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

tel-00703797, version 2 - 7 Jun 2012
(a) PC agent - price ratio smooth function
s(priceratio,diff2tech,18.04)
2.0
1.5
1.0
0.5
0.0
-0.5
1.5
priceratio
1.0
Figure 1.2: GAM smooth functions
1.6. Other regression models
0.4
0.2
0.0
diff2tech
-0.4
-0.2
(b) FC broker - bivariate smooth function
the price elasticity of the lapse decision is negative. Fortunately, this business inconsistency is
small and located. If we had market variables for this dataset, it could be of interest to check
whether this anomaly vanishes.
Discussion on predictions
As for the GLM analysis, we turn to the analysis of the distribution channel and the
coverage type by looking at the lapse rate predictions. We also consider an average lapse rate
function defined as
ˆπn(p) = 1
n
g
n
−1
⎛
⎝ˆµ + xi(p) T β−p
ˆ + zi(p) T ⎞
p
β+p
ˆ × p + ˆfj(˜zi(p), p) ⎠ , (1.5)
i=1
where (ˆµ, ˆ β−p, ˆ β+p) are the fitted parameters, ˆ fj are the fitted smooth functions, (xi, zi, ˜zi)
are parts of explanatory variables of the ith individual and g is the logit link function. What
differentiates Equation (1.5) with Equation (1.1) is the inclusion of additive terms in the
predictor.
On Figure 1.3, we plot the usual bubble plot to compare GAMs and GLMs. We observe
that GAM delta lapse rate predictions are higher than GLM ones in most cases. This is
especially true for PC agent or FC broker: there is a high jump upward. Only two channelcovers
have a lower delta lapse rate ∆1+(5%) with GAMs: the FC direct case, a case where
the dataset is small (so the GAM model selection was hard) and the FC agent case where the
difference is limited.
In terms of central lapse rates, most of predictions ˆπn(1) are higher, i.e. shift to the right
on Figure 1.3. It means that the customers in the portfolio are more price-sensitive even if we
propose exactly the same premium as last year. On a private motor insurance, most people
expect a better bonus-malus from year to another, hence a premium decrease.
j=1
71