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We study a non-concave optimization problem in which a financial companymaximizes the expected utility of the surplus under a risk-based regulatoryconstraint. For this problem, we consider four different prevalent riskconstraints (Expected Shortfall, Expected Discounted Shortfall, Value-at-Risk,and Average Value-at-Risk), and investigate their effects on the optimalsolution. Our main contributions are in obtaining an analytical solution undereach 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, whichdiffers from previous conclusions obtained from the corresponding concaveoptimization problem under a risk constraint. Compared with the benchmark(unconstrained) non-concave utility maximization problem, all four riskconstraints effectively and equivalently reduce the set of zero terminalwealth, but do not fully eliminate this set, indicating the success and failureof the respective financial regulations.

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