Notes:

[1] SBBI-1999, p. 140

[2] SBBI-1999, p. 164

T A B L E 4-1 (continued)

NYSE Data by Decile and Statistical Analysis: 1926“1998

J K L M N O P Q

M2 P2

4 Note [1] Note [5] B L B O

6 CAPM Regr #2 Regr #2

7 Decile Beta CAPM E(R) Error Sq Error Estimate Error Sq Error

8 1 0.90 12.40% 0.29% 0.0008% 11.19% 0.92% 0.0085%

9 2 1.04 13.52% 0.14% 0.0002% 13.42% 0.24% 0.0006%

10 3 1.09 13.92% 0.19% 0.0004% 14.45% 0.34% 0.0011%

11 4 1.13 14.24% 0.52% 0.0027% 15.15% 0.39% 0.0015%

12 5 1.16 14.48% 1.04% 0.0107% 15.76% 0.24% 0.0006%

13 6 1.18 14.64% 0.96% 0.0092% 16.27% 0.68% 0.0046%

14 7 1.23 15.04% 0.95% 0.0091% 16.76% 0.77% 0.0060%

15 8 1.27 15.36% 1.69% 0.0285% 17.26% 0.21% 0.0004%

16 9 1.34 15.92% 1.93% 0.0373% 18.10% 0.25% 0.0006%

17 10 1.44 16.72% 4.31% 0.1859% 19.32% 1.72% 0.0294%

Totals ’

19 0.2848% 0.0533%

Standard error ’

20 1.89% 0.82%

21 Std error-CAPM/std error-log size model 231.11%

23 Std error” 60 year model 0.34%

Notes

[1] Derived from SBBI-1999 pages 130, 131.*

[2] SBBI-1999, page 138**

[3] These averages derived from SBBI-1999, pages 200“201.* Beginning of year 1926 yield was not available.

[4] Betas were not available for the 1939“1998 time period.

[5] SBBI-1999, page 140*

[6] CAPM Equation: Rf (Beta Equity Premium) 5.2% (Beta 8.0%). The equity premium is the simple difference of historical arithmetic mean returns for large company stocks and the risk free rate per SBBI 1999 p. 164. The risk

free rate of 5.2% is the 73 year arithmetic mean income return component of 20 year government bonds per SBBI-1999, page 140.*

* Used with permission. 1999 Ibbotson Associates, Inc. All rights reserved. [Certain portions of this work were derived from copyrighted works of Roger G. Ibbotson and Rex Sinque¬eld.]

** Used with permission. 1999 Ibbotson Associates, Inc. All rights reserved. [Certain portions of this work were derived from copyrighted works of Roger G. Ibbottson and Rex Sinque¬eld.] Source: CRSP University of Chicago. Used

with permission. All rights reserved.

123

F I G U R E 4-1

1926“1998 Arithmetic Mean Returns as a Function of Standard Deviation

10

20%

8

9

6

1926-1998 Arithmetic Mean Returns

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

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

Harrison said that even with 300 years of history showing similarity in

the distribution of returns, he would be inclined to label the years in

question as an outlier that should probably be excluded from the regres-

9. A fascinating story that I remember from an economic history course is that Baron Rothschild,

having placed men with carrier pigeons at the Battle of Waterloo, was the ¬rst

nonparticipant to know the results of the battle. He ¬rst paid a visit to inform the King of

the British victory, and then he proceeded to the stock market to make 100 million

pounds”many billions of dollars in today™s money”a tidy sum for having insider

information. He struck a blow for market ef¬ciency. Even his method of making a fortune in

the market that day is a paradigm of the extent of market ef¬ciency then. He knew that he

was being observed. He began selling, and others followed him in a panic. Later, he sent his

employees to do a huge amount of buying anonymously. The markets were indeed

ef¬cient”at least they were by the end of the day!

PART 2 Calculating Discount Rates

134

sion.10 Thus, we eliminate the years 1926“1937 from the ¬nal regression.

The superior adjusted R 2 and 95% con¬dence intervals of the past 60

years, coupled with Harrison™s results and Ibbotson™s general principle of

using more rather than less data, lead us to conclude that the past 60

years provide the best guide for the future.

Recalculation of the Log Size Model Based on 60 Years

Based on our previous discussion, NYSE data from the past 60 years are

likely to be the most relevant for use in forecasting the future. This time

frame contains numerous data points but excludes the decade of highest

volatility, attributed to nonrecurring historical events, i.e., the Roaring

Twenties and the Depression years. Therefore, we repeat all three regres-

sions for the 60-year time period from 1939“1998, as shown in Column

E of Table 4-1. Regression #1 for this time period is:

r 8.90% (30.79% S) (4-3)

where S is the standard deviation. The adjusted R 2 in this case falls to

95.31% (E30) from the 98.82% (D30) obtained from the 73-year equation,

but is still indicative of a strong relationship. On average, returns were

exceptionally high and volatile during the ¬rst 13 years of the NYSE,

especially in the small ¬rms. It appears that including those years im-

proves the relationship of returns to standard deviation of returns, even

as it worsens the relationship between returns and log size.

The log size equation (regression #2) for the 60-year period is:

r 37.50% [1.039% ln (FMV)] (E42, E48) (4-4)

The regression statistics indicate an excellent ¬t, with an adjusted R 2 of

96.95% (E45).11

APPLICATION OF THE LOG SIZE MODEL

Equation (4-4) is the most appropriate for calculating current discount

rates and will be used for the remainder of the book. In the next sections

we will use it to calculate discount rates for various ¬rm sizes and dem-

onstrate its use in a simpli¬ed discounted cash ¬‚ow analysis.

Discount Rates Based on the Log Size Model

Table 4-3 shows the implied equity discount rate for ¬rms of various sizes

using the log size model (regression equation #2) for the past 60 years.

The implied equity discount rate for a $10 billion ¬rm is 13.6% (B7), and

for a $50 million ¬rm it is 19.1% (B10), based on 60-year average market

returns for deciles #1“#10. While those values and all values in between

are interpolations based on the model, the discount rates for ¬rm val-

10. Related in a personal conversation.

11. For 1938“1997 data, adjusted R 2 was 99.54%. The ˜˜perverse™™ results of 1998 caused a

deterioration in the relationship.

CHAPTER 4 Discount Rates as a Function of Log Size 135

T A B L E 4-3

Table of Stock Market Returns Based on FMV”60-Year Model

A B

5 Regression Results Implied Discount

6 Mktable Min FMV Rate (R)

7 $10,000,000,000 13.6%

8 $1,000,000,000 16.0%

9 $100,000,000 18.4%

10 $50,000,000 19.1%

11 $10,000,000 20.8%

12 $5,000,000 21.5%

13 $3,000,000 22.0%

14 $1,000,000 23.2%

15 $750,000 23.5%

16 $500,000 23.9%

17 $400,000 24.1%

18 $300,000 24.4%

19 $200,000 24.8%

20 $150,000 25.1%

21 $100,000 25.5%

22 $50,000 26.3%

23 $30,000 26.8%

24 $10,000 27.9%

25 $1,000 30.3%

26 $1 37.5%

ues below that are extrapolations because they lie outside the original

data set.

Using equation (4-4), the Excel formula for cell B7 is: 0.3750

(0.01039 * ln(A7)). In Lotus 123, the formula would be: 0.3750

(0.01039 * @ ln(A7)).

Regression #2 (equation [4-4]) tells us that the discount rate is a con-

stant minus another constant multiplied by ln (FMV). Since ln (FMV) has

a characteristic upwardly sloping shape, as seen in Figure 4-6, subtracting

a curve of that shape from a constant leads to a discount rate function

that is a mirror image of Figure 4-6. Figure 4-7 is the graph of that rela-

tionship, and the reader can see that the result is a downward sloping

curve. Again, this curve depicts the rate of return, i.e., the discount rate,

as a function of the absolute dollar value of the ¬rm. Note that this is not

on a log scale. Since the regression equation is r 37.50% [1.0309%

ln (FMV)], we begin at the extreme left with a return of 37.5% for a ¬rm

worth $1 and subtract the fraction of the ln FMV dictated by the equation.

ln y.12

An important property of logarithms is that ln xy ln x

Since regression equation #2 has the form r a b ln FMV, where a

0.3750 and b 0.01039, we can ask how the discount rate varies with

differing orders of magnitude in value. First, however, we will work

12. That is because e x ey e x y. Taking logs of both sides of that equation is the proof.

PART 2 Calculating Discount Rates

136

F I G U R E 4-6

The Natural Logarithm

20.00

18.00

16.00

14.00

12.00

Ln(FMV)

10.00

8.00

6.00

4.00

2.00

0.00

0 5 10 15 20 25 30

FMV ($Millions)

through some general equations where we vary the value of the ¬rm by

a factor of K.

Let r1 the discount rate for Firm #1, whose value FMV1

F I G U R E 4-7

Discount Rates as a Function of FMV

40%

35%

30%

25%

Discount Rate

20%

15%

10%

5%

0%

0 20 40 60 80 100 120

Fair Market Value, Marketable Minority ($ Millions)

For scaling reasons, we eliminate values above $100 million

CHAPTER 4 Discount Rates as a Function of Log Size 137

F I G U R E 4-8

1939“1998 Decile Standard Deviations as a Function of Ln(FMV)

100%

90%

80%

70%

Standard Deviation of Returns

60%

50%

40%

10

30% 9

Std Dev = -3.13% x Ln FMV + 87.77% 8

7 3

R2 = 0.9894

4

6

20%

2

5

3

1

10%

0%

0 5 10 15 20 25 30

Ln(FMV)

Standard deviations of yearly returns are derived from the CRSP Deciles. Data labels are decile numbers. The Y intercept is

the regression intercept, not an actual data point.

r2 the discount rate for Firm #2, whose value FMV2 K

FMV1

r1 a b ln FMV1 (4-6)

regression equation #2 applied to Firm #1

r2 a b ln (K FMV1) (4-7)

regression equation #2 applied to Firm #2

r2 a b [ln K ln FMV1] (4-8)

r2 a b ln FMV1 b ln K (4-9)

r2 r1 b ln K (4-10)

In words, the discount rate of a ¬rm K times larger (smaller) than Firm

#1 is always b ln K smaller (larger) than r1.

Let™s illustrate the nature of this relationship with some speci¬c ex-

amples. First, let™s examine what happens with orders of magnitude of

PART 2 Calculating Discount Rates

138

10. Ln 10 2.302535, so b ln 10 0.01039 2.302585 .02391, or

2.4%. This means that if Firm #2 is 10 times larger (smaller) than Firm

#1, its discount rate should be 2.4% lower (higher) than the Firm #1 dis-

count rate. This result can be seen in Table 4-3. The $10 billion ¬rm has

a discount rate of 13.6%, while the $1 billion ¬rm has a discount rate of

16.0%, which is 2.4% higher. The $100 million ¬rm has a discount rate of

18.4%, which is 2.4% higher than the $1 billion ¬rm. Because of the math-

ematical properties of logarithms, the same percentage change in FMV will

always result in the same absolute change in the discount rate. This phe-

nomenon is also seen in graphs containing log scales. Equal distances on

a log scale are equal percentage changes, not absolute changes.

Let™s try one more useful calculation”an order of magnitude 2. Ln

2 0.6931, so that b ln K 0.01039 0.6931 0.72%. Doubling

(halving) the value of the ¬rm reduces (increases) the discount rate by

0.72%. You can see that in going from a $10 million ¬rm to a $5 million

¬rm, the discount rate has increased from 20.8% to 21.5%, a 0.7% differ-

ence (see Table 4-3).

Now it is possible to construct your own table. All you need to know

is your starting FMV and discount rate. The rest follows easily from the

above formulas. Also, we can easily interpolate the table. Suppose you

wanted to know the discount rate for a $25 million ¬rm. Simply start

with the $50 million ¬rm, where r 19.1%, and add 0.7% 19.8%.

Need for Annual Updating

Tables 4-1 through 4-3 should be updated annually, as the Ibbotson av-

erages change, and new regression equations should be generated. This

becomes more crucial when shorter historical time periods are used, be-

cause changes will have a greater impact on the average values.

Additionally, it is important to be careful to match the regression

equation to the year of the valuation. If the valuation assignment is ret-

roactive and the valuation date is 1994, then one should use a regression

equation for 1939“1994.13

Computation of Discount Rate Is an Iterative Process

In spite of the straightforwardness of these relationships, we have a prob-

lem of circular reasoning when it comes to computing of the discount

rate. We need FMV to obtain the discount rate, which is in turn used to

discount cash ¬‚ows or income to calculate the FMV! Hence, it is necessary

to make sure that our initial estimate of FMV is consistent with the ¬nal

result. If it is not, then we have to use the calculated FMV from the end

of iteration #1 as our new assumed FMV in iteration #2. Using either

equation (4-4) or Table 4-3, that will imply a new discount rate, which

we use to value the ¬rm. We keep repeating the process until the results

are consistent.

It is extremely rare to require more than two iterations to achieve

consistency in the ex ante and ex post values. The reason is that even if

13. Alternatively, one could either use the regression equation in the original article, run one™s own

regression on the Ibbotson data, or contact the author to provide the right equation.

CHAPTER 4 Discount Rates as a Function of Log Size 139

we guess the value of the ¬rm incorrectly by a factor of 10, we will only

be 2.4% off in our discount rate. By the time we come to the second

iteration, we usually are consistent. The reason behind this is that the

discount rate is based on the logarithm of the value. As we saw earlier,

there is not much difference between the log of $10 billion and the log of

$10 million, and multiplying that by the x-coef¬cient of 0.01039 further

reduces the effects of an initial incorrect estimate of value. This is a con-

vergent system 99% of the time with any kind of reasonable initial guess

of value and even most unreasonable guesses.

The need for iteration arises because of the mathematical properties

of the equations we use in valuing a ¬rm. The simplest type of valuation

is that of a ¬rm with constant growth to perpetuity, where we simply

apply the Gordon growth model (˜˜Gordon model™™) to our forecast of cash

¬‚ow for the coming year. For simplicity, we will use the end-year Gordon

model formula, although it is not as accurate as the midyear formula.

We use the following de¬nitions:

CF cash ¬‚ow (available to equity) in year t 1 (the ¬rst

forecast year)

a 0.3750, the regression constant from regression #2

b 0.01039, the x-coef¬cient from regression #2

V fair market value (FMV) of the ¬rm

r the discount rate

Using the Gordon model and ignoring valuation discounts and pre-

miums, the FMV of the ¬rm is:

CF

V (4-11)

r g

Per equation (4-6), our log size equation for the discount rate is:

r a b ln V (4-12)

Substituting (4-12) into (4-11), we get:

CF

V (4-13)

a b ln V g

Equation (4-13) is a transcendental equation with no analytic

solution.14 Therefore, successive approximation is the only method of de-

termining an answer. The simple iterative procedure in Tables 4-4A, 4-4B,

and 4-4C is very easy to use and works in almost all situations.

Practical Illustration of the Log Size Model: Discounted

Cash Flow Valuations

Let™s illustrate how the iterative process works with a speci¬c example.

The assumptions in Tables 4-4A, 4-4B and 4-4C are identical, except for

the discount rate. Table 4-4A is a very simple discounted cash ¬‚ow (DCF)

14. I thank my friend William Scott, Jr., a physicist, for the terminology and the de¬nitive word

that there is no analytic solution.

PART 2 Calculating Discount Rates

140

T A B L E 4-4A

Discounted Cash Flow Analysis Using 60-Year Model”First Iteration

A B C D E F G

5 Description: 1999 2000 2001 2002 2003 Total

6 Assumptions:

7 Base adjusted cash ¬‚ow $100,000

8 Growth rate in adj cash ¬‚ow 12% 10% 9% 8% 7%

9 Discount rate R 20%

10 Growth rate to perpetuity G 6%

11 Control premium 40%

12 Discount-lack of marketability 35%

14 5 Year Forecasts

16 Forecast cash ¬‚ow $112,000 $123,200 $134,288 $145,031 $155,183

17 Present value factor 0.9129 0.7607 0.6339 0.5283 0.4402

18 PV of cash ¬‚ow $102,242 $93,721 $85,130 $76,617 $68,317 $426,028

20 Calculation of Fair Market Value:

21 Formula

22 Forecast cash ¬‚ow 2003 $164,494 (1 G) * F16

23 Gordon model cap rate 7.8246 SQRT (1 R) / (R G)

24 FMV 2003-in¬nity as of 1/1/2003 $1,287,103 B22 B23

25 Present value factor-5 Yrs 0.4019 1/(1 R) 5 [Where 5 is # yrs from 1/1/98 to 1/1/2003]

26 PV of 2003-in¬nity cash ¬‚ow $517,258 B24 B25

27 Add PV of 1998“2002 cash ¬‚ow 426,028 Total of row 18

28 FMV-marketable minority $943,285 B26 B27

29 Control premium 377,314 B11 B28

30 FMV-marketable control interest 1,320,599 B28 B29

31 Disc-lack of marketability (462,210) B12 B30

32 Fair market illiquid control $858,390 B30 B31

33 Calc of Disc Rate-Regr Eq #2

34 Ln (FMV-marketable minority) 13.7571 Ln(B28)

35 * X coef¬cient of .01039 0.1429 B34 * X coef¬cient-regr #2

36 Constant 0.3750 Constant-regression #2

37 Discount rate (rounded) 23% B35 B36

analysis of a hypothetical ¬rm. The basic assumptions appear in B7“B12.

We assume the ¬rm had $100,000 cash ¬‚ow in 1998. We forecast annual

growth through the year 2003 in B8 through F8 and perpetual growth at

6% thereafter in B10. In B9 we assume a 20% discount rate.

The DCF analysis in B22“B32 is standard and requires little expla-

nation. The present value factors are midyear, and the value in B28 is a

marketable minority interest.15 It is this value ($943,285) that we use to

compare the consistency between the assumed discount rate of 20% (B9)

and the calculated discount rate according to the log size model.

We begin calculating the discount rate using the log size model in

B34, where we compute ln (943,285) 13.7571. This is the natural log of

the marketable minority value of the ¬rm. In B35 we multiply that result

15. See Chapter 7 for explanation of the levels of value and valuation discounts and premiums.

CHAPTER 4 Discount Rates as a Function of Log Size 141

T A B L E 4-4B

Discounted Cash Flow Analysis Using 60-Year Model”Second Iteration

A B C D E F G

5 Description: 1999 2000 2001 2002 2003 Total

6 Assumptions:

7 Base adjusted cash ¬‚ow $100,000

8 Growth rate in adj cash ¬‚ow 12% 10% 9% 8% 7%

9 Disc rate R (Table 4-4A, row 37) 23%

10 Growth rate to perpetuity G 6%

11 Control premium 40%

12 Discount-lack of marketability 35%

14 5 Year Forecasts

16 Forecast cash ¬‚ow $112,000 $123,200 $134,288 $145,031 $155,183

17 Present value factor 0.9017 0.7331 0.5960 0.4845 0.3939

18 PV of cash ¬‚ow $100,987 $90,314 $80,034 $70,274 $61,132 $402,741

20 Calculation of Fair Market Value:

21 Formula

22 Forecast cash ¬‚ow 2003 $164,494 (1 G) * F16

23 Gordon model cap rate 6.5238 SQRT (1 R)/(R G)

24 FMV 2003-in¬nity as of 1/1/2003 $1,073,135 B22 B23

25 Present value factor-5 yrs 0.3552 1/(1 R) 5 [where 5 is # yrs from 1/1/98 to 1/1/2003]

26 PV of 2003-in¬nity cash ¬‚ow $381,179 B24 B25

27 Add PV of 1998-2002 cash ¬‚ow 402,741 Total of row 18

28 FMV-marketable minority $783,919 B26 B27

29 Control premium 313,568 B11 B28

30 FMV-marketable control interest 1,097,487 B28 B29

31 Disc-lack of marketability (384,121) B12 B30

32 Fair market value illiquid control $713,367 B30 B31

33 Calc of Disc Rate-Regr Eq #2

34 Ln (FMV-marketable minority) 13.5721 Ln(B28)

35 * X coef¬cient of .01039 0.1410 B34 * X coef¬cient-regr #2

36 Constant 0.3750 Constant-regression #2

37 Discount rate (rounded) 23% B35 B36

Note: We have achieved consistency in the discount rate assumed (Row 9) and the implied discount rate (Row 37). Also the discount rates match Table 4-3 as we interpolate between

$500k and $750k.

by the x-coef¬cient from the regression, or 0.01039, to come to 0.1429.

We then add that product to the regression constant of 0.3750, which

appears in B36, to obtain an implied discount rate of 23% (rounded, B37).

Comparison of the two discount rates (assumed and calculated) re-

veals that we initially assumed too high a discount rate, meaning that we

undervalued the ¬rm. B29“B31 contain the control premium and discount

for lack of marketability. Because the discount rate is not yet consistent,

ignore these numbers in this table, as they are irrelevant. These topics are

explained in depth in Chapter 7. While the magnitude of the control pre-

mium has been the subject of hot debate, it is merely a parameter in the

spreadsheet and does not affect the logic of the analysis.

PART 2 Calculating Discount Rates

142

T A B L E 4-4C

Discounted Cash Flow Analysis Using 60-Year Model”Final Valuation

A B C D E F G

5 Description: 1999 2000 2001 2002 2003 Total

6 Assumptions:

7 Base adjusted cash ¬‚ow $100,000

8 Growth rate in adj cash ¬‚ow 12% 10% 9% 8% 7%

9 Disc rate R [1] 25%

10 Growth rate to perpetuity G 16%

11 Control premium 40%

12 Discount-lack of marketability 35%

14 5 Year Forecasts

16 Forecast cash ¬‚ow $112,000 $123,200 $134,288 $145,031 $155,183

17 Present value factor 0.8944 0.7155 0.5724 0.4579 0.3664

18 PV of cash ¬‚ows $100,176 $88,155 $76,871 $66,416 $56,853 $388,471

20 Calculation of Fair Market Value:

21 Formula

22 Forecast cash ¬‚ow 2003 $164,494 (1 G) * F16

23 Gordon model cap rate 5.8844 SQRT (1 R)/(R G)

24 FMV 2003-in¬nity as of 1/1/2003 $967,948 B22 B23

25 Present value factor-5 yrs 0.3277 1/(1 R) 5 [where 5 is # yrs from 1/1/98 to 1/1/2003]

26 PV of 2003-in¬nity cash ¬‚ow $317,177 B24 B25

27 Add PV of 1998“2002 cash ¬‚ow 388,471 Total of row 18

28 FMV-marketable minority $705,648 B26 B27

29 Control premium 282,259 B11 B28

30 FMV-marketable control interest 987,907 B28 B29

31 Disc-lack of marketability (345,767) B12 B30

32 Fair market value illiquid control $642,139 B30 B31

[1] Disc Rate 23% (from Table 4-4B, B37) 2% for Speci¬c Company Adjustments 25%

The Second Iteration: Table 4-4B

Having determined that a 20% discount rate is too low, we revise our

assumption to a 23% discount rate (B9) in Table 4-4B. In this case, we

arrive at a marketable minority FMV of $ 783,919 (B28). When we perform

the discount rate calculation with this value (B34“B37), we obtain a

matching discount rate of 23%, indicating that no further iterations are

necessary.

Consistency in Levels of Value

In calculating discount rates, it is important to be consistent in the level

of fair market value that we are using. Since the log size model is based

on returns from the NYSE, the corresponding values generated are on a

marketable minority basis. Consequently, it is this level of value that is

we should use for the discount rate calculations.

Frequently, however, the marketable minority value is not the ulti-

mate level of fair market value that we are calculating. Therefore, it is

crucial to be aware of the differing levels of FMV that occur as a result

CHAPTER 4 Discount Rates as a Function of Log Size 143

of valuation adjustments. For example, if our valuation assignment is to

calculate an illiquid control interest, we will add a control premium and

subtract a discount for lack of marketability from the marketable minority

value. Nevertheless, we use only the marketable minority level of FMV

in iterating to the proper discount rate.

Adding Speci¬c Company Adjustments to the DCF Analysis:

Table 4-4C

The ¬nal step in our DCF analysis is performing speci¬c company ad-

justments. Let™s suppose for illustrative purposes that there is only one

owner of this ¬rm. She is 62 years old and had a heart attack three years

ago. The success of the ¬rm depends to a great extent on her personal

relationships with customers, which may not be easily duplicated by a

new owner. Therefore, we decide to add a 2% speci¬c company adjust-

ment to the discount rate to re¬‚ect this situation.16 If there is no speci¬c

company adjustment, then we would proceed with the calculations in

B22“B32.

Prior to adding a speci¬c company adjustment, it is important to

achieve internal consistency in the ex ante and ex post marketable mi-

nority values, as we did in Table 4-4B. Next, we merely add the 2% to

get a 25% discount rate, which we place in B9. The remainder of the table

is identical to its predecessors, except that we eliminate the ex post cal-

culation of the discount rate in B34“B37, since we have already achieved

consistency.

It is at this point in the valuation process that we make adjustments

for the control premium and discount for lack of marketability, which

appear in B29 and B31. Our ¬nal fair market value of $642,139 (B32) is

on an illiquid control basis.

In a valuation report, it would be unnecessary to show Table 4-4A.

One should show Tables 4-4B and 4-4C only.

Total Return versus Equity Premium

CAPM uses an equity risk premium as one component for calculating

return. The discount rate is calculated by multiplying the equity premium

by beta and adding the risk free rate. In my ¬rst article on the log size

model (Abrams 1994), I also used an equity premium in the calculation

of the discount rate. Similarly, Grabowski and King (1995) used an equity

risk premium in the computation of the discount rate.

16. A different approach would be to take a discount from the ¬nal value, which would be

consistent with key person discount literature appearing in a number of articles in Business

Valuation Review (see the BVR index for cites). Another approach is to lower our estimate of

earnings to re¬‚ect our weighted average estimate of decline in earnings that would follow

from a change in ownership or the decreased capacity of the existing owner, whichever is

more appropriate, depending on the context of the valuation. In this example I have already

assumed that we have done that. There are opinions that one should lower earnings

estimates and not increase the discount rate. It is my opinion that we should de¬nitely

increase the discount rate in such a situation, and we should also decrease the earnings

estimates if that has not already been done.

PART 2 Calculating Discount Rates

144

The equity premium form of the log size model is:

r RF size-based equity premium (4-14)

The size-based equity premium is equal to the return, as calculated by

the log size model, minus the historical average risk-free rate.17

Equity Premium a b ln FMV RF (4-15)

where RF is the historical average risk-free rate. Substituting equation

(4-15) into (4-14), we get:

r RF a b ln FMV RF (4-16)

Rearranging terms, we get:

r a b ln FMV (RF RF) (4-17)

Note that the ¬rst two terms in equation (4-17) are the sole terms

included in the total return version of the log size model. Therefore, the

only difference in calculation of discount rates between the two models

is RF RF, the last two terms appearing in equation [4-17]. Consequently,

the total return of the log size model will exceed the equity premium

version of the model whenever current bond yields exceed historical av-

erage yields and vice versa.

The equity premium term was eliminated in Abrams™ second article

(1997) in favor of total return because of the low correlation between stock

returns and bond yields for the past 60 years. The actual correlation was

6.3%”an amount small enough to ignore.

Bond yields were in the 2“3% range before 1960, under 5% until 1968,

and over 7% from 1975“1993; in 1982 they were as high as 13%. During

the 60-year period from 1939“1998, the low bond yields prevalent in the

1950s and 1960s are balanced by higher subsequent rates, resulting in

little difference in the results obtained using the two models. The 60-year

mean bond yield is 5.64%, as compared with 1998 yields that have ranged

from 5.5% to 6.0%. Thus, current yields are comparable with the 60 year

average yields.

Therefore, it is reasonable to simplify the procedure of calculating

discount rates and eliminate the bifurcation of the discount rate into the

risk-free rate and equity premium components.

17. In CAPM, the latter term is a beta-adjusted equity risk premium, equal to ( equity risk

premium). The equity risk premium (ERP) itself is the arithmetic average of the annual

1998

market returns in excess of the risk-free rate. Mathematically, that is ERP [rmt

t 1926

rFt)/73], where r return and the subscripts m market and F risk-free rate. However,

1998

we can rearrange the equation to ERP [(rmt/73) (rFt/73)] rm rF. This is

t 1926

appropriate for the market as a whole. To calculate a discount rate for a particular ¬rm, in

CAPM we scale the ERP up or down according to the systematic risk as measured by beta.

In log size, we replace the average return on the market with the size-based return for the

¬rm. There is no algebraic scaling, as the log size equation accomplishes the adjustment of

the ERP directly by size.

CHAPTER 4 Discount Rates as a Function of Log Size 145

Adjustments to the Discount Rate

Is Table 4-3 the last word in calculating discount rates? No, but it is the

best starting point based on the available data. Table 4-3 is an extrapo-

lation of NYSE data to privately held ¬rms. While the results appear very

reasonable to me, it would be preferable to perform a similar regression

for NASD data. Unfortunately, the data are not readily obtainable.

Privately held ¬rms are generally owned by people who are not well

diversi¬ed. Table 4-3 was derived from portfolios of stocks that were di-

versi¬ed in every sense except for size, as size itself was the method of

sorting the deciles. In contrast, the owner of the local bar is probably not

well diversi¬ed, nor is the probable buyer. The appraiser may want to

add a speci¬c company adjustment of, say, 2% to 5% to the discount rate

implied by Table 4-3 to account for that. On the other hand, a $100 million

FMV ¬rm is likely to be bought by a well diversi¬ed buyer and may not

merit increasing the discount rate.

Another common adjustment to Table 4-3 discount rates would be

for the depth and breadth of management of the subject company com-

pared to other ¬rms of the same size. In general, Table 4-3 already incor-

porates the size effect. No one expects a $100,000 FMV ¬rm to have three

Harvard MBAs running it, but there is still a difference between a com-

plete one-man show and a ¬rm with two talented people. In general, this

methodology of calculating discount rates will increase the importance of

comparing the subject company to its peers via RMA Associates or similar

data. Differences in leverage between the subject company and its RMA

peers could well be another common adjustment.

Discounted Cash Flow or Net Income?

Since the market returns are based on the cash dividends and the market

price at which one can sell one™s stock, the discount rates obtained with

the log size model should be properly applied to cash ¬‚ow, not to net

income. We appraisers, however, sometimes work with clients who want

a ˜˜quick and dirty valuation,™™ and we often don™t want to bother esti-

mating cash ¬‚ow. I have seen suggestions in Business Valuation Review

(Gilbert 1990, for example) that we can increase the discount rate and

thereby apply it to net income, and that will often lead to reasonable

results. Nevertheless, it is better to make an adjustment from net income

based on judgment to estimate cash ¬‚ow to preserve the accuracy of the

discount rate.

DISCUSSION OF MODELS AND SIZE EFFECTS

The size effects described by Fama and French (1993), Abrams (1994,

1997), and Grabowski and King (1995) strongly suggest that the tradi-

tional one-factor CAPM model is obsolete. As Fama and French (1993, p.

54) say, ˜˜Many continue to use the one-factor Sharpe“Lintner model to

evaluate portfolio performance and to estimate the cost of capital, despite

the lack of evidence that it is relevant. At a minimum, these results here

and in Fama and French (1992) should help to break this common habit.™™

PART 2 Calculating Discount Rates

146

CAPM

Consider the usual way we calculate discount rates using CAPM. We

average the betas of many different ¬rms in the industry, which vary

considerably in size, and apply the resulting beta to a ¬rm that is prob-

ably 0.1% to 1% of the industry average, without correction for size, and

hence risk. Ignoring the size effect corrupts the CAPM results.

This ¬‚aw also applies to the guideline public company method. The

usual approach is to average price earnings multiples (and/or price cash

¬‚ow multiples, etc.) for the various ¬rms in the industry without cor-

recting for size and apply the multiple to a small private ¬rm. A better

method is to perform a regression analysis of market capitalization

(value) as a function of earnings (or cash ¬‚ow) and forecast growth, when

available. I also recommend using another form of the regression with

P/E or P/CF as the dependent variable and market capitalization and

forecast growth as the independent variables.

The beta used in CAPM is usually calculated by running a regression

of the equity premium for an individual company versus the market pre-

mium. As previously discussed, the inability of the resulting beta to ex-

plain the size effect has called into question the validity of CAPM. An

alternative method of calculating beta has been proposed which attempts

to capture the size effect and better correlate with market equity returns,

possibly ameliorating this problem.

Sum Beta

Ibbotson et al. (Peterson, Kaplan, and Ibbotson 1997) postulated that con-

ventional estimates of beta are too low for small stocks due to the higher

degree of auto-correlation in returns exhibited by smaller ¬rms. They cal-

culated a beta using a multiple regression model for both the current and

the prior period, which they call ˜˜sum beta.™™ These adjusted estimates of

beta helped to account for the size effect and showed positive correlation

with future returns.

This improved method of calculating betas will reduce will reduce

some of the downward bias in CAPM discount rates, but it still will not

account for the size effect differences between the large ¬rms in the

NYSE”where even the smallest ¬rms are large”and the smaller pri-

vately held ¬rms that many appraisers are called upon to value. Size

should be an explicit variable in the model to accomplish that.

It may be possible to combine the models. One could use the log size