A tail-shape actuarial index based on equal level relationships between Value at Risk and Expected Shortfall

Abstract

We introduce a new actuarial tail-shape index, the $\theta$-index, based on a probability equal level relationship between Value at Risk and Expected Shortfall. The index is defined at each tail probability level as the parameter value for which Value at Risk coincides with Flexible Expected Shortfall, that is a convex mixture of Expected Shortfall and the mean. This yields a level-dependent, scale-free measure of upper tail behaviour. We study basic theoretical properties of the $\theta$-index and introduce a partial order for comparing loss distributions, characterized by the monotonicity of right-tail spread ratios. Additionally, the index leads to characterizations of the tail behaviour of a loss distribution as consistent to the generalized Pareto model, through a direct connection to the mean excess function. Moreover, we derive Euler risk contributions for the $\theta$-index and use probability equal level relationships to compute Value at Risk allocations in a more stable way. Finally, the $\theta$-index is examined as a diagnostic tool for distinguishing tail regimes and its capabilities are illustrated using the Danish fire insurance dataset.