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An Extreme Value Approach for Modeling Operational Risk Losses Depending on Covariates
Author(s) -
ChavezDemoulin Valérie,
Embrechts Paul,
Hofert Marius
Publication year - 2016
Publication title -
journal of risk and insurance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.055
H-Index - 63
eISSN - 1539-6975
pISSN - 0022-4367
DOI - 10.1111/jori.12059
Subject(s) - covariate , generalized pareto distribution , statistics , extreme value theory , smoothing , poisson distribution , mathematics , generalized additive model , econometrics , confidence interval , smoothing spline , bilinear interpolation , spline interpolation
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 with a nonhomogeneous Poisson process with 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 with a nonstationary generalized Pareto distribution (alternatively, a generalized extreme value distribution) depending on the covariates. Since spline smoothing cannot directly be applied in this case, an efficient algorithm based on orthogonal parameters is suggested. The methodology is applied both to simulated loss data and a database of operational risk losses collected from public media. Estimates, including confidence intervals, for risk measures such as Value‐at‐Risk as required by the Basel II/III framework are computed. Furthermore, an implementation of the statistical methodology in R is provided.