Equilibrium intergenerational risk-sharing design for a target benefit pension plan

In this paper, we develop a risk-sharing pension design for a target benefit pension plan to minimize the income instability for all future retirees within a Black-Scholes market setting and a stable population. In contrast to the existing literature, we explicitly consider the difference between individual and intergenerational discount functions. This distinction, motivated by the fact that individual time preferences and societal preferences for different generations are fundamentally different, leads to time-inconsistent preferences for pension sponsors. By using the benefit structure as a control variable and solving a system of extended Hamilton-Jacobi-Bellman equations, we derive an intergenerational Nash equilibrium design that implicitly balances the benefit-risk across different generations. Compared to several conventional designs, we find that the equilibrium design is more robust to the choices of generational weights and time preferences. Consequently, it fosters stronger intergenerational solidarity in the risk-sharing structure, enhancing the stability and continuity of the pension plan. Additional sensitivity tests, including different individual and generational discount functions as well as dynamic investment strategies, are performed.