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Zero‐modified discrete distributions for operational risk modelling

Younès Mouatassim (Actuarial Research, Zurich Insurance, Casablanca, Morocco)

Journal of Risk Finance

ISSN: 1526-5943

Article publication date: 2 November 2012



The purpose of this paper is to introduce the zero‐modified distributions in the calculation of operational value‐at‐risk.


This kind of distributions is preferred when excess of zeroes is observed. In operational risk, this phenomenon may be due to the scarcity of data, the existence of extreme values and/or the threshold from which banks start to collect losses. In this article, the paper focuses on the analysis of damage to physical assets.


The results show that basic Poisson distribution underestimates the dispersion, and then leads to the underestimation of the capital charge. However, zero‐modified Poisson distributions perform well the frequency. In addition, basic negative binomial and its related zero‐modified distributions, in their turn, offer a good prediction of count events. To choose the distribution that suits better the frequency, the paper uses the Vuong's test. Its results indicate that zero‐modified Poisson distributions, basic negative binomial and its related zero‐modified distributions are equivalent. This conclusion is confirmed by the capital charge calculated since the differences between the six aggregations are not significant except that of basic Poisson distribution.


Recently, the zero‐modified formulations are widely used in many fields because of the low frequency of the events. This article aims to describe the frequency of operational risk using zero‐modified distributions.



Mouatassim, Y. (2012), "Zero‐modified discrete distributions for operational risk modelling", Journal of Risk Finance, Vol. 13 No. 5, pp. 476-490.



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