Table Of ContentA Probability Metrics Approach to
Financial Risk Measures
A Probability Metrics
Approach to
Financial Risk Measures
Svetlozar T. Rachev
Stoyan V. Stoyanov
Frank J. Fabozzi
Thiseditionfirstpublished2011
©2011SvetlozarT.Rachev,StoyanV.StoyanovandFrankJ.Fabozzi
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LibraryofCongressCataloging-in-PublicationData
Rachev,S.T.(SvetlozarTodorov)
Aprobabilitymetricsapproachtofinancialriskmeasures/SvetlozarT.Rachev,
StoyanV.Stoyanov,FrankJ.Fabozzi,CFA.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-1-4051-8369-7(hardback)
1. Financialriskmanagement.2. Probabilities. I. Stoyanov,StoyanV.
II. Fabozzi,FrankJ.III. Title.
HD61.R332010
332.01’5192–dc22
2010040519
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
Setin10.5/13.5ptPalatinobyThomsonDigital,Noida,India
PrintedinMalaysia
01 2011
STR
TomygrandchildrenIliana,Zoya,andZari
SVS
TomyparentsVeselinandEvgeniyaKoleviand
mybrotherPavelStoyanov
FJF
TomywifeDonnaand
mychildrenFrancesco,Patricia,andKarly
Contents
Preface xiii
AbouttheAuthors xv
1 Introduction 1
1.1 ProbabilityMetrics 1
1.2 ApplicationsinFinance 2
2 ProbabilityDistancesandMetrics 7
2.1 Introduction 9
2.2 SomeExamplesofProbabilityMetrics 9
2.2.1 Engineer’smetric 10
2.2.2 Uniform(orKolmogorov)metric 10
2.2.3 Le´vymetric 11
2.2.4 Kantorovichmetric 14
2.2.5 L -metricsbetweendistributionfunctions 15
p
2.2.6 KyFanmetrics 16
2.2.7 L -metric 17
p
2.3 DistanceandSemidistanceSpaces 19
2.4 DefinitionsofProbabilityDistancesandMetrics 24
2.5 Summary 28
2.6 TechnicalAppendix 28
2.6.1 Universallymeasurableseparablemetricspaces 29
2.6.2 Theequivalenceofthenotionsof
p.(semi-)distanceonP andonX 35
2
vii
CONTENTS
3 ChoiceunderUncertainty 40
3.1 Introduction 41
3.2 ExpectedUtilityTheory 44
3.2.1 StPetersburgParadox 44
3.2.2 ThevonNeumann–Morgensternexpected
utilitytheory 46
3.2.3 Typesofutilityfunctions 48
3.3 StochasticDominance 51
3.3.1 First-orderstochasticdominance 52
3.3.2 Second-orderstochasticdominance 53
3.3.3 Rothschild–Stiglitzstochasticdominance 55
3.3.4 Third-orderstochasticdominance 56
3.3.5 Efficientsetsandtheportfoliochoiceproblem 58
3.3.6 Returnversuspayoff 59
3.4 ProbabilityMetricsandStochasticDominance 63
3.5 CumulativeProspectTheory 66
3.6 Summary 70
3.7 TechnicalAppendix 70
3.7.1 Theaxiomsofchoice 71
3.7.2 Stochasticdominancerelationsofordern 72
3.7.3 Returnversuspayoffandstochasticdominance 74
3.7.4 Otherstochasticdominancerelations 76
4 AClassificationofProbabilityDistances 83
4.1 Introduction 86
4.2 PrimaryDistancesandPrimaryMetrics 86
4.3 SimpleDistancesandMetrics 90
4.4 CompoundDistancesandMomentFunctions 99
4.5 IdealProbabilityMetrics 105
4.5.1 Interpretationandexamplesofidealprobability
metrics 107
4.5.2 Conditionsforboundednessofidealprobability
metrics 112
4.6 Summary 114
4.7 TechnicalAppendix 114
4.7.1 Examplesofprimarydistances 114
4.7.2 Examplesofsimpledistances 118
4.7.3 Examplesofcompounddistances 131
4.7.4 Examplesofmomentfunctions 135
viii
CONTENTS
5 RiskandUncertainty 146
5.1 Introduction 147
5.2 MeasuresofDispersion 150
5.2.1 Standarddeviation 151
5.2.2 Meanabsolutedeviation 153
5.2.3 Semi-standarddeviation 154
5.2.4 Axiomaticdescription 155
5.2.5 Deviationmeasures 156
5.3 ProbabilityMetricsandDispersionMeasures 158
5.4 MeasuresofRisk 159
5.4.1 Value-at-risk 160
5.4.2 ComputingportfolioVaRinpractice 165
5.4.3 Back-testingofVaR 172
5.4.4 Coherentriskmeasures 175
5.5 RiskMeasuresandDispersionMeasures 179
5.6 RiskMeasuresandStochasticOrders 181
5.7 Summary 182
5.8 TechnicalAppendix 183
5.8.1 Convexriskmeasures 183
5.8.2 Probabilitymetricsanddeviationmeasures 184
5.8.3 Deviationmeasuresandprobability
quasi-metrics 187
6 AverageValue-at-Risk 191
6.1 Introduction 192
6.2 AverageValue-at-Risk 193
6.2.1 AVaRforstabledistributions 200
6.3 AVaREstimationfromaSample 204
6.4 ComputingPortfolioAVaRinPractice 207
6.4.1 Themultivariatenormalassumption 207
6.4.2 Thehistoricalmethod 208
6.4.3 Thehybridmethod 208
6.4.4 TheMonteCarlomethod 209
6.4.5 Kernelmethods 211
6.5 Back-testingofAVaR 218
6.6 SpectralRiskMeasures 220
6.7 RiskMeasuresandProbabilityMetrics 223
6.8 RiskMeasuresBasedonDistortionFunctionals 226
6.9 Summary 227
ix
CONTENTS
6.10 TechnicalAppendix 228
6.10.1 Characteristicsofconditionallossdistributions 228
6.10.2 Higher-orderAVaR 232
6.10.3 TheminimizationformulaforAVaR 234
6.10.4 ETLvsAVaR 237
6.10.5 Kernel-basedestimationofAVaR 242
6.10.6 Remarksonspectralriskmeasures 245
7 ComputingAVaRthroughMonteCarlo 252
7.1 Introduction 253
7.2 AnIllustrationofMonteCarloVariability 256
7.3 AsymptoticDistribution,ClassicalConditions 259
7.4 RateofConvergencetotheNormalDistribution 262
7.4.1 Theeffectoftailthickness 263
7.4.2 Theeffectoftailtruncation 268
7.4.3 Infinitevariancedistributions 271
7.5 AsymptoticDistribution,Heavy-tailedReturns 277
7.6 RateofConvergence,Heavy-tailedReturns 283
7.6.1 StableParetiandistributions 283
7.6.2 Student’stdistribution 286
7.7 OntheChoiceofaDistributionalModel 290
7.7.1 Tailbehaviorandreturnfrequency 290
7.7.2 Practicalimplications 295
7.8 Summary 297
7.9 TechnicalAppendix 298
7.9.1 Proofofthestablelimitresult 298
8 StochasticDominanceRevisited 304
8.1 Introduction 306
8.2 MetrizationofPreferenceRelations 308
8.3 TheHausdorffMetricStructure 310
8.4 Examples 314
8.4.1 The Le´vy quasi-semidistance and first-order
stochasticdominance 315
8.4.2 Higher-orderstochasticdominance 317
8.4.3 TheH-quasi-semidistance 320
8.4.4 AVaRgeneratedstochasticorders 322
8.4.5 Compoundquasi-semidistances 324
8.5 Utility-typeRepresentations 325
x
CONTENTS
8.6 AlmostStochasticOrdersandDegreeofViolation 328
8.7 Summary 330
8.8 TechnicalAppendix 332
8.8.1 Preferencerelationsandtopology 332
8.8.2 Quasi-semidistancesandpreferencerelations 334
8.8.3 Constructionofquasi-semidistancesonclasses
ofinvestors 335
8.8.4 Investorswithbalancedviews 338
8.8.5 Structuralclassificationofprobabilitydistances 339
Index 357
xi
Preface
The theory of probability metrics is a branch of probability theory.
It finds application in different theoretical and applied fields such
asprobabilitytheory,queuingtheory,insurancerisktheory,andfi-
nance.Thetheoryofprobabilitymetricslooksforanswerstothefol-
lowingbasicquestion:Howcanonemeasurethedifferencebetween
randomquantities?Infinance,forexample,weassumeastochastic
modelforassetreturndistributionsand,inordertoestimatetherisk
ofaportfolioofassets,wesamplefromthefitteddistribution.Then,
weusethegeneratedsimulationstocalculateportfoliorisk.Inthis
context,therearetwoissuesarisingontwodifferentlevels.First,the
assumed stochastic model should be “close” to the empirical data.
Inthissense,wesaythatweneedarealisticmodelinthefirstplace.
Second,sincetheriskestimateisessentiallycomputedfromrandom
scenarios,wehavetobeawareofthevariabilityoftheestimatorand
howitdependsontheassumedassetreturndistributions.
Although based on universal principles and ideas, the field of
probability metrics is very specialized. Most of the literature is
highlytechnicalandisaccessiblemostlytospecialistsinprobability
theory. As far as applications are concerned, apart from our book
Advanced Stochastic Models, Risk Assessment, and Portfolio Optimiza-
tion:IdealRisk,Uncertainty,andPerformanceMeasures(JohnWiley&
Sons, 2008), we are unaware of other literature describing applica-
tionsinfinance.
xiii
