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 a CRRA (i.e., constant relative risk aversion ) utility function . In our non-concave optimization problem, we find that  for any VaR-constraint with an arbitrary risk level, there exists an ES-constraint leading to the same investment strategy, assuming that the regulation only protects the debt holders' benefit to a certain level . This differs from the conclusion drawn in Basak and Shapiro (2001) for the concave optimization problem, where VaR and ES lead to different solutions  .
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 .
D
a CRRA (i.e.,
R
) utility function
(CRRA) utility functions
A
with a not too strict regulation
R
assuming that the regulation only protects the debt holders' benefit to a certain level
which shows on some level the ineffectiveness of the ES-based regulation
A
and ES provides a better loss protection