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

Credibility Models for Claim Counts 161
Multivariate credibility models may be considered for several lines of business, or several
types of claims. Multivariate credibility models are discussed, e.g., in Frees (2003). This
topic will be treated in Chapter 6.
3.6.6 Time Dependent Random Effects
The vast majority of the papers which have appeared in the actuarial literature considered
time-independent heterogeneous models. This chapter is restricted to the case of static random
effects: in the classical credibility construction A1–A2 of Definition 3.1, the risk parameter
i relating to policyholder i is assumed to be constant over time. This is of course rather
unrealistic since driving ability may vary during the driving career (because of the learning
effect, or modification in the risk characteristics). In automobile insurance, an unknown
underlying random parameter that develops over time expresses the fact that the abilities
of a driver are not constant. Moreover, the hidden exogeneous variables revealed by claims
experience may vary with time, as do observable ones.
Another reason to allow for random effects that vary with time relates to moral
hazard. Indeed, individual efforts to prevent accidents are unobserved and feature temporal
dependence. The policyholders may adjust their efforts for loss prevention according to their
experience with past claims, the amount of premium and awareness of future consequences
of an accident (due to experience rating schemes). The effort variable determines the moral
hazard and is modelled by a dynamic unobserved factor.
Of course, it is hopeless in practice to discriminate between residual heterogeneity due to
unobservable characteristics of drivers that significantly affect the risk of accident, and their
individual efforts to prevent such accidents. Both effects get mixed in the latent process.
Since the observed contagion between annual claim numbers is always positive, the effect
of omitted explanatory variables seems to dominate moral hazard. Anyway, this issue has no
practical implication since predictions depend on observed contagion, but not on its nature.
Hence, instead of assuming that the risk characteristics are given once and for all by a
single risk parameter, we might suppose that the unknown risk characteristics of each policy
are described by dynamic random effects. In the terminology of Jewell (1975), evolutionary
credibility models allow for random effects to vary in successive periods. Now, the ith
policy of the portfolio, i = 1 2n, is represented by a double sequence i N i where
i is a positive random vector with unit mean representing the unexplained heterogeneity.
Specifically, the model is based on the following assumptions:
B1
Given i = i , the random variables N it , t = 1 2T i , are independent and conform
to the Poisson distribution with mean it it , i.e.
PrN it = k it = it = exp− it it it it k
k!
k= 0 1
B2
with it = d it exp T˜x it .
At the portfolio level, the i s are assumed to be independent. Moreover, i1 iTi
is distributed as 1 Ti
where = 1 Tmax
T is a stationary random
vector (with T max = max T i ). It is further assumed that E it = 1 for all i, t. The unit
mean condition is imposed for identification (otherwise, the mean could be absorbed