GLM III - The Matrix Reloaded

Size: px

Start display at page:

Download "GLM III - The Matrix Reloaded"

Leonard Nichols
6 years ago
Views:

2 Agenda "Quadrant Saddles" The Tweedie Distribution "Emergent Interactions" Dispersion Modeling Modelling sparse claim types Driver Averaging Model Validation Man (with GLM) vs machine 2

3 Interactions Policyholder Age

4 Interactions Policyholder Age

5 Why are interactions present? Because that's how the factors behave Because the multiplicative model can go wrong at the edges 1.5 * 1.4 * 1.7 * 1.5 * 1.8 * 1.5 * 1.8 = 26! 5

6 Interactions

7 Interactions

8 Interactions Vehicle group b b - b b b b b b b Vehicle group b b - b b b b b b b Vehicle group b b - b b b b b b b b b b Age b b b b b b b b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b Age b b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b b b b Age b b b b b b b

9 Example

10 Example

11 Example

12 Saddles

13 Saddles

14 Saddles

15 Saddles

16 Saddles

17 Saddles

18 Saddles

19 Saddles

20 Saddles

21 Saddles

22 Saddles

23 23

24 Saddles - model comparison Motor frequency - out of sample 20% 400,000 18% 350,000 16% 300,000 14% 250,000 12% 200,000 10% 150,000 8% 100,000 6% 50,000 4% 50% 70% 90% 94% 98% 102% 106% 110% 130% 150% 0 Exposure Saddle Original

25 Saddles - model comparison Motor frequency - out of sample 20% 400,000 18% 350,000 16% 300,000 14% 250,000 12% 200,000 10% 150,000 8% 100,000 6% 50,000 4% 50% 70% 90% 94% 98% 102% 106% 110% 130% 150% 0 Exposure Observed Saddle Original

26 Saddles - model comparison Motor frequency - out of sample 2.6% % % % % % % % 1.0% % % < 80% 80% - 86% 83% - 89% 86% - 92% 89% - 95% 92% - 98% 95% - 101% 98% % 10 1% % 104 % % 10 7% - 113% 110 % - 116% 113% % 116% % 119 % % 12 2% % 125 % - 131% 128 % % > 130% 0 Exposure Observed Saddle Original

27 Saddles - model comparison Motor frequency - out of time < 80% 80% - 84% 82% - 86% 84% - 88% 86% - 90% 88% - 92% 90% - 94% 92% - 96% 94% - 98% 96% % 98% % 10 0% % 102 % % 10 4% % 106 % % 108 % % 110 % - 114% 112% % 114 % - 118% 116% % 118 % % > 120% 0 Exposure Observed Saddles Original

28 Agenda "Quadrant Saddles" The Tweedie Distribution "Emergent Interactions" Dispersion Modeling Modelling sparse claim types Driver Averaging Model Validation Man (with GLM) vs machine 28

29 Tweedie GLMs Consider the following empirical probability distribution function 29

Tweedie GLMs Raw pure premiums Incurred losses have a point mass at zero and then a continuous distribution Poisson and gamma not suited to this Tweedie distribution has point mass at

30 Tweedie GLMs Raw pure premiums Incurred losses have a point mass at zero and then a continuous distribution Poisson and gamma not suited to this Tweedie distribution has point mass at zero a parameter which changes shape above zero f Y ( y; q, l, a) = å n= 1 1-a n {( lw) ka (-1/ y) } exp{ lwq [ y-k ( q )]} G( -na) n! y { ( q )} p( Y = 0) = exp - lwk a 0 0 a 0 fory > 0 30

31 Formulization of GLMs Generally accepted standards for link functions and error distribution Observed Response Most Appropriate Link Function Most Appropriate Error Structure Variance Function Normal µ 0 Claim Frequency Log Poisson µ 1 Claim Severity Log Gamma µ 2 Claim Severity Log Inverse Gaussian µ 3 Raw Pure Premium Log Tweedie µ T Retention Rate Logit Binomial µ (1-µ) Conversion Rate Logit Binomial µ(1- µ) 31

32 Formulization of GLMs More formally: Var(Y) = φv(μˆ) ω Variance Function Scale Parameter Prior Weights Tweedie s Variance function: V(m) = m p p=1 Poisson p=2 Gamma 1<p<2 Poisson/Gamma process Other concerns Need to estimate both j and p when fitting models Typically p»1.5 for incurred claims 32

33 Example 1 Vehicle age - frequency Exposure Model Prediction at Base levels Model Prediction +/- 2 Standard Errors 33

34 Example 1 Vehicle age - amounts Exposure Model Prediction at Base levels Model Prediction + 2 Standard Errors Model Prediction - 2 Standard Errors 34

35 Example 1 Vehicle age - pure premium Exposure Model Prediction +/- 2 Standard Errors Traditional 35

36 Example 1 Vehicle age - pure premium Exposure Tweedie Model Prediction +/- 2 Standard Errors Traditional 0 36

37 Example 2 Gender - frequency Male Female Exposure Model Prediction at Base levels Model Prediction +/- 2 Standard Errors 0 37

38 Example 2 Gender - frequency Male Female Exposure (1998) Exposure (1999) Policyholder Exposure (2000) Sex Exposure (2001) Exposure (2002) Loss Year (1998) Loss Year (1999) Loss Year (2000) Loss Year (2001) Loss Year (2002) 0 38

39 Example 2 Gender - amounts Male Female Exposure Model Prediction at Base levels Model Prediction +/- 2 Standard Errors 0 39

40 Example 2 Gender - amounts Male Female Exposure (1998) Exposure (1999) Policyholder Exposure Sex (2000) Exposure (2001) Exposure (2002) Loss Year (1998) Loss Year (1999) Loss Year (2000) Loss Year (2001) Loss Year (2002) 0 40

41 Example 2 Gender pure premium Male Female Model Prediction at Base levels Model Prediction + 2 Standard Errors Model Prediction - 2 Standard Errors Combined Tweedie 0 41

42 Tweedie GLMs Helpful when it's important to fit to incurred costs directly Similar results to frequency/severity traditional approach if frequency and amounts effects are clearly weak or clearly strong Distorted by large insignificant effects Removes understanding of what is driving results Smoothing harder 42

43 Agenda "Quadrant Saddles" The Tweedie Distribution "Emergent Interactions" Dispersion Modeling Modelling sparse claim types Driver Averaging Model Validation Man (with GLM) vs machine 43

44 Combining models Collision Frequency x Collision Severity Overall rates

45 Combining models BI Frequency x BI Severity + PD Frequency x PD Severity Overall rates + Collision Frequency x Collision Severity

46 Combining models PI Severity TP Frequency x PI Propensity x + PD Severity Overall rates Collision Frequency x Collision Severity Apply e.g. trends, case reserves adjustments, target loss ratio etc.

47 Combining models BI Frequency x BI Severity + PD Frequency x PD Severity Overall rates + Collision Frequency x Collision Severity Apply e.g. trends, case reserves adjustments, target loss ratio etc.

48 Combining models Take models Take relevant mix of business eg current in force policies For each record calculate expected frequencies and severities according to the models For each record, calculate expected total cost of claims "C" Fit a GLM to "C" using all available factors 48

49 Combining models PD Numbers PD Amounts BI Numbers BI Amounts Intercept 32% $ % $4860 Sex Male Female Area Town Country Policy Sex Area NUM1 AMT1 NUM2 AMT2 CC1 CC2 RISKPREM M T 32% % F T 24% % M C 40% 700 9% F C 30% 840 6%

50 Except Policy Sex Area NUM1 AMT1 NUM2 AMT2 CC1 CC2 RISKPREM M T 32% % F T 24% % M C 40% 700 9% F C 30% 840 6% The global risk premium is not multiplicative In the town, women have a modelled claim cost 29% lower than men /903.20=0.706 In the country, women have a modelled claim cost 27% lower than men /644.50=0.730

51 To solve Policy Sex Area NUM1 AMT1 NUM2 AMT2 CC1 CC2 RISKPREM M T 32% % F T 24% % M C 40% 700 9% F C 30% 840 6% We can capture this result exactly with an interaction Total risk premium Intercept $ Sex Male Female Area Town Country

52 Example "emergent" interaction

53 "Emergent" interactions In the above examples the interaction "emerged" from the risk premium step Emergent interactions are not risk insights, there is no subtle risk effect we have just discovered The different behaviour is by peril, and the rating factors are just bad proxies for the peril effects Emergent interactions are corrections to fix problems we have introduced Best solution is by peril pricing Reflects true behaviour Underlying models simple to understand and implement If not, check for emergent interactions in the risk premium 53

54 54

55 Agenda "Quadrant Saddles" The Tweedie Distribution "Emergent Interactions" Dispersion Modeling Modelling sparse claim types Driver Averaging Model Validation Man (with GLM) vs machine 55

56 >.2 >.5 >.7 > 1.4 > 2.2 > 2.9 > 3.6 > 4.3 > 5.0 > 5.7 > 6.5 > 7.2 > 7.9 > 8.6 > 9.3 > 10.1 > 10.8 > 11.5 > 12.2 > 12.9 > 13.6 > 14.4 > 15.1 > 15.8 > 16.5 > 17.2 > 17.9 > 18.7 > 19.4 > 20.1 > 20.8 > 21.5 > 22.3 > 23.0 > 23.7 > 24.4 > 25.1 > 25.8 > 26.6 > 27.3 Modeling the Insurance Risk ISSUE: Heterogeneous exposure bases Different policies within the same line can cover entirely different structures (i.e. commercial property) Goal of a predictive model Ideally would like to separate the heterogenous exposure bases Joint-modeling techniques and quasi-likelihood functions allow for analysis of heterogeneous environment without separation Studentized Standardized Deviance Residuals , ,000 2,00 0 3,000 4,00 0 5,000 6,0 00 7,0 00 8,000 9,000 Fitte d Value Two concentrations suggests two perils: split or use joint modeling 56

57 Heterogeneous Exposure Bases If possible should be modeled separately If model together, exposures with high variability may mask patterns of less random risks If loss trends vary by exposure class, the proportion each represents of the total will change and may mask important trends Independent predictors can have different effects on different perils If cannot, use joint modeling techniques to improve overall fit 57

58 Generalized Linear Models Formulation of deviance logarithm of a ratio of likelihoods Where: D a Act = f ( j) Y æ Act ö = ln ç Exp è ø ~ 2 ~ ( y; θ, j) ' E( Y) = y = b' ( q) Exp= f Y ( y;θˆ, j) ' E( Y) = m ˆ = b' ( qˆ ) Then: D a ( j) æ ç f = ln è f Y Y ~ (y; θ, j) ö (y;θˆ, j) ø 2 é ~ yθ - yθˆ -b = 2 ê ë a ( j) ( θ ~ ) + b( θˆ ) ù ú û 58

59 Generalized Linear Models Analyzing the scale parameter When modeling homogeneous data j a ( j) = Þ j = ω D dof Heterogeneous data requires a more rigorous definition of the scale function Scale parameter could vary in a systematic way with other predictors Construct and fit formal models for the dependence of both the mean and the scale 59

60 Dispersion Model Form Double generalized linear models Response model Y~f Y E(Y) Var(Y) ( y;θ; j) = b' ( θ) jb'' ( θ) = ω Dispersion model D~f D E(D) ( d; x, t) = b' ( x) Var(D) = tb'' ω ( x) Where d = ( Y-m) V( m) 2 60

61 Dispersion Model Form Dispersion adjustments Pearson residual has excess variability (deviance residual has bias) Distribution Adjustment Normal 0 Poisson f/(2m) Gamma 3f Parameter in the adjustment term is the scale parameter from the original response model 61

62 Dispersion Model Results Dispersion model is integrated with original response model Initial Response Model Dispersion Model Final Response Model Response Loss / Exposure Squared Pearson Residual Loss / Exposure Weight Exposure Exposure/ (Exposure + Adjustment) Exposure/ Squared Pearson Residual 62

63 Agenda "Quadrant Saddles" The Tweedie Distribution "Emergent Interactions" Dispersion Modeling Modelling sparse claim types Driver Averaging Model Validation Man (with GLM) vs machine 63

64 Amplification of the BI signal using PD experience Fit straight to BI Use PD model as a guide in free fitting BI More Data Use PD model structure Offset PD relativities onto BI data as starting point BI/PD proportion model: BI frequency = BI/PD proportion * PD frequency Less Data 64

65 Reference models BI Frequency x BI Severity PD Frequency x PD Severity BI Severity TP Frequency x BI Propensity x PD Severity

66 Reference models A B C D E

67 Reference models A B C D E

68 Reference models A B C D E

69 Reference models A B C D E

70 Reference models A B C D E

71 Reference models A B C D E

72 Reference models A B C D E

73 Reference models E[Y i ] = m i = g -1 (SX ij.b j +x i ) Offset term When modeling BI set PD fitted values to be offset term GLM will seek effects over and above assumed PD effect

74 Experiment (1) GLM on BI claims on all the data - the "correct" answer Real large company 10% random sample Pretend small company (2) Traditional GLM on BI claims on the "small company" (3) Propensity reference model on BI claims cf PD claims

75 Example result , , , , , , , , , , Vehicle Group Exposure "Correct" Traditional Reference 0

76 Example result 200, , , , , , ,000 60, ,000 20, Provisional Licence Age Exposure "Correct" Traditional Reference 0

77 Agenda "Quadrant Saddles" The Tweedie Distribution "Emergent Interactions" Dispersion Modeling Modelling sparse claim types Driver Averaging Model Validation Man (with GLM) vs machine 77

78 Household Averaging Historically companies assigned operators to vehicles for the purpose of rating More recently driver averaging strategies are deployed to capture household Straight vs. geometric average Weighted average Modified Average/assignment hybrid Modeling data needs to mimic the transaction 78

79 Model Design In all modeling projects, it is imperative that the data set up mimic the rating Consider the following example Vehicle Operator Vehicle Rate V1 Dad $500 V2 Mom $450 Operator Class Factor Dad 0.80 Mom 0.85 Junior 2.80 Assume Mom had a $1000 claim in Dad s car 79

80 Assignment Actual assignment methodology each record represents a single vehicle with one assigned operator Veh Op Sym MYR Age Sex Type Yths Drvrs Vehs Exp Clm Losses Prem V1 Junior M OO ,000 1, V2 Mom F PO Operator characteristics based on assigned operator Vehicle characteristics based on vehicle Policy characteristics catch other drivers Losses assigned to vehicle 80

81 Straight Average Straight average methodology: h Which can be deconstructed: h h h

82 Straight Average Straight average methodology each record represents a single vehicle and operator combination Veh Op Sym MYR Age Sex Yths Drvrs Vehs Exp Clm Losses Prem V1 Dad M / V1 Mom F /3 1 1, V1 Junior M / V2 Dad M / V2 Mom F / V2 Junior M / Policy characteristics are same, but less predictive Exposure split amongst the vehicle Losses assigned to vehicle/operator combination iid is a major concern What about Comprehensive? 82

83 Geometric Average Geometric average methodology: h / No direct decomposition 83

84 Geometric Average Geometric methodology each record represents a single vehicle Veh Sym MYR # of Dads # of Moms # of Juniors Exp Clm Losses Prem V /3 1/3 1/ , V /3 1/3 1/ Policy characteristics are same, but less predictive Predictors are translated to counts Losses assigned to vehicle More challenging to add operator interactions or variates 84

85 Weighted Average Weighted average methodology for a straight average approach Veh Op Sym MYR Age Sex Type Yths Drvrs Vehs Exp Clm Losses Prem V1 Dad M PO / V1 Mom F OC /3 1 1, V1 Junior M OC / V2 Dad M OC / V2 Mom F PO / V2 Junior M OC / Creates a relationship between the vehicle and the operator Uses the model to determine the weights More accurate as it requires more information h

86 Agenda "Quadrant Saddles" The Tweedie Distribution "Emergent Interactions" Dispersion Modeling Modelling sparse claim types Driver Averaging Model Validation Man (with GLM) vs machine 86

87 Validate Models Holdout samples Holdout samples are effective at validating model Determine estimates based on part of data set Uses estimates to predict other part of data set Full Test/Training for Large Data Sets Partial Test/Training for Smaller Data Sets Train Data Build Models All Data Build Models Data Split Data Data Train Data Refit Parameters Test Data Compare Predictions to Actuals Split Data Test Data Compare Predictions to Actuals Predictions should be close to actuals for heavily populated cells 87

88 Validate Models Gains curves Compare predictiveness of models 88

89 Validate Models Lift curves Compare predictiveness of models Model Validation Model Validation , , , , ,000 Data ,000 Data 20,000 20,000 We i g h te d a vera g e ,000 16,000 14,000 12,000 10,000 Weights Current model Weights We i g h te d a vera g e ,000 16,000 14,000 12,000 10,000 Weights Current model Weights 8,000 8, ,000 6, , , , , > E-02, <= E-02 > E-02, <= E-02 Absolute value: Current model > E-02, <= E-02 > E-02, <= E-02 Absolute value: Current model 0 More intuitive but difficult to assess performance 89

90 Validate Models X-Graphs 90

Validate Models Residual analysis Recheck residuals to ensure appropriate shape 10 Studentized Standardized Deviance Residuals by Policyholder Age 8 6 4 2 0 Is the contour plot

91 Validate Models Residual analysis Recheck residuals to ensure appropriate shape 10 Studentized Standardized Deviance Residuals by Policyholder Age Is the contour plot symmetric? lt Does the Box-Whisker show symmetry across levels?

92 Agenda "Quadrant Saddles" The Tweedie Distribution "Emergent Interactions" Dispersion Modeling Modelling sparse claim types Driver Averaging Model Validation Man (with GLM) vs machine 92

93 Machine vs man vs

94 Machine vs man vs

95 Machine vs man vs

96 Machine vs man What is the underlying process?

97 Machine vs man Underwriting Claims 1.25 Cover Level Fraud Likelihood <= 4.5 > 4.5, <= 5 > 5, <= 5.5 > 5.5, <= 6 > 6, <= 6.5 > 6.5, <= 7 > 7, <= 7.5 > 7. 5, <= 8 > 8, <= 8.5 > 8. 5, <= 9 > 9, <= 9.5 > New Business Historic New Business Recent Relativity New Business Historic New Business Recent Relativity

98 Machine vs man What are the underlying drivers?

99 Drivers of elasticity Affordability Need Elasticity Alternatives Brand Affinity Shopping Preferences

100 Machine vs man I reckon that lots of recent bodily injury changes are down to new types of claims

101 Claims management companies 101

102 BI models - "insurance" and "compensation" risk 1.20 Modelled Relativities - Accident Year Exposure Compensation Risk Insurance Risk 0

103 1.20 Modelled Relativities - ABI20 Vehicle Group Proportion TPD with BI - Rated Driver Age Exposure Compensation Risk Insurance Risk Proportion TPD with BI - Car Age Exposure Compensation Risk Insurance Risk Exposure Compensation Risk Insurance Risk Unknown 0

HOUSEHOLD AVERAGING. CAS Annual Meeting Alice Gannon November 2007

HOUSEHOLD AVERAGING. CAS Annual Meeting Alice Gannon November 2007 HOUSEHOLD AVERAGING CAS Annual Meeting 2007 Alice Gannon November 2007 Household Averaging PURPOSE: To discuss household averaging and how it is used to reflect the different relationships between operators