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Joint extreme Value-at-Risk and Expected Shortfall dynamics with a single integrated tail shape parameter

Author

Listed:
  • D’Innocenzo, Enzo

    (University of Bologna)

  • Lucas, André

    (Vrije Universiteit Amsterdam and Tinbergen Institute)

  • Schwaab, Bernd

    (European Central Bank)

  • Zhang, Xin

    (Research Department, Central Bank of Sweden)

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 four exchange rate series, we illustrate how the new model captures the dynamics of extreme VaR and ES.

Suggested Citation

  • D’Innocenzo, Enzo & Lucas, André & Schwaab, Bernd & Zhang, Xin, 2025. "Joint extreme Value-at-Risk and Expected Shortfall dynamics with a single integrated tail shape parameter," Working Paper Series 446, Sveriges Riksbank (Central Bank of Sweden).
    • Handle: RePEc:hhs:rbnkwp:0446
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    More about this item

    Keywords

    dynamic tail risk; integrated score-driven models; extreme value theory;
    All these keywords.

    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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