A Continuous Granular Model for Stochastic Reserving with Individual Information

Abstract

This paper works on the claims data generated by individual policies which are randomly exposed to a period of continuous time. The main aim is to model the occurrence times of individual claims, as well as their developments given the feature information and exposure periods of individual policies, and thus project the outstanding liabilities. In this paper, we also propose a method to compute the moments of outstanding liabilities in an analytic form. It is significant for a general insurance company to more accurately project outstanding liabilities in risk management. It is well-known that the features of individual policies have effects on the occurrence of claims and their developments and thus the projection of outstanding liabilities. Neglecting the information can unquestionably decrease the prediction accuracy of stochastic reserving, where the accuracy is measured by the mean square error of prediction (MSEP), whose analytic form is computed according to the derived moments of outstanding liabilities. The parameters concerned in the proposed model are estimated based on likelihood and quasi-likelihood and the properties of estimated parameters are further studied. The asymptotic behavior of stochastic reserving is also investigated. The asymptotic distribution of parameter estimators is multivariate normal distribution which is a symmetric distribution and the asymptotic distribution of the deviation of the estimated loss reserving from theoretical loss reserve also follows a normal distribution. The confidence intervals for the parameter estimators and the deviation can be easily obtained through the symmetry of the normal distribution. Some simulations are conducted in order to support the main theoretical results.