# Definition and estimation of $\beta$: raw or excess returns?

The CAPM is a single-period model that says $$\mathbb{E}(R^*_i)=\beta\mathbb{E}(R^*_m)$$ where $$R^*_i:=R_i-r_f$$ is an asset's excess return, $$R^*_i:=R_m-r_f$$ is the market's excess return and $$\beta:=\frac{\text{Cov}(R^*_i,R^*_m)}{\text{Var}(R^*_m)}=\frac{\text{Cov}(R_i,R_m)}{\text{Var}(R_m)}$$. The latter equality holds since $$r_f$$ is just a constant, and shifting random variables by a constant does not change their covariance or their variances.

Now, the CAPM and $$\beta$$ are usually estimated from multiple periods of data (think Fama-MacBeth or GMM estimation). There, the risk-free rate $$r_{f,t}$$ is time varying, and so $$\frac{\text{Cov}(R^*_{i,t},R^*_{m,t})}{\text{Var}(R^*_{m,t})}\color{red}{\neq}\frac{\text{Cov}(R_{i,t},R_{m,t})}{\text{Var}(R_{m,t})}$$. Which of the two expression defines $$\beta$$ then?

(For simplicity, let us assume $$\beta$$, $$\mathbb{E}(R^*_{i,t})$$ and $$\mathbb{E}(R^*_{m,t})$$ are constant over time. I think this is a common assumption in the more basic applications; correct me if I am wrong.)

Related threads: "Definitions of Beta" and "Beta using only price returns?".

• A comment by @nbbo2 at the second of the linked threads seems to suggest use of excess returns. I would like a confirmation (or a refutation) of that. Feb 14 at 12:21
• Prom a practical point of view, I always thought the idea of using a constant $r_f$ was very weird. Like, if you're currently estimating $\beta$ over a 5 year window for US stocks, what risk free rate do you take? Could be any value between 0% and 5%. Feb 16 at 8:42
• @oronimbus, I do not think the time window matters that much. You should match the horizon of the risk-free investment with the one of the risky investment. So if you are working with monthly returns on some assets, you should consider one-month risk-free rate. And that certainly varies from month to month. Feb 16 at 8:46

Short answer: for equity asset pricing, working with returns in excess of the risk free rate (i.e. $$r - r_f$$) tends to make more sense.