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

Efficiency and Bonus Hunger 255
5.5 Further Reading and Bibliographic Notes
5.5.1 Modelling Claim Amounts in Related Coverages
In this chapter, only techniques for motor third party liability have been described. Besides
the compulsory motor third party liability insurance, a number of related coverages are
proposed to the drivers (like medical benefits, uninsured or underinsured motorist coverage,
theft and collision and other than collision insurance). The problem caused by the late
settlement of large claims generally disappears when optional coverages are considered. The
annual claim amount S i produced by policyholder i is then represented as
N
∑ i
S i =
C ik
k=1
where N i is the number of claims, and the C ik s are the corresponding claim sizes. The S i s are
assumed to be mutually independent, the C ik s to be independent and identically distributed
for fixed i, and independent of N i .
The analysis of the N i s usually starts with a Poisson regression model. Then, the
residual heterogeneity is taken into account by the inclusion of a random effect. The
impact of the deductibles has to be carefully assessed for optional coverages. In case
deductibles are specified in the insurance policies, the actuary has to keep in mind that the
statistical summaries relate to conditional distributions (given that the claim costs exceed the
corresponding deductibles).
According to the type of coverage, different models can be used for the C ik s. The claim
size is usually expressed as a percentage of the sum insured. For collision insurance, the
C ik s can be decomposed as
C ik = ( J ik + 1 − J ik P ik
)
vi
where v i is the sum insured for policy i (the value of the vehicle fixed according to the rules
contained in the policy); J ik = 1ifthekth claim generates a total loss (that is, a loss that
exhausts the sum insured, i.e. C ik = v i ), and 0 otherwise; and 0