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

Transient Maximum Accuracy Criterion 295
8.1.3 Agenda
The transient regime is discussed in Section 8.2, where the convergence of bonus-malus
systems is analysed. The modified criterion with a quadratic loss function is presented in
Section 8.3. The exponential loss function is briefly discussed in Section 8.4.
In Section 8.5, we give the results obtained on the examples studied in the preceding
chapters (former compulsory Belgian bonus-malus scale, −1/top scale and −1/+2 scale)
when using the transient maximum accuracy criterion. All the results have been computed
with the formulas taking the a priori ratemaking into account. We first examine the
convergence to the steady state. Then we give the transient probability distributions and
the transient relativities obtained when using a uniform initial distribution and a uniform
distribution of the age of policy. We also compare the relativities computed using the
transient maximum accuracy criterion to the relativities obtained with the help of the
asymptotic maximum accuracy criterion. Finally, we give the evolution of the expected
financial income.
Many EU insurers recently started to compete on the basis of bonus-malus systems.
Because of marketing and competition for market shares, several insurers now offer the best
level ‘for life’: provided the insured drivers reach level 0, they are allowed to stay in that
level whatever the claims reported to the company. Note however that insurance companies
remain free to cancel the policy after each claim. There is thus a super bonus level: the
driver reaching level 0 of the scale is then allowed to ‘claim for free’. These gifts to the
best drivers are in contradiction to the actuarial and economic purposes of the a posteriori
ratemaking systems. They are nevertheless very efficient from the marketing point of view,
to keep the best drivers in the portfolio. There is thus an absorbing state in the Markov
model describing the trajectory of the driver in the bonus-malus scale. Consequently, the
stationary distribution is degenerated, placing a unit probability mass in level 0. Making
level 0 absorbing for the Markov chain thus forbids the use of the stationary distribution.
This particular case will be considered in Section 8.6.
All the numerical illustrations of this chapter are based on Portfolio A.
8.2 Transient Behaviour and Convergence of Bonus-Malus Scales
A method of computation of the convergence rate based on the eigenvalues of the transition
matrix has been discussed in Chapter 4. Here, we examine the evolution of the total variation
distance between the transient distribution p n
l
l = 0 1s and the stationary
distribution l l = 0 1s:
d TV p n =
s∑
l=0
p n
l
− l n= 0 1 2
for some given expected annual claim frequency . Considering a policyholder, picked at
random in the portfolio, let us denote as L n the level occupied by this policyholder in the
bonus-malus scale after n years, and as L the level occupied once the stationary regime has
been reached. The convergence can thus be assessed with
d TV p n =
s∑
∣ PrL n = l − PrL = l∣
l=0