Alpha | Logan Nielsen

Conceptually, alpha is the riskless return to an asset.

The alpha of asset $z$ to portfolio $p$ is:

\begin{align} \alpha_{z,p} = \underbrace{E[r_z] - r_f}_{\text{excess return}} - \underbrace{\beta_{z,p}(E[r_p] - r_f)}_{\text{risky return}} \end{align}

Where

\begin{align} \beta_{z,p}= \cfrac{Cov(r_p, r_z)}{Var(r_p)} \end{align}

Alpha is the solution to the following regression:

\begin{align} r &= r_f + \beta(r_m - r_f) + \alpha \\ \implies \alpha &= r - r_f - \beta(r_m - r_f)\end{align}

Where $r_m$ is the benchmark. Thus, alpha is the return to the asset that cannot be explained by exposure to benchmark returns.

If markets are efficient, then all assets relative to the market portfolio have an alpha of zero.