Non-life insurance pricing with generalized linear models

181 Pages·2010·1.26 MB·English

by  Esbjörn Ohlsson

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EAA series/EAA Lecture Notes
Wüthrich,M.V.;Bühlmann,H.;Furrer,H.Market-ConsistentActuarialValuation2007
Lütkebohmert,E.ConcentrationRiskinCreditPortfolios2009
Sundt,B.;Vernic,R.RecursionsforConvolutionsandCompoundDistributions
withInsuranceApplications2009
Ohlsson,E.;Johansson,B.Non-LifeInsurancePricingwithGeneralizedLinear
Models2010
Esbjörn Ohlsson (cid:2) Björn Johansson
Non-Life Insurance
Pricing
with Generalized
Linear Models
Dr.EsbjörnOhlsson Dr.BjörnJohansson
LänsförsäkringarAlliance LänsförsäkringarAlliance
10650Stockholm 10650Stockholm
Sweden Sweden
[email protected]
ISSN1869-6929 e-ISSN1869-6937
ISBN978-3-642-10790-0 e-ISBN978-3-642-10791-7
DOI10.1007/978-3-642-10791-7
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2010923536
MathematicsSubjectClassification(2000):91Gxx,62Jxx,97M30
©Springer-VerlagBerlinHeidelberg2010
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Toour families
Preface
Non-lifeinsurancepricingistheartofsettingthepriceofaninsurancepolicy,tak-
ingintoconsiderationvariouspropertiesoftheinsuredobjectandthepolicyholder.
The main source on which to base the decision is the insurance company’s own
historicaldataonpoliciesandclaims,sometimessupplementedwithdatafromex-
ternalsources.Inatariffanalysis,theactuaryusesthisdatatofindamodelwhich
describes how the claim cost of an insurance policy depends on a number of ex-
planatory variables. In the 1990’s British actuaries introduced generalized linear
models (GLMs) as a tool for tariff analysis and this has now become the standard
approachinmanycountries.
ThisbookfocusesonmethodsbasedonGLMsthatwehavefoundusefulinour
actuarialpractice,andintendtoprovideasetoftoolsthatfillsmostneedsforatariff
analysis.Successiveversionsofthetexthavebeenthebasisforacourseinnon-life
insurance mathematics at Stockholm University since 2001. This course is part of
thecurriculumsetupbytheSwedishActuarialSocietytomeettheEuropeanCore
Syllabusforactuarialeducation.
The aim is to present the basic theory of GLMs in a tariff analysis setting, and
also to give some useful extensions of standard GLM theory that are not in com-
monuse,viz.theincorporationofrandomeffectsandtheuseofsmoothingsplines.
Randomeffectmodelscanbeusedtogreatadvantageforhandlingcategoricalvari-
ableswithalargenumberofpossiblelevels,andthereisaninterestingconnection
withthetraditionalactuarialfieldofcredibilitytheorythatwillbedevelopedhere.
Smoothingsplinesisapowerfulmethodformodelingtheeffectofcontinuousvari-
ables;suchanalysesareoftenpresentedunderthenamegeneralizedadditivemodels
(GAMs).WhileGAMshavebeenusedinbiostatisticsforseveralyears,theyhave
notyetfoundtheirwayintotheactuarialdomaintoanylargeextent.
Thetextisintendedfor practicingactuariesandactuarialstudentswitha back-
ground in mathematics and mathematical statistics: proofs are included whenever
possible without going into asymptotic theory. The prerequisites are basic univer-
sitymathematics—includingagoodknowledgeoflinearalgebraandcalculus—and
basicknowledgeofprobabilitytheory,likelihood-basedstatisticalinferenceandre-
gressionanalysis.Asecondcourseinprobability,sayatthelevelofGut[Gu95],is
useful.
vii
viii Preface
Inordertoprovidestudentswiththepossibilitytoworkwithrealdataofsome
complexity,wehavecompiledacollectionofdatasetstobeusedinconnectionwith
anumberofcasestudies.ThedatawasprovidedbyLänsförsäkringarAllianceand
isavailableatwww.math.su.se/GLMbook.
Forworkingthroughthecasestudies,asuitablesoftwareisneeded.InAppendix
wehavegivensomehintsonhowtoproceedusingtheSASsystem,sincethisisa
standardsoftwareatmanyinsurancecompanies.MatlabandRareotherpossibili-
tiesforacoursebasedonthistext.Therearealsosomegoodspecializedsoftware
packages for tariff analysis with GLMs on the market, but these are not generally
availabletostudents;wealsobelievethathavingtowriteyourownprogramshelps
inunderstandingthesubject.
Somesectionsthatarenotnecessaryforunderstandingtherestofthetexthave
beenindicatedbyastar.
We wish to acknowledge the support from Stockholm University, Division of
MathematicalStatistics,wherewebothhavebeenseniorlecturersforseveralyears,
and the Non-life actuarial group at Länsförsakringar Alliance, were we are now
working as actuaries. Special thanks to Professor Rolf Sundberg for providing the
justification for the pure premium confidence intervals. Finally, we wish to thank
ViktorGrgic´ forvaluablecommentsandhelpwiththegraphics.
Stockholm EsbjörnOhlsson
BjörnJohansson
Contents
1 Non-LifeInsurancePricing . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 RatingFactorsandKeyRatios. . . . . . . . . . . . . . . . . . . . 2
1.2 BasicModelAssumptions . . . . . . . . . . . . . . . . . . . . . . 6
1.2.1 MeansandVariances . . . . . . . . . . . . . . . . . . . . 8
1.3 MultiplicativeModels . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 TheMethodofMarginalTotals . . . . . . . . . . . . . . . 11
1.3.2 OneFactorataTime? . . . . . . . . . . . . . . . . . . . . 12
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 TheBasicsofPricingwithGLMs . . . . . . . . . . . . . . . . . . . . 15
2.1 ExponentialDispersionModels . . . . . . . . . . . . . . . . . . . 16
2.1.1 ProbabilityDistributionoftheClaimFrequency . . . . . . 18
2.1.2 AModelforClaimSeverity . . . . . . . . . . . . . . . . . 20
2.1.3 Cumulant-GeneratingFunction,ExpectationandVariance . 21
2.1.4 TweedieModels . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 TheLinkFunction . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.1 CanonicalLink* . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 ParameterEstimation . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3.1 TheMultiplicativePoissonModel. . . . . . . . . . . . . . 30
2.3.2 GeneralResult . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.3 MultiplicativeGammaModelforClaimSeverity . . . . . . 33
2.3.4 ModelingthePurePremium . . . . . . . . . . . . . . . . . 34
2.4 CaseStudy:MotorcycleInsurance . . . . . . . . . . . . . . . . . 35
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3 GLMModelBuilding . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 HypothesisTestingandEstimationofφ . . . . . . . . . . . . . . . 39
3.1.1 Pearson’sChi-SquareandtheEstimationofφ . . . . . . . 42
3.1.2 TestingHierarchicalModels . . . . . . . . . . . . . . . . . 43
3.2 ConfidenceIntervalsBasedonFisherInformation . . . . . . . . . 44
3.2.1 FisherInformation . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 ConfidenceIntervals . . . . . . . . . . . . . . . . . . . . . 45
ix
x Contents
3.2.3 NumericalEquationSolving* . . . . . . . . . . . . . . . . 49
3.2.4 DotheMLEquationsReallyGiveaMaximum?* . . . . . 50
3.2.5 AsymptoticNormalityoftheMLEstimators* . . . . . . . 51
3.3 Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Overdispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 EstimationWithoutDistributionalAssumptions . . . . . . . . . . 58
3.5.1 EstimatingEquations . . . . . . . . . . . . . . . . . . . . 58
3.5.2 TheOverdispersedPoissonModel . . . . . . . . . . . . . 60
3.5.3 DefiningDeviancesfromVarianceFunctions* . . . . . . . 60
3.6 Miscellanea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.6.1 ModelSelection . . . . . . . . . . . . . . . . . . . . . . . 61
3.6.2 Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.6.3 Offsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6.4 PolynomialRegression . . . . . . . . . . . . . . . . . . . 63
3.6.5 LargeClaims. . . . . . . . . . . . . . . . . . . . . . . . . 63
3.6.6 Deductibles* . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.6.7 DeterminingthePremiumLevel. . . . . . . . . . . . . . . 65
3.7 CaseStudy:ModelSelectioninMCInsurance . . . . . . . . . . . 66
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4 Multi-LevelFactorsandCredibilityTheory . . . . . . . . . . . . . . 71
4.1 TheBühlmann-StraubModel . . . . . . . . . . . . . . . . . . . . 74
4.1.1 EstimationofVarianceParameters . . . . . . . . . . . . . 78
4.1.2 ComparisonwithOtherNotation*. . . . . . . . . . . . . . 81
4.2 CredibilityEstimatorsinMultiplicativeModels . . . . . . . . . . 81
4.2.1 EstimationofVarianceParameters . . . . . . . . . . . . . 84
4.2.2 TheBackfittingAlgorithm. . . . . . . . . . . . . . . . . . 84
4.2.3 ApplicationtoCarModelClassification . . . . . . . . . . 86
4.2.4 MorethanOneMLF . . . . . . . . . . . . . . . . . . . . . 87
4.3 ExactCredibility* . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4 HierarchicalCredibilityModels . . . . . . . . . . . . . . . . . . . 90
4.4.1 EstimationofVarianceParameters . . . . . . . . . . . . . 94
4.4.2 CarModelClassification,theHierarchicalCase . . . . . . 95
4.5 CaseStudy:BusInsurance. . . . . . . . . . . . . . . . . . . . . . 96
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5 GeneralizedAdditiveModels . . . . . . . . . . . . . . . . . . . . . . 101
5.1 PenalizedDeviances . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 CubicSplines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3 Estimation—OneRatingVariable . . . . . . . . . . . . . . . . . . 108
5.3.1 NormalCase . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3.2 PoissonCase . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.3 GammaCase . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4 Estimation—SeveralRatingVariables . . . . . . . . . . . . . . . . 114
5.4.1 NormalCase . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.4.2 PoissonCase . . . . . . . . . . . . . . . . . . . . . . . . . 117