P1:OTE/OTE/SPH P2:OTE fm JWST022-Promislow October13,2010 13:19 PrinterName:YettoCome Fundamentals of Actuarial Mathematics Fundamentals of Actuarial Mathematics: Second Edition S. David Promislow © 2011 John Wiley & Sons, Ltd. ISBN: 978-0-470-68411-5 P1:OTE/OTE/SPH P2:OTE fm JWST022-Promislow October13,2010 13:19 PrinterName:YettoCome Fundamentals of Actuarial Mathematics Second Edition S. David Promislow YorkUniversity,Toronto,Canada A John Wiley and Sons, Ltd., Publication P1:OTE/OTE/SPH P2:OTE fm JWST022-Promislow October13,2010 13:19 PrinterName:YettoCome Thiseditionfirstpublished2011 ©2011JohnWiley&SonsLtd Registeredoffice JohnWiley&SonsLtd,TheAtrium,SouthernGate,Chichester,WestSussex,PO198SQ,UnitedKingdom Fordetailsofourglobaleditorialoffices,forcustomerservicesandforinformationabouthowtoapplyfor permissiontoreusethecopyrightmaterialinthisbookpleaseseeourwebsiteatwww.wiley.com. 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PrintISBN:978-0-470-68411-5 ePDFISBN:978-0-470-97784-2 ePubISBN:978-0-470-97807-8 TypesetbyAptaraInc.,NewDelhi,India P1:OTE/OTE/SPH P2:OTE fm JWST022-Promislow October13,2010 13:19 PrinterName:YettoCome ToMichael,Corinne,NatalieandRuth P1:OTE/OTE/SPH P2:OTE fm JWST022-Promislow October13,2010 13:19 PrinterName:YettoCome 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 P1:OTE/OTE/SPH P2:OTE fm JWST022-Promislow October13,2010 13:19 PrinterName:YettoCome 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 P1:OTE/OTE/SPH P2:OTE fm JWST022-Promislow October13,2010 13:19 PrinterName:YettoCome 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 P1:OTE/OTE/SPH P2:OTE fm JWST022-Promislow October13,2010 13:19 PrinterName:YettoCome x CONTENTS 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 P1:OTE/OTE/SPH P2:OTE fm JWST022-Promislow October13,2010 13:19 PrinterName:YettoCome 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 P1:OTE/OTE/SPH P2:OTE fm JWST022-Promislow October13,2010 13:19 PrinterName:YettoCome xii CONTENTS 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