Expected Shortfall Estimation for Apparently Infinite-Mean Models of Operational Risk
Quantitative Finance, Volume 16, 2016 - Issue 10
23 Pages Posted: 27 Oct 2015 Last revised: 12 Jan 2017
Date Written: October 27, 2015
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 (ES), 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.
Keywords: Operational risk, Expected Shortfall, Infinite Mean, VaR
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