tNST.
AUG
8
1973
J-'BRARIE-S
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
THE MAEKET MODEL APPLIED TO EUROPEAN
COMMON STOCKS: SOME EMPIRICAL RESULTS
Gerald A. Pogue and Bruno H. Solnik
6tvai
MWM ^^^
^^'^"''^ '~'^^^'
Revised May 1973
657-73
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS
i
^\v^»<
THE MARKET MODEL APPLIED TO EUROPEAN
COMMON STOCKS: SOME EMPIRICAL RESULTS
Gerald A. Pogue and Bruno H. Solnik
Giy»j^
MWM
^:Je
-S^^'"^ Hf^^'
Revised May 1973
657-73
Not for Quotation
Comments Invited
-^\v^\<
no.
45'7-73
RECEIVED
MAY 31 1973
M.
1.
T.
LIBRARIES
THE MARKET MODEL APPLIED TO EUROPEAN
COMMON STOCKS:
SOME EMPIRICAL RESULTS
Gerald A. Pogue and Bruno H. Solnik
I.
Introduction
The stock price literature abounds with applications of the
Markowitz
[1
]
- Sharpe
[15] market model to
American stock price data.
2
There is a lack of corresponding studies for non-American securities, due
primarily to the absence of generally available machine readable data bases
(see however [13] and [16]).
The purpose of this paper is to present the
results of some initial tests of the market m.odel for a broad cross-section
of the European common stocks.
Our data base consists of daily price and divi-
dend data for 228 stocks from seven European countries.
3
In addition, for com-
parison purposes we have included a sample of 65 American securities.
Assuming the market model to be the stochastic process generating
security returns, regression analysis was used to estimate the model's parameters for various return measurement intervals and test periods.
sis focuses
The analy-
on the measures of systematic risk (beta), proportion of varia-
tion explained by market movements (R
2
)
,
excess return (alpha) and the
statistical significance of the excess return measures (T-alpha)
.
The esti-
mated parameters were tested for robustness to changes in measurement
interval and stability over time.
Finally, the results were examined for
their implications for relative efficiency of the various markets studied.
The paper is organized as follows:
part
II presents a brief review
of the market model, part III describes the data base, part IV the methodology,
part V the empirical results and part VI the implications for market efficiency.
071S204
- 2 -
II.
The Market Model
The model is based on the hypothesis that the risk premium on
security
j
premium,
(R
during interval
)
t,
(R.
is a linear
),
function of the market risk
The risk premiums are formed by subtracting the riskless
.
rate from the respective security and market returns.
The relationship is
given by
R_
jt
where a. and
6.
=
+
a.
J
R,^
J
are parameters and e.
Jt
3
2
6.
of security risk premium.
The e
+e
>lt
(1)
.^,
jt'
is the non-market related component
variables are usually assumed to have the
following properties: zero expected values; uncorrelated with the market
^
return;
7
fi
palrwise
and serially uncorrelated; finite variance,
Using these assumptions, the beta parameter is
Cov(R
R
9
O..
given by
)
The usual interpretation of 3. is as a measure of the systematic risk of
security
j
relative to that of the market index; that is, the numerator of
the right hand side of equation
2
represents the systematic or non-diversi-
fiable security risk, the denominator the systematic risk of the market
index.
To gain insight into the nature of the a
.
parameter, we rely on
the equilibrium predictions of the Sharpe [14] - Lintner [9] Capital Asset
Pricing Model (CAPM)
.
The CAPM relates the expected security risk pre-
miums to their systematic risk coefficients, 3..
That is.
- 3 -
E(R^^)
wh ere E(R.
and E(S,^
)
Comparing equations
on security
j
)
(1)
=
B.
•
E(l^^)
(3)
are the expected security and market risk premiums.
and (3), a. is seen to be a measure of the return
in excess of that predicted by the CAPM.
the expected value of a. is equal to zero.
Thus,
Under CAPM assumptions
realized values, while not
necessarily zero should tend to be small and serially uncorrelated through
time.
III.
The Data Base
The data base consists of daily prices and dividend data for 228 com-
mon stocks of seven European countries.
1966 to April 1971.
comparison purposes.
The time period covered is from March
In addition, a sample of 65 American stocks was used for
The American data covered the same period and was taken
from the Standard and Poor's I.S.L. tape of New York Exchange securities.
The distribution of the sample by country is shown in table
1.
Within each European country, the companies in our sample tend to be the largest
in terms of market value of shares outstanding.
The 30 Italian stocks, for
example, comprise about three-fourths of the market value of all listed
Italian shares.
For the United Kingdom, France and Germany, the number is not
as high but still in excess of 50 percent in each case.
Fifty of the 65
American stocks were randomly selected from the population of all NYSE stocks
in existence as of March 1966.
The remainder of the sample was composed of
15 corporations among those with the largest total equity market value listed
on the NYSE.
Security risk premiums were computed on a daily, weekly, bi-weekly
and monthly basis, as follows:
- 4 -
P
where R
R
P
P
d
+
d
- P
,
=
the risk premium during interval t
=
risk free rate during interval
=
the stock price at the end of period t
=
the stock price at the end of the previous period
=
dividends paid during the interval (assuming payments on
ex-dividend dates)
t
The stock price and dividend data were corrected for all capital adjustments
(splits, rights, etc.)-
This feature can be very important since firms pay
out most of their earnings this way in some countries.
Dividend data were
not readily available for two of the countries, the Netherlands, and
Switzerland; thus risk
premiums are measured in these cases by proportionate
change in stock price.
For each country, the risk premiums on a market index were computed
on a comparable basis.
For the six countries
(including the USA) for which
stock dividend data were available, the risk premiums on the index include
dividends.
For the reamining two where security dividend data was not
available, dividends were not included in the index.
The names of the mar-
ket indexes used are given in Exhibit 1.
It was not practical to attempt to collect short-term rates for each
return interval for each country.
As an approximation we used the same short-
term rate for each return interval (adjusted to duration of interval).
It
is not felt that this approximation will introduce any significant distortions
in the results.
For all of the European countries, with the exception of
-
5 -
the United Kingdom, we used the short-term prime bank rate.
risk-free rates used are given in Exhibit
IV.
The names of the
1.
Methodological Issues
The purpose of our empirical tests is to measure the relative mag-
nitude and stability of various market model parameters during the five years
covered by our data base.
A principal question involves the choice of an interval for measurement of security and market returns.
Since daily data is available, it would
perhaps seem logical to use daily returns, as opposed to, say, weekly or monthly
values.
However, this question is complicated by difficulties resulting from
measurement errors and adjustment lags in the rates of return.
We will
briefly discuss each of these issues in turn,
(i)
Estimator Efficiency
As shown in Appendix A,
any grouping of the daily data into longer
intervals will result in an increase in the standard errors associated with
the estimated market model parameters.
Thus, if no other considerations were
present, the daily returns would provide the most efficient estimates of the
coefficients,
(ii)
Measurement Errors
Errors in reporting security prices will produce noise in the se-
curity return measures
.
This will result in a higher variance of the residual
term in equation (1) than when correct returns are available.
will be a reduction in the R
2
The effect
between the security and market returns.
How-
ever, there will be no expected attenuation in the estimated beta coefficient
as long as the measurement errors and not correlated with the market returns
cu
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.
- 7
(a plausible assumption)
Other than simple reporting errors or lags, a common source of error
results from the rounding of prices to fractional values (e.g. quarters).
small source of error can be important for short return intervals.
Th:Ls
It should
also
he recognized that the European Markets do not have the same information gathering and dissemination facilities as the major United States markets.
Thus,
short term price changes are much more likely to be in error than their
American counterparts.
The effect of measurement errors will diminish as the length
of the return interval increases.
Thus, measurement errors alone would sug-
gest the use of an interval of maximum length,
(iii)
Adjustment Lags
J
Adjustment lags result from the failure of stock prices to fully
adjust to market changes during the trading day.
be spread over two or more days.
Instead the adjustment may
This can be caused either by the lack of
trading in the stock, or failure to comprehend market conditions when delays
in index reporting exist.
The result is that the price will "catch up" for
previous market activity during later days.
This phenomenon would imply a
distributed lag model for explaining security returns; for example,
\
where R^
,R,^^
.
.
=
.
^ ^
^^It
-^
^A,t-1
" ^2^1,
t-2
•
•
+ ^t
(5)
are the market returns on the current and previous
days
When daily returns are used to estimate the market model parameters
- 8 -
(equation
1)
the price lag effect will result in a reduction of the percent-
age of security return variation explained by the market and the estimated beta
g
will be a downward biased estimate of the true beta,
p.
This problem would
also tend to disappear as the measurement interval increased.
The adjustment
lags will be more important in the European markets, where trading volume
is typically much lower than for U.S.
stocks, and where reliable market in-
formation is more difficult to obtain.
An additional problem results from errors in the market index itself.
In a number of European markets the market index is not computed on daily clos-
ing prices, but on prices at some convenient time during the day.
The result
is that the stock returns might well lead rather than lag the index.
would imply that we should add a term 3_iR
to reflect this effect.
attenuation of
3
,i
This
to equation (5)
Any index lag would also result in a lower R
as in the stock price lag case.
2
and
The importance of this
effect will diminish as the interval is increased.
In summary, the above discussion indicates that optimal Interval
length for measuring returns is by no means obvious.
Consequently, we have
presented summary results in part V for each of four intervals; daily, weekly,
bi-weekly and monthly.
The results for the different intervals are compared
in part VI for their market efficiency implications.
* * *
A second major emphasis of the paper is to examine the stability
of model parameters over time.
This was accomplished by subdividing the five
years and measuring model parameters for each sub-period.
The first period
extends from March 1966 to November 1968, the second from December 1968 to
- 9 -
April 1971.
parameter estimates
To test for stability of the parameters,
values for the first period.
for the second period were regressed on
The regression equation is given by:
where
Y
p^
.
and p
are the subperiod parameter estimates for security j,
and Yi ^^e regression coefficients and M the number of securities.
The parameters chosen for testing were alpha, the
alpha, beta and the coefficient of determination.
represents the systematic risk.
t
statistic for
Beta was chosen since it
Alpha and t-alpha were chosen to examine the
predictability and significance of the excess returns earned and R
2
measures
the percentage of security variation explained by market movements.
Using cross-sectional correlation coefficients to measure parameter
stability is complicated by measurement errors; the estimates
not the true parameter values, p
p^
.
and p„
.
are
and p«., but are only estimated values.
As shown in appendix B, these errors will cause the estimated cross-section
correlation coefficient
p
and regression coefficient
Y-i
to be attenuated.
degree to which the estimates p and f^ understate the true values
p
and
The
Y-i
depends on the size of the measure error variation relative to the dispersion
of the true parameters P^ and P_.
This attenuation bias must be kept in
mind when evaluating the empirical results in part V.
For example, it would
not be unexpected to find higher cross-sectional correlations for parameters
based on daily returns since their measurement errors (as given by the standard errors of the estimates) would be the smallest.
- 10 -
V.
Empirical Results
The empirical results are summarized in tables 1 through
6.
In
some cases parameter estimates are given for each of the four measurement
intervals; however, for brevity, when all observation frequencies lead to
similar conclusions, only the bi-weekly results are given.
The robustness
tests, which examine the sensitivity of parameter estimates to changes in the
return measurement interval, are based on the total five-year period; the
stability tests are based on the two
A.
2
1/2 year subperiods.
Robustness of Market Model Parameters
Table
1
gives the average security return and standard deviation
for each country sample.
Corresponding data are given for the market indexes
Q
used.
The security averages are unweighted, and thus can differ substan-
tially from those of the market weighted indexes.
The average returns vary
from 0.015 percent for the U.S. to 0.466 percent per two-week period for
Germany.
It is interesting to note that the U.S.
average standard deviations.
stocks had the highest
However, this may reflect the broader composi-
tion of the U.S. sample rather than inter-market differences.
Table
country sample.
frequencies.
2
presents average estimated beta, t-beta and R
2
for each
The results are presented for each of the four observation
The most striking observation is the extent to which the aver-
age parameter values depend on return interval.
The estimated betas are lowest for daily returns, highest for
monthly returns.
As discussed in part IV,
this effect most likely results
from lags in the adjustment of stock prices to changes in market levels.
As
such, the effect would be expected to diminish for longer return intervals.
- 11 -
In an efficient market, where such lags were absent, the expected
value of beta should be invariant to the return measurement interval.
Thus,
differences in average betas between daily and monthly returns can be used to
measure the relative importance of adjustment lags in the various markets.
The slower the price adjustment speed, the greater the range between beta
values based on daily and monthly returns.
The following gives the ratios
of average betas based on these return intervals.
Exhibit
2
Effect of Lags in Stock Price Adjustment
Country
- 12 -
The average
t
values are typically reduced by half between daily and monthly
observations.
Finally, the average R
stock, returns explained by
2
figures show the percentage of variation in
market movements in the various countries.
The
numbers display the same general pattern as the beta estimates, rising from
a low for
daily to a high for monthly return observations.
below the spreads between daily and monthly R
Exhibit
2
are larger than for beta.
3
Effect of Lags and Measurement Errors
y
Country
United States
=
However, as shown
—2
R monthly
^
z^
R daily
- 13 -
The average R
2
for the Small Three (Belgium, the Netherlands and Switzerland),
while typically smaller for daily returns, have values similar to the other
European countries for monthly returns.
Table
3
presents data on the bi-weekly excess returns achieved in
Under the CAPM assumptions, the
each country; that is average alpha values.
The table shows
expected average alpha value is zero for each market.
the mean alphas and t-alphas for each country.
The
t
statistics to a first
approximation allow us to identify significant individual alpha values.
table gives the frequency distribution for
of
t
t
for each country.
The
The range
values is larger for all but one of the European countries than for
the U.S.
For the U.S., 6.1 percent of the
t
values exceed 1.5 in absolute
value; for Europe the percentage are typically higher, ranging from a low
of 5.9 (Switzerland) to a high of 46.6 percent (Italy).
To summarize,
the estimated beta and R
tive to the return interval used.
U.S; the effect on R
2
2
parameters are sensi-
The effect on beta is smallest in the
is similar in the U.S.
and the four larger European
countries, and larger in the smaller three.
B.
Parameter Stability
In this section we will focus on cross-sectional an-
alysis of market model parameters estimated in two sub-periods
to November 1968 and December 1968 to March 1971.
—
March 1966
Our purpose is to compare
parameter stability across the eight markets, and thus gain insight into the
degree of predictability of key parameters.
The cross-sectional regressions
were run for parameters obtained for each of the four return intervals.
Table 4 presents average bi-weekly rates of return for each subperiod.
The data illustrate the low degree of intermarket correlation re-
- 14 -
Additionally, for each country the
ported by several previous authors.
subperiod returns show little correlation as well, while average standard
deviations are much more predictable.
Table
R
2
,
a and
t
presents average subperiod parameter estimates for beta,
5
The statistics are based on bi-weekly data.
.
a
The beta results
show that for seven of the eight countries the average value was larger in
The R
the second period.
The average R
2
2
results do not present such a consistent pattern.
increased for four countries (including the U.S.) and de-
clined for four.
Similarly, the alpha and
to change in any systematic way.
t
alpha statistics do not appear
The sample standard deviations
associated with the averages in the table show
the range of estimates
differs considerably among markets.
Table
6
presents the results of interperiod cross-sectional
The coefficient of cross-sectional determination
regressions.
measures the magnitude of parameter correlation (i.e. the portion of
variation in P„
.
explained by
ship is indicated by the
meter (t ^).
t
P, .).
The significance of the relation-
statistic associated with the "slope" para-
The regression results are presented for all four sets of
subperiod parameters.
Cross-sectional comparisons are complicated by two factors, differing sample sizes and dispersion of sub-period parameters.
For example,
the subperiod German betas are relatively tightly grouped, and thus (as
predicted in part IV) the inter-period correlations will be more sensitive
to parameter measurement error.
The intra-country results for standard deviation, beta and R
2
show
.
- 15 -
a consistent pattern for most countries; correlation decreases as the return
observation frequency decreases.
Daily estimates have the highest degree
of predictability, but as discussed above the estimates are biased.
As pre-
dicted the correlations are most quickly attenuated as the return interval
increases for those countries with low parameter dispersions.
of Germany,
for example, the cross sectional R
2
In the case
for beta decreases from
0.718 for daily observations to 0.00 for bi-weekly intervals.
12
When inter-country comparisons are made, the three parameters
are seen to be more stable in several of the European countries (e.g. Great
Gritain, France, Italy) than in the U.S. market.
This result is most evi-
dent for daily observations, less so for monthly data.
The results for the
small Three show a lower correlation of daily parameters than the Big Five;
however, the results are not significantly different for monthly sub-parameters.
Interestingly, the cross-sectional correlation
of
B
and R
2
actually
increase with interval size for one of the smaller countries (The Netherlands)
As for the alpha excess return measures, little evidence for their
predictability can be found.
Exceptions perhaps are Great Britain and Italy.
For Great Britain the correlation between sub-period alpha values is significant at the five percent level for all intervals.
ship between the subperiod
VI.
t
For Italy the relation-
values is significant at the five percent level
Implications for Market Efficiency
Our final task is to review our results for their implications
regarding relative market efficiency.
While our tests are not extensive
enough to permit definitive conclusions, they permit a number of interesting
observations.
13
- 16 -
The first evidence comes from the sensitivity of the average beta
values to the length of the return interval.
In an efficient market, beta
would not be sensitive to the interval size.
This, however, was not the
case.
As reported in part V, average monthly betas exceeded average daily
betas in all eight countries, the ratios ranging from 1.08 for the U.S.
Thus, price adjustment lags appear least significant
to 3.20 for Belgium.
in the U.S. market.
On the whole, the four major European countries had
lower ratios than the three smaller countries, indicating a somewhat shorter
adjustment period for the larger markets.
The next issue involves the extent to which measures of total and
systematic risk are predictable.
If securities are to be properly priced,
Our subperiod cor-
investors must be able to predict their future riskiness.
relation analysis sheds some light on this question.
Both standard devia-
tions and beta values were found to be as predictable in the European
markets as in the U.S.
Under the CAPM assumptions applied to each market, the average alpha
values for each country should average to zero and be uncorrelated across
time.
The results in table 6 support this hypothesis in most of the countries,
with the possible exception of Great Britain where there is some evidence
of significant inter-period correlation.
Similarly, the
alpha should be uncorrelated between subperiods.
t
statistics for
This hypothesis is again
generally true, with the exception of Great Britain and Italy.
In these
countries it would have been possible to earn subperiod returns relative
to the CAPM standard by selection stocks with significant first period t-
alpha values.
- 17 -
The distribution of
t
a
statistics show the European sample to have
larger proportions of outlier values.
for Italy where nearly 1/2 the
This result is particularly striking
values exceeded 1.5 in absolute value.
t
However, given our lack of knowledge concerning the degree of statistical
dependence between intra country
t
values, it is difficult to draw very
strong conclusions from these results.
On the whole our evidence does not show substantial differences
between the U.S. and four major European markets.
Some cases can be made
for the three smaller markets being less efficient.
ing efficiency must be viewed as tentative in nature.
Our conclusions regardWe have dealt with
only a five-year period, using pre-selected security samples.
More defini-
tive conclusions must await more extensive and statistically powerful tests.
.
Appendix
Effect of Grouping Procedure on Estimator Efficiency
A:
Consider the loss of efficiency associated with grouping k daily
The market model for the k daily
returns into H N-day intervals (k = NH)
returns is given by
where R
.
Tt
R,
and £
are the return on day
J
MTt
Tt
interval T (t=1,
.
.
.
,
H)
(t=l,
^
>
t
.
.
.
within
N)y
,
y
.
Now group the security and market returns into the N day values
by averaging the N daily returns in each of the H intervals.
Equation (A
1)
can be restated for the N day returns
R_
T
wh ere R^,
Let a
tively.
/\
and a,
R^^^
a + 3Rj^^ + e^
t=1,
.
and e^ are averages of R^^, R^^^ and G^^ (t=l,
*
.,
.
.
.
H
.
(A 2)
,
N)
respec-
n*
and B be the regression estimates based on daily returns
/\
B
the estimates based on N day return.
If the least squared assump-
tions hold for equation (A 1), they also hold for equation (A 2).
and B* are unbiased estimators, as are a and B.
a are given by
The variances of a
Thus a*
and
14
Var(e
Var(a*)
=
Z
Var(a)
=
)
-.^—^
_
(R
Z
T=l t=l
^^
—
1
Z
T=l
^
(A 4)
-
- R
(R
^
(A 3)
- R )^
2
)
^
2
Appendix A.
wh ere R
the average of the H R^ values (note that R
is
to the average of the K R
variance of e
returns)
is equal to the
Therefore,
divided by
^ N.
Tt
The variance of e
.
is also equal
Var(e
-
Var(a)
)
'-^
"
N
-
(R
Z
.
(A 5)
R
2
)
T=l
Comparison of the two variance is equivalent to comparison of denominators.
Expanding the denominator of Var(a
H
I
N
),
_
_
-R)=ZZ(R
-R+R-R)
„
Z(RTt
Tt
T'^
^
T=l t=l
T
=
Z S
(R
T t
Thus, the denominator of Var(a
The ratio of Z
Z
T
t
- R
(R^
2
)
)
^
Tt
*
3
T
- R )^ + N Z (R - R )^
T
T
T
T
(A 6)
is always greater than that of Var(a).
^
to N Z
(R.^
-
—
R
2
)
is a measure of the loss of
T
precision resulting from the grouping.
variances of
T
T
t
^
and 6.
The results apply equally to the
Appendix B.
Attenuation of Cross-Sectional Correlation Coefficients
Assume the M estimated parameter values P,
to the true values P,
.
are related
and P„. as follows
.
Ij
Tlie
and P.
.
2j
P^.
=
P^. + u.
?,.
-
P^
measurement errors
u.
j=l.
.
.
..
M
+ V.
.
(B
1)
and v. are assumed to be distributed independently
from each other and from the true parameter values.
The measured correlation coefficient between P.. and P..
.
.
(designated p
M)
.,
Plim
where
p
is
=
P»,
will approach a limiting value, given by
a
(
cr
the true correlation coefficient, a
2
(B 2)
I
2
u
15
and o
2
V
are the variance of
12
2
are the cross-sectional variances of P^ and P ,
"
^
true
coris not an unbiased estimate of p, but will understate the
u and V, and a
Thus, p
)
(i=l,
and a
relation between the parameters by an amount depending on the size of the
measurement error variation relative to the dispersion of the true parameters.
(6)
Similarly, the estimated regression coefficient y^ In equation
will be downward biased.
The estimated values of Yt will tend to a
limit which is less than the true value.
plim (Yj^)
i
=
The limit is given by
(B 4)
a
a
Pi
Appendix B.2
is 10 percent of the vari-
Thus, if the variance of the error term for P
ance of the true values (as estimated by a
)
,
then least squares would
1
underestimate
Y-,
by about 10 percent, even for very large sample sizes.
FOOTNOTES
1.
Respectively, Associate Professor of Finance, Sloan School of Management,
M.I.T. and Assistant Professor of Finance, Graduate School of Business,
Stanford University.
for example, Blume
[1,2], Cohen and Pogue [3], Fama [4], King [7].
2.
See,
3.
We wish to express our appreciation to the Research Department of
Eurofinance, Paris, for providing the data on which the study is based.
4.
The parameter values
a.
and
B.
do not have time subscripts since they
are assumed to remain stationary through time.
Of course, one of our
main concerns in this paper is to test the validity of this assumption.
5.
This assumption can only be precisely true in the limit as the propor-
tion of security
6.
j
in the market portfolio approaches zero.
This assumption rules out industry effects.
King
However, these effects (see
typically account for only about ten percent of variation
[7])
in security returns, so that to a first approximation they can be
be ignored.
a
.
and
3.
This assumption has no effect on the estimation of the
parameters, but makes cross-sectional comparisons more dif-
ficult.
7.
This assumption presupposes the existence of variances for the
ables.
e
vari-
The work of Mandelbrot [11] and Fama [4] suggests, however, that
time series values of
e. ^
conform more closely to a non-normal stable
paretian distribution for which the variance does not exist than to the
usually assumed normal distirubtion.
Nonetheless, in this paper we
will make the more common assumption of the existence of these statistics
since Fama [5] has shown that insights into the effects of diversification
Footnotes
2
on dispersion of returns that are derived from the mean-standard
deviation model remain valid when the model is generalized to include the entire stable family.
8.
The market returns shown in the table are averages, computed in the
same way as the stock return averages.
The figures equal the calen-
dar returns only for those countries for which the all-stock return
series were complete over the March 1966-March 1971 period.
includes the U.S. and the major European countries.
This
For countries
where some of the series began after March 1966, the market return
reflects the average market return faced by the sample of stocks;
that is
-
\
where M
R^.
=
1
M
""
.^,
J=l
^j
number stocks in sample
= average market return during period of complete data for
^
9.
_
stock
j
.
The measurement error tends to be submerged with longer return
intervals.
For example, using monthly returns, only two of the approxi-
mately 22 daily returns in the monthly figure would be affected by measurement error.
For daily observations, on the other hand, both the be-
ginning and ending prices of the interval would be subject to error.
10.
A previous study by King [7] examined the relationship between market
and security returns for NYSE stocks.
Sixty-three common stocks were
analyzed, with returns computed on a monthly basis from June 1927
to December 1960.
The average R
2
value decreased monitorically, from 0.584
for the 1927-1935 period to 0.307 for the 1965-1960
interval.
More
recently Blume [1] replicated these tests using a larger sample of stocks
and found results consistent with those of King.
The mean proportion
of return variation explained by the market was 0.280 for the period
1961-1968.
Footnotes
11.
3
For reasons of brevity we have not presented the details of the
cross-sectional regression results.
coefficients can be computed by combining
the estimated y^ and y
results from tables
5
and 6.
^
=
Yi
^
However, for bi-weekly intervals
P
'
pi
^^
2
(P)
where
/
is given in table
^0
6,0 pi
and a
„
^2 - ^1^1
=
where the average subperiod estimates, (P
12.
Therefore,
in table 5.
p2
and P«)
are given in table
»
5.
The stability of common stock beta parameters has been studied by
Blume [1] and Levy [8].
Using monthly returns and seven year estimation
periods, Blume found an average product moment correlation between
stock betas of 0.618 (R
2
= 0.37).
More recently Robert Levy replicated
these tests using weekly returns and various short estimation periods
(e.g.
26,
52 weeks).
For the period from 1962 through 1970 the average
correlation between annual beta values was 0.486 (R
13.
2
= 0.25).
An additional factor which complicates comparisons between U.S. and
European results is the potential lack of compatibility between data
samples.
The U.S. sample, due to its random content, is more representa-
tive of the complete cross-section of NYSE stocks.
The European
samples were pre-selected, and mainly represent the major stocks in
their respective markets.
Further, the U.S. daily price tapes used
for the study may well have a higher error rate than the carefully
screemed European data file.
lA.
See Malinvaud [10], pp. 282-283.
Footnotes 4
15.
the plim(p
for plim(Y
16.
)
expression is derived in an analogous manner to that
)•
See equation BA and footnote 16,
See Johnston [6], equation (9-43), page 282.
t^
(0
>
u
c
s
u
3
i)
CO
n)
<u
c
u
p
u
(fl
3
O
•H
Vj
03
>
1-1
O
U
0)
H
4J
01
e
to
M
tfl
B
to
w
TD
O
•H
>-i
0)
o
H
(U
to
>
<:
O
in
in
J
•
e^ Al
A
tfl
>
u
Qi
in
o
C
.-I
C
lO
C
o o
M
3
4J
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oo
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(U
0)
&
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in
rH
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CQ
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in
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00
n)
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CN
,-1
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-a
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V
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CO
00 ro
x:
PL4
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4J
<-\
<
OO
o
H
0)
00
n)
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>
tfl
x;
1^
1-1
in
oo
0^
>
u
0)
c
c
u
3
Pi
3
PQ
u-1
0)
r-l
u
0)
4-)
a
u
C
0)
«
}^
3
4-1
Pi
CO
3
O
c
•H
+
1-1
CO
>
1—)
.H
CL
rH
o
+
O
Q-
CO
0)
o
c
o
cd
S
3
CO
w
wD'
c
o
•H
CO
Cd
a
u
cd
Ph
•a
o
•H
)-l
<U
Pj
X>
3
c/l
C
0)
0)
PQ
C
o
•H
4-1
to
r-l
o
u
oc
a
REFERENCES
"On the Assessment of Risk," Journal of Finance, March
Blume, M.
1971, pp. 1-10.
,
"Portfolio Theory: A Step Toward Its Practical
Application," Journal of Business, April 1970, pp. 152-173,
,
and G. Pogue, "An Empirical Evaluation of Alternative
Portfolio-Selection Models," Journal of Business, April 1967,
Cohen, K.
pp.
,
166-193.
Fama, E. , "The Behavior of Stock Market Prices," Journal of Business ,
January 1965, pp. 34-104.
"Risk Return and Equilibrium," Journal of Political Economy
,
January-February 1971, pp. 30-55.
Johnston, J.,
Econometric Methods,
New York, 1972^
,
Second Edition, McGraw-Hill,
King, B. F., "Market and Industry Factors in Stock Price Behavior,"
Journal of Business , January 1966, pp. 139-190.
Levy, R. A., "Stationarity of Beta Coefficients," Financial Analysts
Journal , November-December 1971, pp. 55-63.
Lintner, "Security Prices, Risk and Maximal Gains from Diversification,"
Journal of Finance December 1965, pp. 387-615.
,
Malinvaud, E.,
Statistical Methods of Econometrics
North-Holland Publishers, 1970.
,
Second edition
Mandelbrot, B., "The Variation of Certain Speculative Prices,"
Journal of Business , October 1963, pp. 394-419.
Markowitz, H., Portfolio Selection, Efficient Diversification of
Investments , Wiley, New York, 1959
Modigliani, F., G. A. Pogue, M. Scholes and B. Solnik, "Efficiency
of European Capital Markets and a Comparison with the American Market,"
Presented at First World Congress on the Stock Exchange, Milan, Italy,
March 1972. Published in Conference Proceedings.
Sharpe, W. F. , "A Simplified Model for Portfolio Analysis," Management
Science , January 1962, pp. 277-293.
References
2
[15]
Sharpe, W. F., "Capital Asset Prices: A Theory of Market Equilibrium
under Conditions of Risk," Journal of Finance , September 196A,
pp. 425-442.
[16]
Solnik, B. H., "European Capital Markets: Towards a Theory of an
International Capital Market," unpublished Ph.D. Thesis, M.I.T.
June 1972.
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