Personal Lines Actuarial Research Department

Generalized Linear ModelsCAGNY

Wednesday, November 28, 2001

Keith D. Holler Ph.D., FCAS, ASA, ARM, MAAA

Personal Lines Actuarial Research Department

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High Level

e.g. Eye ColorAgeWeight 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

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Personal auto or H.O. class plansDeductible or ILF severity models Liability non-economic claim settlement amountHurricane 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

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Example – Personal Auto

Log (Loss Cost) = Intercept + Driver + Car Age Size Factor i Factor j

Driver Age Car Size

Intercept Young Older Small Medium Large

6.50 .75 0 .50 .20 0

e.g. Young Driver, Large CarLoss Cost = exp (6.50 + .75 + 0) = $1,408

Parameters

Personal Lines Actuarial Research Department

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Technical Bits

1. Exponential families – gamma, poisson, normal, binomial2. Fit parameters via maximum likelihood3. Solve MLE by IRLS or Newton-Raphson4. Link Function (e.g. Log Loss Cost)

i. 1-1 functionii. 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

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Personal Auto Class Plan Issues:

1. Territories or other many level variables2. Deductibles and Limits3. Loss Development4. Trend5. Frequency, Severity or Pure Premium6. Exposure7. Model Selection – penalized likelihood an option

Personal Lines Actuarial Research Department

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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

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Software and References

Software: SAS, GLIM, SPLUS, EMBLEM, GENSTAT, MATLAB, STATA, SPSS

References: Part 9 paper bibliographyGreg Taylor (Recent Astin)Stephen Mildenhall (1999)Hosmer and LemeshowFarrokh Guiahi (June 2000)Karl P. Murphy (Winter 2000)