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Joint extreme value-at-risk and expected shortfall dynamics with a single integrated tail shape parameter

Author

Listed:
  • Lucas, André
  • Schwaab, Bernd
  • Zhang, Xin
  • D’Innocenzo, Enzo

Abstract

We propose a robust semi-parametric framework for persistent time-varying extreme tail behavior, including extreme Value-at-Risk (VaR) and Expected Shortfall (ES). The framework builds on Extreme Value Theory and uses a conditional version of the Generalized Pareto Distribution (GPD) for peaks-over-threshold (POT) dynamics. Unlike earlier approaches, our model (i) has unit root-like, i.e., integrated autoregressive dynamics for the GPD tail shape, and (ii) re-scales POTs by their thresholds to obtain a more parsimonious model with only one time-varying parameter to describe the entire tail. We establish parameter regions for stationarity, ergodicity, and invertibility for the integrated time-varying parameter model and its filter, and formulate conditions for consistency and asymptotic normality of the maximum likelihood estimator. Using two cryptocurrency exchange rates, we illustrate how the simple single-parameter model is competitive in capturing the dynamics of VaR and ES, particularly in the extreme tail. JEL Classification: C22, G11

Suggested Citation

  • Lucas, André & Schwaab, Bernd & Zhang, Xin & D’Innocenzo, Enzo, 2026. "Joint extreme value-at-risk and expected shortfall dynamics with a single integrated tail shape parameter," Working Paper Series 3166, European Central Bank.
    • Handle: RePEc:ecb:ecbwps:20263166
    Note: 955417
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    References listed on IDEAS

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    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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