Threshold-Dependent Dominance in Tail Risk Approximation

Regulatory risk measurement under Basel III’s Fundamental Review of the Trading Book places Expected Shortfall (ES) at the center of market risk capital, yet the fourth-order Edgeworth expansion, still widely used for Value-at-Risk (VaR) and ES calculations, can produce negative densities in the tail regions where these measures concentrate, while saddlepoint approximations preserve positivity but face their own limits in heavy-tailed and sub-Gaussian settings. Whether either method delivers reliable tail estimates in the rare-disaster regimes documented in the empirical consumption-disaster literature therefore remains an open question. We address it by comparing the two approximations across 648 rare-disaster parameter combinations and five additional distributional families (Student- t , Hansen skewed- t , generalised error distribution (GED), two-sided jump mixture, and generalised hyperbolic), and by deriving a closed-form characterisation of the Edgeworth validity envelope. We establish three core findings. First, the validity envelope is bounded above by a sharp kurtosis ceiling at γ 2 = 4 and laterally by a non-monotone skewness boundary peaking at | γ 1 ∗ , max | ≈ 0.685 at γ 2 ≈ 2.533 ; 87.5 % of the rare-disaster grid falls outside it. Second, accuracy is threshold-dependent: Edgeworth dominates at moderate quantiles, saddlepoint at extreme quantiles, with negative-density regions inflating Edgeworth ES error from 6.20 % inside the envelope to 47.04 % outside it. Third, these results reconcile only when point probability, density validity, and integrated-tail accuracy are treated as distinct accuracy criteria. The findings have direct implications for ES-based regulatory capital in heavy-tailed regimes and motivate a regime-conditional rather than universal approximation choice.