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
delta lapse rate (pts)
0 1 2 3 4
FC agent
PC agent
FC broker
TPL agent
PC broker
PC direct
TPL broker
FC direct
TPL direct
Customer behaviors (GAM)
2 4 6 8 10 12
central lapse rates (%)
(a) GAM
delta lapse rate (pts)
0 1 2 3 4
FC agent
PC agent
FC broker
TPL agent
PC broker
PC direct
TPL broker
FC direct
TPL direct
Customer behaviors (GLM)
2 4 6 8 10 12
central lapse rates (%)
(b) GLM
Figure 1.3: GAM vs. GLM - comparison of distribution channels and cover types
Now, we stop the GAM analysis and conclude on the pros and cons of GAMs. GAMs
are less known tools than GLMs in actuarial science. But since their introduction in the
90’s, GAMs are well studied and use state-of-the-art fitting procedures. There are two ways
to perform model selections: prediction errors vs. likelihoods. In this paper, we follow the
Wood’s rule to select variables based on the restricted maximum likelihood. We tested other
statistical quantities, but the impact remains limited.
As for GLMs, GAMs allow us to assess an overall estimated price elasticity (via ˆπn(1)
and ∆1+(5%)) taking into account the individual features of each policy. The additional
complexity coming with additive modelling compared to GLMs permit to really fit the data.
Especially for broker lines, we get a more cautious view of customer price sensitivity. For small
datasets, GAM predictions may lead to irrelevant results. Furthermore, as already noticed for
GLMs, GAMs predictions are reliable for with a small range of price change: extrapolating
outside observed price ratio range leads to doubtful results.
Finally, GAMs need a longer time to fit than GLMs and require a better computing power.
This is a limitation for GAMs to be used easily by everyone. In addition, some user judgement
is needed to select, to linearize or to reject explanatory variables in order to get the final model
for GAMs. Even with Wood’s rules, newcomers may find it hard to choose between two GAM
models with the same “score”, i.e. with the same likelihood or prediction errors.
1.6.3 Other regression models
GLMs and GAMs are static models. One option to take into account for dynamics would
be to use time serie models on regression coefficients of GLMs. But this was impossible with
our datasets due to a limited number of years and it is rather a trick than an appropriate
solution. Generalized Linear Mixed Models (GLMM), where the linear predictor becomes the
sum of a (unknown deterministic) fixed term and a random term, are a natural extension of
GLMs to deal with heterogeneity across time.
Among many others, Frees (2004) presents GLMMs in the context of longitudinal and
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