In this paper, we investigated an optimal investment problem of a defined contribution (DC) pension plan under a joint Value-at-Risk (VaR) and an expected shortfall (ES) constraint. By using a martingale method, we transformed a dynamic optimization problem to a static pointwise optimization problem and derived the closed-form representations of the optimal wealth and portfolio processes in terms of the state price density. Numerical results showed that in comparison to only an ES constraint or a VaR constraint, the joint VaR-ES constraint can not only improve risk management for the bad economic states but also lower the volatility of the optimal terminal wealth.

We consider the portfolio selection problem of maximizing a performance measure of the terminal wealth faced by a manager with a stochastic benchmark. We transform the non-linear fractional optimization problem into a non-fractional optimization problem based on the fractional programming method. When the penalty and reward functions are both power functions, the stochastic benchmark we consider allows us to derive the explicit form of the optimal investment strategy by combining the linearization method, the martingale method, the change of measure, and the concavification method. Theoretical and numerical results show that the optimal terminal relative performance ends up with zero from a certain value of the price density, which reflects the moral hazard problem.