows return as a function of the natural logarithm of the FMV of the

¬rm. The regression equation for the period 1926“1998, which comes from

cells D42 and D48, is as follows:

r 42.24% [1.284% ln (FMV) ] (4-2)

The adjusted R 2 is 89.2% (D45), the t-statistic is 8.7 (D50), and the

p-value is less than 0.01% (D51), meaning that these results are statistically

robust. The standard error of the Y-estimate is 0.82% (D43). As discussed

in Chapters 2 and 11, we can form an approximate 95% con¬dence in-

terval around the regression estimate by adding and subtracting two stan-

dard errors. Thus, we can be 95% con¬dent that the regression forecast

1.6%.5

is approximately 2 0.82%

Figure 4-2 is a graph of arithmetic mean returns over the past 73

years (1926“1998) versus the natural log of FMV. As in Figure 4-1, the

numbered nodes are the actual data for each decile, while the straight

line is the regression estimate. While Figure 4-1 shows that returns are

positively related to risk, Figure 4-2 shows they are negatively related to

size.

Regression #3: Return versus Beta

The third regression in Table 4-1 shows the relationship between the dec-

ile returns and the decile betas for the period 1926“1998 (D56“D65). Ac-

cording to the capital asset pricing model (CAPM) equation, the y-

4. SBBI-1999, p. 140 uses this measure as the risk-free rate for CAPM. Arguably, the average bond

yield is a better measure of the risk-free rate, but the difference is immaterial.

5. This is true near the mean value of our data. Uncertainty increases gradually as we move from

the mean.

PART 2 Calculating Discount Rates

126

F I G U R E 4-2

1926“1998 Arithmetic Mean Returns as a Function of Ln(FMV)

45%

40%

35%

1926-1998 Arithmetic Mean Returns

30%

25%

10

20%

8

9 5

43

7

15%

6

1

2

10%

5%

0%

0 5 10 15 20 25 30

Ln(FMV)

These are arithmetic mean returns for the CRSP Deciles. Data labels are decile numbers.

Y intercept is regression data, not actual

Regression #2: r = 42.24% - [1.284% x Ln(FMV)]

intercept should be the risk-free rate and the x-coef¬cient should be the

long-run equity premium of 8.0%.6 Instead, the y-intercept at 2.78%

(D56) is a country mile from the historical risk-free rate of 5.20%, as is

the x-coef¬cient at 15.75% from the equity premium of 8.0%, demonstrat-

ing the inaccuracy of CAPM.

While the equation we obtain is contrary to the theoretical CAPM, it

does constitute an empirical CAPM, which could be used for a ¬rm

whose capitalization is at least as large as a decile #10 ¬rm. Merely select

the appropriate decile, use the beta of that decile, possibly with some

adjustment, and use regression equation #3 to generate a discount rate.

While it is possible to do this, it is far better to use regression #2.

The second page of Table 4-1 compares the log size model to CAPM.

Columns L and O show the regression estimated return for each decile

using both models”Column L for CAPM and O for log size. The CAPM

expected return was calculated using the CAPM equation: r RF

( Equity Premium) 5.20% ( 8.0%).

Columns M and N show the error and squared error for CAPM,

whereas columns P and Q contain the same information for the log size

6. SBBI-1999, p. 164.

CHAPTER 4 Discount Rates as a Function of Log Size 127

model. Note that the CAPM standard error of 1.89% (N20) is 230% larger

than the log size standard error of 0.82% (Q20). Later in this chapter we

use only the last 60 years of NYSE data, and its standard error for the

log size model is 0.34% (Q23), only 18% of the CAPM error.

The differences in the log size versus CAPM calculations for the 60

years of stock market data ending in 1997 were far more pronounced.

The reason is that for 1995“1998, returns to large cap stocks were higher

than small cap stocks, with 1998 being the most extreme example. For the

four years, the arithmetic mean return to decile #1 ¬rms was 31.2%, and

for decile #10 ¬rms it was 11.1%”contrary to long-term trends. In 1998,

returns to decile #1 ¬rms were 28.5%, and returns to decile #10 ¬rms were

15.4%. Thus, the regression equation was much better at the end of 1997

than at the end of 1998. The 1938“1997 adjusted R 2 was 99.5% (versus

97.0% for 1939“1998), and the standard error of the y-estimate was 0.14%

(versus 0.34% for 1939“1998).

Market Performance

Regression #1 shows that return is a linear function of risk, as measured

by the standard deviation of returns. Regression #2 shows that return

declines linearly with the logarithm of ¬rm size. The logic behind this is

that investors demand and receive higher returns for higher risk. Smaller

¬rms have more volatile (risky) returns, so return is therefore negatively

related to size.

Figure 4-3 shows the relationship between volatility and size, with

the y-axis being the standard deviation of returns for the value-weighted

F I G U R E 4-3

Decade Standard Deviation of Returns versus Decade Average FMV per Company on NYSE 1935“1995

35%

1935

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