# CHAPTERTWO

# Fundamentals of Extreme Value Theory for OpRisk

## 2.1 Introduction

In practical scenarios, it is standard practice to consider basic two-parameter models in the operational risk (OpRisk) modeling of a single-loss process severity distribution under a loss distribution approach (LDA) approach. The most common of these models is based around a LogNormal distribution. Part of the reason for this is the inherent simplicity with which the estimation of the model parameters in this model can be performed. However, because of the extreme quantile requirements that are required to be obtained in regulatory reporting of capital under the Basel II and Basel III accords, the use of models such as LogNormal may not adequately capture the tail features of the loss process under study at such high quantile levels. Therefore, the intention of this section involves motivation of extreme value theory (EVT) concepts to study and understand such extreme loss behaviour.

In this regard, we also note that the concept of what constitutes a heavy-tailed distribution for severity modeling in OpRisk can be defined in several different ways from a statistical perspective. It is common to consider the probabilistic definition of a heavy-tailed distribution as those probability distributions whose tails are not exponentially bounded. That is, they have heavier tails than the exponential distribution. In many applications, it is the right tail of the distribution that is of interest, but a distribution ...