Two definitions of Beta

Question

I have seen two definitions of Beta one is $$\beta = \rho\dfrac{\sigma_{asset}}{\sigma_{market}}$$ Here $\rho$ is the correlated coeffient

another one is $$\beta = \dfrac{r_{expect} - r_{risk\ free}}{r_{market} - r_{risk\ free}}$$ I don't know which one is correct or they are equivalent? By the way, here $\sigma_{asset}$ is the volatility of historical return or expected return?

Answer 1

As @Rosetta states in the comment above, I think the difference between the two formulas you represent can be explained by either focusing on estimating the coefficient $\beta$ or by taking into consideration expectations based on CAPM.

Consider the well-known framework

$$r_a = r_f +\beta (r_m-rf)+\varepsilon$$

If you take expectations you obtain $$E(r_a-r_f) = \beta E(r_m-rf)+E(\varepsilon) \\ \text{ or } \frac{E(r_a)-r_f}{E(r_m)-r_f} = \beta $$

If you want to draw some inference and estimate $\hat{\beta}$, then you are probably going to take some time-series of past observations of the returns $r_a, r_f, r_m$ and compute the OLS estimate $\hat{\beta}$ which boils down to $$\hat{\beta}=\hat{\rho}\frac{\hat{\sigma}^2 _a}{\hat{\sigma}^2 _m}.$$