Efficient portfolios and extreme risks: a Pareto–Dirichlet approach

This paper solves the mean variance skewness kurtosis (MVSK) portfolio optimization problem by introducing a general Pareto–Dirichlet method. We approximate the feasible portfolio set with a calibrated Dirichlet distribution, where a portfolio is MVSK efficient if its profile in terms of the first four moments is not dominated by any other portfolio. Compared to existing higher order portfolio optimization methods, the Pareto–Dirichlet approach cannot misclassify inefficient portfolios as efficient and produces the efficient set in a very quick way. Coupling the Pareto–Dirichlet approach with a new criterion that generalizes the Sharpe ratio, we are able to produce optimal portfolios in a quick way also. We illustrate our approach with Fama-French 30 Industry Portfolios, where we show that the optimal portfolios derived with our method are preferred to those derived with other optimization schemes by all tested classic performance measures.