Alpha

Conceptually, alpha is the riskless return to an asset.

The alpha of asset zz to portfolio pp is:

αz,p=E[rz]−rf⏟excess return−βz,p(E[rp]−rf)⏟risky return\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

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

Alpha is the solution to the following regression:

r=rf+β(rm−rf)+α  ⟹  α=r−rf−β(rm−rf)\begin{align} r &= r_f + \beta(r_m - r_f) + \alpha \\ \implies \alpha &= r - r_f - \beta(r_m - r_f)\end{align}

Where rmr_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.