Open-loop equilibrium strategy for mean–variance asset–liability management portfolio selection problem with debt ratio

In this study, we consider a time-consistent mean–variance asset–liability management portfolio selection problem in which the liability is controllable. The objective is to find an equilibrium investment strategy and an equilibrium debt ratio in the financial market consisting of one risk-free asset and one risky asset. By using forward backward stochastic differential equations (FBSDEs), we derive a sufficient condition and a necessary condition for the open-loop equilibrium strategies. The uniqueness of the strategies is also provided. Furthermore, to illustrate our results, we provide numerical examples to show how the parameters impact on the equilibrium strategies and the corresponding efficient frontier.