Question

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 = βi1F1 + βi2F2 + ...βinFn for an n factor model.

Here Fk is the value of factor k and βik 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.

Answer 1

In a 2 factor model as below;

E[Ri ] − rf = βi1F1 + βi2F2 ; if we fix one of the factors as market factor as shown;

E[Ri ] − rf = beta*E[M] + βi2F2 ; where beta*E[M] is the market factor.

A positive alpha is shown in capm formula because mainly of its inefficieny. ie the model is not incorporating all the factors affecting a stock returns and is only focusing on its relation with market. Suppose if we include some factors that in the new model that can affect the return of the asset, (for example any macro economic indicator, p/E ratio, p/BV etc) we can create a better model;

Using this model the predicted returns of a stock will be close to the actual value compared to capm. As alpha is defined as the difference between actual return and predicted return, the new model can show little or zero alpha. Whereas the CAPM model will show a positive alpha since it have incorporated only the returns based on 1 fator.