Extreme value theory for operational risk in insurance: a case study

Michal Vyskočil and Jiří Koudelka

Abstract

This case study focuses on modeling the real, unique data set of 4245 operational risk claims of an anonymous Central and Eastern European insurance company from 2010 to 2018. We apply extreme value theory to build a more complex model, estimating losses from operational risk events using available historical claims. Low frequency, high-severity claims are identified using the extreme value theory peak-over-threshold method, and modeled via the generalized Pareto distribution, and compared with with Frechet, Weibull and Gumbel distributions. The compound lognormal distribution is used for high-frequency, low-severity claims. Using the bootstrapping principle, many one-year claims portfolio predictions are generated to calculate value-at-risk. The aim of this paper is to examine the sufficiency of the standard formula approach as defined in Solvency II in the case of a company with a specific risk profile. Based on the calculations for the real, unique data set we develop a new approach for estimating the solvency capital requirement for operational risk. Our findings show that the application of extreme value theory to internal data provides a more conservative capital requirement estimation than the standard formula calculation. Thus, the analyzed insurance company may be vulnerable to potential losses when the standard formula is applied.