Expected shortfall estimation for apparently infinite-mean models of operational risk

Statistical analyses on actual data depict operational risk as an extremely heavy-tailed phenomenon, able to generate losses so extreme as to suggest the use of infinite-mean models. But no loss can actually destroy more than the entire value of a bank or of a company, and this upper bound should be considered when dealing with tail-risk assessment. Introducing what we call the dual distribution, we show how to deal with heavy-tailed phenomena with a remote yet finite upper bound. We provide methods to compute relevant tail quantities such as the Expected Shortfall, which is not available under infinite-mean models, allowing adequate provisioning and capital allocation. This also permits a measurement of fragility. The main difference between our approach and a simple truncation is in the smoothness of the transformation between the original and the dual distribution. Our methodology is useful with apparently infinite-mean phenomena, as in the case of operational risk, but it can be applied in all those situations involving extreme fat tails and bounded support.