Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

Risk Classification 81
The unobserved heterogeneity (when correlated to observable characteristics) thus modifies
the regression coefficients: the true effect of X i on N i becomes an apparent effect ˜.
Hence, the estimated jth regression coefficient does not only represent the effect of the
jth covariate on the number of claims, but also accounts for the effect of all the hidden
characteristics Z i correlated with the jth observable one. This is why the ̂ j s may strongly
depend on which covariates are included in the model. Moreover, there remains an error
term ˜ i representing the influence of the hidden variables on N i , corrected for the effect of
the observed risk factors X i .
For these reasons, we now consider a mixed Poisson model
( (
))
p∑
N i ∼ oi exp 0 + j x ij + i i= 1 2n (2.10)
j=1
where the random variable i represents the residual effect of the hidden characteristics.
Therefore, the heterogeneity is taken into account by assuming that the number of accidents
is Poisson distributed with mean varying from one policyholder to another.
Note that some hidden characteristics are correlated to those in X i (e.g. the hidden annual
mileage and the observable use of the vehicle). The random variable i in (2.10) models the
effect of hidden characteristics that is not already explained by X i . Since i accounts for a
residual effect, we will consider in the remainder of this book that i is independent from
X i . The price to pay is that the estimated regression coefficient j does not only express the
effect of the jth regressor, but also the effect of all the hidden characteristics correlated with
the jth regressor. This is important when the actuary tries to interpret the resulting price list.
The policyholders have different accident proneness because of observable characteristics
taken into account in the price list and hidden characteristics to be corrected a posteriori.
The heterogeneity is taken into account by assuming that the number of accidents is Poisson
distributed with mean varying from one policyholder to another. The annual claim frequency
becomes a random variable i i where i = exp i models the oscillations around the grand
mean i (with E i = 1). We can now write N i ∼ Poi i i where i = d i exp˜x T i .
As in (1.29), we have
VN i = i + 2 i V i> i = EN i (2.11)
so that any mixed Poisson regression model induces overdispersion.
2.4.5 Detecting Overdispersion
Residual heterogeneity remains considerable in the risk classes despite the use of many a
priori variables. Indeed, many explanatory variables are unknown to the insurance company
and cannot be incorporated in the price list. Let us denote as ̂m k the empirical mean claim
number of risk class k, and ̂
k
2 the associated variance. In order to graphically test the
Poisson mixture assumption, we have plotted the points ̂m k ̂
k 2 and also the bisecting line.
The plot is displayed in Figure 2.9. We observe that ̂m k < ̂
k
2 in numerous risk classes,
indicating that the homogeneous Poisson model is inappropriate. We also see that most of
the observed pairs ̂m k ̂
k 2 lie above the bisecting line, thus supporting overdispersion. The
three points below the 45-degree line correspond to classes with just a few policies (36, 13
and 12, respectively).