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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
Article

Abstract

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.

Keywords

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

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References

  1. [1]
    V.S. Bawa, Optimal rules for ordering uncertain prospects, J. Fin. Econ. 10 (1975) 849–857.Google Scholar
  2. [2]
    R.S. Dembo, Scenario optimization, Ann. Oper. Res. 30 (1991) 63–80.Google Scholar
  3. [3]
    R.S. Dembo and A.J. King, Tracking models and the optimal regret distribution in asset allocation, IBM Research Report RC 17156 (1991).Google Scholar
  4. [4]
    P.C. Fishburn, Mean-risk analysis with risk associated with below-target returns, Amer. Econ. Rev. 67 (1977) 116–125.Google Scholar
  5. [5]
    W.V. Harlow and R.K.S. Rao, Asset pricing in a generalized mean-lower partial moment framework: Theory and evidence, J. Fin. Quantit. Anal. 24 (1989) 285–311.Google Scholar
  6. [6]
    W.V. Harlow, Asset allocation in a downside risk framework, Equity Portfolio Analysis Report, Salomon Brothers Inc. (1991).Google Scholar
  7. [7]
    P.J. Huber,Robust Statistics (Wiley, New York, 1981).Google Scholar
  8. [8]
    International Business Machines Corporation,Optimization Subroutine Library Guide and Reference (IBM Corp, Document SC23-0519-1, 1990).Google Scholar
  9. [9]
    D.L. Jensen and A.J. King, Frontier: a graphical interface for portfolio optimization in a piecewise linear-quadratic risk framework, IBM Syst. J. 31 (1992) 62–70.Google Scholar
  10. [10]
    A.J. King and D.L. Jensen, Linear-quadratic efficient frontiers for portfolio optimization, Appl. Stoch. Models Data Anal. 8 (1992) 195–207.Google Scholar
  11. [11]
    H. Konno, Piecewise linear risk function and portfolio optimization, J. Oper. Res. Soc. Japan 33 (1990) 139–156.Google Scholar
  12. [12]
    H. Levy and H.M. Markowitz, Approximating expected utility by a function of mean and variance, Amer. Econ. Rev. (June 1979) 308–317.Google Scholar
  13. [13]
    H.M. Markowitz,Portfolio Selection: Efficient Diversification of Investments (Wiley, New York, 1959).Google Scholar
  14. [14]
    T.J. Nantell, K. Price and B. Price, Mean-lower partial moment asset pricing model: Some empirical evidence, J. Fin. Quantit. Anal. 17 (1982) 763–782.Google Scholar
  15. [15]
    J.W. Pratt, Risk aversion in the small and in the large, Econometrica 32 (1964) 122–137.Google Scholar

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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