1. A New Empirical Perspective on the CAPM
Author(s): Marc R. Reinganum
Reviewed work(s):
Source: The Journal of Financial and Quantitative Analysis, Vol. 16, No. 4, Proceedings of
16th Annual Conference of the Western Finance Association, June 18-20, 1981, Jackson Hole,
Wyoming (Nov., 1981), pp. 439-462
Published by: University of Washington School of Business Administration
Stable URL: http://www.jstor.org/stable/2330365 .
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2. JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS
Volume XVI, No. 4, November 1981
A NEW EMPIRICAL PERSPECTIVE ON THE CAPM
Marc R. Reinganum*
Introduction
The adequacy of the capital asset pricing models (CAPM) of Sharpe [27],
Lintner [17], and Black [4] as empirical representations of capital market equili?
brium is now seriously challenged (for example, see Ball [1], Banz [2], Basu [3],
Cheng and Graver [8], Gibbons [15], Marsh [18], Reinganum [22], and Thompson [20]).
Yet, the influence of earlier empirical studies (such as Black, Jensen, and Scholes
[5] and Fama and MacBeth [11]) still remains; the current consensus seems to be
that a security's beta is still an important economic determinant of equilibrium
pricing even though it may not be the sole determinant. In light of the recent
empirical evidence, however, the claim that a security's beta is an important
determinant of equilibrium pricing should be reexamined.
The purpose of this paper is to investigate empirically whether securities
with different estimated betas systematically experience different average rates
of return. While the statistical tests are designed to assess the cross-sectional
importance of beta, cross-sectional regressions are not employed, so that some
of the problems which plagued earlier research are avoided. The test results
demonstrate that estimated betas are not systematically related to average returns
across securities. The average returns of high beta stocks are not reliably dif?
ferent from the average returns of low beta stocks. That is, portfolios with
widely different estimated betas possess statistically indistinguishable average
returns. Thus, estimated betas based on standard market indices do not appear to
reliably measure a "risk which is priced in the market." These findings, along
with the evidence on empirical "anomalies," suggest that the CAPM may lack sig?
nificant empirical content.
University of Southern California. The author wishes to thank Fischer
Black, Victor Canto, Kim Dietrich, Doug Joines, Terry Langetieg, Dick Roll, and
Alan Shapiro. Any errors that remain are the author's responsibility.
439

3. II. The Beta Hypothesis and Test Design
The development of the CAPM is well known and can be found elsewhere (for
example, see Fama [10]). Depending on the particular set of assumptions, the
pricing relationships which emerge from the CAPM can be expressed as either:
(1) E(R.)
i
= E(R ) + 3. [E(R ) - E(R )]
om i m om
(2) E(R.) = R + 3. [E(R ) - Rl
l F l m F
where:
E(R.) = expected return on asset i;
E(R ) E expected return on an asset whose return is uncorrelated
with the market return;
E(R ) E expected return on the market portfolio;
E cov(R.,R E the beta of asset and
3.
i i m)/var(R m) i;
R? E risk-free rate of interest.
F
The two forms of the CAPM share an important implication. Namely, two assets
with different betas possess different expected returns. Thus, a necessary con?
dition for the data to be consistent with the CAPM is that variations in esti?
mated betas must be systematically related to variations in average returns.
While Roll [23] questions the testability of the theoretical CAPM, the concern
of this paper is the common empirical representation of the paradigm. The beta
hypothesis is that assets with different estimated betas experience different
average rates of return. Confirmation of the hypothesis would offer evidence
that supports the contention that betas matter in equilibrium pricing. Evidence
that rejected the hypothesis would seem to indicate that the risk premia associated
with betas are economically insignificant.
A straightforward, two-step strategy is employed to test the beta hypothesis.
First, in period A, individual security betas are estimated, and securities are
placed into one of ten portfolios based upon the relative rank of their estimated
beta. Then, in period B, the returns of the ten beta portfolios are calculated
by combining with equal-weights the returns of the component securities within
each portfolio. With the time-series of ten portfolio returns in hand, a multi?
variate statistical procedure is invoked to test whether or not the ten port?
folios possess significantly different average returns.
440

4. The composition of each beta portfolio is periodically updated. The fre?
quency of the revisions depends upon the data base being analyzed. When analyzing
the daily returns of the New York Stock Exchange and American Stock Exchange
firms (1963-1979), the beta portfolios are revised annually. Thus, the 1964 beta
portfolios are created based upon security betas estimated with 1963 daily re?
turns. Similarly, betas estimated with 1964 data are used to identify the secu?
rities within the 1965 beta portfolios. With monthly return data for NYSE firms
(1930-1979), the beta portfolios are updated every five years. For example,
security betas estimated with data from 1930-1934 are used to form the member?
ship of the 1935-1939 beta portfolios. Regardless of the frequency of updates,
betas are estimated in the period prior to the one in which portfolio returns
are measured.
Three different estimators are used to compute beta estimates. First, for
both daily and monthly return data, betas are calculated using ordinary least
squares. Security returns are regressed against the CRSP value-weighted market
returns, and the computed coefficient on the market is the estimated beta. Recent
research, however, indicates that this "market model" estimator may be inappro?
priate for daily returns because of nonsynchronous trading problems. To assess
the impact of this potential problem, the estimators proposed by Scholes and
Williams [25] and Dimson [9] also are used to calculate security betas. Hence,
with daily data, the sensitivity of the results to different beta estimators can
be investigated.
III. Empirical Tests of the Beta Hypothesis
This section reports the results of tests designed to determine if port?
folios with different estimated beta experience statistically different average
returns. The section is divided into three parts. In the first part, the data
and sample selection criteria are described. In the next part, the test results
based on the daily returns of NYSE and AMEX companies during 1964 through 1979
are presented. The final part contains evidence based on 45 years of monthly
returns for NYSE companies.
A. The Data and Sample Selection Criteria
Stock return data used in this analysis are gathered from the University of
Chicago's Center for Research in Security Prices (CRSP) monthly and daily stock
return files as of December 1979. The daily file contains the daily stock re?
turns (capital gains plus dividends) of all companies that have traded on the
New York Stock Exchange or the American Stock Exchange from July 1962 through
December 1979. Unlike the daily file, the monthly file contains information
only on NYSE companies; however, the stock return information on the monthly file
441

5. dates back to January 1926.
Each time security betas are estimated and the composition of the ten beta
portfolios is revised, the sample of firms changes. With the daily data, the
sample changes yearly. The only restriction placed on securities is that they
have at least 100 one-day returns during the beta estimation period. No other
restriction, such as survival through the portfolio holding period, is imposed.
If a firm is delisted during the holding period, any funds returned are held in
cash until the end of the year. In any one given year, the number of NYSE and
AMEX firms that qualified for inclusion in the sample ranged between 2,000 and
2,700.
The selection criterion with monthly data differs from the above criterion
only because portfolios are updated every five years. During the beta estima?
tion period, a firm is excluded only if it fails to have at least 40 one-month
returns. The number of NYSE firms included in the monthly sample ranged from
678 in the 1930's to 1296 in the 1970's.
B. The Test Results with Daily Returns: 1964-1979
The years from 1964 through 1979 represent a good period in which to study
the cross-sectional relationship between returns and estimated betas for at least
two reasons. First, these years are primarily outside the time periods of the
pivotal studies by Black, Jensen and Scholes [5], and Fama and MacBeth [11],
which ended in December, 1965 and June, 1968, respectively. Second, unlike the
earlier studies, the hypotheses can be tested with AMEX firms as well as with
NYSE companies.
The first stage of the test involves estimating betas and placing securi?
ties into one of ten beta portfolios. Three different beta estimates are used
to create three sets of ten beta portfolios. As explained above, these betas
are computed using the "market model," Scholes-Williams, and Dimson estimators
with a value-weighted NYSE-AMEX market index. In the next period, the daily
returns of the ten beta portfolios are calculated by combining with equal-weights
the daily returns of the component securities within the portfolios. If betas
matter in the way the theory suggests, then one ought to observe a positive rela?
tionship between betas and returns and be able to reject the hypothesis that the
mean returns of the ten beta portfolios are equal.
Tables 1 through 3 present the daily return statistics for the ten beta
portfolios created with the different beta estimates. Table 1 contains information
For the Scholes-Williams and OLS estimators, betas were also calculated
using an equal-weighted market index. The results were not significantly dif?
ferent from those reported in the text. Dimson betas were not calculated with
the equal-weight index.
442

6. TABLE 1
DAILY RETURN STATISTICS FOR THE TEN BETA
PORTFOLIOS WITH BETAS ESTIMATED USING DAILY RETURNS,
A VALUE-WEIGHTED INDEX, AND THE "MARKET MODEL" ESTIMATOR
Estimated Autocorrelations
Portfolio
Beta Skewness Kurtosis 1 2 3
.05 -.101 5.601 .48 .28 .25
.33 .131 6.601 .52 .24 .21
.50 -.074 6.125 .47 .19 .18
.64 -.098 5.864 .47 .15 .16
.79 -.105 4.810 .45 .14 .14
.95 .056 6.025 .42 .11 .13
1.13 .012 5.569 .40 .09 .10
1.34 .177 5.730 .38 .07 .09
1.64 .166 5.764 .33 .06 .08
2.25 .314 5.788 .26 .02 .06
A mean return is calculated using 4009 trading day returns from 1964
through 1979. Mean daily returns are multiplied by 1000 for reporting purposes.
Standard errors are in parentheses. Skewness and kurtosis measures are based on
moments of the normal distribution.
2
The estimated portfolio beta is just the linear combination (equal weights)
of security betas. These betas are estimated in the year prior to the portfolio
holding period.
443

7. on the beta portfolios formed with betas computed using the "market model" esti?
mator. The null hypothesis that the mean returns of the ten portfolios are
identical can be formally tested using Hotelling's T-squared test. This test
takes into account contemporaneous correlations between the ten portfolio re?
turns. The test statistic has an F(9,4000) distribution under the null hypothe?
sis. At the one and five percent levels, the values of F(9,??) are 2.41 and 1.88,
respectively. For the data in Table 1, the computed value of this test statis?
tic is 2.99. Thus, even at the one percent level, the hypothesis of identical
mean returns would be rejected. This rejection should not be interpreted as evi?
dence in support of the CAPM because the average daily return of the low beta
portfolio actually exceeds the average daily return of the high beta portfolio
by .03 percent.
One must be cautious in interpreting the exact statistical significance of
the results because of the apparent departures from normality. In particular,
one observes that the portfolio returns seemed to be both skewed and leptokurtic.
The skewness and kurtosis measures, however, are particularly sensitive to out?
liers. Examination of the daily returns revealed that on May 27, 1970, the mar?
ket experienced about a six percent gain; the returns of the high beta portfolios
were about ten standard deviations above their means. If this one observation
is deleted from the sample, the skewness and kurtosis measures for the high beta
portfolios are vitually the same as those associated with the normal distribu?
tion; the low beta portfolios remain somewhat leptokurtic. One also observes
in Table 1 that the daily returns of the ten beta portfolios are positively auto-
correlated. With autocorrelation, the variance-covariance matrix of portfolio
returns is estimated consistently, but not efficiently. There is no reason,
however, to suspect that the tests are biased in favor of rejecting the null
hypothesis of identical mean returns.
Despite the potential statistical problems associated with constructing an
appropriate confidence region, the results in Table 1 are not consistent with
the predictions of the CAPM: low beta portfolios actually experience greater
average returns than those of the high beta portfolios during the period 1964-
1979. While one might be able to accept this result in any one year, the fact
that it can be detected during a 16-year period reduces the probability that
this phenomenon is a fluke. After all, 16 years represents nearly 30 percent
of the time for which CRSP has collected data. Furthermore, this is the only
period in which computer readable data are systematically available for all Ameri?
can Stock Exchange companies as well as those that trade on the New York Stock
Exchange. Thus, the data analyzed in these tests would not seem to constitute
a "small" sample.
444

8. One potential criticism of the results presented in Table 1 is that the
beta portfolios are created using ordinary least-squares estimates of security
betas based on daily data. If nontrading is a serious problem, then this might
lead to biases in estimation which could affect the results. This possibility
is now explored. Table 2 contains the daily returns statistics for portfolios
constructed with Scholes-Williams estimated betas. The numbers presented in
this table are similar to those reported in Table 1. Even using the Scholes-
Williams estimator, the low beta portfolios experience higher average returns
than do the high beta portfolios. If one tests the hypothesis of identical mean
returns using Hotelling's T-squared technique, the appropriate F-test takes on
a value of 1.85. Hence, one would not reject the null hypothesis at the five
percent level. Of course, the statistical caveats discussed above apply here
too. The results in Table 2 seem to indicate that, at best, the average returns
of the ten beta portfolios are indistinguishable from each other. This corro-
borates the evidence in Table 1; positive differences in estimated betas are not
reliably associated with positive differences in average returns.
Dimson recently argued that even the Scholes-Williams estimator might be
biased and inconsistent if nontrading is a serious enough problem. Dimson sug?
gested that one use an aggregated coefficients method for estimating security
betas with daily data. The idea behind this estimation technique is to regress
lagged and leading (as well as the contemporaneous) market returns on security
returns. Thus, instead of a simple regression, one runs a multiple regression.
The estimated security beta is simply the sum of the estimated slope coefficients.
In this paper, regressions are calculated using 20 lagged and five leading mar?
ket returns. This procedure is virtually identical to those used in Roll [24]
and Reinganum [21].
In Table 3, the ten portfolios are constructed with estimated betas based
on Dimson's aggregated coefficients methodology. One observes that, except for
portfolio P9, the mean daily returns of all the portfolios are between .06 per?
cent and .08 percent. Furthermore, the mean daily return of the high beta port?
folio exceeds the mean daily return of the low beta portfolio by only .001 per?
cent with an associated t-value of 0.09. As with the portfolio returns reported
in the previous tables, there is no immediately evident association between Dim?
son betas and average portfolio returns. One would reject, however, the hypo?
thesis of equality between means for all ten portfolios jointly considered at
the .01 level, but the average returns of the two extreme beta portfolios are
statistically indistinguishable. Furthermore, for the intermediate portfolios,
higher estimated betas are not always associated with higher average returns.
The evidence analyzed in Tables 1 through 3 is based on 16 years of daily
445

9. TABLE 2
DAILY RETURN STATISTICS FOR THE TEN BETA
PORTFOLIOS WITH BETAS ESTIMATED USING DAILY RETURNS,
A VALUE-WEIGHTED INDEX, AND THE SCHOLES-WILLIAMS ESTIMATOR
Autocorrelations
Kurtosis 1 2 3
7.228 .48 .25 .23
6.205 .49 .20 .20
5.668 .48 .19 .17
5.265 .44 .15 .15
4.484 .43 .14 .14
5.354 .41 .12 .12
5.945 .39 .09 .11
6.152 .37 .06 .09
6.345 .35 .06 .09
5.935 .30 .05 .08
lA mean return is calculated using 4009 trading day returns from 1964
through 1979. Mean daily returns are multiplied by 1000 for reporting purposes.
Standard errors are in parentheses. Skewness and kurtosis measures are based on
moments of the normal distribution.
2The estimated portfolio beta is just the linear combination (equal weights)
of security betas. These betas are estimated in the year prior to the portfolio
holding period.
446

10. TABLE 3
DAILY RETURN STATISTICS FOR THE TEN BETA PORTFOLIOS WITH
BETAS ESTIMATED USING DAILY RETURNS, A VALUE-WEIGHTED,
AND THE DIMSON ESTIMATOR
Autocorrelations
.43 .16 .16
.43 .13 .15
.43 .13 .13
.43 .13 .13
.42 .12 .13
.41 .11 .13
.41 .11 .12
.40 .09 .11
.08 .10
.35 .09 .11
A mean return is calculated using 4009 trading day returns from 1964 through
1979. Mean daily returns are multiplied by 1000 for reporting purposes. Standard
errors are in parentheses. Skewness and kurtosis measures are based on moments
of the normal distribution.
The estimated portfolio beta is just the linear combination (equal weights)
of security betas. These betas are estimated in the year prior to the portfolio
holding period.
447

11. return data. One potential problem with drawing inferences from such a time
series could be that the statistical distribution may not be sufficiently station?
ary, especially since the portfolios are revised yearly; however, the year-by-
year portfolios'results can be examined to gauge whether this is a serious prob?
lem. Perhaps the most succinct way to convey the results of this analysis is
to present the differences in average returns between the high and low beta port?
folios. Table 4 reports these differences for security betas calculated with
the different beta estimators.
The year-by-year results also reveal no significant relationship between
estimated portfolio betas and average returns. For example, when security betas
are calculated with the "market model" estimator, the average return of the high
beta portfolio exceeds the average return of the low beta by two standard errors
in only one of the 16 years. For the other 15 years, the differences in average
returns between the high and low beta portfolios are not statistically signifi?
cant. In nine of these years, the point estimate of the mean return of the low
beta portfolio exceeds that of the high beta portfolio.
Using the Scholes-Williams estimator, none of the differences in average
returns between the high and low beta portfolio are more than two standard er?
rors from zero. In seven of the 16 years, the average return of the low beta
portfolio is greater than the average return of the high beta portfolio.
For portfolios created with Dimson betas, the average return of the high
beta portfolio exceeds the average return of the low beta portfolio by two
standard errors in two of the 16 years. The average return, however, of the
low beta portfolio exceeds the average return of the high beta portfolio in one
of the years as well. In the remaining years, the differences are not statisti?
cally significant. Hence, the year-by-year results corroborate the findings
presented in Tables 1 through 3. Thus, the danger of interpreting the results
in Tables 1 through 3 as illustrating the average effects throughout the 16-
year period does not seem great.
Another possible explanation for the above results is that the portfolio
betas are really not different in the year in which portfolio returns are meas?
ured. Recall that portfolios are formed based on security betas estimated in
the prior year. Since these estimated betas are ranked, the extreme beta port?
folios in particular may contain securities whose betas are estimated with the
largest error. This possibility can be investigated by computing the betas of
the ten portfolios in the year in which average returns are measured.
Table 5 compares the grouping period betas with the holding period betas
for the three estimators during the entire 16-year sample. For each set of ten
portfolios, holding period betas are computed with the "market model," Scholes-
448

12. TABLE 4
MEAN DIFFERENCES IN DAILY RETURNS BETWEEN THE HIGH
AND LOW BETA PORTFOLIOS ON A YEARLY BASIS
(Betas Computed with Daily Returns)
Beta Estimator
Mean differences in daily returns are multiplied by 1000 for reporting pur?
poses. T-values are in parentheses. Each year contains approximately 250 trad?
ing days.
449

13. TABLE 5
COMPARISON OF NYSE-AMEX PORTFOLIO BETAS ESTIMATED
IN GROUPING PERIODS AND HOLDING PERIODS
GROUPING PERIOD ESTIMATOR
Aggregated
Scholes-Williams Coefficients
GP MM SW AC GP MM SW AC
.07 .43 .53 1.07 -.50 .72 .82 1.26
.41 .53 .64 1.06 .29 .71 .80 1.14
.59 .65 .77 1.21 .62 .76 .86 1.20
.75 .76 .88 1.28 .89 .81 .91 1.25
.91 .84 .96 1.36 1.15 .87 .97 1.35
6 .95 .95 1.07 1.46 1.07 .96 1.07 1.46 1.41 .93 1.03 1.40
7 1.13 1.06 1.17 1.54 1.24 1.07 1.17 1.53 1.71 .99 1.09 1.51
8 1.34 1.19 1.28 1.63 1.44 1.18 1.28 1.64 2.06 1.08 1.18 1.59
9 1.64 1.37 1.42 1.75 1.72 1.34 1.40 1.76 2.56 1.18 1.27 1.73
High Beta 2.25 1.69 1.67 1.95 2.25 1.60 1.62 1.95 3.77 1.32 1.39 1.89
GP stands for the estimated beta of the portfolio during the grouping
period. Grouping period betas are shown for portfolios created with "market
model" estimates, Scholes-Williams estimates, and Dimson's aggregated coefficients
estimates. For each set of portfolios, three estimated holding period betas are
shown: MM ("market model"); SW (Scholes-Williams); and AC (Dimson's aggregated
coefficients method). In a grouping period, a portfolio beta is just the equal-
weighted combination of estimated security betas within that portfolio. The
grouping period betas reported above are the averages of the portfolio betas over
the 16 grouping periods from 1963 through 1978. Holding period betas are calcu?
lated by analyzing 16 years of daily portfolio and market returns (1964-1979).
450

14. Williams, and Dimson's aggregated coefficient estimators, even though each set
of portfolios is created with betas based on only one of these estimators. Thus,
for example, "market model," Scholes-Williams, and Dimson holding period betas
are presented for the portfolios formed on the basis of "market model" betas alone.
In Table 5 one observes attenuation in the estimated betas of the high and low
beta portfolios. For example, the estimated beta of the lowest "market model"
beta portfolio rises from .05 to .40; similarly, the estimated beta of the highest
"market model" beta portfolio drops from 2.25 to 1.69. The attenuation in esti?
mated betas of the Scholes-Williams beta portfolios is similar to the attenuation
exhibited by the "market model" beta portfolios. The estimated betas of the ag?
gregated coefficients portfolios, however, reveal severe attenuation. For example,
the beta of the lowest AC portfolio rises from -.50 to 1.26; the beta of the high?
est AC portfolio drops from 3.77 to 1.89. Thus, the spread in Ac betas between
the AC portfolios is smaller than the spread in "market model" and Scholes-Williams
holding period betas for portfolios created with those two estimators.
Table 5 also reveals that each estimator almost perfectly preserves the rank
ordering of estimated betas for each set of ten portfolios during the holding
periods, regardless of the estimator used to create the ten portfolios. Consider
portfolios formed on the basis of "market model" betas. During the holding periods,
the Scholes-Williams estimates of the betas of these portfolios are perfectly rank
ordered with the "market model" estimates. Furthermore, the spread between the
high and low beta portfolios during the holding periods is 1.16 based on the Scholes-
Williams estimates, and 1.29 based on the "market model" estimates. The spread
for these portfolios based on Dimson betas is .88. Thus, the other estimators
not only tend to preserve the rank ordering of estimated betas, but also seem to
exhibit spreads roughly equivalent to those of the "market model" estimator which
is used to form these portfolios. One discovers in Table 5 that similar conclu?
sions can be drawn for portfolios created on the basis of Scholes-Williams betas
and Dimson's aggregated coefficients betas. Hence, one may feel confident in con-
cluding that the portfolios analyzed in Tables 1 through 4 possess widely differ?
ent estimated betas during the portfolio holding periods.
C. The Test Results with Monthly Returns: 1935-1979
The purpose of this section is to investigate whether the "beta does not
matter" result is specific to the 1964-1979 period or whether, in fact, it appears
to hold over a longer time horizon. Indeed, evidence from the work of Black,
Jensen, and Scholes [5] may be consistent with the proposition that portfolios
with widely different estimated betas possess statistically indistinguishable
average returns. For example, in their Table 2, the mean excess return, (R -R ),
of the low beta portfolio is within the two standard error confident interval
451

15. about the mean excess return of the high beta portfolio. Furthermore, Black,
Jensen, and Scholes note that the intercepts in their "market model" regressions
are negative for portfolios with high estimated betas (3 > 1) and positive for
portfolios with low estimated betas (3 < 1). This inverse relationship between
betas and the intercepts is precisely what one would expect if portfolios with
different estimated betas had statistically indistinguishable average returns.
The two-stage test procedure used with the monthly data is similar to the
one employed in the previous section except that the initial estimation and port?
folio holding periods are five years instead of one year. In the first period,
security betas are estimated using ordinary least squares and membership in the
ten beta portfolios is established. In the next period, the monthly returns of
the ten beta portfolios are computed by combining with equal-weights the monthly
returns of the securities within the portfolio. The first grouping period is
from 1930 through 1934; the last portfolio holding period is from 1975 through
1979. If estimated betas matter, then the ten beta portfolios ought to have sig?
nificantly different mean returns.
Table 6 presents the monthly return statistics for the ten beta portfolios
formed by grouping securities on the basis of their OLS betas estimated with the
CRSP equal-weighted NYSE market index. The statistics in these tables are based
upon 45 years of monthly return data from 1935 through 1979. At first glance,
the evidence seems to indicate that estimated betas matter. For example, the
average return of the high beta portfolio is about 1.5 percent per month, whereas
the average return of the low beta portfolio is only .9 percent. In addition,
the rank ordering of average returns corresponds to the rank ordering of estimated
betas. One cannot, however, draw inferences from point estimates alone. The
fact that the high beta portfolios possess higher average returns than the low
beta portfolios does not mean that the differences are reliable or statistically
significant. Indeed, while the mean difference between the returns of the high
and low beta portfolios in Table 6 is .580 percent per month, the standard error
of the difference is .294 percent. Thus, the mean difference between the average
returns of the high and low beta portfolios is less than two standard errors from
zero. Furthermore, since these statistics are computed with 540 observations,
one could argue that, taking into account the power of the test, a three standard
error confidence region might be more appropriate than the conventional two stan?
dard error interval. One can also formally test whether the ten beta portfolios
possess identical mean returns using Hotelling's T-squared test. Under the null
hypothesis of identical mean returns, the test statistic assumes an F (9,531) dis?
tribution. Based on the data analyzed in Table 6, the value of the test statistic
is 1.22; one clearly cannot reject the null hypothesis of identical mean returns
452

16. TABLE 6
MONTHLY RETURN STATISTICS FOR THE TEN BETA
PORTFOLIOS BASED ON BETAS ESTIMATED USING AN
EQUAL-WEIGHTED NYSE INDEX AND THE "MARKET MODEL" ESTIMATOR
?2 Autocorrelations
Skewness Kurtosis 1 2 3
-.287 4.961 .11 -.00 .04
-.388 4.521 .03 .02 .03
-.111 5.131 .00 .07 .04
.206 6.300 .02 .08 .00
.375 7.860 -.01 .10 -.02
.321 7.022 .03 .11 .00
.827 10.000 .02 .08 -.00
.453 6.290 .02 .10 .01
.876 8.422 .03 .08 -.01
.948 5.642 .03 .09 -.02
Mean returns are multiplied by 100 for reporting purposes. A 1.0 equals
1%. Standard errors are in parentheses. The statistics are based on 540 monthly
observations from 1935 through 1979.
2The estimated portfolio beta is the equal-weighted combination of security
betas. These betas are estimated in the five-year periods prior to the portfolio
holding periods.
453

17. even at the .05 level.
The data analyzed in Table 7 are similar to the data analyzed in Table 6
except that security OLS betas are computed with a value-weighted NYSE market in?
dex. Again, the high beta portfolio experienced an average return greater than
the low beta portfolio, but in this case the difference is about .4 percent per
month rather than .85 percent; the t-statistic associated with this difference
is only 1.59. In addition, the null hypothesis of identical mean returns for the
ten beta portfolios still would not be rejected at the .05 level, although the
value of the test statistic, 1.88, is just slightly less than the critical value
for the F(9,531) distribution. Unlike the data in Table 6, however, the average
portfolio returns and estimated betas are not perfectly rank correlated in Table
7. Thus, based on the evidence in these two tables, there does not appear to be
a statistically reliable relationship between average portfolio returns and esti?
mated portfolio betas.
While one might tentatively conclude that betas computed with standard methods
and market indices do not seem to be reliably related to average portfolio returns,
three additional issues should be addressed. First, are the results within the
subperiods consistent with the findings based on the analysis of 45 years of monthl
data? Secondly, are the estimated betas of the ten portfolios during the holding
periods similar to the grouping period betas? Finally, given that the empirical
distribution of monthly returns appears nonnormal (refer to the skewness and kur?
tosis measures in Table 6 and 7), are the conclusions drawn from test statistics
based upon multivariate normality still valid?
An analysis of the subperiod results for the data summarized in Tables 6 and
7 is important because the returns distributions of the ten beta portfolios are
probably not stationary over the entire 45-year period, especially since the com?
position of each beta portfolio changes every five years. Table 8 contains the
mean differences between the monthly returns of the high and low beta portfolios
in each of the nine five-year subperiods. These data corroborate the finding
that a strong systematic relationship between estimated betas and average port?
folio returns does not exist. For example, with portfolios formed using betas
estimated with the equal-weighted index, the mean difference is more than two
standard errors from zero in only one subperiod. In three of the other eight
subperiods, the average return of the low beta portfolio exceeds the average
return of the high beta portfolio. When grouping is based on security betas
estimated with the value-weighted index, the mean difference in average returns
between the high and low beta portfolios does not exceed two standard errors in
any of the nine subperiods.
One possible explanation for the above results is that the holding period
454

18. TABLE 7
MONTHLY RETURN STATISTICS FOR THE TEN BETA
PORTFOLIOS BASED ON BETAS ESTIMATED USING A
VALUE-WEIGHTED NYSE INDEX AND THE "MARKET MODEL" ESTIMATOR
Estimated Autocorrelations
Portfolio
Beta Skewness Kurtosis 1 2 3
.44 .175 7.818 .11 .01 .05
.69 -.171 4.259 .03 .03 .04
.84 -.166 5.585 .02 .06 .02
.98 .190 6.444 .00 .08 .01
1.10 .605 8.797 .03 .10 -.01
1.23 .568 8.087 .01 .10 .01
1.36 .064 4.588 .04 .08 -.00
1.52 1.401 14.888 .02 .07 -.00
1.71 .614 6.134 .03 .10 -.01
2.13 .680 5.240 .02 .10 -.01
Mean returns are multiplied by 100 for reporting purposes. A 1.0 equals
1%. Standard errors are in parentheses. The statistics are based on 540 monthly
observations from 1935 through 1979.
2
The estimated portfolio beta is the equal-weighted combination of security
betas. These betas are estimated in the five-year periods prior to the portfolio
holding periods.
455

19. TABLE 8
MEAN DIFFERENCES IN MONTHLY RETURNS
BETWEEN THE HIGH AND LOW BETA PORTFOLIOS
DURING THE FIVE-YEAR PERIODS FROM 1935 THROUGH 1979
Beta Estimator, NYSE Market Index
Market Model, Market Model,
Period Equal-Weighted Value-Weighted
Overall .579 .416
(1.97) (1.59)
1/35 - 12/39 1.297 1.172
(0.78) (0.76)
1/40 - 12/44 1.690 1.364
(1.42) (1.34)
1/45 - 12/49 .295 .287
(0.42) (0.45)
1/50 - 12/54 .688 .692
(1.32) (1.45)
1/55 - 12/59 -.049 .062
(-.11) (0.14)
1/60 - 12/64 -.384 -.408
(-.95) (-1.12)
1/65 - 12/69 .757 .529
(1.47) (1.12)
1/70 - 12/74 -.853 -1.009
(-1.06) (-1.38)
1/75 - 12/79 1.775 1.053
(2.30) (1.88)
Mean differences in monthly returns are multiplied by 100 for reporting
purposes. T-values are in parentheses. Results for the overall period are
based on 540 months.
456

20. betas of the ten portfolios do not differ from each other; however, this possibility
seems ruled out by evidence contained in Table 9. This table presents a compari?
son of the grouping period betas with the holding period betas. One observes at?
tenuation in the estimated betas of the high and low beta portfolios; that is, the
holding period betas of these portfolios are closer to 1.0 than are the grouping
period betas. Nonetheless, the difference in estimated holding period betas be?
tween the two extreme portfolios is still greater than .9. Furthermore, the hold?
ing period betas preserve the rank ordering established by the grouping period
betas. Thus, this evidence indicates that there are significant differences in
the holding period betas of the ten portfolios.
Since the empirical distributions of monthly returns do not appear to con-
form to the normal distribution, a proper concern is whether test statistics based
on normality might lead to inappropriate interpretations of the data. One notices
in Tables 6 and 7 that the monthly portfolio returns of the ten beta portfolios
tend to be skewed and leptokurtic relative to the normal distribution; one also
observes that, unlike daily portfolio returns, the monthly returns do not suffer
from severe autocorrelation. A nonparametric test can be employed to test for a
beta effect if one believes that the skewness and kurtosis in monthly returns
might seriously affect Hotelling's T-squared test. To avoid the assumption of
normality, Friedman's [13] rank test for a beta effect is performed. Under the
null hypothesis, any one ranking of the ten portfolio returns (from 1 through 10)
in a given month is assumed to be as likely as any other ranking. The null hypo?
thesis does not imply that each set of ten monthly returns is drawn from the same
population; however, independence between monthly returns is assumed. It is im?
portant to note that each set of monthly observations may differ tremendously with
respect to location, dispersion, or both. Hence, skewness and kurtosis relative
to the normal distribution will not invalidate this test. The test is only de?
signed to detect any systematic tendency for the monthly returns of one portfolio
to exceed or be smaller than the same-month returns of other portfolios. Under
the null hypothesis, the appropriate test statistic is distributed approximately
as chi-square with nine degrees of freedom.
Table 10 presents the chi-square test statistics for the two sets of ten
portfolios during the overall period as well as during each of the nine five-
year subperiods. At the .01 significance level, the null hypothesis of identical
returns could not be rejected for either set of ten beta portfolios during the
overall period. Indeed, with 540 observations, the .01 level may not be too
a criterion against which to test the hypothesis. Furthermore, at the
stringent
.05 significance level, the hypothesis of identical returns could not be rejected
for the portfolios created with security betas estimated with the value-weighted
457

21. TABLE 9
COMPARISON OF GROUPING PERIOD AND HOLDING PERIOD BETAS
FOR THE TEN BETA PORTFOLIOS OF NYSE STOCKS
(Betas computed with Monthly Returns using the "Market Model" Estimator)
NYSE Market Index
In a grouping period, a portfolio beta is just the equal-weighted combina?
tion of estimated security betas within that portfolio. The grouping period
betas reported above are the averages of portfolio betas over the nine five-year
grouping periods from 1930 through 1974.
2
The holding period betas are calculated by regressing monthly portfolio
returns against market returns from 1935 through 1979. Standard errors, which
are rounded to two significant digits, are reported in parentheses.
458

22. TABLE 10
CHI-SQUARE STATISTICS BASED ON FRIEDMAN'S
NONPARAMETRICRANK TEST FOR A BETA EFFECT
Beta Estimator, NYSE Market Index
Market Model Market Model
Period Equal-Weighted Value-Weighted
Overall 19.74 16.23
1/35 - 12/39 4.94 6.88
1/40 - 12/44 10.70 10.81
1/45 - 12/49 2.81 4.89
1/50 - 12/54 13.66 19.71
1/55 - 12/59 13.43 7.57
1/60 - 12/64 8.58 12.21
1/65 - 12/69 29.32 18.96
1/70 - 12/74 22.69 23.45
1/75 - 12/79 26.75 18.89
The chi-square statistics presented in this table are distributed with
nine degrees of freedom. The values of the 1 and 5 percent limits for this
distribution are 21.65 and 16.93, respectively.
459

23. NYSE market index. The subperiod results seem to indicate that a systematic rela?
tionship between estimated betas and portfolio returns was not present. For port?
folios formed with betas computed with the equal-weighted index, the hypothesis
of identical returns would be rejected at the .01 level in three of the nine sub?
periods. In one of these three subperiods, however, the average return of the
low beta portfolio actually exceeded the average return of the high beta port?
folio. For portfolios formed with betas calculated against a value-weighted in?
dex, the null hypothesis would be rejected at the .01 level in only one of the
nine subperiods, but in this subperiod the low beta portfolio experienced higher
returns than the high beta portfolio.
The nonparametric tests do not seem to detect a strong, persistent and sys?
tematic relationship between estimated betas and portfolio returns. Yet these
tests do yield insights into the nature of the data analyzed in Tables 6 and 7.
In those tables, one could not help but notice a monotonic relationship between
average portfolio returns and estimated betas that appeared to be consistent with
the CAPM. But the average returns turned out to be deceptive to the extent that
they masked the great variability associated with the time-series of portfolio
returns. While the average returns exhibited a rank ordering consistent with the
CAPM, the hypothesis test based on Friedman's nonparametric rank test indicated
that the month-by-month rankings of portfolio returns could not be distinguished
from random rankings. This variability in the time-series of portfolio returns
is also the reason why the parametric procedure, Hotelling's T-squared test, did
not reject the hypothesis of identical mean returns.
IV. Conclusion
This paper investigates whether differences in estimated portfolio betas are
reflected in differences in average portfolio returns. During 1964 through 1979,
the evidence indicates that NYSE-AMEX stock portfolios with widely difrerent esti?
mated betas possess statistically indistinguishable average returns. Evidence
based on NYSE stock portfolios dating back to 1935 corroborates this result. Of
course, this finding should not be construed to mean that all securities possess
identical average returns. Indeed, during this time period, Banz [2] and Reinganum
[22] report that portfolios of small firms experienced average returns nearly 20
percent higher than portfolios of large firms. The findings of this study demon?
strate that cross-sectional differences in portfolio betas estimated with common
market indices are not reliably related to differences in average portfolio re?
turns; that is, the returns of high beta portfolios are not significantly differ?
ent from the returns of low beta portfolios. In this cross-sectional sense, the
risk premia associated with these betas do not seem to be of economic or empirical
importance for securities traded on the New York and American Stock Exchanges.
460

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