Provisions for claims outstanding, incurred but not reported, with generalized linear models: prediction error formulated according to calendar year

[EN] In the current context of Solvency II, insurance companies are required to implement demanding business risk management systems. An important aspect of this risk management is the problem of technical provisions in non-life insurance and, as such, it is in the interest of insurers to calculate the prediction error that has occurred when using methodology to estimate a company’s future payments. Furthermore, the predictive distribution of the fitted values, which is descriptive of the risk, allows us to estimate, for example, its Value at Risk at a given confidence level. In this paper we focus on the application of generalized linear models to the amounts of claim losses of a run-off triangle. In order to achieve error distribution, a parameter dependent parametric family is assumed, along with the logarithmic link function. The parametric family has as particular cases the Poisson, the Gamma and the Inverse Gaussian distributions. The particular case which assumes an (over-dispersed) Poisson distribution with the logarithmic link is widely known because it offers the same provision estimation as the deterministic Chain-Ladder method. In this study we develop formulas of the prediction error of future payments by calendar years for the general parametric family. This allows us to perform calculations that consider a financial environment, whether employing analytical formulation or bootstrap estimation. In practice, the presented formulations allow a determination to be made of the present value of the incurred but not reported claim of future payments including a risk margin with statistical significance.