Estimating the Lognormal-Gamma Model of Operational Risk Using the MCMC Method

Abstract

The lognormal-gamma distribution, being a heavy-tailed distribution, is very attractive from the operational risk modeling perspective because historical operational losses also exhibit heavy tails. Unfortunately, fitting this model requires two severe challenges to be properly addressed. First, the density function of the lognormal-gamma distribution is expressed in the form of a Lebesgue integral. Second, if the information contained in a sample of losses is insufficient to accurately estimate the shape of the distributions tail, the capital estimates become extremely volatile. We address both challenges using the Markov chain Monte Carlo (MCMC) method and imposing prior assumptions about the models unknown parameters. As a result, we were able to reduce statistical uncertainty around capital estimates substantially. Our results also indicate that there is no need to reduce the currently accepted 99.9% quantile level for regulatory capital as suggested elsewhere in the operational risk literature.