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

Bonus-Malus Scales 185
4.5.2 Bayesian Relativities
Predictive accuracy is a useful measure of the efficiency of a bonus-malus scale. The idea
behind this notion is as follows: A bonus-malus scale is good at discriminating among the
good and the bad drivers if the premium they pay is close to their ‘true’ premium. According
to Norberg (1976), once the number of classes and the transition rules have been fixed,
the optimal relativity r l associated with level l is determined by maximizing the asymptotic
predictive accuracy.
Let us pick at random a policyholder from the portfolio. Both the a priori expected
claim frequency and relative risk parameter are random in this case. Let us denote as the
(random) a priori expected claim frequency of this randomly selected policyholder, and as
the residual effect of the risk factors not included in the ratemaking. The actual (unknown)
annual expected claim frequency of this policyholder is then . Since the random effect
represents residual effects of hidden covariates, the random variables and may
reasonably be assumed to be mutually independent. Let w k be the weight of the kth risk
class whose annual expected claim frequency is k . Clearly, Pr = k = w k .
Let L be the level occupied by this randomly selected policyholder once the steady state
has been reached. The distribution of L can be written as
PrL = l = ∑ k
∫ +
w k l k dF (4.12)
0
Here, PrL = l represents the proportion of the policyholders in level l.
Our aim is to minimize the expected squared difference between the ‘true’ relative premium
and the relative premium r L applicable to this policyholder (after the steady state has been
reached), i.e. the goal is to minimize
The solution is given by
]
E
[ − r L 2 =
=
s∑
∣ ]
E
[ − r l 2 ∣∣L = l PrL = l
l=0
∫ +
s∑
l=0
= ∑ k
0
w k
∫ +
0
− r l 2 PrL = l = dF
s∑
− r l 2 l k dF
l=0
r l = EL = l
[
]
∣
= E EL = l ∣L = l
= ∑ k
= ∑ k
EL = l = k Pr = k L = l
∫ +
0
PrL = l = = kw k
dF
PrL = l = k Pr = kL= l
PrL = l