Modelling of left-truncated heavy-tailed data with application to catastrophe bond pricing

In this article, we concentrate on modelling heavy-tailed data which can be subjected to left-truncation. We modify an existing procedure for modelling left-truncated data via a compound non-homogeneous Poisson process to make it systematically applicable in the context heavy-tailed data. The introduced procedure can be applied when the underlying severities of the process follow Burr type XII, Generalised Pareto and Generalised Extreme Value distributions by using the Maximum Product of Spacings (MPS) parameter estimation technique. As a natural consequence of the MPS technique, we consider how Moran’s log spacings statistic for testing goodness-of-fit of the severity distributions can be adapted to suit left-truncated data. Thereafter, we compare the performance of this new fitting procedure against traditional maximum likelihood estimation in the context of natural catastrophe loss data, and evidence in favour of MPS is found. Within the context of these data, we also compare our procedure to a one that does not account for left-truncation. We end our contribution by proposing, for our modelling procedure, a Monte Carlo importance sampling algorithm which ensures that large losses are satisfactorily simulated. In closing, we illustrate the potential usage of both the new fitting and simulation procedures by presenting catastrophe bond prices with a trigger based on the analysed heavy-tailed data.