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On the equivalence between Value-at-Risk- and Expected Shortfall-based risk measures in non-concave optimization

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  • An Chen
  • Mitja Stadje
  • Fangyuan Zhang

Abstract

We study a non-concave optimization problem in which a financial company maximizes the expected utility of the surplus under a risk-based regulatory constraint. For this problem, we consider four different prevalent risk constraints (Expected Shortfall, Expected Discounted Shortfall, Value-at-Risk, and Average Value-at-Risk), and investigate their effects on the optimal solution. Our main contributions are in obtaining an analytical solution under each of the four risk constraints, in the form of the optimal terminal wealth. We show that the four risk constraints lead to the same optimal solution, which differs from previous conclusions obtained from the corresponding concave optimization problem under a risk constraint. Compared with the benchmark (unconstrained) non-concave utility maximization problem, all four risk constraints effectively and equivalently reduce the set of zero terminal wealth, but do not fully eliminate this set, indicating the success and failure of the respective financial regulations.

Suggested Citation

  • An Chen & Mitja Stadje & Fangyuan Zhang, 2020. "On the equivalence between Value-at-Risk- and Expected Shortfall-based risk measures in non-concave optimization," Papers 2002.02229, arXiv.org, revised Jun 2022.
    • Handle: RePEc:arx:papers:2002.02229
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    References listed on IDEAS

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    Cited by:

    1. Marcos Escobar-Anel & Michel Kschonnek & Rudi Zagst, 2022. "Portfolio optimization: not necessarily concave utility and constraints on wealth and allocation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(1), pages 101-140, February.

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