The portfolio optimization process determines the set of portfolios that achieve the highest expected returns at given risk levels. In this context, risk is usually measured by the standard deviation of the portfolio’s returns. But many problems associated with return dispersion can be expressed in terms of a shortfall constraint—the minimum return that must be exceeded with a given probability. This “confidence limit” approach may provide a more meaningful description of risk for many investment situations.

Consider investors who are judged relative to a benchmark portfolio, such as an index or basket of indexes. The return and risk characteristics of potential investment portfolios and the benchmark portfolio can be used to determine a distribution of deviations from the benchmark. The portfolio optimization process can then be limited to portfolios that have a given probability (e.g., 95 per cent) of not underperforming the benchmark by more than a specified amount (e.g., 5 per cent). Constrained optimization may be especially helpful for investors who are subject to multiple, and potentially conflicting, objectives.