Measuring the Tail Risk: An Asymptotic Approach

The risk exposure of a business line could be perceived in many ways and is sensitive to the exercise that is performed. One way is to understand the effect of some common/reference risk over the performance of the business line in question, but irrespective of the modelling exercise, the exposure is evaluated under the presence of some suitable adverse scenarios. That is, measuring the tail risk is the main aim. We choose to evaluate the performance via an expectation, which is the most acceptable risk measure amongst academics, practitioners and regulators. In contrast to the common practice where the extreme region is chosen such that only the common/reference risk is explicitly allowed to be large, we assume in this paper an extreme region where both the business line in question and common/reference risks are explicitly allowed to be large. The advantage of this tail risk measure is that the asymptotic approximations are meaningful in all cases, especially in the asymptotic independence case, which helps in understanding the risk exposure in any possible setting. Our numerical examples illustrate these findings and provide a discussion about the sensitivity analysis of our approximations, which is a standard way of checking the importance of parameter estimation of the risk model. The numerical analysis shows strong evidence that our proposed tail risk measure has a lower sensitivity than the standard tail risk measure.