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 F j x(z, ϕ(z)) = 0, it leads to
∂F j x
nj −1
(z, ϕ(z)) = < 0.
∂z n z
Thus, the function βj ↦→ x ⋆ j (βj) is decreasing. In such a case of an inactive constraint, the
premium equilibrium is independent of the initial portfolio size nj.
i.e.
When the solvency constraint is active, the premium equilibrium x ⋆ j verifies g1 j (x⋆ j
) = 0,
x ⋆ j = πj + k995σ(Y ) √ nj − Kj
. (2.9)
nj(1 − ej)
Here, the implicit function theorem is not necessary since x⋆ j does not depend on x⋆−j . We
deduce that x⋆ j is an increasing function of πj, k995, σ(Y ), ej and a decreasing function Kj.
The function nj ↦→ x⋆ j (nj) is not necessarily monotone. Let z be nj. Differentiating
Equation (2.9) with respect to z, we get
ϕ ′ (z) =
1
z3/2
kσ(Y ) Kj
− + √z ,
(1 − ej) 2
whose sign depends on the value of the other parameters.
2.2.7 Numerical illustration
All numerical applications are carried out with the R software, R Core Team (2012). Please
refer to Appendix 2.6.1 for computation details.
Base parameters
We consider a three-player game operating a 10 000-customer insurance market, i.e. n =
10000, I = 3. To ensure that insurers already have underwritten business, we provide d-year
history for portfolio size, where d = 3. Although we provide a 3-year history in this section,
we only consider the one-period equilibrium. So only the value at year 0 matters.
Insurer portfolio size nj(t)’s are given in Table 2.2. The portfolio size is chosen such that
player 1 is the leader, player 2 the challenger and player 3 the outsider with 45%, 32% and
23% market shares, respectively.
time P1 P2 P3
-2 4200 3800 2000
-1 4700 3200 2100
0 4500 3200 2300
Table 2.2: Insurer portfolio sizes
We consider two types of loss model: (i) loss E(Y ) = 1, σ(Y ) = 4.472, Poisson-Lognormal
model, (ii) loss E(Y ) = 1, σ(Y ) = 10, Negative Binomial-Lognormal model. The loss history is
such that the actuarially based premiums āj,0’s are given in Table 2.3 and the market premium
¯m0 is 1.190, 1.299 for PLN and NBLN, respectively.
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