Optimal management of DC pension fund under the relative performance ratio and VaR constraint

This paper investigates the optimal management of defined contribution pension plan under the Omega ratio and Value-at-Risk (VaR) constraint. Interest and inflation risks are considered, and the financial market consists of cash, a zero-coupon bond, an inflation-indexed zero-coupon bond, and a stock. The goal of the pension manager is to maximize the performance ratio of the real terminal wealth under the VaR constraint. An auxiliary process is introduced to transform the original problem into a self-financing problem. We obtain the optimal terminal wealth under different cases by combining the linearization method, the Lagrange dual method, the martingale method, and the concavification method. There are fourteen cases for the convex penalty function, and there are six cases for the concave penalty function. Besides, when the penalty and reward functions are both power functions, the explicit forms of the optimal investment strategies are obtained. Numerical examples are shown to illustrate the impacts of the performance ratio and VaR constraint.