Practical GLM Modeling of Deductibles

1. Practical GLM Modelingof Deductibles David Cummings State Farm Insurance Companies

2. Overview • Traditional Deductible Analyses • GLM Approaches to Deductibles • Tests on simulated data

3. Empirical Method All losses at $500 deductible $1,000,000 Losses eliminated by $1000 deductible $ 100,000 Loss Elimination Ratio 10%

4. Empirical Method • Pros • Simple • Cons • Need credible data at low deductible • No $1000 deductible data is used to price the $1000 deductible

5. Loss Distribution Method • Fit a severity distribution to data

6. Loss Distribution Method • Fit a severity distribution to data • Calculate expected value of truncated distribution

7. Loss Distribution Method • Pros • Provides framework to relate data at different deductibles • Direct calculation for any deductible • Cons • Need to reflect other rating factors • Framework may be too rigid

8. Complications • Deductible truncation is not clean • “Pseudo-deductible” effect • Due to claims awareness/self-selection • May be difficult to detect in severity distribution

9. GLM Modeling Approaches • Fit severity distribution using other rating variables • Use deductible as a variable in severity/frequency models • Use deductible as a variable in pure premium model

10. GLM Approach 1– Fit Distribution w/ variables • Fit a severity model • Linear predictor relates to untruncated mean • Maximum likelihood estimation adjusted for truncation • Reference: • Guiahi, “Fitting Loss Distributions with Emphasis on Rating Variables”, CAS Winter Forum, 2001

11. GLM Approach 1– Fit Distribution w/ variables X = untruncated random variable ~ Gamma Y = loss data, net of deductible d

12. GLM Approach 1– Fit Distribution w/ variables • Pros • Applies GLM within framework • Directly models truncation • Cons • Non-standard GLM application • Difficult to adapt to rate plan • No frequency data used in model

13. Not a member of Exponential Family of distributions Practical Issues • No standard statistical software • Complicates analysis • Less computationally efficient

14. Practical Issues • No clear translation into a rate plan • Deductible effect depends on mean • Mean depends on all other variables • Deductible effect varies by other variables

15. Practical Issues • No use of frequency information • Frequency effects derived from severity fit • Loss of information

16. GLM Approach 2-- Frequency/Severity Model • Standard GLM approach • Fit separate frequency and severity models • Use deductible as independent variable

17. GLM Approach 2-- Frequency/Severity Model • Pros • Utilizes standard GLM packages • Incorporates deductible effects on frequency and severity • Allows model forms that fit rate plan • Cons • Potential inconsistency of models • Specification of deductible effects

18. Test Data • Simulated Data • 1,000,000 policies • 80,000 claims • Risk Characteristics • Amount of Insurance • Deductible • Construction • Alarm System • Gamma Severity Distribution • Poisson Frequency Distribution

19. Conclusions from Test Data– Frequency/Severity Models • Deductible as categorical variable • Good overall fit • Highly variable estimates for higher or less common deductibles • When amount effect is incorrect, interaction term improves model fit

20. Severity RelativitiesUsing Categorical Variable

21. Conclusions from Test Data– Frequency/Severity Models • Deductible as continuous variable • Transformations with best likelihood • Ratio of deductible to coverage amount • Log of deductible • Interaction terms with amount improve model fit • Carefully examine the results for inconsistencies

22. Frequency Relativities

23. Severity Relativities

24. Pure Premium Relativities

25. GLM Approach 3 – Pure Premium Model • Fit pure premium model using Tweedie distribution • Use deductible as independent variable

26. GLM Approach 3 – Pure Premium Model • Pros • Incorporates frequency and severity effects simultaneously • Ensures consistency • Analogous to Empirical LER • Cons • Specification of deductible effects

27. Conclusions from Test Data – Pure Premium Models • Deductible as categorical variable • Good overall fit • Some highly variable estimates • Good fit with some continuous transforms • Can avoid inconsistencies with good choice of transform

28. Extension of GLM – Dispersion Modeling • Double GLM • Iteratively fit two models • Mean model fit to data • Dispersion model fit to residuals • Reference Smyth, Jørgensen, “Fitting Tweedie’s Compound Poisson Model to Insurance Claims Data: Dispersion Modeling,” ASTIN Bulletin, 32:143-157

29. Double GLM in Modeling Deductibles • Gamma distribution assumes that variance is proportional to µ2 • Deductible effect on severity • Mean increases • Variance increases more gradually • Double GLM significantly improves model fit on Test Data • More significant than interactions

30. Pure Premium Relativities Tweedie Model – $500,000 Coverage Amount

31. Conclusion • Deductible modeling is difficult • Tweedie model with Double GLM seems to be the best approach • Categorical vs. Continuous • Need to compare various models • Interaction terms may be important