Portfolio optimization with quantile-based risk measures

Abstract

In this thesis we analyze Portfolio Optimization risk-reward theory, a generalization of the mean-variance theory, in the cases where the risk measures are quantile-based (such as the Value at Risk (V aR) and the shortfall). We show, using multicriteria theory arguments, that if the measure of risk is convex and the measure of reward concave with respect to the allocation vector, then the expected utility function is only a special case of the risk-reward framework. We introduce the concept of pseudo-coherency of risk measures, and analyze the mathematics of the Static Portfolio Optimization when the risk and reward measures of a portfolio satisfy the concepts of homogeneity and pseudo-coherency. We also implement and analyze a sub-optimal dynamic strategy using the concept of consistency which we introduce here, and achieve a better mean-V aR than with a traditional static strategy. We derive a formula to calculate the gradient of quantiles of linear combinations of random variables with respect to an allocation vector, and we propose the use of a nonparametric statistical technique (local polynomial regression - LPR) for the estimation of the gradient. This gradient has interesting financial applications where quantile-based risk measures like the V aR and the shortfall are used: it can be used to calculate a portfolio sensitivity or to numerically optimize a portfolio. In this analysis we compare our results with those produced by current methods. Using our newly developed numerical techniques, we create a series of examples showing the properties of efficient portfolios for pseudo-coherent risk measures. Based on these examples, we point out the danger for an investor of selecting the wrong risk measure and we show the weaknesses of the Expected Utility Theory.