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

Credibility Models for Claim Counts 127
Definition 3.1 In the Poisson credibility model, the ith policy of the portfolio, i =
1 2n, is represented by a sequence i N i where i is a positive random variable
with unit mean representing the unexplained heterogeneity. Moreover,
A1 given i = , the random variables N it , t = 1 2, are independent and conform
to the oi it distribution;
A2 at the portfolio level, the sequences i N i , i = 1 2n, are assumed to be
independent.
It is essential to understand the meaning of this classical actuarial construction. In
Definition 3.1, dependence between annual claim numbers is a consequence of the
heterogeneity of the portfolio (i.e. of i ); the dependence is only apparent. If we had a
complete knowledge of policy characteristics then i would become deterministic and there
would be no more dependence between the N it s for fixed i. The unexplained heterogeneity
(which has been modelled through the introduction of the risk parameter i for policyholder
i) is then revealed by the claims and premiums histories in a Bayesian way. These histories
modify the distribution of i and hence modify the premium.
Let
T i
T
∑
∑ i
N i• = N it and i• = it (3.1)
t=1
be the total observed and expected claim numbers for policyholder i during the T i observation
periods; the statistic N i• is a convenient summary of past claims history.
Let us prove that in the credibility model of Definition 3.1, N i• is an exhaustive summary
of the past claims history.
Property 3.1 In the Poisson credibility model of Definition 3.1, the predictive distribution
of i only depends on N i• , i.e. the equality
holds true whatever t ≥ 0.
t=1
Pr i ≤ tN i1 N i2 N iTi = Pr i ≤ tN i•
Proof Let f ·k i1 k iTi be the conditional probability density function of i given
that N i1 = k i1 N iTi = k iTi , and let
T
∑ i
k i• =
k it
t=1
be the total number of accidents reported by policyholder i to the company. We can then
write
f k i1 k iTi = PrN i1 = k i1 N iTi = k iTi i = f
PrN i1 = k i1 N iTi = k iTi
exp−
=
i• k i• f
∫ +
exp−
0 i• k i• f d
which depends only on k i• . This ends the proof.