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
Date Written: February 7, 2020
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: Suggested Citation