Generalized Linear Models II: Applying GLMs in Practice Duncan Anderson MA FIA Watson Wyatt LLP W W W. W A T S O N W Y A T T.

Transcription

1 Generalized Linear Models II: Applying GLMs in Practice Duncan Anderson MA FIA Watson Wyatt LLP W W W. W A T S O N W Y A T T. C O M

2 Agenda Introduction / recap Model forms and model validation Aliasing and "near aliasing" Interactions Smoothing, Combining models, Restrictions Tweedie GLMs Applications and interpreting the results

3 The premium rating process Rate level adjustments Expense loadings 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 Current Rates Competitor Model Risk Model Compare Lapse/takeup Model New Rates Model office

4 The premium rating process Rate level adjustments Expense loadings 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 Current Rates Competitor Model Risk Model Compare Lapse/takeup Model New Rates Model office

5 Generalized linear models E[Y i ] = i = g 1 ( X ij j + i ) Var[Y i ] =.V( i )/ i Consider all factors simultaneously Allow for nature of random process Robust and transparent Increasingly a global industry standard

6 Generalized linear models E[Y] = = g 1 (X. + ) Var[Y] =.V( )/

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

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

9 Generalized linear models Var[Y] =.V( )/ Scale parameter Variance function Prior weights Usually assume exponential family, eg = 2 (estimated), V(x) = 1 Var[Y i ] = 2 Normal = 1 (specifed), V(x) = x Var[Y i ] = i Poisson = k (estimated), V(x) = x 2 Var[Y i ] = k i2 Gamma

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

11 Maximum likelihood estimation Seek parameters which give highest likelihood function given data Likelihood Parameter 1 Parameter 2

12 Example of GLM output (real UK data).25 22% % % 7% 14 Log of multiplier % 5% 19% 4% 16% 2% 1 5% 19% 17% Exposure (policy years) Factor Expos ure On ew ay re la tiv ities Approx 2 SE from estimate Smoothed GL M estimate

13 "A Practitioner's Guide to Generalized Linear Models" CAS 24 Discussion Paper Programme Copies available here and at and

14 Data required Minimum 1, earned exposures Model separately by claim type Key fields for each risk period of earned exposure rating factors applicable at time (start of period) other "dummy" factors (area, time) incurred claim count (by type) incurred loss amount (by type), based on most recent estimates (optionally) earned premium on current basis

15 Preliminary analyses Oneway analyses Twoway analyses Correlation analyses Number of policies Distribution analyses High Vehicle Value Low New Old Vehicle Age Claim type 1 Third party property damage Vehicle type (Type) 25 2 Demonstration job Cla im type 2 Thir d par ty materi al damag e Wh ere Itpm^= an d Ntp m^= Level Number of records Exposure Premium Number of claims Incurred losses Claim frequency Average cost per claim Pure premium Loss ratio A 27,661 24,757 1,584,626 1,87 8,457,28 7.3% 4, % B 22,89 19,777 9,623,698 1,598 6,957, % 4, % C 13,768 12,334 6,35,96 1,11 4,245,92 8.2% 4, % D 19,662 17,592 9,382,767 1,584 6,7,943 9.% 3, % E 11,235 1,76 5,676, ,262, % 3, % F 5,67 5,37 3,118, ,858, % 3, % Numbe r o f claims ,22 89,572 44,691,424 7,532 3,852, % 4, % Average claim size

16 Agenda Introduction / recap Model forms and model validation Aliasing and "near aliasing" Interactions Smoothing, Combining models, Restrictions Tweedie GLMs Applications and interpreting the results

17 Interesting properties Poisson multiplicative parameter estimates unchanged if group by unique combination of rating factor invariant to measures of time Gamma multiplicative parameter estimates unchanged by grouping but standard errors are not generally do not group except for multiple claims on a risk in a policy period invariant to measures of currency Logistic (binomial / logit) maps (,1) to (, ) invariant to whether measuring of success / failure (eg same if model lapse / renew) more appropriate for retention/conversion analyses, but harder to communicate

18 Typical model forms Y Claim frequency Claim number Average claim amount Probability (eg lapses) g(x) ln(x) ln(x) ln(x) ln(x/(1x)) Error Poisson Poisson Gamma Binomial V(x) 1 x 1 x estimate x 2 1 x(1x) exposure 1 # claims 1 ln(exposure)

19 Example of effect of changing assumed error Data Normal

20 Example of effect of changing assumed error Data Normal Poisson

21 Example of effect of changing assumed error Data Normal Poisson Gamma

22 Example of effect of changing assumed error 2 Example portfolio with five rating factors, each with five levels A, B, C, D, E Typical correlations between those rating factors Assumed true effect of factors Claims randomly generated (with Gamma) Random experience analysed by three models

23 Example of effect of changing assumed error Log of multiplier A B C D E.2 True effect

24 Example of effect of changing assumed error Log of multiplier A B C D E.2 True effect One way

25 Example of effect of changing assumed error Log of multiplier A B C D E.2 True effect One way GLM / Normal

26 Example of effect of changing assumed error Log of multiplier A B C D E.2 True effect One way GLM / Normal GLM / Gamma

27 Example of effect of changing assumed error Log of multiplier A B C D E.2 True effect GLM / Gamma + 2 SE 2 SE

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

29 Standard errors Roughly speaking, for a parameter p: SE = 1 / ( 2 / p 2 Likelihood) Likelihood Parameter 1 Parameter 2

30 GLM output (significant factor) % 138% % 14 Log of multiplier % 39% 45% 58% 84% 73% 72% 93% Exposure (years).2 6 % 5% Vehicle symbol Onew ay relativities Approx 95% confidence interval Parameter es timate P value =.%

31 GLM output (insignificant factor) % 14 Log of multiplier.5.5 % 4% 1% 3% 5% % 5% 1% 3% 1% 4% 1% Exposure (years) Vehicle symbol Onew ay relativities Approx 95% confidence interval Par ameter estimate P value = 52.5%

32 Awkward cases Log of multiplier % 5% 16% 14% 11% 1% 22% % 42% 65% 92% 11% 123% 159% 2%

33 Awkward cases Log of multiplier % 35% 28% 22% 16% 11% 5% % 5% 1% 14% 18% 22% 26% 3%

34 Deviance type III tests Single figure measure of goodness of fit Try model with & without a factor Statistical tests show the theoretical significance given the extra parameters Age Sex Vehicle Zone Model A Fitted value Deviance = 9585 df = Excess NCD? Age Sex Vehicle Model B Fitted value Deviance = 964 df = Excess NCD

35 GLM output (significant factor) % 138% % 14 Log of multiplier % 39% 45% 58% 84% 73% 72% 93% Exposure (years).2 6 % 5% Vehicle symbol Onew ay relativities Approx 95% confidence interval Parameter es timate P value =.%

36 GLM output (insignificant factor) % 14 Log of multiplier.5.5 % 4% 1% 3% 5% % 5% 1% 3% 1% 4% 1% Exposure (years) Vehicle symbol Onew ay relativities Approx 95% confidence interval Par ameter estimate P value = 52.5%

37 Consistency over time A B C D A B C D

38 Consistency over time % 1 Log of multiplier % 27% 45% 38% 34% 51% 44% 4% 63% 56% 97% 81% 73% 78% 76% 64% 81% 69% 65% 14% 89% 86% 18% 14% 15% 15% 149% 118% 14% Exposure (years).2 4% % 4% 1% 5% 2% Vehicle symbol.year of exposure Approx 95% confidence interval, Year of exposure: 2 Approx 95% confidence interval, Year of exposure: 21 Approx 95% confidence interval, Year of exposure: 22 Parameter estimate, Year of exposure: 2 Parameter estimate, Year of exposure: 21 Parameter estimate, Year of exposure: 22

39 Consistency over time % % 8 Log of multiplier % 21% 23% 2% 13% 2% 29% 15% 25% 3% 4% % 26% 14% 9% 16% 15% 3% 22% Exposure (years) Territory.Year of exposure Approx 95% confidence interval, Year of exposure: 2 Approx 95% confidence interval, Year of exposure: 21 Approx 95% confidence interval, Year of exposure: 22 Smoothed estimate, Year of exposure: 2 Smoothed estimate, Year of exposure: 21 Smoothed estimate, Year of exposure: 22

40 Intuition Are ordered categorical variables well behaved? Can you believe it, given correlations with other factors? Can the underwriters believe it? How different is it to the oneway? What does this factor do in other frequency/amounts models and for other claim types?

41 Practical model iteration Start with all factors if possible In theory reject one at a time In practice be more brutal if oneways are similar to GLM parameter estimates Recheck excluded factors at end Interactions Stepwise algorithms can be useful if used with care This is what takes time

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

43 Residuals Residuals Fitted values Data

44 Residuals Several forms, eg standardized deviance sign (Y u u ) / ( (1h u ) ) ½ 2 u ( Y u ) / V( d standardized Pearson Y u u (.V( ).(1h ) / ) u u u Standardized deviance Normal (,1) Numbers/frequency residuals problematical ½ Y u u

45 Residuals Histogram of Deviance Residuals Run 12 (Final models with analysis) Model 8 (AD amounts) Frequency Size of deviance residuals Pretium 8/1/24 12:24

46 Residuals Residual Pretium 4/5/24 11:3 Log of fitted value

47 Residuals

48 Gamma data, Gamma error Plot of deviance residual against fitted value Run 12 (All claim types, final models, N&A) Model 6 (Own damage, Amounts) 2 1 Deviance Residual Pretium 8/1/24 12:32 Fitted Value

49 Gamma data, Normal error Plot of deviance residual against fitted value Run 12 (All claim types, final models, N&A) Model 7 (Own damage, Amounts) Deviance Residual Pretium 8/1/24 12:32 Fitted Value

50 Leverage Plot of leverage against fitted value Run 12 (All claim types, final models, N&A) Model 6 (Own damage, Amounts) Leverage Pretium 8/1/24 12:32 Fitted Value

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

52 BoxCox link function investigation Motor third party property frequencies Likelihood Inverse Multiplicative Additive

53 BoxCox link function investigation Motor third party property average amounts Likelihood Inverse Multiplicative Additive

54 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

55 Agenda Introduction / recap Model forms and model validation Aliasing and "near aliasing" Interactions Smoothing, Combining models, Restrictions Tweedie GLMs Applications and interpreting the results

56 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

57 Example Suppose we wanted a model of the form: = + if age < if age if age > if sex male 1 + if sex female 2

58 Form of X. in this case Age Sex <3 34 >4 M F

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

60 X. having adjusted for base levels Age Sex <3 34 >4 M F

61 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

62 Extrinsic aliasing If a perfect correlation exists, one factor can alias levels of another Eg if doors declared first: Selected base Exposure: # Doors Unknown Colour 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)

63 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

64 "Near aliasing" If two factors are almost perfectly, but not quite aliased, convergence problems can result and/or results can become hard to interpret Selected base Exposure: # Doors Unknown Colour 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 2 Unknown 3,242 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 colour

65 Agenda Introduction / recap Model forms and model validation Aliasing and "near aliasing" Interactions Smoothing, Combining models, Restrictions Tweedie GLMs Applications and interpreting the results

66 No interaction age Example job Run 12 Model 3 Small interaction Third party material damage, Numbers %.8 15% % Log of multiplier % 24% % 6% 11% 15 1 Exposure.2 2% Age of driver Approx 2 SEs from estimate Unsmoothed estimate Smoothed estimate P level =.% Rank 7/7

67 No interaction sex Example job Run 12 Model 3 Small interaction Third party material damage, Numbers.3 % Log of multiplier Exposure % Female Male Sex of driver Approx 2 SEs from estimate Unsmoothed estimate Smoothed estimate P level =.% Rank 2/7

68 Age sex interaction Example job Run 5 Model 3 Small interaction Third party material damage, Numbers 1 155% 138% % 63% Log of multiplier % 4% 28% 19% 24% 2% 2% % 6% 6% 13% 11% Exposure.2 18% 19% Age of driver.sex of driver P level =.% Rank 6/6 Approx 2 SEs from estimate, Sex of driver: Female Approx 2 SEs from estimate, Sex of driver: Male Unsmoothed estimate, Sex of driver: Female Unsmoothed estimate, Sex of driver: Male Smoothed estimate, Sex of driver: Female Smoothed estimate, Sex of driver: Male

69 Interactions W x X Y x Z x No interaction A B C D x x x Full interaction A B C D W x x x x X x x x Y x x x x Z x x x x Marginal interaction A B C D x x x W x x x x X Y x x x x Z x x x x

70 Interactions Factor 1: A B C D Full interaction Factor 2: W X Y Z Factor 1: A B C D Marginal interaction Factor 2: W X Y Z Consider D/Z: 2.1 = 1.2 * 1.4 * 1.25

71 Age sex interaction Example job Run 5 Model 3 Small interaction Third party material damage, Numbers 1 155% 138% % 63% Log of multiplier % 4% 28% 19% 24% 2% 2% % 6% 6% 13% 11% Exposure.2 18% 19% Age of driver.sex of driver P level =.% Rank 6/6 Approx 2 SEs from estimate, Sex of driver: Female Approx 2 SEs from estimate, Sex of driver: Male Unsmoothed estimate, Sex of driver: Female Unsmoothed estimate, Sex of driver: Male Smoothed estimate, Sex of driver: Female Smoothed estimate, Sex of driver: Male

72 Marginal interaction: Age effect Example job Run 16 Model 3 Small interaction Third party material damage, Numbers % 138%.8 2 Log of multiplier x 63% 28% 24% 15 1 Exposure (years).2 % 19% 6% 11% Age of driver Onew ay relativities Approx 95% confidence interval Unsmoothed estimate Smoothed estimate

73 Marginal interaction: Age.Sex (ie additional female multipliers) Example job Run 16 Model 3 Small interaction Third party material damage, Numbers Log of multiplier x.64 = 1.63 as before % % % % % 3% 7% 14% % 2% 1% % 13% % 27% % Exposure (years) % 38% Age of driver.sex of driver Approx 95% confidence interval, Sex of driver: Female Unsmoothed estimate, Sex of driver: Female Unsmoothed estimate, Sex of driver: Male Smoothed estimate, Sex of driver: Female Smoothed estimate, Sex of driver: Male

74 An example of no interaction.8 1% 45 91% 82% % 56% 6% 46% 66% 51% 46% 39% 5% 35% 51% 35 3 Log of multiplier.2 22% 2% 17% 7% % % 5% 25 2 Exposure (years) 6% 7% 7% 6% % 31% Complete interaction of Age of driver and Payment frequency P value =.% Approx 95% confidence interval, Payment frequency: Yearly Approx 95% confidence interval, Payment frequency: Halfyearly Approx 95% confidence interval, Payment frequency: Quarterly Parameter estimate, Payment frequency: Yearly Parameter estimate, Payment frequency: Halfyearly Parameter estimate, Payment frequency: Quarterly

75 An example of no interaction % 46% 51% 2 Log of multiplier.2 22% 2% % 7% 7% 15 1 Exposure (years).2 22% Age of driver Oneway relativities Approx 95% confidence interval Parameter estimate P value =.%

76 An example of no interaction % Log of multiplier % 3 2 Exposure (years).5 % 1.5 Yearly Halfyearly Quarterly Payment frequency Oneway relativities Approx 95% confidence interval Parameter estimate P value =.%

77 An example of no interaction % 35 Log of multiplier % 2% % 6% 2% % 8% 3% % 7% 2% % 5% 6% % % 12% 9% % 5% % % 6% Exposure (years) % Marginal interaction of Age of driver and Payment frequency P value = 61.9% Approx 95% confidence interval, Payment frequency: Halfyearly Approx 95% confidence interval, Payment frequency: Quarterly Parameter estimate, Payment frequency: Yearly Parameter estimate, Payment frequency: Halfyearly Parameter estimate, Payment frequency: Quarterly

78 Why marginal interactions can be hard to interpret Exposure ('s) A B C D A 98 1,31 1,42 87 B 1,3 1,84 1,34 1 C 2,62 3,58 2,11 2,43 D 4,8 4,76 2,3 1,68 With low frequency Marginal interaction parameter estimates A B C D A B C D

79 Interactions Age Factor Group Factor

80 Interactions Group > Age v

81 Interactions Group Factor Age Factor

82 Interactions testing consistency with time Testing consistency with time Run 1 Model 1 Interaction with time Third party property damage (numbers) Log of multiplier Exposure (years) < Policyholder age.calendar year P value = 72.2% Rank 3/1 Approx 95% confidence interval, Calendar year: 21 Approx 95% confidence interval, Calendar year: 22 Approx 95% confidence interval, Calendar year: 23

83 Agenda Introduction / recap Model forms and model validation Aliasing and "near aliasing" Interactions Smoothing, Combining models, Restrictions Tweedie GLMs Applications and interpreting the results

84 Smoothing Models can often 15 be improved by 1 smoothing raw 5 statistical parameter 5 estimates where factors have a natural order Log of multiplier Best done at frequency / severity level Artificial constraints and commercial smoothing come later

85 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

86 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

87 Calculation of risk premium TPPD Numbers TPPD Amounts TPBI Numbers TPBI 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%

88 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 interpretting such approximations since they do not reflect model error, eg deciding to exclude a marginal factor

89 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

90 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

91 Restricted models E[Y] = = g 1( X + ) 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

92 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

93 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

94 Agenda Introduction / recap Model forms and model validation Aliasing and "near aliasing" Interactions Smoothing, Combining models, Restrictions Tweedie GLMs Applications and interpreting the results

95 .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 has point mass and parameters which can alter the shape to be like Poisson and gamma above zero f Y ( y;,, ) n 1 1 ( ) ( 1/ y) ( n ) n! y p( Y ) exp ( ) n.exp [ y ( )] for y

96 Tweedie distributions Tweedie = k, V(x) = x p Var[Y] = k 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

97 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 v alue =.% Rank 12/12

98 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 v alue = 5.9% Rank 4/12

99 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

100 Example 2: 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

101 Example 3: 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

102 Example 3: 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

103 Example 3: traditional RP vs Tweedie Comparison of Tweedie model with traditional frequency/amounts approach Run 11 Model 1 Tweedie Models Log of multiplier % 23% 11% 6% 2% 1% % 2% 4% 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

104 Agenda Introduction / recap Model forms and model validation Aliasing and "near aliasing" Interactions Smoothing, Combining models, Restrictions Tweedie GLMs Applications and interpreting the results

105 Comparison with actual rates Demonstration job Run 1 Model 2 Third party material, standard risk premium run Unsmoothed standard risk premium model.8 89% % 67% 2.4 Log of multiplier.2 22% 22% 11% 3% % 22% % % % % % 1% 15 1 Exposure.2 22% 1% 18% MAGE Age of driver Approx 2 SEs from unsmoothed estimate Unsmoothed unr estricted estimate Unsmoothed restricted estimate Current rating structure

106 Comparison with actual rates Demonstration job Run 1 Model 2 Third party material, standard risk premium run Unsmoothed standard risk premium model % Log of multiplier % 15% 18% 7% 1% % % 22% 8% 49% 11% 31% Exposure % to to 17 MGROUP Group of vehicle Approx 2 SEs from unsmoothed estimate Unsmoothed unr estricted estimate Unsmoothed restricted estimate Current rating structure

107 Comparison with actual rates Demonstration job Run 1 Model 2 Third party material, standard risk premium run Unsmoothed standard risk premium model % 49% 2 Log of multiplier % 5% 4% 2% % % 4% 5% 1% 7% 13% 11% 15% 16% Exposure.2 A B C D E F G H MAREA Area of garage Approx 2 SEs from unsmoothed estimate Unsmoothed unrestricted estimate Unsmoothed restricted estimate Curr ent r ating str ucture

108 Comparison with actual rates Demonstration job Run 1 Model 2 Third party material, standard risk premium run Unsmoothed standard risk premium model % 5 4 Log of multiplier % 3 2 Exposure.5 % % 5% 5% 1.5 Yearly Halfyearly Quarterly MPFREQ Payment frequency Approx 2 SEs from unsmoothed estimate Unsmoothed unrestricted estimate Unsmoothed restricted estimate Current rating structure

109 Impact analysis 7 Currently profitable business Example job 6 5 Count of records Currently unprofitable business Ratio: Risk Premium / Current tariff

110 Impact analysis Example job 7 18% 17% Count of records % 15% 14% 13% 12% 11% 1% 9% 8% 7% 6% 5% 4% Loss ratio % Ratio: Risk Premium / Current tariff Yearly Claims / Earnedprem

111 Impact analysis Example job Age of driver 7 18% 17% Count of records % 15% 14% 13% 12% 11% 1% 9% 8% 7% 6% 5% 4% Loss ratio % Ratio: Risk Premium / Current tariff Claims / Earnedprem

112 Impact analysis Example job Area of garage 7 18% 17% Count of records % 15% 14% 13% 12% 11% 1% 9% 8% 7% 6% 5% 4% Loss ratio % Ratio: Risk Premium / Current tariff A B C D E F G H Claims / Earnedprem

113 Impact analysis Example job Class of vehicle 7 18% 17% Count of records % 15% 14% 13% 12% 11% 1% 9% 8% 7% 6% 5% 4% Loss ratio % Ratio: Risk Premium / Current tariff A & Y & Z B C D & E F & G & H & J Claims / Earnedprem

114 Impact analysis Example job Payment frequency 7 18% 17% 6 16% 15% 5 14% 13% Count of records Good proof of effectiveness of model 12% 11% 1% 9% 8% 7% 6% 5% Loss ratio 4% % Ratio: Risk Premium / Current tariff Yearly Halfyearly Quaterly Claims / Earnedprem

115 The premium rating process Rate level adjustments Expense loadings 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 Current Rates Competitor Model Risk Model Compare Lapse/takeup Model New Rates Model office

116 Considering the competitive position Example of competitor analysis Third party cover Log of multiplier % 5% 45% 31% 43% 4% 28% 4% 35% 24% 28% 3% 2% 23% 25% 18% 19% 2% 1% 17% 15% 3% 4% 1% 5% 9% 5% % % 11% 7% 5% 28% 15% 1% 35% 18% 15% 48% 31% 2% 62% 3% 25% 7% 48% 3% 8% 41% 35% 91% 44% 4% 17% 65% 45% 125% 82% 5% Exposure (years) Vehicle group Current tariff Approx 95% confidence interval Third cheapest market quote Smoothed estimate P value =.% Rank 9/11

117 Profitability (theoretical premium/current premium) Profitable Considering the competitive position Competitiveness (market premium/current premium) Competitive

118 Modeling retention / conversion Age Sex Vehicle age Premium Claims Premium / Competitors' premium GLM Probability Model normal factors other products held payment method change in cover discount expectation plus source change in premium claims history competitiveness

119 The premium rating process Rate level adjustments Expense loadings 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 Current Rates Competitor Model Risk Model Compare Lapse/takeup Model New Rates Model office

120 Profitability scoring Construct profitability score based on expected loss ratio Profitability score can then be used to target sections of a portfolio Expected loss ratio can be modeled using a risk premium model offset by current premium rates Expected loss ratio can be banded into discrete bands if desired

121 Profitability scoring Distribution of score 25 16% 14% 2 12% Number of policies % 8% 6% Actual loss ratio 5 4% 2% Score based on expected loss ratio % Number of policies Actual loss ratio

122 Generalized Linear Models II: Applying GLMs in Practice Duncan Anderson MA FIA Watson Wyatt LLP W W W. W A T S O N W Y A T T. C O M