Appendix 5.2: Risk-Adjusted Performance

The basic CAPM result states that the excess return on any asset (portfolio) is proportional to the excess return on the benchmark portfolio where the factor of proportionality is the asset's beta:

The Sharpe ratio is the excess return per unit risk. Thus, we rewrite the CAPM as follows:

Noting that and making this substitution, we have:

Collecting terms and assuming ,

Equivalently,

And finally,

Thus, the return on a weighted average of the risky asset and the risk-free asset mimics the return to the benchmark portfolio. The weight scales the asset's return to reflect the risk difference between the benchmark and the asset. Thus, if benchmark risk is higher, then the asset's return is adjusted upward to reflect this difference. This is the basic result of the Modigliani risk-adjusted performance result (Modigliani 1997). What is interesting is that the risk on this portfolio is the same as the benchmark risk. This is particularly easy to estimate since the risk-free asset has zero risk. Therefore: