Rank-based methods for modeling dependence between loss triangles

Abstract

In order to determine the risk capital for their aggregate portfolio, property and casualty insurance companies must fit a multivariate model to the loss triangle data relating to each of their lines of business. As an inadequate choice of dependence structure may have an undesirable effect on reserve estimation, a two-stage inference strategy is proposed in this paper to assist with model selection and validation. Generalized linear models are first fitted to the margins. Standardized residuals from these models are then linked through a copula selected and validated using rank-based methods. The approach is illustrated with data from six lines of business of a large Canadian insurance company for which two hierarchical dependence models are considered, i.e., a fully nested Archimedean copula structure and a copula-based risk aggregation model.

Modeling Dependence Between Loss Triangles With Hierarchical Archimedean Copulas

“…It is interesting to note that, unlike the slightly negative pairwise association obtained by the PWD model in Table 5 and also displayed for these two lines of business in Table 4 of Côté et al (2015), hierarchical models generate positive dependence between loss triangles with the same dataset.…”

“…On the other hand, the Bodily Injury (BI) coverage provides compensation to the insured if he is injured or killed through the fault of a motorist who has no insurance, or by an unidentified vehicle. Côté et al (2015) demonstrate that a gamma distribution provides a good fit for the two lines of business. We work with their conclusion and then continue with the same continuous distribution for each line of business.…”

“…This is a limitation of the joint estimation of the marginal and dependence parameters. This undesirable effect will be addressed in a future work within a two-stage inference strategy; see Côté et al (2015) for more details.…”

“…To better emphasize the features of the hierarchical model, we work with two other runoff triangles that were recently used in Côté et al (2015), which come from a Canadian property-casualty insurer. The two lines of business comprise personal and commercial auto insurance.…”

“…We know that y * ( ) Given that the Gumbel copula generates the best fit for many models in this paper, we have decided to focus on this copula for the bootstrap. Below, the bootstrap study is performed with the datasets of Côté et al (2015).…”