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

Question

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?".}

Answer 1

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.

• By subtracting off the risk free rate, practically what you're doing is setting the dynamics of the risk free rate aside as a separate problem.
  • e.g. it makes returns from high inflation periods like the 1970s in some sense more comparable to numbers from low inflation periods like 1990s.
• A number of financial economic theories give you formulas with excess returns. It tends to come out in theory more.

It's not something though where financial economic theory is so settled that there's a definitively right or wrong answer. You can find papers where people did all kinds of different things: beta estimated off of regular returns or excess returns? beta estimated off of log returns? beta estimated of a two-year rolling window (makes more sense for individual companies)? beta estimated off of the full sample (makes more sense for portfolios constructed with some consistent criteria)? etc...

Note: the CAPM doesn't work, but there are still a number of reasons why one may want to estimate market betas.