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

274 Actuarial Modelling of Claim Counts
6.3.3 Determination of the relativities
The relativity associated with level l is denoted as r l , as before. The meaning is that an
insured occupying that level pays an amount of premium equal to r l % of the reference
premium determined on the basis of his observable characteristics.
As in Chapter 4, 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 stationary state has been reached), i.e. the goal is to minimize
The solution is given by
]
E
[ − r L 2 =
s∑
l=0
= ∑ k
∣ ]
E
[ − r l 2 ∣∣L = l PrL = l
w k
∫ +
0
s∑
− r l 2 l k q k dF
l=0
∫ + ∑k w k
0 l k q k dF
r l = EL = l = ∫ + ∑k w k
0 l k q k dF (6.9)
It is easily seen that Er L = 1, resulting in financial equilibrium once steady state is reached.
6.3.4 Numerical Illustrations
− 1/ + 2/ + 3 Bonus-Malus Scale
Let us now consider the scale with six levels (numbered from 0 to 5) already used in the
previous chapters. But now, instead of considering one type of claim, we penalize differently
claims with bodily injuries and claims with material damage only. If no claims have been
reported then the policyholder moves one level down. Claims with material damage only
are penalized by two levels whereas claims with bodily injuries entail a penalty of three
levels. If n 1 claims with bodily injuries and n 2 claims with material damage only are reported
during the year then the policyholder moves 3n 1 + 2n 2 levels up. This system is abbreviated
as −1/ + 2/ + 3, in obvious notations.
The transition matrix for a policyholder with annual mean claim frequency and vector
probability q = q 1 q 2 T is given by
⎛
⎞
P 0 0 P 1 P 2 P 3 1 − 1
P 0 0 0 P 1 P 2 1 − 2
P q =
0 P 0 0 0 P 1 1 − 3
⎜ 0 0 P 0 0 0 1− 4
⎟
⎝ 0 0 0 P 0 0 1− 4
⎠
0 0 0 0 P 0 1 − 4
where
P 0 = exp−
P 1 = q 2 exp−q 2 exp−q 1