Annals of Operations Research

, Volume 45, Issue 1, pp 165–177 | Cite as

Asymmetric risk measures and tracking models for portfolio optimization under uncertainty

  • Alan J. King


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. We extend the Levy-Markowitz argument to account for asymmetric risk by basing the local approximation onpiecewise linear-quadratic risk measures, which can be tuned to express a wide range of preferences and adjusted to reject outliers in the data. The implications of this argument lead us to reject 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.


Portfolio optimization asymmetric risk lower partial moments tracking models quadratic programming stochastic programming 


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Copyright information

© J.C. Baltzer AG, Science Publishers 1993

Authors and Affiliations

  • Alan J. King
    • 1
  1. 1.IBM Thomas J. Watson Research CenterYorktown HeightsUSA

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