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

132 Actuarial Modelling of Claim Counts
Let us briefly comment on this a posteriori correction:
• We see that the a posteriori corrections will be more severe as the residual heterogeneity,
measured by V i = 1/a, increases.
• Considering two policyholders (numbered i 1 and i 2 ) in the portfolio during T i1
= T i2
periods
such that i 1 is a priori a better driver than i 2 , that is, i1 • < i2 •: if these policyholders
do not report any claim (i.e., N i1 • = N i2 • = 0) then the corrections to be applied to these
policyholders satisfy
a a
>
a + i1 • a + i2 •
so that the a priori worse driver receives more discount.
• If these policyholders report k ≥ 1 claims (i.e., N i1 • = N i2 • = k) then the penalties are such
that
a + k
a + i1 •
> a + k
a + i2 •
Hence, the penalty for the a priori bad driver is less severe than for the good one.
3.3.5 Predictive Distribution and Bayesian Credibility Premium
From (3.4) we know that the posterior distribution of i given past claims history is still
Gamma, with parameters a+N i• and a+ i• . Therefore, the predictive distribution of N iTi +1
is Negative Binomial, that is,
( ) ( ) k (
) a+k•
a + k• + k − 1 iTi +1
a +
PrN iTi +1 = kN i• = k • =
i•
k a + i• + iTi +1 a + i• + iTi +1
Furthermore, the Bayesian credibility premium is given by
a + k
EN iTi +1N i• = k • = iTi +1E i N i• = k • = •
iTi +1
a + i•
Remark 3.1 As time goes on, the Bayesian credibility premium tends to 0 for a policyholder
reporting no claim. This can be seen as an unrealistic feature. An easy way to avoid this
problem is to decompose N i into two parts: a component N 1
i distributed according to a pure
Poisson distribution that represents the claims occurring purely at random, and a component
N 2
i that is Negative Binomial and influenced by the driver’s abilities. This introduces a
lower bound on the a posteriori claim frequency, which is no more allowed to vanish in the
long term.
It is worth mentioning that the Bayesian credibility premium can be cast into
( a
EN iTi +1N i• = k • = E
a + i +
)
i• k •
EN
i• a + i• iTi +1
i•