Modelling of Left-Truncated Heavy-Tailed Data with Application to Catastrophe Bond Pricing

In this article, we build on Chernobai et al. [1]'s procedure for modelling left-truncated data via a compound non-homogeneous Poisson process. The contribution we make is that we modify the fitting process introduced so that it is systematically applicable in the context of data that is not only left-truncated, but heavy-tailed as well. Principally, we show how the procedure can be applied when the underlying severities of the process follow a Burr type XII, Generalised Pareto and Generalised Extreme Value distribution 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 the Property Claims Services (PCS) data, and evidence in favour of MPS is found. Within the context of this 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 PCS index