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

176 Actuarial Modelling of Claim Counts
4.3.4 Ergodicity and Regular Transition Matrix
A Markov chain with transition matrix P is said to be ergodic if P is regular, that is, if there
exists some n 0 ≥ 1 such that all entries of P n 0 are strictly positive. This condition means
that it is possible, with a strictly positive probability, to go from one level i to another level
j in a finite number of transitions or, in other words, that all states of the Markov chain are
accessible from any initial state in a finite number of steps.
All bonus-malus scales in practical use have a ‘best’ level, with the property that a policy
in that level remains in the same level after a claim-free period. In our framework, the best
level is level 0 and, for any level, it is possible to reach the superbonus level 0 after a
sufficiently large number of claim-free years, resulting in p n
l0
> 0 for all sufficiently
large n. In the following, we restrict our attention to such non-periodic bonus rules. The
transition matrix P associated with such a bonus-malus scale is regular, i.e. there exists
some integer n 0 ≥ 1 such that all entries of the n 0 th power P n 0 of the one-step transition
matrix are strictly positive.
4.4 Long-Term Behaviour of Bonus-Malus Systems
4.4.1 Stationary Distribution
A natural question that arises concerns the long term behaviour of a bonus-malus system.
Intuitively, we expect that the system will stabilize in the long run. Since the annual claim
numbers have been assumed to be independent and identically distributed, each policyholder
will ultimately stabilize around an equilibrium level correponding to the expected annual
claim frequency , and will gravitate around this level.
To formalize this intuitive idea, let us compute the powers of the transition matrix P
for = 01 inthe−1/top and −1/ + 2 bonus-malus scales. This is done in the following
examples.
Example 4.7 (−1/Top Scale)
⎛
P01 =
⎜
⎝
we get
⎛
P 2 01 =
⎜
⎝
Starting from
0904837 0 0 0 0 0095163
0904837 0 0 0 0 0095163
0 0904837 0 0 0 0095163
0 0 0904837 0 0 0095163
0 0 0 0904837 0 0095163
0 0 0 0 0904837 0095163
0818731 0 0 0 0086107 0095163
0818731 0 0 0 0086107 0095163
0818731 0 0 0 0086107 0095163
0 0818731 0 0 0086107 0095163
0 0 0818731 0 0086107 0095163
0 0 0 0818731 0086107 0095163
⎞
⎟
⎠
⎞
⎟
⎠