On the Equivalence Between Value-at-Risk and Expected Shortfall in Non-Concave Optimization

39 Pages Posted: 4 Mar 2020 Last revised: 25 Mar 2020

See all articles by An Chen

An Chen

University of Ulm

Mitja Stadje

University of Ulm - Department of Mathematics and Economics

Fangyuan Zhang

University of Ulm - Department of Mathematics and Economics

Date Written: February 7, 2020

Abstract

This paper studies a non-concave optimization problem under a Value-at-Risk (VaR) or an Expected Shortfall (ES) constraint. The non-concavity of the problem stems from the non-linear payoff structure of the optimizing investor. We obtain the closed-form optimal wealth with an ES constraint as well as with a VaR constraint respectively, and explicitly calculate the optimal trading strategy for constant relative risk aversion (CRRA) utility functions. In our non-concave optimization problem, we find that with a not too strict regulation for any VaR-constraint with an arbitrary risk level, there exists an ES-constraint leading to the same investment strategy, which shows on some level the ineffectiveness of the ES-based regulation. This differs from the conclusion drawn in Basak and Shapiro (2001) for the concave optimization problem, where VaR and ES lead to different solutions and ES provides a better loss protection.

Keywords: Value-at-Risk, Expected Shortfall, Optimal investment strategy, Non-concave utility maximization

JEL Classification: C61, G11

Suggested Citation

Chen, An and Stadje, Mitja and Zhang, Fangyuan, On the Equivalence Between Value-at-Risk and Expected Shortfall in Non-Concave Optimization (February 7, 2020). Available at SSRN: https://ssrn.com/abstract=3533948 or http://dx.doi.org/10.2139/ssrn.3533948

An Chen

University of Ulm ( email )

Helmholtzstrasse 20
Ulm, D-89081
Germany

HOME PAGE: http://www.uni-ulm.de/mawi/ivw/team

Mitja Stadje

University of Ulm - Department of Mathematics and Economics ( email )

Helmholzstrasse
Ulm, D-89081
Germany

Fangyuan Zhang (Contact Author)

University of Ulm - Department of Mathematics and Economics ( email )

Helmholzstrasse
Ulm, D-89081
Germany

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