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Extreme Value Inference for CoVaR and Systemic Risk

Xiaoting Li, Harry Joe

Abstract

We develop an extreme value framework for CoVaR centered on $v(q \mid p ; C)$, the copula-adjusted probability level, or equivalently, the CoVaR on the uniform (0,1) scale. We characterize the possible tail regimes of $v(q \mid p ; C)$ through the limit behavior of the copula conditional distribution and show that these regimes are determined by the joint tail expansions of the copula. This leads to tractable conditions for identifying the tail regime and deriving the asymptotic behavior of $v(q | p ; C)$. Building on this characterization, we propose a minimum-distance estimation approach for CoVaR that accommodates multiple tail regimes. The methodology links CoVaR and $Δ$CoVaR to the underlying joint tail behavior, thereby providing a clear interpretation of these measures in systemic risk analysis. An empirical analysis across U.S. sectors demonstrates the practical value of the approach for assessing systemic risk contributions and exposures with important implications for macroprudential surveillance and risk management.

Extreme Value Inference for CoVaR and Systemic Risk

Abstract

We develop an extreme value framework for CoVaR centered on , the copula-adjusted probability level, or equivalently, the CoVaR on the uniform (0,1) scale. We characterize the possible tail regimes of through the limit behavior of the copula conditional distribution and show that these regimes are determined by the joint tail expansions of the copula. This leads to tractable conditions for identifying the tail regime and deriving the asymptotic behavior of . Building on this characterization, we propose a minimum-distance estimation approach for CoVaR that accommodates multiple tail regimes. The methodology links CoVaR and CoVaR to the underlying joint tail behavior, thereby providing a clear interpretation of these measures in systemic risk analysis. An empirical analysis across U.S. sectors demonstrates the practical value of the approach for assessing systemic risk contributions and exposures with important implications for macroprudential surveillance and risk management.

Paper Structure

This paper contains 15 sections, 12 theorems, 44 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic definitions and properties of CoVaR
  3. Tail regimes of the copula-adjusted level in CoVaR
  4. Tail regimes of $v(q\mid p;C)$ for fixed $q$
  5. Tail regimes of $v(q\mid p;C)$ for extreme-level $q$
  6. Tail limits of $\Delta$CoVaR and systemic risk implications
  7. Estimation of CoVaR and its asymptotic properties
  8. Empirical estimators of tail dependence
  9. Minimum distance estimation of $\operatorname{CoVaR}$
  10. Empirical systemic risk analysis
  11. CoVaR and $\Delta$CoVaR in the Copula-AR-GARCH model
  12. Systemic risk impact of the financial sector
  13. Cross-sectional contributions to systemic risk
  14. Systemic risk exposures and hedging assets
  15. Discussion

Key Result

Proposition 2.3

Figures (6)

  • Figure 1: Mean estimates of $r_{S\mid i}(p)$ (left) and $\Delta \operatorname{CoVaR}_{S \mid i}(p)$(right) for large U.S. financial institutions with $p=0.05$, segmented by depository, broker-dealers, insurance companies, and real estate firms. The estimates for $r_{S\mid i}(p)$ and $\Delta \operatorname{CoVaR}_{S \mid i}(p)$ are calculated based on the daily log return data from June 2000 to June 2025 using five-year rolling window.
  • Figure 2: Path of the estimated adjustment factor $r_{S \mid i}(p)$ at $p=0.05$ for 35 institutions that are consistently among the top 100 firms by market capitalization throughout 2000-2025.
  • Figure 3: $\operatorname{CoVaR}_{S \mid i}(p)$ versus the adjustment factor $r_{S \mid i}(p)$ at $p = 0.05$ for major U.S. financial institutions, large technology firms, and equity ETFs, based on daily returns for two sample periods: 2012–2018 (left panel) and 2018–2025 (right panel).
  • Figure 4: $\Delta \operatorname{CoVaR}_{S \mid i}(p)$ versus the adjustment factor $r_{S \mid i}(p)$ at $p=0.05$ for firms across consumer, energy, health care, real estate, and utilities sectors, based on daily returns for two sample periods: 2012–2018 (left panel) and 2018–2025 (right panel).
  • Figure 5: $\Delta \operatorname{CoVaR}_{i \mid S}(p)$ versus the adjustment factor $r_{i \mid S}(p)$ at $p=0.05$ for fixed income securities, cryptocurrency, commodities, computed based the daily returns from 2012--2018 (left panel) and 2018--2025 (right panel).
  • ...and 1 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Definition 2.2: $\Delta$CoVaR
  • Proposition 2.3
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Remark 3.6
  • Example 3.7
  • ...and 7 more