# Two definitions of Beta

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?

• i'm not a pro in finance but beta has no "absolute" formula. Its job is to measure volatility and one of the problem of beta is difficulty to calculate it. So i'd say both formula are correct; which one to use depends on what kind of data you have. – Rosetta Feb 7 '17 at 9:06
• when you use 'historical' data, you are doing a review of performance of the past. when you use 'expected' information, u try to make sense of the future. Again, which one to use depends on what you are evaluating. – Rosetta Feb 7 '17 at 9:11

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.
$$r_a = r_f +\beta (r_m-rf)+\varepsilon$$
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 $$\hat{\beta}=\hat{\rho}\frac{\hat{\sigma}^2 _a}{\hat{\sigma}^2 _m}.$$