Sparsifying defaults: Optimal bailout policies for financial networks in distress

We propose a quantitative framework for constructing optimal policies to manage systemic risk in financial networks. We analyze borrower-lender networks where all the loan amounts and cash flows are known, and where some nodes may default in the absence of external intervention. Given a fixed amount of cash to be injected into the system, we address the problem of allocating it among the nodes to minimize the overall amount of unpaid liabilities. We show that this problem is equivalent to a linear program. 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 an approximate algorithm which uses reweighted ℓ1 minimization. We illustrate this algorithm using two synthetic network structures for which the optimal solution can be calculated exactly. We show through numerical simulations that the solutions calculated by our algorithm are close to optimal.