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
Using Theorem 3.A.23 of Shaked and Shanthikumar (2007), except that for all φ convex,
E(φ(uwj(x, n))) is a decreasing function of n and Nk(x) ≤st Nj(x), we can show
2.4.4 Numerical illustration
UWj = uwj(x, Nj(x)) ≤icx uwk(x, Nk(x)) = UWk.
In this subsection, we present numerical illustrations of the repeated game. As explained at
the beginning of this section, objective and solvency constraint functions depend on parameters
evolving over time: the portfolio size nj,t, the capital Kj,t, the break-even premium πj,t. Doing
this, we want to mimic the real economic actions of insurers on a true market: in private motor
insurance, each year insurers and mutuals update their tariff depending on last year experience
of the whole company and the claim experience of each particular customer through bonusmalus.
In our game, we are only able to catch the first aspect.
In addition to this parameter update, we want to take into account the portfolio size
evolution over time. As nj,t will increase or decrease, the insurer j may become a leader or
lose leadership. Hence, depending on market share (in terms of gross written premium), we
update the lapse, the expense and the sensitivity parameters αj,t, µj,t, ej,t and βj,t, respectively.
Before the game starts, we define three sets of parameters for the leader, the outsider and the
challenger, respectively. At the beginning of each period t, each insurer has its parameter
updated according to its market share GWPj,t−1/GWPt−1. When the market share is above
40% (respectively 25%), insurer j uses the “leader” parameter set (respectively the “outsider”
parameter set), otherwise insurer j uses the “challenger” parameter set. There is only one
parameter not evolving over time: the credibility factor ωj which is set to a common value of
ωj = 9/10 in our numerical experiments.
For the following numerical application, the parameters are summarized in Table 2.9. Lapse
parameters ∗ are identical as in Subsection 2.2.7, but we change the expense and sensitivity
parameters to get realistic outputs.
Random paths
αj µj ej βj
leader -12.143 9.252 0.2 3.465
challenger -9.814 7.306 0.18 4.099
outsider -8.37 6.161 0.16 4.6
Table 2.9: Parameter sets
In order to compare the two different loss models (PLN, NBLN) and the sensitivity functions
¯ fj, ˜ fj, we fix the seed for losses and insured moves. That’s why we observe similar
patterns for the four situations. On Figures 2.1 and 2.2, we plot the individual premium
x ⋆ j,t , the gross written premium GWPj,t, the loss ratio LRj,t and the solvency coverage ratio
Kj,t/SCRj,t.
∗. We give here only the lapse parameter for the price sensitivity function ¯ fj, but there are also three
parameter sets for ˜ fj.
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