# GLM III: Advanced Modeling Strategy 2005 CAS Seminar on Predictive Modeling Duncan Anderson MA FIA Watson Wyatt Worldwide

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1 GLM III: Advanced Modeling Strategy 25 CAS Seminar on Predictive Modeling Duncan Anderson MA FIA Watson Wyatt Worldwide W W W. W A T S O N W Y A T T. C O M

2 Agenda Introduction Testing the link function The Tweedie distribution Splines Reference models Aliasing / near aliasing Combining models across claim types Restricted models

3 "A Practitioner's Guide to GLMs" 24 CAS Discussion Paper Program Discusses testing the link function the Tweedie distribution aliasing / near aliasing combining models across claim types restricted models Copies available here

4 Generalized linear models E[Y i ] = µ i = g 1 (ΣX ij β j + ξ i ) Var[Y i ] = φ.v(µ i )/ω i Consider all factors simultaneously Provide statistical diagnostics Allow for nature of random process Robust and transparent Increasingly a global industry standard

5 Generalized linear models E[Y] = µ = g 1 (X.β + ξ) Var[Y] = φ.v(µ)/ω

6 Generalized linear models E[Y] = µ = g 1 (X.β + ξ) Link function Yvariate Design matrix Parameter estimates Offset term

7 Generalized linear models E[Y] = µ = g 1 (X.β + ξ) Observed thing (data) Some function (user defined) Some matrix based on data (user defined) Parameters to be estimated (the answer!) Known effects

8 Generalized linear models Var[Y] = φ.v(µ)/ω Scale parameter Variance function Prior weights Usually assume exponential family, eg φ = σ 2 (estimated), V(x) = 1 Var[Y i ] = σ 2 Normal φ = 1 (specified), V(x) = x Var[Y i ] = µ i Poisson φ = k (estimated), V(x) = x 2 Var[Y i ] = kµ i 2 Gamma

9 Agenda Introduction Testing the link function The Tweedie distribution Splines Reference models Aliasing / near aliasing Combining models across claim types Restricted models

10 Link function g(x) Offset ξ Error Structure V(µ) Linear Predictor Form X.β = α i + β j + γ k + δ l Data Y Scale Parameter φ Prior Weights ω Numerical MLE Parameter Estimates α, β, γ, δ Diagnostics

11 Model testing Use only those factors which are predictive standard errors of parameter estimates F tests / χ 2 tests on deviances stepwise approach (helpful if used with care) consistency over time human intuition Make sure the model is reasonable variance function: residual plots (histograms / QQ / residual vs fitted value etc) outliers: leverage / Cook's distance link function: BoxCox

12 BoxCox link function investigation GLM structure is E[Y] = µ = g 1 (X.β + ξ) Var[Y] = φ.v(µ) / ω Box Cox transforms defines g(x) = ( x λ 1 ) / λ for λ, ln(x) for λ= λ = 1 g(x) = x 1 additive (with base level shift) λ g(x) ln(x) multiplicative (via maths) λ = 1 g(x) = 1 1/x inverse (with base level shift) Try different values of λ and measure goodness of fit to see which fits experience best

13 BoxCox link function investigation Auto third party property damage frequencies Likelihood Inverse Multiplicative Additive λ

14 BoxCox link function investigation Auto third party property damage average amounts Likelihood Inverse Multiplicative Additive λ

15 BoxCox link function investigation Comparing fitted values of different link functions 35 Not much 3 25 Count of records Ratio of fitted values for Lambda = (mult) to Lambda =.3

16 Agenda Introduction Testing the link function The Tweedie distribution Splines Reference models Aliasing / near aliasing Combining models across claim types Restricted models

17 Standard approach BI Freq x Amt = Cost 1 PD Freq x Amt = Cost 2 MED Freq x Amt = Cost 3 COL Freq x Amt = Cost 4 OTC Freq x Amt = Cost 5

18 .2 Tweedie distributions Incurred losses have a point mass at zero and then a continuous distribution Poisson and gamma not suited to this Tweedie distribution can have point mass at zero and parameters which can alter the shape to be like Poisson and gamma above zero f Y ( y ; θ, λ, α ) = n = 1 1 α n {( λω ) κ α ( 1 / y )}. exp { λω [ θ y κ ( θ )]} Γ ( n α ) n! y α for y > { ( θ )} p( Y = ) = exp λωκ α

19 Tweedie distributions Tweedie: φ = k, V(x) = x p Var[Y] = kµ p p=1 corresponds to Poisson, p=2 to gamma Defines a valid distribution for p<, 1<p<2, p>2 Can be considered as Poisson/gamma process for 1<p<2 Need to estimate both k and p when fitting models often estimate a where p = (2a)/(1a) Typical values of p for insurance incurred claims around, or just under, 1.5

20 Generalised linear models Var[Y] = φ.v(µ) / ω Normal: φ = σ 2, V(x) = 1 Var[Y] = σ 2.1 Poisson: φ = 1, V(x) = x Var[Y] = µ Gamma: φ = k, V(x) = x 2 Var[Y] = kµ 2 Tweedie: φ = k, V(x) = x p Var[Y] = kµ p

21 Example 1: frequency Comparison of Tweedie model with traditional frequency/amounts approach Run 7 Model 2 Frequency % 446% 1 Log of multiplier % 128% 94% 96% 44% % 23% 15% 21% 17% Exposure (years) % A B C D E F G H I J K L M Bonus Malus Onew ay relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate P value =.% Rank 12/12

22 Example 1: amounts Comparison of Tweedie model with traditional frequency/amounts approach Run 7 Model 6 Amounts Log of multiplier % 26% 6% 1% 5% 3% 7% % 2% 3% 2% 4% 1% Number of claims A B C D E F G H I J K L M Bonus Malus EXCLUDED FACTOR Onew ay relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate P value = 5.9% Rank 4/12

23 Example 1: traditional RP vs Tweedie Comparison of Tweedie model with traditional frequency/amounts approach Run 11 Model 2 Tweedie Models % 568% 443% 446% 1 Log of multiplier % 128% 94% 96% 44% % 23% 15% 21% 17% Exposure (years) % A B C D E F G H I J K L M Bonus Malus Tw eedie SE Tw eedie RPSE RP

24 Example 2: frequency Comparison of Tweedie model with traditional frequency/amounts approach Run 7 Model 1 Frequency Log of multiplier % 84% 69% 75% 68% 67% 61% 58% 58% 45% 48% 54% 47% 38% Exposure (years).1 % >13 Age of vehicle Onew ay relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate P value =.% Rank 12/12

25 Example 2: amounts Comparison of Tweedie model with traditional frequency/amounts approach Run 7 Model 5 Amounts Log of multiplier % 1% 11% 17% 14% 11% 24% 8% 11% 14% 13% 1% 4% Number of claims % 1 5% >13 Age of vehicle Onew ay relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate P value =.% Rank 5/7

26 Example 2: traditional RP vs Tweedie Comparison of Tweedie model with traditional frequency/amounts approach Run 11 Model 1 Tweedie Models 1 1 Log of multiplier % 83% 17% 11% 9% 87% 16% 15% 94% 92% 9% 86% 14% 99% 73% 75% 71% 67% 7% 73% 65% 67% 7% 55% 53% 34% 31% Exposure (years) % >13 Age of vehicle Tw eedie SE Tw eedie RPSE RP Numbers Amounts

27 Example 3: traditional RP vs Tweedie Comparison of Tweedie model with traditional frequency/amounts approach Run 11 Model 1 Tweedie Models.1 45 Log of multiplier % 9% % Exposure (years) Category 1 Category 2 Occupation Tw eedie SE Tw eedie RPSE RP Numbers Amounts

28 Example 4: frequency Comparison of Tweedie model with traditional frequency/amounts approach Run 7 Model 1 Frequency.4 8 Log of multiplier % 6% 2% % 2% 2% 5% 8% 3% 8% 8% 3% 17% 2% Exposure (years) < Unknown 39% 1 Age of driver Onew ay relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate P value =.% Rank 5/12

29 Example 4: amounts Comparison of Tweedie model with traditional frequency/amounts approach Run 7 Model 5 Amounts.18 16% 6 Log of multiplier % 3% 2% % 1% 1% 6% 9% 5% 3% 5% % 2% 4% Number of claims < Unknown Age of driver EXCLUDED FACTOR Onew ay relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate P value = 5.6% Rank 4/9

30 Example 4: traditional RP vs Tweedie Comparison of Tweedie model with traditional frequency/amounts approach Run 11 Model 1 Tweedie Models Log of multiplier % 21% 11% 6% 2% 1% % 4% 2% 2% % 13% 5% 2% 8% 3% 1% 16% 8% 15% 8% 2% 3% 14% 17% 16% 2% 28% Exposure (years).4 39% 1.6 < Unknown Age of driver Tw eedie SE Tw eedie RPSE RP Numbers

31 Agenda Introduction Testing the link function The Tweedie distribution Splines Reference models Aliasing / near aliasing Combining models across claim types Restricted models

32 Spline definition 1 A series of polynomial functions, with each function defined over a short interval Intervals are defined by k+2 knots two exterior knots at extremes of data variable number (k) of interior knots At each interior knot the two functions must join "smoothly"

33 Cubic splines Each polynomial is a cubic a + bx + cx 2 + dx 3 "Smoothness" at interior knots is defined as: continuous continuous first derivative continuous second derivative

34 Regression splines The position of the knots is specified by the user Standard GLMs can be used by careful definition of variates Pros fits easily into existing structures no complex resampling needed Cons position of knots can effect final answer

35 Smoothing splines One knot at each unique data value Additional curvature penalty prevents over fitting Curvature penalty selected by repeatedly sampling subsets and optimising generalised goodness of fit measure such as AIC Pros allows data to guide final result Cons 1s of knots required optimisation process is timeconsuming difficult to produce new fitted values

36 "Easy" regression splines Fit a cubic over the whole range simply define x, x 2 and x 3 as variates and include in the model Fit additional cubic "correction" variates for each interval, defined as if x<k r ((x k r+1 )/(k r k r+1 )) 3 otherwise

37 "Easy" regression splines

38 "Easy" regression splines

39 "Easy" regression splines

40 "Easy" regression splines "Correction" variates get large quickly In practice GLM process can struggle with these large numbers Alternate basis is clearly desirable

41 BSplines Set of basis functions usually covering four segments (defined by five knots) Each function is itself a cubic spline Each basis function has the same shape, except for the three basis functions at each extreme which occupy fewer than four segments

42 BSplines

43 BSplines

44 BSplines

45 BSplines

46 Example Factor

47 Example Factor Polynomial

48 Example Spline Factor Polynomial

49 Example Spline Factor Spline SE Spline SE

50 Extrapolation Can combine the three boundary basis functions to achieve linear extrapolation Sum of three functions tends to 1 at the exterior knot, and continues as 1 when extrapolated Can combine the last two functions to give function that tends to a straight line at the exterior knot, and can be extrapolated as such

51 BSplines

52 BSplines

53 BSplines

54 Example Spline Factor Spline SE Spline SE Polynomial

55 Further example 1.2 Comparison of factor with spline Urban density 28% % 5 81%.6 44% 53% 4 Log of multiplier.3 9% 15% 2% % 1% 18% 3 Exposure (years).3 33% 37% 34% 35% 35% 34% 28% 25% 28% Population density Factor SE Factor P value =.% Rank 7/11

56 1.2 Further example Comparison of factor with spline Urban density Log of multiplier Exposure (years) Population density Factor SE Factor Spline P value =.% Rank 7/11

57 1.2 Further example Comparison of factor with spline Urban density Log of multiplier Exposure (years) Population density Spline P value =.% Rank 7/11

58 Further example 1.2 Comparison of factor with spline Urban density 182% % 139% % Log of multiplier.3 15% 8% 3% % 5% 17% 4% 4 3 Exposure (years).3 38% 37% 37% 36% 35% 33% 31% 29% 24% Population density Factor Spline SE Spline P value =.% Rank 7/11

59 Splines Practical way of modeling continuous variables Often better than polynomials Increases complexity, therefore best used when it is important that rates vary continuously with a variable when modeling elasticity to be used in price optimization analyses

60 Agenda Introduction Testing the link function The Tweedie distribution Splines Reference models Aliasing / near aliasing Combining models across claim types Restricted models

61 Standard approach BI Freq x Amt = Cost 1 PD Freq x Amt = Cost 2 MED Freq x Amt = Cost 3 COL Freq x Amt = Cost 4 OTC Freq x Amt = Cost 5

62 Binomial reference models Amt = Cost 1 TP Freq BI / PD? Amt = Cost 2 MED Freq x Amt = Cost 3 COL Freq x Amt = Cost 4 OTC Freq x Amt = Cost 5

63 Offset reference model

64 Offset reference model

65 Offset reference model

66 Offset reference model

67 Offset reference model

68 Offset reference model

69 Offset reference model

70 Offset reference model

71 Testing the reference model approach (1) Fit to BI claims on all data the "correct answer" 1% of large company 1% Random sample to emulate small company (2) Model BI claims with standard approach (3) Model BI claims referencing PD experience on this small sample

72 Example of reference model method working Example PI models Green: BI 1% ("correct answer") Blue: Standard BI GLM on 1% Purple: Referencing PD on 1% 27% Log of multiplier % 14% 19% 1 8 Exposure (years) 6.6 % Standard BI GLM factor rejected Level 1 Level 2 Level 3 Factor X Approx 2 s.e. from estimate Full model Unsmoothed estimate Full model PD model

73 Example of reference model method working Example Green: BI PI 1% models ("correct answer") % 182% Blue: Standard BI GLM on 1% Purple: Referencing PD on 1% Log of multiplier.6.3 1% 54% 37% 9% 66% 54% 38% 42% 29% 24% 34% 2% 12% 12% 11% 6% 34% 14% 1% 1% 3% 38% 6% 7% % Exposure (years) 2.3 Level A Level B Level C Level D Level E Level F Level G Level H Level I Level J Factor Y Approx 2 s.e. from estimate Full model Unsmoothed estimate Full model Unsmoothed estimate 1% model Approx 2 s.e. from estimate 1% model PD model

74 Agenda Introduction Testing the link function The Tweedie distribution Splines Reference models Aliasing / near aliasing Combining models across claim types Restricted models

75 Aliasing and "near aliasing" Aliasing the removal of unwanted redundant parameters Intrinsic aliasing occurs by the design of the model Extrinsic aliasing occurs "accidentally" as a result of the data

76 Example Suppose we wanted a model of the form: µ = α + β if age < β if age β if age > γ if sex male 1 + γ if sex female 2

77 Form of X.β in this case Age Sex <3 34 >4 M F α β β β γ γ

78 Example Suppose we wanted a model of the form: µ = α + β if age < β if age "Base levels" + β if age > γ if sex male 1 + γ if sex female 2

79 X.β having adjusted for base levels Age Sex <3 34 >4 M F α β β β γ γ

80 X.β having adjusted for base levels Age Sex <3 >4 F α β 1 β 3 γ 2

81 Intrinsic aliasing Example job Run 16 Model 3 Small interaction Third party material damage, Numbers % 138%.8 2 Log of multiplier % 28% 24% 15 1 Exposure (years).2 % 19% 6% 11% Age of driver Onew ay relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate

82 Extrinsic aliasing If a perfect correlation exists, one factor can alias levels of another Eg if doors declared first: Selected base Exposure: # Doors Unknown Color Selected base Red 13,234 12,343 13,432 13,432 Green 4,543 4,543 13,243 2,345 Blue 6,544 5,443 15,654 4,565 Black 4,643 1,235 14,565 4,545 Further aliasing Unknown 3,242 This is the only reason the order of declaration can matter (fitted values are unaffected)

83 Extrinsic aliasing Example job Run 16 Model 3 Small interaction Third party material damage, Numbers 1 135% 6 Log of multiplier % 39% 45% 57% 82% 72% 7% 91% 13% Exposure (years) % % Malus No claim discount Onew ay relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate

84 "Near aliasing" If two factors are almost perfectly, but not quite aliased, convergence problems can result and/or results can become hard to interpret Exposure: # Doors Unknown Color Selected base Red 13,234 12,343 13,432 13,432 Eg if the 2 black, unknown doors policies had no claims, GLM would try to estimate a very large negative number for unknown doors, and a very large positive number for unknown color Selected base Green 4,543 4,543 13,243 2,345 Blue 6,544 5,443 15,654 4,565 Black 4,643 1,235 14,565 4,545 2 Unknown 3,242

85 "Near aliasing" solution 1. Spot it 2. Fix the data! Exposure: # Doors Unknown Colour Red 13,234 12,343 13,432 13,432 Green 4,543 4,543 13,243 2,345 Blue 6,544 5,443 15,654 4,565 Black 4,643 1,235 14,565 4,545 2 Unknown 3,242

86 Agenda Introduction Testing the link function The Tweedie distribution Splines Reference models Aliasing / near aliasing Combining models across claim types Restricted models

87 Combining claim elements I BI Freq x Amt = Cost 1 PD Freq x Amt = Cost 2 MED Freq x Amt = Cost 3 Multiply factors for frequencies and amounts Calculate risk premium as sum of claim elements COL Freq x Amt = Cost 4 OTC Freq x Amt = Cost 5

88 Combining claim elements II Consider current exposure BI Freq x Amt = Cost 1 PD Freq x Amt = Cost 2 MED Freq x Amt = Cost 3 COL Freq x Amt = Cost 4 OTC Freq x Amt = Cost 5 Calculate expected frequency and amount for each claim type for each record Combine to give expected total cost of claims for each record Fit model to this expected value

89 Calculation of risk premium TPPD TPPD TPBI TPBI Numbers Amounts Numbers Amounts Intercept 32% 1 12% 486 Sex Male Female Area Town Country Policy Sex Area WWNUM1 WWAMT1 WWNUM2 WWAMT2 WWCC1 WWCC2 WWRSKPRM M T 32% 1 12% F T 24% 12 8% M C 4% 7 9% F C 3% 84 6%

90 Risk premium standard errors Risk premium model standard errors are small owing to the smoothness of the expected value It is possible to approximate standard error of risk premium parameter estimates based on standard errors of parameter estimates in underlying models Care needed in interpreting such approximations since they do not reflect model error, eg deciding to exclude a marginal factor

91 Risk premium standard errors failings Numbers Amounts Risk premium

92 Risk premium standard errors failings Numbers Amounts Risk premium

93 Agenda Introduction Testing the link function The Tweedie distribution Splines Reference models Aliasing / near aliasing Combining models across claim types Restricted models

94 Restricted models E[Y] = µ = g 1 ( X.β + ξ ) Offset Offset term used for known effects, eg exposure in a numbers model Can also be used to constrain model (eg claim free years / payment frequency / amount of cover) Other factors adjusted to compensate

95 Restricted models Age Sex <3 >4 F α β 1 β 3 γ 2

96 Restricted models E[Y] = µ = g 1 ( X.β ) Age Sex <3 >4 F α β 1 β 3 γ 2

97 Restricted models E[Y] = µ = g 1 ( X.β + ξ ) Age <3 > α β 1 β

98 Restricted models Age Sex <3 >4 F α β 1 β 3.1

99 Restricted models Log of multiplier Restriction Log of multiplier BonusMalus Vehicle group Log of multiplier.5 1 Log of multiplier A B C D E F G H I J K L M N O Geographical zone Age of driver

100 Restricted models Log of multiplier Restriction Log of multiplier BonusMalus Vehicle group Log of multiplier Log of multiplier 1.5 Model compensates (as best it can) to allow for restriction A B C D E F G H I J K L M N O Geographical zone Age of driver

101 Testing the effectiveness of restrictions 12 1 Effect of restricting Bonus Malus Other factors not very correlated with Bonus Malus 8 Count of records Ratio A B C D E F G H I J K L M N O P Q R S

102 Testing the effectiveness of restrictions 12 1 Effect of restricting Bonus Malus Adding in a measure of claim free years 8 Count of records Ratio A B C D E F G H I J K L M N O P Q R S

103 Restrictions Only use to "get around" restrictions A commercial smoothing is a commercial smoothing Apply at risk premium stage

104 GLM III: Advanced Modeling Strategy 25 CAS Seminar on Predictive Modeling Duncan Anderson MA FIA Watson Wyatt Worldwide W W W. W A T S O N W Y A T T. C O M

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