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

tel-00703797, version 2 - 7 Jun 2012
Chapitre 1. Sur la nécessité d’un modèle de marché
continuous variable and d − 1 parameters for a categorical variable with d categories. Cutting
the driver age, for example, into three values ]18, 35], ]35, 60] and ]60, 99] enables to test for
the significance of the different age classes.
The numerical application reveals that a GLM with categorical data is better in terms of
deviance and AIC. Hence, we only report this model in Appendix 1.8.2, first column is the
coefficient estimates ˆµ, ˆ β−p and ˆ β+p. The GLM with continuous variables also has business
inconsistent fitted coefficients, e.g. the coefficient for the price ratio was negative. This also
argues in favor of the GLM with categorized variables. We also analyze (but do not report)
different link functions to compare with the (default) logit link function. But the fit gives
similar estimate for the coefficients ˆµ, ˆ β−p and ˆ β+p, as well as similar predictions.
To test our model, we want to make lapse rate predictions and to compare against observed
lapse rates. From a GLM fit, we get the fitted probabilities ˆπi for 1 ≤ i ≤ n. Plotting those
probabilities against the observed price ratios does not help to understand the link between
a premium increase/decrease and the predicted lapse rate. Recall that we are interested in
deriving a portfolio elasticity based on individual policy features, we choose to use an average
lapse probability function defined as
ˆπn(p) = 1
n
n
i=1
−1
g ˆµ + xi(p) T β−p
ˆ + zi(p) T
β+p
ˆ × p , (1.1)
where (ˆµ, ˆ β−p, ˆ β+p) are the fitted parameters, xi price-noncross explanatory variables, zi pricecross
explanatory variables ∗ and g the logit link function, i.e. g −1 (x) = 1/(1 + e −x ). Note
that this function applies a price ratio constant to all policies. For example, ˆπn(1) the average
lapse rate, called central lapse rate, if the premium remains constant compared to last year
for all our customers.
Computing this sum for different values of price ratio is quite heavy. We could have use a
prediction for a new obsveration (˜x, ˜y, ˜p),
−1
g ˆµ + ˜x T β−p
ˆ + ˜y T
β+p
ˆ × ˜p ,
where the covariate (˜x, ˜y, ˜p) corresponds to the average individual. But in our datasets, the
ideal average individual is not the best representative of the average behavior. Equation (1.1)
has the advantage to really take into account portfolio specificities, as well as the summation
can be done over a subset of the overall data.
In Table 1.4, we put the predicted lapse rates, i.e. ˆπn(1). We also present a measure of
price sensitivity, the delta lapse rate defined as
∆1−(δ) = ˆπn(1 − δ) − ˆπn(1) and ∆1+(δ) = ˆπn(1 + δ) − ˆπn(1), (1.2)
where δ represents a premium change, for example 5%.
As mentioned in the introduction, this measure has many advantages compared to the price
elasticity † (er(p) = dr(p)
dp
× p
r(p) ): it is easier to compute, more robust ‡ , easier to interpret.
∗. Both xi and yi may depend on the price ratio, e.g. if xi represents the difference between the proposed
premium and a technical premium.
†. It is the customer’s sensitivity to price changes relative to their current price. A price elasticity of e
means that an increase by 1% of p increase the lapse rate by e%.
‡. Price elasticity interpretation is based on a serie expansion around the point of computation. So, price
elasticity is not adapted for large δ.
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