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

172 Actuarial Modelling of Claim Counts
4.3 Transition Probabilities
4.3.1 Definition
Let us now assume that N 1 N 2 are independent and oi distributed. The trajectory
will be denoted as L 1 L 2 to emphasize the dependence upon the annual
expected claim frequency . Note however that the argument in L k does not mean
that the L k s are functions of the parameter , but only that their distribution depends
on .
Let p l1 l 2
be the probability of moving from level l 1 to level l 2 for a policyholder with
annual mean claim frequency , that is,
p l1 l 2
= PrL k+1 = l 2 L k = l 1
with l 1 l 2 ∈ 0 1s. Clearly, the p l1 l 2
s satisfy
p l1 l 2
≥ 0 for all l 1 and l 2 , and
s∑
p l1 l 2
= 1 (4.1)
l 2 =0
Moreover, the transition probabilities can be expressed using the t ij ·s introduced above.
To see this, it suffices to write
p l1 l 2
=
=
+∑
n=0
∑
n=0
PrL k+1 = l 2 N k+1 = n L k = l 1 PrN k+1 = nL k = l 1
n
n! exp −t l 1 l 2
n
Note that we have used the fact that N k+1 and L k are independent (since L k depends
on N 1 N k ), so that
PrN k+1 = nL k = l 1 = PrN k+1 = n = n
exp −
n!
The transition probabilities allow the actuary to compute the probability of any trajectory
in the scale. Specifically, since the probability that a certain policyholder with expected
annual claim frequency is in level l 1 l n at time 1n is simply the probability
of going from l 0 to l n via the intermediate levels l 1 l n−1 , we have
PrL 1 = l 1 L n = l n L 0 = l 0 = p l0 l 1
···p ln−1 l n
(4.2)
Furthermore, it is enough to know the current position in the scale to determine the probability
of being transferred to any other level in the bonus-malus scale. Formally,
PrL n = l n L n−1 = l n−1 L 0 = l 0 = p ln−1 l n
whenever PrL n−1 = l n−1 L 0 = l 0 >0.