Predicting Future Claims Among High Risk Policyholders Using Random Effects

Abstract

Insurance claims are often modelled by a standard Poisson model with fixed effects. With such a model, no individual adjustments are made to account for unobserved heterogeneity between policyholders. A Poisson model with random effects makes it possible to detect policyholders with a high or low individual risk. The premium can then be adjusted accordingly. Others have applied such models without much focus on the model’s prediction performance. As the usefulness of an insurance claims model typically is measured by its ability to predict future claims, we have chosen to focus on this aspect of the model. We model insurance claims with a Poisson random effects model and compare its performance with the standard Poisson fixed effects model. We show that the random effects model both fits the data better and gives better predictions for future claims for high risk policy holders than the standard model.

Appendix

We will prove the relations in Eq.  11.9 between the observed data and the fitted values from the mixed effects model. By Eq.  11.7 the loglikelihood becomes

$$\begin{aligned} l=\sum _{i=1}^N \left[ \sum _{t=1}^{t_i} \left\{ y_{\textit{it}}\log \lambda _{\textit{it}}^{\text{ RE }} + y_{\textit{it}}\log e_{\textit{it}} -\log (y_{\textit{it}} !)\right\} +\log \left\{ (-1)^{y_{i,\text{ tot }}}L^{(y_{i,\text{ tot }})}(\Lambda _i^{\text{ RE }})\right\} \right] . \end{aligned}$$

Now $\lambda _{\textit{it}}^{\text{ RE }} =\mathrm{{exp}}\left\{ \beta _0^{\text{ RE }}+{\varvec{x}}_{\textit{it}}^\textit{T} \, {\beta }^{\text{ RE }}\right\}$ and $\Lambda _i^{\text{ RE }}=\sum _{t=1}^{t_i}e_{\textit{it}}\lambda _{\textit{it}}^{\text{ RE }}$. Using this and Eq.  11.5, we find that score functions for the $\beta$’s take the form

$$\begin{aligned} \nonumber \frac{\partial l}{\partial \beta _0^{\text{ RE }}}&= \sum _{i=1}^N \left[ \sum _{t=1}^{t_i} y_{\textit{it}} + \frac{L^{(y_{i,\text{ tot }}+1)}(\Lambda _i^{\text{ RE }})}{L^{(y_{i,\text{ tot }})}(\Lambda _i^{\text{ RE }})} \frac{\partial \Lambda _i^{\text{ RE }}}{\partial \beta _0^{\text{ RE }}} \right] \\[0.2cm]&= \sum _{i=1}^N \sum _{t=1}^{t_i} y_{\textit{it}} - \sum _{i=1}^N \sum _{t=1}^{t_i} \hat{\theta }_i e_{\textit{it}}\lambda _{\textit{it}}^{\text{ RE }} \nonumber \end{aligned}$$

and

$$\begin{aligned} \nonumber \frac{\partial l}{\partial {\beta }^{\text{ RE }}}&= \sum _{i=1}^N \left[ \sum _{t=1}^{t_i} y_{\textit{it}}{\varvec{x}}_{\textit{it}} + \frac{L^{(y_{i,\text{ tot }}+1)}(\Lambda _i^{\text{ RE }})}{L^{(y_{i,\text{ tot }})}(\Lambda _i^{\text{ RE }})} \frac{\partial \Lambda _i^{\text{ RE }}}{\partial {\beta }^{\text{ RE }}} \right] \\[0.2cm]&= \sum _{i=1}^N \sum _{t=1}^{t_i} y_{\textit{it}} {\varvec{x}}_{\textit{it}}- \sum _{i=1}^N \sum _{t=1}^{t_i} \hat{\theta }_i e_{\textit{it}}\lambda _{\textit{it}}^{\text{ RE }}{\varvec{x}}_{\textit{it}}. \nonumber \end{aligned}$$

The maximum likelihood estimates are obtained by setting the score functions equal to zero. From this and Eq. 11.8 the relations in Eq. 11.9 follow.