Over time academics and practitioners have shown that CAPM does not fully describe the returns on stocks. They showed this by finding large persistent alphas on the returns.

Specifically, if we let Fm = E[Rm] rf , we then regress the excess returns of portfolios, E[Rp] rf on Fm.

In these regressions, we find large persistent alphas. To resolve this many now use factor models. In these models the excess return of a stock (or portfolio) can be written as E[Ri ] rf = i1F1 + i2F2 + ...inFn for an n factor model.

Here Fk is the value of factor k and ik is the loading or beta of stock i on factor k.

We can think of these Fk as measures of different risk.

1. Suppose a two-factor model described reality, where one of the factors was a market factor, Fm. Explain how an investor could earn positive alpha on their stock portfolio if we measured alpha using the CAPM, even if they earned no alpha if we measured returns using the two-factor model.