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

tel-00703797, version 2 - 7 Jun 2012
estimated periods: average period is around 10.
market premium m_t
1.3 1.4 1.5 1.6 1.7 1.8
market premium
0 5 10 15 20
time t
Q5%
Q50%
Q95%
path 1
path 2
Frequency
0 1000 2000 3000 4000 5000
Figure 2.3: Market premium
Histogram of cycle periods
Period (years)
2.5. Conclusion
6 8 10 12 14 16 18 20
Finally, on a short time horizon, T = 5, we want to assess the impact of initial portfolio
size nj,0 and capital value Kj,0 on the probability to be leader at time T . We consider Insurer
1 as the leader and Insurers 2 and 3 as identical competitors. We take different values of
K1,0 and n1,0 for Insurer 1 and deduce capital values and portfolio sizes for Insurers 2 and 3
as K2,0 = K3,0 = k0σ(Y ) √ n2,0 and n2,0 = n3,0 = (n − n1,0)/2, where k0 is a fixed solvency
coverage ratio. The sensitivity analysis consists in increasing both market shares and capital
values of Insurer 1 while other competitors have a decreasing market share and a constant
coverage ratio.
We look at the probability for insurer i to be a leader in terms of gross written premium
at period T = 5, i.e.
pi = P (∀k = i, GWPi,T > GWPk,T |Ni,0 = ni, Ki,0 = ki).
We test two loss models NBLN and PGLN, for which the margingal claim distribution is a
compound negative binomial distribution with lognormal distribution, but for PGLN, the loss
frequency among insurers is comonotonic, see Subsection 2.2.2.
On Figures 2.4, we observe the probability to a leader after five periods is an increasing
function of the initial market share. The initial capital does not seem to have any influence,
which can be explained by the high profit per policy. As one could expect, the comonotonic
loss model (Figure 2.4b) is favorable to Insurer 1 than the independent case (Figure 2.4a).
2.5 Conclusion
This paper assesses the suitability of noncooperative game theory for insurance market
modelling. The game-theoretic approach proposed in this paper gives first answers of the
effect of competition on the insurer solvency whose a significant part is linked to the ability
of insurers to sell contracts. The proposed game models a rational behavior of insurers in
setting premium taken into account other insurers. The ability of an insurer to sell contracts
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