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Measuring risk contagion in financial networks with CoVaR

Bikramjit Das, Vicky Fasen-Hartmann

TL;DR

This work develops a tail-risk framework for contagion in financial networks modeled as a bipartite graph of banks and assets, where asset returns are heavy-tailed and tail dependence is captured by Gaussian and Marshall-Olkin copulas. It leverages multivariate regular variation on cones to derive asymptotic CoVaR expressions for linear network mappings $\boldsymbol X=\boldsymbol A\boldsymbol Z$, and introduces the Extreme CoVaR Index (ECI) to quantify contagion strength under asymptotic independence. The authors provide explicit CoVaR formulas and ECI values for the Gaussian and Marshall-Olkin models, and extend the results from pairs of portfolios to aggregates and to networks with many portfolios. The approach highlights how network structure and tail dependence jointly shape systemic risk, offering tractable tools for regulatory risk assessment and potential extensions to MES, MME, and SRISK in diverse copula settings.

Abstract

The stability of a complex financial system may be assessed by measuring risk contagion between various financial institutions with relatively high exposure. We consider a financial network model using a bipartite graph of financial institutions (e.g., banks, investment companies, insurance firms) on one side and financial assets on the other. Following empirical evidence, returns from such risky assets are modeled by heavy-tailed distributions, whereas their joint dependence is characterized by copula models exhibiting a variety of tail dependence behavior. We consider CoVaR, a popular measure of risk contagion and study its asymptotic behavior under broad model assumptions. We further propose the Extreme CoVaR Index (ECI) for capturing the strength of risk contagion between risk entities in such networks, which is particularly useful for models exhibiting asymptotic independence. The results are illustrated by providing precise expressions of CoVaR and ECI when the dependence of the assets is modeled using two well-known multivariate dependence structures: the Gaussian copula and the Marshall-Olkin copula.

Measuring risk contagion in financial networks with CoVaR

TL;DR

This work develops a tail-risk framework for contagion in financial networks modeled as a bipartite graph of banks and assets, where asset returns are heavy-tailed and tail dependence is captured by Gaussian and Marshall-Olkin copulas. It leverages multivariate regular variation on cones to derive asymptotic CoVaR expressions for linear network mappings , and introduces the Extreme CoVaR Index (ECI) to quantify contagion strength under asymptotic independence. The authors provide explicit CoVaR formulas and ECI values for the Gaussian and Marshall-Olkin models, and extend the results from pairs of portfolios to aggregates and to networks with many portfolios. The approach highlights how network structure and tail dependence jointly shape systemic risk, offering tractable tools for regulatory risk assessment and potential extensions to MES, MME, and SRISK in diverse copula settings.

Abstract

The stability of a complex financial system may be assessed by measuring risk contagion between various financial institutions with relatively high exposure. We consider a financial network model using a bipartite graph of financial institutions (e.g., banks, investment companies, insurance firms) on one side and financial assets on the other. Following empirical evidence, returns from such risky assets are modeled by heavy-tailed distributions, whereas their joint dependence is characterized by copula models exhibiting a variety of tail dependence behavior. We consider CoVaR, a popular measure of risk contagion and study its asymptotic behavior under broad model assumptions. We further propose the Extreme CoVaR Index (ECI) for capturing the strength of risk contagion between risk entities in such networks, which is particularly useful for models exhibiting asymptotic independence. The results are illustrated by providing precise expressions of CoVaR and ECI when the dependence of the assets is modeled using two well-known multivariate dependence structures: the Gaussian copula and the Marshall-Olkin copula.
Paper Structure (16 sections, 23 theorems, 109 equations, 2 figures)

This paper contains 16 sections, 23 theorems, 109 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Multivariate heavy tails
  3. Preliminaries: multivariate regular variation
  4. Regular variation under Gaussian copula
  5. Regular variation under Marshall-Olkin copula
  6. Measuring CoVaR
  7. Measuring CoVaR under different model assumptions
  8. Risk contagion in a bipartite network with asymptotically independent objects
  9. Risk contagion between two portfolios
  10. Risk contagion between various aggregates
  11. Risk contagion in more than two portfolios
  12. Conclusion
  13. Proofs of Section \ref{['sec:mrvwithasyind']}
  14. Proofs of Section \ref{['sec:systemicrisk']}
  15. Proofs of Section \ref{['sec:bipnet']}
  16. ...and 1 more sections

Key Result

Lemma 2.4

Let $\Sigma\in \mathbb{R}^{d\times d}$ be a positive-definite correlation matrix. Then the quadratic programming problem has a unique solution $\boldsymbol e^*=\boldsymbol e^*(\Sigma)\in\mathbb{R}^d$ such that Moreover, there exists a unique non-empty index set $I:=I(\Sigma)\subseteq \{1,\ldots, d\}=:\mathbb{I}_d$ with $J:=J(\Sigma):=\mathbb{I}_d\setminus I$ such that the unique solution $\bolds

Figures (2)

  • Figure 1: (1) Left network: Bank-asset bipartite network of the Mexican financial system on a particular day. Nodes in the network represent banks (blue) and assets (red). Links between an asset and a bank exist if the bank holds the asset in its portfolio. (Courtesy: poledna:etal:2021); (2) Right network: Bipartite network structure of common assets held by Chinese banks in 2021. The numbers and letters in the circles respectively represent identifiers for banks and industries (Courtesy: fan:2024)
  • Figure 2: A bipartite network with $q=3$ banks (or agents) and $d=5$ assets (or objects). We model inter-dependencies via a multivariate model for the assets and the bipartite network (which may be assumed to be random or fixed) connecting assets to banks.

Theorems & Definitions (61)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Example 2.7
  • Proposition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 51 more