Portfolio Optimization within Mixture of Distributions

The recent financial crisis has highlighted the necessity to introduce mixtures of probability distributions in order to improve the estimation of asset returns and in particular to better take account of risks. Since Pearson (1894), these mixtures have been intensively used in many scientific fields since they provide very convenient mathematical tools to examine various statistical data and to approximate many probability distributions. They are typically introduced to model the choice of probability distributions among a given parametric family. The coefficients of the mixture usually correspond to the relative frequencies of each possible parameter. In this framework, we examine the single-period portfolio choice model, which has been addressed in the partial equilibrium framework, by Brennan and Solanki (1981), Leland (1980) and Prigent (2006). We consider an investor who wants to maximize the expected utility of the value of his portfolio consisting of one risk-free asset and one risky asset. We provide and analyze the solution for log return with mixture distributions, in particular for the mixture Gaussian case. The optimal portfolio is characterized for arbitrary utility functions. Our results show that mixture of distributions can have significant implications on the portfolio management.