Measuring systemic importance with capital injections

Abstract

Financial networks consist of agents connected through various financial transactions, which can increase diversification but also spread the effects of shocks to individual agents. This can result in systemic risk, where the default of certain agents can cause the default of other agents connected to them. In this study, we examine these effects in a simplified system of exogenously determined payment obligations, where uncollateralized debt obligations act as a channel of contagion and risky outside endowments are the source of initial shocks. We use a clearing mechanism to settle these obligations and determine the outcome of contagion.

To assess each agent's contribution to systemic risk, we use the framework of cooperative games and develop a concept called the systemic risk capital allocation game. This framework defines the minimum amount of cash injections needed to rescue a given coalition of agents, after shocks have been realized but before the clearing mechanism. Our key contribution is introducing this type of capital injection as a tool for measuring the systemic importance of coalitions of agents. We prove that for rescuing a given coalition with a minimal amount of cash, it suffices to inject cash into agents within that group. We also show how this result enables the practical application of our method, with the computational cost of calculating the capital injection being the same as the cost of calculating the clearing payment for the system. Lastly, we demonstrate how the systemic risk allocation game can be combined with Monte Carlo simulation to analyze a network of obligations of any structure, and how capital injections can better capture the role of key agents compared to other metrics.