Default Resilience and Worst-Case Effects in Financial Networks

TL;DR

This paper addresses how financial networks with shared external asset exposures respond to bounded asset-price shocks and how defaults propagate under worst-case scenarios. It extends the Eisenberg–Noe clearing framework to include asset commonality, defines the default resilience margin $\epsilon^*$ and the insolvency resilience margin $\epsilon_{\mathrm{ub}}$, and derives efficient linear-programming methods to compute the worst-case systemic loss for perturbations beyond the resilience threshold. The core contributions include precise formulas for $\epsilon^*$ under $\ell_\infty$ and $\ell_1$ norms, a duality-based characterization of the worst-case loss curve $\eta_{\mathrm{wc}}(\epsilon)$, and numerical demonstrations on small and core–periphery networks showing how defaults emerge and how loss grows with perturbation magnitude. The results provide a rigorous, tractable framework for quantifying resilience and systemic risk in networks with asset commonality, with implications for stress testing and risk management under bounded external shocks.

Abstract

In this paper we analyze the resilience of a network of banks to joint price fluctuations of the external assets in which they have shared exposures, and evaluate the worst-case effects of the possible default contagion. Indeed, when the prices of certain external assets either decrease or increase, all banks exposed to them experience varying degrees of simultaneous shocks to their balance sheets. These coordinated and structured shocks have the potential to exacerbate the likelihood of defaults. In this context, we introduce first a concept of {default resilience margin}, $ε^*$, i.e., the maximum amplitude of asset prices fluctuations that the network can tolerate without generating defaults. Such threshold value is computed by considering two different measures of price fluctuations, one based on the maximum individual variation of each asset, and the other based on the sum of all the asset's absolute variations. For any price perturbation having amplitude no larger than $ε^*$, the network absorbs the shocks remaining default free. When the perturbation amplitude goes beyond $ε^*$, however, defaults may occur. In this case we find the worst-case systemic loss, that is, the total unpaid debt under the most severe price variation of given magnitude. Computation of both the threshold level $ε^*$ and of the worst-case loss and of a corresponding worst-case asset price scenario, amounts to solving suitable linear programming problems.}

Figures (6)

• Figure 1: An illustrative example with $n=4$ nodes and one external asset.
• Figure 2: A schematic network with 8 nodes.
• Figure 3: Worst-case losses vs. losses under random shocks, $\ell_1$ and $\ell_\infty$ norms. The dashed lines mark the default resilience margin $\epsilon^*$ and the insolvency resilience margin $\epsilon _{ {\mathrm{ub} } }$.
• Figure 4: Index $i^*$ of the worst-case case perturbation $\delta^*$ in the $\ell_1$ case. Using Proposition \ref{['prop:uniqueness']}, we have checked that $\delta^*$ is unique for all $\epsilon>\epsilon^*$.
• Figure 5: Core-periphery random graph: Worst-case losses vs. losses under random shocks for the $\ell_1$ and $\ell_\infty$ norms. The dashed lines mark the default resilience margin $\epsilon^*$ and the insolvency resilience margin $\epsilon _{ {\mathrm{ub} } }$.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Proposition 1
• Proposition 2
• Proposition 3
• Definition 3: Default resilience margin
• Theorem 1
• Theorem 2
• Proposition 4: Uniqueness of the worst-case perturbation scenario
• Definition 4: Insolvency resilience margin