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rather than time series. Column A lists the entire NYSE divided into dif-
ferent groups (known as deciles) based on market capitalization as a
proxy for size, with the largest ¬rms in decile #1 and the smallest in decile
10.3 Columns B through F contain market data for each decile which is
described below.
Note that the 73-year average market return in Column B rises with
each decile. The standard deviation of returns (Column C) also rises with
each decile. Column D shows the 1998 market capitalization of each dec-
ile, with decile #1 containing 189 ¬rms (Column F) with a market capi-
talization of $5.986 trillion (D8). Market capitalization is the price per
share times the number of shares. We use it as a proxy for the fair market
value (FMV).
Dividing Column D (FMV) by Column F (the number of ¬rms in the
decile), we obtain Column G, the average capitalization, or the average
fair market value of the ¬rms in each decile. For example, the average
company in decile #1 has an FMV of $31.670 billion (G8, rounded), while
the average ¬rm in decile #10 has an FMV of $56.654 million (G17,
rounded).
Column H shows the percentage difference between each successive
decile. For example, the average ¬rm size in decile #9 ($146.3 million;
G16) is 158.2% (H16) larger than the average ¬rm size in decile #10 ($56.7
million; G17). The average ¬rm size in decile #8 is 92.5% larger (H15)
than that of decile #9, and so on.
The largest gap in absolute dollars and in percentages is between
decile #1 and decile #2, a difference of $26.1 billion (G8“G9), or 468.9%
(H8). Deciles #9 and #10 have the second-largest difference between them
in percentage terms (158.2%, per H16). Most deciles are only 45% to 70%
larger than the next-smaller one.
The difference in return (Column B) between deciles #1 and #2 is
1.6% and between deciles #9 and #10 is 3.2%, while the difference between


3
All of the underlying decile data in Ibbotson originate with the University of Chicago™s Center for
Research in Security Prices (CRSP), which also determines the composition of the deciles.




PART 2 Calculating Discount Rates
120
T A B L E 4-1

NYSE Data by Decile and Statistical Analysis: 1926“1998


A B C D E F G H I

4 Note [1] Note [1] Note [2] Note [2] Note [2] D/F
5 Y X1 X2
6 Recent Mkt % Change
7 Decile Mean Arith Return Std Dev Capitalization % Cap # Co.s Avg Cap FMV in Avg FMV Ln(FMV)
8 1 12.11% 18.90% 5,985,553,146,000 72.60% 189 31,669,593,365 468.9% 24.1786
9 2 13.66% 22.17% 1,052,131,226,000 12.76% 189 5,566,831,884 121.8% 22.4401
10 3 14.11% 23.95% 476,920,534,000 5.78% 190 2,510,108,074 73.2% 21.6436
11 4 14.76% 26.40% 273,895,749,000 3.32% 189 1,449,183,857 60.3% 21.0943
12 5 15.52% 27.24% 170,846,605,000 2.07% 189 903,950,291 49.2% 20.6223
13 6 15.60% 28.23% 114,517,587,000 1.39% 189 605,913,159 46.5% 20.2222
14 7 15.99% 30.58% 78,601,405,000 0.95% 190 413,691,605 46.9% 19.8406
15 8 17.05% 34.36% 53,218,441,000 0.65% 189 281,579,053 92.5% 19.4559
16 9 17.85% 37.02% 27,647,937,000 0.34% 189 146,285,381 158.2% 18.8011
17 10 21.03% 45.84% 10,764,268,000 0.13% 190 56,654,042 N/A 17.8525
18 Std deviation 2.48% 1,893
19 Value wtd index 12.73% NA 8,244,096,898,000 100.00%

23 1st Regression: Return F(Std Dev. of Returns)

25 1926“1998 1939“1998

26 Constant 6.56% 8.90%
27 72/60 year mean T-bond yield [Note 3] 5.28% 5.70%
28 Std err of Y est 0.27% 0.42%
29 R squared 98.95% 95.84%
30 Adjusted R squared 98.82% 95.31%
31 No. of observations 10 10
32 Degrees of freedom 8 8
33 X coef¬cient(s) 31.24% 30.79%
34 Std err of coef. 1.14% 2.27%
35 T 27.4 13.6
36 P .01% .01%
121
122
T A B L E 4-1 (continued)

NYSE Data by Decile and Statistical Analysis: 1926“1998


A B C D E F G H I

39 2nd Regression: Return F[LN(Mkt Capitalization)]
41 1926“1998 1939“1998
42 Constant 42.24% 37.50%
43 Std err of Y est. 0.82% 0.34%
44 R squared 90.37% 97.29%
45 Adjusted R squared 89.17% 96.95%
46 No. of observations 10 10
47 Degrees of freedom 8 8
48 X coef¬cient(s) 1.284% 1.039%
49 Std err of coef. 0.148% 0.061%
50 T 8.7 16.9
51 P .01% .01%

53 3rd Regression: Return F[Decile Beta]
54 Note [4]
55 1926“1998 1939“1998
56 Constant 2.78% NA
57 Std err of Y est 0.57% NA
58 R squared 95.30% NA
59 Adjusted R squared 94.71% NA
60 No. of observations 10 NA
61 Degrees of freedom 8 NA
62 X coef¬cient(s) 15.75% NA
63 Std err of coef. 1.24% NA
64 T 12.7 NA
65 P .01% NA

68 Assumptions:
69 Long-term gov™t bonds arithmetic mean income 1926“1998 [1] 5.20%
return
70 Long horizon equity premium [2] 8.0%

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


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