Time-Consistent Strategies for DC Pension Plans with the Return of Premiums Clauses in Stochastic Environments

This paper studies a defined contribution (DC) pension plan with stochastic interest rate and stochastic volatility in a mean–variance framework. In the DC pension plans, a part of premiums are often returned to the members who died during the accumulation phase in order to protect the rights of the plan members, while the survival members can share the difference between the accumulated wealth and the returned premiums. From the survival members’ point of view, they hope to maximize the expectation of terminal wealth and to minimize the volatility of terminal wealth. In this paper, we formulate this problem as a continuous-time mean–variance model in a game theoretical framework. By using the extended stochastic optimal control theory, we obtain the time-consistent equilibrium strategy and the efficient frontier in explicit form. In addition, the characteristics of the demand for stocks and bonds are discussed and some special cases are also derived in detail. Our theoretical results display that the characteristic and structure of the time-consistent equilibrium strategy and the efficient frontier are considerably different from those obtained with constant interest rate and constant volatility. Finally, our results are illustrated by a numerical simulation and some economic implications are revealed.