In this talk, we discuss a mean-field model of a banking system with defaults and government interventions. The banks are represented by their capital buffer which is modelled as a diffusion process on the real line. Whenever the capital buffer becomes negative, the bank has a positive default intensity and a default occurs once the cumulative intensity exceeds an exponential clock. Defaults feed back into the remaining system by lowering the capital buffer of other banks. This contagion can lead to systemic events, where a large number of banks default at once. A government intervenes in the system through capital injections with the goal of limiting the contagion. We show that the optimal level of capital provided by the government only depends on the current level of the capital buffer and not on the accumulated default intensity. This result is relevant for numerical simulations of the model as it reduces the effective dimension of the control problem. Proving it leads us to the study of a novel type of nonlocal semilinear BSPDE.