Table Of ContentInsuranceRiskandRuin
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
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Firstpublished2005
Reprinted2006
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
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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
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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
Description:Based on the author's experience of teaching final-year actuarial students in Britain and Australia, and suitable for a first course in insurance risk theory, this book focuses on the two major areas of risk theory - aggregate claims distributions and ruin theory. For aggregate claims distributions,
