We propose a sensitivity testing framework suitable for insurance losses that follow heavy tailed distributions, including the family of generalised Pareto distributions. Starting from a baseline probability measure, e.g., a parametric model estimated from data, we solve the problem of finding the perturbed probability measure that exceeds a given risk tolerance and has smallest Rényi divergence to the baseline measure. We study the optimisation problem with risk tolerances given by the Value-at-Risk, Expected Shortfall, and expectiles, and prove that the perturbed probability measures exist and are uniqueness. We provide semi-analytical solutions and characterise the perturbed measures via the so-called lambda-exponential functions.

Our findings are illustrated on a Canadian insurance loss dataset stemming from natural catastrophes.