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Portfolio optimization with two coherent risk measures

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  • Tahsin Deniz Akturk
  • c{C}au{g}{i}n Ararat

Abstract

We provide analytical results for a static portfolio optimization problem with two coherent risk measures. The use of two risk measures is motivated by joint decision-making for portfolio selection where the risk perception of the portfolio manager is of primary concern, hence, it appears in the objective function, and the risk perception of an external authority needs to be taken into account as well, which appears in the form of a risk constraint. The problem covers the risk minimization problem with an expected return constraint and the expected return maximization problem with a risk constraint, as special cases. For the general case of an arbitrary joint distribution for the asset returns, under certain conditions, we characterize the optimal portfolio as the optimal Lagrange multiplier associated to an equality-constrained dual problem. Then, we consider the special case of Gaussian returns for which it is possible to identify all cases where an optimal solution exists and to give an explicit formula for the optimal portfolio whenever it exists.

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  • Tahsin Deniz Akturk & c{C}au{g}{i}n Ararat, 2019. "Portfolio optimization with two coherent risk measures," Papers 1903.10454, arXiv.org, revised Jul 2020.
    • Handle: RePEc:arx:papers:1903.10454
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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.
    2. Zinoviy Landsman & Udi Makov, 2016. "Minimization of a Function of a Quadratic Functional with Application to Optimal Portfolio Selection," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 308-322, July.
    3. Merton, Robert C., 1972. "An Analytic Derivation of the Efficient Portfolio Frontier," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 7(4), pages 1851-1872, September.
    4. Rockafellar, R. Tyrrell & Uryasev, Stanislav, 2002. "Conditional value-at-risk for general loss distributions," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1443-1471, July.
    5. Levy, Haim, 1978. "Equilibrium in an Imperfect Market: A Constraint on the Number of Securities in the Portfolio," American Economic Review, American Economic Association, vol. 68(4), pages 643-658, September.
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    Cited by:

    1. c{C}au{g}{i}n Ararat, 2020. "Portfolio optimization with two quasiconvex risk measures," Papers 2012.06173, arXiv.org.

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