Modelling Operational Risk Using Extreme Value Theory and Skew t-Copulas via Bayesian Inference

Operational risk losses are heavy tailed and likely to be asymmetric and extremely dependent among business lines and event types. We propose a new methodology to assess, in a multivariate way, the asymmetry and extreme dependence between severity distributions and to calculate the capital for operational risk. This methodology simultaneously uses several parametric distributions and an alternative mix distribution (the lognormal for the body of losses and the generalized Pareto distribution for the tail) via the extreme value theory using SAS®; the multivariate skew t-copula applied for the first time to operational losses; and the Bayesian inference theory to estimate new n-dimensional skew t-copula models via Markov chain Monte Carlo (MCMC) simulation. This paper analyzes a new operational loss data set, SAS® Operational Risk Global Data (SAS OpRisk Global Data), to model operational risk at international financial institutions. All of the severity models are constructed in SAS® 9.2. We implement PROC SEVERITY and PROC NLMIXED and this paper describes this implementation. INTRODUCTION Operational Risk has played a decisive role over the last decade across the bank industry given the significant losses due to operational failures that the financial sector has suffered. Thus, operational risk has become as important as credit risk or market risk. The Basel II accord (2004) allows banks to estimate the regulatory capital that covers their annual operational risk exposure (total Operational Value at Risk OpVaR) using their own models via the advanced measurement approach (AMA). Under the AMA, the loss distribution approach (LDA) has emerged as one of the most convenient statistical methods to calculate OpVaR. However, significant problems have arisen for this standard approach. First Operational losses almost never fit a parametric distribution; the main reason of this is their inherently elusive nature: highfrequency low-severity and low-frequency high-severity (McNeil et al. 2005; Racheddi and Fantazzini 2009). Moscadelli (2004) and De Fontnouvelle et al. (2006) show that the tails of the loss distributions functions are in first approximation heavy-tailed Pareto-type. However, because of the high quantile level requirement (99.9%) for the OpR capital charge, precise modelling of extremely high losses is critical. Second, publicly available operational losses datasets possesses several issues. One of the most significant is that the data is collected from a minimum threshold. In this sense the resulting dataset is incomplete and left-truncated. Fitting an unconditional distribution to the observed (incomplete, truncated data) losses would lead to biased estimates of the parameters of both severity and frequency distributions, as shown in Chernobai et al. (2005a). In brief, several difficulties still remain concerning these advanced modelling techniques. The main points of these recognized problems are as follows: (i) the distribution of such losses has a heavy tail and a major issue is to choose the modelling strategy that accounts for shape and truncation and (ii) the unrealistic and deficient aggregation of event types (ET) or business lines (BL) through a simple conservative sum if the dependence between ET is not identified. To address these problems, this paper attempts to develop a new methodology that integrates the use of extreme value theory (EVT) for modelling the loss severities distribution and skew t-copulas functions for undertaking the dependence structure between ET in ndimensions. This paper presents a complete procedure using SAS® to fill the first gap. All of the models related to the construction of severities are made in SAS® 9.2 and the specific implemented procedures are PROC SEVERITY and PROC NLMIXED.