Limiting Distributional Fixed Points in Systemic Risk Graph Models

We analyse the equilibrium behaviour of a large network of banks in presence of incomplete information, where inter-bank borrowing and lending is allowed, and banks suffer shocks to assets. In a two time period graphical model, we show that the equilibrium wealth distribution is the unique fixed point of a complex, high dimensional distribution-valued map. Fortunately, there is a dimension collapse in the limit as the network size increases, where the equilibriated system converges to the unique fixed point involving a simple, one dimensional distribution-valued operator, which, we show, is amenable to simulation. Specifically, we develop a Monte-Carlo algorithm that computes the fixed point of a general distribution-valued map and derive sample complexity guarantees for it. We numerically show that this limiting one-dimensional regime can be used to obtain useful structural insights and approximations for networks with as low as a few hundred banks.