Financial Modeling, Actuarial Valuation and Solvency in Insurance

437 Pages·2013·6.76 MB·English

by  Mario V. Wüthrich

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
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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
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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