Optimal investment strategy for the DC plan with the return of premiums clauses in a mean–variance framework

In this paper, we study the optimal investment strategy in the DC pension plan during the accumulation phase. During the accumulation phase, a pension member contributes a predetermined amount of money as premiums and the management of the pension plan invests the premiums in equities and bonds to increase the value of the accumulation. In practice, most of the DC pension plans have return of premium clauses to protect the rights of the plan members who die during the accumulation phase. In the model, the members withdraw their premiums when they die and the difference between the premium and the accumulation (negative or positive) is distributed among the survival members. From the surviving members’ point of view, when they retire, they want to maximize the fund size and to minimize the volatility of the accumulation. We formalize the problem as a continuous-time mean–variance stochastic optimal control problem. The management of the pension plan chooses the optimal investment strategy, i.e., the proportions invested in equities and bonds, to maximize the mean–variance utility of the pension member at the time of retirement. Using the variational inequalities methods in Björk and Murgoci (2009), we transform the mean–variance stochastic control into Markovian time inconsistent stochastic control, then establish a verification theorem, which is similar to one of He and Liang (2008, 2009) and Zeng and Li (2011), to find the optimal strategy and the efficient frontier of the pension member. The differences of the optimal strategies between the Pension plans with and without the return of premium clauses are studied via the Monte Carlo methods. The impacts of the risk averse level on the optimal strategies is also explored by the numerical methods.