An extreme value approach for modeling Operational Risk losses depending on covariates

A general methodology for modeling loss data depending on covariates is developed. The parameters of the frequency and severity distributions of the losses may depend on covariates. The loss frequency over time is modeled via a non-homogeneous Poisson process with integrated rate function depending on the covariates. This corresponds to a generalized additive model which can be estimated with spline smoothing via penalized maximum likelihood estimation. The loss severity over time is modeled via a nonstationary generalized Pareto model depending on the covariates. Whereas spline smoothing can not be directly applied in this case, an efficient algorithm based on orthogonal parameters is suggested. The methodology is applied to a database of (mostly banking) operational risk losses. Estimates, including confidence intervals, for risk measures such as Value-at-Risk or Expected-Shortfall as required by the Basel II/III framework are computed. We provide links to a detailed R implementation of the statistical methodology.