Table Of ContentBestMasters
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Simona Roccioletti
Backtesting Value at Risk
and Expected Shortfall
Simona Roccioletti
Guilianova, Italy
Master Thesis, University of Applied Sciences (b(cid:191) ) Vienna, Austria, 2015
BestMasters
ISBN 978-3-658-11907-2 ISBN 978-3-658-11908-9 (eBook)
DOI 10.1007/978-3-658-11908-9
Library of Congress Control Number: 2015958558
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Acknowledgements
Foremost,IwouldliketoexpressmysinceregratitudetomyadvisorProf. Chris-
tianCechforhisexcellentguidance, hisadvicesandcorrectionsthatgreatlyim-
provedthework.
I am also grateful to Prof. Umberto Cherubini, who prompted me to study this
subjectandguidedmerightfromthestart.
IwishtoexpressmysincerethankstoDr. CarloAcerbi,whoprovidedinsightand
expertisethatgreatlyassistedtheresearch,althoughhemaynotagreewithallof
theinterpretations/conclusionsofthisthesis.
ItakethisopportunitytoexpressgratitudetoalloftheDepartmentfacultymem-
bersfortheirhelpandkindness.
I would like to thank my QF and ARIMA collegues, who shared with me the
bestmomentsofthiscourseofstudy.
Still more I am grateful to my family, who gave me the opportunity to pursue
acollegecareerandwhohasalwayssupportedme.
Then I would like to express my gratitude to my lifelong friends, who never had
anydoubtsaboutmy“finalsuccess”andwhoalwaysencouragedmetodothebest.
Finally,IwouldliketothankthepersonIlove,foralwaysbeingwithme...
whereverIgo...
SimonaRoccioletti
v
Contents
Acknowledgements v
Contents vii
List of Figures xi
List of Tables xiii
Abbreviations xv
Symbols xvii
Abstract xix
1 Introduction 1
2 Risk Measures and their Properties 5
2.1 Definitionofriskmeasure . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 ValueatRisk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 ExpectedShortfall . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Expectiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Coherentriskmeasures . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.1 CoherenceofVaR . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5.2 CoherenceofES . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5.3 CoherenceofExpectiles . . . . . . . . . . . . . . . . . . . . 18
2.6 RiskMeasures: adeeperview . . . . . . . . . . . . . . . . . . . . . 19
2.6.1 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6.2 ComonotonicAdditivity . . . . . . . . . . . . . . . . . . . . 21
2.6.3 LawInvariance . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6.4 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Elicitability 27
3.1 EvaluatePointForecasts . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.2 BacktoElicitability . . . . . . . . . . . . . . . . . . . . . . 33
vii
viii Contents
3.2 ElicitabilityofVaR . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 ElicitabilityofES . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 ElicitabilityofExpectiles . . . . . . . . . . . . . . . . . . . . . . . . 39
4 Backtesting 43
4.1 TheBacktestingIdea . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 BacktestingVaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2.1 UnconditionalCoverageTests . . . . . . . . . . . . . . . . . 48
4.2.2 ConditionalCoverageTests . . . . . . . . . . . . . . . . . . 50
4.2.3 BacktestingwithInformationVariables . . . . . . . . . . . . 54
4.2.4 RegulatoryFramework . . . . . . . . . . . . . . . . . . . . . 55
4.3 BacktestingES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 Test1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.2 Test2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.3.3 Test3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3.4 Powerofthetests . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Empirical Analysis 71
5.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2.1 NormalDistribution . . . . . . . . . . . . . . . . . . . . . . 73
5.2.2 Student’st-distribution. . . . . . . . . . . . . . . . . . . . . 74
5.2.3 KernelDensityEstimation . . . . . . . . . . . . . . . . . . . 74
5.2.4 GARCHModels. . . . . . . . . . . . . . . . . . . . . . . . . 77
5.3 Backtestingresults . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.1 VaRresults . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3.2 ESresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6 Conclusions 99
A MATLAB Code 101
A.1 MATLAB variables . . . . . . . . . . . . . . . . . . . . . . . . . 101
A.2 ESTIMATION OF RISK MEASURES . . . . . . . . . . . . . 104
A.2.1 Normal model . . . . . . . . . . . . . . . . . . . . . . . . 104
A.2.2 Student’s t model . . . . . . . . . . . . . . . . . . . . . . 105
A.2.3 Kernel model . . . . . . . . . . . . . . . . . . . . . . . . . 106
A.2.4 Garch with normal innovations . . . . . . . . . . . . . . 107
A.2.5 Garch with Student’s t innovations . . . . . . . . . . . 109
A.3 Value at Risk Tests . . . . . . . . . . . . . . . . . . . . . . . . . 110
A.4 Expected Shortfall Tests . . . . . . . . . . . . . . . . . . . . . . 114
A.5 Monte Carlo p-values . . . . . . . . . . . . . . . . . . . . . . . . 116
B Figures 119
B.1 DAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Contents ix
B.2 FTSE100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.3 NIKKEI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B.4 EUROSTOXX50 . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Bibliography 141
List of Figures
2.1 Empiricalsensitivity(inpercentage)ofthehistoricalVaRandhis-
toricalES,asinContetal.[16] . . . . . . . . . . . . . . . . . . . . 24
2.2 Empiricalsensitivity(inpercentage)oftheES0.01estimatedwith
differentmethods,asinContetal.[16] . . . . . . . . . . . . . . . 25
5.1 S&P500log-returns&Histogram.
(Lossesarepositivenumbers) . . . . . . . . . . . . . . . . . . . . . 72
5.2 QQplot-S&P500vsStandardNormal. . . . . . . . . . . . . . . . 73
5.3 S&P500vsFittedNormal&Student-t . . . . . . . . . . . . . . . . 75
5.4 S&P500vsFittedGaussianKernel . . . . . . . . . . . . . . . . . . 77
5.5 ConditionalstandarddeviationsestimatedbytheGARCH(1,1)model. 79
5.6 S&P500: VaRandESestimates . . . . . . . . . . . . . . . . . . . 80
5.7 S&P500: VaRestimatesfordifferentmodels(2007-2010). . . . . . 81
5.8 S&P500-PercentageofVaRexceptions . . . . . . . . . . . . . . . 83
5.9 S&P500: Log-returnsandexceptionsofVaR97.5% (orangecircles)
andES97.5% (redcircles).. . . . . . . . . . . . . . . . . . . . . . . . 89
5.10 MCdistributionsforZ1 . . . . . . . . . . . . . . . . . . . . . . . . 92
5.11 S&P500-MCdistributionforZ2 . . . . . . . . . . . . . . . . . . . 94
5.12 OverestimationanUnderestimationAreas . . . . . . . . . . . . . . 95
5.13 S&P500-Z1 andZ2 ineachcalendaryear . . . . . . . . . . . . . . 96
5.14 S&P500-Z1 andZ2 cumulativeinyears . . . . . . . . . . . . . . . 97
B.1 DAXlog-returns(Lossesarepositivenumbers). . . . . . . . . . . . 119
B.2 DAX-QQplot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
B.3 DAXvsFittedDistributions. . . . . . . . . . . . . . . . . . . . . . 120
B.4 DAX-σ GARCHmodels . . . . . . . . . . . . . . . . . . . . . . . 120
B.5 DAX-VaRandESestimates. . . . . . . . . . . . . . . . . . . . . . 121
B.6 DAX-VaRexceptions . . . . . . . . . . . . . . . . . . . . . . . . . 121
B.7 DAX-Z1 Z2 percalendaryear. . . . . . . . . . . . . . . . . . . . . 122
B.8 DAX-Z1 Z2 cumulativeinyears. . . . . . . . . . . . . . . . . . . . 122
B.9 FTSE100log-returns(Lossesarepositivenumbers). . . . . . . . . 124
B.10FTSEvsFittedDistributions. . . . . . . . . . . . . . . . . . . . . . 125
B.11FTSE100-σ GARCHmodels. . . . . . . . . . . . . . . . . . . . . 125
B.12FTSE100-VaRandESestimates. . . . . . . . . . . . . . . . . . . 126
B.13FTSE100-VaRexceptions . . . . . . . . . . . . . . . . . . . . . . 126
B.14FTSE100-Z1 Z2 percalendaryear. . . . . . . . . . . . . . . . . . 127
xi
