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

tel-00703797, version 2 - 7 Jun 2012
Chapitre 2. Theorie des jeux et cycles de marché
Figure 2.2: A random path for PLN loss model and ˜ fj
Ruin before Ruin before Leader Leader Leader
t = 10 t = 20 at t = 5 at t = 10 at t = 20
Insurer 1 6.1e-05 6.1e-05 0.593 0.381 0.331
Insurer 2 0 0 0.197 0.308 0.329
Insurer 3 0.000244 0.000244 0.21 0.312 0.34
Table 2.10: Ruin and leadership probabilities
results are rather marginal. In fact, the probability of negative underwriting results are
(0.0352, 0.0378, 0.0358) for Insurers 1, 2 and 3, respectively.
Min. 1st Qu. Median Mean 3rd Qu. Max.
Insurer 1 -0.7905 0.2309 0.3617 0.3563 0.4869 1.2140
Insurer 2 -0.4340 0.2279 0.3600 0.3555 0.4869 1.1490
Insurer 3 -0.4730 0.2308 0.3627 0.3563 0.4871 1.0950
Table 2.11: Summary statistics of underwriting result by policy at t = 20
On the left-hand plot of Figure 2.3, we analyze the (individual) market premium mt. We
plot quantiles at 5%, 50% and 95% as well as two random paths. The two plotted random
paths show a cyclic behavior, whereas the three quantiles are stable in time. On each random
path, we can fit an AR(2) model and estimate the cycle period. Only in 240 random paths, the
fitted AR(2) is not periodic, i.e. a2 ≥ 0 or a 2 1 + 4a2 ≥ 0. Otherwise, the period is computed
as p = 2π arccos
122
a1
2 √ −a2
. On the right-hand plot of Figure 2.3, we plot the histogram of