1. Practical GLM Modeling of 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 %

4. Empirical Method Pros Cons Simple 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
Not a member of Exponential Family of distributions

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 Risk Characteristics
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 Relativities Using 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:

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