Systemic values-at-risk and their sample-average approximations

TL;DR

This paper studies the convergence of sample-average approximations for set-valued systemic risk measures, focusing on theoretical properties and practical computation within network models, including sensitivity analysis.

Contribution

It develops a general convergence theory for SAA of systemic risk measures and provides mixed-integer programming formulations for specific network-based cases.

Findings

Convergence of SAA under Wijsman and Hausdorff topologies established.

Mixed-integer programming formulations for Eisenberg-Noe network models provided.

Sensitivity analysis demonstrates applicability to financial networks with economic disruptions.

Abstract

This paper investigates the convergence properties of sample-average approximations (SAA) for set-valued systemic risk measures. We assume that the systemic risk measure is defined using a general aggregation function with some continuity properties and value-at-risk applied as a monetary risk measure. We focus on the theoretical convergence of its SAA under Wijsman and Hausdorff topologies for closed sets. After building the general theory, we provide an in-depth study of an important special case where the aggregation function is defined based on the Eisenberg-Noe network model. In this case, we provide mixed-integer programming formulations for calculating the SAA sets via their weighted-sum and norm-minimizing scalarizations. To demonstrate the applicability of our findings, we conduct a comprehensive sensitivity analysis by generating a financial network based on the preferential attachment model and modeling the economic disruptions via a Pareto distribution.