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...