Personal Lines Actuarial Research Department Generalized Linear Models CAGNY Wednesday, November 28, 2001 Keith D. Holler Ph.D., FCAS, ASA, ARM, MAAA.

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Personal Lines Actuarial Research Department Generalized Linear Models CAGNY Wednesday, November 28, 2001 Keith D. Holler Ph.D., FCAS, ASA, ARM, MAAA

Personal Lines Actuarial Research Department 2 High Level e.g.Eye Color Age Weight Coffee Size Given Characteristics: Predict Response: e.g. Probability someone takes Friday off, given it’s sunny and 70°+ e.g. Expected amount spent on lunch

Personal Lines Actuarial Research Department 3 Personal auto or H.O. class plans Deductible or ILF severity models Liability non-economic claim settlement amount Hurricane damage curves* Direct mail response and conversion* Policyholder retention* WC transition from M.O. to L.T.* Auto physical damage total loss identification* Claim disposal probabilities* Insurance Examples * Logistic Regression

Personal Lines Actuarial Research Department 4 Example – Personal Auto Log (Loss Cost) = Intercept + Driver + Car Age Size Factor i Factor j Driver AgeCar Size InterceptYoungOlderSmallMediumLarge e.g. Young Driver, Large Car Loss Cost = exp ( ) = $1,408 Parameters

Personal Lines Actuarial Research Department 5 Technical Bits 1.Exponential families – gamma, poisson, normal, binomial 2.Fit parameters via maximum likelihood 3.Solve MLE by IRLS or Newton-Raphson 4.Link Function (e.g. Log Loss Cost) i.1-1 function ii.Range Predicted Variable  ( - ,  ) iii.LN  multiplicative model, id  additive model logit  binomial model (yes/no) 5.Different means, same scale

Personal Lines Actuarial Research Department 6 Personal Auto Class Plan Issues: 1.Territories or other many level variables 2.Deductibles and Limits 3.Loss Development 4.Trend 5.Frequency, Severity or Pure Premium 6.Exposure 7.Model Selection – penalized likelihood an option

Personal Lines Actuarial Research Department 7 Why GLMS? 1.Multivariate – adjusts for presence of other variables. No overlap. 2.For non-normal data, GLMS better than OLS. 3.Preprogrammed – easy to run, flexible model structures. 4.Maximum likelihood allows testing importance of variables. 5.Linear structure allows balance between amount of data and number of variables.

Personal Lines Actuarial Research Department 8 Software and References Software:SAS, GLIM, SPLUS, EMBLEM, GENSTAT, MATLAB, STATA, SPSS References:Part 9 paper bibliography Greg Taylor (Recent Astin) Stephen Mildenhall (1999) Hosmer and Lemeshow Farrokh Guiahi (June 2000) Karl P. Murphy (Winter 2000)