Risk, Return, and Equilibrium

Risk, Return, and EquilibriumRisk, Return, and EquilibriumRisk, Return, and EquilibriumRisk, Return, and EquilibriumEmpirical Tests

presented by Yuchen Zhang, Kaichuang Shu and Yinqian Shi

IntroductionIntroduction

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 BackgroundTheoretical 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 portf

olios 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 choo

se among portfolios on the basis of maximum expected utility.


• 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.


In the portfolio model the investor looks at individual assets only in terms of their contributions to the expected value and dispersion, or risk.


• 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 i

s• The number of assets is N


• 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.


• 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.


• 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 ImplicationsTestable 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.


• The intercept in (2),

…………(4)

• Then,

…………………(5)

……(6)


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.


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


• C. A Stochastic Model for Returns To use observed average returns to test the expected-return condi

tions 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.


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.


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.


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 gam

es.

Previous WorkPrevious 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.

MethodologyMethodology

TargetTarget

• Beta• Non-beta risk

General Approach General Approach Calculate BetaCalculate 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 General Approach Calculate BetaCalculate 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.

DetailsDetails

• 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.

DetailsDetails

Portfolio 1 2 … 9 10

Beta Lowest … … … Highest

Number of Securities

101 100 … 100 102

Suppose N=1003

DetailsDetails

• 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.

DetailsDetails

1930 1931 1932 1933 1934 1935 1936 1937 1938

beta 0 period 0

beta 1 period 1

beta 2 period 2

beta 3 period 3

DetailsDetails

• 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

DetailsDetails

• Finally, put all the pieces together and for each portfolio, run the regression:

DetailsDetails

• 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.

DetailsDetails

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.

DetailsDetails

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.

RESULTSRESULTS

RESULTSRESULTS

• 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 DistributionThick 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 t-statistics under the assumption that the underlying variables are normal, the probability or significance levels obtained are likely to be overestimate.

Thick Tailed Distribution ConclusionThick Tailed Distribution Conclusion

If these hypotheses cannot be rejected when t-statistics 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 HypothesesTests 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 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 Tests of the Major Hypotheses 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 Tests of the Major HypothesesHypotheses

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 MarketThe Behavior of the Market

• Some perspective on the behavior of the market during different periods and on the interpretation of the coefficients in the risk-return regressions can be obtained from the following table.

The Behavior of the Market 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 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 CoefficiErrors and True Variation in the Coefficientsents

• 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 through time due to variation in ?

※ To what extent is the variation in through time due to ?

Errors and True Variation in the CoefficiErrors and True Variation in the Coefficients ents

• 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 due entirely to

and to variation in ?


• 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.


• 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 HypothesisTests 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 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 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 Tests of the Sharpe-Lintner Hypothesis 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 .

ConclusionConclusion

• 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.

ConclusionConclusion

• 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.

THANK YOUTHANK YOU

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Risk, Return, and Equilibrium