Practical GLM Analysisof Homeowners

David CummingsState Farm Insurance Companies

Overview

• How are GLMs different?– Practical Implications

• Modeling Deductibles in GLMs

How are GLMs different?

• Multivariate Analysis• Statistical Framework• Flexible Modeling Tool

Multivariate Analysis

• Multivariate analyses reduce bias

• Practical implications– Requires analysis of all rate factors– Ensures consistency in analysis–May change your processes

Consistent Analysis

• Consistent Exposure BasePure Premium Relativities

0

1

2

3

4

5

0 200,000 400,000 600,000 800,000Amount of Insurance

Earned Policies Earned Exposure

Statistical Framework

• Enhances the analysis

• Practical Implications– Re-learn hypothesis testing and

analysis of standard errors– Different application of “Credibility”

Flexible Modeling Tool

• Allows for many analyses

• Practical Implications– Freq/Severity vs. Pure Premium vs.

Loss Ratio– Design an analysis process– Easily accommodates new data– Fight the urge to overanalyze

Modeling Deductibles

• 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– Simple

• Cons– Need credible data at low deductible– No $1000 deductible data is used to

price the $1000 deductible

0 2000 4000 6000 8000 10000

Loss Distribution Method

• Fit a severity distribution to data

0 2000 4000 6000 8000 10000

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

0 2000 4000 6000 8000 10000

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

1. Fit severity distribution using other rating variables

2. Use deductible as a variable in severity/frequency models

3. 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 ~ GammaY = loss data, net of deductible d

);(1);()(

)log( 110

XX

XXY

nnX

dFdyfyf

vv

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

Practical Issues

• No standard statistical software– Complicates analysis– Less computationally efficient

);(1);()(

)log( 110

XX

XXY

nnX

dFdyfyf

vv

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

);(1);()(

)log( 110

XX

XXY

nnX

dFdyfyf

vv

Practical Issues

• No use of frequency information– Frequency effects derived from

severity fit

– Loss of information

);(1 XX dyF

GLM Approach 2-- Frequency/Severity Model• Standard GLM approach• Fit separate frequency and

severity models• Use deductible as independent

variable

• 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

GLM Approach 2-- Frequency/Severity Model

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

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 RelativitiesUsing Categorical Variable

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0.5

1

1.5

2

2.5

3

3.5

0 2000 4000 6000 8000 10000

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

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0.2

0.4

0.6

0.8

1

1.2

0 1000 2000 3000 4000 5000

Deductible

100,000500,000

CoverageAmount

Severity Relativities

0

0.2

0.4

0.6

0.8

1

1.2

0 1000 2000 3000 4000 5000

Deductible

100,000500,000

CoverageAmount

Pure Premium Relativities

0

0.2

0.4

0.6

0.8

1

1.2

0 1000 2000 3000 4000 5000

Deductible

100,000500,000

CoverageAmount

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

• ReferenceSmyth, 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

0.8

0.9

1

1.1

0 1000 2000 3000 4000 5000

Deductible

Constant Dispersion Double GLM

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