Anderson

WWW.WATSONWYATT.COM

GLM III: Advanced Modeling Strategy2005CASSeminaronPredictiveModeling

DuncanAndersonMAFIA

WatsonWyattWorldwide

CopyrightWatsonWyattWorldwide.Allrightsreserved.

Agenda

l Introduction

l Testingthelinkfunction

l TheTweediedistribution

lSplines

lReferencemodels

lAliasing/nearaliasing

lCombiningmodelsacrossclaimtypes

lRestrictedmodels


"A Practitioner's Guide to GLMs"

l 2004CASDiscussionPaperProgram

lDiscusses testingthelinkfunction theTweediedistribution aliasing/nearaliasing combiningmodelsacrossclaimtypes

restrictedmodels

lCopiesavailablehere


Generalized linear models

E[Yi]= m i=g1(SXij b j+ x i)

Var[Yi]= f.V(m i)/w ilConsiderallfactorssimultaneously

lProvidestatisticaldiagnostics

lAllowfornatureofrandomprocess

lRobustandtransparent

l Increasinglyaglobalindustrystandard



E[Y]= m =g1(X.b + x)

Var[Y]= f.V(m)/w



E[Y]= m =g1(X.b + x)

Yvariate

Linkfunction

Designmatrix Parameterestimates

Offsetterm



E[Y]= m =g1(X.b + x)

Observedthing(data)

Somefunction(userdefined)

Somematrixbasedondata

(userdefined)

Parameterstobe

estimated(theanswer!)

Knowneffects



Var[Y]= f.V(m)/w

Scaleparameter

Variancefunction

Priorweights

l Usuallyassumeexponentialfamily,eg

l f = s 2 (estimated),V(x)=1 Var[Yi]= s 2 Normal

l f =1 (specified), V(x)=x Var[Yi]= m i Poisson

l f =k (estimated),V(x)=x2 Var[Yi]=km i2 Gamma


Agenda

l Introduction



lSplines

lReferencemodels



lRestrictedmodels

NumericalMLE

ParameterEstimates Diagnostics

Data

LinearPredictorForm

Linkfunctiong(x) ErrorStructure V(m)

ScaleParameter

PriorWeights

f

Offset x

w Y

a, b, g, d

X.b = a i+ b j+ g k+ d l


Model testing

lUseonlythosefactorswhicharepredictive standarderrorsofparameterestimates

Ftests/ c 2testsondeviances stepwiseapproach(helpfulifusedwithcare) consistencyovertime humanintuition

lMakesurethemodelisreasonable variancefunction:residualplots

(histograms/QQ/residualvsfittedvalueetc) outliers:leverage/Cook'sdistance linkfunction:BoxCox


BoxCox link function investigation

lGLMstructureisE[Y]= m =g1(X.b + x) Var[Y]= f.V(m)/ w

lBoxCoxtransformsdefinesg(x)=(x l 1)/ l for l0,ln(x)for l=0

l l =1 g(x)=x1 additive(withbaselevelshift)

l l 0 g(x) ln(x) multiplicative(viamaths)

l l =1 g(x)=11/x inverse(withbaselevelshift)

l Trydifferentvaluesof l andmeasuregoodnessoffittoseewhichfitsexperiencebest


BoxCox link function investigation Auto third party property damage frequencies

137910

137870

137830

137790

0.6 0.4 0.2 0 0.2 0.4 0.6

l

Likelihoo

d

MultiplicativeInverse Additive


BoxCox link function investigation Auto third party property damage average amounts

268569

268569

268568

268568

268567

268567

268566

0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3

l

Likelihoo

d

Inverse Multiplicative Additive


BoxCox link function investigation Comparing fitted values of different link functions

0

50000

100000

150000

200000

250000

300000

350000

0.792

0.803

0.814

0.824

0.835

0.846

0.857

0.868

0.878

0.889

0.900

0.911

0.921

0.932

0.943

0.954

0.965

0.975

0.986

0.997

1.008

1.018

1.029

1.040

1.051

1.061

1.072

1.083

1.094

1.105

1.115

1.126

1.137

1.148

1.158

1.169

1.180

1.191

1.202

1.212

1.223

1.234

1.245

1.255

RatiooffittedvaluesforLambda=0(mult)toLambda=0.3

Cou

ntofrec

ords

Notmuch


Agenda

l Introduction



lSplines

lReferencemodels



lRestrictedmodels


Standard approach

AmtFreq

AmtFreq

AmtFreq

AmtFreq

AmtFreq

BI x =Cost1

PD x =Cost2

COL x =Cost4

MED x =Cost3

OTC x =Cost5


Tweedie distributions

l Incurredlosseshaveapointmassatzeroandthenacontinuousdistribution

l Poissonandgammanotsuitedtothis

l TweediedistributioncanhavepointmassatzeroandparameterswhichcanaltertheshapetobelikePoissonandgammaabovezero

{ } { }

=

-

> - - G

- =

100

1

0for)]([exp.!)(

)/1()(),,(

n

n

Y yyynny

yf q k q lw a

k lw a l q a a a

0

0.2

{ })(exp)0( 0 q lwk a - = =Yp


Tweedie distributions

l p=1correspondstoPoisson,p=2togamma

l Definesavaliddistributionforp


Var[Y]= f.V(m)/ w Normal: f = s 2,V(x)=1 Var[Y]= s 2.1

Poisson: f =1,V(x)=x Var[Y]= m

Gamma: f =k,V(x)=x2 Var[Y]=km 2

Generalised linear models

Tweedie: f =k,V(x)=xp Var[Y]=km p


Example 1: frequency

ComparisonofTweediemodelwithtraditionalfrequency/amountsapproachRun7Model2Frequency

75%

17%

21%

15%23%

0%

44%

96%94%128%

156%

446%568%

1.8

1.2

0.6

0

0.6

1.2

1.8

2.4

BonusMalus

Logofm

ultip

lier

0

20000

40000

60000

80000

100000

A B C D E F G H I J K L M

Exp

osure(yea

rs)

Onew ayrelativities Approx95%confidenceinterval Unsmoothedestimate Smoothedestimate

Pvalue=0.0%Rank12/12


Example 1: amounts

ComparisonofTweediemodelwithtraditionalfrequency/amountsapproachRun7Model6Amounts

10%

4%

2%3%

2%0%

7%

3%5%

10%

6%

26%24%

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

BonusMalus

Logofm

ultip

lier

0

200

400

600

800

1000

1200

1400

1600


Num

berofclaim

s

Onew ayrelativities Approx95%confidenceinterval Unsmoothedestimate SmoothedestimateEXCLUDEDFACTOR



Example 1: traditional RP vs Tweedie

ComparisonofTweediemodelwithtraditionalfrequency/amountsapproachRun11Model2TweedieModels

75%

17%

21%

15%23%

0%

44%

96%94%128%

156%

443%567%

75%

17%

21%

15%23%

0%

44%

96%94%128%

156%

446%568%

1.8

1.2

0.6

0

0.6

1.2

1.8

2.4

BonusMalus

Logofm

ultip

lier

0

20000

40000

60000

80000

100000


Exp

osure(yea

rs)

Tw eedieSE Tw eedie RPSE RP




62%

84%

69%75%

68% 67%61%

58% 58%

45% 48%54%

47%

38%

0%

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ageofvehicle

Logofm

ultip

lier

0

20000

40000

60000

80000

100000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 >13

Exp

osure(yea

rs)




Example 2: amounts


0%

5%

4%

10%

13%14%

11%8%

24%

11%

14%

17%

11%10%

13%

0.06

0

0.06

0.12

0.18

0.24

0.3

0.36

Ageofvehicle

Logofm

ultip

lier

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 1 2 3 4 5 6 7 8 9 10 11 12 13 >13

Num

berofclaim

s


Pvalue=0.0%Rank5/7




88%

107%

90%

106%94% 90%

104%

73% 75%67% 70%

73%

55%

34%

0%

83%

101%

87%

105%

92%86%

99%

71% 75%65% 67%

70%

53%

31%

0%

0.2

0

0.2

0.4

0.6

0.8

1

Ageofvehicle

Logofm

ultip

lier

0

20000

40000

60000

80000

100000

0 1 2 3 4 5 6 7 8 9 10 11 12 13 >13

Exp

osure(yea

rs)

Tw eedieSE Tw eedie RPSE RP Numbers Amounts




0%

9%

0%

10%

0.2

0.15

0.1

0.05

0

0.05

0.1

Occupation

Logofm

ultip

lier

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

Category1 Category2

Exp

osure(yea

rs)

Tw eedieSE Tw eedie RPSE RP Numbers Amounts




39%

20%17%

3%

8%8%3%

8%5%

2%2%0%2%

6%

21%

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

Ageofdriver

Logofm

ultip

lier

0

10000

20000

30000

40000

50000

60000

70000

80000


Example 4: amounts


16%

4%2%

0%

5%3%

5%

9%

6%

1%1%0%2%

3%3%

0.24

0.18

0.12

0.06

0

0.06

0.12

0.18

Ageofdriver

Logofm

ultip

lier

0

1000

2000

3000

4000

5000

6000




28%

16%14%

2%

15%16%

1%

20%13%

0%4%

0%1%

11%

23%

39%

20%17%

3%

8%8%3%

8%5%

2%2%0%2%

6%

21%

0.6

0.4

0.2

0

0.2

0.4

0.6

Ageofdriver

Logofm

ultip

lier

0

10000

20000

30000

40000

50000

60000

70000

80000


Agenda

l Introduction



lSplines

lReferencemodels



lRestrictedmodels


Spline definition

0

10

20

30

40

50

60

70

80

90

100

0 25 50 75 100 125 150 175 200

l Aseriesofpolynomialfunctions,witheachfunctiondefinedoverashortinterval

l Intervalsaredefinedbyk+2knots twoexteriorknotsatextremesofdata

variablenumber(k)ofinteriorknots

l Ateachinteriorknotthetwofunctionsmustjoin"smoothly"


Cubic splines

lEachpolynomialisacubic a+bx+cx2+dx3

l "Smoothness"atinteriorknotsisdefinedas: continuous continuousfirstderivative continuoussecondderivative


Regression splines

l Thepositionoftheknotsisspecifiedbytheuser

lStandardGLMscanbeusedbycarefuldefinitionofvariates

lPros fitseasilyintoexistingstructures

nocomplexresamplingneeded

lCons positionofknotscaneffectfinalanswer


Smoothing splines

lOneknotateachuniquedatavalue

lAdditionalcurvaturepenaltypreventsoverfitting

lCurvaturepenaltyselectedbyrepeatedlysamplingsubsetsandoptimisinggeneralisedgoodnessoffitmeasuresuchasAIC

lPros allowsdatatoguidefinalresult

lCons 100sofknotsrequired optimisationprocessistimeconsuming

difficulttoproducenewfittedvalues


"Easy" regression splines

l Fitacubicoverthewholerange simplydefinex,x2andx3asvariatesandincludeinthemodel

l Fitadditionalcubic"correction"variatesforeachinterval,definedas 0ifx



0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 25 50 75 100



0

5

10

15

20

25

0 25 50 75 100



0

10

20

30

40

50

60

70

0 25 50 75 100



l "Correction"variatesgetlargequickly

l InpracticeGLMprocesscanstrugglewiththeselargenumbers

lAlternatebasisisclearlydesirable


BSplines

l Setofbasisfunctionsusuallycoveringfoursegments(definedbyfiveknots)

l Eachfunctionisitselfacubicspline

l Eachbasisfunctionhasthesameshape,exceptforthethreebasisfunctionsateachextremewhichoccupyfewerthanfoursegments

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 25 50 75 100


BSplines

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 25 50 75 100


Example

0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75

Factor


Example

0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75

Factor Polynomial


Example

0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75

Spline Factor Polynomial


Example

0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75

Spline Factor SplineSE SplineSE


Extrapolation

lCancombinethethreeboundarybasisfunctionstoachievelinearextrapolation

lSumofthreefunctionstendsto1attheexteriorknot,andcontinuesas1whenextrapolated

lCancombinethelasttwofunctionstogivefunctionthattendstoastraightlineattheexteriorknot,andcanbeextrapolatedassuch


BSplines

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 25 50 75 100


Example

0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78

Spline Factor SplineSE SplineSE Polynomial


Further example

ComparisonoffactorwithsplineUrbandensity

149%

208%

81%

53%44%

18%

1%0%2%

15%9%

28%25%28%34%35%35%34%37%

33%

0.9

0.6

0.3

0

0.3

0.6

0.9

1.2

Populationdensity

Logofm

ultip

lier

0

10000

20000

30000

40000

50000

60000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Exp

osure(yea

rs)

FactorSE FactorPvalue=0.0%Rank7/11


Further example


0.9

0.6

0.3

0

0.3

0.6

0.9

1.2

Populationdensity

Logofm

ultip

lier

0

10000

20000

30000

40000

50000

60000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Exp

osure(yea

rs)

FactorSE Factor SplinePvalue=0.0%Rank7/11


Further example


0.9

0.6

0.3

0

0.3

0.6

0.9

1.2

Populationdensity

Logofm

ultip

lier

0

10000

20000

30000

40000

50000

60000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Exp

osure(yea

rs)

Spline

Pvalue=0.0%Rank7/11


Further example


182%

139%

101%

74%

40%

17%

5%0%3%

8%15%

24%29%

31%33%35%36%37%37%38%

0.9

0.6

0.3

0

0.3

0.6

0.9

1.2

Populationdensity

Logofm

ultip

lier

0

10000

20000

30000

40000

50000

60000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Exp

osure(yea

rs)

Factor SplineSE SplinePvalue=0.0%Rank7/11


Splines

lPracticalwayofmodelingcontinuousvariables

lOftenbetterthanpolynomials

l Increasescomplexity,thereforebestusedwhenitisimportantthatratesvarycontinuouslywithavariable

whenmodelingelasticitytobeusedinpriceoptimizationanalyses


Agenda

l Introduction



lSplines

lReferencemodels



lRestrictedmodels


Standard approach

AmtFreq

AmtFreq

AmtFreq

AmtFreq

AmtFreq

BI x =Cost1

PD x =Cost2

COL x =Cost4

MED x =Cost3

OTC x =Cost5


Binomial reference models

Amt

Amt

AmtFreq

AmtFreq

AmtFreq

=Cost1

=Cost2

COL x =Cost4

MED x =Cost3

OTC x =Cost5

Freq BI/PD?TP


0.6

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5

Offset reference model


Testing the reference model approach

100%oflargecompany

(1)FittoBIclaimsonalldatathe"correctanswer"

10%Randomsampletoemulatesmallcompany

(2)ModelBIclaimswithstandardapproach

(3)ModelBIclaimsreferencingPDexperienceonthissmallsample


Example of reference model method working

ExamplePImodels

27%

14%

0%

19%

14%

0%

0.06

0

0.06

0.12

0.18

0.24

0.3

0.36

FactorX

Logofm

ultip

lier

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

Level1 Level2 Level3

Expos

ure(yea

rs)

Approx2s.e.fromestimateFullmodel UnsmoothedestimateFullmodel PDmodel

StandardBIGLMfactorrejected

Green:BI100%("correctanswer")

Blue:StandardBIGLMon10%

Purple:ReferencingPDon10%


Example of reference model method working

ExamplePImodels

0%6%

10%

1%

12%

20%

42%

54%

66%

54%

0%

38%

10%

34%

6%

34%29%

182%191%

100%

0%7%

3%

14%11%12%

24%

38%

90%

37%

0.3

0

0.3

0.6

0.9

1.2

FactorY

Logofm

ultip

lier

0

200000

400000

600000

800000

1000000

1200000

1400000

1600000

LevelA LevelB LevelC LevelD LevelE LevelF LevelG LevelH LevelI LevelJ

Exp

osure(yea

rs)

Approx2s.e.f romestimateFullmodel UnsmoothedestimateFullmodel Unsmoothedestimate10%model Approx2s.e.f romestimate10%model PDmodel

Green:BI100%("correctanswer")

Blue:StandardBIGLMon10%

Purple:ReferencingPDon10%


Agenda

l Introduction



lSplines

lReferencemodels



lRestrictedmodels


Aliasing and "near aliasing"

lAliasing theremovalofunwantedredundantparameters

l Intrinsicaliasing occursbythedesignofthemodel

lExtrinsicaliasing occurs"accidentally"asaresultofthedata


Example

l Supposewewantedamodeloftheform:

m = a + b ifage40

+ g ifsexmale

+ g ifsexfemale

2

3

1

2

1 m = a + b ifage40

+ g ifsexmale

+ g ifsexfemale

2

3

1

2

1


1 010 101 100 101 100 011 001 101 010 01....................................................

Age Sex

ab b b g g

1

2

1

3

2

.

40 MF

1

2

3

4

5

Form ofX.b in this case


Example

l Supposewewantedamodeloftheform:

m = a + b ifage40

+ g ifsexmale

+ g ifsexfemale

2

3

1

2

1 m = a + b ifage40

+ g ifsexmale

+ g ifsexfemale

2

3

1

2

1

"Baselevels"


X.b having adjusted for base levels

1 010 101 100 101 100 011 001 101 010 01....................................................

Age Sex

ab b b g g

1

2

1

3

2

.

40 MF

1

2

3

4

5


X.b having adjusted for base levels

1 00 01 10 01 10 11 01 01 00 1..........................................

Age Sex

a

b 1

g 2

b 3

.

40 F

1

2

3

4

5


Intrinsic aliasing

ExamplejobRun16Model3SmallinteractionThirdpartymaterialdamage,Numbers

11%6%

19%

0%

24%28%

63%

138%155%

0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

Ageofdriver

Logofm

ultip

lier

0

50000

100000

150000

200000

250000

1721 2224 2529 3034 3539 4049 5059 6069 70+

Exp

osure(yea

rs)

Onew ayrelativities Approx95%conf idenceinterval Unsmoothedestimate Smoothedestimate


Extrinsic aliasing

l Ifaperfectcorrelationexists,onefactorcanaliaslevelsofanother

l Egifdoorsdeclaredfirst:

l Thisistheonlyreasontheorderofdeclarationcanmatter(fittedvaluesareunaffected)

Exposure:#Doors Color

2 3 4 5Unknown

Red 13,234 12,343 13,432 13,432 0

Green 4,543 4,543 13,243 2,345 0

Blue 6,544 5,443 15,654 4,565 0

Black 4,643 1,235 14,565 4,545 0

Unknown 0 0 0 0 3,242

Selectedbase

Selectedbase

Furtheraliasing


Extrinsic aliasing

ExamplejobRun16Model3SmallinteractionThirdpartymaterialdamage,Numbers

0%

135%

103%

91%

70%72%82%

57%

45%39%

30%

0%

0.2

0

0.2

0.4

0.6

0.8

1

Noclaimdiscount

Logofm

ultip

lier

0

100000

200000

300000

400000

500000

600000

50 5559 6064 6569 7074 7579 8084 8589 9094 9599 100 Malus

Exp

osure(yea

rs)

Onew ayrelativities Approx95%conf idenceinterval Unsmoothedestimate Smoothedestimate


"Near aliasing"

l Iftwofactorsarealmostperfectly,butnotquitealiased,convergenceproblemscanresultand/orresultscanbecomehardtointerpret

l Egifthe2black,unknowndoorspolicieshadnoclaims,GLMwouldtrytoestimateaverylargenegativenumberforunknowndoors,andaverylargepositivenumberforunknowncolor

Exposure:#Doors Color

2 3 4 5Unknown

Red 13,234 12,343 13,432 13,432 0

Green 4,543 4,543 13,243 2,345 0

Blue 6,544 5,443 15,654 4,565 0

Black 4,643 1,235 14,565 4,545 2

Unknown 0 0 0 0 3,242

Selectedbase

Selectedbase


"Near aliasing" solution

1. Spotit

2. Fixthedata!

Exposure:#Doors Colour

2 3 4 5Unknown

Red 13,234 12,343 13,432 13,432 0

Green 4,543 4,543 13,243 2,345 0

Blue 6,544 5,443 15,654 4,565 0

Black 4,643 1,235 14,565 4,545 2

Unknown 0 0 0 0 3,242


Agenda

l Introduction



lSplines

lReferencemodels



lRestrictedmodels


lMultiplyfactorsforfrequenciesandamounts

lCalculateriskpremiumassumofclaimelements

AmtFreqPD x =Cost2

AmtFreqMED x =Cost3

AmtFreqCOL x =Cost4

AmtFreqBI x =Cost1

AmtFreqOTC x =Cost5

Combining claim elements I


l Considercurrentexposure

l Calculateexpectedfrequencyandamountforeachclaimtypeforeachrecord

l Combinetogiveexpectedtotalcostofclaimsforeachrecord

l Fitmodeltothisexpectedvalue

Combining claim elements II

AmtFreq

AmtFreq

AmtFreq

AmtFreq

AmtFreq

BI x =Cost1

PD x =Cost2

COL x =Cost4

MED x =Cost3

OTC x =Cost5


Calculation of risk premium

Policy Sex Area WWNUM1 WWAMT1 WWNUM2 WWAMT2 WWCC1 WWCC2 WWRSKPRM

82155654 M T 32% 1000 12% 4860 320 583.20 903.2082168746 F T 24% 1200 8% 4374 288 349.92 637.9282179481 M C 40% 700 9% 4050 280 364.50 644.5082186845 F C 30% 840 6% 3645 252 218.70 470.70

TPPDNumbers

TPPDAmounts

TPBINumbers

TPBIAmounts

Intercept 32% 1000 12% 4860Sex Male 1.000 1.000 1.000 1.000

Female 0.750 1.200 0.667 0.900Area Town 1.000 1.000 1.000 1.000

Country 1.250 0.700 0.750 0.833


Risk premium standard errors

lRiskpremiummodelstandarderrorsaresmallowingtothesmoothnessoftheexpectedvalue

l Itispossibletoapproximatestandarderrorofriskpremiumparameterestimatesbasedonstandarderrorsofparameterestimatesinunderlyingmodels

lCareneededininterpretingsuchapproximationssincetheydonotreflectmodelerror,egdecidingtoexcludeamarginalfactor


Risk premium standard errors failings

Amounts

Numbers

Riskpremium


Agenda

l Introduction



lSplines

lReferencemodels



lRestrictedmodels


E[Y]= m =g(X .b + x )1

l Offsettermusedforknowneffects,egexposureinanumbersmodel

l Canalsobeusedtoconstrainmodel(egclaimfreeyears/paymentfrequency/amountofcover)

l Otherfactorsadjustedtocompensate

Offset

Restricted models


Restricted models

1 00 01 10 01 10 11 01 01 00 1..........................................

Age Sex

a

b 1 b 1

g 2

g 2

b 3

b 3

.

40 F

1

2

3

4

5

1

2

3

4

5


E[Y]= m =g(X .b )1Restricted models

1 00 01 10 01 10 11 01 01 00 1..........................................

Age Sex

a

b 1 b 1

g 2

g 2

b 3

b 3

.

40 F

1

2

3

4

5

1

2

3

4

5


E[Y]= m =g(X .b + x )1Restricted models

000.100.1......

+

1 001 101 101 011 00..........................................

Age

a

b 1

b 3

.

40

1

2

3

4

5

1

2

3

4

5


Restricted models

1 00 01 10 01 10 11 01 01 00 1..........................................

Age Sex

a

b 1

0.1

b 3

.

40 F

1

2

3

4

5

1

2

3

4

5


Restricted models

Restriction

0 1 2 3 4 5 6 7 8 9 10 11 12 13 141.5

1

0.5

0

0.5

1

1.5

2

BonusMalus

Logofm

ultip

lier

1 2 3 4 5 6 7 8 9 10 11 12 13 14 151.5

1

0.5

0

0.5

1

1.5

Vehiclegroup

Logofm

ultip

lier

GeographicalzoneA B C D E F G H I J K L M N O

2

1.5

1

0.5

0

0.5

1

1.5

Logofm

ultip

lier

18 19 20 22 24 26 28 30 35 40 45 50 55 60 651

0.5

0

0.5

1

1.5

Ageofdriver

Logofm

ultip

lier


0 1 2 3 4 5 6 7 8 9 10 11 12 13 141.5

1

0.5

0

0.5

1

1.5

2

BonusMalus

Logofm

ultip

lier

1 2 3 4 5 6 7 8 9 10 11 12 13 14 151.5

1

0.5

0

0.5

1

1.5

Vehiclegroup

Logofm

ultip

lier

GeographicalzoneA B C D E F G H I J K L M N O

2

1.5

1

0.5

0

0.5

1

1.5

Logofm

ultip

lier

18 19 20 22 24 26 28 30 35 40 45 50 55 60 651

0.5

0

0.5

1

1.5

Ageofdriver

Logofm

ultip

lier

Restricted models

Restriction

Modelcompensates(asbestitcan)

toallowforrestriction


Testing the effectiveness of restrictions

0

2000

4000

6000

8000

10000

12000

0.800

0.810

0.830

0.840

0.860

0.870

0.890

0.900

0.920

0.930

0.950

0.960

0.980

0.990

1.010

1.020

1.040

1.050

1.070

1.080

1.100

1.110

1.130

1.140

1.160

1.170

1.190

1.200

1.220

1.230

1.250

1.260

1.280

1.290

1.310

1.320

1.340

1.350

1.370

1.380

1.400

1.410

1.430

1.440

1.460

1.470

1.490

1.500

Ratio

Cou

ntofreco

rds

A B C D E F G H I J K L M N O P Q R S

l EffectofrestrictingBonusMalus

l OtherfactorsnotverycorrelatedwithBonusMalus


Testing the effectiveness of restrictions

0

2000

4000

6000

8000

10000

12000

0.800

0.810

0.830

0.840

0.860

0.870

0.890

0.900

0.920

0.930

0.950

0.960

0.980

0.990

1.010

1.020

1.040

1.050

1.070

1.080

1.100

1.110

1.130

1.140

1.160

1.170

1.190

1.200

1.220

1.230

1.250

1.260

1.280

1.290

1.310

1.320

1.340

1.350

1.370

1.380

1.400

1.410

1.430

1.440

1.460

1.470

1.490

1.500

Ratio

Cou

ntofreco

rds

A B C D E F G H I J K L M N O P Q R S

l EffectofrestrictingBonusMalus

l Addinginameasureofclaimfreeyears


Restrictions

lOnlyuseto"getaround"restrictions

lAcommercialsmoothingisacommercialsmoothing

lApplyatriskpremiumstage

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