A Copula-Extreme Value Theory Approach for Modelling Operational Risk

The recent disposal of the Basel Committee (Basel Committee on Banking Supervision, 2001) about operational risk recommend the development of advanced modeling techniques in measuring the banking capital requirement. Following the Loss distribution Approach, the distribution of the losses due to operational risk over a specified time horizon – we suppose one year – is explicitly modeled. The model we present in this work is a classical actuarial frequency/severity model. Since the loss data for operational risks are generally collected for business line and event type, the loss distribution and the relative risk measures are first calculated for each business line/event type. The total requirement is then assessed assuming a certain hypothesis about the dependence structure among losses of different business lines/event types. The frequency of a loss event over a one-year time horizon for each business line/event type is assumed to be a Poisson or a negative binomial distribution. In order to model the loss severity distribution we use a distribution lognormal in the centre and in the left tail and Extreme Value Theory (EVT) distributed in the right tail. In this way we may model more efficiently just that portion of the distribution (the right tail) that most affects the assessment of the risk measures. Having estimated the loss distribution for each business line/event type, we assess the total capital requirement for operational risk. Instead of simply summing the capital at risk of each business line/event type, we explicitly model the dependence structure among the losses of different business lines/event types by using copula functions. In this way, the financial institution may save precious regulatory capital. The remainder of this chapter is organized as follows. In the next section, we present the framework of the frequency/severity actuarial model. Then, we give some insight into the concepts of EVT and copula function useful for the development of the model in the following two sections respectively. In the section “An application to catastrophe insurance loss data”, we implement the model to a set of catastrophe insurance loss data. Finally, we draw some concluding remarks in the final section