Computation of Operational Value at Risk Using the Severity Distribution Model Based on Bayesian Method with Gibbs Sampler

Abstract

Under Basel III the minimum capital requirement due to operational risk is computed as the 99th quantile of the annual total loss distribution. This annual loss distribution is a result of the convolution between the loss frequency and the loss severity distributions. The estimation of parameters of these two distributions i.e. frequency and severity distributions is not only essential but crucial to obtaining reliable estimates of operational risk measures. In practical applications, Poisson and lognormal distributions are used to fit these two distributions respective. The maximum likelihood method, the method of moments as well as the probability-weighted moments used to obtain the parameters of these distributions can sometimes produce nonsensical estimates due to estimation risk and sample bias. This paper proposes a different calibration of the frequency and the severity distributions based on Bayesian method with Gibbs sampler. Further to that, the paper models the severity distribution by making use of the lognormal and the generalised Pareto distribution simultaneously. Simulated results suggest that computed operational value at risk estimates based of this new method are unbiased with minimum variance.