Optimal Clearing Payments in a Financial Contagion Model

TL;DR

The paper analyzes clearing payments in financial networks under the Eisenberg–Noe framework, deriving necessary and sufficient conditions for the uniqueness of pro-rata clearing payments across arbitrary topologies and introducing a convex-optimization-based unconstrained clearing framework. It shows that the dominant clearing vector minimizes system losses and provides a graph-theoretic, topology-based procedure to test uniqueness, along with a two-stage method to obtain a unique, minimal-norm clearing when pro-rata is lifted. Numerical experiments demonstrate that relaxing pro-rata substantially reduces systemic loss and containment failures, improving isolation of defaults in both schematic and random networks. The work highlights practical challenges to implementation—requiring ex-ante contractualization and transparency of interbank liabilities—and points toward potential distributed algorithms to realize centralized clearing benefits. Overall, the results offer a principled path to enhance systemic resilience by reconciling fair clearing with system-wide performance targets.

Abstract

Financial networks are characterized by complex structures of mutual obligations. These obligations are fulfilled entirely or in part (when defaults occur) via a mechanism called clearing, which determines a set of payments that settle the claims by respecting rules such as limited liability, absolute priority, and proportionality (pro-rated payments). In the presence of shocks on the financial system, however, the clearing mechanism may lead to cascaded defaults and eventually to financial disaster. In this paper, we first study the clearing model under pro-rated payments of Eisenberg and Noe, and we derive novel necessary and sufficient conditions for the uniqueness of the clearing payments, valid for an arbitrary topology of the financial network. Then, we argue that the proportionality rule is one of the factors responsible for cascaded defaults, and that the overall system loss can be reduced if this rule is lifted. The proposed approach thus shifts the focus from the individual interest to the overall system's interest to control and contain adverse effects of cascaded failures, and we show that clearing payments in this setting can be computed by solving suitable convex optimization problems.

Figures (6)

• Figure 1: Examples of possible shapes of ${\@fontswitch{}{\mathcal{}} D}$ and the dominant clearing vector in the case of two nodes.
• Figure 1: A four-nodes network with a shock on the cash inflow at node 3. Panel (a) shows the pro-rated clearing payments, panel (b) shows the optimal matrix clearings.
• Figure 2: Network with a sink node and three strongly connected components.
• Figure 2: The cost of introducing the pro-rata rule.
• Figure 3: The sets $I_s$ (encircled by blue lines) and $I_s'$ (encircled by red lines) for a special financial network with $n=15$ nodes with $\bar{p}_i>0$, $\forall i\ne 0$.

Theorems & Definitions (21)

• Remark 1
• Definition 3.1
• Definition 4.1
• Definition 4.2
• Lemma 4.3
• Remark 2
• Lemma 4.4
• Lemma 4.5
• Remark 3
• Theorem 4.6