Multivariate modelling of multiple guarantees in motor insurance of a household

Abstract

Actuarial risk classification is usually performed at a guarantee and policyholder level: for each policyholder, the claim frequencies corresponding to each guarantee are modelled in isolation, without accounting for the correlation between the different guarantees and the different policyholders from the same household. However, sometimes, a common event will trigger both guarantees at the same time. Moreover, the claim frequencies for policyholders from the same household appear to be correlated. This paper aims to supplement the standard actuarial approach by combining two guarantees and the policyholders from the household, which allows to refine the prediction on the claim frequencies and account for the common shocks on multiple guarantees. Some possible cross-selling opportunities can also be identified.

Appendix

1.1 A priori model

We aim here to give more details on the a priori model that is used to estimate the a priori claim frequencies.

As explained in Sect. 3, three separate models are considered, one for each count variable. The GAMs were chosen to model the claim frequencies, using a Poisson regression with a $\log$ link function. See Wood [21] for an introduction on GAMs. In practice, the mgcv package available in R was used. After careful variable selection, the final models were as follows:

Let us make a few comments on the models. The power variable was not significant in the two first models, probably due to the association between the covered value and the power of the vehicle. However, the covered value did not appear to be significant for the MD:TPL claims, resulting in the power variable to be significant, once the covered value variable was removed. Note that litigation24 corresponds to the variable litigation with the levels 2 and 4 merged. This has been done after a likelihood ratio test suggested both levels were not significantly different in the regression model used for the claim frequency in MD : TPL. Also, note that the functions s(.) mean that a smooth function is estimated. Finally, notice that we included the variable gender, as this modelling concerns the technical claim frequencies (not the commercial ones, for which the usage of the gender variable has been prohibited in the EU).

Due to confidentiality reasons, we cannot display the estimated claim frequencies. We can however show the relative impact of some selected variables. We show on Fig. 7 the impact of the age of the policyholder, which was estimated for both genders. Also, on Fig. 8, the geographic effect is displayed. Moreover, in what relates the categorical variables, in TPL, policyholders with new cars have about 13.6% less claims and professional usage increases by 37% the claim frequency. Litigation also increased the claim frequency.

In MD, new cars, as opposed to TPL, appeared to have a multiplicative effect of 1.59, while the professional usage increased the claim frequency by 8.5%. Again, the litigation variable showed an increase of the claim frequency when the policyholder once failed to pay its premium in due time.

In MD:TPL, policyholders with new cars also appear to have more claims (multiplicative effect of 1.15), as well as professional usage (1.15) and the litigation variable. Finally, cars with higher power have also an increase of the claim frequency (1.23).

Fig. 7

Estimation of the effect of the age on the claim frequency. The effect is shown by gender and by type of claim on the score scale

Fig. 8

Geographic effect (additive on the scale of the score) for each of the three count variables

1.2 Numerical integration using GHQ

We provide hereafter a short sensitivity analysis of the numerical integration using the Gauss-Hermite quadrature. Using the quadrature involves choosing a number of nodes per dimension which are used to compute the integral.

On Table 7, we show the nodes per dimension (remember that for a two policyholders’ household, the contribution to the likelihood involves the computation of a six-dimensional integral), and the obtained maximum likelihood estimators. For the three-dimensional integrals (i.e., for the households with only one policyholder), we fixed the number of nodes per dimension to 10, meaning that the three-dimensional integrals are computed with 1,000 nodes. We observe that there is a rapid convergence. The results highlighted in this paper have been computed with 10 nodes per dimension, regardless of the number of policyholders.

Table 7 Sensitivity analysis on the maximum likelihood estimates with respect to the number of nodes for six-dimensional integrals in the Gauss–Hermite quadrature