Global optimization of higher order moments in portfolio selection

We discuss the global optimization of the higher order moments of a portfolio of financial assets. The proposed model is an extension of the celebrated mean variance model of Markowitz. Asset returns typically exhibit excess kurtosis and are often skewed. Moreover investors would prefer positive skewness and try to reduce kurtosis of their portfolio returns. Therefore the mean variance model (assuming either normally distributed returns or quadratic utility functions) might be too simplifying. The inclusion of higher order moments has therefore been proposed as a possible augmentation of the classical model in order to make it more widely applicable. The resulting problem is non–convex, large scale, and highly relevant in financial optimization. We discuss the solution of the model using two stochastic algorithms. The first algorithm is Differential Evolution (DE). DE is a population based metaheuristic originally designed for continuous optimization problems. New solutions are generated by combining up to four existing solutions plus noise, and acceptance is based on evolutionary principles. The second algorithm is based on the asymptotic behavior of a suitably defined Stochastic Differential Equation (SDE). The SDE consists of three terms. The first term tries to reduce the value of the objective function, the second enforces feasibility of the iterates, while the third adds noise in order to enable the trajectory to climb hills.