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

2. 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

3. 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

4. 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 6.50.750.50.200 e.g. Young Driver, Large Car Loss Cost = exp (6.50 +.75 + 0) = $1,408 Parameters

5. 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

6. 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

7. 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.

8. 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)