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

tel-00703797, version 2 - 7 Jun 2012
1.4. Incorporating new variables in the regression
choose to insert that variable in the regression via the relative difference compared to the
proposed premium. We consider
mi = marketi − premiumi ,
premiumi where marketi and premium i denote the market premium and the proposed premium for the
ith policy, respectively. In Table 1.9, we give a basic cross-table of lapse and relative market
premium variables. Among the lapsed policies, 65% of them have a higher premium than the
market proxy, whereas for renewed policies it drops to 57%.
m (-0.75,-0.5] (-0.5,-0.25] (-0.25,0] (0,0.25] (0.25,0.5] (0.5,0.75] (0.75,1]
Renew 0.69 18.484 33.248 28.254 9.735 0.571 0.066
Lapse 0.079 1.326 4.158 2.637 0.327 0.032 0.006
Table 1.9: Percentage of policies (%)
However, we cannot conclude that lapses result from a higher premium compared to the
market, just based on this table. In fact, the market proxy is just a proxy for the third-part
liability coverage, computed as the tenth lowest premium. Moreover, the market proxy is a
theoretical premium based on the risk characteristics. If a client goes to another company, it
may have a lower or a higher premium depending if he get a rebate or choose an add-on cover.
As the indemnification procedure also varies between two insurers, the market proxy should
be seen as a probe of the market level rather than a true premium.
Now, that we have described the new explanatory variable, we turn our attention to the
GLM regression. The residual deviance and Akaike Information Criterion (AIC) have slightly
decreased with the addition of the market proxy (8866 to 8728 and 8873 to 8735, respectively).
Regression summary for the GLM with market variable is available on request to the author.
The most instructive results are the average lapse prediction. Comparing the Table 1.10
with Table 1.8 reveals that the addition of the market proxy has a major impact of the delta
lapse rate ∆1+(5%), cf. bolded figures. For the TPL agent subset, it goes from 0.918 to 1.652
pts, while for the TPL direct subset, from 1.490 to 2.738. Central lapse rates before and after
the market proxy inclusion are consistent.
ˆπn(1) ∆1+(5%) ˆπn(1) ∆1+(5%) ˆπn(1) ∆1+(5%)
Agent 7.278 0.482 8.486 0.896 8.548 1.652
Broker 10.987 2.888 9.754 2.776 10.972 3.437
Direct 12.922 1.154 11.303 1.263 11.958 2.738
Full Comp. Part. Comp. TPL
Table 1.10: Central lapse rates (%) and deltas (pts)
The predicted results are plotted on Figure 1.1, where the x-axis represents central lapse
rates (ˆπn(1)), the y-axis delta lapse rates for a 5% premium increase (∆1+(5%)). The bubble
radius are determined by the proportion of the subet in the whole dataset. The text order in
the legends is the decreasing order of bubble radius.
On Figure 1.1, we clearly observe the difference between those two channels both in terms
of central lapse rates and delta lapse rates. These two differences can be explained again by
59