Optimal Monitoring and Mitigation of Systemic Risk in Financial Networks

TL;DR

This paper develops optimization models and algorithms for allocating cash injections in financial networks to minimize systemic risk, considering deterministic and stochastic scenarios, and proposes heuristics for default minimization.

Contribution

It introduces a linear programming framework for optimal cash injection allocation and develops distributed, stochastic, and heuristic algorithms for systemic risk mitigation.

Findings

Linear program formulation for cash injection allocation

Distributed duality-based algorithm for decentralized computation

Heuristic algorithms for minimizing defaults

Abstract

This paper studies the problem of optimally allocating a cash injection into a financial system in distress. Given a one-period borrower-lender network in which all debts are due at the same time and have the same seniority, we address the problem of allocating a fixed amount of cash among the nodes to minimize the weighted sum of unpaid liabilities. Assuming all the loan amounts and asset values are fixed and that there are no bankruptcy costs, we show that this problem is equivalent to a linear program. We develop a duality-based distributed algorithm to solve it which is useful for applications where it is desirable to avoid centralized data gathering and computation. We also consider the problem of minimizing the expectation of the weighted sum of unpaid liabilities under the assumption that the net external asset holdings of all institutions are stochastic. We show that this problem is a two-stage stochastic linear program. To solve it, we develop two algorithms based on: Benders decomposition algorithm and projected stochastic gradient descent. We show that if the defaulting nodes never pay anything, the deterministic optimal cash injection allocation problem is an NP-hard mixed-integer linear program. However, modern optimization software enables the computation of very accurate solutions to this problem on a personal computer in a few seconds for network sizes comparable with the size of the US banking system. In addition, we address the problem of allocating the cash injection amount so as to minimize the number of nodes in default. For this problem, we develop two heuristic algorithms: a reweighted l1 minimization algorithm and a greedy algorithm. We illustrate these two algorithms using three synthetic network structures for which the optimal solution can be calculated exactly. We also compare these two algorithms on three types random networks which are more complex.