Confusion regarding hedge calculation

this might not be the most advanced question, but hopefully this is the right community.

Suppose that we have an asset with return $$R_a$$, a respective benchmark with return $$R_b$$ and the risk-free return $$R_{rf}$$. I'm looking at an example that hedges our asset with respect to the benchmark (today) in the following way:

Suppose the current time is $$0$$, and that we want to calculate the hedge with lag $$l$$ (i.e. that we use information from the time steps $$-l,-l+1,\dots,-1$$ for our calculation). We then calculate the beta-coefficient as $$\beta = \frac{corr(R_a,R_b)\sigma_{R_a}}{\sigma_{R_b}}$$ (which is simply the regression coefficient from $$R_a$$ as depending on $$R_b$$, correct?). The hedged return is then calculated as $$H = (R_a - R_{rf}) - \beta(R_b - R_{rf})$$.

I don't really understand the last step. Why do we remove the risk-free return here, but not when we calculate $$\beta$$)? To me it feels like we calculate the regression coefficient for one scenario, and then use it in another. What is it that we arrive at in the end, more explicitely?

Any help is appreciated!

• With $R_f$ so close to zero nowadays, perhaps we should not worry about this unduly. But conceptually you are correct, it would be more logical to use $\rho(R_a-R_f,R_b-R_f)$ in the first step if you are going to use excess returns $R_.-R_f$ in the second step – noob2 Mar 25 at 17:49