# Excess, Residual and Active Return

in CAPM. What's the difference between these different types of returns?

• Active return
• Excess return
• Residual return

Active return: $R-R_m$ i.e. your security (or portfolio) compared to the market portfolio. Used to judge performance before the CAPM was invented

Excess return: $R-R_f$ the security compared to the risk free rate, appears on the left hand side of the CAPM equation.

Excess return on the market: $R_m-R_f$, appears on the right hand side

In words the CAPM says that "there is a linear relationship between the excess return and the excess return on the market" $R-R_f=\beta(R_m-R_f)+\epsilon_i$

Residual return: $R-R_f-\beta(R_m-R_f)$ i.e. the part of your return which the CAPM does not explain, or your outperformance versus the CAPM; its estimated value over a period of time is called Alpha. This is what is advocated to measure performance under the CAPM.

• Thanks, reading Grinold & Kahn I find the following claim repeated throughout the text: "The CAPM states that the expected residual return on all stocks and any portfolio is equal to 0" i.e. that $E\left[\theta_p\right] = 0$ (also, I presume $\epsilon_i = \theta_p$ in your notation above). But you mentioned that the residual return is what we call Alpha (i.e. "your outperformance vs CAPM" ). How do I reconcile these statements? Does CAPM assume that you can't outperfom the market? Something else?
– Josh
May 24, 2016 at 0:06
• Yes, the CAPM says in the long run there is no Alpha $E[\epsilon_i]=0$, but people still try to accomplish positive alpha, and some do get some in the short run, but then the rejoinder from the efficient market folks is that these are just a small minority who got lucky... May 24, 2016 at 12:19