Practical GLM Modeling of Deductibles David Cummings State Farm Insurance Companies

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

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

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

Loss Distribution Method Fit a severity distribution to data

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Severity Relativities Using Categorical Variable

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

Frequency Relativities

Severity Relativities

Pure Premium Relativities

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

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

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

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

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

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

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