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.