1
$\begingroup$

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?".

$\endgroup$
3
  • $\begingroup$ 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. $\endgroup$ Feb 14, 2023 at 12:21
  • $\begingroup$ 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%. $\endgroup$
    – oronimbus
    Feb 16, 2023 at 8:42
  • $\begingroup$ @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. $\endgroup$ Feb 16, 2023 at 8:46

1 Answer 1

1
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Thank you! That is helpful! $\endgroup$ Feb 15, 2023 at 14:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.