Scandinavian Working Papers in Economics

Working Paper Series,
Sveriges Riksbank (Central Bank of Sweden)

No 446: Joint extreme Value-at-Risk and Expected Shortfall dynamics with a single integrated tail shape parameter

Enzo D’Innocenzo (), André Lucas (), Bernd Schwaab () and Xin Zhang ()
Additional contact information
Enzo D’Innocenzo: University of Bologna, Postal: Piazza Antonio Scaravilli 2, 40122 Bologna, Italy
André Lucas: Vrije Universiteit Amsterdam and Tinbergen Institute, Postal: De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands
Bernd Schwaab: European Central Bank, Postal: Sonnemannstrasse 22, 60314 Frankfurt, Germany
Xin Zhang: Research Department, Central Bank of Sweden, Postal: Sveriges Riksbank, SE-103 37 Stockholm, 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.

Keywords: dynamic tail risk; integrated score-driven models; extreme value theory

JEL-codes: C22; G11

Language: English

65 pages, February 1, 2025

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