dealing with continuous variables and geographical information in non life insurance ratemaking...
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Dealing with continuous variables and geographical information in non life
insurance ratemaking
Maxime Clijsters
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Introduction
Tariff ?
Professional use (Y/N)
Postal code
Age of the permit
Kilowatt of the vehicle
Age of the vehicle
Vehicle type(4x4 Y/N)
Policyholder’s Age
Categorical variableContinuous variableMulti-Level Factor
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• GLMs remain a very important statistical regression technique for pricing car insurance products
• GAMs provide interesting insights in the underlying dependency structure, but come at a high computational cost
• GAM as a complementary modelling tool
Introduction
GLM = Generalized Linear ModelGAM = Generalized Additive Model
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AGENDA
• Binning continuous variables– GAM to explore nonlinear effects– GAM and regression trees for binning
• Modelling geographical information
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• GLM is satisfying modelling tool• Industry-wide standard
• Only categorical variables
• Continuous variables
• High computational cost• No parametric functional form
Binning continuous variables
GLM
GAM
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Binning continuous variablesGAM to explore nonlinear effects
• We fit a GAM for a continuous variable , with the observed number of claims a Poisson distributed random variable
• The GAM estimate:
with the exposure corresponding to policyholder and the nonparametric GAM estimate
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Binning continuous variablesGAM to explore nonlinear effects
(a) Nonparametric prediction
(b) Total prediction
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3 +∑𝑘=1
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�̂�𝑘 (𝑥 𝑖−𝑥𝑘 )+¿3¿
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Binning continuous variablesGAM to explore nonlinear effects
Often not desirable to keep the continuous effect in the tariff
» GAM has a high computational cost (iterative method)
» GAM lacks a parametric functional form
GAMs provide insight in defining risk homogeneous
groupings of variables
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Binning continuous variablesGAM for binning
• Results of the GAM as a starting point for binning– Broader categories where the risk is similar– More categories when the risk varies a lot
• Defining boundaries by means of regression trees
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Binning continuous variables Regression tree
• Divide variables into groups based on GAM estimate• Find splits that minimize overall sum of squared errors • Grow tree with desired number of classes
Figure: The black coloured nodes correspond to the regression tree used, the blue coloured nodes are the following splits, and the light blue nodes are the subsequent splits
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Binning continuous variables Binning results
Figure: Visualization of the classes suggested by the regression tree
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AGENDA
• Binning continuous variables
• Geographical information–Modelling• GLM without geographical information• GAM with geographical information
– Visualizing and binning
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Geographical informationIntroduction
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Geographical information Introduction
Latit
ude
Longitude
Bree:51°07'08.8"N 5°38'32.5"E
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Geographical informationStep 1: GLM without geographical information
• We fit a Poisson GLM, ignoring any geographical information, to model the claim frequency
• with the non-spatial categorical variables and the exposure corresponding to policyholder i.
• Aggregate the predicted number of claims per district (INS code)
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Geographical informationStep 1: GLM without geographical information
Predicted number of claims per district
Observed number of claims per district
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Geographical informationStep 2: GAM with geographical information
• Calculate the residual effect • Visualization of by means of quantile binning:
– < 1: number of claims overestimated– > 1: number of claims underestimated
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• Add the longitude and latitude coordinates of the center of each district j.
• We fit a GAM to estimate the geographical effect:
with a two-dimensional smooth function, capturing the geographical effects.
Geographical informationStep 2: GAM with geographical information
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Geographical informationStep 2: GAM with geographical information
• The GAM estimate
which is the geographic effect on top of all other effects included in the GLM prediction
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• Create zones similar in terms of risk– Bin the estimates using classification methods
• Include resulting zones in claim frequency model
Geographical informationVisualizing and binning the geographic effect
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Geographical informationVisualizing and binning the geographic effect
• Problematic issue– Different classification methods can yield dissimilar classes– Maps are very sensitive to the classification method used– Visualization of the same data can convey different
impressions
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Geographical informationVisualizing and binning the geographic effect
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Conclusion
• GLMs remain a very important statistical regression technique for pricing car insurance products.
• GAMs provide interesting insights in the underlying dependency structure, but come at a high computational cost.
• Care is needed when reading and interpreting choropleth maps– Different classification techniques produce different
results.– Classification strongly affects the visual impressions
readers obtain.
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Thank you