Time-lagged marginal expected shortfall

TL;DR

The paper tackles the problem of measuring dynamic, time-lagged contributions to systemic risk by introducing TMES, a time-lagged extension of MES. It defines TMES via $\delta(h)=\lim_{u\to\infty} E[X_t\mid Y_{t-h}>u]$ within a partial regular variation framework and develops a model-free estimator $\widehat{\delta}(h)$ with rigorous asymptotic theory under mixing, plus a stationary bootstrap to form confidence bands. The authors establish existence conditions for TMES, provide two illustrative examples (regularly varying random fields and copula-based time series), and validate the methodology through extensive simulations and two real-data applications in finance and hydrology. The results show TMES effectively captures time-lagged systemic risk and supports reliable inference via bootstrap bands, with practical utility across domains requiring dynamic extreme risk assessment.

Abstract

Marginal expected shortfall (MES) is an important measure when assessing and quantifying the contribution of the financial institution to a systemic crisis. In this paper, we propose time-lagged marginal expected shortfall (TMES) as a dynamic extension of the MES, accounting for time lags in assessing systemic risks. A natural estimator for the TMES is proposed, and its asymptotic properties are studied. To address challenges in constructing confidence intervals for the TMES in practice, we apply the stationary bootstrap method to generate confidence bands for the TMES estimator. Extensive simulation studies were conducted to investigate the asymptotic properties of empirical and bootstrapped TMES. Two practical applications of TMES, supported by real data analyses, effectively demonstrate its ability to account for time lags in risk assessment.

Figures (16)

• Figure 1: A simulation of $(X_t^{(1)}, Y_t^{(1)})$. The blue solid line represents $(X_t^{(1)})$ and the orange dotted line represents $(Y_t^{(1)})$.
• Figure 2: The TMES $\delta(h)$ at lags $h=0,1,\ldots,9$ (blue solid line with circle marks) and the empirical TMES $\widehat{\delta}(h)$ at lags $h=0,1,\ldots, 9$ (orange solid line with square marks) of $(X_t^{(1)}, Y_t^{(1)})$ along with the bootstrapped $90\%$-confidence interval (orange dotted line) and the simulated $90\%$-confidence interval (green dashed line).
• Figure 3: QQ-plots of the bootstrapped and simulation distributions against the standard normal distribution for $(X_t^{(1)}, Y_t^{(1)})$. Up-left: The simulated distribution for $\widehat{\delta}(0)$. Up-right: The simulated distribution for $\widehat{\delta}(3)$. Low-left: The bootstrapped distribution of $\delta^{\star}(0)$. Low-right: The bootstrapped distribution of $\delta^{\star}(3)$.
• Figure 4: A simulation of $(X_t^{(2)}, Y_t^{(2)})$. The blue solid line represents $(X_t^{(2)})$ and the orange dotted line represents $(Y_t^{(2)})$.
• Figure 5: The TMES $\delta(h)$ at lags $h=0,1,\ldots,9$ (blue solid line with circle marks) and the empirical TMES $\widehat{\delta}(h)$ at lags $h=0,1,\ldots, 9$ (orange solid line with square marks) of $(X_t^{(2}, Y_t^{(2)})$ along with the bootstrapped $90\%$-confidence interval (orange dotted line) and the simulated $90\%$-confidence interval (green dashed line).

Theorems & Definitions (29)

• Theorem 2.1
• Remark 1
• Theorem 2.2
• Remark 2
• Theorem 2.3
• proof
• Remark 3
• Remark 4
• Proposition 2.4
• proof : Proof of Proposition \ref{['prop:latent']}