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

Paper: arXiv 2408.08511

Authors: Wissam AlAli, Çağın Ararat

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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework extending Sample-Average Approximations (SAA) to set-valued systemic risk measures using topological convergence theory. While it offers mixed-integer programming formulations and synthetic sensitivity analysis with network models, it lacks backtesting on real-world empirical financial datasets.

Research Flowchart

  flowchart TD
    A[Research Goal: Convergence of SAA for Set-Valued Systemic VaR] --> B[Develop SAA Theory under Wijsman & Hausdorff Topologies]
    B --> C[Apply Theory to Eisenberg-Noe Network Model]
    D[Inputs: Preferential Attachment Network & Pareto Disruptions] --> E[Formulate MIP for Weighted-Sum & Norm-Minimizing Scalarizations]
    C --> E
    E --> F[Conduct Sensitivity Analysis via Simulations]
    F --> G[Outcomes: Theoretical Convergence Proofs & Computational MIP Formulations]