Risk, Return, and Equilibrium

1. Risk, Return, and Equilibrium Empirical Tests presented by Yuchen Zhang, Kaichuang Shu and Yinqian Shi

2. 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.

3. 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.

4. 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.

5. 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.

6. 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

7. 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.

8. 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.

9. Theoretical Background Subject to constraints Lagrangian methods can then be used to show that the weights , mustbe 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)

10. 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.

11. Testable Implications The intercept in (2), …………(4) Then, …………………(5) ……(6)

12. 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.

13. 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

14. 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.

15. 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.

16. 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.

17. 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.

18. 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.

19. Methodology

20. Target Beta Non-beta risk

21. 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.

22. 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.

23. 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.

24. Details Suppose N=1003

25. 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.

26. Details

27. Details Total Risk =0 Beta Risk • 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:

28. Details Finally, put all the pieces together and for each portfolio, run the regression:

29. 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.

30. 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.

31. 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.

32. RESULTS

33. 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

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

37. Thick 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 intoaccount the thick tails of empirical return distributions.

38. 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.

39. 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.

40. 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. Forthe overall period 1935-6/68,is large for all models. and the values of are also systematically positive in thesubperiods.

41. Tests of the Major Hypotheses Test of Market Efficiency The behavior through time of is also consistent withhypothesis ME that the capital market is efficient. As for statistical significance, underthe hypothesis that the true serial correlation is zero, the standard deviation of the sample coefficient can be approximated by

42. 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 risk-return regressions can be obtained from the following table.

44. 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 .

45. 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.

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

47. 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 due entirely to and to variation in ?

49. 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.