Portfolio optimization when asset returns have the Gaussian mixture distribution

Portfolios of assets whose returns have the Gaussian mixture distribution are optimized in the static setting to find portfolio weights and efficient frontiers using the probability of outperforming a target return and Hodges’ modified Sharpe ratio objective functions. The sensitivities of optimal portfolio weights to the probability of the market being in the distressed regime are shown to give valuable diagnostic information. A two-stage optimization procedure is presented in which the high-dimensional non-linear optimization problem can be decomposed into a related quadratic programming problem, coupled to a lower-dimensional non-linear problem.