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