1. Empirical Tests of CAPM - Fama&Macbeth Group member: Ruize Ge Yuan Zhang Shaojie Wu

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 Three Assumptions 1. Capital market is perfect Investors are price takers and there are neither transactions costs nor information costs. 2. Investors are risk averse Investors are assumed to be risk averse and to behave as if they choose on the basis of maximum expected utility. 3. Two-parameter return distributions is normal Distributions of returns are assumed to be normal or to conform to some other two-parameter member of the symmetric stable class. Conclusion The optimal portfolio must be efficient: no other portfolio with the same or higher expected return has lower dispersion of return.

4. Theoretical Background The definition of risk of an asset in the portfolio In the portfolio model the investor looks at individual assets only in terms of their contributions to the expected value and risk. With normal return distributions the risk of portfolio p is measured by the standard deviation, σ( ), of its return,,and the risk of an asset for an investor who holds p is the contribution of the asset to σ( ). is the proportion of portfolio funds invested in asset i is the covariance between the return on assets i and j, and N is the number of assets, then

5. Theoretical Background From last equation, the risk of asset i in the portfolio p ---- is proportional to Note that since the weight vary from portfolio to portfolio, the risk of an asset is different for different portfolio.

6. Theoretical Background If an investor chooses the portfolio m, the fact that m is efficient means that the weight, i = 1,2, …, N, maximize expected portfolio return Subject to constraints

7. Theoretical Background Lagrangian methods now can be used to show the weight must be chosen in such a way that for any asset i in m (1) Where S m is the rate of change of with respect to a change in at the point on the efficient set corresponding to portfolio m.

8. Testable Implications A. Expected Returns From last equation, we can get the equation: (2) Where (3) β i can be interpreted as the risk of asset i in the portfolio m, measured relative to the risk of m

9. Testable Implications From the equation (2), we can get the intercept: (4) E(R o )is the expected return on a security whose return is uncorrelated with, a zero-β security. Since β = 0 implies that a security contributes nothing to, it is appropriate to say that it is riskless in this portfolio. But β = 0 does not mean security i has zero variance of return. Then, (5) So, we can get the equation (6)

10. Testable Implications the expected return on security i is, the expected return on a security that is riskless in the portfolio m, plus a risk premium that is β i times the difference between and. Last equation has three testable implications: C1: The relationship between the expected return on a security and its risk in any efficient portfolio m is linear. C2: β i is a complete measure of the risk of security i in the efficient portfolio m. C3: In a market of risk-averse investors, higher risk should be associated with higher expected return.

11. Testable Implications B. Market Equilibrium and the Efficiency of the Market Portfolio Assume that 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.” Black(1972) showed that in a market equilibrium, weights Is always efficient.

12. Testable Implications C. A Stochastic Model for Return To test those three testable implications, we construct a stochastic model for returns (7) 1. The variable is included in (7) to test linearity. 2. S i in (7), which is meant to be some measure of the risk of security i that is not deterministically related to β. 3. The expected value of the risk premium, which is the slope in (6), is positive.

13. 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 ηit are fair games. γ jt → ﹛ E( +1 )= 0 E { +1 - [E( +1 )–E( +1 )] }=0 E( +1 )= 0 E( +1 )=0 ﹜

14. 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 Sharp and Lintner. Since and market efficiency requires that be a fair game.

15. 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( ) = R ft. ME(market efficiency)-----the stochastic coefficients and the disturbances are fair games.

16. 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: 1.Condition C1 has been largely overlooked. 2. The previous empirical work on the two-parameter model has not been concerned with tests of market efficiency.

17. Methodology A. General Approach B. Details

18. Model S-LC3C1C2

19. General Approach Unavoidable problem: “errors-in-variables” True : Estimate : Fisher’s Arithmetic Index: an equally weighted average of the returns on all stocks in NYSE. Blume (1970):

20. * Using portfolio is better than individual securities. Reducing the loss of information. Using ranked,but causing bunch of positive and negative sampling errors within portfolio. Large is overstate, low is underestimate. Using a subsequent period to obtain the.

21. Period T Period T+1...................... Low High Random error...................... + underestimate + overestimate P1 P2 P19 P20 P1 P2 P19 P20 Low High

22. Methodology-Detail Steps: 1.Data resource: Monthly percentage return for all common stocks traded on NYSE during 1926.1-1968.6; 2.Using the first 4 years(1926-29) of monthly return data to calculate the of each stock in NYSE; 3.Rank those from each company, divided them as 20 portfolios. 4.Using the following 5 years(1930-34) of data to calculate new, those still in the same portfolio as the same ranking order.

23. * Steps: 5.Calculate the 20 portfolio. 6.Set next 4 years(1935-38), as the testing period. 7. Update each monthly portfolio, and update each yearly in those 4 years. 8. Using the to measure non- risk; Calculate the residual of the market model: 9. Calculate the standard deviation of residuals.

24. * Step: 10. Calculate each portfolio’s standard deviation of residual. 11. Set all the parameters together, then run the regression to get one interception and three coefficients. ( is the average of for securities in portfolio p) Question: why can measure non- risk? Because and.

25. * Steps: 12. The same steps of method to calculate the results from following period years repeat the above.( e.g. divide 1927-33 as a new group, calculate the new to form new portfolios; the next 5 years are used to compute initial values of independent variables; and then run the risk-return regression are fit month by month for the following 4-year period.) See table 1:

27. V. Results ▪T-statistics & “thick-tailed” distribution ▪Tests of Major Hypotheses of Two- Parameter Model ▪The Behavior of the Market ▪Errors and True variation in the Coefficients ▪Tests of the S-L Hypothesis ▪Conclusions

28. To keep separate the pre- and post-WWII periods, results data are presented for 10 periods: I.The overall period 1935-6/68 II.Three long sub-periods, 1935-45, 1946-55 and 1956-6/68 III.Six subperiods which cover 5 years each except for the first and last.

29. To analysis and compare different variables, results are presented for four different versions of the risk-return regression equation: Panel D is based on equation(10) itself, but in panel A-C, some of the variables is suppressed. t-statistics for testing the hypothesis that are presented.

32. ▪Fama (1965a) & Blume (1970) which suggests that distributions of common stock returns are “thick-tailed” relative to the normal distribution. ▪While interpreting large t-statistics under the assumption that the underlying variables are normal, the significance levels obtained are likely to be overestimates. Thus, if these hypotheses cannot be rejected when t-statistics are interpreted under the assumption of normality, we have more statistical power to accept the hypotheses.

33. Tests of the Major Hypotheses Test of C1 In panel B, the value of for overall period(1935-6/68) is only -.29. In the 5-year subperiods, for 1951-55 is approximately - 2.7,other value of are much closer to zero. Results in panels B and D of table 3 do not reject condition C1, which says the relationship between expected return and is linear. Results

34. Test of C2 Consider condition C2, which says no measure of risk, in addition to,systematically affects expected returns. Results: in panels C & D, The values of are small, signs of it are randomly positive and negative. Thus, C2 is not rejected. Results

35. Test of C3 Consider condition C3, which says there is on average a positive tradeoff between risk and return. If the critical condition C3 is rejected, all we have done before is for naught.

36. Results: for the overall period 1935-6/68, is large for all models. Results Except for the period 1956-60, the values of are also systematically positive in the subperiods, but not so systematically large. How does it happen?

37. In panel A, for period 1935-40, =.0109, which means bearing risk had substantial rewards. However, since is 11.6 percent per month in this period-- is only.79.panel A At least with the sample of the overall period, achieves values to support C3.

38. Test of Market Efficiency (ME) The serial correlations are always low in terms of explanatory power and generally low in terms of statistical significance. Results:Results: the value of in table 3 are generally statistically close to zero.

39. 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 table 4.

41. To test whether is equal to we use the equation of one-variable regression of panel A, since E(Rpt)=E(Rmt), and E(beta0)=1, we got: So far, one can replace with Rft and the question transfers to test whether = Rft

42. 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 1950-60, is substantially less than.1935-40 Cause: is greater than (we’ll test it later) The tradeoff of average return for risk between common stocks and short-term bonds has been more consistently large through time than the tradeoff of average return for risk among common stock.

43. Errors and True Variation in the Coefficients Each cross-sectional regression coefficient has two components : the true and the estimation error The question is: To what extent is the variation in through time due to variation in and to what extent is it due to ?

44. So far, we have test the null hypotheses of C1, C2 and C3, which says E( )=0. However, we don’t know whether the specific data of is equal to zero in every month. First, test whether & month by month.

47. In table 5, data shows for each,, the sample variance of the month-by-month ;, the average of the month-by-month value of,where In table 5 is the standard error of from the cross- sectional risk-return regression for month t; and the F-statistic: which is relevant for testing the hypothesis,.

48. ▪The F-statistics for both and are large. Month by month, but it doesn’t imply rejection of C1 & C2 that is ▪The F-statistics for are also in general large. has substantial variation through time, since is always directly related to Rmt which changes through time.

49. Tests of the S-L Hypothesis In the Sharpe-Lintner two-parameter model of market equilibrium one has the hypothesis that 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-WWII period, estimates of seem to be significantly greater than Rft.

50. Each of the four models in table 3 can be used to test S-L hypothesis. The most efficient tests are provided by one- variable model in panel A, since the values of are substantially smaller than those for other models.

51. Ambiguity of S-L tests (Negative conclusion)Negative conclusion Except for the most recent period 1961-6/68, the value of in panel A are all positive and generally greater than 0.4 percent per m. The value of for the overall period is 2.55, and the t-statistics for the subperiods 1946-55,1951-55, and 1956-60 are likewise large.

52. (Positive conclusion)Positive conclusion However, in the two-variable model of panel B, table 3 and especially in the three-variables model of panel D, are closer to zero. (since value of that are closer to zero)

55. For test the S-L model, consider new model with, the above model is equal to regression equation(10). Although both equations are statistically indistinguishable, tests of S-L hypothesis do not yield the same results as tests of the hypothesis from the new equation, where is systematically positive for all periods but 1961-6/68 and statistically very different from zero for the overall period and for the 1946-55,1951-55, and 1956-60.

56. Conclusions Positive tradeoff between return and risk (C3) Relationship between a security’s portfolio risk and its expected return is linear (C1) There’s no measure of risk, in addition to beta.(C2) Cannot prove Sharpe-Lintner Hypothesis The market portfolio is efficient—or, more specifically, approximately efficient. Fair game properties in regressions consistent with ECM.

57. Subsequence Study

58. Other studies after 1973 Fama and French, 1992; He and Ng, 1994; Davis, 1994; Miles and Timmermann, 1996: provide weak empirical evidence on these relationships. Roll, 1977; Ross, 1977:The single-factor CAPM is rejected when the portfolio used as a market proxy is inefficient. Roll and Ross, 1994; Kandel and Stambaugh, 1995:Even very small deviations from efficiency can produce an insignificant relationship between risk and expected returns. Bos and Newbold, 1984); Faff et al.,1992; Brooks et al., 1994; Faff and Brooks, 1998: Beta is unstable over time. Kim (1995) and Amihud et al.(1993) : errors-in-the-variables problem impact on the empirical research. Kan and Zhang (1999) : a time-varying risk premium. Jagannathan and Wang (1996) : specifying a broader market portfolio can affect the results. Clare et al. (1998) : failing to take into account possible correlations between idiosyncratic returns may have an impact on the results.

62. Challenges on CAPM Market beta is not suffice to explain the expected return: 1. earning-price ratio (E/P ratio)-Basu(1977) 2.size: size effect-Banz(1981) 3.leverage-Bhandari(1988) 4.book-to-market equity ratio (B/M ratio)- Fama & French (1992)

63. References

65. THE END THANK YOU!