Regression Beta and CAPM Expected-Return Beta

Summary

The document compares two ways of describing beta: as a relationship between an asset’s and market’s returns, and as the ratio of expected excess returns implied by CAPM. It connects them through a return model in which the asset return depends on the risk-free rate, market excess return, beta, and a residual. Taking expectations in that model yields the CAPM expression when the residual has zero expectation.

For estimation, the answer describes using historical time-series observations and an ordinary least squares regression. In standard form, the estimated slope equals the sample correlation multiplied by the ratio of the asset’s return standard deviation to the market’s return standard deviation. The document’s displayed estimator instead uses a ratio of variances, which does not match that standard identity. The distinction between an estimated historical coefficient and an expected-return relation is useful, but the assumptions and estimation uncertainty are not explored.

Key ideas

• CAPM relates expected asset excess return to beta times expected market excess return.
• A time-series regression estimates beta from observed asset and market returns.
• The standard regression slope can be expressed using correlation and standard deviations.
• The document’s displayed variance-ratio estimator appears inconsistent with the standard regression identity.

Tags

# Two definitions of Beta


# 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`?

## Answer by Stefan Voigt (score 3)

https://quant.stackexchange.com/a/32276

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 $$\hat{\beta}=\hat{\rho}\frac{\hat{\sigma}^2 _a}{\hat{\sigma}^2 _m}.$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.