Reinsurance : actuarial and statistical aspects

357 Pages·2017·10.25 MB·English

by  Albrecher

Reinsurance:ActuarialandStatisticalAspects
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Reinsurance: Actuarial and Statistical Aspects
HansjörgAlbrecher
UniversityofLausanne
Switzerland
JanBeirlant
KULeuven,BelgiumandUniversityoftheFreeState,
SouthAfrica
JozefL.Teugels
KULeuven
Belgium
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Names:Albrecher,Hansjörg,author.|Beirlant,Jan,author.|Teugels,
JozefL.,author.
Title:Reinsurance:actuarialandstatisticalaspects/byHansjörg
Albrecher,UniversityofLausanne,Switzerland;JanBeirlant,Katholieke
UniversiteitLeuven,BE,UniversityoftheFreeState,SouthAfrica;
JozefL.Teugels,KatholiekeUniversiteitLeuven,BE.
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Subjects:LCSH:Reinsurance.|Actuarialscience.
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correspondinginsuredsumsQiforQSreinsurancewithLxsMreinsurance
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         
v
Contents
Preface ix
1 Introduction 
. WhatisReinsurance? 
. WhyReinsurance? 
. ReinsuranceData 
.. CaseStudyI:MotorLiabilityData 
.. CaseStudyII:DutchFireInsuranceData 
.. CaseStudyIII:AustrianStormClaimData 
.. CaseStudyIV:EuropeanFloodRiskData 
.. CaseStudyV:GroningenEarthquakes 
.. CaseStudyVI:DanishFireInsuranceData 
. NotesandBibliography 
2 ReinsuranceFormsandtheirProperties 
. Quota-shareReinsurance 
.. SomePracticalConsiderations 
. SurplusReinsurance 
. Excess-of-lossReinsurance 
.. MomentCalculations 
.. Reinstatements 
.. FurtherPracticalConsiderations 
. Stop-lossReinsurance 
. LargeClaimReinsurance 
. CombinationsofReinsuranceFormsandGlobalProtections 
. FacultativeContracts 
. NotesandBibliography 
3 ModelsforClaimSizes 
. TailsofDistributions 
. LargeClaims 
. CommonClaimSizeDistributions 
.. Light-tailedModels 
.. Heavy-tailedModels 
. MeanExcessAnalysis 
vi Contents
. FullModels:Splicing 
. MultivariateModellingofLargeClaims 
4 StatisticsforClaimSizes 
. HeavyorLightTails:QQ-andDerivativePlots 
. LargeClaimsModellingthroughExtremeValueAnalysis 
.. EVAforPareto-typeTails 
.. GeneralTailModellingusingEVA 
.. EVAunderUpper-truncation 
. GlobalFits:Splicing,Upper-truncationandIntervalCensoring 
.. Tail-mixedErlangSplicing 
.. Tail-mixedErlangSplicingunderCensoringandUpper-truncation 
. IncorporatingCovariateInformation 
.. Pareto-typeModelling 
.. GeneralizedParetoModelling 
.. RegressionExtremeswithCensoredData 
. MultivariateAnalysisofClaimDistributions 
.. TheMultivariatePOTApproach 
.. MultivariateMixturesofErlangs 
. EstimationofOtherTailCharacteristics 
. FurtherCaseStudies 
. NotesandBibliography 
5 ModelsforClaimCounts 
. GeneralTreatment 
.. MainPropertiesoftheClaimNumberProcess 
. ThePoissonProcessanditsExtensions 
.. TheHomogeneousPoissonProcess 
.. InhomogeneousPoissonProcesses 
.. MixedPoissonProcesses 
.. DoublyStochasticPoissonProcesses 
. OtherClaimNumberProcesses 
.. TheNearlyMixedPoissonModel 
.. InfinitelyDivisibleProcesses 
.. TheRenewalModel 
.. MarkovModels 
. DiscreteClaimCounts 
. StatisticsofClaimCounts 
.. ModellingYearlyClaimCounts 
.. ModellingtheClaimArrivalProcess 
. ClaimNumbersunderReinsurance 
.. NumberofClaimsunderExcess-lossReinsurance 
. NotesandBibliography 
6 TotalClaimAmount 
. GeneralFormulasforAggregatingIndependentRisks 
. ClassicalApproximationsfortheTotalClaimSize 
Contents vii
.. ApproximationsbasedontheFirstFewMoments 
.. AsymptoticApproximationsforLight-tailedClaims 
.. AsymptoticApproximationsforHeavy-tailedClaims 
. PanjerRecursion 
. FastFourierTransform 
. TotalClaimAmountunderReinsurance 
.. ProportionalReinsurance 
.. Excess-lossReinsurance 
.. Stop-lossReinsurance 
. NumericalIllustrations 
. AggregationforDependentRisks 
. NotesandBibliography 
7 ReinsurancePricing 
. ClassicalPrinciplesofPremiumCalculation 
. SolvencyConsiderations 
.. TheRuinProbability 
.. One-yearTimeHorizonandCostofCapital 
. PricingProportionalReinsurance 
. PricingNon-proportionalReinsurance 
.. ExposureRating 
.. ExperienceRating 
.. AggregatePurePremium 
. TheAggregateRiskMargin 
. LeadingandFollowingReinsurers 
. NotesandBibliography 
8 ChoiceofReinsurance 
. DecisionCriteria 
. ClassicalOptimalityResults 
.. Pareto-optimalRiskSharing 
.. StochasticOrdering 
.. MinimizingRetainedVariance 
.. MaximizingExpectedUtility 
.. MinimizingtheRuinProbability 
.. CombiningReinsuranceTreatiesoverSubportfolios 
. SolvencyConstraintsandCostofCapital 
. MinimizingOtherRiskMeasures 
. CombiningReinsuranceTreaties 
. ReinsuranceChains 
. DynamicReinsurance 
. BeyondPiecewiseLinearContracts 
. NotesandBibliography 
9 Simulation 
. TheMonteCarloMethod 
. VarianceReductionTechniques 
viii Contents
.. ConditionalMonteCarlo 
.. ImportanceSampling 
.. ControlVariates 
. Quasi-MonteCarloTechniques 
. NotesandBibliography 
10 FurtherTopics 
. MoreonLargeClaimReinsurance 
.. TheOrderedClaims 
.. LargeClaimReinsurance 
.. ECOMOR 
. AlternativeRiskTransfer 
.. NotesandBibliography 
. ReinsuranceandFinance 
. CatastrophicRisk 
References 
Index 
ix
Preface
Reinsurance is a fascinating field. Several of the challenges of classical insurance are
amplifiedforreinsurance,particularlywhenitcomestodealingwithextremesituations
likelargeclaimsandrareevents.Thisposesparticularchallengesforthemodellingof
claims and their occurrence, which often needs to be based on only few data points.
Inaddition,intermsofbetterdiversificationofusual-scaleriskonthelocalandglobal
levelaswellasintermsofthedevelopmentofinnovativeandsustainabletechniquesto
dealwithrisksofanunusualkind,reinsurersplayacrucialroleintheinsuranceprocess.
Thisalsoreflectsonpractitionersandresearchersinvolvedinsuchtopics,astheyhave
torethinkclassicalmodelsinordertocopesuccessfullywiththerespectivechallenges.
Overtheyears,therehasbeenenormousresearchactivityonproblemsconnectedto
reinsurance. Close to % of the references in our literature list have appeared over
the last  years, with a steep upward gradient over the last – years. While there
exist some excellent classical textbooks on reinsurance either from the academic or
the practitioner’s side, our impression was that there was no modern reference book
availablethatgaveanoverviewoftheacademicresearchlandscapeinthisfieldandalso
puts it in perspective with the practical viewpoint. The main reason for writing this
bookwastotrytoaddressthisgap,atleastforactuarialandstatisticalmatters.Asall
the authors are from academia, there naturally remains a bias towards the academic
angle.However,numerousandenlighteningdiscussionswithinsuranceandreinsurance
practitionersoverthelastfewyearshavemotivatedustoproducethecurrentaccount,
hoping and trying to further bridge the two worlds. The focus of the book is on
modelling together with the statistical challenges that go along with it. We illustrate
the discussed statistical approaches alongside six case studies of insurance loss data
sets,rangingfromMTPLoverfiretostormandfloodlossdata.Someofthepresented
material also contains new results that have not yet been published in the research
literature.Wehopethatthematerialpresentedcantriggernewresearchquestionsand
fosterthecommunicationbetween(re)insurancepractitionersandacademicsworking
inthesefields.Oneofourmaingoalswastogiveanup-to-dateoverviewoftherelevant
researchliteratureandtoframeittoquestionsthatmatterinreinsurancepractice.Since
this a vast topic, we naturally had to take various compromises and we apologize for
possibleomissionsoneitherside.
The book is written for researchers with an interest in reinsurance problems, for
graduate students with a basic knowledge of probability and statistics as well as for
practitionersinthefield.
x Preface
WestartwithageneralintroductiontothefieldinChapter,presentingsomebasic
factsandmotivationsforreinsuranceactivities.Wealsointroducethesixreal-lifecase
studies that will accompany the considerations throughout the book. In Chapter ,
we discuss the most common reinsurance forms and their properties, together with
somepracticalaspectsoftheirimplementation.Chapterisdedicatedtomotivating
anddevelopingmodelsforclaimsizedistributionsthatarecommonly used.Herewe
emphasizethoseaspectsfromactuarialmathematicsthatarerelevantforreinsurance.
Reinsurance is often invoked in the presence of large claims, therefore we need a
thorough discussion of models capable of catching the essentials of what actuaries
would call large. Chapter  contains detailed guidelines on how to proceed in the
modelchoicewhenactuallyfacingdata.Throughoutthetext,weillustratethepresented
procedures for our case studies. Chapter  proceeds with models for claim numbers,
both from a conceptual and a practical viewpoint. We also provide guidelines for a
statisticalanalysisofdatasetsinthiscontext.Thetwoingredients(claimnumbersand
claim sizes) are then used in Chapter  for the aggregation of the claims. Emphasis
is put on the aggregation of independent risks, and we describe both numerical and
asymptotic methods in detail. The case of dependence in the aggregation process is
alsodiscussedbriefly,althoughnotindetail,astheresultstypicallyareverysensitive
to the particular dependence structure used in the modeling process, and often the
number of data points does not allow one such model to be decisively favored over
another. It is beyond the scope of this book to discuss all such approaches. Chapter
treatsimportantactuarialaspectsofreinsurancepricing,onceadistributionforthe
individual (or aggregate) risk is available (or, rather, decided upon). In Chapter  we
discuss some guidelines on possible criteria for the choice of reinsurance forms and
the respective consequences on the optimal choice of contracts. The identification
of optimal reinsurance forms has been a very active research field recently and it is
impossibletoreflectallthesecontributionsinonebookchapterinanexhaustiveway.
We instead provide an overview of some of the main approaches and contributions
alongside a structure in terms of decision criteria, with an emphasis on the intuition
behind the results. Since stochastic simulation is an essential tool in many models
relevant in reinsurance, we cover this topic in Chapter  and discuss some variance
reductiontechniquesthatcanhelptoconsiderablyspeedupcalculations.Chapter
thenexaminessomefurthertopics.Wefirstprovidemoreinformationonlargeclaim
analysis,andcontinuewithanoverviewofalternativerisktransferproducts,whichcan
serveasacomplementtotraditionalreinsurance.Wealsohighlighttheroleoffinance
inreinsuranceandfinishwithasectiononcatastropheinsurance.Withinthechapters
andinparticularattheendofthechaptersweprovidelinksforfurtherreading.
Manyofthetopicsdealtwithinthebookapplytobothnon-lifeandlifeinsurance.
Even when there is a clear emphasis on non-life insurance throughout, we hope that
some of our attempts may help to also be of service to life insurance. As the title
suggests, thisbook isabout (traditional)actuarialaswell as statisticalaspectsarising
in reinsurance. As is outlined in Chapter , reinsurance also serves financial and
managementpurposesinpractice.Correspondingly,theroleofcapitalisnowadaysan
important ingredient in managing and steering reinsurance companies, and financial
pricing techniques for reinsurance contracts as well as general capital management
tools eventually have to complement the actuarial approach. While we do consider
such aspects when discussing the pricing and the possible choice of contracts in