Risk, Return, and Equilibrium
Empirical Tests
presented by Yuchen Zhang, Kaichuang Shu and Yinqian Shi
Introduction
This paper tests the relationship between
average return and risk for New York Stock
Exchange common stocks. The theoretical
basis of the tests is the "two-parameter"
portfolio model and models of market
equilibrium derived from the two-parameter
portfolio model.
Theoretical Background
• A perfect capital market
The capital market is assumed to be perfect in the sense that investors
are price takers and there are neither transactions costs nor
information costs.
• Two-parameter return distributions is normal
Distribution, of one-period percentage returns on all assets and
portfolios are assumed to be normal or to conform to some other
two-parameter member of the symmetric stable class.
• Investor risk aversion
Investors are assumed to be risk averse and to behave as if they
choose among portfolios on the basis of maximum expected utility.
Theoretical Background
• Optimal Portfolio
The optimal portfolio for any investor must be
efficient in the sense that no other portfolio
with the same or higher expected return has
lower dispersion of return.
Theoretical Background
In the portfolio model the investor looks at
individual assets only in terms of their
contributions to the expected value and
dispersion, or risk.
Theoretical Background
• The standard deviation is σ( )
the risk of an asset for an investor who holds p is the
contribution of the asset to σ(
).
• The proportion of portfolio funds invested in asset is
• The covariance between the returns on assets i and j
is
• The number of assets is N
Theoretical Background
• The risk of asset i in the portfolio p is proportional to
• Note that since the weights
, vary from portfolio
to portfolio, the risk of an asset is different for
different portfolios.
Theoretical Background
• For an individual investor the relationship between
the risk of an asset and its expected return is implied
by the fact that the investor's optimal portfolio is
efficient.
• Portfolio m
m is efficient means that the weight
,
i = 1,2, …, N, maximize expected portfolio return.
Theoretical Background
• Subject to constraints
• Lagrangian methods can then be used to show that the weights
, must be chosen in such a way that for any asset i in m.
is
the rate of change of
, with respect to a change in
at
the point on the efficient set corresponding to portfolio m.
----- (1)
Testable Implications
A . Expected Returns
……….(2)
Where
…(3)
can be interpreted as the risk of asset i in the portfolio
m, measured relative to the
, the total risk of m.
Testable Implications
• The intercept in (2),
…………(4)
• Then,
…………………(5)
……(6)
Testable Implications
Equation (6) has three testable implications:
• C1: Linear relationship exists.
• C2: No risk factors other than βi exists.
• C3: Higher return means higher risk,
i.e. Rm > R0.
Testable Implications
B. Market Equilibrium and the Efficiency of the
Market Portfolio
Assume the capital market is perfect. Suppose that from the
information available without cost all investors derive the
same and correct assessment of the distribution of the future
value of any asset or portfolio-----an assumption usually
called “homogeneous expectations.”
Then assume that short selling of all assets is allowed. Black
(1972) has shown that in a market equilibrium, the so-called
market portfolio, defined by the weights
Testable Implications
• C. A Stochastic Model for Returns
To use observed average returns to test the expected-return
conditions C1-C3:
……….. (7)
• The variable is included in (7) to test linearity.
•
in (7), which is meant to be some measure of the risk of
security i that is not deterministically related to β.
• The expected value of the risk premium
, which is the slope
in (6), is positive.
Testable Implications
D. Capital Market Efficiency: The Behavior of
Returns through Time
Market efficiency in the two-parameter model
requires that
, nonlinearity
coefficient
, non-β risk coefficient
and the
time series of return disturbances
are fair games.
Testable Implications
E. Market Equilibrium with Riskless Borrowing
and Lending
If we add to the model as presented thus far the
assumption that there is unrestricted riskless
borrowing and lending at the known rate, then one
has the market setting of the original two-parameter
“CAPM” of Sharpe (1963) and Lintner (1965).
Since
and market efficiency
requires that
be a fair game.
Testable Implications
F. The Hypotheses
• C1 (linearity) ----- E(
)=0
• C2 (no systematic effects of non-β risk) ----- E(
)=0
• C3 (positive expected return-risk tradeoff) ----E(
) = E(
) – E(
)>0
• Sharpe – Lintner (S-L) Hypothesis----- E( ) = Rft.
• ME(market efficiency)-----the stochastic coefficients
and the disturbances
are fair games.
Previous Work
Douglas (1969)
Refute condition C2
Miller and Scholes (1972)
Support Douglas’s test
Friend and Blume (1970), Black, Jensen, and Scholes (1972)
Average
is systematically greater than
.
Insufficiency:
• Condition C1 has been largely overlooked.
• The previous empirical work on the two-parameter model has
not been concerned with tests of market efficiency.
Methodology
Target
• Beta
• Non-beta risk
General Approach
Calculate Beta
• Calculate Beta
– Single stock: use sample covariance and variance to
estimate actual ones.
• Bias exists between estimated and real beta
– Solution: using portfolio beta to give a more accurate
estimation.
– Portfolio betas are calculated as value-weighted
average of individual betas.
General Approach
Calculate Beta
• To account for the influence of portfolios, portfolios are
forms by ranking of individual beta.
• By naively doing so, since
high-observed betas tend to be bigger than true betas and
vice visa.
• Solution: rank beta in one period and calculate portfolio
beta in another.
Details
• In period 1 (1926-29, 4 years), rank beta and form
portfolios.
– Beta is calculated using
– Let N be total securities, 20 be total portfolios to be
formed.
– The middle 18 portfolio has int(N/20) securities.
– If what left is of odd number, then the last portfolio
has one more security.
Details
Suppose N=1003
Portfolio
1
2
…
9
10
Beta
Lowest
…
…
…
Highest
Number of
Securities
101
100
…
100
102
Details
• Initial portfolio betas are computed with the data
from period 2 (1930-34, 5 years).
• Portfolio betas are updated monthly in period 3
(1935-38, 4 years)
– These betas are computed as simple average
of individual stocks, which automatically
adjust for delisting of securities monthly.
– Individual betas are updated yearly.
Details
1930
1931
1932
1933
1934
beta 0
1935
1936
1937
1938
period 0
beta 1
period 1
beta 2
beta 3
period 2
period 3
Details
• Measure the non-Beta risk
– Take the regression above.
– Compute the standard deviation of the error terms of the
same time period as that of computing beta.
– This error term measures the non-beta risk as:
Total Risk
Beta Risk
=0
Details
• Finally, put all the pieces together and for each
portfolio, run the regression:
Details
• We have generate the regression for 1935-38.
Repeating this work, we get a set of regressions
of 1939-42, 1943-46, …, 1963-68 and 1967-68.
Details
Why choose a 7-5-4 pattern
• Choose of 4-year test periods
– Computing costs – higher when longer
– Desire to update the data – better when higher
• Choose of portfolio formation period (4-7) and Beta/std
computing period
– Longer time period eliminate disturbance from other
factors.
– Longer time period requires statistical labor.
Details
Some Observations on the Approach
• The variance of portfolio beta is generally 1/3 to 1/7 of
individual beta. So estimating beta using portfolio are
more precise than using individual stocks.
• Portfolio methods increase the accuracy of estimation
more efficiently when beta is not extreme.
• The model is initially developed as a normative theory,
but this paper test it as a positive theory.
RESULTS
RESULTS
•
•
•
•
•
•
Thick-tailed Distribution & t-statistics
Tests of major hypotheses of the model
The behavior of the market
Variation in coefficient
Tests of Sharpe-Lintner Hypotheses
Conclusion
Thick Tailed Distribution
• Using t-statistics for testing the hypothesis:
• Fama and Blume suggests that distributions of common stock
returns are "thick-tailed" relative to the normal distribution.
• Fama and Babiak suggests that when one interprets large tstatistics under the assumption that the underlying variables
are normal, the probability or significance levels obtained are
likely to be overestimate.
Thick Tailed Distribution Conclusion
If these hypotheses cannot be rejected when tstatistics are interpreted under the assumption of
normality, the hypotheses are on even firmer ground
when one takes into account the thick tails of
empirical return distributions.
Tests of the Major Hypotheses
 Test of C1
Results in panels B and D of the table do not reject condition C1
of the two-parameter model, which says that the relationship
between expected return and β is linear.
Tests of the Major Hypotheses
 Test of C2
This hypothesis is not rejected by the results in panels C and
D. The values of
are small, and the signs of the
are randomly positive and negative.
Tests of the Major Hypotheses
 Test of C3
• C3 suggests that there is on average a positive tradeoff between
risk and return.
• If the critical condition C3 is rejected, then all is for naught.
• For the overall period 1935-6/68,
is large for all models.
and the values of
are also systematically positive in the
subperiods.
Tests of the Major Hypotheses
 Test of Market Efficiency
• The behavior through time of
is also
consistent with hypothesis ME that the capital market is efficient.
• As for statistical significance, under the hypothesis that the true
serial correlation is zero, the standard deviation of the sample
coefficient can be approximated by
The Behavior of the Market
• Some perspective on the behavior of the market during different
periods and on the interpretation of the coefficients in the riskreturn regressions can be obtained from the following table.
The Behavior of the Market
• If the two-parameter model is valid, then
• Sharpe-Lintner two-parameter model of market equilibrium.
• In the period 1935-40 and in the most recent period 1961-6/68,
is close to
and the t-statistics for the two averages are
similar. In other periods, and especially in the period 1951-60,
is substantially less than
.
The Behavior of the Market
 Conclusion
Trade-off of average return for risk between common
stocks and short-term bonds has been more
consistently large through time than the trade-off of
average return for risk among common stocks.
Errors and True Variation in the
Coefficients
•
Each cross-sectional regression coefficient
in equation 10
has two components: the true
and the estimation
error
.
------- (10)
•
Question:
※ To what extent is the variation in
to variation in
?
through time due
※ To what extent is the variation in
to
?
through time due
Errors and True Variation in the
Coefficients
• Alternative Question:
※ Can we reject the hypothesis that for all t,
※ Can we reject the hypothesis that month-by-month
?
※ Is the variation through time in
and to variation in
?
due entirely to
?
Errors and True Variation in the
Coefficients
• Results:
※ There is a substantial decline in the reliability of the
coefficients
and
.
※ F-statistics for
are also in general large.
※ F-statistics for
also indicate that
has substantial variation through time.
Errors and True Variation in the
Coefficients
• Results 2 and 3:
※ F-statistics for
and
are generally large for
the models of panels B and C and for the model of
panel D which includes both variables.
Tests of the Sharpe-Lintner Hypothesis
• S-L two-parameter model of Market Equilibrium:
• Friend and Blume (1970) and Black, Jensen, and Scholes (1972)
suggests that the S-L hypothesis is not upheld by the data. At
least in the post-World War II period, estimates of
seem
to be significantly greater than
.
• The S-L Hypthesis is ambiguous.
Tests of the Sharpe-Lintner Hypothesis
• Positive Evidence:
The hypothesis seems to do somewhat better in the two-variable
quadratic model of panel B and especially in the three-variable
model of panel D.
Tests of the Sharpe-Lintner Hypothesis
• Negative Evidence:
One-variable model of panel A provides the most efficient tests,
since values of
for this model are substantially smaller
than those for other models.
Tests of the Sharpe-Lintner Hypothesis
• Results:
Given that the S-L hypothesis is not supported by the
data, tests of the market efficiency hppothesis that
is a fair game are difficult since we no
longer have a specific hypothesis about
.
Conclusion
• Results support the important testable implications of
the two-parameter model.
• We cannot reject the hypothesis that average returns
on New York Stock Exchange common stocks reflect
the attempts of risk-averse investors to hold efficient
portfolios.
• Positive tradeoff between return and risk.
Conclusion
• Condition 1:
Relationship between a security's portfolio risk and its
expected return is linear.
• Condition 2:
No measure of risk, in addition to beta.
• Condition 3:
Positive trade-off between risk and return.
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