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

This paper studies an optimal asset allocation problem for a surplus-driven financial institution facing a Value-at-Risk (VaR) or an Expected Shortfall (ES) constraint corresponding to a non-concave optimization problem under constraints. We obtain the closed-form optimal wealth with the ES constraint as well as with the VaR constraint respectively, and explicitly calculate the optimal trading strategy for constant relative risk aversion (CRRA) utility functions. We find that both VaR and ES-based regulation can effectively reduce the probability of default for a surplus-driven financial institution. However, the liability holders' benefits cannot be fully protected under either VaR- or ES-based regulation. In addition, we show that the VaR and ES-based regulation can induce the same optimal portfolio choice for a surplus-driven financial institution. This differs from the conclusion drawn in Basak and Shapiro 2001 where the financial institution aims at maximizing the expected utility of the total assets, and ES provides better loss protection.