Bayesian Extreme Value Theory with Hawkes-AR-Gumbel Dependence for Extreme CVaR Estimation in Operational Risk

Abstract: Operational risk capital estimation under Basel II/III requires quantifying aggregate losses at extreme confidence levels of 99.9% and beyond, yet the standard Loss Distribution Approach (LDA) assumes independence between loss frequency and severity, an assumption frequently violated during stress episodes. Furthermore, MLE of tail parameters ignores parameter uncertainty, leading to overconfident risk estimates at extreme quantiles. We propose a Bayesian framework that combines Extreme Value Theory with a dynamic dependence architecture, the Hawkes-AR-Gumbel model, for operational risk Conditional Value-at-Risk (CVaR) estimation at confidence levels up to 99.995%. The model integrates three mechanisms that capture empirically documented features of operational losses: an autoregressive latent stress process that captures persistence of crisis regimes, a Hawkes selfexcitation component for frequency that generates event clustering and overdispersion, and a Gumbel copula for upper-tail dependence that links frequency and severity innovations through an asymmetric copula concentrating dependence in the extreme tail. Inference is performed via Hamiltonian Monte Carlo using PyMC, yielding full posterior distributions for all parameters, and CVaR at arbitrary confidence levels is estimated through posterior predictive Monte Carlo simulation. We compare three models on simulated operational risk data: the independent model (standard LDA), a shared latent factor model with symmetric dependence, and the proposed Hawkes-AR-Gumbel model. The independent model underestimates CVaR at 99.995% by approximately 40%, while the shared factor model fails to capture temporal persistence, event clustering, and upper-tail asymmetry. The HawkesAR-Gumbel model recovers the true dependence structure and correctly estimates CVaR at extreme levels.