Practical GLM Modelingof 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– 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 ~ Gamma

Y = loss data, net of deductible d

);(1

);()(

)log( 110

XX

XXY

nnX

dF

dyfyf

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

dF

dyfyf

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

dF

dyfyf

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

100,000

500,000

CoverageAmount

Severity Relativities

0

0.2

0.4

0.6

0.8

1

1.2

0 1000 2000 3000 4000 5000

Deductible

100,000

500,000

CoverageAmount

Pure Premium Relativities

0

0.2

0.4

0.6

0.8

1

1.2

0 1000 2000 3000 4000 5000

Deductible

100,000

500,000

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