# Interpretation of Excess Return

How is excess return defined for a given asset?

There are altogether two different definitions for excess return used in the calculation of alpha and beta and I'm unable to understand which one should we apply to generate valid statistical factors.

Definition 1: Given by the difference between return rate of the stock and risk-asset (such as Treasury Bill) or $ER^{(1)} = R_i - R_f$

Definition 2: Given by the difference between return rate of the stock and return rate of the market index with same risk or $ER^{(2)} = R_i - R_M$

They are both excess returns even though the standard convention is to talk about risk premium for $R_{t+1}-R^f$ and excess return on the market for $R_{t+1}-R_M$. If you believe in CAPM, then you need the former to compute: $$\alpha= (\bar{R}-R_f) - \beta(\bar{R}_M-R_f)$$