Nonlife Actuarial Models: Theory, Methods and Evaluation (International Series on Actuarial Science)

542 Pages·2009·3.16 MB·English

This page intentionally left blank
NonlifeActuarialModels
Actuaries must pass exams, but more than that: they must put knowledge
intopractice.ThiscoherentbookgivescompletesyllabuscoverageforExam
C of the Society of Actuaries (SOA) while emphasizing the concepts and
practical application of nonlife actuarial models. Ideal for those approaching
their professional exams, it is also a class-tested textbook for undergraduate
universitycoursesinactuarialscience.
AllthetopicsthatstudentsneedtoprepareforExamCarehere,including
modelingoflosses,risk,andruintheories,credibilitytheoryandapplications,
andempiricalimplementationoflossmodels.Thebookalsocoversmorerecent
topics,suchasriskmeasuresandbootstrapping.Readersareassumedtohave
studiedstatisticalinferenceandprobabilityattheintroductoryundergraduate
level.
Numerousexamplesandexercisesareprovided,withmanyexercisesadapted
from past Exam C questions. Computational notes on the use of Excel are
included.Teachingslidesareavailablefordownload.
InternationalSeriesonActuarialScience
ChristopherDaykin,IndependentConsultantandActuary
AngusMacdonald,Heriot-WattUniversity
The International Series on Actuarial Science, published by Cambridge
UniversityPressinconjunctionwiththeInstituteofActuariesandtheFaculty
of Actuaries, contains textbooks for students taking courses in or related to
actuarial science, as well as more advanced works designed for continuing
professional development or for describing and synthesizing research. The
seriesisavehicleforpublishingbooksthatreflectchangesanddevelopmentsin
thecurriculum,thatencouragetheintroductionofcoursesonactuarialscience
in universities, and that show how actuarial science can be used in all areas
wherethereislong-termfinancialrisk.

NONLIFE ACTUARIAL MODELS
Theory, Methods and Evaluation
YIU-KUEN TSE
SingaporeManagementUniversity
CAMBRIDGEUNIVERSITYPRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521764650
© Y.-K. Tse 2009
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2009
ISBN-13 978-0-511-65198-4 eBook (NetLibrary)
ISBN-13 978-0-521-76465-0 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
ToVicky

Contents
Prefacepagexiii
Notationandconvention xv
PartI Lossmodels 1
1 Claim-frequencydistribution 3
1.1 Claimfrequency,claimseverity,andaggregateclaim 4
1.2 Reviewofstatistics 4
1.3 Somediscretedistributionsforclaimfrequency 6
1.3.1 Binomialdistribution 7
1.3.2 Geometricdistribution 8
1.3.3 Negativebinomialdistribution 9
1.3.4 Poissondistribution 11
1.4 The(a,b,0)classofdistributions 15
1.5 Somemethodsforcreatingnewdistributions 20
1.5.1 Compounddistribution 21
1.5.2 Mixturedistribution 31
1.6 Excelcomputationnotes 34
1.7 Summaryandconclusions 34
Exercises 36
2 Claim-severitydistribution 41
2.1 Reviewofstatistics 42
2.1.1 Survivalfunctionandhazardfunction 42
2.1.2 Mixeddistribution 44
2.1.3 Expectedvalueoffunctionofrandomvariable 45
2.1.4 Distributionoffunctionofrandomvariable 46
2.2 Somecontinuousdistributionsforclaimseverity 49
2.2.1 Exponentialdistribution 49
vii
viii Contents
2.2.2 Gammadistribution 50
2.2.3 Weibulldistribution 51
2.2.4 Paretodistribution 51
2.3 Somemethodsforcreatingnewdistributions 52
2.3.1 Transformationofrandomvariable 53
2.3.2 Mixturedistribution 54
2.3.3 Splicing 58
2.4 Tailpropertiesofclaimseverity 59
2.5 Effectsofcoveragemodifications 66
2.5.1 Deductible 66
2.5.2 Policylimit 72
2.5.3 Coinsurance 73
2.5.4 Effectsofinflation 76
2.5.5 Effectsofdeductibleonclaimfrequency 77
2.6 Excelcomputationnotes 79
2.7 Summaryandconclusions 81
Exercises 82
3 Aggregate-lossmodels 86
3.1 Individualriskandcollectiveriskmodels 87
3.2 Individualriskmodel 88
3.2.1 Exactdistributionusingconvolution 89
3.2.2 ExactdistributionusingtheDePrilrecursion 92
3.2.3 Approximationsoftheindividualriskmodel 94
3.3 Collectiveriskmodel 96
3.3.1 Propertiesofcompounddistributions 96
3.3.2 Panjerrecursion 98
3.3.3 Approximationsofthecollectiveriskmodel 100
3.3.4 CompoundPoissondistributionandindividual
riskmodel 102
3.4 Coveragemodificationsandstop-lossreinsurance 103
3.5 Summaryandconclusions 108
Exercises 108
PartII Riskandruin 113
4 Riskmeasures 115
4.1 Usesofriskmeasures 116
4.2 Somepremium-basedriskmeasures 117
4.3 Axiomsofcoherentriskmeasures 118
4.4 Somecapital-basedriskmeasures 120
4.4.1 Value-at-Risk(VaR) 120