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Copyright 2001 The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
Sum Beta
The Fama“French Cost of Equity Model
Log Size Models
Heteroscedasticity
INDUSTRY EFFECTS
SATISFYING REVENUE RULING 59-60 WITHOUT A GUIDELINE
PUBLIC COMPANY METHOD
SUMMARY AND CONCLUSIONS
APPENDIX A: AUTOMATING ITERATION USING
NEWTON™S METHOD
APPENDIX B: MATHEMATICAL APPENDIX
APPENDIX C: ABBREVIATED REVIEW AND USE




PART 2 Calculating Discount Rates
118
PRIOR RESEARCH
Historically, small companies have shown higher rates of return when
compared to large ones, as evidenced by data for the New York Stock
Exchange (NYSE) over the past 73 years of its existence (Ibbotson Asso-
ciates 1999). The relationship between ¬rm size and rate of return was
¬rst published by Rolf Banz in 1981 and is now universally recognized.
Accordingly, company size has been included as a variable in several
models used to determine stock market returns.
Jacobs and Levy (1988) examined small ¬rm size as one of 25 vari-
ables associated with anomalous rates of return on stocks. They found
that small size was statistically signi¬cant both in single-variable and
multivariate form, although size effects appear to change over time, i.e.,
they are nonstationary. They found that the natural logarithm (log) of
market capitalization was negatively related to the rate of return.
Fama and French (1993) found they could explain historical market
returns well with a three-factor multiple regression model using ¬rm size,
the ratio of book equity to market equity (BE/ME), and the overall market
factor Rm Rf , i.e., the equity premium. The latter factor explained overall
returns to stocks across the board, but it did not explain differences from
one stock to another, or more precisely, from one portfolio to another.2
The entire variation in portfolio returns was explained by the ¬rst
two factors. Fama and French found BE/ME to be the more signi¬cant
factor in explaining the cross-sectional difference in returns, with ¬rm size
next; however, they consider both factors as proxies for risk. Furthermore,
they state, ˜˜Without a theory that speci¬es the exact form of the state
variables or common factors in returns, the choice of any particular ver-
sion of the factors is somewhat arbitrary. Thus detailed stories for the
slopes and average premiums associated with particular versions of the
factors are suggestive, but never de¬nitive.™™
Abrams (1994) showed strong statistical evidence that returns are
linearly related to the natural logarithm of the value of the ¬rm, as mea-
sured by market capitalization. He used this relationship to determine the
appropriate discount rate for privately held ¬rms. In a follow-up article,
Abrams (1997) further simpli¬ed the calculations by relating the natural
log of size to total return without splitting the result into the risk-free
rate plus the equity premium.
Grabowski and King (1995) also described the logarithmic relation-
ship between ¬rm size and market return. They later (Grabowski and
King 1996) demonstrated that a similar, but weaker, logarithmic relation-
ship exists for other measures of ¬rm size, including the book value of
common equity, ¬ve-year average net income, market value of invested
capital, ¬ve-year average EBITDA, sales, and number of employees. Their
latest research (Grabowski and King 1999) demonstrates a negative log-
arithmic relationship between returns and operating margin and a posi-


2. The regression coef¬cient is essentially beta controlled for size and BE/ME. After controlling for
the other two systematic variables, this beta is very close to 1 and explains only the market
premium overall. It does not explain any differentials in premiums across ¬rms or
portfolios, as the variation was insigni¬cant.




CHAPTER 4 Discount Rates as a Function of Log Size 119
tive logarithmic relationship between returns and the coef¬cient of vari-
ation of operating margin and accounting return on equity.
The discovery that return (the discount rate) has a negative linear
relationship to the natural logarithm of the value of the ¬rm means that
the value of the ¬rm decays exponentially with increasing rates of return.
We will also show that ¬rm value decays exponentially with the standard
deviation of returns.


TABLE 4-1: ANALYSIS OF HISTORICAL STOCK RETURNS
Columns A“F in Table 4-1 contain the input data from the Stocks, Bonds,
Bills and In¬‚ation 1999 Yearbook (Ibbotson Associates 1999) for all of the
regression analyses as well as the regression results. We use the 73-year
average arithmetic returns in both regressions, from 1926 to 1998. For
simplicity, we have collapsed 730 data points (73 years 10 deciles) into
73 data points by using averages. Thus, the regressions are cross-sectional
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

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