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Application of a general risk management model to portfolio optimization problems with elliptical distributed returns for risk neutral and risk averse decision makers

Author

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  • Kaynar, B.
  • Birbil, S.I.
  • Frenk, J.B.G.

Abstract

We discuss a class of risk measures for portfolio optimization with linear loss functions, where the random returns of financial instruments have a multivariate elliptical distribution. Under this setting we pay special attention to two risk measures, Value-at-Risk and Conditional-Value-at-Risk and differentiate between risk neutral and risk averse decision makers. When the so-called disutility function is taken as the identity function, the optimization problem is solved for a risk neutral investor. In this case, the optimal solutions of the two portfolio problems using the Value-at-Risk and Conditional-Value-at-Risk measures are the same as the solution of the classical Markowitz model. We adapt an existing less known finite algorithm to solve the Markowitz model. Its application within finance seems to be new and outperforms the standard quadratic programming procedure quadprog within MATLAB. When the disutility function is taken as a convex increasing function, the problem at hand is associated with a risk averse investor. If the Value-at-Risk is the choice of measure we show that the optimal solution does not differ from the risk neutral case. However, when Conditional-Value-at-Risk is preferred for the risk averse decision maker, the corresponding portfolio problem has a different optimal solution. In this case the used objective function can be easily approximated by Monte Carlo simulation. For the actual solution of the Markowitz model, we modify and implement the less known finite step algorithm and explain its core idea. After that we present numerical results to illustrate the effects of two disutility functions as well as to examine the convergence behavior of the Monte Carlo estimation approach.

Suggested Citation

  • Kaynar, B. & Birbil, S.I. & Frenk, J.B.G., 2007. "Application of a general risk management model to portfolio optimization problems with elliptical distributed returns for risk neutral and risk averse decision makers," Econometric Institute Research Papers EI 2007-12, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    • Handle: RePEc:ems:eureir:9412
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    References listed on IDEAS

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    1. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
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    Cited by:

    1. Krajina, A., 2009. "A Method of Moments Estimator of Tail Dependence in Elliptical Copula Models," Discussion Paper 2009-42, Tilburg University, Center for Economic Research.
    2. Jamie Fairbrother & Amanda Turner & Stein W. Wallace, 2018. "Scenario Generation for Single-Period Portfolio Selection Problems with Tail Risk Measures: Coping with High Dimensions and Integer Variables," INFORMS Journal on Computing, INFORMS, vol. 30(3), pages 472-491, August.
    3. Krajina, A., 2010. "An M-estimator of multivariate tail dependence," Other publications TiSEM 66518e07-db9a-4446-81be-c, Tilburg University, School of Economics and Management.

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