Asymmetric risk measures and tracking models for portfolio optimization under uncertainty

Traditional asset allocation of the Markowitz type defines risk to be the variance of the return, contradicting the common-sense intuition that higher returns should be preferred to lower. An argument of Levy and Markowitz justifies the mean/variance selection criteria by deriving it from a local quadratic approximation to utility functions. The paper extends the Levy-Markowitz argument to account for asymmetric risk by basing the local approximation on piecewise linear-quadratic risk measures, which can be turned to express a wide range of preferences and adjusted to reject outliers in the data. The implications of this argument led to the rejection of the commonly proposed asymmetric alternatives, the mean/lower partial moment efficient frontiers, in favor of the ‘risk tolerance’ frontier. An alternative model that allows for asymmetry is the tracking model, where a portfolio is sought to reproduce a (possibly) asymmetric distribution at lowest cost.