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On Extending the LP Computable Risk Measures to Account Downside Risk

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Abstract

A mathematical model of portfolio optimization is usually quantified with mean-risk models offering a lucid form of two criteria with possible trade-off analysis. In the classical Markowitz model the risk is measured by a variance, thus resulting in a quadratic programming model. Following Sharpe’s work on linear approximation to the mean-variance model, many attempts have been made to linearize the portfolio optimization problem. There were introduced several alternative risk measures which are computationally attractive as (for discrete random variables) they result in solving linear programming (LP) problems. Typical LP computable risk measures, like the mean absolute deviation (MAD) or the Gini’s mean absolute difference (GMD) are symmetric with respect to the below-mean and over-mean performances. The paper shows how the measures can be further combined to extend their modeling capabilities with respect to enhancement of the below-mean downside risk aversion. The relations of the below-mean downside stochastic dominance are formally introduced and the corresponding techniques to enhance risk measures are derived.

The resulting mean-risk models generate efficient solutions with respect to second degree stochastic dominance, while at the same time preserving simplicity and LP computability of the original models. The models are tested on real-life historical data.

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Authors and Affiliations

  1. Institute of Control and Computation Engineering, Warsaw University of Technology, Nowowiejska 15/19, 00-665, Warsaw, Poland

    Adam Krzemienowski & Włodzimierz Ogryczak

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  1. Adam Krzemienowski
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  2. Włodzimierz Ogryczak
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Additional information

The research was supported by the grant PBZ-KBN-016/P03/99 from The State Committee for Scientific Research.

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Krzemienowski, A., Ogryczak, W. On Extending the LP Computable Risk Measures to Account Downside Risk. Comput Optim Applic 32, 133–160 (2005). https://doi.org/10.1007/s10589-005-2057-4

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  • DOI: https://doi.org/10.1007/s10589-005-2057-4

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