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Springer Finance Mario V. Wüthrich Michael Merz Financial Modeling, Actuarial Valuation and Solvency in Insurance Springer Finance EditorialBoard MarcoAvellaneda GiovanniBarone-Adesi MarkBroadie MarkH.A.Davis EmanuelDerman ClaudiaKlüppelberg WalterSchachermayer Springer Finance SpringerFinanceisaprogrammeofbooksaddressingstudents,academicsandprac- titioners working on increasingly technical approaches to the analysis of financial markets. It aims to cover a variety of topics, not only mathematical finance but foreignexchanges,termstructure,riskmanagement,portfoliotheory,equityderiva- tives,andfinancialeconomics. Forfurthervolumes: http://www.springer.com/series/3674 Mario V. Wüthrich (cid:2) Michael Merz Financial Modeling, Actuarial Valuation and Solvency in Insurance MarioV.Wüthrich MichaelMerz RiskLab FacultyforEconomicandSocialStudies DepartmentofMathematics DepartmentofBusinessAdministration ETHZurich UniversityofHamburg Zurich,Switzerland Hamburg,Germany ISSN1616-0533 ISSN2195-0687(electronic) ISBN978-3-642-31391-2 ISBN978-3-642-31392-9(eBook) DOI10.1007/978-3-642-31392-9 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013936252 MathematicsSubjectClassification: 62P05,91G30 JELClassification: G22,D52,D53,D82,E43,G12,G17,G32,G38 ©Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub- lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect tothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Acknowledgements This book is the product of an ongoing project we have been working on for sev- eral years. As such it was not really defined as a project but it is rather the re- sult of many activities we have been involved in. These include our own prac- tical experience; discussions with regulators, scientists, practitioners, politicians, other decision-makers, colleagues and students; continuing education at confer- ences, workshops, working groups and our own lectures. We are deeply grateful toETHZürichandtoUniversityofHamburg.Duringallthesetimeswewerevery generously supported by our departments at these universities, and we have and continue to experience these environments as stimulating and motivating. Special thank-you’s are reserved for Prof. Hans Bühlmann and Prof. Paul Embrechts for theircontinuedsupport. We greatly appreciate that the present manuscript profits from various inspir- ingdiscussions,continuativethoughts,helpfulcontributionsandcriticalcomments with and by several people: Hansjörg Albrecher, Peter Antal, Philipp Arbenz, Manuela Baumann, Hans Bühlmann, Bikramjit Das, Catherine Donnelly, Karl- Theodor Eisele, Paul Embrechts, Peter England, Vicky Fasen, Damir Filipovic´, Alois Gisler, Sebastian Happ, Enkelejd Hashorva, Frank Häusler, John Hibbert, LaurentHuber,PhilippKeller,RogerLaeven,AlexanderMcNeil,ChristophMöhr, Antoon Pelsser, Enrico Perotti, Eckhard Platen, Simon Rentzmann, Robert Salz- mann,MarcSarbach,UrsSchubiger,PavelShevchenko,WernerStahel,DavidSte- fanovits,JosefTeichmann,AndreasTsanakas,RichardVerrall,FrankWeber,Armin Wolf and Hans Peter Würmli. We especially thank Manuela Baumann for coding Example3.21.Moreover,weappreciatethatseveralanonymousreviewershaveread previousversionsofthismanuscriptverycarefully.Theyhaveapproachedthesub- jectfromseveraldifferentangleswhichhasledustoprovideamorecomprehensive andcompletedescriptionofthetopicandhelpedusbridgeafewgapsinprevious versionsofthemanuscript. Special thanks go to Alessia, Luisa, Anja, Rosmarie, Valo, Coral, Jürg, Gior- gio, Matthias, Stephan, Ted, Juvy, Ursin, Francesco, Peter and Peter, Fritz, Reini. v vi Acknowledgements Last but not least we thank Dave, Martin and Andy for endlessly enjoying the silence. Zurich,Switzerland MarioV.Wüthrich Hamburg,Germany MichaelMerz February2013 Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 FullBalanceSheetApproach . . . . . . . . . . . . . . . . . . . . 3 1.2 SolvencyConsiderations . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 FurtherModelingIssues . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 OutlineofThisBook . . . . . . . . . . . . . . . . . . . . . . . . 6 PartI FinancialValuationPrinciples 2 StatePriceDeflatorsandStochasticDiscounting . . . . . . . . . . . 11 2.1 ZeroCouponBondsandTermStructureofInterestRates . . . . . 11 2.1.1 MotivationforDiscounting . . . . . . . . . . . . . . . . . 11 2.1.2 SpotRatesandTermStructureofInterestRates . . . . . . 12 2.1.3 EstimatingtheYieldCurve . . . . . . . . . . . . . . . . . 15 2.2 BasicDiscreteTimeStochasticModel . . . . . . . . . . . . . . . 18 2.2.1 ValuationatTime0 . . . . . . . . . . . . . . . . . . . . . 19 2.2.2 InterpretationofStatePriceDeflators . . . . . . . . . . . . 22 2.2.3 ValuationatTimet>0 . . . . . . . . . . . . . . . . . . . 23 2.3 EquivalentMartingaleMeasure . . . . . . . . . . . . . . . . . . . 26 2.3.1 BankAccountNumeraire . . . . . . . . . . . . . . . . . . 26 2.3.2 MartingaleMeasureandtheFTAP . . . . . . . . . . . . . 27 2.4 MarketPriceofRisk . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 SpotRateModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1 GeneralGaussianSpotRateModels. . . . . . . . . . . . . . . . . 35 3.2 One-FactorGaussianAffineTermStructureModels . . . . . . . . 38 3.3 DiscreteTimeOne-FactorVasicekModel . . . . . . . . . . . . . . 41 3.3.1 SpotRateDynamicsonaYearlyGrid . . . . . . . . . . . . 42 3.3.2 SpotRateDynamicsonaMonthlyGrid. . . . . . . . . . . 45 3.3.3 ParameterCalibrationintheOne-FactorVasicekModel . . 47 3.4 ConditionallyHeteroscedasticSpotRateModels . . . . . . . . . . 56 3.5 Auto-RegressiveMovingAverage(ARMA)SpotRateModels. . . 60 3.5.1 AR(1)SpotRateModel . . . . . . . . . . . . . . . . . . . 61 vii viii Contents 3.5.2 AR(p)SpotRateModel . . . . . . . . . . . . . . . . . . . 62 3.5.3 GeneralARMASpotRateModels . . . . . . . . . . . . . 63 3.5.4 ParameterCalibrationinARMAModels . . . . . . . . . . 64 3.6 DiscreteTimeMultifactorVasicekModel . . . . . . . . . . . . . . 65 3.6.1 MotivationforMultifactorSpotRateModels . . . . . . . . 65 3.6.2 MultifactorVasicekModel(withIndependentFactors) . . . 67 3.6.3 ParameterEstimationandtheKalmanFilter . . . . . . . . 72 3.7 One-FactorGammaSpotRateModel . . . . . . . . . . . . . . . . 87 3.7.1 GammaAffineTermStructureModel . . . . . . . . . . . . 87 3.7.2 ParameterCalibrationintheGammaSpotRateModel . . . 90 3.8 DiscreteTimeBlack–KarasinskiModel . . . . . . . . . . . . . . . 92 3.8.1 Log-NormalSpotRateDynamics . . . . . . . . . . . . . . 92 3.8.2 ParameterCalibrationintheBlack–KarasinskiModel . . . 93 3.8.3 ARMAExtendedBlack–KarasinskiModel . . . . . . . . . 95 4 StochasticForwardRateandYieldCurveModeling . . . . . . . . . 97 4.1 GeneralDiscreteTimeHJMFramework . . . . . . . . . . . . . . 98 4.2 GaussianDiscreteTimeHJMFramework . . . . . . . . . . . . . . 100 4.2.1 GeneralGaussianDiscreteTimeHJMFramework . . . . . 100 4.2.2 Two-FactorGaussianHJMModel . . . . . . . . . . . . . . 102 4.2.3 Nelson–SiegelandSvenssonHJMFramework . . . . . . . 105 4.3 YieldCurveModeling . . . . . . . . . . . . . . . . . . . . . . . . 106 4.3.1 DerivationsfromtheForwardRateFramework . . . . . . . 106 4.3.2 StochasticYieldCurveModeling . . . . . . . . . . . . . . 109 Appendix ProofsofChap.4 . . . . . . . . . . . . . . . . . . . . . . . 125 5 PricingofFinancialAssets . . . . . . . . . . . . . . . . . . . . . . . . 131 5.1 PricingofCashFlows . . . . . . . . . . . . . . . . . . . . . . . . 132 5.1.1 GeneralCashFlowValuationintheVasicekModel . . . . 132 5.1.2 DefaultableCouponBonds . . . . . . . . . . . . . . . . . 135 5.2 FinancialMarket . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.2.1 ALog-NormalExampleintheVasicekModel . . . . . . . 139 5.2.2 AFirstAsset-and-LiabilityManagementProblem . . . . . 143 5.3 PricingofDerivativeInstruments . . . . . . . . . . . . . . . . . . 146 Appendix ProofsofChap.5 . . . . . . . . . . . . . . . . . . . . . . . 149 PartII ActuarialValuationandSolvency 6 ActuarialandFinancialModeling. . . . . . . . . . . . . . . . . . . . 155 6.1 FinancialMarketandFinancialFiltration . . . . . . . . . . . . . . 155 6.2 BasicActuarialModel . . . . . . . . . . . . . . . . . . . . . . . . 157 6.3 ImprovedActuarialModel . . . . . . . . . . . . . . . . . . . . . . 164 7 ValuationPortfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.1 ConstructionoftheValuationPortfolio . . . . . . . . . . . . . . . 170 7.1.1 FinancialPortfoliosandCashFlows . . . . . . . . . . . . 171 7.1.2 ConstructionoftheVaPo . . . . . . . . . . . . . . . . . . 171 Contents ix 7.1.3 Best-EstimateReserves . . . . . . . . . . . . . . . . . . . 174 7.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.2.1 ExamplesinLifeInsurance . . . . . . . . . . . . . . . . . 177 7.2.2 ExampleinNon-lifeInsurance . . . . . . . . . . . . . . . 181 7.3 ClaimsDevelopmentResultandALM . . . . . . . . . . . . . . . 187 7.3.1 ClaimsDevelopmentResult . . . . . . . . . . . . . . . . . 187 7.3.2 HedgeableFiltrationandALM . . . . . . . . . . . . . . . 188 7.3.3 ExamplesRevisited . . . . . . . . . . . . . . . . . . . . . 192 7.4 ApproximateValuationPortfolio . . . . . . . . . . . . . . . . . . 197 8 ProtectedValuationPortfolio . . . . . . . . . . . . . . . . . . . . . . 205 8.1 ConstructionoftheProtectedValuationPortfolio . . . . . . . . . . 205 8.2 Market-ValueMargin . . . . . . . . . . . . . . . . . . . . . . . . 207 8.2.1 Risk-AdjustedReserves . . . . . . . . . . . . . . . . . . . 207 8.2.2 ClaimsDevelopmentResultofRisk-AdjustedReserves . . 209 8.2.3 Fortuin–Kasteleyn–Ginibre(FKG)Inequality. . . . . . . . 211 8.2.4 ExamplesinLifeInsurance . . . . . . . . . . . . . . . . . 213 8.2.5 ExampleinNon-lifeInsurance . . . . . . . . . . . . . . . 223 8.2.6 FurtherProbabilityDistortionExamples . . . . . . . . . . 230 8.3 NumericalExamples . . . . . . . . . . . . . . . . . . . . . . . . . 234 8.3.1 Non-lifeInsuranceRun-Off . . . . . . . . . . . . . . . . . 234 8.3.2 LifeInsuranceExamples . . . . . . . . . . . . . . . . . . 244 9 Solvency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 9.1 RiskMeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 9.1.1 Definitionof(Conditional)RiskMeasures . . . . . . . . . 261 9.1.2 ExamplesofRiskMeasures . . . . . . . . . . . . . . . . . 265 9.2 SolvencyandAcceptability . . . . . . . . . . . . . . . . . . . . . 268 9.2.1 DefinitionofSolvencyandAcceptability . . . . . . . . . . 268 9.2.2 FreeCapitalandSolvencyTerminology . . . . . . . . . . 274 9.2.3 Insolvency . . . . . . . . . . . . . . . . . . . . . . . . . . 277 9.3 NoInsuranceTechnicalRisk . . . . . . . . . . . . . . . . . . . . 278 9.3.1 TheoreticalALMSolutionandFreeCapital . . . . . . . . 278 9.3.2 GeneralAssetAllocations . . . . . . . . . . . . . . . . . . 283 9.3.3 LimitedLiabilityOption . . . . . . . . . . . . . . . . . . . 286 9.3.4 MargrabeOption. . . . . . . . . . . . . . . . . . . . . . . 291 9.3.5 HedgingMargrabeOptions . . . . . . . . . . . . . . . . . 296 9.4 InclusionofInsuranceTechnicalRisk . . . . . . . . . . . . . . . . 299 9.4.1 InsuranceTechnicalandFinancialResult . . . . . . . . . . 300 9.4.2 TheoreticalALMSolutionandSolvency . . . . . . . . . . 302 9.4.3 GeneralALMProblemandInsuranceTechnicalRisk . . . 309 9.4.4 Cost-of-CapitalLoadingandDividendPayments . . . . . . 313 9.4.5 RiskSpreadingandLawofLargeNumbers. . . . . . . . . 321 9.4.6 LimitationsoftheVasicekFinancialModel . . . . . . . . . 325 9.5 PortfolioOptimization . . . . . . . . . . . . . . . . . . . . . . . . 326 9.5.1 StandardDeviationBasedRiskMeasure . . . . . . . . . . 327 9.5.2 EstimationoftheCovarianceMatrix . . . . . . . . . . . . 333

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Risk management for financial institutions is one of the key topics the financial industry has to deal with. The present volume is a mathematically rigorous text on solvency modeling. Currently, there are many new developments in this area in the financial and insurance industry (Basel III and Solve
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