Quantitative Risk Management - Concepts, Techniques and Tools

554 Pages·2005·6.66 MB·english

by  Alexander J. McNeil

Quantitative Risk Management
QuantitativeRiskManagement:Concepts,TechniquesandTools
isapartofthe
PrincetonSeriesinFinance
SeriesEditors
DarrellDuffie StephenSchaefer
StanfordUniversity LondonBusinessSchool
Finance as a discipline has been growing rapidly. The numbers of researchers in
academyandindustry,ofstudents,ofmethodsandmodelshaveallproliferatedin
the past decade or so. This growth and diversity manifests itself in the emerging
cross-disciplinaryaswellascross-nationalmixofscholarshipnowdrivingthefield
offinanceforward.Theintellectualrootsofmodernfinance,aswellasthebranches,
willberepresentedinthePrincetonSeriesinFinance.
Titles in this series will be scholarly and professional books, intended to be read
by a mixed audience of economists, mathematicians, operations research scien-
tists, financial engineers, and other investment professionals. The goal is to pro-
videthefinestcross-disciplinaryworkinallareasoffinancebywidelyrecognized
researchersintheprimeoftheircreativecareers.
OtherBooksinThisSeries
FinancialEconometrics:Problems,Models,andMethodsbyChristianGourieroux
andJoannJasiak
CreditRisk:Pricing,Measurement,andManagementbyDarrellDuffieandKenneth
J.Singleton
MicrofoundationsofFinancialEconomics:AnIntroductiontoGeneralEquilibrium
AssetPricingbyYvanLengwiler
CreditRiskModeling:TheoryandApplicationsbyDavidLando
Quantitative Risk Management
Concepts, Techniques and Tools
AlexanderJ.McNeil
Ru¨digerFrey
PaulEmbrechts
PrincetonUniversityPress
PrincetonandOxford
Copyright(cid:1)c 2005byPrincetonUniversityPress
PublishedbyPrincetonUniversityPress,
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Allrightsreserved
LibraryofCongressCataloguing-in-PublicationData
McNeil,AlexanderJ.,1967–
Quantitativeriskmanagement:concepts,techniques,andtools/AlexanderJ.
McNeil,Ru¨digerFrey,PaulEmbrechts
p.cm.—(Princetonseriesinfinance)
Includesbibliographicalreferencesandindex.
ISBN0-691-12255-5(cloth:alk.paper)
1.Riskmanagement—Mathematicalmodels.2.Finance—Mathematical
models.3.Insurance—Mathematicalmodels.4.Mathematicalstatistics.
I.Frey,Ru¨diger.II.Embrechts,Paul.III.Title.IV.Series.
HD61.M3952005
(cid:2) (cid:2)
658.1550151—pcc22 2005049603
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ThisbookhasbeencomposedinTimesandtypesetbyT&TProductionsLtd,London
Printedonacid-freepaper(cid:1)∞
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10987654321
To Janine,Alexander and Calliope
Alexander
Fu¨r Catharina und Sebastian
Ru¨diger
Voor Gerda, Rita en Guy
Paul

Contents
Preface xiii
1 RiskinPerspective 1
1.1 Risk 1
1.1.1 RiskandRandomness 1
1.1.2 FinancialRisk 2
1.1.3 MeasurementandManagement 3
1.2 ABriefHistoryofRiskManagement 5
1.2.1 FromBabylontoWallStreet 5
1.2.2 TheRoadtoRegulation 8
1.3 TheNewRegulatoryFramework 10
1.3.1 BaselII 10
1.3.2 Solvency2 13
1.4 WhyManageFinancialRisk? 15
1.4.1 ASocietalView 15
1.4.2 TheShareholder’sView 16
1.4.3 EconomicCapital 18
1.5 QuantitativeRiskManagement 19
1.5.1 TheNatureoftheChallenge 19
1.5.2 QRMfortheFuture 22
2 BasicConceptsinRiskManagement 25
2.1 RiskFactorsandLossDistributions 25
2.1.1 GeneralDefinitions 25
2.1.2 ConditionalandUnconditionalLossDistribution 28
2.1.3 MappingofRisks:SomeExamples 29
2.2 RiskMeasurement 34
2.2.1 ApproachestoRiskMeasurement 34
2.2.2 Value-at-Risk 37
2.2.3 FurtherCommentsonVaR 40
2.2.4 OtherRiskMeasuresBasedonLossDistributions 43
2.3 StandardMethodsforMarketRisks 48
2.3.1 Variance–CovarianceMethod 48
2.3.2 HistoricalSimulation 50
2.3.3 MonteCarlo 52
2.3.4 LossesoverSeveralPeriodsandScaling 53
2.3.5 Backtesting 55
2.3.6 AnIllustrativeExample 55
viii Contents
3 MultivariateModels 61
3.1 BasicsofMultivariateModelling 61
3.1.1 RandomVectorsandTheirDistributions 62
3.1.2 StandardEstimatorsofCovarianceandCorrelation 64
3.1.3 TheMultivariateNormalDistribution 66
3.1.4 TestingNormalityandMultivariateNormality 68
3.2 NormalMixtureDistributions 73
3.2.1 NormalVarianceMixtures 73
3.2.2 NormalMean-VarianceMixtures 77
3.2.3 GeneralizedHyperbolicDistributions 78
3.2.4 FittingGeneralizedHyperbolicDistributionstoData 81
3.2.5 EmpiricalExamples 84
3.3 SphericalandEllipticalDistributions 89
3.3.1 SphericalDistributions 89
3.3.2 EllipticalDistributions 93
3.3.3 PropertiesofEllipticalDistributions 95
3.3.4 EstimatingDispersionandCorrelation 96
3.3.5 TestingforEllipticalSymmetry 99
3.4 DimensionReductionTechniques 103
3.4.1 FactorModels 103
3.4.2 StatisticalCalibrationStrategies 105
3.4.3 RegressionAnalysisofFactorModels 106
3.4.4 PrincipalComponentAnalysis 109
4 FinancialTimeSeries 116
4.1 EmpiricalAnalysesofFinancialTimeSeries 117
4.1.1 StylizedFacts 117
4.1.2 MultivariateStylizedFacts 123
4.2 FundamentalsofTimeSeriesAnalysis 125
4.2.1 BasicDefinitions 125
4.2.2 ARMAProcesses 128
4.2.3 AnalysisintheTimeDomain 132
4.2.4 StatisticalAnalysisofTimeSeries 134
4.2.5 Prediction 136
4.3 GARCHModelsforChangingVolatility 139
4.3.1 ARCHProcesses 139
4.3.2 GARCHProcesses 145
4.3.3 SimpleExtensionsoftheGARCHModel 148
4.3.4 FittingGARCHModelstoData 150
4.4 VolatilityModelsandRiskEstimation 158
4.4.1 VolatilityForecasting 158
4.4.2 ConditionalRiskMeasurement 160
4.4.3 Backtesting 162
4.5 FundamentalsofMultivariateTimeSeries 164
4.5.1 BasicDefinitions 164
4.5.2 AnalysisintheTimeDomain 166
4.5.3 MultivariateARMAProcesses 168
4.6 MultivariateGARCHProcesses 170
4.6.1 GeneralStructureofModels 170
4.6.2 ModelsforConditionalCorrelation 172
4.6.3 ModelsforConditionalCovariance 175
Contents ix
4.6.4 FittingMultivariateGARCHModels 178
4.6.5 DimensionReductioninMGARCH 179
4.6.6 MGARCHandConditionalRiskMeasurement 182
5 CopulasandDependence 184
5.1 Copulas 184
5.1.1 BasicProperties 185
5.1.2 ExamplesofCopulas 189
5.1.3 MetaDistributions 192
5.1.4 SimulationofCopulasandMetaDistributions 193
5.1.5 FurtherPropertiesofCopulas 195
5.1.6 PerfectDependence 199
5.2 DependenceMeasures 201
5.2.1 LinearCorrelation 201
5.2.2 RankCorrelation 206
5.2.3 CoefficientsofTailDependence 208
5.3 NormalMixtureCopulas 210
5.3.1 TailDependence 210
5.3.2 RankCorrelations 215
5.3.3 SkewedNormalMixtureCopulas 217
5.3.4 GroupedNormalMixtureCopulas 218
5.4 ArchimedeanCopulas 220
5.4.1 BivariateArchimedeanCopulas 220
5.4.2 MultivariateArchimedeanCopulas 222
5.4.3 Non-exchangeableArchimedeanCopulas 224
5.5 FittingCopulastoData 228
5.5.1 Method-of-MomentsusingRankCorrelation 229
5.5.2 FormingaPseudo-SamplefromtheCopula 232
5.5.3 MaximumLikelihoodEstimation 234
6 AggregateRisk 238
6.1 CoherentMeasuresofRisk 238
6.1.1 TheAxiomsofCoherence 238
6.1.2 Value-at-Risk 241
6.1.3 CoherentRiskMeasuresBasedonLossDistributions 243
6.1.4 CoherentRiskMeasuresasGeneralizedScenarios 244
6.1.5 Mean-VaRPortfolioOptimization 246
6.2 BoundsforAggregateRisks 248
6.2.1 TheGeneralFre´chetProblem 248
6.2.2 TheCaseofVaR 250
6.3 CapitalAllocation 256
6.3.1 TheAllocationProblem 256
6.3.2 TheEulerPrincipleandExamples 257
6.3.3 EconomicJustificationoftheEulerPrinciple 261
7 ExtremeValueTheory 264
7.1 Maxima 264
7.1.1 GeneralizedExtremeValueDistribution 265
7.1.2 MaximumDomainsofAttraction 267
7.1.3 MaximaofStrictlyStationaryTimeSeries 270
7.1.4 TheBlockMaximaMethod 271
x Contents
7.2 ThresholdExceedances 275
7.2.1 GeneralizedParetoDistribution 275
7.2.2 ModellingExcessLosses 278
7.2.3 ModellingTailsandMeasuresofTailRisk 282
7.2.4 TheHillMethod 286
7.2.5 SimulationStudyofEVTQuantileEstimators 289
7.2.6 ConditionalEVTforFinancialTimeSeries 291
7.3 TailsofSpecificModels 293
7.3.1 DomainofAttractionofFre´chetDistribution 293
7.3.2 DomainofAttractionofGumbelDistribution 294
7.3.3 MixtureModels 295
7.4 PointProcessModels 298
7.4.1 ThresholdExceedancesforStrictWhiteNoise 299
7.4.2 ThePOTModel 301
7.4.3 Self-ExcitingProcesses 306
7.4.4 ASelf-ExcitingPOTModel 307
7.5 MultivariateMaxima 311
7.5.1 MultivariateExtremeValueCopulas 311
7.5.2 CopulasforMultivariateMinima 314
7.5.3 CopulaDomainsofAttraction 314
7.5.4 ModellingMultivariateBlockMaxima 317
7.6 MultivariateThresholdExceedances 319
7.6.1 ThresholdModelsUsingEVCopulas 319
7.6.2 FittingaMultivariateTailModel 320
7.6.3 ThresholdCopulasandTheirLimits 322
8 CreditRiskManagement 327
8.1 IntroductiontoCreditRiskModelling 327
8.1.1 CreditRiskModels 327
8.1.2 TheNatureoftheChallenge 329
8.2 StructuralModelsofDefault 331
8.2.1 TheMertonModel 331
8.2.2 PricinginMerton’sModel 332
8.2.3 TheKMVModel 336
8.2.4 ModelsBasedonCreditMigration 338
8.2.5 MultivariateFirm-ValueModels 342
8.3 ThresholdModels 343
8.3.1 NotationforOne-PeriodPortfolioModels 344
8.3.2 ThresholdModelsandCopulas 345
8.3.3 IndustryExamples 347
8.3.4 ModelsBasedonAlternativeCopulas 348
8.3.5 ModelRiskIssues 350
8.4 TheMixtureModelApproach 352
8.4.1 One-FactorBernoulliMixtureModels 353
8.4.2 CreditRisk+ 356
8.4.3 AsymptoticsforLargePortfolios 357
8.4.4 ThresholdModelsasMixtureModels 359
8.4.5 Model-TheoreticAspectsofBaselII 362
8.4.6 ModelRiskIssues 364
8.5 MonteCarloMethods 367
8.5.1 BasicsofImportanceSampling 367
8.5.2 ApplicationtoBernoulli-MixtureModels 370