Extreme Events: Robust Portfolio Construction in the Presence of Fat Tails (The Wiley Finance Series)

336 Pages·2011·4.1 MB·English

by  Malcolm Kemp

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Extreme Events
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Extreme Events
Robust Portfolio Construction in the
Presence of Fat Tails
Malcolm H. D. Kemp
A John Wiley and Sons, Ltd., Publication
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Contents
Preface xv
Acknowledgements xvii
Abbreviations xix
Notation xxi
1 Introduction 1
1.1 Extremeevents 1
1.2 Theportfolioconstructionproblem 2
1.3 Copingwithreallyextremeevents 2
1.4 Riskbudgeting 3
1.5 Elementsdesignedtomaximisebenefittoreaders 4
1.6 Bookstructure 4
2 FatTails–InSingle(i.e.,Univariate)ReturnSeries 7
2.1 Introduction 7
2.2 Afattailrelativetowhat? 7
2.3 Empiricalexamplesoffat-tailedbehaviourinreturnseries 11
2.3.1 Introduction 11
2.3.2 Visualisingfattails 11
2.3.3 Behaviourofindividualbondsandbondindices 17
2.3.4 Behaviourofequityindices 17
2.3.5 Currenciesandotherassettypes 22
2.4 Characterisingfat-taileddistributionsbytheirmoments 23
2.4.1 Introduction 23
2.4.2 Skewandkurtosis 25
2.4.3 The(fourth-moment)Cornish-Fisherapproach 26
2.4.4 WeaknessesoftheCornish-Fisherapproach 27
2.4.5 ImprovingontheCornish-Fisherapproach 28
2.4.6 Statisticaltestsfornon-Normality 29
2.4.7 HigherordermomentsandtheOmegafunction 31
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2.5 Whatcausesfattails? 32
2.5.1 Introduction 32
2.5.2 TheCentralLimitTheorem 33
2.5.3 WaysinwhichtheCentralLimitTheoremcanbreakdown 34
2.6 Lackofdiversification 35
2.7 Atime-varyingworld 36
2.7.1 Introduction 36
2.7.2 Distributionalmixtures 37
2.7.3 Time-varyingvolatility 38
2.7.4 Regimeshifts 40
2.8 Stabledistributions 41
2.8.1 Introduction 41
2.8.2 Definingcharacteristics 42
2.8.3 TheGeneralisedCentralLimitTheorem 44
2.8.4 Quantile–quantileplotsofstabledistributions 45
2.9 Extremevaluetheory(EVT) 45
2.9.1 Introduction 45
2.9.2 Extremevaluedistributions 46
2.9.3 Tailprobabilitydensities 47
2.9.4 Estimationofandinferencefromtailindexvalues 47
2.9.5 Issueswithextremevaluetheory 47
2.10 Parsimony 50
2.11 Combiningdifferentpossiblesourcemechanisms 52
2.12 Thepractitionerperspective 53
2.12.1 Introduction 53
2.12.2 Time-varyingvolatility 53
2.12.3 Crowdedtrades 53
2.12.4 Liquidityrisk 54
2.12.5 ‘Rational’behaviourversus‘boundedrational’behaviour 54
2.12.6 Ourowncontributiontothepicture 55
2.13 Implementationchallenges 55
2.13.1 Introduction 55
2.13.2 Smoothingofreturnseries 56
2.13.3 Timeclocksandnon-constanttimeperiodlengths 57
2.13.4 Priceorotherdataratherthanreturndata 58
2.13.5 Economicsensitivitiesthatchangethroughtime 58
3 FatTails–InJoint(i.e.,Multivariate)ReturnSeries 61
3.1 Introduction 61
3.2 Visualisationoffattailsinmultiplereturnseries 61
3.3 Copulasandmarginals–Sklar’stheorem 64
3.3.1 Introduction 64
3.3.2 Fractile–fractile(i.e.,quantile–quantilebox)plots 66
3.3.3 Time-varyingvolatility 72
3.4 Exampleanalyticalcopulas 75
3.4.1 Introduction 75
3.4.2 TheGaussiancopula 76
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3.4.3 Thet-copula 76
3.4.4 Archimedeancopulas 78
3.5 Empiricalestimationoffattailsinjointreturnseries 78
3.5.1 Introduction 78
3.5.2 Disadvantagesofempiricallyfittingthecopula 79
3.5.3 Multi-dimensionalquantile–quantileplots 80
3.6 Causaldependencymodels 83
3.7 Thepractitionerperspective 83
3.8 Implementationchallenges 85
3.8.1 Introduction 85
3.8.2 Seriesofdifferentlengths 85
3.8.3 Non-coincidentlytimedseries 86
3.8.4 Clusteranalysis 87
3.8.5 Relativeentropyandnonlinearclusteranalysis 88
4 IdentifyingFactorsThatSignificantlyInfluenceMarkets 91
4.1 Introduction 91
4.2 Portfolioriskmodels 92
4.2.1 Introduction 92
4.2.2 Fundamentalmodels 94
4.2.3 Econometricmodels 95
4.2.4 Statisticalriskmodels 96
4.2.5 Similaritiesanddifferencesbetweenriskmodels 96
4.3 Signalextractionandprincipalcomponentsanalysis 97
4.3.1 Introduction 97
4.3.2 Principalcomponentsanalysis 99
4.3.3 Thetheorybehindprincipalcomponentsanalysis 100
4.3.4 Weightingschemas 102
4.3.5 Idiosyncraticrisk 103
4.3.6 Randommatrixtheory 103
4.3.7 Identifyingprincipalcomponentsoneatatime 106
4.4 Independentcomponentsanalysis 106
4.4.1 Introduction 106
4.4.2 Practicalalgorithms 107
4.4.3 Non-Normalityandprojectionpursuit 109
4.4.4 Truncatingtheanswers 110
4.4.5 Extractingalltheun-mixingweightsatthesametime 110
4.4.6 Complexitypursuit 111
4.4.7 Gradientascent 112
4.5 Blendingtogetherprincipalcomponentsanalysisandindependent
componentsanalysis 112
4.5.1 Introduction 112
4.5.2 Includingbothvarianceandkurtosisintheimportancecriterion 113
4.5.3 Eliminatingsignalsfromtheremainingdataset 114
4.5.4 Normalisingsignalstrength 115
4.6 Thepotentialimportanceofselectioneffects 116
4.6.1 Introduction 116
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4.6.2 Quantifyingthepossibleimpactofselectioneffects 116
4.6.3 Decompositionoffat-tailedbehaviour 119
4.7 Marketdynamics 120
4.7.1 Introduction 120
4.7.2 Linearregression 120
4.7.3 Differenceequations 121
4.7.4 Thepotentialrangeofbehavioursoflineardifferenceequations 122
4.7.5 Multivariatelinearregression 124
4.7.6 ‘Chaotic’marketdynamics 125
4.7.7 Modellingmarketdynamicsusingnonlinearmethods 127
4.7.8 Locallylineartimeseriesanalysis 128
4.8 Distributionalmixtures 129
4.8.1 Introduction 129
4.8.2 Gaussianmixturemodelsandtheexpectation-maximisation
algorithm 129
4.8.3 k-meansclustering 131
4.8.4 Generaliseddistributionalmixturemodels 132
4.8.5 Regimeshifts 132
4.9 Thepractitionerperspective 133
4.10 Implementationchallenges 134
4.10.1 Introduction 134
4.10.2 Localextrema 135
4.10.3 Globalextrema 136
4.10.4 Simulatedannealingandgeneticportfoliooptimisation 137
4.10.5 Minimisation/maximisationalgorithms 138
4.10.6 Runtimeconstraints 138
5 TraditionalPortfolioConstructionTechniques 141
5.1 Introduction 141
5.2 Quantitativeversusqualitativeapproaches? 141
5.2.1 Introduction 141
5.2.2 Viewinganyprocessthroughthewindowofportfolio
optimisation 142
5.2.3 Quantitativeversusqualitativeinsights 143
5.2.4 Thecharacteristicsofpricing/returnanomalies 144
5.3 Risk-returnoptimisation 145
5.3.1 Introduction 145
5.3.2 Mean-varianceoptimisation 146
5.3.3 Formalmathematicalnotation 148
5.3.4 TheCapitalAssetPricingModel(CAPM) 149
5.3.5 Alternativemodelsofsystematicrisk 152
5.4 Moregeneralfeaturesofmean-varianceoptimisation 152
5.4.1 Introduction 152
5.4.2 Monotonicchangestoriskorreturnmeasures 152
5.4.3 Constraint-lessmean-varianceoptimisation 153
5.4.4 Alpha-betaseparation 153