Fundamentals of Actuarial Mathematics

554 Pages·2014·2.79 MB·English

by  S. David Promislow

Fundamentals
of Actuarial
Mathematics
Third Edition
S. David Promislow
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Fundamentals of
Actuarial Mathematics
i
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Fundamentals of
Actuarial Mathematics
Third Edition
S. David Promislow
YorkUniversity,Toronto,Canada
iii
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Thiseditionfirstpublished2015
©2015JohnWiley&Sons,Ltd
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LibraryofCongressCataloging-in-PublicationData
Promislow,S.David.
Fundamentalsofactuarialmathematics/S.DavidPromislow.–Thirdedition.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-1-118-78246-0(hardback)
1.Insurance–Mathematics. 2.Businessmathematics. I.Title.
HG8781.P76 2014
368′.01–dc23
2014027082
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN:9781118782460
Setin10/12ptTimesbyAptaraInc.,NewDelhi,India
1 2015
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ToGeorgia andGriffith
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Contents
Preface xix
Acknowledgements xxiii
Aboutthecompanionwebsite xxiv
PartI THEDETERMINISTICLIFECONTINGENCIESMODEL 1
1 Introductionandmotivation 3
1.1 Riskandinsurance 3
1.2 Deterministicversusstochasticmodels 4
1.3 Financeandinvestments 5
1.4 Adequacyandequity 5
1.5 Reassessment 6
1.6 Conclusion 6
2 Thebasicdeterministicmodel 7
2.1 Cashflows 7
2.2 Ananalogywithcurrencies 8
2.3 Discountfunctions 9
2.4 Calculatingthediscountfunction 11
2.5 Interestanddiscountrates 12
2.6 Constantinterest 12
2.7 Valuesandactuarialequivalence 13
2.8 Vectornotation 17
2.9 Regularpatterncashflows 18
2.10 Balancesandreserves 20
2.10.1 Basicconcepts 20
2.10.2 Relationbetweenbalancesandreserves 22
2.10.3 Prospectiveversusretrospectivemethods 23
2.10.4 Recursionformulas 24
2.11 Timeshiftingandthesplittingidentity 26
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viii CONTENTS
∗2.11 Changeofdiscountfunction 27
2.12 Internalratesofreturn 28
∗2.13 Forwardpricesandtermstructure 30
2.14 Standardnotationandterminology 33
2.14.1 Standardnotationforcashflowsdiscountedwithinterest 33
2.14.2 Newnotation 34
2.15 Spreadsheetcalculations 34
Notesandreferences 35
Exercises 35
3 Thelifetable 39
3.1 Basicdefinitions 39
3.2 Probabilities 40
3.3 Constructingthelifetablefromthevaluesofq 41
x
3.4 Lifeexpectancy 42
3.5 Choiceoflifetables 44
3.6 Standardnotationandterminology 44
3.7 Asampletable 45
Notesandreferences 45
Exercises 45
4 Lifeannuities 47
4.1 Introduction 47
4.2 Calculatingannuitypremiums 48
4.3 Theinterestandsurvivorshipdiscountfunction 50
4.3.1 Thebasicdefinition 50
4.3.2 Relationsbetweeny forvariousvaluesofx 52
x
4.4 Guaranteedpayments 53
4.5 Deferredannuitieswithannualpremiums 55
4.6 Somepracticalconsiderations 56
4.6.1 Grosspremiums 56
4.6.2 Genderaspects 56
4.7 Standardnotationandterminology 57
4.8 Spreadsheetcalculations 58
Exercises 59
5 Lifeinsurance 61
5.1 Introduction 61
5.2 Calculatinglifeinsurancepremiums 61
5.3 Typesoflifeinsurance 64
5.4 Combinedinsurance–annuitybenefits 64
5.5 Insurancesviewedasannuities 69
5.6 Summaryofformulas 70
5.7 Ageneralinsurance–annuityidentity 70
5.7.1 Thegeneralidentity 70
5.7.2 Theendowmentidentity 71