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Probability equivalent level for CoVaR and VaR in bivariate Student-\textit{t} copulas with application to foreign exchange risk monitoring

Daniela I. Flores-Silva, Miguel A. Sordo, Alfonso Suárez-Llorens

TL;DR

The paper generalizes the probability-equivalent level of VaR and CoVaR (PELCoV) to bivariate risks governed by a Student-t copula, enabling tail-dependent spillover analysis beyond SSI-based frameworks. It provides analytic characterizations of the PELCoV set, including when there is a unique level or up to two levels, via the derivative of the copula ∂1C(u,v) and its limits L0,L1. The authors implement a dynamic version by allowing the copula parameter ρ to evolve over time, and apply the method to a foreign-exchange risk pair (USD/GBP) with USD/EUR as an auxiliary indicator, using ARMA-GARCH marginals and a two-stage MLE estimation approach. Empirically, the framework yields time-varying PELCoV_v signals that offer early warnings of risk underestimation, demonstrated across major stress periods and emphasizing the practical value for risk monitoring in financial markets.

Abstract

We extend the "probability-equivalent level of VaR and CoVaR" (PELCoV) methodology to accommodate bivariate risks modeled by a Student-t copula, relaxing the strong dependence assumptions of earlier approaches and enhancing the framework's ability to capture tail dependence and asymmetric co-movements. While the theoretical results are developed in a static setting, we implement them dynamically to track evolving risk spillovers over time. We illustrate the practical relevance of our approach through an application to the foreign exchange market, monitoring the USD/GBP exchange rate with the USD/EUR series as an auxiliary early warning indicator over the period 1999-2024. Our results highlight the potential of the extended PELCoV framework to detect early signs of risk underestimation during periods of financial stress.

Probability equivalent level for CoVaR and VaR in bivariate Student-\textit{t} copulas with application to foreign exchange risk monitoring

TL;DR

The paper generalizes the probability-equivalent level of VaR and CoVaR (PELCoV) to bivariate risks governed by a Student-t copula, enabling tail-dependent spillover analysis beyond SSI-based frameworks. It provides analytic characterizations of the PELCoV set, including when there is a unique level or up to two levels, via the derivative of the copula ∂1C(u,v) and its limits L0,L1. The authors implement a dynamic version by allowing the copula parameter ρ to evolve over time, and apply the method to a foreign-exchange risk pair (USD/GBP) with USD/EUR as an auxiliary indicator, using ARMA-GARCH marginals and a two-stage MLE estimation approach. Empirically, the framework yields time-varying PELCoV_v signals that offer early warnings of risk underestimation, demonstrated across major stress periods and emphasizing the practical value for risk monitoring in financial markets.

Abstract

We extend the "probability-equivalent level of VaR and CoVaR" (PELCoV) methodology to accommodate bivariate risks modeled by a Student-t copula, relaxing the strong dependence assumptions of earlier approaches and enhancing the framework's ability to capture tail dependence and asymmetric co-movements. While the theoretical results are developed in a static setting, we implement them dynamically to track evolving risk spillovers over time. We illustrate the practical relevance of our approach through an application to the foreign exchange market, monitoring the USD/GBP exchange rate with the USD/EUR series as an auxiliary early warning indicator over the period 1999-2024. Our results highlight the potential of the extended PELCoV framework to detect early signs of risk underestimation during periods of financial stress.
Paper Structure (17 sections, 6 theorems, 54 equations, 5 figures, 2 tables)

This paper contains 17 sections, 6 theorems, 54 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Motivation
  4. Aim of the paper
  5. PELCoV$_v$'s in Student-t copulas
  6. Proof:
  7. Proof:
  8. Proof:
  9. Application to Exchange Rate Risk Monitoring
  10. The model
  11. Estimation and Testing
  12. Data analysis
  13. Results
  14. Existence, uniqueness and calculation of PELCoV$_v$
  15. Interpretation of PELCoV$_v$
  16. ...and 2 more sections

Key Result

Theorem 2

Let $\left( X, Y\right)$ be a random vector satisfying the regularity conditions with copula $C$ and let $v \in (0,1).$ Then, (a) CoVaR$_{v,u}[Y\mid X]\ge \text{VaR}_v[Y]$ (respectively $\le,=$) if, and only if $\partial_1 C(u,v)\le v$ (respectively $\ge,=$). (b) $CoVaR_{v,u}[Y\mid X]$ is continuou

Figures (5)

  • Figure 1: Function $v\to\partial_1C(u,v)$, where $C(u,v)$ is the Student t-copula with parameters $\rho=0.4, n=2$ and fixed values of $u=0.95,0.55$.
  • Figure 2: Scatter-plot of the (time-invariant) empirical copula of the bivariate time series $(X_t,Y_t),$ given by $(F_{t}(x_t ; \hat{\lambda}_1), G_{t} (y_t ; \hat{\lambda}_2)).$
  • Figure 3: Evolution of $L^t_0$ as a function of $\rho_t$. Within the range of $\rho_t$ values observed in our data (blue dashed lines), $L_0$ remains above the threshold level $v = 0.99$ (red line) for all $t$.
  • Figure 4: The upper panel shows the time series of the negative log-returns of the U.S. Dollar to Euro Spot Exchange Rate (EXUSEU), along with the PELCoV$_{0.99}$ under the assumption of a time-varying Student-t copula. The lower panel displays the time series of the negative log-returns of the U.S. Dollar to British Pound (EXUSUK), together with the corresponding Value at Risk at the 0.99 level.
  • Figure 5: The upper panel shows the time series of the negative log-returns of the U.S. Dollar to Euro Spot Exchange Rate (EXUSEU), along with the PELCoV$_{0.95}$ under the assumption of a time-varying Student-t copula. The lower panel displays the time series of the negative log-returns of the U.S. Dollar to British Pound (EXUSUK), together with the corresponding Value at Risk at the 0.95 level.

Theorems & Definitions (9)

  • Definition 1
  • Theorem 2
  • Remark 3
  • Lemma 4
  • Lemma 5
  • Theorem 6
  • Remark 7
  • Theorem 8
  • Lemma 9