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

Bonus-Malus Scales 173
4.3.2 Transition Matrix
Further, P is the one-step transition matrix, i.e.
⎛
p 00 p 01 ··· p 0s
p 10 p 11 ··· p 1s
P = ⎜
⎝
p s0 p s1 ··· p ss
⎞
⎟
⎠
From (4.1), we see that the matrix P is a stochastic matrix. As already mentioned, the
future level of a policyholder is independent of its past levels and only depends on its present
level (and also on the number of claims reported during the present year).
In matrix form, we can write P as
P =
∑
k=0
k
exp −Tk
k!
provided the N t s are independent and oi distributed.
Example 4.5 (−1/Top Scale)
system is given by
⎛
P =
⎜
⎝
The transition matrix P associated with this bonus-malus
exp− 0 0 0 0 1− exp−
exp− 0 0 0 0 1− exp−
0 exp− 0 0 0 1− exp−
0 0 exp− 0 0 1− exp−
0 0 0 exp− 0 1− exp−
0 0 0 0 exp− 1 − exp−
Example 4.6 (−1/+2 Scale)
system is given by
⎞
⎟
⎠
The transition matrix P associated with this bonus-malus
⎛
exp− 0 exp− 0
2
exp− 1 − ⎞
2 1
⎜ exp− 0 0 exp− 0 1− 2
⎜⎜⎜⎜⎝ 0 exp− 0 0 exp− 1 − P =
3
0 0 exp− 0 0 1− exp−
⎟
0 0 0 exp− 0 1− exp− ⎠
0 0 0 0 exp− 1 − exp−
where i represents the sum of the elements in columns 1 to 5 in row i, i = 1 2 3, that is,
)
1 = exp−
(1 + + 2
2
2 = 3 = exp− 1 +