Abstract
This paper deals with portfolio optimization under different risk constraints. We use a set of hedge funds where departures from normality are significant. We optimize the expected return under standard deviation, semivariance, value-at-risk (VAR) and expected shortfall (or CVAR) constraints. As far as the VAR is concerned, we compare different estimators. While the optimization with respect to VAR constraints appears to be difficult and lengthy, there are very fast optimization algorithms for the other risk constraints. We find that the choice of a particular VAR estimator is less discriminant than the choice of the risk constraint itself. We provide financial interpretations of the optimal portfolios associated with a decomposition of risk measures.