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
0
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
0
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
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