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

tel-00703797, version 2 - 7 Jun 2012
P1 P2 P3
PLN 1.066 1.159 0.972
NBLN 1.079 1.189 1.035
Table 2.3: Actuarially based premium āj,0
2.2. A one-period model
The weight parameters (ωj)j used in the computation of the insurer break-even premium
are ω = (1/3, 1/3, 1/3). Before giving the sensitivity parameters βj’s, we present the lapse
models. For customer behavior, we have two parameters µj, αj per player given a price
sensitivity function. At first, we consider the price function based on the premium ratio
¯fj(xj, xl) = µj + αj
The central lapse rate parameters (i.e. lapse rate when every insurers use the same premium)
are set to 10%, 14% and 18% for j = 1, 2 or 3, respectively. In addition to this first constraint,
we also impose that on increase of 5% compared to other players increases the total lapse rate
by 5 percentage points.
Let x 1 = (1, 1, 1) and x 1.05 = (1.05, 1, 1). The two constraints are equivalent to
lg 2 1(x 1 ) + lg 3 1(x 1 ) = 10% and lg 2 1(x 1.05 ) + lg 3 1(x 1.05 ) = 15%
for Insurer 1. We get µ1 = −12.14284 and α1 = 9.25247. With this central lapse rate
parameters, the expected numbers of lost policies when all insurers propose the same premium
are 450.1, 448.0 and 414.0.
Secondly, we consider the price function based on the premium difference
xj
xl
˜fj(xj, xl) = ˜µj + ˜αj(xj − xl).
Calibration is done similarly as for fj. In Figure 2.5 in Appendix 2.6.1, we plot the total lapse
ratio function of each player for the two different price function fj (left graph) and ˜ fj (right
graph). In a grey dot-dash horizontal line, we highlight the central rates at 10%, 14% and
18% (the premium of other players is set to 1.4). In the central graph, we plot the total lapse
rate function of player 1 with the two different price functions.
Price sensitivity parameters βj
of objective functions are fitted in the following way 1 −
xj
βj mj(x) − 1 ≈ lg j
j (x). Using x = (1.05, 1, 1), we get
1 − lgj j
βj = (x)
.
0.05
Using the premium ratio function ¯ fj, we have (β1, β2, β3) = (3.0, 3.8, 4.6).
Last parameters are capital values and the expense rates. Capital values (K1, K2, K3)
are set such that the solvency coverage ratio is 133%. Expense rates are (e1, e2, e3) =
(10%, 10%, 10%).
Results and sensitivity analysis
Since we consider two loss models (PLN, NBLN) and two price sensitivity functions ¯ fj, ˜ fj
(denoted by ‘ratio’ and ‘diff’, respectively), we implicitly define four sets of parameters, which
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