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

We study a non-concave optimization problem in which an insurance company maximizes the expected utility of the surplus under a risk-based regulatory constraint. The non-concavity does not stem from the utility function, but from non-linear functions related to the terminal wealth characterizing the surplus. 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 utility maximization problem, all the 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.¹