Necessary and sufficient conditions for open-loop equilibrium portfolio for a DC pension plan with piecewise linear state-dependent risk tolerance

Wang, Chen, and Yang (2023) investigated the equilibrium behavioral strategy for a defined contribution (DC) pension plan during the accumulation phase and obtained the explicit expressions for the suboptimal equilibrium behavioral strategy. However, the authors failed to obtain the optimal solution and did not completely solve the pension investment problem with piecewise linear state-dependent risk tolerance. The primary objective of this article is to solve an improved version of the above DC pension plan problem thoroughly. We investigate the open-loop equilibrium portfolio for a DC pension plan and derive necessary and sufficient conditions for the open-loop equilibrium portfolio through coupled forward-backward stochastic differential equations (FBSDEs). The uniqueness of an equilibrium portfolio is obtained by decoupling FBSDEs. Finally, the economic significance and implications of the obtained open-loop equilibrium portfolio are illustrated from the theoretical perspective.