The Matrix Inverted A Primer in GLM Theory and Practical Issues. March 11-12, 2004 CAS Ratemaking Seminar Roosevelt Mosley, FCAS, MAAA

Transcription

1 The Matrix Inverted A Primer in GLM Theory and Practical Issues March 1112, 2004 CAS Ratemaking Seminar Roosevelt Mosley, FCAS, MAAA

2 Practical Issues Data Analysis Implementation

3 Data

4 Data Topics How much? Data sources Data cleansing Homogeneity

5 How much data is needed? Overall dataset size Rule of thumb: 50,000 exposure years May achieve by adding additional years Considerations Holdback for testing Data quality Number of explanatory factors Potential segmentation of analysis

6 How much data is needed? Explanatory factors No theoretical limit Practical limitation Practical issues Model fit times Interpretation Overlap

7 Data Sources Internal Ratemaking Underwriting Marketing Agency Other Lines External Credit Demographic Vehicle information Fire station Data Considerations Robustness Cost Time Availability

8 Data Cleansing Watch out for: Bad exposure, premium & claim fields Tables that do not match Null data values Miscoded variables Midterm changes Use: Database analysis Oneway analysis

9 Midterm Changes Claim on 5/1/2000 Policy Number Age Territory Symbol Effective Date Expiration Date Transaction Date Exposure /1/2000 7/1/2000 1/1/ /1/2000 7/1/2000 4/1/ /1/2000 7/1/2000 4/1/ Policy Number Age Territory Symbol Effective Date Expiration Date Transaction Date Exposure /1/2000 7/1/2000 1/1/ /1/2000 7/1/2000 4/1/ Policy Number Age Territory Symbol Effective Date Expiration Date Transaction Date Exposure /1/2000 3/30/2000 1/1/ /1/2000 7/1/2000 4/1/

10 Midterm Changes Problems Reversals don t always match original entry Matching claims Exposure weights Solutions Adjust dates by reviewing individual transactions Fix characteristics at beginning of policy year or calendar year

11 Nonhomogeneity Studentized Standardized Deviance Residuals Transformed Fitted Value

12 Homogeneity Model will fit better when underlying data is more homogenous Auto By coverage By market segment Homeowners By peril By coverage

13 Analysis

14 Analysis Topics Oneway analysis Modeling Analysis tools

15 OneWay Analysis Oneway summary of: Premium Exposure Losses Claim Counts Purpose Initial understanding of importance Understanding of data levels Potential factor elimination Facilitates comparison to multivariate results Can alert you to problems with the data

16 One Way Analysis Understanding Data dictionary Systems representatives Data input representatives One way summary statistic reviews Exposures, losses, premium, loss ratio, etc.

17 Modeling Dependent and independent variables Automated analysis Interactions Mixed models Evaluation

18 Modeling Dependent Variables Frequency Severity Combined into pure premium Loss ratio? Pure premium directly?

19 Modeling Independent Variables Include all in initial analysis Evaluation Standard error of parameter differences Chisquare and Ftests Simplify Exclude nonsignificant variables Grouped factors Variates Interactions

20 Automated Analysis 100 s of potential variables Problems Fit times Convergence Interpretation Stepwise regression Type 1 Type 3

21 Interactions Test all possible combinations in GLM Intuition Comparison of 1way results in 2way tables Credit Score < Credit Score < Prior Claims Prior Claims

22 Mixed Models Problem GLM assumes a completely multiplicative or completely additive structure Many US rating structures are mixed Potential solutions Use multiplicative results to estimate additive factors Hybrid model Multiplicative model with interactions

23 Estimate Additive Factors Rate = Base Rate x (Class Factor + Driv Rec Surch) x etc. Driving Record Points Additive Surcharge Multiplicative Indication Additive Indication Problems Additive factors correct for base level Error increases as indications for other factors change significantly from base level

24 Hybrid Mixed GLM Process Fit GLM to multiplicative factors Include multiplicative model factor indications as offsets in additive model Fit GLM to additive factors Include additive factors as offset to multiplicative model Etc. Problems GLM s not designed for this approach Convergence

25 Multiplicative Factors With Interactions Rate = Base Rate x (Class Factor + Driv Rec Surch) x etc. Driving Record Points Class

26 Model Evaluation Example Impact Residuals 1.4 3,000 Testing dataset Loss ratio normalization % 84% 117% 93% 93% 83% 86% 82% 90% 76% 81% 73% 80% 64% 78% 61% 59% 51% 57% 46% 55% 51% 50% 47% 46% 44% 44% 43% 61% 44% 55% 44% 69% 44% 41% 38% 45% 42% 41% 38% 2,500 2,000 1,500 27% 1, Bonus Malus (Mbonus) 0 Exposure in years Theoretical loss ratio Actual loss ratio

27 Analysis Tool Considerations Knowledge of GLM Size of dataset Programming knowledge Budget

28 Implementation

29 Implementation Topics System restrictions Impacts Competition

30 System Restrictions Addition of new variables Expansion of current variables Changing of rating algorithm Homeowners rating by peril Tier rating

31 Impacts Example Impact Graph new tariff/old tariff Count of records Current Loss Ratio Total Current Loss Ratio

32 Competition Company ABC State X Accident Surcharge Number of Accidents Present Factor Indicated Factor Competitive Index Proposed Factor