Bayesian Analysis of Reserving Models and Applications

Abstract

This thesis focuses on developing models for loss reserving in insurance applications. In the first chapter, a Bayesian approach is presented in order to model heavy tail loss reserving data using the generalized beta distribution of the second kind (GB2) with dynamic mean functions and mixture model representation. It is shown through both simulation study and forecasting that model parameters are estimated with high accuracy. Apart from predicting the expected loss in the future, risk margin estimation is another important aspect of loss reserving. We propose to develop quantile functions from regression models to derive risk margin and evaluate capital in non-life insurance applications. Two modeling frameworks are considered based around parametric and nonparametric quantile regression models which we contrast specifically in this insurance setting. Furthermore, we consider 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. In the process we extend the independent log-normal model by incorporating different dependence structures using a Data-Augmented mixture Copula Paid-Incurred claims model. Inference in such models is developed via a class of adaptive Markov chain Monte Carlo (MCMC) sampling algorithms. These incorporate a data-augmentation framework utilised to efficiently evaluate the likelihood for the copula based PIC model in the loss reserving triangles.