Pre-commitment and equilibrium investment strategies for the DC pension plan with regime switching and a return of premiums clause

This paper studies an optimal investment problem for a defined-contribution (DC) pension plan during the accumulation phase, where a pension member contributes a predetermined amount of money as a premium and then the manager of the pension fund invests the premium in a financial market to increase the value of the accumulation. To protect the rights of pension members who die before retirement, a return of premiums clause is introduced, under which a member who dies before retirement can withdraw all the premiums she has contributed. We assume that the financial market consists of one risk-free asset and multiple risky assets, the returns of the risky assets depend on the market states, the evolution of the market states is described by a Markov chain, and the transition matrixes are time-varying. The pension fund manager aims to maximize the expected terminal wealth of each surviving member at retirement and to minimize the risk measured by the variance of her terminal wealth, which are two conflicting objectives. We formulate the investment problem as a discrete-time mean-variance model. Since the model is time-inconsistent, we seek its pre-commitment and equilibrium strategies. Using the embedding technique and the dynamic programming method, we obtain the pre-commitment strategy and the corresponding efficient frontier in closed form. Applying the game theory and the extended Bellman equation, we derive the analytical expressions of the equilibrium strategy and the corresponding efficient frontier. For the two obtained investment strategies and their corresponding efficient frontiers, as well as the impact of regime switching and the return of premiums clause on them, some interesting theoretical and numerical results are found.