Etude des marchés d'assurance non-vie à l'aide d'équilibres de ...

tel-00703797, version 2 - 7 Jun 2012
2.4. Dynamic framework
An interesting feature of these random paths is that a cyclic pattern is observed for the
market individual premium mt, strong correlations of gross written premiums GWPj,t and
loss ratio LRj,t for each insurer j. We fit a basic second-order autoregressive model on market
premium (i.e. Xt − m = a1(Xt−1 − m) + a2(Xt−2 − m) + ɛt) ∗ . Estimation on the serie (mt)t
leads to period of 11.01 and 9.82 years, respectively for Figures 2.1 and 2.2.
Figure 2.1: A random path for NBLN loss model and ˜ fj
Furthermore, this numerical application shows that insurers set premium well above the
pure premium E(Y ) = 1. Thus, the insurer capitals tend to infinite as we observe on the
(bottom right) plot of the solvency coverage ratio. We also do the computation of PLN/NBLN
loss models with the sensitivity function ¯ fj. Similar comments apply, see Appendix 2.6.2.
Some Monte-Carlo estimates
In this subsection, we run the repeated game a certain of times with the NBLN loss model
and the price sensitivity function ˜ fj in order to assess certain indicators by a Monte-Carlo
method. We choose a sample size of 214 ≈ 16000 and a time horizon T = 20. Our indicators
are: (i) the ruin probability of insurer j before time T and (ii) the probability a leader at time
T , where T = T/2 or T .
Results are given in Table 2.10. Estimates of ruin probabilities are extremely low, because
the safety loadings of equilibrium premium are very high, see previous Figures. Leadership
probabilities are more interesting. Recalling that Insurer 1 is the leader at time 0, the probability
for Insurer 1 to be leader after t periods decreases quickly as t increases. After only 20
periods, Insurer 1 has losed its initial advantage.
Then, we look at the underwriting result by policy to see if some insurers underwrite a
deliberate loss. As the first quartile is above zero, we observe that negative underwriting
∗. When a2 < 0 and a 2 1 + 4a2 < 0, the AR(2) is p-periodic with p = 2π arccos
a1
2 √ −a2
.
121