A New Empirical Perspective on the CAPM

A New Empirical Perspective on the CAPM
Author(s): Marc R. Reinganum
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: Cambridge University Press on behalf of the University of Washington School of
Business Administration
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JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS
Volume XVI, No. 4, November 1981
ON THE CAPM
A NEW EMPIRICAL PERSPECTIVE
Marc R. Reinganum*
Introduction
of the capital
The adequacy
Lintner
and Black
[17],
brium is
now seriously
Cheng and Graver
[5]
a security's
beta
even though it
pricing
determinant
of equilibrium
with different
estimated
While
importance
of beta,
demonstrate
betas
systematically
that
The average
securities.
ferent
from the average
widely
different
returns
Thus,
estimated
reliably
measure
a "risk
with the evidence
empirical
betas
an important
is
beta
are
different
on standard
suggest
results
portfolios
market indices
that
returns
These
dif?
not reliably
indistinguishable
in the market."
"anomalies,"
are
That is,
stocks.
some
so that
to average
related
stocks
rates
average
The test
avoided.
are
securities
the cross-sectional
not employed,
statistically
priced
whether
to assess
of high beta
possess
based
of the recent
not systematically
of low beta
which is
on empirical
research
returns
betas
estimated
returns.
are
betas
estimated
of equilibrium
In light
experience
regressions
earlier
seems to be
consensus
empirically
are designed
tests
and Scholes
Jensen,
Black,
determinant
a security's
to investigate
cross-sectional
across
nificant
is
which plagued
of the problems
economic
[3],
be reexamined.
should
paper
the statistical
of return.
as
Basu
[2],
and Thompson [20]).
[22],
Reinganum
(such
Banz
[1],
determinant.
that
the claim
Ball
the current
remains;
an important
pricing
of this
The purpose
[18],
studies
may not be the sole
however,
evidence,
empirical
still
still
is
Marsh
empirical
[11])
see
[27],
market equili?
of capital
representations
(for example,
[15],
of earlier
(CAPM) of Sharpe
models
pricing
empirical
Gibbons
and Fama and MacBeth
that
as
[4]
challenged
[8],
the influence
Yet,
asset
with
average
do not appear
findings,
the CAPM may lack
along
sig?
content.
The author wishes to thank Fischer
of Southern California.
University
and
Dick Roll,
Kim
Victor
Joines,
Dietrich,
Terry Langetieg,
Canto,
Doug
Black,
Alan Shapiro.
responsibility.
Any errors that remain are the author's
439
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to
II.
The development
pricing
Fama [10]).
see
example,
The Beta
and Test
Hypothesis
of the CAPM is
well
Design
known and can be found elsewhere
on the particular
Depending
set
which emerge from the CAPM can be expressed
relationships
(for
of assumptions,
as
the
either:
) + 3. [E(R ) - E(R
)]
i
m
om
om
(1)
E(R.)i
= E(R
(2)
E(R.)
l
= R + 3. [E(R ) - Rl
F
l
m
F
where:
= expected
E(R.)
return
E(R
E expected
)
return
R?
F
E risk-free
rate
betas
for the data
of this
average
[23]
to be consistent
is
rates
assets
related
of the hypothesis
supports
the contention
that
that
rejected
the hypothesis
would seem to indicate
with betas
are
economically
A straightforward,
First,
in period
placed
into
beta.
Then,
each
folios
possess
based
B, the returns
procedure
significantly
is
that
returns.
The beta
different
experience
would offer
evidence
Evidence
pricing.
the risk
estimated,
of the ten beta
associated
premia
the beta
hypothesis.
and securities
rank of their
portfolios
are
to test
average
returns
are
estimated
calculated
of the component securities
of ten portfolio
invoked
different
are
to test
upon the relative
the returns
is
employed
betas
security
With the time-series
statistical
in esti?
CAPM, the concern
in equilibrium
matter
strategy
with equal-weights
portfolio.
variate
two-step
one of ten portfolios
by combining
betas
con?
insignificant.
A, individual
in period
variations
in average
betas
that
a necessary
of the paradigm.
representation
Confirmation
Thus,
that
to variations
estimated
two assets
Namely,
of the theoretical
with different
of return.
and
i;
returns.
with the CAPM is
the common empirical
that
of asset
implication.
expected
the testability
questions
is
paper
hypothesis
an important
must be systematically
Roll
uncorrelated
of interest.
different
possess
is
on the market portfolio;
E cov(R.,R
E the beta
i m)/var(R m)
The two forms of the CAPM share
While
whose return
3.i
with different
mated betas
i;
) E expected return on an asset
with the market return;
E(R
dition
on asset
in hand,
within
a multi?
whether or not the ten port?
returns.
440
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The composition
of each
of the revisions
quency
the daily
firms
turns.
are
created
within
the beta
(1930-1979),
betas
security
of the 1935-1939
ship
are
betas
estimated
are
with data
beta
analyzed.
Exchange
annually.
Thus,
the 1964 beta
with 1963 daily
estimated
are
to identify
used
With monthly return
every five
updated
from 1930-1934
portfolios.
in the period
being
When analyzing
Stock
with 1964 data
portfolios.
portfolios
estimated
betas
The fre?
updated.
and American
revised
upon security
estimated
the 1965 beta
periodically
base
Exchange
are
portfolios
based
betas
Similarly,
rities
upon the data
depends
the beta
(1963-1979),
is
portfolio
of the New York Stock
returns
portfolios
beta
are
data
the secu?
for NYSE firms
For example,
years.
to form the member?
used
of the frequency
Regardless
of updates,
to the one in which portfolio
prior
re?
returns
are measured.
Three
different
both daily
estimators
and monthly return
returns
Security
squares.
are
betas
data,
are
for daily
priate
the impact
Williams
returns
of this
potential
data,
[9]
"market model"
the estimators
also
used
are
estimator
III.
Tests
To assess
by Scholes
and
betas.
security
to different
Recent
may be inappro?
problems.
proposed
to calculate
of the results
market
beta.
the estimated
trading
problem,
the sensitivity
least
the CRSP value-weighted
against
for
First,
ordinary
using
on the market is
this
estimates.
calculated
of nonsynchronous
because
and Dimson
[25]
with daily
that
indicates
however,
research,
are
regressed
and the computed coefficient
returns,
to compute beta
used
beta
Hence,
can
estimators
be investigated.
This
section
estimated
and sample
selection
are
three
contains
part
different
statistically
In the first
parts.
evidence
during
part,
results
1964 through
1979
of monthly
on 45 years
based
average
the data
the test
In the next part,
described.
if port?
to determine
designed
experience
into
Hypothesis
of NYSE and AMEX companies
returns
The final
are presented.
of tests
beta
divided
criteria
on the daily
based
is
The section
returns.
of the Beta
the results
reports
with different
folios
for NYSE companies.
returns
A.
Empirical
The Data
Stock
and Sample
return
Center
Chicago's
return
files
turns
(capital
1979.
used
for Research
as of December
New York Stock
December
data
gains
plus
Exchange
Criteria
Selection
in this
Prices
in Security
1979.
dividends)
the daily
only on NYSE companies;
however,
file
The daily
of all
file,
(CRSP)
Exchange
return
monthly and daily
contains
the monthly file
the stock
from the University
gathered
companies
Stock
or the American
Unlike
are
analysis
that
the daily
have
traded
of
stock
stock
re?
on the
from July 1962 through
contains
information
information
on the monthly file
441
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to January
back
dates
time security
Each
is
portfolios
sample
If a firm is
delisted
AMEX firms that
of firms changes.
returns
survival
during
during
the holding
In any one given
they
No other
is
imposed.
are
held
in
the number of NYSE and
year,
in the sample
that
period.
period,
holding
any funds returned
period,
for inclusion
qualified
estimation
the
data,
is
on securities
placed
the beta
of the ten beta
With the daily
through the portfolio
the end of the year.
until
and the composition
estimated
The only restriction
100 one-day
such as
are
the sample
yearly.
least
restriction,
cash
betas
revised,
changes
have at
1926.
between
ranged
and
2,000
2,700.
are updated
only because
portfolios
tion
a firm is
period,
678 in the 1930's
B.
excluded
five
every
only
if it
The Test
to 1296
Results
the cross-sectional
two reasons.
studies
pivotal
which ended
earlier
with Daily
relationship
these
First,
in December,
the beta
estima?
40 one-month
to have at least
in the monthly sample
from
ranged
between
[11],
the
unlike
Second,
respectively.
can be tested
of the
and Fama and MacBeth
[5],
1968,
for at least
betas
the time periods
outside
are primarily
1965 and June,
and estimated
returns
and Scholes
in which to study
a good period
1979 represent
years
the hypotheses
studies,
During
1964-1979
Returns:
Jensen
by Black,
years.
criterion
in the 1970's.
from 1964 through
The years
from the above
fails
The number of NYSE firms included
returns.
differs
with monthly data
criterion
The selection
with AMEX firms as well
as with
and placing
securi?
NYSE companies.
The first
ties
into
one of ten beta
to create
are
of the test
stage
with a value-weighted
returns
returns
the daily
matter
tionship
between
mean returns
Tables
portfolios
betas
and returns
of the ten beta
1 through
created
suggests,
and be able
are
the daily
with the different
beta
used
betas
the daily
with equal-weights
the portfolios.
then one ought to observe
portfolios
3 present
by combining
within
of the component securities
in the way the theory
these
above,
In the next period,
calculated
are
estimates
and Dimson estimators
Scholes-Williams,
are
portfolios
beta
As explained
NYSE-AMEX market index.
of the ten beta
betas
different
portfolios.
the "market model,"
computed using
estimating
Three
portfolios.
of ten beta
sets
three
involves
a positive
the hypothesis
to reject
If betas
that
rela?
the
equal.
return
statistics
estimates.
Table
for the ten beta
1 contains
information
betas were also calculated
For the Scholes-Williams
and OLS estimators,
dif?
were not significantly
The results
market index.
using an equal-weighted
with
Dimson betas were not calculated
ferent from those reported in the text.
the equal-weight
index.
442
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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
Portfolio
Beta
.05
Autocorrelations
Skewness
-.101
.33
The estimated
of security
betas.
holding period.
1
2
3
5.601
.48
.28
.25
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
Mean daily returns
through 1979.
Standard errors are in parentheses.
moments of the normal distribution.
2
.131
Kurtosis
using 4009 trading day returns from 1964
are multiplied
by 1000 for reporting purposes.
measures are based on
Skewness and kurtosis
combination
beta is just the linear
portfolio
(equal weights)
in the year prior to the portfolio
These betas are estimated
443
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on the beta
mator.
The null
into
account
The test
turns.
tested
tic
is
2.99.
Thus,
the ten portfolio
re?
under the null
hypothe?
of F(9,??)
and 1.88,
of this
test
statis?
of identical
as evi?
of the low beta
return
daily
of the high beta
return
daily
2.41
not be interpreted
should
rejection
are
the hypothesis
level,
the average
the average
exceeds
actually
portfolio
This
of the CAPM because
in support
dence
between
the values
the one percent
would be rejected.
mean returns
test
1, the computed value
in Table
even at
This
test.
T-squared
distribution
levels,
percent
For the data
respectively.
an F(9,4000)
has
esti?
are
of the ten portfolios
Hotelling's
correlations
contemporaneous
statistic
At the one and five
sis.
using
the "market model"
computed using
the mean returns
that
hypothesis
can be formally
identical
takes
formed with betas
portfolios
portfolio
by .03 percent.
the results
of the apparent
because
one observes
that
The skewness
and kurtosis
of the daily
a six
about
ket experienced
measures,
were about
ten standard
is
from the sample,
deleted
are
portfolios
the daily
1 that
in Table
is
estimated
to suspect
however,
confidence
that
it
can be detected
this
phenomenon is
Exchange
Exchange.
Thus,
a "small"
sample.
the null
during
a fluke.
a 16-year
After
companies
the data
16 years
data.
collected
as well
analyzed
are
as
in these
nearly
Furthermore,
this
that
tests
trade
1964-
the fact
the probability
represents
systematically
those
the period
in any one year,
reduces
period
all,
data
result
this
greater
experience
during
portfolios
an
with
not consistent
actually
portfolios
to accept
in which computer readable
can Stock
no reason,
with constructing
1 are
in Table
of the high beta
of the time for which CRSP has
period
is
of portfolio
of rejecting
associated
problems
low beta
While one might be able
1979.
There
in favor
biased
the results
region,
than those
returns
average
statistical
of the CAPM:
the predictions
are
the tests
the potential
Despite
matrix
but not efficiently.
auto-
are positively
portfolios
mean returns.
of identical
hypothesis
appropriate
that
observes
One also
somewhat leptokurtic.
the variance-covariance
consistently,
for the high beta
with the normal distribu?
associated
of the ten beta
With autocorrelation,
correlated.
returns
returns
portfolios
one observation
measures
and kurtosis
those
remain
If this
means.
their
the mar?
1970,
of the high beta
the returns
gain;
the skewness
portfolios
on May 27,
that
to out?
sensitive
are particularly
revealed
above
the same as
vitually
the low beta
tion;
returns
percent
of
significance
In particular,
seemed to be both skewed and leptokurtic.
however,
deviations
statistical
from normality.
departures
returns
the portfolio
Examination
liers.
the exact
in interpreting
One must be cautious
that
30 percent
is
available
the only
for all
Ameri?
on the New York Stock
would not seem to constitute
444
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One potential
beta
to biases
data.
Table
table
similar
are
returns
using
a value
of 1.85.
percent
level.
too.
Of course,
The results
borates
in Table
associated
Dimson recently
and inconsistent
gested
that
betas
one use
with daily
and leading
lagged
returns.
In this
security
ket returns.
This
and Reinganum
[21].
on Dimson's
folio
beta
aggregated
is
exceeds
coefficients
return
t-value
in the previous
there
and average
son betas
of equality
thesis
the
.01
higher
between
indistinguishable.
estimated
The evidence
betas
are
analyzed
is
returns.
returns
not always
in Tables
used
One observes
return
evident
in Roll
betas
based
for
.06 per?
returns
jointly
beta
Dim?
between
the hypo?
portfolios
at
are
portfolios,
average
on 16 years
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per?
reported
considered
445
port?
.001
by only
however,
with higher
based
[24]
except
between
for the intermediate
3 is
mar?
leading
association
of the two extreme
1 through
coefficients.
of the high beta
portfolio
ten portfolios
associated
regression.
that,
are
One would reject,
Furthermore,
on security
with estimated
the portfolios
no immediately
security
to regress
slope
As with the portfolio
means for all
but the average
level,
statistically
portfolio
of 0.09.
Dimson sug?
and five
to those
of the low beta
not
might be
is
technique
20 lagged
the mean daily
Furthermore,
are
estimator
market returns
constructed
of all
corro-
betas
one runs a multiple
methodology.
returns
cent with an associated
tables,
are
This
enough problem.
identical
virtually
returns
returns.
estimation
using
here
apply
the average
the sum of the estimated
calculated
is
the mean daily
this
on
at the five
method for estimating
regression,
simply
are
the mean daily
a serious
the contemporaneous)
of a simple
procedure
and .08 percent.
cent
as
behind
3, the ten portfolios
P9,
portfolio
well
regressions
paper,
In Table
The idea
instead
Thus,
The estimated
(as
is
coefficients
mean
takes
F-test
other.
even the Scholes-Williams
if nontrading
of identical
in estimated
in average
returns
average
above
from each
in
the Scholes-
hypothesis
at best,
that,
differences
an aggregated
data.
higher
discussed
differences
1; positive
that
argued
biased
caveats
indistinguishable
with positive
Even using
the null
2 seem to indicate
in Table
1.
the hypothesis
technique,
the statistical
are
portfolios
the evidence
reliably
T-squared
possibility
for portfolios
the appropriate
one would not reject
Hence,
might
The numbers presented
experience
If one tests
portfolios.
Hotelling's
of the ten beta
portfolios
This
statistics
betas.
the
that
of security
then this
problem,
the results.
returns
in Table
reported
the low beta
estimator,
affect
estimated
to those
than do the high beta
a serious
the daily
1 is
estimates
least-squares
is
which could
2 contains
in Table
presented
ordinary
If nontrading
with Scholes-Williams
constructed
Williams
using
in estimation
now explored.
this
of the results
created
on daily
based
lead
is
are
portfolios
betas
criticism
returns.
of daily
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
lA mean return is calculated
Mean daily returns
through 1979.
Standard errors are in parentheses.
moments of the normal distribution.
2The estimated
betas.
of security
holding period.
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
using 4009 trading day returns from 1964
are multiplied
by 1000 for reporting purposes.
measures are based on
Skewness and kurtosis
combination
beta is just the linear
(equal weights)
portfolio
in the year prior to the portfolio
These betas are estimated
446
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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
.09
.11
.35
A
1979.
errors
of the
mean return is calculated
using 4009 trading day returns from 1964 through
Standard
Mean daily returns are multiplied
by 1000 for reporting purposes.
measures are based on moments
Skewness and kurtosis
are in parentheses.
normal distribution.
The estimated
betas.
of security
holding period.
combination
beta is just the linear
(equal weights)
portfolio
in the year prior to the portfolio
These betas are estimated
447
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return
data.
series
could
One potential
be that
since
especially
ary,
can be examined
year portfolios'results
lem.
the most succinct
Perhaps
the differences
to present
beta
the different
are
these
results
exceeds
beta
Using
rors
is
portfolio
portfolio
presented
in Tables
1 through
in Tables
1 through
3 as
does
are
really
the prior
in particular
folios
largest
This
error.
the ten portfolios
Table
portfolios,
the danger
portfolio
by two
in one
portfolio
not statisti?
are
the findings
corroborate
the results
of interpreting
effects
of the
however,
return,
throughout
these
betas
estimated
are
the 16-
possibility
can be investigated
in which average
period
during
betas
period
the entire
are
betas
that
are meas?
returns
betas
in
estimated
are
port?
with the
estimated
the betas
by computing
returns
of
are measured.
with the holding
16-year
the portfolio
the extreme beta
ranked,
whose betas
may contain
the grouping
is
on security
securities
estimators
holding
results
in which portfolio
formed based
are
in the year
5 compares
for the three
results
in the year
portfolios
Since
year.
er?
of the high
the differences
years,
for the above
explanation
not different
that
Recall
ured.
in average
portfolio.
of the high beta
the average
illustrating
signifi?
of the low
return
the average
return
Thus,
in average
not seem great.
Another possible
betas
3.
errors
of the low beta
return
of the high beta
The average
the year-by-year
Hence,
of the high
return
by two standard
of the low beta
return
betas
are more than two standard
with Dimson betas,
In the remaining
between
of the mean return
the average
return
the average
exceeds
significant.
year period
with
portfolio.
portfolio
of the 16 years,
the average
as well.
of the years
port?
when security
none of the differences
estimator,
in two of the 16 years.
low beta
is
calculated
not statistically
are
estimate
of the high beta
created
exceeds
errors
prob?
analysis
the differences
15 years,
portfolios
than the average
greater
standard
cally
years,
In seven
portfolio
a serious
relationship
of the low beta
return
the high and low beta
For portfolios
beta
betas
the average
estimator,
the point
the Scholes-Williams
from zero.
is
of this
For example,
returns.
For the other
that
exceeds
between
returns
no significant
the average
of these
portfolio
the year-by-
the high and low beta
between
reveal
the high and low beta
between
In nine
cant.
this
for security
also
in only one of the 16 years.
returns
whether
differences
with the "market model"
portfolio
however,
the results
returns
and average
betas
portfolio
calculated
beta
to gauge
yearly;
estimators.
The year-by-year
estimated
revised
way to convey
in average
4 reports
Table
folios.
are
the portfolios
station?
may not be sufficiently
distribution
the statistical
from such a time
inferences
with drawing
problem
sample.
betas
period
For each
computed with the "market model,"
448
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set
of ten
Scholes-
TABLE 4
MEAN DIFFERENCES IN DAILY RETURNS BETWEEN THE HIGH
AND LOW BETA PORTFOLIOS ON A YEARLY BASIS
(Betas Computed with Daily Returns)
Beta
Mean differences
are
T-values
poses.
ing days.
Estimator
in daily returns are multiplied
pur?
by 1000 for reporting
250 trad?
in parentheses.
Each year contains
approximately
449
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TABLE 5
COMPARISON OF NYSE-AMEX PORTFOLIO BETAS ESTIMATED
IN GROUPING PERIODS AND HOLDING PERIODS
GROUPING PERIOD ESTIMATOR
Aggregated
Coefficients
GP
MM SW
AC
.72
.82
1.26
.29
.71
.80
1.14
1.21
.62
.76
.86
1.20
.88
1.28
.89
.81
.91
1.25
.84
.96
1.36
1.15
.87
.97
1.35
Scholes-Williams
MM SW
GP
AC
.07
.43
.53
1.07
.41
.53
.64
1.06
.59
.65
.77
.75
.76
.91
-.50
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
2.25
1.69
1.67
1.95
2.25
1.60
1.62
1.95
3.77
1.32
1.39
1.89
High Beta
GP stands for the estimated
beta of the portfolio
during the grouping
created with "market
period.
Grouping period betas are shown for portfolios
model" estimates,
Scholes-Williams
and Dimson's aggregated
coefficients
estimates,
For each set of portfolios,
estimates.
three estimated
holding period betas are
shown:
MM ("market model");
SW (Scholes-Williams);
and AC (Dimson's
aggregated
In a grouping period,
a portfolio
beta is just the equalcoefficients
method).
of estimated
betas within that portfolio.
The
security
weighted combination
betas over
of the portfolio
grouping period betas reported above are the averages
the 16 grouping periods
from 1963 through 1978.
Holding period betas are calcu?
16 years of daily portfolio
lated by analyzing
and market returns
(1964-1979).
450
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and Dimson's
Williams,
of portfolios
for example,
"market model,"
are presented
for the portfolios
beta
created
portfolio
"market model"
beta
portfolio
gregated
coefficients
the beta
of the lowest
AC portfolio
is
5 also
Table
of estimated
periods,
regardless
of the estimator
the Scholes-Williams
estimates.
with the "market model"
for these
and 1.29
estimates,
exhibit
spreads
is
to form these
roughly
the rank ordering
equivalent
and Dimson's
cluding
the portfolios
that
betas
ent estimated
The Test
is
result
to hold
over
Jensen,
and Scholes
average
specific
a longer
returns.
of the low beta
section
is
For example,
portfolio
is
betas
within
period
Indeed,
betas
confident
4 possess
whether
in condiffer?
widely
the "beta
or whether,
evidence
Table
statistically
in fact,
it
not
appears
that portfolios
indistinguishable
2, the mean excess
the two standard
does
from the work of Black,
with the proposition
possess
in their
of Scholes-Williams
1935-1979
to the 1964-1979
estimated
which
periods.
to investigate
may be consistent
seem to
estimator
conclu?
1 through
holding
estimators
similar
5 that
one may feel
Hence,
in Tables
the portfolio
time horizon.
[5]
different
betas.
with Monthly Returns:
of this
matter"
analyzed
during
Results
The purpose
with widely
coefficients
The spread
but also
betas,
"market model"
on the basis
created
rank
the
on the Scholes-
based
1.16
the other
in Table
One discovers
portfolios.
aggregated
of the
periods,
between
estimates.
Thus,
of estimated
to those
can be drawn for portfolios
sions
is
Consider
are perfectly
the spread
periods
.88.
is
the holding
the holding
portfolios
on the "market model"
based
on Dimson betas
based
portfolios
not only tend to preserve
used
during
portfolios
the rank
preserves
During
Furthermore,
the holding
and Scholes-Williams
during
betas.
of these
of the betas
estimates
high and low beta
C.
of "market model"
between
the ten portfolios.
to create
used
in Ac betas
perfectly
of ten portfolios
set
For example,
of the high?
two estimators.
with those
estimator
of the ag?
betas
the beta
in "market model"
almost
in esti?
attenuation.
to 1.26;
the spread
Thus,
created
for each
betas
formed on the basis
portfolios
to 1.89.
for portfolios
severe
reveal
from -.50
of the highest
to the attenuation
similar
The estimated
portfolios.
rises
each
is
alone.
"market model"
beta
The attenuation
to 1.69.
than the spread
that
the estimated
portfolios
however,
from 3.77
reveals
ordering
beta
AC portfolio
smaller
betas
period
Williams
from 2.25
betas
of the high and low
of the lowest
beta
similarly,
beta
portfolios,
drops
the AC portfolios
ordered
.40;
drops
by the "market model"
exhibited
holding
to
betas
Thus,
betas
period
of "market model"
in the estimated
of the Scholes-Williams
mated betas
est
formed on the basis
set
estimators.
and Dimson holding
the estimated
from .05
rises
based
attenuation
even though each
estimators,
on only one of these
Scholes-Williams,
For example,
portfolios.
beta
with betas
5 one observes
In Table
coefficient
aggregated
is
error
return,
confident
451
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(R -R ),
interval
about
the mean excess
and Scholes
Jensen,
are
and the intercepts
different
estimated
folio
test
security
betas
ten beta
portfolios
the ten beta
are
estimated
is
different
nificantly
seems to indicate
of average
the rank ordering
betas.
One cannot,
that
is
of the difference
zero.
one could
dard
argue
that,
confidence
error
possess
tribution.
1.22;
these
of identical
one clearly
Thus,
are
statistics
account
test
using
Hotelling's
mean returns,
the test
cannot
analyzed
reject
in Table
the null
of estimated
The
than the low
or statistically
of the high
the returns
error
per month, the standard
less
between
than two standard
the average
errors
from
computed with 540 observations,
the power of the test,
formally
the
alone.
returns
reliable
the mean difference
is
glance,
In addition,
estimates
between
percent
might be more appropriate
on the data
are
based
per month, whereas
percent
average
are
For example,
.9 percent.
from point
higher
.580
1.5
with the
tables
to the rank ordering
the differences
portfolios
into
mean returns
Based
sig?
portfolios
estimated
At first
1979.
matter.
only
corresponds
6 is
One can also
interval.
portfolio
percent.
taking
region
identical
hypothesis
is
since
Furthermore,
error
.294
about
the mean difference
of the high and low beta
returns
ought to have
in these
betas
is
possess
in Table
portfolios
is
draw inferences
portfolios
while
Indeed,
and low beta
is
period
from 1975 through
OLS betas
The statistics
portfolio
not mean that
does
portfolios
grouping
is
of
the monthly
for the ten beta
of their
estimated
returns
however,
the high beta
significant.
that
of the low beta
return
the average
period,
the monthly returns
portfolios
from 1935 through
data
of the high beta
return
average
to the
and port?
and membership in the
period
statistics
on the basis
of monthly return
the evidence
similar
with equal-weights
holding
the monthly return
NYSE market index.
CRSP equal-weighted
upon 45 years
returns.
estimation
The first
then the ten beta
matter,
securities
formed by grouping
portfolio
is
with
mean returns.
6 presents
Table
the portfolio.
for
between
In the first
squares
computed by combining
the last
betas
least
and positive
average
of one year.
ordinary
regressions
if portfolios
the initial
In the next period,
within
1934;
through
If estimated
1979.
beta
are
portfolios
that
Black,
relationship
with the monthly data
instead
years
using
inverse
indistinguishable
except
established.
of the securities
from 1930
section
five
(3 > 1)
what one would expect
used
procedure
"market model"
betas
This
had statistically
are
periods
in their
(3 < 1).
precisely
in the previous
holding
returns
the intercepts
betas
Furthermore,
portfolio.
with high estimated
is
betas
The two-stage
one employed
fact
that
with low estimated
portfolios
betas
note
for portfolios
negative
of the high beta
return
a three
two stan?
than the conventional
whether
the ten beta
test.
T-squared
statistic
assumes
6, the value
hypothesis
standard
portfolios
Under the null
an F (9,531)
of the test
of identical
452
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dis?
statistic
mean returns
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
3
-.00
.04
Kurtosis
1
-.287
4.961
.11
-.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
.827
10.000
.02
.08
.453
6.290
.02
.10
.876
8.422
.03
.08
-.01
.948
5.642
.03
.09
-.02
Mean returns are multiplied
by 100 for reporting
The statistics
Standard errors are in parentheses.
1%.
from 1935 through 1979.
observations
2The estimated
These betas
betas.
holding periods.
2
Skewness
.00
-.00
.01
A 1.0 equals
purposes.
are based on 540 monthly
of security
combination
beta is the equal-weighted
portfolio
in the five-year
are estimated
prior to the portfolio
periods
453
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even at the
.05
The data
that
except
dex.
level.
OLS betas
security
the high beta
Again,
the low beta
portfolio,
month rather
than
is
ten beta
portfolio
7.
based
Thus,
and estimated
betas
on the evidence
a statistically
reliable
mated portfolio
betas.
in these
additional
three
data?
be addressed.
First,
the estimated
to the grouping
similar
in Table
tosis
measures
based
upon multivariate
probably
not stationary
position
of each
folio
returns
does
subperiods,
from zero
the average
of the high beta
estimated
the high and low beta
any of the nine
One possible
during
that
within
the
of monthl
the holding
the empirical
drawn from test
These
between
index,
and kur?
statistics
corroborate
data
estimated
betas
portfolios
are
since
the com?
In three
is
the mean difference
does
not exceed
betas
eight
the average
on security
based
port?
more than two
of the other
exceeds
portfolio
portfolios
the finding
and average
is
the
8 contains
formed using
the mean difference
When grouping
portfolios
Table
years.
6 and
in Tables
especially
with portfolios
of the low beta
index,
summarized
of the high and low beta
in only one subperiod.
with the value-weighted
returns,
to the skewness
period,
every five
For example,
portfolio.
methods
portfolio
of the ten beta
45-year
subperiods.
return
and esti?
of 45 years
given
(refer
for the data
distributions
changes
not exist.
to be
valid?
results
relationship
in Table
returns
the results
are
the conclusions
the monthly returns
five-year
value
the average
not appear
to average
Finally,
nonnormal
the entire
portfolio
systematic
errors
standard
between
over
with the equal-weighted
estimated
return
beta
of the nine
a strong
still
between
mean differences
that
normality
the returns
because
important
in each
are
of the subperiod
An analysis
betas?
appears
6 and 7),
does
of the ten portfolios
betas
the
although
6, however,
on the analysis
based
period
of monthly returns
distribution
7 is
should
with the findings
are
Secondly,
periods
related
issues
for the
computed with standard
betas
that
level,
portfolio
average
per
difference
rank correlated
there
do not seem to be reliably
consistent
subperiods
between
.4 percent
than the critical
less
in Table
two tables,
than
mean returns
.05
not perfectly
conclude
While one might tentatively
and market indices
the data
are
relationship
at the
slightly
just
Unlike
distribution.
returns
is
1.88,
about
greater
with this
of identical
hypothesis
NYSE market in?
return
is
associated
would not be rejected
statistic,
for the F(9,531)
the difference
case
the t-statistic
the null
still
portfolios
an average
experienced
6
in Table
analyzed
computed with a value-weighted
but in this
.85 percent;
of the test
value
are
to the data
similar
portfolio
In addition,
1.59.
only
7 are
in Table
analyzed
in average
two standard
betas
returns
errors
subperiods.
explanation
for the above
results
is
that
the holding
454
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period
in
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
Portfolio
Beta
Autocorrelations
Skewness
Kurtosis
1
2
3
.175
7.818
.11
.01
.05
.44
.69
-.171
4.259
.03
.03
.04
.84
-.166
5.585
.02
.06
.02
.01
.98
.190
6.444
.00
.08
1.10
.605
8.797
.03
.10
1.23
.568
8.087
.01
.10
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
The statistics
1%. Standard errors are in parentheses.
from 1935 through 1979.
observations
2
betas.
holding
The estimated
These betas
periods.
-.01
.01
A 1.0 equals
purposes.
are based on 540 monthly
of security
combination
beta is the equal-weighted
portfolio
in the five-year
are estimated
prior to the portfolio
periods
455
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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,
Equal-Weighted
Period
Market Model,
Value-Weighted
Overall
.579
(1.97)
.416
(1.59)
1/35 - 12/39
1.297
(0.78)
1.172
(0.76)
1/40 - 12/44
1.690
(1.42)
1.364
(1.34)
1/45 - 12/49
.295
(0.42)
.287
(0.45)
1/50 - 12/54
.688
(1.32)
.692
(1.45)
1/55 - 12/59
-.049
(-.11)
.062
(0.14)
1/60 - 12/64
-.384
(-.95)
-.408
(-1.12)
1/65 - 12/69
.757
(1.47)
.529
(1.12)
1/70 - 12/74
-.853
(-1.06)
-1.009
(-1.38)
1/75 - 12/79
1.775
(2.30)
1.053
(1.88)
in monthly returns
Mean differences
are
in parentheses.
T-values
purposes.
based on 540 months.
are multiplied
by 100 for reporting
for the overall
Results
period are
456
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do not differ
of the ten portfolios
betas
seems ruled
out by evidence
son of the grouping
tenuation
holding
with the holding
betas
of the high and low beta
betas
of these
the holding
evidence
betas
period
Since
is
might lead
a proper
to inappropriate
affect
might seriously
null
month is
thesis
however,
population;
to note
respect
to location,
each
dispersion,
will
to the normal distribution
to detect
signed
the null
as
Table
year
returns
overall
stringent
.05
could
degrees
the overall
.01
At the
not be rejected
a criterion
against
level,
significance
for the portfolios
is
Under the
The null
It
assumed.
hypo?
is
im?
tremendously
is
The test
of other
10)
drawn from the same
with
relative
and kurtosis
test.
of
only de?
of one portfolio
Under
portfolios.
distributed
approximately
of freedom.
period
as during
as well
which to test
the
level
betas
estimated
of identical
the
may not be too
Furthermore,
returns
five-
during
portfolios
the hypothesis.
of identical
with security
.01
of the nine
hypothesis
of ten beta
set
for either
each
the null
level,
significance
of ten
for the two sets
statistics
test
the hypothesis
created
is
may differ
is
statistic
with 540 observations,
Indeed,
period.
this
for a
(from 1 through
for the monthly returns
test
to test
ranking.
skewness
Hence,
not invalidate
the chi-square
10 presents
subperiods.
or both.
do not suffer
the assumption
returns
monthly returns
between
tendency
one also
performed.
any other
than the same-month returns
with nine
during
as
likely
is
effect
of ten monthly returns
the appropriate
hypothesis,
chi-square
portfolios
any systematic
or be smaller
to exceed
for a beta
of monthly observations
set
portfolios
in monthly returns
To avoid
of the ten portfolio
set
One notices
of the data.
and kurtosis
test.
based
statistics
can be employed
test
T-squared
independence
that
portant
each
test
the monthly returns
the skewness
to be as
assumed
not imply that
does
in
to con-
do not appear
to the normal distribution;
returns,
rank test
[13]
any one ranking
hypothesis,
in a given
that
Hotelling's
Friedman's
normality,
period
differences
of the ten beta
returns
A nonparametric
if one believes
effect
beta
be?
the hold?
Furthermore,
whether
interpretations
relative
portfolio
daily
autocorrelation.
from severe
the
is,
betas
period
significant
is
concern
the monthly portfolio
tend to be skewed and leptokurtic
unlike
are
at?
the grouping
by the grouping
of monthly returns
distributions
6 and 7 that
.9.
established
there
than are
holding
than
greater
that
indicates
form to the normal distribution,
that,
still
that
of the ten portfolios.
the empirical
on normality
in estimated
One observes
portfolios;
to 1.0
possibility
a compari?
presents
betas.
period
closer
the rank ordering
preserve
this
Thus,
are
portfolios
the difference
Nonetheless,
betas
ing period
observes
table
betas
tween the two extreme portfolios
in Tables
This
period
period
betas.
9.
in Table
contained
this
however,
other;
in the estimated
betas.
period
from each
could
at the
not be rejected
with the value-weighted
457
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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
combina?
a portfolio
In a grouping period,
beta is just the equal-weighted
The grouping period
tion of estimated
betas within that portfolio.
security
betas reported above are the averages
of portfolio
betas over the nine five-year
from 1930 through 1974.
grouping periods
2
The holding period betas
market returns
returns against
are rounded to two significant
are calculated
monthly portfolio
by regressing
which
Standard errors,
from 1935 through 1979.
are reported in parentheses.
digits,
458
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TABLE 10
CHI-SQUARE STATISTICS BASED ON FRIEDMAN'S
NONPARAMETRICRANK TEST FOR A BETA EFFECT
Beta
Estimator,
Market Model
Equal-Weighted
Period
NYSE Market Index
Market Model
Value-Weighted
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
Overall
with
in this table are distributed
statistics
The chi-square
presented
for this
of the 1 and 5 percent limits
The values
nine degrees of freedom.
are 21.65 and 16.93,
distribution
respectively.
459
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The subperiod
NYSE market index.
between
tionship
formed with betas
folios
of identical
returns
low beta
folio.
dex,
the null
nine
subperiods,
but in this
do yield
In those
one could
tables,
they masked the great
While
returns.
test
This
is
why the parametric
the reason
also
in differences
reflected
mated betas
course,
this
identical
[22]
returns.
that
portfolios
higher
that
that
is,
are
the returns
premia
importance
for securities
during
to mean that
this
betas
average
betas
are
In this
Of
result.
securities
possess
and Reinganum
[2]
returns
with common
re?
portfolio
not significantly
cross-sectional
460
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All use subject to JSTOR Terms and Conditions
differ?
sense,
do not seem to be of economic
Stock
20
nearly
study demon?
estimated
on the New York and American
esti?
Evidence
this
in average
1979,
difrerent
of this
The findings
portfolios
portfolios.
1964 through
Banz
time period,
to differences
related
all
are
betas
returns.
average
in portfolio
of high beta
traded
did
test,
portfolio
to 1935 corroborates
firms.
of large
with these
returns
T-squared
with widely
firms experienced
differences
of low beta
associated
back
of small
not reliably
ent from the returns
risk
Indeed,
indicated
of portfolio
During
portfolios
indistinguishable
not be construed
than portfolios
cross-sectional
market indices
turns;
should
finding
rank test
not be distinguished
in estimated
returns.
portfolio
dating
portfolios
with the
consistent
Hotelling's
differences
NYSE-AMEX stock
statistically
average
report
percent
strate
possess
on NYSE stock
based
that
indicates
the evidence
in average
of portfolio
Conclusion
whether
investigates
paper
that
mean returns.
IV.
This
with
to the extent
could
returns
procedure,
of identical
the hypothesis
not reject
appeared
in the time-series
variability
between
to be consistent
nonparametric
of portfolio
the month-by-month rankings
relationship
a rank ordering
on Friedman's
based
6 and 7.
with the time-series
exhibited
Yet these
in Tables
analyzed
out to be deceptive
associated
returns
that
betas
and sys?
persistent
returns.
a monotonic
but notice
turned
variability
from random rankings.
higher
experienced
portfolio
of the data
the nature
returns
the average
CAPM, the hypothesis
in?
in only one of the
level
and portfolio
betas
and estimated
But the average
the CAPM.
estimated
not help
returns
portfolio
average
port?
a value-weighted
a strong,
do not seem to detect
into
insights
.01
of the
return
of the high beta
against
at the
sub?
portfolio.
between
relationship
the average
calculated
For port?
of the nine
in three
return
rela?
the hypothesis
index,
level
the low beta
subperiod
tests
The nonparametric
tematic
the average
would be rejected
hypothesis
.01
however,
subperiods,
exceeded
than the high beta
returns
that
three
formed with betas
For portfolios
was not present.
returns
at the
would be rejected
actually
portfolio
and portfolio
a systematic
that
computed with the equal-weighted
In one of these
periods.
tests
betas
estimated
seem to indicate
results
the
or empirical
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