A Copula Based Bayesian Approach for PaiddIncurred Claims Models for Non-Life Insurance Reserving

Abstract

Our article considers the class of recently developed stochastic models that combine claims payments and incurred losses information into a coherent reserving methodology. In particular, we develop a family of hierarchical Bayesian paid-incurred-claims models, combining the claims reserving models of Hertig and Gogol. In the process we extend the independent log-normal model of Merz and Wuethrich by incorporating different dependence structures using a Data-Augmented mixture Copula paid-incurred claims model. The usefulness of incorporating both payment and incurred losses into estimating of the full predictive distribution of the outstanding loss liabilities and the resulting reserves is demonstrated in the following cases: (i) an independent payment data model; (ii) the independent payment and incurred claims data model of Merz and Wuethrich; (iii) a novel dependent lag-year telescoping block diagonal Gaussian copula payment and incurred claim data model incorporating conjugacy via transformation; (iv) a novel data-augmented mixture Archimedean copula dependent payment and incurred claim data model. Inference in such models is developed by adaptive Markov chain Monte Carlo sampling algorithms. These incorporate a data-augmentation framework utilised to efficiently evaluate the likelihood for the copula based payment and incurred claim model in the loss reserving triangles. The adaptation strategy of the Markov chain Monte Carlo is based on two components. The first component uses an adaptive strategy for learning the posterior structures for the parameters defined over a Euclidean space and the second component deals with an adaptive learning of the posterior for the covariance matrices restricted to the Riemann manifold corresponding to the space of positive definite matrices for the linear dependence structure specified for the payment and incurred claim model.