Background

CAPM model defines the risk-returns relation as

$E(R_i)=R_f+\beta_i[E(R_m)-R_f]$

Two approaches to test CAPM

• Cross-sectional regression

  If CAPM model is correct, we would expect that market betas completely explain the cross sectional difference in expected returns. So the expected stock returns should be linear to the market betas.The cross-sectional test can be conducted in two steps:

  1. Estimate the systematic betas

     $R_{it}=\alpha_i+\beta_i R_{mt}+\epsilon_{it}$

  2. Stock returns are regressed on the estimates of the market beta $\hat\beta_i$

     $R_{i}=\lambda_0+\lambda_i \hat\beta_i+v_{i}$

One approach is to first calculate the mean return for each stock over the sample period and then regress the mean rturns on the market betas esitmated over the sample period. This approach is problematic, however, since stock returns are offen corss-sectionally correlated and heteroscedastic.

why? explain more

The other approach is Fama-Macbeth regression. They first estimate the cross-sectional regression for each month in the sample period, and compute the same mean of the etimated slope coefficients.

• Mean-variance efficiency