Mixed Distributions for Loss Severity Modelling with zeros in the Operational Risk losses

Abstract

Many events have contributed to the recognition of operational risk as a major problem by banks and their regulators. Basel Committee on Banking Supervision (BCBS) adopted two innovative features in its accord of 2004. First, it recognized operational risk as a distinct risk beside credit and market risks. Second, the methodologies proposed to calculate capital charges for the three types of risk become more risk-sensitive. Indeed, the advanced measurement approach aims to permit to banks to take into account their proper exposure to risks.
In developing their proper models to quantify their exposure to operational risk, banks should take into account the fact that there are huge differences between the behaviour of the central part and the tail of the distribution of losses. This is especially true in the case of losses characterized by the so-called low frequency-high severity losses.
In this paper, mixture models with a probability concentration for the zeros losses are fitted to our dataset. We used separately a lognormal distribution and a gamma distribution in the mixture models. Such models capture differently tail behaviour and allow to take into account the features that are important for operational risk modelling; that is: the existence of zeros, positive skewness and heavy tailedness of data. Both fits are done with a data on the damages of physical assets incurred by a Moroccan bank during two years. Results show significant differences when the lognormal or the gamma mixture models are used to evaluate capital at risk or equivalently return period of a given loss and confirm the discussion on distributions tails related to the importance of model selection.