Fundamentals of Actuarial Mathematics, Second Edition

464 Pages·2011·5.73 MB·English

by  S. David Promislow(auth.)

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Fundamentals of
Actuarial Mathematics
Fundamentals of Actuarial Mathematics: Second Edition S. David Promislow
© 2011 John Wiley & Sons, Ltd. ISBN: 978-0-470-68411-5
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Fundamentals of
Actuarial Mathematics
Second Edition
S. David Promislow
YorkUniversity,Toronto,Canada
A John Wiley and Sons, Ltd., Publication
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Thiseditionfirstpublished2011
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LibraryofCongressCataloging-in-PublicationData
Promislow,S.David.
Fundamentalsofactuarialmathematics/S.DavidPromislow.–2nded.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-470-68411-5(cloth)
1.Insurance—Mathematics. 2.Businessmathematics. I.Title.
HG8781.P762010
368(cid:1).01—dc22
2010029552
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
PrintISBN:978-0-470-68411-5
ePDFISBN:978-0-470-97784-2
ePubISBN:978-0-470-97807-8
TypesetbyAptaraInc.,NewDelhi,India
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ToMichael,Corinne,NatalieandRuth
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Contents
Preface xvii
Acknowledgements xxi
Notationindex xxiii
PartI THEDETERMINISTICMODEL 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 Regularpatterncashflows 17
2.9 Balancesandreserves 19
2.9.1 Basicconcepts 19
2.9.2 Relationshipbetweenbalancesandreserves 21
2.9.3 Prospectiveversusretrospectivemethods 22
2.9.4 Recursionformulas 23
2.10 Timeshiftingandthesplittingidentity 24
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viii CONTENTS
*2.11 Changeofdiscountfunction 26
*2.12 Internalratesofreturn 27
*2.13 Forwardpricesandtermstructure 29
2.14 Standardnotationandterminology 31
2.14.1 Standardnotationforcashflowsdiscountedwithinterest 31
2.14.2 Newnotation 32
2.15 Spreadsheetcalculations 33
2.16 Notesandreferences 33
2.17 Exercises 33
3 Thelifetable 37
3.1 Basicdefinitions 37
3.2 Probabilities 38
3.3 Constructingthelifetablefromthevaluesofq 39
x
3.4 Lifeexpectancy 40
3.5 Choiceoflifetables 42
3.6 Standardnotationandterminology 42
3.7 Asampletable 43
3.8 Notesandreferences 43
3.9 Exercises 43
4 Lifeannuities 45
4.1 Introduction 45
4.2 Calculatingannuitypremiums 46
4.3 Theinterestandsurvivorshipdiscountfunction 48
4.3.1 Thebasicdefinition 48
4.3.2 Relationsbetween y forvariousvaluesofx 50
x
4.3.3 Tontines 51
4.4 Guaranteedpayments 52
4.5 Deferredannuitieswithannualpremiums 53
4.6 Somepracticalconsiderations 54
4.6.1 Grosspremiums 54
4.6.2 Genderaspects 55
4.7 Standardnotationandterminology 55
4.8 Spreadsheetcalculations 56
4.9 Exercises 57
5 Lifeinsurance 60
5.1 Introduction 60
5.2 Calculatinglifeinsurancepremiums 60
5.3 Typesoflifeinsurance 63
5.4 Combinedinsurance–annuitybenefits 63
5.5 Insurancesviewedasannuities 67
5.6 Summaryofformulas 68
5.7 Ageneralinsurance–annuityidentity 69
5.7.1 Themaintheorem 69
5.7.2 Theendowmentidentity 69
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CONTENTS ix
5.8 Standardnotationandterminology 71
5.8.1 Singlepremiumnotation 71
5.8.2 Annualpremiumnotation 72
5.8.3 Identities 72
5.9 Spreadsheetapplications 72
5.10 Exercises 73
6 Insuranceandannuityreserves 76
6.1 Introductiontoreserves 76
6.2 Thegeneralpatternofreserves 79
6.3 Recursion 80
6.4 Detailedanalysisofaninsuranceorannuitycontract 81
6.4.1 Gainsandlosses 81
6.4.2 Therisk–savingsdecomposition 83
6.5 Interestandmortalitybasesforreserves 84
6.6 Nonforfeiturevalues 86
6.7 Policiesinvolvinga‘returnofthereserve’ 87
6.8 Premiumdifferenceandpaid-upformulas 88
6.8.1 Premiumdifferenceformulas 88
6.8.2 Paid-upformulas 89
6.8.3 Levelendowmentreserves 89
*6.9 Universallifeandvariableannuities 89
6.9.1 Universallife 90
6.9.2 Variableannuities 93
6.10 Standardnotationandterminology 94
6.11 Spreadsheetapplications 95
6.12 Exercises 96
7 Fractionaldurations 101
7.1 Introduction 101
7.2 Cashflowsdiscountedwithinterestonly 102
7.3 Lifeannuitiespaidmthly 104
7.3.1 Uniformdistributionofdeaths 104
7.3.2 Presentvalueformulas 105
7.4 Immediateannuities 106
7.5 Approximationandcomputation 107
*7.6 Fractionalperiodpremiumsandreserves 109
7.7 Reservesatfractionaldurations 110
7.8 Notesandreferences 112
7.9 Exercises 112
8 Continuouspayments 115
8.1 Introductiontocontinuousannuities 115
8.2 Theforceofdiscount 116
8.3 Theconstantinterestcase 117
8.4 Continuouslifeannuities 118
8.4.1 Basicdefinition 118
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8.4.2 Evaluation 119
8.4.3 Lifeexpectancyrevisited 120
8.5 Theforceofmortality 121
8.6 Insurancespayableatthemomentofdeath 122
8.6.1 Basicdefinitions 122
8.6.2 Evaluation 123
8.7 Premiumsandreserves 125
8.8 Thegeneralinsurance–annuityidentityinthecontinuouscase 126
8.9 Differentialequationsforreserves 127
8.10 Someexamplesofexactcalculation 128
8.10.1 Constantforceofmortality 128
8.10.2 Demoivre’slaw 129
8.10.3 Anexampleofthesplittingidentity 130
8.11 Standardactuarialnotationandterminology 131
8.12 Notesandreferences 131
8.13 Exercises 132
9 Selectmortality 136
9.1 Introduction 136
9.2 Selectandultimatetables 137
9.3 Changesinformulas 138
9.4 Projectionsinannuitytables 140
9.5 Furtherremarks 141
9.6 Exercises 141
10 Multiple-lifecontracts 143
10.1 Introduction 143
10.2 Thejoint-lifestatus 143
10.3 Joint-lifeannuitiesandinsurances 145
10.4 Last-survivorannuitiesandinsurances 146
10.5 Momentofdeathinsurances 147
10.6 Thegeneraltwo-lifeannuitycontract 149
10.7 Thegeneraltwo-lifeinsurancecontract 150
10.8 Contingentinsurances 151
10.8.1 First-deathcontingentinsurances 151
10.8.2 Second-deathcontingentinsurances 152
10.8.3 Moment-of-deathcontingentinsurances 153
10.8.4 Generalcontingentprobabilities 153
10.9 Durationproblems 154
10.10 Applicationstoannuitycreditrisk 157
10.11 Standardnotationandterminology 158
10.12 Spreadsheetapplications 159
10.13 Notesandreferences 159
10.14 Exercises 159
11 Multiple-decrementtheory 164
11.1 Introduction 164
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CONTENTS xi
11.2 Thebasicmodel 164
11.2.1 Themultiple-decrementtable 165
11.2.2 Quantitiescalculatedfromthemultiple-decrementtable 166
11.3 Insurances 167
11.4 Determiningthemodelfromtheforcesofdecrement 168
11.5 Theanalogywithjoint-lifestatuses 169
11.6 Amachineanalogy 169
11.6.1 Method1 170
11.6.2 Method2 171
11.7 Associatedsingle-decrementtables 173
11.7.1 Themainmethods 173
11.7.2 Forcesofdecrementintheassociated
single-decrementtables 174
11.7.3 Conditionsjustifyingthetwomethods 175
11.7.4 Otherapproaches 178
11.8 Notesandreferences 179
11.9 Exercises 179
12 Expenses 182
12.1 Introduction 182
12.2 Effectonreserves 184
12.3 Realisticreserveandbalancecalculations 185
12.4 Notesandreferences 187
12.5 Exercises 187
PartII THESTOCHASTICMODEL 189
13 Survivaldistributionsandfailuretimes 191
13.1 Introductiontosurvivaldistributions 191
13.2 Thediscretecase 192
13.3 Thecontinuouscase 193
13.3.1 Thebasicfunctions 194
13.3.2 Propertiesofµ 194
13.3.3 Modes 195
13.4 Examples 195
13.4.1 Theexponentialdistribution 195
13.4.2 Theuniformdistribution 195
13.4.3 TheGompertz–Makehamdistribution 196
13.5 Shifteddistributions 197
13.6 Thestandardapproximation 198
13.7 Thestochasticlifetable 199
13.8 Lifeexpectancyinthestochasticmodel 201
13.9 Stochasticinterestrates 202
13.10 Notesandreferences 202
13.11 Exercises 203
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14 Thestochasticapproachtoinsuranceandannuities 205
14.1 Introduction 205
14.2 Thestochasticapproachtoinsurancebenefits 206
14.2.1 Thediscretecase 206
14.2.2 Thecontinuouscase 206
14.2.3 Approximation 207
14.2.4 Endowmentinsurances 208
14.3 Thestochasticapproachtoannuitybenefits 209
14.3.1 Discreteannuities 209
14.3.2 Continuousannuities 212
*14.4 Deferredcontracts 213
14.5 Thestochasticapproachtoreserves 214
14.6 Thestochasticapproachtopremiums 215
14.6.1 Theequivalenceprinciple 215
14.6.2 Percentilepremiums 216
14.6.3 Aggregatepremiums 217
14.6.4 Generalpremiumprinciples 220
14.7 Thevarianceof L 220
r
14.8 Standardnotationandterminology 223
14.9 Notesandreferences 223
14.10 Exercises 224
15 Simplificationsunderlevelbenefitcontracts 228
15.1 Introduction 228
15.2 Variancecalculationsinthecontinuouscase 228
15.2.1 Insurances 228
15.2.2 Annuities 229
15.2.3 Prospectivelosses 229
15.2.4 Usingequivalenceprinciplepremiums 229
15.3 Variancecalculationsinthediscretecase 230
15.4 Exactdistributions 231
15.4.1 Thedistributionof Z¯ 231
15.4.2 ThedistributionofY¯ 231
15.4.3 Thedistributionof L 232
15.4.4 ThecasewhereT isexponentiallydistributed 232
15.5 Non-levelbenefitexamples 233
15.5.1 Terminsurance 233
15.5.2 Deferredinsurance 234
15.5.3 Anannualpremiumpolicy 234
15.6 Exercises 235
16 Theminimumfailuretime 238
16.1 Introduction 238
16.2 Jointdistributions 238
16.3 ThedistributionofT 240
16.3.1 Thegeneralcase 240
16.3.2 Theindependentcase 240