Testing the Capital Asset Pricing Model free essay sample

Testing the Capital Asset Pricing Model And the Fama-French Three-Factor Model By Jiaxin Ling (Cindy) March 19, 2013 Key words: Asset Pricing, Statistical Methods, CAPM, Fama-French Three-Factor Model Abstract: This paper examines the Capital Asset Pricing Model(CAPM) and the Fama-French three-factor model(FF) and the Fama-MacBeth model(FM) for the 201211 CRSP database using monthly returns from 25 portfolios for 2 periods July 1931 to June 2012 and July 1631 to June 2012. The theory’s prediction is that the intercept should equal to zero the slope should be the excess return on the market portfolio. The findings of this study are not substantiating the theory’s claim for the fact that in some portfolios the alpha is statistically significant with non zero value and in some regression models, the slope is not statistically significant. 1. The BJS time-series test of the CAPM Black, Jensen and Scholes introduced a time series test of CAPM which is based on time series regression of the portfolio’s excess return on excess market return. [pic] (2) The intercept is known as Jensen’s alpha, which is a coefficient that is proportional to the excess return of a portfolio over its expected return, for its expected risk as measured by beta. Hence, alpha is determined by the fundamental values of the company in contrast to beta, which measures the return due to its volatility. If CAPM holds, by definition the intercept of all portfolios (Jensen’s alpha) are zero. Also note that, if the alpha is negative, then the portfolio underperforms the market. Table One presents the estimated alpha coefficients and the p values. Sample size for the regression is 972 for time period one and 588 for time period two respectively and period two is a sub period of period one. The following conclusions can be drawn from the table one and two available in appendix: 1) In period one, all estimated alphas are non zero ranging from -0. 53(portfolio 1) to 0. 57 (portfolio 5). Six out of twenty-five are negative which suggests that these portfolios cannot reach the expected return of market level. Twelve of twenty-five have p value smaller than 0. 05 which implies in those cases, it rejects the null hypothesis of zero alpha. The results of other thirteen portfolios confirm that the intercept is statistically insignificant upholding the CAPM theory that alpha is zero. 2) In period two, most of alphas are positive yet 5 out of 25 are negative, which indicates that they underperform the market. Besides, it is observed that the p value of alpha in fourteen portfolios is smaller than 0. 05 which implies that it rejects the null hypothesis of zero alpha. Similarly, the results of the intercepts of the remaining 11 portfolios show us that it cannot reject the hypothesis of zero alpha. Also in Table 1 and Table 2, interesting remarks can be derived from the following evidence based on the estimated beta coefficients and their p values: 1) In time period one, majority portfolios have beta larger than 1, which indicates that the return of those portfolios tends to be more volatile than the market level. All betas are statistically significant (p value less than 0. 05). In CAPM, it points out that higher systematic risk (beta) would lead to higher level of return. However, in this study, higher estimated beta portfolios are not associated with higher excess returns. Portfolio one for example, has the highest beta (1. 65) with -0. 53 excess return. In contrast, portfolio 13 has relatively lower beta (1. 17). But it produces a higher and positive excess return (0. 24). 2) In time period two, betas are statistically significant. Portfolio 4 has an estimated beta of the unit value which implies it has the same volatility as the market, other 17 having a superior to one, which therefore has a higher volatility than the market. Similarly, in this case, higher estimated beta is not necessarily correlated to higher excess return (which can be shown more clearly in the following plots). This evidence can be seen in portfolio 1 (with beta 1. 43 and alpha -0. 45) and portfolio 25 (with beta 1. 02 and alpha 0. 56). 3) One may observe that the data in period two is a sub period of period one and the beta is not stable over time for more portfolios have less than unit value beta in period two and some portfolios tend to be more volatile in the whole period (July 1931 to June 2012) but in sub period (July 1963 to June 2012) is less volatile than market level, take portfolio 24 as an example: its beta is 1. 4 in period one and in period two its beta is 0. 83. 2. The OLS cross-sectional test of the CAPM The CAPM states that the securities plot on the Security Market Line (SML) in equilibrium. We do cross-sectional test is to identify whether the above statement is true with our two data set and whether or not it rejects the hypothesis that the slope is zero. In the equation 3, the gamma 0 is the excess return on a zero beta portfolio and gamma 1 (the slope of the regression ) is the market portfolios average risk premium. [pic] (3) We perform the OLS cross-sectional test of equation (3) for both two periods. The results have shown in Table 3that gamma1 in time period 1 is positive (0. 55) and it is statistically significant for its p value is 0. 05, which implies that it rejects the null hypothesis of zero slope of the model. The gamma0 is also positive (0. 26) which suggests that the cross-sectional return of 25 sample portfolios during July 1931 to June 2012 outperformed the market level. In time period 2, gamma 0 is positive (1. 13) and gamma1 is negative (-0. 41) as shown in Table 4. In this case, the p value of gamma 1 is 0. 18, which indicates that it cannot reject the null hypothesis of zero gamma 1. Therefore, the model is interpreted as being false. The CAPM is rejected by the fact that the slope in period two is not statistically significant. We plot the values of Rj-Rf’s against betaj’s in both time periods. The Fig. 1 demonstrates the excess return of 25 portfolios in period one and their beta accordingly. These two curves are both wave-like pattern but not exactly follow the same movement. As discussed in previous section, higher stimated beta is not always necessarily correlated to higher excess return. These graphs are more vivid to illustrate this point. Majority portfolios follows the rule that higher beta is linked to bigger excess return. But special cases may be seen as portfolio one, six etc. Similarly, in Fig. 2 which is for period two, most of the 25 portfolios obey the rule that higher risk more return. But portfolio one seems to be an extreme case as well as it has the highest beta among the 25 portfolios with the lowest excess return (discussed in previous beta section). Also in portfolio six, it generates second highest beta (1. 4) but the excess return (-0. 20, portfolio 6) falls behind portfolio five (0. 61), which has much smaller beta (1. 08). 3 The time-series test of the Fama-French three-factor model [pic] (7) Table 5 presents basic statistics of the FF factors. We perform this test is to identify whether alpha is zero in each portfolio in equation 6. It is observed that these alphas ranges from -0. 87(portfolio 1) to 0. 09(portfolio 14). 12 out of 25 alphas are positive, but none of them exceed the value of 0. 1. Seven alphas are statistically significant with their absolute value of t statistics bigger than 1. 96. As a result, in these seven portfolios, it rejects the null hypothesis of zero alpha. Other eighteen ones cannot reject the zero alpha hypothesis. In time period two, the estimated alphas fall inside the limits between -0. 52(portfolio1) and 0. 18(portfolio 21) which seems to have narrower range comparing to period one in Table 6. Twelve out of twenty-five portfolios have negative alpha. Also in seven portfolios, each alpha is statistically significant and the null hypothesis of zero alpha is rejected. Interestingly, the seven portfolios in period one, which reject zero alpha hypothesis, are not the same as in period two (which is a sub period of period one). This result may show that the performance of these portfolios is not stable for the time being between July 1931 and June 2012. Besides, three zero alpha hypotheses of FF models remain rejected in both two periods (and they are portfolio 1, 6 and 21). 4 The OLS cross-sectional test of the Fama-French three-factor model. [pic] (8) The cross-sectional test of the FF three-model parallels the cross-sectional test of the CAPM. We mainly examine whether the intercept and the slope coefficients are equal to zero. In time period one, the results of gamma 0, gamma 1, gamma 2 and gamma 3 are 1. 98(t statistics 6. 04), -1. 25(t statistics-3. 96), 0. 19(t statistics 4. 01) and 0. 41(t statistics 6. 23) respectively. The null hypothesis of zero intercept can be rejected. And all estimated coefficients can be regarded as statistically significant. In the sub period (period two), the gammas are 1. 17, -0. 69, 0. 19 and 0. 41 respectively. The zero intercept is still rejected and this result is consistent with period one. Apart from gamma 1 with t statistics -1. 64, other coefficients are also statistically significant. It cannot reject zero gamma hypothesis. 5 The Fama-MacBeth(FM) month-by-month test of the CAPM in the July 1931 to June 2012 period [pic] (4b) The CAPM states that securities’ expected excess return and betas plot on the same line that passes through the origin. And it is also known that any unit-weight portfolio also plots on this line. It follows that a zero-weight portfolio’s expected return and beta would plot on a certain line too. In equation 4b, the gamma 0t is interpreted as the excess return on a zero-beta portfolio and the gamma 1t is a zero-weight portfolio return on the market in excess of the zero-beta rate. Since beta jt equals to beta j, the average of the gamma 0(0. 26) and gamma 1(0. 55) in our data set are equal to the results estimated from equation 3 in previous part. The estimated gamma 0 ranges from -102. 20 to 44. 08 and 546 out of 972 are positive. Among these 972 portfolios, 415 gamma 0 are shown statistically significant with their absolute values of t statistics larger than 1. 96, which indicates that the hypothesis of zero intercept of the FM model is rejected. That is to say, almost 43% of monthly return in this time series data set rejects the null hypothesis. 47. 5% of the estimated gamma 1 is positive. The minimum estimated gamma 1 is -35. 46 in the 20th month and it produces some what much bigger estimated value (127. 94) in the 23rd month. There are 499 gamma 1 out of 972 months are statistically significant. In these 499 portfolios, it rejects zero gamma hypothesis. Conclusion The CAPM null hypothesis of zero alpha is not always true in our findings. It is also shown that in real financial world, higher estimated beta portfolios are not always associated with higher excess returns. It happens that lower beta would generate higher return. Note that beta is also not stable through periods of time. Besides, in one of our FF model, the CAPM is rejected because the slope is not statistically significant in one sub period(period two) but different result is drawn in a longer time period(period one).