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
differ on loss model and price sensitivity functions. In Table 2.4, we report premium equilibria
of the four models (PLN-ratio, PLN-diff, NBLN-ratio and NBLN-diff), differences between
equilibrium vector x ⋆ and actuarial and average market premium, and expected difference in
portfolio size (∆ ˆ N1 negative means insurer 1 expects to lose customers).
x ⋆ 1 x ⋆ 2 x ⋆ 3 ||x ⋆ − ā|| 2 ||x ⋆ − ¯m|| 2 ∆ ˆ N1 ∆ ˆ N2 ∆ ˆ N3
PLN-ratio 1.612 1.583 1.531 11.064 13.199 -258.510 -43.479 301.989
PLN-diff 1.659 1.621 1.566 11.191 13.297 -382.879 -38.401 421.280
NBLN-ratio 1.727 1.697 1.648 11.994 15.645 -239.465 -35.301 274.766
NBLN-diff 1.777 1.738 1.685 12.152 15.752 -385.404 -29.645 415.049
Table 2.4: Base premium equilibrium
The premium equilibrium vector x ⋆ is quite similar between the four different tested models.
The change between price sensitivity functions ¯ fj, ˜ fj from an insurer point of view is a
change in sensitivity parameter βj in its objective function. The change between ¯ fj, ˜ fj results
in a slight increase of premium equilibrium whereas the change between PLN or NBLN loss
models is significantly higher. Unlike the sensitivity function change, a change in loss models
does not impact the objective function but the constraint function (an increase in σ(Y )).
In Tables 2.5, 2.6, we perform a sensitivity analysis considering the NBLN-ratio model
as the base model. Table 2.5 reports the analysis with respect to capital (K3 decreases) and
sensitvity parameter (βj increases). Table 2.6 focuses on actuarially based premiums (āj,0
increases), average market premium ( ¯m0 increases) and credibility factors (ωj increases).
x ⋆ 1 x ⋆ 2 x ⋆ 3 ||x ⋆ − ā|| 2 ||x ⋆ − ¯m|| 2 ∆ ˆ N1 ∆ ˆ N2 ∆ ˆ N3
base 1.727 1.697 1.648 11.994 15.645 -239.465 -35.301 274.766
capital down 1.797 1.764 1.79 12.185 15.678 -96.943 87.126 9.817
x ⋆ 1 x ⋆ 2 x ⋆ 3 ||x ⋆ − ā|| 2 ||x ⋆ − ¯m|| 2 ∆ ˆ N1 ∆ ˆ N2 ∆ ˆ N3
base 1.727 1.697 1.648 11.994 15.645 -239.465 -35.301 274.766
sensitivity up 1.643 1.62 1.575 11.736 15.479 -207.466 -42.836 250.302
Table 2.5: Base premium equilibrium
x ⋆ 1 x ⋆ 2 x ⋆ 3 ||x ⋆ − ā|| 2 ||x ⋆ − ¯m|| 2 ∆ ˆ N1 ∆ ˆ N2 ∆ ˆ N3
base 1.727 1.697 1.648 11.994 15.645 -239.465 -35.301 274.766
actuarial up 1.766 1.714 1.665 13.015 15.706 -325.31 4.752 320.558
market up 1.91 1.874 1.822 12.72 20.503 -240.803 -29.484 270.287
x ⋆ 1 x ⋆ 2 x ⋆ 3 ||x ⋆ − ā|| 2 ||x ⋆ − ¯m|| 2 ∆ ˆ N1 ∆ ˆ N2 ∆ ˆ N3
base 1.727 1.697 1.648 11.994 15.645 -239.465 -35.301 274.766
credibility up 1.68 1.657 1.599 11.839 15.545 -232.155 -68.27 300.425
106
Table 2.6: Base premium equilibrium