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

Transient Maximum Accuracy Criterion 297
8.3 Quadratic Loss Function
8.3.1 Transient Maximum Accuracy Criterion
In this chapter, a policyholder picked at random from the portfolio is now characterized by
three random variables: , and A. As before, is the expected claim frequency derived
from the a priori ratemaking and the residual effect due to the heterogeneity remaining
inside each risk class. The integer-valued random variable A then represents the age of the
policy in the portfolio and will enable us to take the transient behaviour of a bonus-malus
system into account. The probability mass function of A is denoted as
PrA = n = a n n= 1 2
In words, this means that a proportion a n of the policies in the portfolio have been in force
for n years.
We have already seen that the assumption of the independence between and is
reasonable. As for and A, it seems that they are likely to be correlated. Indeed, it seems
probable that the age of the policyholder, which is included in the a priori ratemaking, is
correlated with the age of the policy. However, in order to simplify the computation, we will
further assume that and A are independent. We also assume the independence between
and A.
Recall from Chapter 4 that the sequence of levels occupied by the policyholder in the
bonus-malus scale is denoted as L 0 L 1 L 2 . Here, we denote as L A the level occupied
by a policyholder subject to the bonus-malus system for A years. Let us assume that the
relativity applied to a policyholder picked at random from the portfolio is r A
L A
. For a
policyholder with age of policy n, this relativity becomes r n
L n
. As a result, for each group of
policyholders with age of policy n, the goal is then to minimize the expected squared rating
error
[ ∣ ] [ ]
Q n = E − r A
L A
2 ∣∣A = n = E − r n
L n
2
The solution is given by
= ∑ k
w k
∫ +
0
s∑
− r n
l
2 p n
l kdF (8.3)
l=0
r n
l
= EL A = l A = n
∫ + ∑k w k p n
0 l
=
kdF
∫ + ∑k w k p n
0 l kdF (8.4)
We see that (4.13) is a limiting form of equation (8.4) when n tends to infinity.
The asymptotic criterion Q seems reasonable when a majority of the risks are close to
the steady state. In practice, however, real portfolios will often have a substantial fraction of
comparatively young policies. Then it is desirable to obtain a solution for the relativities not