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Dynamic CoVaR Modeling and Estimation

Timo Dimitriadis, Yannick Hoga

TL;DR

The paper develops a dynamic, semiparametric framework for jointly forecasting VaR and CoVaR through CoCAViaR models and a two-step M-estimator grounded in multi-objective elicitability. It proves consistency and asymptotic normality of the estimators, and validates finite-sample performance via simulations under CCC--GARCH dynamics. An empirical application to US G-SIBs demonstrates that CoCAViaR forecasts often outperform benchmark DCC--GARCH CoVaR predictions, especially for CoVaR, with robust backtests. The work offers a practical, tail-focused alternative to fully specified multivariate volatility models, with potential implications for macroprudential risk analysis and Growth-at-Risk-type studies.

Abstract

The popular systemic risk measure CoVaR (conditional Value-at-Risk) and its variants are widely used in economics and finance. In this article, we propose joint dynamic forecasting models for the Value-at-Risk (VaR) and CoVaR. The CoVaR version we consider is defined as a large quantile of one variable (e.g., losses in the financial system) conditional on some other variable (e.g., losses in a bank's shares) being in distress. We introduce a two-step M-estimator for the model parameters drawing on recently proposed bivariate scoring functions for the pair (VaR, CoVaR). We prove consistency and asymptotic normality of our parameter estimator and analyze its finite-sample properties in simulations. Finally, we apply a specific subclass of our dynamic forecasting models, which we call CoCAViaR models, to log-returns of large US banks. A formal forecast comparison shows that our CoCAViaR models generate CoVaR predictions which are superior to forecasts issued from current benchmark models.

Dynamic CoVaR Modeling and Estimation

TL;DR

The paper develops a dynamic, semiparametric framework for jointly forecasting VaR and CoVaR through CoCAViaR models and a two-step M-estimator grounded in multi-objective elicitability. It proves consistency and asymptotic normality of the estimators, and validates finite-sample performance via simulations under CCC--GARCH dynamics. An empirical application to US G-SIBs demonstrates that CoCAViaR forecasts often outperform benchmark DCC--GARCH CoVaR predictions, especially for CoVaR, with robust backtests. The work offers a practical, tail-focused alternative to fully specified multivariate volatility models, with potential implications for macroprudential risk analysis and Growth-at-Risk-type studies.

Abstract

The popular systemic risk measure CoVaR (conditional Value-at-Risk) and its variants are widely used in economics and finance. In this article, we propose joint dynamic forecasting models for the Value-at-Risk (VaR) and CoVaR. The CoVaR version we consider is defined as a large quantile of one variable (e.g., losses in the financial system) conditional on some other variable (e.g., losses in a bank's shares) being in distress. We introduce a two-step M-estimator for the model parameters drawing on recently proposed bivariate scoring functions for the pair (VaR, CoVaR). We prove consistency and asymptotic normality of our parameter estimator and analyze its finite-sample properties in simulations. Finally, we apply a specific subclass of our dynamic forecasting models, which we call CoCAViaR models, to log-returns of large US banks. A formal forecast comparison shows that our CoCAViaR models generate CoVaR predictions which are superior to forecasts issued from current benchmark models.
Paper Structure (44 sections, 15 theorems, 192 equations, 2 figures, 6 tables)

This paper contains 44 sections, 15 theorems, 192 equations, 2 figures, 6 tables.

Table of Contents

  1. Motivation
  2. Modeling and Estimation
  3. The CoVaR
  4. Dynamic Models for VaR and CoVaR
  5. Parameter Estimation
  6. Asymptotic Properties of the Estimators
  7. Consistency
  8. Additional Asymptotic Results
  9. Simulations
  10. Applied Systemic Risk Forecasting for US G-SIBs
  11. CoVaR Forecasting with CoCAViaR Models
  12. Comparison with DCC--GARCH Models
  13. Conclusion
  14. Proof of Theorem \ref{['CoQR_arXiv_v4:thm:cons']}
  15. Proofs of Equations \ref{['CoQR_arXiv_v4:eq:un id VaR']} and \ref{['CoQR_arXiv_v4:eq:un id CoVaR']}
  16. ...and 29 more sections

Key Result

Theorem 1

Suppose Assumption CoQR_arXiv_v4:ass:cons holds. Then, as $n\to\infty$, $\widehat{\bm \theta}_{n}^{v}\overset{\mathbb{P}}{\longrightarrow}\bm \theta_0^{v}$ and $\widehat{\bm \theta}_{n}^{c}\overset{\mathbb{P}}{\longrightarrow}\bm \theta_0^{c}$.

Figures (2)

  • Figure 1: Out-of-sample VaR and CoVaR forecasts from the CoCAViaR-SAV-fullA model where JPMorgan Chase's log-losses are used for $X_t$ and the S&P 500 for $Y_t$. Log-losses on days with a VaR exceedance of JPMorgan Chase are displayed in black in both panels.
  • Figure 2: This figure graphically illustrates the one and a half-sided forecast comparison tests for the VaR and CoVaR of FH24 based on their multi-objective scoring function in \ref{['CoQR_arXiv_v4:eq:loss']} for a significance level of $10\%$. We use log-losses of JPMorgan Chase for $X_t$ and of the S&P 500 for $Y_t$. The respective CoCAViaR models are given in the captions of the two plots and are compared against the baseline "DCC-t-Chol" model.

Theorems & Definitions (24)

  • Example 1
  • Example 1: continued
  • Remark 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Remark 3
  • Proposition 1
  • Theorem 3
  • Example 2
  • ...and 14 more