Minimization of risks in defined benefit pension plan with time-inconsistent preferences

In this paper, we investigate the defined benefit pension plan, where the object of the manager is to minimise the contribution rate risk and the solvency risk by considering a quadratic performance criterion. To incorporate some well-documented behavioural features of human beings, we consider the situation where the discounting is non-exponential. It leads to a time-inconsistent control problem in the sense that the Bellman optimality principle does no longer hold. In our model, we assume that the benefit outgo is constant, and the pension fund can be invested in a risk-free asset and a risky asset whose return follows a geometric Brownian motion. We characterise the time-consistent strategies and value function in terms of the solution of a system of integral equations. The existence and uniqueness of the solution is verified, and the approximation of the solution is obtained. Some numerical results of the equilibrium contribution rate and equilibrium investment policy are presented for three types of discount functions.