Dealing with continuous variables and geographical information in non life

insurance ratemaking

Maxime Clijsters

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

• 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

AGENDA

• Binning continuous variables– GAM to explore nonlinear effects– GAM and regression trees for binning

• Modelling geographical information

• 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

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

Binning continuous variablesGAM to explore nonlinear effects

(a) Nonparametric prediction

(b) Total prediction

�̂� (𝑥 𝑖 )=�̂�2𝑖 𝑥 𝑖2+ �̂�3 𝑖

3 +∑𝑘=1

𝐾

�̂�𝑘 (𝑥 𝑖−𝑥𝑘 )+¿3¿

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

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

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

Binning continuous variables Binning results

Figure: Visualization of the classes suggested by the regression tree

AGENDA

• Binning continuous variables

• Geographical information–Modelling• GLM without geographical information• GAM with geographical information

– Visualizing and binning

Geographical informationIntroduction

Geographical information Introduction

Latit

ude

Longitude

Bree:51°07'08.8"N 5°38'32.5"E

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)

Geographical informationStep 1: GLM without geographical information

Predicted number of claims per district

Observed number of claims per district

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

• 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

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

• 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

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

Geographical informationVisualizing and binning the geographic effect

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.

Thank you