A Strategy for Including Odd and Even-Numbered Higher Moments in Portfolio Selection

Previous theoretical work by the authors has developed a framework for optimizing portfolio decisions when moments higher than the variance are considered. Apart a significant increase in computational complexity, inclusion of higher order moments implies a careful judgement on which cross-moments to choose as non-zero. The reason for this lies not only in that the number of cross-moments grows exponentially with the order (of the moment), making results more difficult to obtain and interpret, as well as in the fact that solutions can vary widely, depending on the zeros assigned to a given higher-order moments tensor. On the other hand, empirical evidence produced up to now shows that mean-variance solutions are usually not robust, so inclusion of some higher-order moments is a must. We try to outline criteria for setting up an optimal portfolio selection programme that tries to reconcile parsimony and simplicity of interpretation with the robustness acquired with the use of more moments. The criteria combine statistical (sample dependent) and theoretical considerations, to devise nearly-optimal rules to be applied in a concrete case. The rules can be easily translated to a dynamic setting. If the user accepts to include risk behaviour assumptions, the rules can be sharpened and the procedure becomes simpler. Though the framework is general, the paper focuses on the 3rd and 4th moment cases.