Asymptotically unbiased extreme Expected Shortfall and tail risk forecasting in international financial markets

This paper develops an asymptotically unbiased estimator for extreme Expected Shortfall (ES) and evaluates its performance for tail risk forecasting across eight major international financial markets. The proposed estimator builds on a bias-corrected Value-at-Risk (VaR) estimator and the first-order asymptotic relation between ES and VaR, with asymptotic normality established under certain regularity conditions. Monte Carlo simulations show that the new estimator performs well in terms of bias and efficiency relative to standard extreme value theory benchmarks. To validate its practical relevance for international finance, we apply the proposed estimator to forecast extreme ES and compare its out-of-sample performance with various parametric and semi-parametric benchmarks. When combined with an AR-TGARCH filter–denoted as the conditional bias-corrected approach (C-UGH)–our approach exhibits superior out-of-sample predictive performance relative to competing methods, particularly in lower-tail ES forecasting. Notably, the proposed method demonstrates high tolerance for truncation level variations, alleviating the well-documented challenge of optimal truncation level selection in EVT-based approaches. Intriguingly, it outperforms the conditional Weissman (C-W) and conditional Peaks-Over-Threshold (C-POT) models across all eight international financial markets, even when using a randomly selected truncation level. These findings provide useful insights for extreme ES estimation and have practical implications for tail risk management and regulatory capital calculation in global financial markets.