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

tel-00703797, version 2 - 7 Jun 2012
2.3 Refinements of the one-period model
2.3. Refinements of the one-period model
In this section, we propose refinements on the objective and constraint functions of the
previous section.
2.3.1 Objective function
The objective function given in Subsection 2.2.3 is based on an approximation of the true
demand function. For insurer j, the expected portfolio size is given by
(x) +
ˆNj(x) = nj × lg j
j
l=j
nl × lg j
l (x),
where lg l j’s are lapse functions and lg j
j the “renew” function. Note that the expected size ˆ Nj(x)
contains both renewal and new businesses. So, a new objective function could be
Oj(x) = ˆ Nj(x)
n (xj − πj),
where πj is the break-even premium as defined in Subsection 2.2.3. However, we do not consider
this function, since the function xj ↦→ Oj(x) does not verify some generalized convexity
properties, which we will explain in Subsection 2.3.3. And also, the implicit assumption is
that insurer j targets the whole market: this may not be true in most competitive insurance
markets.
Instead, we will test the following objective function
Oj(x) = nj lg j
j (x)
(xj − πj), (2.10)
n
taking into account only renewal business. This function has the good property to be infinitely
in equation (2.1), one can show that the function
differentiable. Using the definition lg j
j
xj ↦→ lg j
j
(x) is a strictly decreasing function, see Appendix 2.6.1. As for the objective function
Oj, maximising Oj is a trade-off between increasing premium for better expected profit and
decreasing premium for better market share.
2.3.2 Constraint function
We also change the solvency constraint function xj ↦→ g 1 j (xj) defined in equation (2.4),
which is a basic linear function of the premium xj. We also integrate other insurer premium
x−j in the new constraint function, i.e. xj ↦→ ˜g 1 j (x). We could use the following constraint
function
k995σ(Y ) ˆNj(x)
˜g 1 j (x) = Kj + ˆ Nj(x)(xj − πj)(1 − ej)
the ratio of the expected capital and the required solvency capital. Unfortunately, this function
does not respect a generalized convexity property, that we will define in the Subsection 2.3.3.
So instead, we consider a simpler version
k995σ(Y ) ˆNj(x)
˜g 1 j (x) = Kj + nj(xj − πj)(1 − ej)
− 1,
− 1, (2.11)
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