Claudio Giorgio Giancaterino
This work has the aim to compare different statistics modelling approaches both used and could be second-hand used in motor third party liability pricing.
Generalized Linear Models are made up by a systematic component ηi = Σnj=1 xijβj linked with a random component Yi = EF(b(θi); φi/ωi) by a link function g(μi). A Generalized Non-linear Model is the same as the GLM model except the link function g(μi) = ηi(xij; βj) where the systematic component is non linear in the parameters βj. Generalized Additive Models extend Generalized Linear Models in the predictor ηi = Σnp=1 xipβp + Σnj=1 fj(xij) made up by one parametric linear part and one non parametric part built by the sum of unknown “smoothing” functions of the covariates.
Mean commercial tariff Tariff requirement Loss Ratio Residuals degrees of freedom Expected Losses Actual Losses Explained Deviance Risk coefficients Uni- GLM 234,4587 1,000490 1,447822 27.501 9.337.547 9.342.125 96,96% 20 Variate GNM 234,4647 1,000476 1,447785 27.501 9.337.683 9.342.125 96,96% 20 Analysis GAM 232,8702 1,001729 1,457698 27.476 9.325.999 9.342.125 96,20% 45 Multi- GLM 234,6486 0,9981246 1,446650 27.505 9.359.678 9.342.125 87,64% 16 Variate GNM 234,6165 0,9979703 1,446848 27.505 9.361.125 9.342.125 87,04% 16 Analysis GAM 248,5732 0,8596438 1,365612 27.265 10.867.438 9.342.125 84,80% 256
GAM approach is flexible to fit data, with realistic values and low level of residual deviance, but quite complex to realize. GLM is easier to use, but sometimes with overestimated coefficients and high values about residual deviance. GNM is an upgrade of GLM model, it grants some elaborations that GLM can’t replicate and with lower values compared to it.
From these three models GAM is able to personalize a premium with more risk coefficients.
Keywords: Actuarial Sciences, Non-Life Insurance Pricing, Statistical Modelling.
Pages: 427 – 481 | Full PDF Paper