A Multivariate Micro-Level Insurance Counts Model With a Cox Process Approach

When calculating the risk margins of a company with multiple Lines of Business–typically, a quantile in the right tail of an aggregate loss, assumptions about the dependence structure between the different Lines are crucial. Many current multivariate reserving methodologies focus on aggregated claims information, typically in the format of claim triangles. This aggregation is subject to some inefficiencies, such as possibly insufficient data points, and potential elimination of useful information. This inefficiency is particularly problematic for the estimation of dependence. So-called 'micro-level models', on the other hand, utilise more granular levels of observations. Such granular data lend themselves naturally to a stochastic process modelling approach. However, the literature interested in the incorporation of a dependency structure with a micro-level approach is still scarce.In this paper, we extend the literature of micro-level stochastic reserving models to the multivariate context. We develop a multivariate Cox process to model the joint arrival process of insurance claims in multiple Lines of Business. This allows for a dependency structure between the frequencies of claims. We also explicitly incorporate known covariates, such as seasonality patterns and trends, which may explain some of the relationship between two insurance processes (or at least help tease out those relationships). We develop a filtering algorithm to estimate the unobservable stochastic intensities. Model calibration is illustrated using real data from the AUSI data set