Extreme conditional tail risk inference in ARMA–GARCH models

Abstract

In this study, we investigate the estimation of extreme conditional Value-at-Risk (CVaR) and conditional Expected Shortfall (CES) within the framework of ARMA-GARCH models, where innovations are assumed to follow a Pareto-type tail distribution and have no finite fourth moments. Building on the two-stage self-weighted estimation procedure proposed by He et al. (2022), we develop a robust methodology for forecasting extreme CVaR and CES. Using extreme value theory, we derive a unified asymptotic theory for the extreme CVaR and CES estimators. Through comprehensive simulation studies, we evaluate the performance of our approach and compare it with several recently proposed estimators in the literature. Additionally, we apply our methodology to forecast extreme CVaR and CES for daily negative log-returns (i.e., losses) of four financial assets, demonstrating its practical applicability in financial risk management.