Insurance Risk and Ruin

306 Pages·2016·1.777 MB·English

by  David C. M. Dickson

InsuranceRiskandRuin
Thefocusofthisbookisonthetwomajorareasofrisktheory:aggregateclaims
distributionsandruintheory.Foraggregateclaimsdistributions,detaileddescriptions
aregivenofrecursivetechniquesthatcanbeusedintheindividualandcollectiverisk
models.Forthecollectivemodel,thebookdiscussesdifferentclassesofcounting
distribution,andpresentsrecursionschemesforprobabilityfunctionsandmoments.
Fortheindividualmodel,thebookillustratesthethreemostcommonlyapplied
techniques.Beyondtheclassicaltopicsinruintheory,thisneweditionfeaturesan
expandedsectioncoveringtimeofruinproblems,Gerber-Shiufunctions,andthe
applicationofDeVylderapproximations.Suitableforafirstcourseininsurancerisk
theoryandextensivelyclassroomtested,thebookisaccessibletoreaderswithasolid
understandingofbasicprobability.Numerousworkedexamplesareincludedandeach
chapterconcludeswithexercisesforwhichcompletesolutionsareprovided.
david dicksonisProfessorofActuarialStudiesattheUniversityofMelbourne.
Hismainresearchinterestisruintheory,andhehaspublishedmanypapersinthe
leadinginternationalactuarialjournals.HeisaFellowoftheInstituteandFacultyof
ActuariesintheUKandtheInstituteofActuariesofAustralia,andhastwicebeen
awardedtheH.M.JacksonPrizeoftheInstituteofActuariesofAustralia,most
recentlyforthefirsteditionofthisbook.Heisalsoaco-authorofActuarial
MathematicsforLifeContingentRisks,therecommendedtextfortheSocietyof
ActuariesMLCexam.
INTERNATIONAL SERIES ON ACTUARIAL SCIENCE
EditorialBoard
ChristopherDaykin(IndependentConsultantandActuary)
AngusMacdonald(Heriot-WattUniversity)
TheInternationalSeriesonActuarialScience,publishedbyCambridgeUniversityPressincon-
junctionwiththeInstituteandFacultyofActuaries,containstextbooksforstudentstakingcourses
inorrelatedtoactuarialscience,aswellasmoreadvancedworksdesignedforcontinuingpro-
fessional development or for describing and synthesizing research. The series is a vehicle for
publishing books that reflect changes and developments in the curriculum, that encourage the
introductionofcoursesonactuarialscienceinuniversities,andthatshowhowactuarialscience
canbeusedinallareaswherethereislong-termfinancialrisk.
Acompletelistofbooksintheseriescanbefoundatwww.cambridge.org/statistics.Recenttitles
includethefollowing:
InsuranceRiskandRuin(2ndEdition)
DavidC.M.Dickson
PredictiveModelingApplicationsinActuarialScience,Volume2:CaseStudiesinInsurance
EditedbyEdwardW.Frees,GlennMeyers&RichardA.Derrig
ComputationandModellinginInsuranceandFinance
ErikBølviken
PredictiveModelingApplicationsinActuarialScience,Volume1:PredictiveModelingTechniques
EditedbyEdwardW.Frees,RichardA.Derrig&GlennMeyers
ActuarialMathematicsforLifeContingentRisks(2ndEdition)
DavidC.M.Dickson,MaryR.Hardy&HowardR.Waters
SolutionsManualforActuarialMathematicsforLifeContingentRisks(2ndEdition)
DavidC.M.Dickson,MaryR.Hardy&HowardR.Waters
RiskModellinginGeneralInsurance
RogerJ.Gray&SusanM.Pitts
FinancialEnterpriseRiskManagement
PaulSweeting
RegressionModelingwithActuarialandFinancialApplications
EdwardW.Frees
Insurance Risk and Ruin
SecondEdition
DAVID C. M. DICKSON
UniversityofMelbourne
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©DavidC.M.Dickson2017
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Firstpublished2017
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LibraryofCongressCataloguinginPublicationdata
Names:Dickson,D.C.M.(DavidC.M.),1959–author.
Title:Insuranceriskandruin/DavidC.M.Dickson,UniversityofMelbourne.
Description:Cambridge,UnitedKingdom:CambridgeUniversityPress,2017.|
Series:Internationalseriesonactuarialscience|Earlieredition:2010.
|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2016019813|ISBN9781107154605(hardback:alk.paper)
Subjects:LCSH:Risk(Insurance)
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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 24
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 39
3.1 Introduction 39
3.2 PropertiesofPremiumPrinciples 39
3.3 ExamplesofPremiumPrinciples 40
3.4 NotesandReferences 50
3.5 Exercises 51
vii
viii Contents
4 TheCollectiveRiskModel 53
4.1 Introduction 53
4.2 TheModel 54
4.3 TheCompoundPoissonDistribution 57
4.4 TheEffectofReinsurance 60
4.5 RecursiveCalculationofAggregateClaimsDistributions 64
4.6 ExtensionsofthePanjerRecursionFormula 73
4.7 TheApplicationofRecursionFormulae 80
4.8 ApproximateCalculationofAggregateClaimsDistributions 84
4.9 NotesandReferences 90
4.10 Exercises 90
5 TheIndividualRiskModel 95
5.1 Introduction 95
5.2 TheModel 95
5.3 DePril’sRecursionFormula 96
5.4 Kornya’sMethod 99
5.5 CompoundPoissonApproximation 103
5.6 NumericalIllustration 107
5.7 NotesandReferences 110
5.8 Exercises 110
6 IntroductiontoRuinTheory 114
6.1 Introduction 114
6.2 ADiscreteTimeRiskModel 115
6.3 TheProbabilityofUltimateRuin 116
6.4 TheProbabilityofRuininFiniteTime 120
6.5 Lundberg’sInequality 122
6.6 NotesandReferences 125
6.7 Exercises 126
7 ClassicalRuinTheory 127
7.1 Introduction 127
7.2 TheClassicalRiskProcess 127
7.3 PoissonandCompoundPoissonProcesses 129
7.4 DefinitionsofRuinProbability 131
7.5 TheAdjustmentCoefficient 132
7.6 Lundberg’sInequality 135
7.7 SurvivalProbability 137
7.8 TheLaplaceTransformofφ 140
7.9 RecursiveCalculation 144
Contents ix
7.10 ApproximateCalculationofRuinProbabilities 153
7.11 NotesandReferences 156
7.12 Exercises 156
8 AdvancedRuinTheory 160
8.1 Introduction 160
8.2 ABarrierProblem 160
8.3 TheSeverityofRuin 161
8.4 TheMaximumSeverityofRuin 166
8.5 TheSurplusPriortoRuin 168
8.6 TheTimeofRuin 174
8.7 Dividends 200
8.8 NotesandReferences 207
8.9 Exercises 208
9 Reinsurance 211
9.1 Introduction 211
9.2 ApplicationofUtilityTheory 211
9.3 ReinsuranceandRuin 215
9.4 NotesandReferences 229
9.5 Exercises 230
Appendix 232
SolutionstoExercises 234
References 289
Index 292