Insurance Risk and Ruin (International Series on Actuarial Science) PDF
Preview Insurance Risk and Ruin (International Series on Actuarial Science)
InsuranceRiskandRuin Thefocusofthisbookisonthetwomajorareasofrisktheory: aggregateclaims distributionsandruintheory. Foraggregateclaimsdistributions,detaileddescriptions aregivenofrecursivetechniquesthatcanbeusedintheindividualandcollectiverisk models. Forthecollectivemodel,differentclassesofcountingdistributionare discussed,andrecursionschemesforprobabilityfunctionsandmomentspresented. Fortheindividualmodel,thethreemostcommonlyappliedtechniquesarediscussed andillustrated. Thebookisbasedontheauthor’sexperienceofteachingfinal-year actuarialstudentsinBritainandAustralia,andissuitableforafirstcourseininsurance risktheory. Carehasbeentakentomakethebookaccessibletoreaderswhohavea solidunderstandingofthebasictoolsofprobabilitytheory. Numerousworked examplesareincludedinthetextandeachchapterconcludeswithasetofexcercises forwhichoutlinesolutionsareprovided. InternationalSeriesonActuarialScience MarkDavis,ImperialCollegeLondon JohnHylands,StandardLife JohnMcCutcheon,Heriot-WattUniversity RagnarNorberg,LondonSchoolofEconomics H.Panjer,WaterlooUniversity AndrewWilson,WatsonWyatt TheInternationalSeriesonActuarialScience,publishedbyCambridgeUniversity PressinconjunctionwiththeInstituteofActuariesandtheFacultyofActuaries,will containtextbooksforstudentstakingcoursesinorrelatedtoactuarialscience,aswell asmoreadvancedworksdesignedforcontinuingprofessionaldevelopmentorfor describingandsynthesisingresearch. Theserieswillbeavehicleforpublishingbooks thatreflectchangesanddevelopmentsinthecurriculum,thatencouragethe introductionofcoursesonactuarialscienceinuniversities,andthatshowhowactuarial sciencecanbeusedinallareaswherethereislong-termfinancialrisk. Insurance Risk and Ruin DAVID C. M. DICKSON CentreforActuarialStudies, DepartmentofEconomics,UniversityofMelbourne PUBLISHEDBYTHEPRESSSYNDICATEOFTHEUNIVERSITYOFCAMBRIDGE ThePittBuilding,TrumpingtonStreet,Cambridge,UnitedKingdom CAMBRIDGE UNIVERSITY PRESS TheEdinburghBuilding,CambridgeCB22RU,UK 40West20thStreet,NewYork,NY10011–4211,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia RuizdeAlarco´n13,28014Madrid,Spain DockHouse,TheWaterfront,CapeTown8001,SouthAfrica http://www.cambridge.org (cid:1)C CambridgeUniversityPress2005 Thisbookisincopyright. Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithout thewrittenpermissionofCambridgeUniversityPress. Firstpublished2005 Reprinted2006 PrintedintheUnitedKingdomattheUniversityPress,Cambridge TypefaceTimes10/13pt. SystemLATEX2ε [TB] AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressCataloguinginPublicationdata Dickson,D.C.M.(DavidC.M.),1959– Insuranceriskandruin/DavidC.M.Dickson. p. cm. –(Theinternationalseriesonactuarialscience) Includesbibliographicalreferencesandindex. ISBN0521846404(alk. paper) 1. Insurance–Mathematics. 2. Risk(Insurance–Mathematicalmodels. I.Title. II.Series. HG8781.D532004 368:01–dc22 2004054520 ISBN0521846404hardback ThepublisherhasuseditsbestendeavourstoensurethattheURLsforexternalwebsites referredtointhisbookarecorrectandactiveatthetimeofgoingtopress. However,the publisherhasnoresponsibilityforthewebsitesandcanmakenoguaranteethatasitewill remainliveorthatthecontentisorwillremainappropriate. ToRobertandJanice Contents Preface pagexi 1 Probabilitydistributionsandinsuranceapplications 1 1.1 Introduction 1 1.2 Importantdiscretedistributions 2 1.3 Importantcontinuousdistributions 5 1.4 Mixeddistributions 9 1.5 Insuranceapplications 11 1.6 Sumsofrandomvariables 18 1.7 Notesandreferences 23 1.8 Exercises 24 2 Utilitytheory 27 2.1 Introduction 27 2.2 Utilityfunctions 27 2.3 Theexpectedutilitycriterion 28 2.4 Jensen’sinequality 29 2.5 Typesofutilityfunction 31 2.6 Notesandreferences 36 2.7 Exercises 36 3 Principlesofpremiumcalculation 38 3.1 Introduction 38 3.2 Propertiesofpremiumprinciples 38 3.3 Examplesofpremiumprinciples 39 3.4 Notesandreferences 50 3.5 Exercises 50 4 Thecollectiveriskmodel 52 4.1 Introduction 52 vii viii Contents 4.2 Themodel 53 4.3 ThecompoundPoissondistribution 56 4.4 Theeffectofreinsurance 59 4.5 Recursivecalculationofaggregateclaimsdistributions 64 4.6 ExtensionsofthePanjerrecursionformula 72 4.7 Theapplicationofrecursionformulae 79 4.8 Approximatecalculationofaggregateclaimsdistributions 83 4.9 Notesandreferences 89 4.10 Exercises 89 5 Theindividualriskmodel 93 5.1 Introduction 93 5.2 Themodel 93 5.3 DePril’srecursionformula 94 5.4 Kornya’smethod 97 5.5 CompoundPoissonapproximation 101 5.6 Numericalillustration 105 5.7 Notesandreferences 108 5.8 Exercises 108 6 Introductiontoruintheory 112 6.1 Introduction 112 6.2 Adiscretetimeriskmodel 113 6.3 Theprobabilityofultimateruin 114 6.4 Theprobabilityofruininfinitetime 118 6.5 Lundberg’sinequality 120 6.6 Notesandreferences 123 6.7 Exercises 123 7 Classicalruintheory 125 7.1 Introduction 125 7.2 Theclassicalriskprocess 125 7.3 PoissonandcompoundPoissonprocesses 127 7.4 Definitionsofruinprobability 129 7.5 Theadjustmentcoefficient 130 7.6 Lundberg’sinequality 133 7.7 Survivalprobability 135 7.8 TheLaplacetransformofφ 138 7.9 Recursivecalculation 142 7.10 Approximatecalculationofruinprobabilities 151 Contents ix 7.11 Notesandreferences 153 7.12 Exercises 154 8 Advancedruintheory 157 8.1 Introduction 157 8.2 Abarrierproblem 157 8.3 Theseverityofruin 158 8.4 Themaximumseverityofruin 163 8.5 Thesurpluspriortoruin 165 8.6 Thetimeofruin 172 8.7 Dividends 180 8.8 Notesandreferences 186 8.9 Exercises 187 9 Reinsurance 190 9.1 Introduction 190 9.2 Applicationofutilitytheory 190 9.3 Reinsuranceandruin 194 9.4 Notesandreferences 205 9.5 Exercises 206 References 208 Solutiontoexercises 211 Index 227
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