Alpha and Beta in CAPM regression

What is another interpretation of $\alpha$ and $\beta$ ?

Answer

Another interpretation of $\beta$ is that it is basically the slope coefficient of the market exposure when we regress the portfolio excess return on the excess market return (the CAPM regression).

Since this is basically a simple linear model,

R_p - R_f = \alpha + \beta (R_m - R_f) + \varepsilon,

we know that the slope coefficient is given by

\frac{\mathrm{Cov}(X,Y)}{\mathrm{Var}(X)}.

In our case, this implies that

\beta=\frac{\mathrm{Cov}(R_p,\;R_m)}{\mathrm{Var}(R_m)}.

which is the known formula for beta.

Also note that $\alpha$ is the intercept in this simple linear regression.