Explicit moments for a class of micro-models in non-life insurance

Abstract

This paper considers properties of the micro-model analysed in Antonio and Plat (2014). The main results are analytical expressions for the moments of the outstanding claims payments subdivided into IBNR claims and individual RBNS claims. These moments are possible to compute explicitly using the discretisation scheme for estimation and simulation used in Antonio and Plat (2014) since the expressions then do not involve any integrals that, typically, would require numerical solutions. Other aspects of the model that are investigated are properties of the maximum likelihood estimators of the model parameters, such as bias and consistency, and a way of computing prediction uncertainty in terms of the mean squared error of prediction that does not require simulations. Moreover, a brief discussion is given on how to compute moments or risk-measures of the claims development result (CDR) using simulations, which based on the results of the present paper can be done without any nested simulations. Based on this it is straightforward to compute the one-year Solvency Capital Requirement, which corresponds to the 99.5% Value-at-Risk of the one-year CDR. A brief numerical illustration is used to show the theoretical performance of the maximum likelihood estimators of the parameters in the claims development process under this model using a realistic set-up based on the case-study of Antonio and Plat (2014). Additionally, the paper ends with a short numerical illustration discussing the model’s robustness under violations of an independence assumption.