Optimal management of DB pension fund under both underfunded and overfunded cases

This paper investigates the optimal management of an aggregated defined benefit pension plan in a stochastic environment. The interest rate follows the Ornstein-Uhlenbeck model, the benefits follow the geometric Brownian motion while the contribution rate is determined by the spread method of fund amortization. The pension manager invests in the financial market with three assets: cash, a zero-coupon bond and a stock. Regardless of the initial status of the plan, we suppose that the pension fund may become underfunded or overfunded in the planning horizon. The optimization goal of the manager is to maximize the expected utility in the overfunded region minus the weighted solvency risk in the underfunded region. By introducing an auxiliary process and related equivalent optimization problems and using the martingale method, the optimal wealth process, optimal portfolio and efficient frontier are obtained under four cases (high tolerance towards solvency risk, low tolerance towards solvency risk, a specific lower bound, and high lower bound). Moreover, we also obtain the probabilities that the optimal terminal wealth falls in the overfunded and underfunded regions. At last, we present numerical analyzes to illustrate the manager's economic behaviors.