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Risk management techniques for decision making in highly uncertain environments

University of Florida Institutional Repository
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Ithankmysupervisor,Prof.StanislavUryasev,andthemembersofmyPh.D.committeefortheirhelpandguidance.Iamalsogratefultomywifeandmyparentsfortheirsupport. iii

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page ACKNOWLEDGMENTS .................................... iii LISTOFTABLES ....................................... vii LISTOFFIGURES ....................................... viii ABSTRACT ........................................... x CHAPTER 1INTRODUCTION .................................... 1 2PORTFOLIOOPTIMIZATIONWITHCONDITIONALVALUE-AT-RISKOBJEC-TIVEANDCONSTRAINTS ............................. 5 2.1Introduction .................................... 5 2.2ConditionalValue-at-Risk ............................ 9 2.3EfcientFrontier:DifferentFormulations .................... 12 2.4EquivalentFormulationswithAuxiliaryVariables ................ 15 2.5DiscretizationandLinearization ......................... 18 2.6OnePeriodPortfolioOptimizationModelwithTransactionCosts ........ 19 2.7CaseStudy:PortfolioofS&P100Stocks ..................... 23 2.7.1EfcientFrontierandPortfolioConguration .............. 24 2.7.2ComparisonwithMean-VariancePortfolioOptimization ........ 25 2.8ConcludingRemarks ............................... 30 3COMPARATIVEANALYSISOFLINEARPORTFOLIOREBALANCINGSTRATE-GIES:ANAPPLICATIONTOHEDGEFUNDS ................... 32 3.1Introduction .................................... 32 3.2LinearPortfolioRebalancingAlgorithms .................... 35 3.2.1ConditionalValue-at-Risk ........................ 37 3.2.2ConditionalDrawdown-at-Risk ...................... 41 3.2.3Mean-AbsoluteDeviation ......................... 44 3.2.4MaximumLoss .............................. 45 3.2.5Market-Neutrality ............................. 45 3.2.6ProblemFormulation ........................... 46 3.2.7ConditionalValue-at-RiskConstraint ................... 47 3.2.8ConditionalDrawdown-at-RiskConstraint ................ 48 3.2.9MADConstraint ............................. 48 3.2.10MaxLossConstraint ............................ 48 3.3CaseStudy:PortfolioofHedgeFunds ...................... 49 3.3.1In-SampleResults ............................. 50 iv

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....................... 54 3.4Conclusions .................................... 68 4OPTIMALPOSITIONLIQUIDATION ......................... 76 4.1Introduction .................................... 76 4.2GeneralDenitionsandProblemStatement ................... 79 4.2.1RepresentationofUncertaintiesbyaSetofPaths ............ 79 4.2.2AGenericProblemFormulation ..................... 81 4.2.3ModelingtheMarketImpact ....................... 82 4.3OptimalPositionLiquidationunderTemporaryMarketImpact ......... 85 4.3.1PathsGroupingandLawn-MowerStrategy .............. 85 4.3.2ApproximationbyConvexandLinearProgramming .......... 90 4.3.3PropertiesofSolutionsoftheNon-ConvexandConvexOptimalLiqui-dationProblems ............................ 97 4.4OptimalLiquidationwithPermanentMarketImpact ............... 100 4.5RiskConstraints .................................. 101 4.6Casestudy:OptimalClosingofLongPositioninaStock ............ 104 4.6.1OptimalClosinginFrictionlessMarket ................. 105 4.6.2OptimalClosingunderTemporaryMarketImpact ............ 107 4.6.3OptimalClosingunderPermanentMarketImpact ............ 109 4.7Conclusions .................................... 112 5ROBUSTDECISIONMAKING:ADDRESSINGUNCERTAINTIESINDISTRIBU-TIONS ......................................... 114 5.1Introduction .................................... 114 5.2TheGeneralApproach .............................. 115 5.2.1RiskManagementUsingConditionalValue-at-Risk ........... 117 5.2.2RiskManagementUsingCVaRinthePresenceofUncertaintiesinDis-tributions ................................ 119 5.3Example:StochasticWeapon-TargetAssignmentProblem ........... 120 5.3.1DeterministicWTAProblem ....................... 120 5.3.2One-StageStochasticWTAProblemwithCVaRConstraints ...... 122 5.3.3Two-StageStochasticWTAProblemwithCVaRconstraints ...... 124 5.4Numericalresults ................................. 127 5.4.1Single-stagedeterministicandstochasticWTAproblems ........ 127 5.4.2Two-StageStochasticWTAProblem ................... 129 5.5Conclusions .................................... 131 6USEOFCONDITIONALVALUE-AT-RISKINSTOCHASTICPROGRAMSWITHPOORLYDEFINEDDISTRIBUTIONS ....................... 132 6.1Introduction .................................... 132 6.2DeterministicWeapon-TargetAssignmentProblem ............... 133 6.3Two-StageStochasticWTAProblem ....................... 137 6.4Two-StageWTAProblemwithUncertaintiesinSpeciedDistributions ..... 139 6.5CaseStudy .................................... 143 6.6Conclusions .................................... 146 v

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..................................... 148 APPENDIX ......................................... 152 AONTHEPRICINGOFAMERICAN-STYLEDERIVATIVESECURITIESUSINGLINEARPROGRAMMING .............................. 152 A.1Black-ScholesApproachtoPricingofOptions .................. 152 A.2SolutionoftheFree-BoundaryProblemforAmericanPutUsingLinearPro-gramming .................................... 154 A.3PricingofPath-DependentOptionsUsingSimulationandStochasticProgramming 161 REFERENCES ......................................... 166 BIOGRAPHICALSKETCH .................................. 173 vi

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Table page 2Portfolioconguration:assets'weights(%)intheoptimalportfoliodependingontherisklevel ..................................... 26 3Instrumentweightsintheoptimalportfoliowithdifferentriskconstraints ...... 53 3Weightsofresidualinstrumentsintheoptimalportfoliowithdifferentriskconstraints 53 3Summarystatisticsfortherealout-of-sampletests ................. 70 3Summarystatisticsforthemixedout-of-sampletests ................ 73 4Optimaltradingstrategyinfrictionlessmarket. .................... 106 4Optimaltradingstrategyunderlineartemporarymarketimpact(b=2). ...... 108 4Optimaltradingstrategyunderlineartemporarymarketimpactthatdependsonpricedynamics. 4Optimaltradingstrategyunderlinearpermanentmarketimpact(b=2). ...... 110 4Optimaltradingstrategyunderpiecewise-linearpermanentmarketimpact. ..... 111 4Optimaltradingstrategyundernonlinearpermanentmarketimpact. ......... 111 5OptimalsolutionofthedeterministicWTAproblem( 5.3a ) .............. 129 5Optimalsolutionoftheone-stagestochasticWTAproblem( 5.6 ),( 5.8 ) ....... 129 5First-stageoptimalsolutionofthetwo-stagestochasticWTAproblem ........ 130 5First-stageoptimalsolutionofthetwo-stagestochasticWTAproblem( 5.11 )fortherstscenario ..................................... 131 5Second-stageoptimalsolutionofthetwo-stagestochasticWTAproblem( 5.11 )forthesecondscenario ................................. 131 6Theexpectedvaluesforthenumberofthesecond-stagetargetsintwocategoriesforscenarioss=1;2;3. ................................ 145 6SolutionoftheMIPproblemandproblem( 6.15a )fordifferentvaluesofcondencelevela ........................................ 145 APricingofAmericanputusinglinearprogramming .................. 160 APricingofAmericanputbysimulationandstochasticprogramming ......... 164 vii

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Figure page 2Efcientfrontier(optimizationwithCVaRconstraints) ............... 25 2EfcientfrontierofoptimalportfoliowithCVaRconstraintsinpresenceoftransac-tioncosts ...................................... 27 2EfcientfrontiersofCVaRandMVoptimalportfolios(a=0:95,Return/CVaRscale) ........................................ 28 2EfcientfrontiersofCVaRandMVoptimalportfolios(a=0:99,Return/CVaRscale) ........................................ 29 2EfcientfrontiersofCVaRandMVoptimalportfolios(a=0:95,Return/StDevscale) ........................................ 29 2EfcientfrontiersofCVaRandMVoptimalportfolios(a=0:99,Return/StDevscale) ........................................ 30 3Lossdistribution,VaR,CVaR,andMaximumLoss. ................. 39 3Portfoliovalueanddrawdown. ............................ 40 3Efcientfrontiersforportfoliowithvariousriskconstraints ............. 50 3Efcientfrontierformarket-neutralportfoliowithvariousriskconstraints ..... 51 3Historicalperformanceandrateofreturndynamicsforresidualassets ....... 55 3HistoricaltrajectoriesofoptimalportfoliowithCVaRconstraints. ......... 57 3HistoricaltrajectoriesofoptimalportfoliowithCDaRconstraints. ......... 57 3HistoricaltrajectoriesofoptimalportfoliowithMADconstraints. .......... 57 3HistoricaltrajectoriesofoptimalportfoliowithMaxLossconstraints. ........ 58 3Historicaltrajectoriesofmarket-neutraloptimalportfoliowithCVaRconstraints .. 60 3Historicaltrajectoriesofmarket-neutraloptimalportfoliowithCDaRconstraints .. 60 3Historicaltrajectoriesofmarket-neutraloptimalportfoliowithMADconstraints .. 61 3Historicaltrajectoriesofmarket-neutraloptimalportfoliowithMaxLossconstraints 61 3Optimalportfoliosvs.benchmarks .......................... 63 3Market-neutraloptimalportfoliosvs.benchmarks .................. 63 3PortfoliowithCVaRconstraints(mixedtest) ..................... 64 viii

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..................... 65 3PortfoliowithMADconstraints(mixedtest) ..................... 65 3PortfoliowithMaxLossconstraints(mixedtest) ................... 65 3Market-neutralportfoliowithCVaRconstraints(mixedtest) ............ 66 3Market-neutralportfoliowithCDaRconstraints(mixedtest) ............ 66 3Market-neutralportfoliowithMADconstraints(mixedtest) ............. 67 3Market-neutralportfoliowithMaxLossconstraints(mixedtest) ........... 67 4Samplepathsandandscenariotrees. ......................... 80 4Impactfunctiondt(). ................................. 83 4Pathgrouping ..................................... 86 4Tree-likegrouping. .................................. 87 4Thelawn-mowerprinciple. ............................. 88 4IllustrationtoproofofProposition 4.3.2 ....................... 93 4IllustrationtoproofofProposition 4.3.5 ....................... 96 4Performanceofoptimaltradingstrategyinfrictionlessmarket. ........... 106 5Lossfunctiondistributionanddifferentriskmeasures. ................ 116 5AvisualizationofVaRandCVaRconcepts. ..................... 117 5Dependencebetweenthecostandefciencyfordifferenttypesofweaponsinone-stageSWTAproblem( 5.8 )deterministicWTAproblem( 5.3a ). .......... 128 5Dependencebetweenthecostandefciencyfordifferenttypesofweaponsintwo-stageSWTAproblem( 5.11 ). ............................ 130 6SolutionoftheMIPproblem ............................. 146 ix

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Thedissertationstudiesmodernriskmanagementtechniquesfordecisionmakinginhighlyuncertainenvironments.Thetraditionalframeworkofdecisionmakingunderuncertaintiesreliesonstochasticprogrammingorsimulationapproachestosurpasssimplerquasi-deterministictech-niques,wheretheuncertaintyismodeledbyrelevantstatisticsofstochasticparameters.Inmanyapplications,however,thementionedmethodologiesintheirconventionalformfailtogenerateef-cientandrobustdecisions.Mathematicalmodelsforsuchclassofapplications,therefore,arereferredtoashighlyuncertainenvironments,withthedeningfeaturessuchas:largenumberofmutuallycorrelatedstochasticfactorswithdynamicallychangingoruncertaindistributions,multi-pletypesofriskexposure,out-of-sampleapplicationofthesolution,etc.Robustdecisionmakinginsuchenvironmentsrequiresanexplicitcontroloftheriskinducedbyuncertainties. Therstpartofthedissertationconsidersriskmanagementapproachesfornancialapplica-tions.Wepresentageneralframeworkofrisk/rewardoptimizationthatestablishestheequivalenceofdifferentformulationsofoptimizationproblemswithriskandrewardfunctions.Asanapplicationofthegeneralresult,weconsideroptimizationofportfolioofstockswithConditionalValue-at-Riskobjectiveandconstraints,andcomparethisapproachwiththeclassicalMarkowitzMean-Variancemethodology.Anextensivestudyofout-of-sampleperformanceoftradingalgorithmsbasedondifferentriskmeasuresisperformedontheexampleofmanagingofahedgefundportfolio.Achapterdedicatedtomulti-stagedecision-makingproblemspresentsanewsample-pathapproach x

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Thesecondpartofthedissertationconsidersriskmanagementtechniquesformilitarydecision-makingproblems.Themainchallengesofmilitaryapplicationsattributetovarioustypesofriskexposure,uncertainprobabilitymeasuresofrisk-inducingfactors,andinapplicabilityofthelongrunconvention.DifferentformulationsforstochasticWeapon-TargetAssignmentproblemwithuncertaintiesindistributionsareconsidered,andrelaxationandlinearizationtechniquesfortheresultingnonlinearmixed-integerprogrammingproblemsaresuggested. xi

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Thedissertationisdevotedtostudyofriskmanagementtechniquesfordecisionmakinginhighlyuncertainenvironments.Thetraditionalframeworkofdecisionmakingunderthepresenceofuncertaintiesreliesonstochasticprogramming( BirgeandLouveaux 1997 ; Prekopa 1995 )orsimulation( Ripley 1987 )approachestosurpasssimplerquasi-deterministictechniques,wheretheuncertaintyismodeledbyrelevantstatisticsofthestochasticparameters,suchasexpectationorvariance. Thestudyofdecision-makingmathematicalprogrammingmodelsthatinvolveuncertainpa-rameterswasoriginatedby DantzigandMadansky ( 1961 ),andreceivedmajordevelopmentinworksofR.J.-B.Wetsandcolleagues( RockafellarandWets 1976a b ; WalkupandWets 1967 1969 ; Wets 1966a b 1974 ).Incontrasttothequasi-deterministicaveragingapproachesinde-cisionmakingunderuncertainties,stochasticprogrammingintroducestheconceptofstagesinthedecisionmaking,wheresomedecisionshavetobemadewithoutpriorknowledgeoftherealizationoftherandomparametersinthesystem,andfollowedbycorrectiveactions,takenaftertheuncer-taintieshavebeenrealized.Foracomprehensivepresentationofstochasticprogrammingtheoryandapplications,see BirgeandLouveaux ( 1997 )or Prekopa ( 1995 ). Useofstochasticprogrammingsolutionsinuncertainenvironmentsensuresincreasedrobust-nessandeffectivenessincomparisontodeterministicallyobtainedsolutions.However,thereex-istapplications,wherethestochasticprogrammingmethodologyinitsconventionalformfailstogenerateefcientandrobustdecisions.Mathematicalmodelsforsuchclassofapplicationsusu-allyfeaturealargenumberofmutuallycorrelatedstochasticfactorswithdynamicallychangingoruncertaindistributions,multipletypesofriskexposure,applicationoftheobtainedsolutioninsituationsthatcannotbedescribedbythescenariosetoftheoriginalproblem(so-calledout-of-sampleapplications),etc.Werefertomathematicalmodelsthatexhibitsuchorsimilarpropertiesashighlyuncertainenvironments.Robustdecisionmakinginsuchenvironmentsrequires,besidesmaximizingtheexpectedperformance,anexplicitcontrolontheriskinducedbyuncertainties. 1

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Theareaofstochasticanalysisandmathematicalprogramming,knownasriskmanagement,focusesontheeffectsofextremelyadverseoutcomesoreventsontheprocessofdecisionmaking.Therstoutstandingcontributiontothiseldisdueto Markowitz ( 1952 1991 ),whoidentiedtheriskinessofaportfolioofnancialinstrumentswiththevolatilityofassets'returns,andproposedthefamousMean-Varianceportfoliooptimizationmodelforconstructingaportfoliowithlowestriskforagivenlevelofexpectedreturn.ThetransparencyandefciencyofMarkowqitz'sapproachmadeitaverypopulartool,whichiswidelyusedinthenanceindustryevennowadays.SinceMarkowitz'sseminalwork,aconsiderableprogresshasbeenachievedintheareaofriskanalysisandmanagement,whichevolvedintoasophisticateddisciplinecombiningrigorousandelegantthe-oreticalresultswithpracticaleffectiveness.ItwasrecognizedthatneithertheMean-Variancemodel,whichutilizesvariancetomeasuretheuncertaintyinthesystem,northeValue-at-Riskconcept(theuppera-quantileofdistribution,see Jorion ( 1996 1997 )),whichbecametheofcialstandardinthenanceindustry,canprovideanadequategureofriskexposure.Thetheoryofcoherentriskmea-sures,developedby Artzneretal. ( 1997 1999 2001 ), Delbaen ( 2000 ),establishedasetofaxiomstobesatisedbyariskmeasure(usually,acertainstatisticofthedistribution)inordertoyieldcor-rectestimationofriskunderdifferentconditions.Atthesametime,aseriesofdevelopmentsofnewriskmeasuressuchasConditionalValue-at-Risk,ExpectedShortfall( Acerbi 2002 ; Acerbietal. 2001 ; AcerbiandTasche 2002 ; RockafellarandUryasev 2000 2002 )demonstratedthetheoreticalandpracticalimportanceandefciencyoftheconceptofcoherentriskmeasures. Therstpartofthedissertation(Chapters2to4)considersriskmanagementapproachesinthescopeofnancialapplications.InChapter2,wepresentageneralframeworkofrisk/rewardop-timizationthatestablishestheequivalenceofdifferentformulationsofoptimizationproblemswithriskandrewardfunctions.Asanapplicationofthegeneralresult,weconsideroptimizationofport-folioofstockswithConditionalValue-at-Riskobjectiveandconstraints,andcomparethisapproachwiththeclassicalMarkowitzMean-Variancemethodology.Chapter3containsanextensivestudyofin-sampleandout-of-sampleperformanceoftradingalgorithmsbasedondifferentriskmeasures(ConditionalValue-at-risk,ConditionalDrawdown-at-Risk,Mean-AbsoluteDeviation,andMaxi-mumLoss)ontheexampleofmanagingaportfolioofhedgefunds.Basedontheresultsofthisandothercasestudies,wemakerecommendationsonhowtoimprovethereal(i.e.,out-of-sample)performanceofportfoliorebalancingstrategies.

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Chapter4introducesanewsample-pathapproachtotheproblemofoptimaltransactionim-plementation.Basedontheideasofmulti-stagestochasticprogramming,andusingasample-pathcollectioninsteadofthetraditionalscenariotree,thisapproachproducesanoptimaltradingstrat-egythatadmitsadifferentiateresponsetorealizedmarketconditionsateachtimestep.Incontrasttootherexistingapproachestooptimalexecution,ourapproachadmitsaseamlessincorporationofdifferenttypesofconstraints,e.g.regulatoryorriskconstraints,intothetradingstrategy.Thegeneralityofthedevelopedapproachallowstouseitforpricingofcomplexderivativesecurities,suchasexoticoptions. Althoughtheeldofriskmanagementisrootedinnanceandnancialapplications,thesuc-cessesofriskmanagementintheoreticaland,especially,appliedcontextshavedrawnattentiontothesetechniquesinothereldswhereuncertaintiesimpactthedecisionmakingprocess.Oneoftheareaswheresystematicriskmanagementapproachwillcontributesignicantlytorobustnessofdecisionsandpoliciesismilitaryapplications,whichtypicallyinvolvedecisionmakingindynamic,distributed,andhighlyuncertainenvironments.Themainchallengesofmilitaryapplicationsat-tributetovarioustypesofriskexposure,uncertainprobabilitymeasuresofrisk-inducingfactors,andinapplicabilityofthelongrunconvention. Indeed,incontrasttotheproblemsofnance,wheretheonlytypeofriskistheriskofnancialloss,militaryincorporatedifferenttypesofrisk,e.g.,riskofnotkillingatarget,riskoffalsetargetattack,riskofbeingdestroyedbyenemy,etc.Inaddition,distributionsofriskfactorsinmilitaryproblemsarerarelyknownwithcertainty.Also,atypicalnancialmodelisexpectedperformwellinrepetitiveapplications,i.e.,itisrequiredtobeeffectiveonaverage,orinalongrun.Obviously,thereisnolongruninmilitaryapplications,thereforethegenerateddecisionmustbebothsafeandeffectivehereandnow. Inthesecondpartofthedissertation(Chapters5to6),weconsiderriskmanagementtech-niquesformilitarydecision-makingproblems.Ageneralapproachformilitarydecision-makingproblemswithuncertaintiesindistributionsisdevelopedinChapter5,and,asanexample,weconsiderastochasticversionoftheWeapon-TargetAssignmentproblem.WedemonstratethatemployingriskmanagementtechniquesbasedontheConditionalValue-at-Riskmeasureleadstosolutionsthatperformrobustlyunderawiderangeofscenarios.

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InChapter6,weconsideranonlinearintegerprogrammingmodelfortheStochasticWeapon-TargetAssignmentproblem,anddevelopalinearprogrammingrelaxation,whichallowsforef-cientincorporationofriskconstraintsintheproblem. Finally,Appendixpresentstheresultsofourongoingresearcheffortsintheareaofpricingofpath-dependentderivativesecuritiesusingthetechniquesofmathematicalprogramming.Wediscussalinearprogramming-basedalgorithmforsolvingfree-boundaryproblemarisinginpricingofAmericanputoptionwithintheclassicalBlack-Scholesframework,aswellasanewapproachforpricingofpath-dependentderivativesecuritiesusingsimulationandstochasticprogramming.

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Inthischapter,weextendtherecentlysuggestedapproach( RockafellarandUryasev 2000 )foroptimizationofConditionalValue-at-Risk(CVaR)tothecaseofoptimizationproblemsCVaRconstraints.Wederiveageneralequivalenceresultfordifferenttypesofoptimizationproblemswithbroadclassofriskandrewardfunctions.Inparticular,theapproachcanbeusedformaximizingtheexpectedreturnsunderCVaRconstraints.Asafurtherextension,multipleCVaRconstraintswithvariouscondencelevelscanbeusedtoshapetheprot/lossdistribution.AcasestudyfortheportfolioofS&P100stocksisperformedtodemonstratehowthenewoptimizationtechniquescanbeimplemented.TheapproachiscomparedwiththeclassicMarkowitzMean-Variancemodelforportfoliooptimization. Markowitz ( 1952 )seminalworkwhichin-troducesreturn/varianceriskmanagementframework.Developmentsinportfoliooptimizationarestimulatedbytwobasicrequirements: Currentregulationsfornancebusinessesformulatesomeoftheriskmanagementrequire-mentsintermsofpercentilesoflossdistributions.AnupperpercentileofthelossdistributioniscalledValue-at-Risk(VaR). 5

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underlyingriskfactorsarenormally(log-normally)distributed.ForcomprehensiveintroductiontoriskmanagementusingVaR,wereferthereaderto Jorion ( 1997 ).However,fornon-normaldistributions,VaRmayhaveundesirableproperties( Artzneretal. 1997 1999 )suchaslackofsub-additivity,i.e.,VaRofaportfoliowithtwoinstrumentsmaybegreaterthanthesumofindividualVaRsofthesetwoinstruments. Rockafellar 1970 )andnon-smoothasafunctionofpositions,andhasmultiplelocalextrema.AnextensivedescriptionofvariousmethodologiesforthemodelingofVaRcanbeseen,alongwithrelatedresources,atURLhttp://www.gloriamundi.org/.Mostly,approachestocalculatingVaRrelyonlinearapproximationoftheportfoliorisksandassumeajointnormal(orlog-normal)distribu-tionoftheunderlyingmarketparameters( DufeandPan ( 1997 ), Jorion ( 1996 ), Pritsker ( 1997 ), RiskMetricsTM 1996 ), Simons ( 1996 ), StubloBeder ( 1995 ), Staumbaugh ( 1996 )).Also,historicalorMonteCarlosimulation-basedtoolsareusedwhentheportfoliocontainsnonlinearinstrumentssuchasoptions( Jorion ( 1996 ), MausserandRosen ( 1991 ), Pritsker ( 1997 ), RiskMetricsTM 1996 ), StubloBeder ( 1995 ), Staumbaugh ( 1996 )).DiscussionsofoptimizationproblemsinvolvingVaRcanbefoundin Litterman ( 1997a b ), Kastetal. ( 1998 ), LucasandKlaassen ( 1998 ). Althoughriskmanagementwithpercentilefunctionsisaveryimportanttopicandinspiteofsignicantresearchefforts( AndersenandSornette ( 2001 ), BasakandShapiro ( 2001 ), Emmeretal. ( 2000 ), GaivoronskiandPug ( 2000 ), Gourierouxetal. ( 2000 ), GrootweldandHallerbach ( 2000 ), Kastetal. ( 1998 ), Puelz ( 1999 ), Tasche ( 1999 )),efcientalgorithmsforoptimizationofpercentilesforreasonabledimensions(overonehundredinstrumentsandonethousandscenarios)arestillnotavailable.Ontheotherhand,theexistingefcientoptimizationtechniquesforportfolioallocation 2

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donotallowfordirectcontrolling KonnoandYamazaki 1991 ),theregretoptimizationapproach( DemboandRosen 1999 ),andtheminimaxapproach( Young 1998 ).Thisfactstimulatedourdevelopmentofthenewoptimizationalgorithmspresentedinthischapter. Thischaptersuggeststouse,asasupplement(oralternative)toVaR,anotherpercentileriskmeasurewhichiscalledConditionalValue-at-Risk.TheCVaRriskmeasureiscloselyrelatedtoVaR.Forcontinuousdistributions,CVaRisdenedastheconditionalexpectedlossunderthecon-ditionthatitexceedsVaR,see RockafellarandUryasev ( 2000 ).Forcontinuousdistributions,thisriskmeasurealsoisknownasMeanExcessLoss,MeanShortfall,orTailValue-at-Risk.However,forgeneraldistributions,includingdiscretedistributions,CVaRisdenedastheweightedaverageofVaRandlossesstrictlyexceedingVaR( RockafellarandUryasev 2002 ).Recently, Acerbietal. ( 2001 ), AcerbiandTasche ( 2002 )redenedexpectedshortfallsimilarlytoCVaR. Forgeneraldistributions,CVaR,whichisaquitesimilartoVaRmeasureofriskhasmoreattractivepropertiesthanVaR.CVaRissub-additiveandconvex( RockafellarandUryasev 2000 ).Moreover,CVaRisacoherentmeasureofriskinthesenseof Artzneretal. ( 1997 1999 ).CoherencyofCVaRwasrstprovedby Pug ( 2000 );seealso RockafellarandUryasev ( 2002 ), Acerbietal. ( 2001 ), AcerbiandTasche ( 2002 ).AlthoughCVaRhasnotbecomeastandardinthenanceindus-try,CVaRisgainingintheinsuranceindustry( Embrechtsetal. 1997 ).SimilartoCVaRmeasureshavebeenintroducedearlierinstochasticprogrammingliterature,butnotinnancialmathemat-icscontext.Theconditionalexpectationconstraintsandintegratedchanceconstraintsdescribedin Prekopa ( 1995 )mayservethesamepurposeasCVaR. NumericalexperimentsindicatethatusuallytheminimizationofCVaRalsoleadstonearop-timalsolutionsinVaRtermsbecauseVaRneverexceedsCVaR( RockafellarandUryasev 2000 ).Therefore,portfolioswithlowCVaRmusthavelowVaRaswell.Moreover,whenthereturn-loss performancefunctions,seeforinstance Duarte ( 1999 ).Thereaderinterestedinvariousapplica-tionsofoptimizationtechniquesinthenanceareacanndrelevantpapersin ZiembaandMulvey ( 1998 ).4

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distributionisnormal,thesetwomeasuresareequivalent( RockafellarandUryasev 2000 ),i.e.,theyprovidethesameoptimalportfolio.Howeverforveryskeweddistributions,CVaRandVaRriskoptimalportfoliosmaybequitedifferent.Moreover,minimizingofVaRmaystretchthetailexceedingVaRbecauseVaRdoesnotcontrollossesexceedingVaR,see Larsenetal. ( 2002 ).Also, GaivoronskiandPug ( 2000 )havefoundthatinsomecasesoptimizationofVaRandCVaRmayleadtoquitedifferentportfolios. RockafellarandUryasev ( 2000 )demonstratedthatlinearprogrammingtechniquescanbeusedforoptimizationoftheConditionalValue-at-Risk(CVaR)riskmeasure.AsimpledescriptionoftheapproachforminimizingCVaRandoptimizationproblemswithCVaRconstraintscanbefoundin Uryasev ( 2000 ).SeveralcasestudiesshowedthatriskoptimizationwiththeCVaRperformancefunctionandconstraintscanbedoneforlargeportfoliosandalargenumberofscenarioswithrelativelysmallcomputationalresources.AcasestudyonthehedgingofaportfolioofoptionsusingtheCVaRminimizationtechniqueisincludedin( RockafellarandUryasev 2000 ).Thisproblemwasrststudiedinthepaperby MausserandRosen ( 1991 )withtheminimumexpectedregretapproach.Also,theCVaRminimizationapproachwasappliedtocreditriskmanagementofaportfolioofbonds,see Anderssonetal. ( 2001 ). ThischapterextendstheCVaRminimizationapproach( RockafellarandUryasev 2000 )tootherclassesofproblemswithCVaRfunctions.Weshowthatthisapproachcanbeusedalsoformaximizingrewardfunctions(e.g.,expectedreturns)underCVaRconstraints,asopposedtomini-mizingCVaR.Moreover,itispossibletoimposemanyCVaRconstraintswithdifferentcondencelevelsandshapethelossdistributionaccordingtothepreferencesofthedecisionmaker.Thesepreferencesarespecieddirectlyinpercentileterms,comparedtothetraditionalapproach,whichspeciesriskpreferencesintermsofutilityfunctions.Forinstance,wemayrequirethatthemeanvaluesoftheworst1%,5%and10%lossesarelimitedbysomevalues.Thisapproachprovidesanewefcientandexibleriskmanagementtool. ThenextsectionbrieydescribestheCVaRminimizationapproachfrom( RockafellarandUryasev 2000 )tolaythefoundationforthefurtherextensions.InSection3,weformulatedageneraltheoremonvariousequivalentrepresentationsofefcientfrontierswithconcaverewardandconvexriskfunctions.Thisequivalenceiswellknownformean-variance,seeforinstance, Steinbach ( 1999 ),andformean-regret,( DemboandRosen 1999 ),performancefunctions.Wehave

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shownthatitholdsforanyconcaverewardandconvexriskfunction,inparticularfortheCVaRriskfunctionconsideredinthischapter.Usingauxiliaryvariables,weformulatedatheoremonreducingtheproblemwithCVaRconstraintstoamuchsimplerconvexproblem.AsimilarresultisalsoformulatedforthecasewhenboththerewardandCVaRareincludedintheperformancefunction.Asitwasearlieridentiedin( RockafellarandUryasev 2000 ),theoptimizationautomaticallysetstheauxiliaryvariabletoVaR,whichsignicantlysimpliestheproblemsolution.Further,whenthedistributionisgivenbyaxednumberofscenariosandthelossfunctionislinear,weshowedhowtheCVaRfunctioncanbereplacedbyalinearfunctionandanadditionalsetoflinearconstraints.Insection 2.7 ,wedevelopedaone-periodmodelforoptimizingaportfolioofstocksusinghistoricalscenariogeneration.AcasestudyontheoptimizationofS&P100portfolioofstockswithCVaRconstraintsispresentedinsection 2.7 .Wecomparedthereturn-CVaRandreturn-varianceefcientfrontiersoftheportfolios. RockafellarandUryasev 2000 )providesthefoundationfortheanalysisconductedinthischapter.First,following( RockafellarandUryasev 2000 ),weformallydeneCVaRandpresentseveraltheoreticalresultswhichareneededforunderstandingthischapter.Letf(x;y)bethelossassociatedwiththedecisionvector Foreachx,thelossf(x;y)isarandomvariablehavingadistributioninRinducedbythatofy.TheunderlyingprobabilitydistributionofyinRmwillbeassumedforconveniencetohavedensity,whichwedenotebyp(y).Thisassumptionisnotcriticalfortheconsideredapproach.Thepaperby RockafellarandUryasev ( 2002 )denesCVaRforgeneraldistributions;however,here,forsimplicity,weassumethatthedistributionhasdensity.Theprobabilityoff(x;y)notexceedinga

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thresholdzisthengivenby Asafunctionofzforxedx,Y(x;z)isthecumulativedistributionfunctionforthelossassociatedwithx.ItcompletelydeterminesthebehaviorofthisrandomvariableandisfundamentalindeningVaRandCVaR. ThefunctionY(x;z)isnondecreasingwithrespectto(w.r.t.)zandweassumethatY(x;z)iseverywherecontinuousw.r.t.z.Thisassumption,likethepreviousoneaboutdensityiny,ismadeforsimplicity.Insomecommonsituations,therequiredcontinuityfollowsfrompropertiesofthelossf(x;y)andthedensityp(y);see Uryasev ( 1995 ). Thea-VaRanda-CVaRvaluesforthelossrandomvariableassociatedwithxandanyspeci-edprobabilitylevelain(0;1)willbedenotedbyza(x)andfa(x).Inoursettingtheyaregivenby and Intherstformula,za(x)comesoutastheleftendpointofthenonemptyinterval Thekeytotheapproachisacharacterizationoffa(x)andza(x)intermsofthefunctionFaonXRthatwenowdeneby where[t]+=maxft;0g.ThecrucialfeaturesofFa,undertheassumptionsmadeabove,areasfollows( RockafellarandUryasev 2000 ).

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Forbackgroundonconvexity,whichisakeypropertyinoptimizationthatinparticulareliminatesthepossibilityofalocalminimumbeingdifferentfromaglobalminimum,see,forinstance, Rock-afellar ( 1970 ).OtherimportantadvantagesofviewingVaRandCVaRthroughtheformulasinTheorem1arecapturedinthenexttheorem,alsoprovedin( RockafellarandUryasev 2000 ) Furthermore,Fa(x;z)isconvexw.r.t.(x;z),andfa(x)isconvexw.r.t.x,whenf(x;y)isconvexwithrespecttox,inwhichcase,iftheconstraintsaresuchthatXisaconvexset,thyjointminimizationisaninstanceofconvexprogramming.

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AccordingtoTheorem 2.2.2 ,itisnotnecessary,forthepurposeofdeterminingavectorxthatyieldstheminimuma-CVaR,toworkdirectlywiththefunctionfa(x),whichmaybehardtodobecauseofthenatureofitsdenitionintermsofthea-VaRvalueza(x)andtheoftentrouble-somemathematicalpropertiesofthatvalue.Instead,onecanoperateonthefarsimplerexpressionFa(x;z)withitsconvexityinthevariablezandeven,verycommonly,withrespectto(x;z). RockafellarandUryasev ( 2000 )consideredminimizingCVaR,whilerequiringaminimumexpectedreturn.Byconsideringdifferentexpectedreturns,wecangenerateanefcientfrontier.Alternatively,wealsocanmaximizereturnswhilenotallowinglargerisks.We,therefore,canswaptheCVaRfunctionandtheexpectedreturnintheproblemformulation(comparedto Rock-afellarandUryasev ( 2000 ),thusminimizingthenegativeexpectedreturnwithaCVaRconstraint.Byconsideringdifferentlevelsofrisks,wecangeneratetheefcientfrontier. Wewillshowinageneralsettingthattherearethreeequivalentformulationsoftheoptimiza-tionproblem.Theyareequivalentinthesensethattheyproducethesameefcientfrontier.Thefollowingtheoremisvalidforgeneralfunctionssatisfyingconditionsofthetheorem.

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TheproofofTheorem 2.3.1 isbasedontheKuhn-Tuckernecessaryandsufcientconditionsstatedinthefollowingtheorem. Pshenichnyi 1971 )).Considertheproblemminy0(x);yi(x)0i=m;:::;1;yi(x)=0i=1;:::;n;x2X:Letyi(x)befunctionalsonalinearspace,E,suchthatyi(x)areconvexfori0andlinearfori1andXissomegivenconvexsubsetofE.Theninorderthaty8(x)achievesitsminimumpointatx2Eitisnecessarythatthereexistsconstantsli;i=m;:::;n,suchthatni=mliyi(x)ni=mliyi(x)forallx2X.Moreover,li0foreachi0,andliyi(x0)=0foreachi6=0.Ifl00,thentheconditionsarealsosufcient. LetuswritedownthenecessaryandsufcientKuhn-Tackerconditionsforproblems(P1),(P2),and(P3).Aftersomeequivalenttransformationstheseconditionscanbestatedasfollows K-Tconditionsfor(P1):(KT1)f(x)m1R(x)f(x)m1R(x);m10;x2X:

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Steinbach ( 1999 ),wecallm1in(KT2)theoptimalrewardmultiplier,andm3in(KT3)theriskmultiplier.Further,usingconditions(KT1)and(KT7),weshowthatasolutionofproblem(P1)isalsoasolutionof(P2)andviceversa,asolutionofproblem(P2)isalsoasolutionof(P1). 2.3.1 Now,letusprovethesecondstatementofLemma 2.3.1 .Supposethatxisasolutionof(P2)and(KT2)issatised.Then,(KT1)issatisedwithparameterm1=m2andxisasolutionof(P1).Lemma 2.3.1 iyproved.

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2.3.2 2.3.2 .Ifxisasolutionof(P1),thenitsatisesthecondition(KT1).Ifm1>0,thenthissolutionxsatises(KT3)withm3=1=m9andw=f(x). Now,letusprovethesecondstatementofLemmaA2.Supposethatxisasolutionof(P3)and(KT6)issatisedwithm3>0.Then,(KT1)issatisedwithparameterm1=1=m3andxisasolutionof(P1).Lemma 2.3.2 isproved. 2.3.1 .Lemma 2.3.1 impliesthattheefcientfrontiersofproblems(P1)and(P2)coincide.Similar,Lemma 2.3.2 impliesthattheefcientfrontiersofproblems(P1)and(P3)coincide.Consequently,efcientfrontiersofproblems(P1),(P2),and(P3)coincide. Steinbach ( 1999 )andmean-regret DemboandRosen ( 1999 )efcientfrontiers.Behaveshownthatitholdsforanyconcaverewardandconvexriskfunctionswithconvexconstraints. Further,weconsiderthatthelossfunctionf(x;y)islinearw.r.t.x,thereforeTheorem2impliesthattheCVaRriskfunctionfa(x)isconvexw.r.t.x.Also,wesupposethattherewardfunction,R(x)islinearandthisconstraintsarelinear.TheconditionsofTheorem3aresatisedfortheCVaRriskfunctionfa(x)andtherewardfunctionR(x).Therefore,maximizingtherewardunderaCVaRconstraint,generatesthesameefcientfrontierastheminimizationofCVaRunderaconstraintonthereward. 2.3.1 impliesthatwecanuseproblemformulations(P1),(P2),and(P3)forgenerat-ingtheefcientfrontierwiththeCVaRriskfunctionfa(x)andtherewardfunctionR(x).Theorem 2.2.2 showsthatthefunctionFa(x;z)canbeusedinsteadoffa(x)tosolveproblem(P2).Further,wedemonstratethat,similarly,thefunctionFa(x;z)canbeusedinsteadoffa(x)inproblems(P1)and(P8).

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2.4.1 Now,letussupposethat(x;z)achievestheminimumof(P4')andm3>0.Forxedx,thepointzminimizesthefunctionR(x)+m3Fa(x;z),and,consequently,thefunctionFa(x;z).

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Then,Theorem 2.2.1 impliesthatz2Aa(x).Further,since(x;z)isasolutionof(P4'),condi-tions(KT3')andTheorem 2.2.1 implythatR(x)+m3fa(x)=R(x)+m3Fa(x;z)R(x)+m3Fa(x;za(x))=R(x)+m3fa(x) 2.4.2

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Now,letusconsiderthat(x;z)isasolutionofproblem(P5').Forthexedpointx,thepointzminimizesthefunctionsFa(x;z)m1R(x)and,consequently,thepointzminimizesthefunctionFa(x;z).Then,Theorem 2.2.1 impliesthatz2Aa(x).Further,since(x;z)isasolutionof(P5'),Theorem 2.2.1 impliesfa(x)m1R(x)=Fa(x;z)m1R(x)Fa(x;za(x))m1R(x)=fa(x)m1R(x);x2X: 2.4.2 2.4 ).Thisoffersarichrangeofpossibilities. TheintegralinFa(x;z)canbeapproximatedinvariousways.Forexample,thiscanbedonebysamplingtheprobabilitydistributionofyaccordingtoitsdensityp(y).Ifthesamplinggeneratesacollectionofvectorsy1;y2;:::;yJ,thenthecorrespondingapproximationtoFa(x;z)=z+(9a)1Zy2Rn[f(x;y)z]+p(y)dy wherepjareprobabilitiesofscenariosyj.Ifthelossfunctionf(x;y)islinearw.r.t.x,thenthefunctionFa(x;z)isconvexandpiecewiselinear. ThefunctionFa(x;z)inoptimizationproblemsinTheorems 2.2.2 2.4.1 ,and 2.4.2 canbeapproximatedbythefunctionFa(x;z).Further,byusingdummyvariableszj;j=1;:::;J,thefunctionFa(x;z)canbereplacedbythelinearfunctionz+(1a)1Jj=1pjzjandthesetoflinearconstraintszjf(x;yj)z;zj0;j=1;:::;J;z2R: 2.4.1 wecanreplacetheconstraintfa(x)w

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inoptimizationproblem(P4)bytheconstraintFa(x;z)w: andreducedtothefollowingsystemoflinearconstraints Similarly,approximationsbylinearfunctionscanbedoneintheoptimizationproblemsinTheorems1and5. Lossandrewardfunctions. TherewardfunctionR(x)istheexpectedvalueoftheportfolioattheendoftheperiod, Evidently,denedinthisway,therewardfunctionR(x)andthelossfunctionf(x;y)arerelatedasR(x)=E[f(x;y)]+qTx0:

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wheretheriskfunctionfa(x)isdenedastheaCVaRforthelossfunctiongivenby( 2.14 ),andwisapercentageoftheinitialportfoliovalueqTx0,allowedforriskexposure.Thelossfunctiongivenby( 2.14 )islinear(andthereforeconvex)inx,therefore,thea-CVaRfunctionfa(x)isalsoconvexinx.Thesetoflinearconstraintscorrespondingto( 2.16 ),is KonnoandWi-jayanayake ( 1999 ).Witheveryinstrument,weassociateatransactioncostci.Whenbuyingorsellinginstrumenti,onepayscitimestheamountoftransaction.Forcashwesetccash=0.Thatis,oneonlypaysforbuyingandsellingtheinstrument,andnotformovingthecashinandoutoftheaccount. Accordingtothat,weconsiderabalanceconstraintthatmaintainsthetotalvalueoftheport-folioincludingtransactioncostsni=4qix4i=ni=1ciqijx0ixij+ni=1qixi:

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Thisequalitycanbereformulatedusingthefollowingsetoflinearconstraints We,also,considerthatthepositionsthemselvescanbebounded subjectto

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Bysolvingthisproblem,wegettheoptimalvectorx,thecorrespondingVaR,whichequals

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wheret1;:::;tJareclosingtimesforJconsecutivebusinessdays.Further,inthenumericalsimu-lations,weconsideratwoweekperiod,Dt=10.Theexpectedend-of-periodpriceofinstrumentiisE[yi]=Jj=1pjyij=J1Jj=1ywj; Thiscasestudyisdesignedasademonstrationofthemethodology,ratherthanapracticalrecommendationforinvestments.Wehaveusedhistoricaldataforscenariogeneration(10-dayhis-toricalreturns).Whilethereissomeestimationerrorintheriskmeasure,thiserrorismuchgreaterforexpectedreturns.Thehistoricalreturnsovera10-dayperiodprovideverylittleinformationontheactualto-be-realizedout-of-samplereturns;i.e.,historicalreturnshavelittleforecast-ingpower.Theseissuesarediscussedinmanyacademicstudies,including( Jorion 1996 2000 ; Michaud 1989 ).TheprimarypurposeofthepresentedcasestudyisthedemonstrationofthenovelCVaRriskmanagementmethodologyandthepossibilitytoapplyittoportfoliooptimization.Thistechnologycanbecombinedwithmoreadequatescenariogenerationproceduresutilizingexpertopinionsandadvancedstatisticalforecastingtechniques,suchasneuralnetworks.Thesuggestedmodelisdesignedasonestageofthemultistageinvestmentmodeltobeusedinarealisticin-vestmentenvironment.However,discussingthismultistageinvestmentmodelandthescenariogenerationproceduresusedforthismodelisbeyondthescopeofthischapter. ThesetofinstrumentstoinvestinwassettothestocksintheS&P100asoftherstofSeptem-ber1999.Duetoinsufcientdata,sixofthestockswereexcluded.

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hundredsoftheoverlappingtwo-weekperiods(July1,1997-July8,1999).Ineffect,thiswasanin-sampleoptimizationusing500overlappingreturnsmeasuredover10businessdays. Theinitialportfoliocontainedonlycash,andthealgorithmshoulddetermineanoptimalin-vestmentdecisionsubjecttoriskconstraints.Thelimitsonthepositionsweresettox 2 showstheefcientfrontieroftheportfoliowiththeCVaRconstraint.ThevaluesontheRiskscalerepresentthetolerancelevelw,i.e.,thepercentageoftheinitialportfoliovaluewhichisallowedforriskexposure.Forexample,settingRisk=10%(w=0:10)anda=0:95impliesthattheaveragelossin5%worstcasesmustnotexceed10%oftheinitialportfoliovalue.Naturally,higherrisktolerancelevelswinCVaRconstraint( 2.21 )allowforachievinghigherexpectedreturns.ItisalsoapparentfromFig. 2 thatforeveryvalueofriskcondencelevelathereexistssomevaluew,afterwhichtheCVaRconstraintbecomesinactive(i.e.,notbinding).Ahigherexpectedreturncannotbeattainedwithoutlooseningotherconstraintsinproblem( 2.20 )( 2.27 ),orwithoutaddingnewinstrumentstotheoptimizationset.Inthisnumericalexample,themaximumrateofreturnthatcanbeachievedforthegivensetofinstrumentsandconstraintsequals2.96%overtwoweeks.However,verysmallvaluesofrisktolerancewcausetheoptimizationproblem( 2.20 )( 2.27 )tobeinfeasible;inotherwords,thereisnosuchcombinationofassetsthatwouldsatisfyCVaRconstraints( 2.21 )( 2.22 )andtheconstraintsonpositions( 2.23 )( 2.27 )simultaneously. Table 2 presentstheportfoliocongurationfordifferentrisklevels(a=0:90).Recallthatweimposedtheconstraintonthepercentagenofthetotalportfoliovaluethatonestockcanconsti-tute( 2.23 ).Wesetn=0:2,i.e.,asingleassetcannotconstitutemorethan20%ofthetotalportfoliovalue.Table1showsthatforhigherlevelsofallowedrisk,thealgorithmreducesthenumberofthe

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Figure2: Efcientfrontier(optimizationwithCVaRconstraints).RateofReturnistheexpectedrateofreturnoftheoptimalportfolioduringa2weekperiod.TheRiskscaledisplaystherisktolerancelevelwintheCVaRriskconstraintasthepercentageoftheinitialportfoliovalue. instrumentsintheportfolioinordertoachieveahigherreturn(duetotheimposedconstraints,theminimalnumberofinstrumentsintheportfolio,includingrisk-freecash,equalsve).Thisconrmsthewell-knownfactthatdiversifyingtheportfolioreducestherisk.Relaxingtheconstraintonriskallowsthealgorithmtochooseonlythemostprotablestocks.Aswetightentherisktolerancelevel,thenumberofinstrumentsintheportfolioincreases,andformoreconservativeinvesting(2%risk),weobtainaportfoliowithmorethan15assets,includingtherisk-freeasset(cash).Theinstrumentsnotshowninthetablehavezeroportfolioweightsforallrisklevels. Transactioncostsneedtobetakenintoaccountwhenemployinganactivetradingstrategy.Transactioncostsaccountforafeepaidtothebroker/market,bid-askspreads,andpoorliquidity.Toexaminetheimpactofthetransactioncosts,wecalculatedtheefcientfrontierwiththefollowingtransactioncosts,c=0%;0:25%,and1%.Figure 2 showsthatthetransactioncostsnonlinearlylowertheexpectedreturn.Sincetransactioncostsareincorporatedintotheoptimizationproblem,theyalsoaffectthechoiceofstocks. RockafellarandUryasev 2000 )thatfornormallydistributedlossfunctionsthesetwomethodologiesareequivalentinthesensethattheygenerate

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Table2: Portfolioconguration:assets'weights(%)intheoptimalportfoliodependingontherisklevel(theinstrumentsnotincludedinthetablehavezeroportfolioweights). Exp.Ret,%1.5081.9622.1952.3842.5652.7192.8382.9152.956St.Dev.0.02200.02900.03330.03850.04390.04860.05320.05860.0637CVaR0.020.030.040.050.060.070.080.090.10 Here,weusedhistoricalreturnsasascenarioinputtothemodel,withoutmakinganyassump-tionsaboutthedistributionofthescenariovariables.WecomparedtheCVaRmethodologywiththeMVapproachbyrunningtheoptimizationalgorithmsonthesamesetofinstrumentsandscenarios.TheMVoptimizationproblemwasformulatedasfollows( Markowitz 1991 ): subjectto

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Figure2: EfcientfrontierofoptimalportfoliowithCVaRconstraintsinpresenceoftransactioncostsc=0%;0:25%,and1%.RateofReturnistheexpectedrateofreturnoftheoptimalportfolioduringa2weekperiod.TheRiskscaledisplaystherisktolerancelevelwintheCVaRriskconstraint(a=0:90)asthepercentageoftheinitialportfoliovalue. wherexiareportfolioweights,unlikeproblem( 2.20 )( 2.27 ),wherexiarenumbersofsharesofcorrespondinginstruments.riistherateofreturnofinstrumenti,andsikisthecovariancebetweenreturnsofinstrumentsiandk:sik=cov(ri;rk).Therstconstraint( 2.29 )isthebudgetconstraint;( 2.30 )requiresportfolio'sexpectedreturntobeequaltoaprescribedvaluerp;nally,( 2.31 )im-posesboundsonportfolioweights,whereniarethesameasin( 2.23 ).Thesetofconstraints( 2.29 )( 2.31 )isidenticalto( 2.23 )( 2.27 ),exceptfortransactioncostconstraints.Theexpectationsandcovariancesin( 2.28 ),( 2.30 )arecomputedusingthe10-dayhistoricalreturns,whichwereusedforscenariogenerationintheCVaRoptimizationmodel:rij=ptj+10i=ptji1;E[ri]=1 2 displaystheCVaRefcientportfoliosinReturn/CVaRscalesfortheriskcon-dencelevela=0:95(continuousline).Also,foreachreturnitdisplaystheCVaRoftheMV

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optimalportfolio(dashedline).Note,thatforagivenreturn,theMVoptimalportfoliohasahigherCVaRrisklevelthantheefcientReturn/CVaRportfolio.Figure 2 displayssimilargraphsfora=0:99.ThediscrepancybetweenCVaRandMVsolutionsishigherforthehighercondencelevel. Figure2: EfcientfrontiersofCVaRandMVoptimalportfolios.TheCVaRoptimalportfoliowasobtainedbymaximizingexpectedreturnssubjecttotheconstraintonportfolio'sCVaRwith95%condencelevel(a=0:95).ThehorizontalandverticalscalesrespectivelydisplayCVaRandexpectedrateofreturnofaportfoliooveratwoweekperiod. Figure 2 displaystheefcientfrontierforReturn/MVefcientportfolios(continuousline).Also,foreachreturnitdisplaysthestandarddeviationoftheCVaRoptimalportfoliowithcon-dencelevela=0:95(dashedline).Asexpected,foragivenreturn,theCVaRoptimalportfoliohasahigherstandarddeviationthantheefcientReturn/MVportfolio.SimilargraphsaredisplayedinFigure 2 fora=0:99.ThediscrepancybetweenCVaRandMVsolutionsishigherforthehighercondencelevel,similartoFigures 2 2 However,thedifferencebetweentheMVandCVaRapproachesisnotverysignicant.Rela-tivelyclosegraphsofCVaRandMVoptimalportfoliosindicatethataCVaRoptimalportfolioisnearoptimalinMVsense,andviceversa,aMVoptimalportfolioisnearoptimalinCVaRsense.Thisagreementbetweenthetwosolutionsshouldnot,however,bemisleadingindecidingthatthediscussedportfoliomanagementmethodologiesarethesame.Theobtainedresultsaredataset-specic,andtheclosenessofsolutionsofCVaRandMVoptimizationproblemsiscausedbyapparentlyclose-to-normaldistributionsofthehistoricalreturnsusedinourcasestudy.Including

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Figure2: EfcientfrontiersofCVaRandMVoptimalportfolios.TheCVaRoptimalportfoliowasobtainedbymaximizingexpectedreturnssubjecttotheconstraintonportfolio'sCVaRwith99%condencelevel(a=0:99).ThehorizontalandverticalscalesrespectivelydisplayCVaRandexpectedrateofreturnofaportfoliooveratwoweekperiod. Figure2: EfcientfrontiersofCVaRandMVoptimalportfolios.TheCVaRoptimalport-foliowasobtainedbymaximizingexpectedreturnssubjecttotheconstraintonportfolio'sCVaRwith95%condencelevel(a=0:95).Thehorizontalandverticalscalesrespectivelydisplaythestandarddeviationandexpectedrateofreturnofaportfoliooveratwoweekperiod.

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Figure2: EfcientfrontiersofCVaRandMVoptimalportfolios.TheCVaRoptimalport-foliowasobtainedbymaximizingexpectedreturnssubjecttotheconstraintonportfolio'sCVaRwith99%condencelevel(a=0:99).Thehorizontalandverticalscalesrespectivelydisplaythestandarddeviationandexpectedrateofreturnofaportfoliooveratwoweekperiod. optionsintheportfolioorcreditriskwithskewedreturndistributionsmayleadtoquitedifferentoptimalsolutionsoftheefcientMVandCVaRportfolios( MausserandRosen 1999 ; Larsenetal. 2002 ). RockafellarandUryasev 2000 ),whichsimultaneouslycalculatesVaRandoptimizesCVaR.Werstshowed(Theorem 2.3.1 )thatforrisk-returnoptimizationproblemswithconvexconstraints,onecanusedifferentoptimizationformulations.ThisistrueinparticularfortheconsideredCVaRoptimizationproblem.Wethenshowed(Theorems 2.4.1 and 2.4.2 )thattheapproachby RockafellarandUryasev ( 2000 )canbeextendedtothereformulatedproblemswithCVaRconstraintsandtheweightedreturn-CVaRper-formancefunction.TheoptimizationwithmultipleCVaRconstrainsfordifferenttimeframesandatdifferentcondencelevelsallowsforshapingdistributionsaccordingtothedecisionmaker'spreferences.WedevelopedamodelforoptimizingportfolioreturnswithCVaRconstraintsusinghistoricalscenariosandconductedacasestudyonoptimizingportfolioofS&P100stocks.Thecasestudyshowedthattheoptimizationalgorithm,whichisbasedonlinearprogrammingtechniques,isverystableandefcient.Theapproachcanhandlelargenumberofinstrumentsandscenarios.

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CVaRriskmanagementconstraints(reducedtolinearconstraints)canbeusedinvariousapplica-tionstoboundpercentilesoflossdistributions.

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Inthepreviouschapter,weconsideredacasestudyofportfoliooptimizationofS&P100stocksunderConditionalValue-at-Riskconstraints.Inthischapter,weperformfurthernumeri-calanalysisoftheperformanceofportfoliooptimizationtechniquesbasedonCVaRriskmeasure,aswellasseveralotherriskmeasures(ConditionalDrawdown-at-Risk,Mean-AbsoluteDeviation,andMaximumLoss).Inparticular,weperformin-sampleandout-of-samplerunsforaportfolioofhedgefunds(fundoffunds).Thecommonpropertyoftheconsideredriskmanagementtechniquesisthattheyadmitformulationofaportfoliooptimizationmodelasalinearprogramming(LP)prob-lem.Thepossibilitytoformulateandsolveportfoliooptimizationproblemasalinearprogrammingproblemleadstoefcientandrobustportfolioallocationalgorithms,whichcansuccessfullyhandleoptimizationproblemswiththousandsofinstrumentsandscenarios. Weusein-sampleandout-of-sampletests,whichsimulateareal-lifeportfoliobehavior,toinvestigatetheperformanceofvariousriskconstraintsintheportfoliomanagementalgorithm.Ournumericalexperimentsshowthatimposingriskconstraintsmayimprovetherealperformanceofaportfoliorebalancingstrategyinout-of-sampleruns.Itisalsobenecialtocombineseveraltypesofriskconstraintsthatcontroldifferentsourcesofrisk. Anderssonetal. ( 2001 ), Chekhlovetal. ( 2000 ), Krokhmaletal. ( 2002 ), RockafellarandUryasev ( 2000 2002 )).Thechoiceofhedgefunds,asasubjectfor 32

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theportfoliooptimizationstrategy,isstimulatedbyastronginteresttothisclassofassetsbybothpractitionersandscholars,aswellasbychallengesrelatedtorelativelysmalldatasetsavailableforhedgefunds. Recentstudies Ziemba ( 2003 )). Hedgefundsareinvestmentpoolsemployingsophisticatedtradingandarbitragetechniquesincludingleverageandshortselling,wideusageofderivativesecuritiesetc.Generally,hedgefundsrestrictshareownershiptohighnetworthindividualsandinstitutions,andarenotallowedtooffertheirsecuritiestothegeneralpublic.Manyhedgefundsarelimitedto99investors.Thisprivatenatureofhedgefundshasresultedinfewregulationsanddisclosurerequirements,comparedforex-ample,withmutualfunds(however,stricterregulationsexistforhedgefundstradingfutures).Also,thehedgefundsmaytakeadvantageofspecialized,risk-seekinginvestmentandtradingstrategies,whichotherinvestmentvehiclesarenotallowedtouse. Therstofcial Ackermannetal. ( 1999 ), AminandKat ( 2001 ), BrownandGoet-zmann ( 2000 ), FungandHsieh ( 1997 2001 2000 ),and Lhabitant ( 2001 )2 ( 2003 )tracesearlyunofcialhedgefunds,suchasKeynesChestFundetc.,thatexistedinthe1920'sto1940's.

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Nowadays,hedgefundsbecomearapidlygrowingpartofthenancialindustry.AccordingtoVanHedgeFundAdvisors,thenumberofhedgefundsattheendof1998was5830,theymanaged311billionUSDincapital,withbetween$800billionand$1trillionintotalassets.Nearly80%ofhedgefundshavemarketcapitalizationlessthan100million,andaround50%aresmallerthan$25million,whichindicateshighnumberofnewentries.Morethan90%ofhedgefundsarelocatedintheU.S. Hedgefundsaresubjecttofarfewerregulationsthanotherpooledinvestmentvehicles,espe-ciallytoregulationsdesignedtoprotectinvestors.Thisappliestosuchregulationsasregulationsonliquidity,requirementsthatfund'ssharesmustberedeemableananytime,protectingconictsofinterests,assuringfairnessofpricingoffundshares,disclosurerequirements,limitingusageofleverage,shortsellingetc.Thisisaconsequenceofthefactthathedgefunds'investorsqualifyassophisticatedhigh-incomeindividualsandinstitutions,whichcanstandforthemselves.Hedgefundsoffertheirsecuritiesasprivateplacements,onindividualbasis,ratherthanthroughpublicadvertisement,whichallowsthemtoavoiddisclosingpubliclytheirnancialperformanceorassetpositions.However,hedgefundsmustprovidetoinvestorssomeinformationabouttheiractivity,andofcourse,theyaresubjecttostatutesgoverningfraudandothercriminalactivities. Asmarket'ssubjects,hedgefundsdosubordinatetoregulationsprotectingthemarketintegritythatdetectattemptsofmanipulatingordominatinginmarketsbyindividualparticipants.Forexam-ple,intheUnitedStateshedgefundsandotherinvestorsactiveoncurrencyfuturesmarkets,mustregularlyreportlargepositionsincertaincurrencies.Also,manyoptionexchangeshavedevelopedLargeOptionPositionReportingSystemtotrackchangesinlargepositionsandidentifyoutsizedshortuncoveredpositions. Inthischapter,weconsiderproblemofmanagingfundoffunds,i.e.,constructingoptimalportfoliosfromsetsofhedgefunds,subjecttovariousriskconstraints,whichcontroldifferenttypesofrisks.However,thepracticaluseofthestrategiesislimitedbyrestrictiveassumptions

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Theobtainedresultscannotbetreatedasdirectrecommendationsforinvestinginhedgefundsmar-ket,butratherasadescriptionoftheriskmanagementmethodologiesandportfoliooptimizationtechniquesinarealisticenvironment.Foranoverviewofthepotentialproblemsrelatedtothedataanalysisandportfoliooptimizationofhedgefunds,see Lo ( 2001 ). Section 3.2 presentsanoverviewoflinearportfoliooptimizationalgorithmsandtherelatedriskmeasures,whichareexploredinthischapter.Section 3.3 containsdescriptionofourcasestudy,resultsofin-sampleandout-of-sampleexperimentsandtheirdetaileddiscussion.Section 3.4 presentstheconcludingremarks. RockafellarandUryasev 2000 2002 ),ConditionalDrawdown-at-Risk( Chekhlovetal. 2000 ),Mean-AbsoluteDeviation( KonnoandYamazaki 1991 ; KonnoandShirakawa 1994 ; KonnoandWijayanayake 1999 ),MaximumLoss( Young 1998 )andthemarket-neutrality(betaoftheportfolioequalszero).

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Alinearportfoliorebalancingalgorithmisatrading(investment)strategywithmathematicalmodelthatcanbeformulatedasalinearprogramming(LP)problem.ThefocusonLPtechniquesinapplicationtoportfoliorebalancingandtradingproblemsisexplainedbyexceptionaleffectivenessandrobustnessofLPalgorithms,whichbecomeespeciallyimportantinnanceapplications.Recentdevelopments(see,forexample, Anderssonetal. ( 2001 ), CarinoandZiemba ( 1998 ), Carinoetal. ( 1998 ), Chekhlovetal. ( 2000 ), ConsigliandDempster ( 1997 1998 ), DemboandKing ( 1992 ), Duarte ( 1999 ), Krokhmaletal. ( 2002 ), RockafellarandUryasev ( 2000 2002 ), Turneretal. ( 1994 ), Zenios ( 1999 ), ZiembaandMulvey ( 1998 ), Young ( 1998 ))showthatLP-basedalgorithmscansuccessfullyhandleportfolioallocationproblemswiththousandsandevenmillionsofdecisionvariablesandscenarios,whichmakesthosealgorithmsattractivetoinstitutionalinvestors. Inthecitedpapers,alongwithConditionalValue-at-RiskandConditionalDrawdown-at-Risk,other,muchearlierestablishedmeasuresofrisk,suchasMaximumLoss,Mean-AbsoluteDeviation,LowPartialMomentwithpoweroneandExpectedRegret ZiembaandVickson ( 1975 )).SomeoftheseriskmeasuresarequitecloselyrelatedtoCVaRconcept. However,theclassoflineartradingorportfoliooptimizationtechniquesisfarfromencom-passingtheentireuniverseofportfoliomanagementtechniques.Forexample,thefamousportfolio Harlow ( 1991 ).Expectedregret(see,forexample, DemboandKing ( 1992 ))isaconceptsimilartothelowerpartialmoment.However,theexpectedregretmaybecalculatedwithrespecttoarandombenchmark,whilethelowpartialmomentiscalculatedwithrespecttoaxedthreshold.6 TesturiandUryasev ( 2000 )showedthattheCVaRconstraintandthelowpartialmomentconstraintwithpoweroneareequivalentinthesensethattheefcientfrontierforportfoliowithCVaRconstraintcanbegeneratedbythelowpartialmomentapproach.Therefore,theriskmanagementwithCVaRandwithlowpartialmomentleadstosimilarresults.However,theCVaRapproachallowsfordirectcontrollingofpercentiles,whilethelowpartialmomentpenalizeslossesexceedingsomexedthresholds.

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optimizationmodelby Markowitz ( 1952 1991 ),whichutilizesthemean-varianceapproach,be-longstotheclassofquadraticprogramming(QP)problems;thewell-knownconstant-proportionruleleadstononconvexmultiextremumproblems,etc. RockafellarandUryasev 2000 2002 )devel-opsandenhancestheideasofriskmanagement,whichhavebeenputintheframeworkofValue-at-Risk(VaR)(see,forexample, DufeandPan ( 1997 ), Jorion ( 1997 ), Pritsker ( 1997 ), Staumbaugh ( 1996 )).Incorporatingsuchmeritsaseasy-to-understandconcept,simpleandconvenientrepresen-tationofrisks(onenumber),applicabilitytoawiderangeofinstruments,VaRhasevolvedintoacurrentindustrystandardforestimatingrisksofnanciallosses.Basically,VaRanswerstheques-tionwhatisthemaximumloss,whichisexpectedtobeexceeded,say,onlyin5%ofthecaseswithinthegiventimehorizon?Forexample,ifdailyVaRfortheportfolioofsomefundXYZisequalto10millionsUSDatthecondencelevel0.95,itmeansthatthereisonlya5%chanceoflossesexceeding10millionsduringatradingday. TheformaldenitionofVaRisasfollows.Consideralossfunctionf(x;y),wherexisadecisionvector(e.g.,portfoliopositions),andyisastochasticvectorstandingformarketuncertain-ties(inthischapter,yisthevectorofreturnsofinstrumentsintheportfolio).LetY(x;z)bethecumulativedistributionfunctionoff(x;y),Y(x;z)=P[f(x;y)z]: 3 ):za(x)=minz2RfY(x;z)ag: Artzneretal. 1997 1999 ; MausserandRosen 1999 ).Non-convexityofVaRmeansthatasafunctionofportfoliopositions,ithasmultiplelocalextrema,whichprecludesusingefcientoptimizationtechniques.

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ThedifcultieswithcontrollingandoptimizingVaRinnon-normalportfolioshaveforcedthesearchforsimilarpercentileriskmeasures,whichwouldalsoquantifydownsiderisksandatthesametimecouldbeefcientlycontrolledandoptimized.Fromthisviewpoint,CVaRisaperfectcandidateforconductingaVaR-styleportfoliomanagement. Forcontinuousdistributions,CVaRisdenedasanaverage(expectation)ofhighlossesre-sidinginthea-tailofthelossdistribution,or,equivalently,asaconditionalexpectationoflossesexceedingthea-VaRlevel(Fig. 3 ).FromthisfollowsthatCVaRincorporatesinformationonVaRandonthelossesexceedingVaR. Forgeneral(non-continuous)distributions, RockafellarandUryasev ( 2002 )deneda-CVaRfunctionfa(x)asthea-tailexpectationofarandomvariablez,fa(x)=Eatail[z]; Acerbietal. ( 2001 ), AcerbiandTasche ( 2002 )redenedtheexpectedshortfallsimilartotheCVaRdenitionpresentedabove. Alongwitha-CVaRfunctionfa(x),thefollowingfunctionscalledupperandlowerCVaR(a-CVaR+anda-CVaR),areconsidered:f+a(x)=E[f(x;y)jf(x;y)>za(x)];fa(x)=E[f(x;y)jf(x;y)za(x)]: ( 2002 )showedthata-CVaRcanbepresentedasaconvexcombinationofa-VaRanda-CVaR+,fa(x)=la(x)za(x)+[1la(x)]f+a(x);

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Figure3: Lossdistribution,VaR,CVaR,andMaximumLoss. wherela(x)=[Y(x;za(x))a][1a];0la(x)1: 1a"jaj=1qja!f(x;yja)+Jj=ja+1qjf(x;yj)#; 3 ). WhileinheritingsomeofthenicepropertiesofVaR,suchasmeasuringdownsiderisksandrepresentingthembyasinglenumber,applicabilitytoinstrumentswithnon-normaldistributionsetc.,CVaRhassubstantialadvantagesoverVaRfromtheriskmanagementstandpoint.Firstofall,CVaRisaconvexfunction Rockafellar ( 1970 ).

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onaconvexset,whichgreatlysimpliescontrolandoptimizationofCVaR.CalculationofCVaR,aswellasitsoptimization,canbeperformedbymeansofaconvexprogrammingshortcut( Rock-afellarandUryasev 2000 2002 ),wheretheoptimalvalueofCVaRiscalculatedsimultaneouslywiththecorrespondingVaR;forlinearorpiecewise-linearlossfunctionstheseprocedurescanbereducedtolinearprogrammingproblems.Also,unlikea-VaR,a-CVaRiscontinuouswithrespecttocondencelevela.AcomprehensivedescriptionoftheCVaRriskmeasureandCVaR-relatedoptimizationmethodologiescanbefoundin RockafellarandUryasev ( 2000 2002 ).Also, Rock-afellarandUryasev ( 2000 )showedthatfornormallossdistributions,theCVaRmethodologyisequivalenttothestandardMean-Varianceapproach.Similarresultalsowasindependentlyprovedforellipticdistributionsby Embrechtsetal. ( 1997 ). Figure3: Portfoliovalueanddrawdown. Accordingto RockafellarandUryasev ( 2000 2002 ),theoptimizationproblemwithmultipleCVaRconstraintsminx2Xg(x)subjecttofai(x)wi;i=1;:::;I; 1akJj=1qjmax0;f(x;yj)zkwk;k=1;:::;K;

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providedthattheobjectivefunctiong(x)andthelossfunctionf(x;y)areconvexinx2X.Whentheobjectiveandlossfunctionsarelinearinxandconstraintsx2Xaregivenbylinearinequalities,thelastoptimizationproblemcanbereducedtoLP,see RockafellarandUryasev ( 2002 2000 ). ExceptforthefactthatCVaRcanbeeasilycontrolledandoptimized,CVaRisamoreadequatemeasureofriskascomparedtoVaRbecauseitaccountsforlossesbeyondtheVaRlevel.ThefundamentaldifferencebetweenVaRandCVaRasriskmeasuresare:VaRistheoptimisticlowboundofthelossesinthetail,whileCVaRgivesthevalueoftheexpectedlossesinthetail.Inriskmanagement,wemayprefertobeneutralorconservativeratherthanoptimistic.Moreover,CVaRsatisesseveralnicemathematicalpropertiesandiscoherentinthesenseof Artzneretal. ( 1999 1997 ). Chekhlovetal. 2000 )closelyrelatedtoCVaR.Bydenition,aportfolio'sdrawdownonasample-pathisthedropoftheuncompounded

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wherexisthevectorofportfoliopositions,andvt(x)istheuncompoundedportfoliovalueattimet.Weassumethattheinitialportfoliovalueisequalto1;therefore,thedrawdownistheuncom-poundedportfolioreturnstartingfromthepreviousmaximumpoint.Figure 3 illustratestherelationbetweentheportfoliovalueandthedrawdown. Thedrawdownquantiesthenanciallossesinaconservativeway:itcalculateslossesforthemostunfavorableinvestmentmomentinthepastascomparedtothecurrent(discrete)moment.Thisapproachreectsquitewellthepreferencesofinvestorswhodenetheirallowedlossesinpercentagesoftheirinitialinvestments(e.g.,aninvestormayconsideritunacceptabletolosemorethan10%ofhisinvestment).Whileaninvestormayacceptsmalldrawdownsinhisaccount,hewoulddenitelystartworryingabouthiscapitalinthecaseofalargedrawdown.Suchdrawdownmayindicatethatsomethingiswrongwiththatfund,andmaybeitistimetomovethemoneytoamoresuccessfulinvestmentpool.Themutualandhedgefundconcernsarefocusedonkeepingexistingaccountsandattractingnewones;therefore,theyshouldensurethatclients'accountsdonothavelargedrawdowns. Onecanconcludethatdrawdownaccountsnotonlyfortheamountoflossesoversomeperiod,butalsoforthesequenceoftheselosses.Thishighlightstheuniquefeatureofthedrawdownconcept:itisalossmeasurewithmemorytakingintoaccountthetimesequenceoflosses. Foraspeciedsample-path,thedrawdownfunctionisdenedforeachtimemoment.How-ever,inordertoevaluateperformanceofaportfolioonthewholesample-path,wewouldliketohaveafunction,whichaggregatesalldrawdowninformationoveragiventimeperiodintoonemeasure.Asthisfunctiononecanpick,forexample,theMaximumDrawdown,MaxDD=max0tTf(x;t);

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Ontheotherhand,theAverageDrawdowntakesintoaccountalldrawdownsinthesample-path.However,smalldrawdownsareacceptable(e.g.,1-2%drawdowns)andaveragingmaymasklargedrawdowns. Chekhlovetal. ( 2000 )suggestedanewdrawdownmeasure,ConditionalDrawdown-at-Risk,thatcombinesboththedrawdownconceptandtheCVaRapproach.Forinstance,0.95-CDaRcanbethoughtofasanaverageof5%ofthehighestdrawdowns.Formally,a-CDaRisa-CVaRwithdrawdownlossfunctionf(x;t)givenby( 3.1 ).Namely,assumethatpossiblerealizationsoftherandomvectorsdescribinguncertaintiesinthelossfunctionisrepresentedbyasample-path(time-dependentscenario),whichmaybeobtainedfromhistoricalorsimulateddata.Inthischapter,itisassumedthatweknowonesample-pathofreturnsofinstrumentsincludedintheportfolio.Letrijbetherateofreturnofi-thinstrumentinj-thtradingperiod(thatcorrespondstoj-thmonthinthecasestudy,seebelow),j=1;:::;J.Supposethattheinitialportfoliovalueequals1.Letxi,i=1;:::;nbeweightsofinstrumentsintheportfolio.Theuncompoundedportfoliovalueattimejequalsvj(x)=ni=11+js=1ris!xi:

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of 1a1 Chekhlovetal. 2000 ). Formally,theMean-AbsoluteDeviation(MAD)V(x)ofportfolio'srateofreturnequals KonnoandYamazaki ( 1991 ).Wesupposethatj=1;:::;Jscenariosofreturnswithprobabilitiesqjareavailable.Letusdenotebyrijthereturnofi-thassetinthescenarioj.Theportfolio'sMADcanbewrittenas

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3 ),seeforinstance,Young(1998).Whenthedistributionoflossesiscontinuous,thisriskmeasuremaybeunbounded,unlessthedistributionistruncated.Forexample,fornormaldistribution,themaximumlossisinnitelylarge.However,fordiscretelossdistributions,especiallyforthosebasedonsmallhistoricaldatasets,theMaxLossisareasonablemeasureofrisk.WealsowouldliketopointoutthattheMaximumLossadmitsanalternativedenitionasaspecialcaseofa-CVaRwithacloseto1. Letussupposethatj=1;:::;Jscenariosofreturnsareavailable(rijdenotesreturnofi-thassetinthescenarioj).TheMaximumLoss(MaxLoss)functionhastheform(seeforinstance,Young(1998))

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wherex1;:::;xndenotetheproportionsinwhichthetotalportfoliocapitalisdistributedamongnassets,andbiarebetasofindividualassets,bi=Cov(ri;rM) Inourcasestudy,weinvestigatetheeffectofconstructingamarket-neutral(zero-beta)portfo-lio,byincludingamarket-neutralityconstraintintheportfoliooptimizationproblem.Wecomparetheperformanceoftheoptimalportfoliosobtainedwithandwithoutmarket-neutralityconstraint. subjectto where

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Theobjectivefunction( 3.2 )representstheexpectedreturnoftheportfolio.Therstconstraint( 3.3 )oftheoptimizationproblemimposeslimitationsontheamountoffundsinvestedinasingleinstrument(wedonotallowshortpositions).Thesecondconstraint( 3.4 )isthebudgetconstraint.Constraints( 3.5 )controlrisksofnanciallosses.Thekeyconstraintinthepresentedapproachistheriskconstraint( 3.5 ).FunctionFRisk(x1;:::;xn)representseitheraCVaRoraCDaRriskmeasure,andrisktolerancelevelwisthefractionoftheportfoliovaluethatisallowedforriskexposure. Constraint( 3.6 ),withbirepresentingmarket'sbetaforinstrumenti,forcestheportfoliotobemarket-neutralinthezero-betasense,i.e.,theportfoliocorrelationwiththemarketisbounded.Thecoefcientkin( 3.6 )isasmallnumberthatsetstheportfolio'sbetaclosetozero.Toinvestigatetheeffectsofimposingazero-betarequirementontheportfolio-rebalancingalgorithm,wesolvedtheoptimizationproblemwithandwithoutthisconstraint.Constraint( 3.6 )signicantlyimprovestheout-of-sampleperformanceofthealgorithm. Theriskmeasuresconsideredinthischapterallowforformulatingtheriskconstraint( 3.5 )intermsoflinearinequalities,whichmakestheoptimizationproblem( 3.2 )( 3.6 )linear,giventhelinearityofobjectivefunctionandotherconstraints.Belowwepresenttheexplicitformoftheriskconstraint( 3.5 )forCVaRandCDaRriskmeasures. wherethevectorofinstruments'returnsy=r=(r1;:::;rn)israndom.Theriskconstraint( 3.5 ),fa(x)w,whereCVaRriskfunctionreplacesthefunctionFRisk(x),is

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whererijisreturnofi-thinstrumentinscenarioj;j=1;:::;J.Sincethelossfunction( 3.7 )islinear,theriskconstraint( 3.8 )canbeequivalentlyrepresentedbythelinearinequalities, 1a1 3.2 )( 3.6 )withtheCVaRcon-strainttoalinearprogrammingproblem. 1a1 4.5 )canbewrittenasmax1jJ(ni=1rijxi)w:

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Similartootherconsideredriskconstraints,itcanbereplacedbyasystemoflinearinequalitiesww;ni=1rijxiw;j=1;:::;J: ThedatasetforconductingthenumericalexperimentswasprovidedtotheauthorsbytheFoun-dationforManagedDerivativesResearch.Itcontainedamonthlydataformorethan5000hedgefunds,fromwhichweselectedthosewithsignicantlylonghistoryandsomeminimumlevelofcapitalization.Topasstheselection,ahedgefundshouldhave66monthsofhistoricaldatafromDecember1995toMay2001,anditscapitalizationshouldbeatleast5millionU.S.dollarsatthebeginningofthisperiod.Thetotalnumberoffunds,whichsatisedthesecriteriaandaccordinglyconstitutedtheinvestmentpoolforouralgorithm,was301.Inthisdataset,theeldwiththenames

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ofhedgefundswasunavailable;therefore,weidentiedthehedgefundswithnumbers,i.e.,HF1,HF2,andsoon.Thehistoricalreturnsfromthedatasetwereusedtogeneratescenariosforal-gorithm( 3.2 )( 3.6 ).Eachscenarioisavectorofmonthlyreturnsforallsecuritiesinvolvedintheoptimization,andallscenariosareassignedequalprobabilities. Weperformedseparaterunsoftheoptimizationproblem( 3.2 )( 3.5 ),withandwithoutcon-straint( 3.6 )withCVaRandCDaRriskmeasureinconstraint( 3.5 ),varyingsuchparametersascondencelevels,risktolerancelevelsetc. Thecasestudyconsistedfromtwosetsofnumericalexperiments.Therstsetofin-sampleexperimentsincludedthecalculationofefcientfrontiersandtheanalysisoftheoptimalportfoliostructureforeachoftheriskmeasures.Thesecondsetofexperiments,out-of-sampletesting,wasdesignedtodemonstratetheperformanceofourapproachinasimulatedhistoricalenvironment. Efcientfrontier. 3.2 )( 3.5 )withdifferentrisktolerancelev-elswinconstraint( 3.5 ),variedfromw=0:005tow=0:25.TheparameterainCVaRandCDaRriskconstraintswassettoa=0:90.TheefcientfrontierispresentedinFigure 3 ,wheretheportfoliorateofreturnmeansexpectedmonthlyrateofreturn.Intheseruns,themarket-neutralityconstraint( 3.6 )isinactive. Figure3: Efcientfrontiersforportfoliowithvariousriskconstraints(k=0:01).Themarket-neutralityconstraintisinactive.

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Figure 3 showsthatthreeCVaR-relatedriskmeasures(CVaR,CDaRandMaxLoss)pro-ducerelativelysimilarefcientfrontiers.However,theMADriskmeasureproducesadistinctivelydifferentefcientfrontier. Foroptimalportfolios,inthesenseofproblem( 3.2 )( 3.5 ),thereexistsanupperbound(equalto48.13%)fortheportfolio'srateofreturn.OptimalportfolioswithCVaR,MADandMaxLossconstraintsreachthisboundatdifferentrisktolerancelevels,buttheCDaR-constrainedportfoliodoesnotachievethemaximalexpectedreturnwithinthegivenrangeofwvalues.CDaRisarelativelyconservativeconstraintimposingrequirementsnotonlyonthemagnitudeofloses,butalsoonthetimesequenceoflosses(smallconsecutivelossesmayleadtolargedrawdown,withoutsignicantincreasingofCVaR,MaxLoss,andMAD). Figure 3 presentsefcientfrontiersofoptimalportfolio( 3.2 )( 3.5 )withtheactivemarket-neutralityconstraint( 3.6 ),wherecoefcientkisequalto0:01.Asoneshouldexpect,imposingtheextraconstraint( 3.6 )causesadecreaseinin-sampleoptimalexpectedreturn.Forexample,thesaturationleveloftheportfolio'sexpectedreturnisnow41.94%,andallportfoliosreachthatlevelatmuchlowervaluesofrisktolerancew.However,themarket-neutralityconstraintalmostdoesnotaffectthecurvesofefcientportfoliosintheleftmostpointsofefcientfrontiers,whichcorrespondtothelowestvaluesofrisktolerancew.Thismeansthatbytighteningtheriskconstraint( 3.5 )onecanobtainanearlymarket-neutralportfoliowithoutimposingthemarket-neutralityconstraint( 3.6 ). Figure3: Efcientfrontierformarket-neutralportfoliowithvariousriskconstraints(k=0.01).

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Quitehighratesofreturnfortheefcientportfolioscanbeexplainedbythefactthat301funds,selectedtoformtheoptimalportfolios,constituteabout6%oftheinitialhedgefundpool,andalreadyarethebestofthebestinourdatasample. 3 ).Table 3 presentsportfolioweightsforthefouroptimalportfolios.Itshowshowaparticularriskmeasureselectsinstrumentsgiventhespeciedexpectedreturn.TheleftcolumnofTable 3 containsthesetoftheassets,whicharechosenbythealgorithm( 3.2 )( 3.5 )underdifferentriskconstraints.Notethatamongthe301availableinstru-ments,onlyafewofthemareusedinconstructingtheoptimalportfolio.Moreover,acloserlookatTable 3 showsthatnearlytwothirdsoftheportfoliovalueforallriskmeasuresisformedbythreehedgefundsHF209,HF219andHF231(thecorrespondinglinesaretypesetinboldface).Thesethreehedgefundshavestableperformance,andeachriskmeasureincludesthemintheop-timalportfolio.Similarly,linestypesetinslantedfont,indicateinstrumentsthatareincludedintheportfoliowithsmallerweights,butstillareapproximatelyevenlydistributedamongtheportfolios.Thus,theinstrumentsHF93,100,209,219,231,258,and259constitutethecoreoftheoptimalportfoliounderallriskconstraints.ThelastrowinTable 3 liststhetotalweightoftheseinstru-mentsincorrespondingoptimalportfolio.Theremainingassets(withouthighlightinginthetable)areresidualinstruments,whicharespecictoeachriskmeasure.Theymayhelpustospotdiffer-encesininstrumentselectionofeachriskconstraint.Table 3 displaystheresidualweightsofHF49,84,106,124,126,169,196,and298.Theresidualweightsarecalculatedastheinstrument'sweightwithrespecttotheresidualpartoftheportfolio.Forexample,intheoptimalportfoliowiththeCDaRconstraint,thehedgefundHF49represents11.02%ofthetotalportfoliovalue,andatthesametimeitrepresents49.06%oftheresidual(1:000:775)100%portfoliovalue.Inotherwords,itoccupiesalmosthalfoftheportfolioassets,notcapturedbyhedgefundsinthegrayedcells.Also,notethatneitheroftheresidualinstrumentsissimultaneouslypresentinallportfolios.

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Table3: Instrumentweightsintheoptimalportfoliowithdifferentriskconstraints HF490.1102160.04386600.191952HF840.041898000.078352HF930.0813940.087540.0452810.062609HF1000.066290.0736590.0238990.068993HF1060000.00191HF124000.0202890HF12600.0086730.0279080HF1690.0542980.01014400HF19600.0156270.0847910HF2090.2146830.2246690.2608240.226262HF2190.1371650.2592540.1692390.111746HF2310.1830330.1692070.1370680.177008HF2580.0340830.0141560.0975970.012606HF2590.0586840.0894030.1331040.068562HF2980.0182570.003800 0.7753310.9178890.8670120.727785 Weightsofresidualinstrumentsintheoptimalportfoliowithdifferentriskconstraints HF490.4905710.53422900.705151HF840.186487000.287833HF93XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXHF100XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXHF1060000.007016HF124000.1525610HF12600.1056240.2098550HF1690.241680.12354400HF19600.190310.6375820HF209XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXHF219XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXHF231XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXHF258XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXHF259XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXHF2980.0812610.04628500

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Figure 3 containsgraphsofthehistoricalreturnandpricedynamicsfortheresidualhedgefunds.Weincludedthesegraphstoillustratedifferencesinriskconstraintsandtomakesomespeculationsonthissubject. Forexample,instrumentHF84isselectedbyCDaRandMaxLossriskmeasures,butisex-cludedbyCVaRandMAD.NotethatgraphofreturnsfortheinstrumentHF84showsnonegativemonthlyreturnsexceeding10%,whichisprobablyacceptablefortheMaxLossriskmeasure.Also,thepricegraphfortheinstrumentHF84showsthatitexhibitsfewdrawdowns;therefore,CDaRpickedthisinstrument.However,MADprobablyexcludedinstrumentHF8becauseithadahighmonthlyreturnof30%(recallthatMADdoesnotdiscriminatebetweenhighpositivereturnsandhighnegativereturns).ItisnotclearfromthegraphswhyCVaRrejectedtheHF84instruments.Probably,otherinstrumentshadbetterCVaR-returncharacteristicsfromtheviewpointoftheover-allportfolioperformance. ThehedgefundHF124hasnotbeenchosenbyanyriskmeasures,withtheexceptionofMAD.Besidesratheraverageperformance,itsufferslong-lastingdrawdowns(CDaRdoesnotlikethis),hasmultiplenegativereturnpeaksof10%magnitude(CVaRdoesnotfavorthat),anditsworstnegativereturnisalmost20%(MaxLossmustprotectfromsuchperformancedrops).ThequestionwhythisinstrumentwasnotrejectedbyMADcannotbeclearlyansweredinthiscase.Donotforgetthatsuchadecisionisasolutionofanoptimizationproblem,anddifferentinstrumentswithproperlyadjustedweightsmaycompensateeachother'sshortcomings.ThismayalsobeanexcuseforMADnotpickingtheHF49fund,whosemeritsareconrmedbyhighresidualweightsofthisfundinCDaR,CVaR,andMaxLossportfolios. FundHF126hasthehighestexpectedreturnamongresidualinstruments,butitalsosuffersthemostseveredrawdownsandhasthehighestnegativereturn(exceeding20%)that's,probably,whyalgorithmswithCDaRandMaxLossmeasuresrejectedthisinstrument. 3.2 )( 4.6 )shedslightontheactualperformanceofthedevelopedapproaches.Inotherwords,thequestionishowwelldothealgorithmswithdifferentriskmeasuresutilizethescenarioinformationbasedonpasthistoryinproducingasuccessfulportfoliomanagementstrategy?Ananswercanbeobtained,forinstance,byinterpretingtheresultsoftheprecedingsectionasfollows:supposewewerebackinMay2001,

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Figure3: Historicalperformanceinpercentsofinitialvalue(ontheright),andtherateofreturndynamicsinpercentageterms(ontheleft)forsomeoftheresidualassetsinoptimalportfolios.

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andwewouldliketoinvestacertainamountofmoneyinaportfolioofhedgefundstodeliverthehighestrewardunderaspeciedrisklevel.Then,accordingtoin-sampleresults,thebestportfoliowouldbetheoneontheefcientfrontierofaparticularrebalancingstrategy.Infact,suchaportfoliooffersthebestreturn-to-riskratioprovidedthatthehistoricaldistributionofreturnswillrepeatinthefuture. Togetanideaabouttheactualperformanceoftheoptimizationapproach,weusedsomepartofthedataforscenariogeneration,andtherestforevaluatingtheperformanceofthestrategy.Thistechniqueisreferredtoasout-of-sampletesting.Inourcasestudy,weperformtheout-of-sampletestingintwosetups:1)Realout-of-sampletesting,and2)Mixedout-of-sampletesting.Eachoneisdesignedtorevealspecicpropertiesofriskconstraintspertainingtotheperformanceoftheportfolio-rebalancingalgorithminout-of-sampleruns. First,weperformedout-of-samplerunsforeachriskmeasureinconstraint( 3.5 )fordifferentvaluesofrisktolerancelevelw(market-neutralityconstraint,( 3.6 ),isinactive).Figures 3 to 3 illustratehistoricaltrajectoriesoftheoptimalportfoliounderdifferentriskconstraints(theportfoliovaluesaregivenin%relativelytotheinitialportfoliovalue).Risktolerancelevelwwassetto0.005,0.01,0.03,0.05,0.10,0.12,0.15,0.17and0.20,butforbetterreadingofgures,wereportonlyresultswithw=0:005,0.01,0.05,0.10,and0.15.TheparameterainCDaRandCVaRconstraints(inthelastcaseastandsforriskcondencelevel)wassettoa=0:90.

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Figure3: HistoricaltrajectoriesofoptimalportfoliowithCVaRconstraints. Figure3: HistoricaltrajectoriesofoptimalportfoliowithCDaRconstraints. Figure3: HistoricaltrajectoriesofoptimalportfoliowithMADconstraints.

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Figure3: HistoricaltrajectoriesofoptimalportfoliowithMaxLossconstraints. Figures 3 to 3 showsthatriskconstraint( 3.5 )hassignicantimpactonthealgorithm'sout-of-sampleperformance.Earlier,wehadalsoobservedthatthisconstrainthassignicantimpactonthein-sampleperformance.Itiswellknownthatconstrainingriskinthein-sampleoptimiza-tiondecreasestheoptimalvalueoftheobjectivefunction,andtheresultsreportedintheprecedingsubsectioncomplywiththisfact.Theriskconstraintsforcethealgorithmtofavorlessprotablebutsaferdecisionsovermoreprotablebutdangerousones.Fromamathematicalviewpoint,im-posingextraconstraintsalwaysreducesthefeasibilityset,andconsequentlyleadstoloweroptimalobjectivevalues.However,thesituationchangesdramaticallyforanout-of-sampleapplicationoftheoptimizationalgorithm.Thenumericalexperimentsshowthatconstrainingrisksimprovestheoverallperformanceoftheportfoliorebalancingstrategyinout-of-sampleruns;tighterin-sampleriskconstraintmayleadtobothlowerrisksandhigherout-of-samplereturns.Forallconsideredriskmeasures,looseningtherisktolerance(i.e.,increasingwvalues)resultsinincreasedvolatilityofout-of-sampleportfolioreturnsand,afterexceedingsomethresholdvalue,indegradationofthealgorithm'sperformance,especiallyduringthelast13months(March2000May2001).Forallriskfunctionsinconstraint( 3.5 ),themostattractiveportfoliotrajectoriesareobtainedforrisktol-erancelevelw=0:005,whichmeansthattheseportfolioshavehighreturns(highnalportfoliovalue),lowvolatility,andlowdrawdowns.Increasingwto0.01leadstoaslightincreaseofthenalportfoliovalue,butitalsoincreasesportfoliovolatilityanddrawdowns,especiallyforthesecondquarterof2001.Forlargervaluesofwtheportfolioreturnsdeteriorate,andforallriskmeasures,a

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portfoliocurvewithw=0:10showsquitepoorperformance.Furtherincreasingtherisktolerancetow=0:15,insomecasesallowsforachievinghigherreturnsattheendof2000,butafterthishighpeaktheportfoliosuffersseveredrawdowns(seeguresforCDaR,MAD,andMaxLossriskmeasures). ThenextseriesofFigures 3 to 3 illustrateseffectsofimposingmarket-neutralitycon-straint( 3.6 )inadditiontoriskconstraint( 3.5 ).Recallthatprimarypurposeofconstraint( 3.6 )ismakingtheportfoliouncorrelatedwithmarket.Themainideaofcomposingamarket-neutralport-folioisprotectingitfrommarketdrawdowns.Figures 3 to 3 comparetrajectoriesofmarket-neutralandwithoutrisk-neutralityoptimalportfolios.Additionalconstrainingresultedinmostcasesinfurtherimprovementoftheportfolio'sout-of-sampleperformance,especiallyforCVaRandCDaR-constrainedportfolios.Toclarifyhowtherisk-neutralitycondition( 3.6 )inuencestheportfolio'sperformance,wedisplayedonlyguresforlowestandhighestvaluesoftherisktoler-anceparameter,namelyforw=0:005andw=0:20.Coefcientkin( 3.6 )wassettok=0:01,andinstruments'betasbiwerecalculatedbycorrelatingwithS&P500index,whichistraditionallyconsideredasamarketbenchmark.Forportfolioswithtightriskconstraints(w=0:005)imposingmarket-neutralityconstraint( 3.6 )straightenedtheirtrajectories(reducedvolatilityanddrawdowns),whichmadethehistoriccurvesalmostmonotonecurveswithpositiveslope.Ontopofthat,CVaRandCDaRportfolioswithmarket-neutralityconstrainthadahighernalportfoliovalue,comparedtothosewithoutmarket-neutrality.Also,forportfolioswithlooseriskconstraints(w=0:20)impos-ingmarket-neutralityconstrainthadapositiveeffectontheformoftheirtrajectories,dramaticallyreducingvolatilityanddrawdowns. Finally,Figures 3 and 3 demonstratetheperformanceoftheoptimalportfoliosversustwobenchmarks:1)S&P500index;2)Best20,representingtheportfoliodistributedequallyamongthebest20hedgefunds.These20hedgefundsincludefundswiththehighestexpectedmonthlyreturnscalculatedwithpasthistoricalinformation.Similarlytotheoptimalportfolios( 3.2 )( 3.6 ),theBest20portfoliowasmonthlyrebalanced(withoutriskconstraints). AccordingtoFigures 3 to 3 ,themarket-neutraloptimalportfoliosaswellasportfolioswithoutthemarket-neutralityconstraintoutperformtheindexunderriskconstraintsofalltypes,whichprovidesanevidenceofhighefciencyoftherisk-constrainedportfoliomanagementalgo-rithm( 3.2 )( 3.6 ).Also,wewouldliketoemphasizethebehaviorofmarket-neutralportfoliosin

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Figure3: HistoricaltrajectoriesofoptimalportfoliowithCVaRconstraints.Lineswithb=0correspondtoportfolioswithmarket-neutralconstraint. Figure3: HistoricaltrajectoriesofoptimalportfoliowithCDaRconstraints.Lineswithb=0correspondtoportfolioswithmarket-neutralconstraint.

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Figure3: HistoricaltrajectoriesofoptimalportfoliowithMADconstraints.Lineswithb=0correspondtoportfolioswithmarket-neutralconstraint. Figure3: HistoricaltrajectoriesofoptimalportfoliowithMaxLossconstraints.Lineswithb=0correspondtoportfolioswithmarket-neutralconstraint.

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downmarketconditions.TwomarksonFig. 3 indicatethepointswhenallfourportfoliosgainedpositivereturns,whilethemarketwasfallingdown.Also,allrisk-neutralportfoliosseemtowithstandthedownmarketin2000,whenthemarketexperiencedsignicantdrawdown.Thisdemonstratestheefciencyandappropriatenessoftheapplicationofmarket-neutralityconstraint( 3.6 )toportfoliooptimizationwithriskconstraints( 3.2 )( 3.5 ). TheBest20benchmarkevidentlylacksthesolidperformanceofitscompetitors.Itnotonlysignicantlyunderperformsalltheportfoliosconstructedwithalgorithm( 3.2 )( 3.6 ),butalsoun-derperformsthemarkethalfofthetime.Unlikeportfolios( 3.2 )( 3.6 ),theBest20portfoliopro-nouncedlyfollowsthemarketdropinthesecondhalfof2000,andmoreover,itsuffersmuchmoreseveredrawdownsthanthemarketdoes.Thisindicatesthattheriskconstraintsinthealgorithm( 3.2 )( 3.6 )playanimportantroleinselectingthefunds. Aninterestingpointtodiscussisthebehaviorofalgorithm( 3.2 )( 3.6 )undertheMADriskconstraint.Figures 3 and 3 showthatatightMADconstraintmakestheportfoliocurveal-mostastraightline,andimposingofmarket-neutralityconstraint( 3.6 )doesnotaddmuchtothealgorithm'sperformance,andevenslightlylowerstheportfolio'sreturn.Atthesametime,portfo-lioswithCVaR-typeriskconstraints(CVaR,CDaRandMaxLoss)donotexhibitsuchremarkablystableperformance,andtakeadvantageofconstraint( 3.6 ).NotethatCVaR,CDaRandMaxLossaredownsideriskmeasures,whereastheMADconstraintsuppressesbothhighlossesandhighreturns.Themarket-neutralityconstraint( 3.6 )byitselfalsoputssymmetricrestrictionsontheport-folio'svolatility;that'swhyitaffectsMADandCVaR-relatedconstraintdifferently.However,here,weshouldpointoutthatwejustscratchedthesurfaceregardingthecombinationofvariousriskconstraints.WehaveimposedCVaRandCDaRconstraintsonlywithonecondencelevel.Wecanimposecombinationsofconstraintswithvariouscondencelevelsincludingconstrainingper-centilesofhighreturnsandaswellaspercentilesofhighlosses.Theseissuesarebeyondthescopeofourstudy. Summarizing,weemphasizethegeneralinferenceabouttheroleofriskconstraintsintheout-of-sampleandin-sampleapplicationofanoptimizationalgorithm,whichcanbedrawnfromourexperiments:riskconstraintsdecreasethein-samplereturns,whileout-of-sampleperformancemaybeimprovedbyaddingriskconstraints,andmoreover,strongerriskconstraintsusuallyensurebetterout-of-sampleperformance.

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Figure3: PerformanceoftheoptimalportfolioswithvariousriskconstraintsversusS&P500indexandbenchmarkportfoliocombinedfrom20besthedgefunds.Risktolerancelevelw=0:005,parametera=0.90.Market-neutralityconstraintisinactive. Figure3: Performanceofmarket-neutraloptimalportfoliowithvariousriskconstraintsversusS&P500indexandbenchmarkportfoliocombinedfrom20besthedgefunds.Risktolerancelevelisw=0:005,parameterisa=0.90.

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3.2 )( 3.6 )usesanalternativesetupforsplittingthedatainin-sampleandout-of-sampleportions.Insteadofutilizingonlypastinformationforgeneratingscenariosforportfoliooptimization,asitwasdonebefore,nowweletthealgorithmusebothpastandfutureinformationforconstructingscenarios.Thedesignofthisexperimentisasfollows.Theportfoliorebalancingprocedurewasperformedeveryvemonths,andthescenariomodelutilizedallthehistoricaldataexceptforthe5-monthperioddirectlyfollowingtherebalancingdate.TheprocedurewasstartedonDecember1995.TheinformationforscenariogenerationwascollectedfromMay1996toMay2001.TheportfoliowasoptimizedusingthesescenariosandinvestedonDecember1995.After5monthsthemoneygainedbytheportfoliowasreinvestedand,atthistime,thescenariomodelwasbuiltoninformationcontainedintheentiretimeinterval12/1995/2001exceptwindow05/1996/1996andsoon. Figures 3 to 3 displaydynamicsoftheoptimalportfolioundervariousriskconstraintsandwithdifferentrisktolerancelevels.Toavoidoverloadingthepresentationwithdetails,wereportresultsonlyforw=0:01,0.05and0.10.Asearlier,wesetparameterainCVaRandCDaRconstraintstoa=0:90. Figure3: Mixedout-of-sampletrajectoriesofoptimalportfoliowithCVaRconstraints Thegeneralpictureoftheseresultsisconsistentwithconclusionsderivedfromrealout-of-sampletests:tighteningofriskconstrainsimproveperformanceoftherebalancealgorithms.Loweroverallperformanceoftheportfoliooptimizationstrategyunderallriskconstraintscomparingto

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Figure3: Mixedout-of-sampletrajectoriesofoptimalportfoliowithCDaRconstraints Figure3: Mixedout-of-sampletrajectoriesofoptimalportfoliowithMADconstraints Figure3: Mixedout-of-sampletrajectoriesofoptimalportfoliowithMaxLossconstraints

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thatinrealout-of-sampletestingisexplainedbylongerrebalancingperiod.Itiswellknownthatmorefrequentrebalancingmaygivehigherreturns(atleast,intheabsenceoftransactioncosts). ThenextfourFigures 3 to 3 ,demonstratetheinuenceofthemarket-neutralitycon-straintontheperformanceoftheportfolio. Figure3: Mixedout-of-sampletrajectoriesofmarket-neutralityoptimalportfoliowithCVaRconstraints Figure3: Mixedout-of-sampletrajectoriesofmarket-neutralityoptimalportfoliowithCDaRconstraints Imposingofmarket-neutralityconstraint( 3.6 )inproblem( 3.2 )( 3.5 )forthemixedout-of-sampletestinghasasimilarimpactasinrealout-of-sampletesting. Wenalizetheout-of-sampletestingofbypresentingsummarystatisticsofbothrealandmixedout-of-sampletestsinTables 3 and 3

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Figure3: Mixedout-of-sampletrajectoriesofmarket-neutralityoptimalportfoliowithMADconstraints Figure3: Mixedout-of-sampletrajectoriesofmarket-neutralityoptimalportfoliowithMaxLossconstraints

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Summarizing,weemphasizethegeneralinferenceabouttheroleofriskconstraintsintheout-of-sampleandin-sampleapplicationofanoptimizationalgorithm,whichcanbedrawnfromourexperiments:riskconstraintsdecreasethein-samplereturns,whileout-of-sampleperformancemaybeimprovedbyaddingriskconstraints,andmoreover,strongerriskconstraintsusuallyensurebetterout-of-sampleperformance. Thenumericalexperimentsconsistofin-sampleandout-of-sampletesting.Wegeneratedef-cientfrontiersandcomparedalgorithmswithvariousconstraints.Theout-of-samplepartofex-perimentswasperformedintwosetups,whichdifferedinconstructingthescenariosetfortheoptimizationalgorithm. Theresultsobtainedaredataset-specicandwecannotmakedirectrecommendationsonport-folioallocationsbasedontheseresults.However,welearnedseverallessonsfromthiscasestudy.Imposingriskconstraintsmaysignicantlydegradein-sampleexpectedreturnswhileimprovingriskcharacteristicsoftheportfolio.In-sampleexperimentsshowedthatfortightrisktolerancelev-els,allriskconstraintsproducerelativelysimilarportfoliocongurations.Imposingriskconstraintsmayimprovetheout-of-sampleperformanceoftheportfolio-rebalancingalgorithmsinthesenseofrisk-returntradeoff.Especiallypromisingresultscanbeobtainedbycombiningseveraltypesofriskconstraints.Inparticular,wecombinedthemarket-neutrality(zero-beta)constraintwithCVaRorCDaRconstraints.Wefoundthattighteningofriskconstraintsgreatlyimprovesportfoliodynamicperformanceinout-of-sampletests,increasingtheoverallportfolioreturnanddecreasingbothlossesanddrawdowns.Inaddition,imposingthemarket-neutralityconstraintaddstothesta-bilityofportfolio'sreturn,andreducesportfoliodrawdowns.BothCDaRandCVaRriskmeasuresdemonstratedasolidperformanceinout-of-sampletests.

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WethanktheFoundationforManagedDerivativesResearchforprovidingthedatasetforcon-ductingnumericalexperimentsandpartialnancialsupportofthiscasestudy.

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70 Table 3-3: Summarystatisticsfortherealout-of-sampletests. Typeof constraint Risk tolerance Zerobeta Aver. Rateof Return Standard Deviation CVaR Drawdown Sharpe Ratio Yearly Aver. Return CDaR 0.005 0.258 0.709 0.818 0.178 0.321 0.231 CVaR 0.005 0.265 0.7078 0.834 0.183 0.332 0.239 MAD 0.005 0.233 0.239 0.11 0.019 0.848 0.218 MaxLoss 0.005 0.273 0.703 0 0.154 0.346 0.245 CDaR 0.01 0.245 0.803 1.035 0.2887 0.267 0.216 CVaR 0.01 0.269 0.804 1.011 0.2541 0.297 0.241 MAD 0.01 0.234 0.38 0.38 0.0526 0.539 0.211 MaxLoss 0.01 0.278 0.742 0 0.154 0.33 0.249 CDaR 0.03 0.32 1.045 1.32 0.338 0.278 0.284 CVaR 0.03 0.3 1.073 1.37 0.393 0.25 0.272 MAD 0.03 0.234 0.752 0.794 0.243 0.271 0.215 MaxLoss 0.03 0.266 0.939 09 0.339 0.2519 0.236 CDaR 0.05 0.27 1.256 1.712 0.465 0.193 0.252 CVaR 0.05 0.327 1.303 1.638 0.468 0.228 0.306 MAD 0.05 0.299 1.221 1.529 0.365 0.22 0.273 MaxLoss 0.05 0.289 1.150 0 0.460 0.224 0.267 CDaR 0.07 0.248 1.41 1.983 0.562 0.155 0.213 CVaR 0.07 0.262 1.473 2.067 0.569 0.158 0.241 MAD 0.07 0.201 1.489 2.234 0.597 0.115 0.175 MaxLoss 0.07 0.284 1.345 0 0.553 0.189 0.265 CDaR 0.1 0.282 1.45 1.98 0.5 0.174 0.255 CVaR 0.1 0.239 1.591 2.401 0.62 0.131 0.211 MAD 0.1 0.264 1.62 2.46 0.651 0.144 0.232 MaxLoss 0.1 0.238 1.487 0 0.605 0.14 0.216 CDaR 0.12 0.237 1.53 2.311 0.6 0.135 0.205 CVaR 0.12 0.234 1.606 2.43 0.634 0.127 0.201 MAD 0.12 0.251 1.624 2.467 0.662 0.136 -0.097 MaxLoss 0.12 0.237 1.547 0 0.61 0.134 0.211 CDaR 0.15 0.163 1.48 2.407 0.594 0.09 0.145 CVaR 0.15 0.222 1.618 2.468 0.662 0.119 0.1803 MAD 0.15 0.231 1.607 2.458 0.658 0.125 0.178

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Risktolerance Zero-beta Aver.RateofReturn StandardDeviation CVaR Drawdown SharpeRatio YearlyAver.Return MaxLoss 0.15 0.234 1.61 0 0.634 0.127 0.201 CDaR 0.17 0.195 1.52 2.399 0.607 0.108 0.162 CVaR 0.17 0.222 1.618 2.468 0.662 0.119 0.1805 MAD 0.17 0.247 1.612 2.458 0.658 0.135 0.208 MaxLoss 0.17 0.231 1.61 0 0.64 0.125 0.196 CDaR 0.2 0.233 1.505 2.397 0.499 0.135 0.219 CVaR 0.2 0.222 1.618 2.468 0.6628 0.119 0.18 MAD 0.2 0.233 1.639 2.562 0.685 0.124 0.193 MaxLoss 0.2 0.227 1.611 0 0.648 0.122 0.189 CDaR 0.005 + 0.251 0.529 0.642 0.089 0.42 0.225 CVaR 0.005 + 0.267 0.543 0.642 0.086 0.437 0.241 MAD 0.005 + 0.22 0.183 0.056 0.011 1.041 0.2078 MaxLoss 0.005 + 0.249 0.537 0 0.092 0.408 0.227 CDaR 0.01 + 0.255 0.581 0.735 0.117 0.387 0.231 CVaR 0.01 + 0.268 0.604 0.712 0.122 0.393 0.242 MAD 0.01 + 0.231 0.301 0.216 0.0295 0.668 0.211 MaxLoss 0.01 + 0.254 0.5870 0 0.108 0.382 0.231 CDaR 0.03 + 0.258 0.757 0.991 0.194 0.302 0.233 CVaR 0.03 + 0.276 0.774 0.923 0.193 0.318 0.248 MAD 0.03 + 0.238 0.611 0.657 0.104 0.341 0.211 MaxLoss 0.03 + 0.253 0.707 0 0.165 0.315 0.233 CDaR 0.05 + 0.266 0.754 0.891 0.184 0.312 0.237 CVaR 0.05 + 0.265 0.841 1.122 0.231 0.279 0.237 MAD 0.05 + 0.261 0.829 1.159 0.241 0.27 0.25 MaxLoss 0.05 + 0.256 0.765 0 0.173 0.295 0.238 CDaR 0.07 + 0.263 0.802 1.08 0.227 0.29 0.244 CVaR 0.07 + 0.252 0.875 1.22 0.232 0.254 0.218 MAD 0.07 + 0.247 0.859 1.272 0.271 0.252 0.232 MaxLoss 0.07 + 0.284 1.345 0 0.553 0.189 0.265 CDaR 0.10 + 0.223 0.848 1.194 0.276 0.228 0.211 CVaR 0.10 + 0.248 0.882 1.268 0.244 0.246 0.214 MAD 0.10 + 0.237 0.861 1.268 0.252 0.241 0.209 MaxLoss 0.10 + 0.258 0.854 0 0.239 0.266 0.230

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Risktolerance Zero-beta Aver.RateofReturn StandardDeviation CVaR Drawdown SharpeRatio YearlyAver.Return CDaR 0.12 + 0.244 0.851 1.215 0.276 0.252 0.224 CVaR 0.12 + 0.247 0.882 1.27 0.244 0.246 0.214 MAD 0.12 + 0.257 0.883 1.268 0.245 0.257 0.233 MaxLoss 0.12 + 0.252 0.865 0 0.244 0.257 0.223 CDaR 0.15 + 0.22 0.858 1.141 0.246 0.221 0.186 CVaR 0.15 + 0.247 0.882 1.268 0.244 0.246 0.214 MAD 0.15 + 0.251 0.88 1.268 0.245 0.251 0.222 MaxLoss 0.15 + 0.247 0.882 0 0.244 0.246 0.214 CDaR 0.17 + 0.243 0.843 1.108 0.223 0.252 0.228 CVaR 0.17 + 0.247 0.882 1.268 0.244 0.246 0.214 MAD 0.17 + 0.251 0.88 1.268 0.245 0.251 0.222 MaxLoss 0.17 + 0.247 0.882 0 0.244 0.246 0.214 CDaR 0.20 + 0.216 0.864 1.248 0.278 0.215 0.186 CVaR 0.20 + 0.247 0.882 1.27 0.244 0.246 0.214 MAD 0.20 + 0.251 0.88 1.268 0.2458 0.251 0.221 MaxLoss 0.20 + 0.247 0.882 0 0.244 0.246 0.214

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73 Table 3-4: Summarystatisticsforthemixedout-of-sampletests. Typeof constraint Risk tolerance Zerobeta Aver. Rateof Return Standard Deviation CVaR Drawdown Sharpe Ratio Yearly Aver. Return CDaR 0.005 0.1790 0.4246 0.5024 0.112 0.3504 0.165 CVaR 0.005 0.1564 0.4175 0.6162 0.1698 0.3027 0.142 MAD 0.005 0.2184 0.0949 -0.026 0.005 1.984 0.212 MaxLoss 0.005 0.177 0.396 0.852 0.129 0.370 0.166 CDaR 0.01 0.19 0.437 0.508 0.120 0.366 0.175 CVaR 0.01 0.143 0.493 0.788 0.218 0.229 0.133 MAD 0.01 0.204 0.182 0.16 0.032 0.96 0.19 MaxLoss 0.01 0.165 0.45 1.01 0.17 0.30 0.155 CDaR 0.03 0.170 0.518 0.649 0.171 0.271 0.161 CVaR 0.03 0.112 0.67 1.19 0.349 0.123 0.104 MAD 0.03 0.174 0.486 0.738 0.230 0.296 0.161 MaxLoss 0.03 0.154 0.628 1.44 0.275 0.198 0.145 CDaR 0.05 0.154 0.592 0.840 0.217 0.21 0.15 CVaR 0.05 0.125 0.807 1.452 0.366 0.117 0.126 MAD 0.05 0.11 0.8556 1.5936 0.4086 0.0936 0.105 MaxLoss 0.05 0.143 0.757 1.886 0.305 0.149 0.141 CDaR 0.07 0.114 0.681 1.042 0.279 0.123 0.117 CVaR 0.07 0.074 0.861 1.689 0.432 0.0510 0.076 MAD 0.07 0.040 0.993 2.05 0.550 0.010 0.025 MaxLoss 0.07 0.114 0.811 2.34 0.361 0.104 0.115 CDaR 0.10 0.132 0.741 1.19 0.331 0.137 0.132 CVaR 0.10 0.001 1.07 2.32 0.647 -0.027 -0.0006 MAD 0.10 -0.027 1.157 2.346 0.69 -0.049 -0.0448 MaxLoss 0.10 0.071 0.943 2.92 0.487 0.043 0.067 CDaR 0.12 0.134 0.774 1.27 0.354 0.134 0.136 CVaR 0.12 -0.012 1.140 2.4256 0.681 -0.037 -0.0184 MAD 0.12 -0.014 1.16 2.346 0.667 -0.038 -0.033 MaxLoss 0.12 0.0576 1.039 3.106 0.545 0.0265 0.049 CDaR 0.15 0.124 0.773 1.36 0.352 0.122 0.124 CVaR 0.15 -0.026 1.153 2.336 0.691 -0.048 -0.044 MAD 0.15 -0.016 1.185 2.346 0.674 -0.039 -0.0349

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Risktolerance Zero-beta Aver.RateofReturn StandardDeviation CVaR Drawdown SharpeRatio YearlyAver.Return MaxLoss 0.15 0.0108 1.152 3.75 0.644 -0.017 -0.0039 CDaR 0.17 0.142 0.84 1.35 0.429 0.134 0.139 CVaR 0.17 -0.0151 1.164 2.356 0.67 -0.039 -0.034 MAD 0.17 -0.017 1.167 2.35 0.67 -0.040 -0.036 MaxLoss 0.17 -0.008 1.19 4.0429 0.682 -0.032 -0.024 CDaR 0.20 0.142 0.84 1.354 0.429 0.134 0.14 CVaR 0.20 -0.0151 1.164 2.356 0.67 -0.039 -0.034 MAD 0.20 -0.0169 1.167 2.356 0.67 -0.04 -0.036 MaxLoss 0.20 -0.008 1.189 4.042 0.682 -0.032 -0.024 CDaR 0.005 + 0.190 0.374 0.416 0.089 0.428 0.176 CVaR 0.005 + 0.18 0.401 0.467 0.128 0.374 0.165 MAD 0.005 + 0.208 0.097 -0.006 0.0031 1.841 0.202 MaxLoss 0.005 + 0.203 0.384 0.815 0.111 0.452 0.19 CDaR 0.01 + 0.207 0.419 0.435 0.078 0.422 0.19 CVaR 0.01 + 0.173 0.453 0.61 0.154 0.315 0.157 MAD 0.01 + 0.227 0.173 0.088 0.022 1.144 0.218 MaxLoss 0.01 + 0.195 0.441 1.11 0.146 0.374 0.181 CDaR 0.03 + 0.210 0.478 0.587 0.109 0.378 0.197 CVaR 0.03 + 0.153 0.629 1.059 0.290 0.196 0.142 MAD 0.03 + 0.166 0.456 0.593 0.179 0.298 0.149 MaxLoss 0.03 + 0.235 0.563 1.175 0.175 0.364 0.223 CDaR 0.05 + 0.205 0.533 0.672 0.136 0.328 0.193 CVaR 0.05 + 0.131 0.701 1.208 0.378 0.143 0.12 MAD 0.05 + 0.103 0.706 1.222 0.368 0.1038 0.088 MaxLoss 0.05 + 0.208 0.639 1.383 0.212 0.278 0.203 CDaR 0.07 + 0.191 0.592 0.747 0.114 0.271 0.182 CVaR 0.07 + 0.118 0.716 1.23 0.375 0.122 0.102 MAD 0.07 + 0.105 0.716 1.23 0.391 0.105 0.090 MaxLoss 0.07 + 0.16 0.677 1.865 0.282 0.192 0.155 CDaR 0.10 + 0.071 0.943 2.92 0.487 0.043 0.067 CVaR 0.10 + 0.1098 0.722 1.23 0.389 0.110 0.094 MAD 0.10 + 0.104 0.721 1.23 0.388 0.102 0.088 MaxLoss 0.10 + 0.142 0.701 1.749 0.324 0.16 0.133

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Risktolerance Zero-beta Aver.RateofReturn StandardDeviation CVaR Drawdown SharpeRatio YearlyAver.Return CDaR 0.12 + 0.161 0.627 0.818 0.209 0.209 0.148 CVaR 0.12 + 0.109 0.723 1.23 0.392 0.109 0.093 MAD 0.12 + 0.104 0.722 1.23 0.388 0.102 0.089 MaxLoss 0.12 + 0.128 0.709 1.66 0.353 0.139 0.117 CDaR 0.15 + 0.142 0.635 0.877 0.241 0.176 0.128 CVaR 0.15 + 0.109 0.723 1.23 0.391 0.109 0.093 MAD 0.15 + 0.105 0.716 1.23 0.391 0.104 0.09 MaxLoss 0.15 + 0.110 0.721 1.798 0.388 0.112 0.0952 CDaR 0.17 + 0.142 0.635 0.877 0.241 0.177 0.128 CVaR 0.17 + 0.109 0.723 1.23 0.39 0.11 0.094 MAD 0.17 + 0.105 0.716 1.23 0.391 0.104 0.09 MaxLoss 0.17 + 0.11 0.721 1.798 0.388 0.112 0.0950 CDaR 0.20 + 0.11 0.647 0.993 0.285 0.123 0.094 CVaR 0.20 + 0.109 0.723 1.23 0.391 0.109 0.093 MAD 0.20 + 0.105 0.721 1.23 0.39 0.104 0.09 MaxLoss 0.20 + 0.109 0.723 1.8 0.391 0.11 0.093

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Weconsidertheproblemofoptimalpositionliquidationwiththeaimofmaximizingtheex-pectedcashowstreamfromthetransactioninthepresenceofatemporaryorpermanentmarketimpact.Weuseastochasticprogrammingapproachtoderivetradingstrategiesthatdifferentiatedecisionswithrespecttodifferentrealizationsofmarketconditions.Thescenariosetconsistsofacollectionofsamplepathsrepresentingpossiblefuturerealizationsofstatevariableprocesses(priceofthesecurity,tradingvolumeetc.)Ateachtimemomentthesetofpathsispartitionedintoseveralgroupsaccordingtospeciedcriteria,andeachgroupiscontrolledbyitsowndecisionvari-able(s),whichallowsforadequaterepresentationofuncertaintiesinmarketconditionsandcircum-ventsanticipativityinthesolutions.Incontrasttotraditionaldynamicprogrammingapproaches,thestochasticprogrammingformulationadmitsincorporationofdifferenttypesofconstraintsinthetradingstrategy,e.g.riskconstraints,regulatoryconstraints,variousdecision-makingpoliciesetc.Weconsiderthelawn-mowerprinciple,whichincreasesstabilityofthesolutionwithrespecttopathspartitioningandsaturationofthescenariopattern,butleadstonon-convexoptimizationproblems.Itisshownthatinthecaseoftemporarymarketimpacttheoptimalliquidationstrategywiththelawn-mowerprinciplecanbeapproximatedbyasolutionofconvexorlinearprogrammingproblems.Implementedasalinearprogrammingproblem,ourapproachiscapableofhandlinglarge-scaleinstancesandproducesrobustoptimalsolutions.Arisk-aversetradingstrategywascon-structedbyincorporatingriskconstraintsinthestochasticprogrammingproblem.Wecontrolledtherisk,associatedwithtrading,usingtheConditionalValue-at-Riskmeasure.Numericalresultsandoptimaltradingpatternsfordifferentformsofmarketimpactarepresented. 76

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Themainchallengeofconstructingoptimaltradingstrategiesconsistsinpreventingormini-mizinglossescausedbytheso-calledmarketimpact,orpriceslippage,thataffectpayoffsduringmarkettransactions.Themarketimpactphenomenonmanifestsitselfbyadversepricemovementsduringtradesandiscausedbydisturbancesofthemarketequilibrium.Asarathersimplisticexpla-nation,consideraninvestorexecutingamarketordertosellablockofshares.Ifatthemomentofexecutionnooneiswillingtobuythisblockforthecurrentmarketprice,thesellerwillbeforcedtoofferalowerpricepershareinordertoaccomplishthetransaction,andconsequentlysufferalossduetothemarketimpact.Inrealitythefunctionofadjustingthepriceduringtransactionsisperformedbymarketintermediaries. Traditionally,twotypesofmarketimpactareconsidered:temporaryandpermanent.Perma-nentmarketimpactdenotesthechangesinpricesthatarecausedbytheinvestor'stradesandpersistduringtheentireperiodofhis/hertradingactivity.Ifthedeviationsinpricescausedbyinvestor'stransactionareunobservablebythetimeofhis/hernexttrade,themarketimpactissaidtobetem-porary.Naturally,themagnitudeofmarketimpactand,consequently,thelossduetoadversepricemovements,dependonthesizeoftrades,aswellasonthetimewindowsbetweentransactions.Foradetaileddiscussionofissuesrelatedtomarketimpact,see,amongothers, ChanandLakonishok ( 1995 ), KeinandMadhavan ( 1995 ), KrausandStoll ( 1972 ).Someofthelatestdevelopmentsontheoptimaltransactionimplementationandoptimaltradingpoliciesarepresentedinpapersby Bertsi-masandLo ( 1998 ), Bertsimasetal. ( 1999 ), RickardandTorre ( 1999 ), AlmgrenandChriss ( 2000 ),and Almgren ( 2001 ). BertsimasandLo ( 1998 )employedadynamicprogrammingapproachfordevisingoptimaltradingstrategiesthatminimizetheexpectedcostoftradingablockofSshareswithinaxednum-berofperiodsT.Theyderivedanalyticalexpressionsforbest-executionstrategiesinthestandardframeworkofdiscreterandomwalkmodels,underassumptionthatmarketimpactislinearinthenumberofsharestraded.Togainaninsightonhowinformationcomponentcaninuencetheop-timalstrategy,authorsintroduceaseriallycorrelatedinformationvariableinthepriceprocessofthesecurity.Extensionofthismethodologytothecaseofmultipleassetsandoptimalexecutionforportfoliosispresentedin Bertsimasetal. ( 1999 ). AlmgrenandChriss ( 2000 )constructedrisk-aversetradingstrategiesusingtheclassicalMarkowitzmean-variancemethodology.Withpermanentandtemporarymarketimpactfunctionsbeingalso

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linearinthenumberofsharestraded,theriskofthetradingstrategyisassociatedwiththevolatilityofthemarketvalueoftheposition.Acontinuous-timeapproximationofthisapproach,aswellasnonlinearandstochastictemporarymarketimpactfunctions,wasconsideredby Almgren ( 2001 ).Anapplicationoffuzzysettheorytooptimaltransactionexecution,whichhelpstomimicnon-rationalhumanbehaviouroftraders,isdiscussedinpaperby RickardandTorre ( 1999 ). However,acommondrawbackofthedescribedmethodologiescanbeseenintheinabilityofatradingstrategytoresponddynamicallytorealizedorchangedmarketconditions.Insteadofhavingaprescribedsequenceoftradesbasedontheparametersofsecurity'spriceprocess,wewouldliketodevelopatradingstrategythatdifferentiatesdecisionswithrespecttoactualrealizationofmarketconditionsateachmomentoftransaction.Also,thetradingstrategyshouldbeabletoincorporatedifferenttypesofconstraintsthatmayreectinvestor'spreferences,includingriskpreferences,institutionalregulations,etc. Inthischapter,weconsidertheoptimalliquidationprobleminthescopeofmaximizingtheexpectedcashowfromsellingablockofsharesinthemarket.Theproblemisformulatedandsolvedinthestochasticprogrammingframework,whichallowsforcreatingmulti-stagedecision-makingalgorithmswithappropriateresponsetodifferentrealizationsofuncertaintiesateachtimemoment.Thekeyfeatureofourapproachisasample-pathscenariomodelthatrepresentstheuncertainpriceprocessofthesecuritybyasetofitspossiblefuturetrajectories(samplepaths).Theapproachadmitsaseamlessincorporationofvarioustypesofconstraintsinthedenitionoftradingstrategy,andisapplicableunderdifferentformsofmarketimpact. Thesample-pathscenariomodelsisarelativelynewtechniqueintheareaofmulti-stagedeci-sionmakingproblems,wherethedominantscenariomodelsaretheclassicalmultinomialtreesorlattices.Sample-pathbasedsimulationmodelshavebeenrecentlyusedforpricingofAmerican-styleoptions(seeTilley(1993), Boyleetal. ( 1997 ), BroadieandGlasserman ( 1997 ), Carriere ( 1996 ), BarraquandandMartineau ( 1995 ),andothers),wheretheoptimaldecisionpolicycontainssinglebinarydecision(i.e.,exercise-or-not-exercisethesecuritycontract).Inthissense,theprob-lemsofoptimaltransactionexecutionaremorecomplex,sincetheoptimalstrategyisasequenceofnon-binarydecisions. Thechapterisorganizedasfollows.Thenextsectionintroducesthegeneralformulationoftheoptimalliquidationproblem,denitionsofmarketimpact,etc.Sections 4.3 and 4.4 dealwith

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optimalpositionliquidationundertemporaryandpermanentmarketimpacts,correspondingly.InSection 4.5 ,wediscussconstructionofrisk-aversetradingstrategiesbyincorporatingConditionalValue-at-Riskconstraintsintheoptimalliquidationproblem.Section 4.6 presentsacasestudyandnumericalresults,andSection 4.7 concludesthechapter. 4 a).Theobjectiveistogenerateanoptimaltradingpolicywhichmaximizestheexpectedcashowstreamincurredfromliquidatingtheposition.Throughoutthechapter,weimplicitlyassumethatthepositiontobeliquidatedisalongposition. 4 b).Theseclassicaltechniqueshaveprovedtobeeffectivetoolsindealingwithmultistageproblems,especiallyinanalyticalframe-work.However,inmanyreal-lifenancialapplicationsthatrequiresolvinglarge-scaleoptimizationproblems,useofthetree-orlattice-basedscenariomodelsmayleadtoconsiderablecomputationaldifculties,knownasthecurseofdimensions.Therefore,manynancialinstitutionsintheirre-searchandinvestmentpracticeadoptscenariomodelsdifferentfromtheclassicalmultinomialtreesandlattices.Oneofthemostpopularalternativeapproachesisrepresentingtheuncertainfutureasacollectionofsamplepaths,eachbeingapossiblefuturetrajectoryofanancialinstrumentorgroupofinstruments(Fig. 4 a).Thistypeofscenariomodelissupposedtoovercomethecurse

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ofdimensionsthatoftendefeatslarge-scaleinstancesofoptimizationmodelsbasedonscenariotreesorlattices. Samplepathsandandscenariotrees. Asignicantdeciencyofmultinomialtreesorlatticesisapoorbalancebetweenrandomnessandthetree/latticesize.Asmallnumberofbranchespernodeinatree/latticeclearlymakesaroughapproximationoftheuncertainfuture,whereasincreasingofthisparameterresultsinexponentialgrowthoftheoverallnumberofscenariosinthetree. Incontrasttothis,asample-pathscenariomodelrepresentstheuncertainvalueofstochasticparameterateachtimebya(large)numberofsamplepoints(nodes)belongingtodifferentsamplepathsofthescenarioset.Increasingthenumberofnodesateachtimestepforbetteraccuracyresultsinalinearincreaseofthenumberofsamplepathsintheset.Similarly,increasingthenumberoftimeperiodsinthemodelleadstoalinearincreaseofthenumberofnodes,asopposedtoexponentialgrowthinthenumberofnodesinscenariotrees. Besidessuperiorscalability,sample-pathconceptallowsforeffortlessincorporationofhistor-icaldataintothescenariomodel,whichisanimportantfeaturefromapracticalpointofview. Therehasbeenanincreasinginteresttouseofsamplepathsindescribinguncertainmarketenvironmentinproblemsofnanceandnancialengineeringduringrecentyears.Forthemostpart,thisapproachwasemployedintheareaofpricingofderivatives(Titley(1993), Boyleetal. ( 1997 ), BroadieandGlasserman ( 1997 ), Carriere ( 1996 ), BarraquandandMartineau ( 1995 )etc.).Recently,asamplepathsframeworkwasappliedinsolvingdynamicassetandliabilitymanagementproblems( Hibiki ( 1999 2001 )).Inthischapter,weconsidertheconceptofsample-pathscenario

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setsandcorrespondingoptimizationtechniquesinapplicationtoaproblemofoptimaltransactionexecutioninpresenceofmarketimperfectionsandfriction. Wedeneatradingstrategy,correspondingtothesample-pathcollectionS,asaset wherexjtisthenormalizedvalueofthepositionattimetonpathj.Theinstantproceedsincurredbytransactionattimetaredeterminedbythepayofffunctionpt(),whosegeneralformispt(xjt;Sjtjtt);j=1;:::;J;t=1;:::;T: Theobjectiveofourtradingstrategyistomaximizetheexpectedcashowstreamincurredfromsellingtheasset: Problem( 4.2 )isastochasticprogrammingformulationoftheoptimalclosingproblembasedonthesample-pathscenariomodel.HereEj2SistheexpectationoperatordenedonS,andXdenotesthesetofallpossibletradingstrategies( 4.1 ).Itwillbeseenlaterthatthegenericformulation( 4.2 )is

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farfrombeingperfect;beforedivingintofurtherdetails,wehavetodiscusstheformandpropertiesofthepayofffunctionpt()in( 4.2 ),whichhasamajorimpactonthepropertiesoftheproblemofoptimalclosingingeneral. wherethepriceSjtisalwayspositive:Sjt>0.Thetermd(Dxjt;Sjtjtt)in( 4.3 )capturestheeffectsofmarketfriction.Itmaydependparametricallyoninformationobservableatpathjuptotimet,e.g.,volumeVjt,pricesS0;:::;Sjt,etc.

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(i) Intheperfectfrictionlessmarket,obviously,d(y)=y,y2[0;1].Weassumethatinthepresenceoftemporarymarketimpact,proportionaltransactioncosts (ii) (iii) (iv) Condition(ii)statesthatpayoffinthemarketwithfrictionisalwayslessthanthatinthefrictionlessmarket.Condition(iii)and(iv)ensurethatlargertradesleadtohigherrevenues,however,themarginalrevenuesbecomesmallerwithincreasingoftradesize(Fig. 4 ). Figure4: Impactfunctiondt(). Theimportanceoftheconcavityrequirement(iv)willbeclearlaterwhenwepresentrenedformulationsfortheoptimizationproblem( 4.2 ).Observealsothat(iv)impliesthatdt()iscontin-uouson[0,1]:dt2C([0;1]). Summarizingtheaforesaid,inthecaseoftemporarymarketimpactwewritethepayofffunc-tionpt()intheobjectiveof( 4.2 )intheform

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whereSjt=const>0,andd()satisesconditions(i)(iv). whereSjt,t=1;:::;tistheundisturbedpricetrajectorythatwouldrealizeintheabsenceofourtrades,andSjtistheactualpriceatmomenttandpathjreceivedbyinvestorforliquidatingportionDxjtoftheposition.Functionsqt(Dx)andqt(Dx)areequaltothepercentagedropinmarketpriceduetothetemporaryandpermanentmarketimpact,correspondingly.Similarlytotheabovenotationdt(),functionsqt()andqt()maycontain,ingeneral,valuesS0;:::;Sjtasparameters:qt(Dxjt),q(Dxjt;Sjtjtt);qt(Dxjt),q(Dxjt;Sjtjtt): 4.5 ),thetotalpayoffattainedoverapath(S0;Sj1;:::;SjT)isTt=1SjtDxjt=Tt=1(SjtDxjtDxjtqt(Dxjt)Dxjtt1t=1Sjtqt(Dxjt))=Tt=1Sjt(DxjtDxjtqt(Dxjt)qt(Dxjt)Tt=t+1Dxjt): Therstterminbracesin( 4.6 )attributestotheprotofsellingtheportionDxjtofthepositioninperfectfrictionlessmarket.Thesecondandthirdsummandsin( 4.6 )representthelossesduetoeffectsoftemporaryandpermanentmarketimpacts,correspondingly. Forconsistencywiththeabovediscussionoftemporarymarketimpact,weassumethatqt()issuchthatthefunctionyyqt(y)

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Similarly,thepermanentmarketimpactfunctionqt()isassumedtobenon-negativeandnon-decreasing.Wesupposethatfunctionqt()satisesqt(Dx)=0;0Dxltforsomelt2(0;1): wherek=const2(0;1]. 4.2 ).Then,thelawn-mowerdecision-makingrule,whichresultsintradingstrategiesofspecializedform,isintroduced.Thepropertiesoflawn-mower-complianttradingstrategiesarediscussed. 4.2.2 ,weintroducedagenericformulation( 4.2 )fortheoptimalclosingproblem,whichisrewrittenheretakingintoaccounttheformofthepayofffunction( 4.4 )undertemporarymarketimpact,andreplacingtheexpectationoperatorbyanaverageoverthesetofpaths: Itturnsout,however,thatthetradingstrategybasedontheoptimalsolutionof( 4.8 )iscontingentonperfectknowledgeofthefutureandthereforeinappropriate.Asanexample,considerasimplecasewhenSjtdt(Dxjt)=SjtDxjt,i.e.,thepayoffequalstothedollarprotofsellingaportionDxjtofpositionforpriceSjtinfrictionlessmarket.Then,itiseasytoseethattheoptimalsolutionof( 4.8 )

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isgivenby Theoptimalstrategy( 4.9 )liquidatesthewholepositiononcethemaximalpriceovertheentiretrajectory(S0;Sj1;:::;SjT)isachieved.Inpracticethiswouldmeanthatthetraderperfectlyforecaststhefuturebyidentifyingsomepathjwiththeactualpricetrajectory(forexample,byobservingpriceS1attimet=1). Fromtheformalpointofview,theanticipativityoftheoptimalsolutionof( 4.2 )isattributabletothepossibilityofmakingseparatedecisionsregardingthepositionvaluexjtateachtandj. Figure4: Pathgrouping 4.2 )istheso-calledpathgrouping(seesimilarapproachesinTitley(1993)and Hibiki ( 1999 2001 )).ThesetofpathsSispartitionedintoKtgroupsGktateachtimet(Fig. 4 ) Onepossibleapproachistomakeidenticaldecisionateachpathinagivengroup: Herexk(j;t)tisthenewdecisionvariable,equaltothevalueofpositionvariablesxjtinthegroupGkt,andk(j;t)isthefunctionthatreturnstheindexkofsetGkt,whichcontainspathjatmomentt.We

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denotethisask(j;t)p(S): Inotherwords,timetpartitionKtSk=1Gktisobtainedfromtimet1partitionKt1Sk=1Gkt1bysplittinggroupsGkt1intosubgroups,similarlytosplittingthescenariotreeintobranchesateachtimestep(Fig. 4 ).Inthissetting,thedecisionrule( 4.11 )isanalogoustowhatislongknowninstochasticprogrammingasnon-anticipativityconstraints(see,forexample, BirgeandLouveaux ( 1997 )). Figure4: Tree-likegrouping. Wegeneralizethepathgroupingmethod,comparedtoTitley(1993)and Hibiki ( 1999 2001 )intworespects.First,werelaxthetreegroupingcondition( 4.12 ),thusallowingforpathinter-mixing(i.e.,pathsfromdifferentgroupscanmergeintothesamegroupatthenextstep)9t;k;j1;j2:j1;j22Gkt;Gk(j1;t1)t16=Gk(j2;t1)t1: Second,weconsidergeneraldecisionrulesinthefollowingform: DependenceofallthepositionvariablesxjtwithinagroupGktonthecommondecisionvariablexk(j;t)tensuresthenon-anticipativityofthesolution.Belowweprovethatwiththetreegrouping( 4.12 )thedecisionpolicy( 4.13 )isidenticaltothesimplerule( 4.11 ).

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4.12 ),thenthegener-alizedgroupdecisionrule( 4.13 )isequivalentto( 4.11 ). 4.13 )thatispertinenttotheconstructionofanoptimalliquidatingstrategywithatemporaryorzeromarketimpact,namely Theintuitionbehindrelation( 4.15 )isasfollows.Firstofall,thedecisionvariablexk(j;t)trepresentsanupperboundforthepositionvaluesxjtinagroupGkt: Moreover,observethatpositionvaluexjt1isallowedtochange(i.e.,decrease)attimetonlybybeingtrimmedtothelevelxk(j;t)t.Ifthepositionvaluexjt1isnothighenoughtobetrimmedbyxk(j;t)t,itremainsunchangedattimet.Thisresembleslawnmowing,withpositionvariablesxjtbeingthegrassanddecisionvariablesxk(j;t)tbeingtheblades(Fig. 4 ). Figure4: Thelawn-mowerprinciple.

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Itfollowsfrom( 4.15 )andthetradingstrategydenition( 4.1 ),whichrequireszeropositionatthenaltimemomentT,thatvariablesxmustsatisfy Byincorporatingthelawn-mowerprinciplewithpathgroupingp(S)intotheformulation( 4.8 ),weobtaintheoptimalclosingproblemintheformmaxX1 4.18 )wendtheoptimalallocationofthresholdsxk(j;t)tdeterminingthefollowingoptimaltradingstrategy:ifscenariojbelongstothegroupGkt,thenselltheportionofpositioninexcessofxkt;ifcurrentpositionvalueislessthanxkt,donothing. 4.18 )iswell-posed,i.e.,foranyallocationofthresholdsxktthecorrespondingtradingstrategyisunique. 4.15 )eachvariablexjtisasingle-valuedfunctionofvariablesxk(j;t)t.Inversestatementdoesnothold. 4.15 )impliesthatthepositionvariablexjtcanbeexpressedas Obviously,theassertionholdsfort=1:xj1=minf1;xk(j;1)1g=xk(j;1)1=minfxk(j;1)1g.Assumethat( 4.19 )holdsforsomet=t,thenxjt+1=minfxjt;xk(j;t+1)t+1g=minnminfxk(j;1)1;:::;xk(j;t)tg;xk(j;t+1)t+1o=minfxk(j;1)1;:::;xk(j;t+1)t+1g; 4.19 )byinduction.

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Toshowthattheinverseisnotgenerallytrue,consideratradingstrategyfxjtgthatclosesthepositiononallpathsjatsomet0
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4.18 )numericallytractable,wetransformitintoaprob-lemwithconvexfeasibleregionandnon-convexobjectivefunction.Toperformthetransformation,letextendthedomainoffunctiondt()to[1;1]: (v) weconstructthefollowingproblemmaxX1 4.1 )oftradingstrategysetXandcondition( 4.17 ),weputuj1=1xk(j;t)t;ujT=xjT1: 4.21 )isthatitsobjectivefunctiondoesnothavesuchatransparentmeaning,asin( 4.18 ).However,thelinearityoftheconstraintsmakestheproblem( 4.21 )muchmoretractablecomparedto( 4.18 ),whoseconstraintsetcontainsthenon-convexlawn-mowerconstraint( 4.15 ). 4.18 )and( 4.21 )areequivalentinthesensethatoptimalvaluesoftheirobjectivesareequal,andthesetsoftheiroptimalsolutionsinvariablesxjtandxk(j;t)tcoincide.

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thusproblem( 4.18 )canberewrittenasmaxX1 4.20 )ofvariablesujtandproperties(i)(v)offunctiondt(),theaboveformulationofproblem( 4.18 )canbepresentedasmaxX1 4.24 )differsfrom( 4.21 )onlybythethirdconstraint.Nowweshowthattheoptimalsolutionoftheproblem( 4.21 )doesalsosatisfytheconstraint whichisanequivalentformofthelawn-mowerprinciple( 4.15 ),giventhedenition( 4.20 )ofujt.

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Figure4: IllustrationtoproofofProposition 4.3.2 Assumethattheoptimalsolutionofproblem( 4.21 )doesnotsatisfythelawn-mowerprinciple( 4.15 ).Obviously,theoptimalsolutionof( 4.21 )cannotsatisfyinequalityxjt>minfxjt1;xk(j;t)tg 4.21 )andthedenitionofthesetoftradingstrategiesX.Therefore,theonlycaseleftiswhenintheoptimalsolutionofproblem( 4.21 )wehaveforsometandjxjt
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Thisprovesthatthesolutionof( 4.21 )thatdoesnotsatisfythelawn-mowerprinciple( 4.15 )isnon-optimal.Consequently,theoptimalsolutionofproblem( 4.21 )doesnotchangeifthelawn-mowerconstraint( 4.22 )isaddedtotheconstraintsetof( 4.21 ),whichwouldmaketheformulationof( 4.21 )identicalto( 4.24 ).Thus,wehaveshownthatbothproblems( 4.18 )and( 4.21 )canberewrittenintheform( 4.24 ),whichimpliesthatoptimalvaluesoftheirobjectives,andtheirsetsofoptimalsolutionsinvariablesxjtandxk(j;t)tcoincide. 4.3.4 4.18 )doesnotguaranteetheuniquenessoftheoptimaltradingstrategy.Asanexample,onecanconsiderthecasewithconstantpricesSjt=S0andidenticalfunctionsdt().Then,theprotwillbethesameforanytradingstrategy. Buteveniftheoptimaltradingstrategy(intermsofthepositionvariablesxjt),derivedasasolutionof( 4.18 ),isunique,theoptimalthresholdvariablesxktmaybenon-single-valued(seeProposition 4.3.1 ). 4.21 )belongstotheclassofproblemswithDC Horstetal. ( 1995 ),or KonnoandWijayanayake ( 1999 )),butallofthemdonotdemonstratesatisfactoryperformanceforlarge-scaleproblems. Inviewofthepresumablylargesizeofproblem( 4.21 ),wesuggesttoapproximatethesolutionoftheDCmaximizationproblem( 4.21 )bysolutionofaconvexprogrammingproblem.Inpartic-ular,weconstructaconcavemaximizationproblemwhosesolutionprovidesalowerboundtotheoptimaltradingstrategygivenbytheoptimalsolutionof( 4.21 ). ThisisaccomplishedbyreplacingDCfunctionsmaxf0;dt(ujt)gintheobjective( 4.21 )byconcavefunctionsdt(ujt),whichcanbefurtherapproximatedbypiece-wiselinearconcavefunctionslt(): providedthatlt(y)satisestheproperties(i)(v).

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TheconcavemaximizationproblemhastheformmaxX1 4.27 ),whendt(x)isapiecewise-linearconcavefunction,admitsalinearpro-grammingrepresentation:maxX1 4.18 )intheexpenseofhavingaDCobjectivefunctionin( 4.21 ).Sincetheobjectiveof( 4.27 )isconcave,wehavetomakesurethattheoptimalsolutionofproblem( 4.27 )complieswiththelawn-mowerprinciple. 4.27 ),variablesujtareeithernon-negative,indicatingthatxjt=xk(j;t)t,ornegative,indicatingthatxjt
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IllustrationtoproofofProposition 4.3.5 Nowsupposethatthresholdsxk(j;t)tarenon-monotonic(Fig. 4 b).Lettbethesmallestpos-siblesuchthatforsomejwehavexk(j;t1)t1
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4.3.5 Theoptimalsolutionofconcaveprogrammingproblem( 4.27 )givesalowerboundoftheoptimalsolutionofDCmaximizationproblem( 4.21 ): 4.3.5 4.12 ),problems( 4.18 )and( 4.27 )areequivalent,andthelowerbound( 4.30 )isexact. 4.3.1 and 4.3.5 implythestatement. Optimaltradinginfrictionlessmarket. Belowweshowthatintheabsenceofmarketimpactandtransactioncoststherealwaysexistsanoptimal0tradingstrategy,compliantwiththelawn-mowerprinciple,eveniftheuniqueoptimalsolutionof( 4.18 )doesnotexist. 4.18 ). 4.25 )ofthelawn-mowerprincipleandexpression( 4.19 )forpositionvariablesxjt,problem( 4.18 )canbewrittenasfollowsmaxj(x)=1

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Usingcontradictionargument,wenowshowthatthereexistsanoptimal0solutiontoproblem( 4.31 ).Forsimplicity,letconsidertheoptimalsolutionof( 4.31 )withonlyonenon-binarycompo-nentxk0t02(0;1).Havingallotherelementsofxxed,considerj()asafunctionofxk0t0only: where1;:::;4denoteappropriatesummationsofSjt:1=jk(j;t0)=k0;8t=1;:::;t01:xk(j;t)t=1J1Sjt0;2=j;t>t0k(j;t0)=k0;8t2f1;:::;t1g=ft0g:xk(j;t)t=1;xk(j;t)t=0J1Sjt;3=jk(j;t0)=k0;9t2f1;:::;t01g:xk(j;t)t=0J1Sjt0;4=j;t>t0k(j;t0)=k0;8t2f1;:::;tg=ft0g:xk(j;t)t=1J1Sjt: 4.32 )equaltozero,so( 4.32 )takestheform If16=2,function( 4.33 )canbeimprovedbyputtingxk0t0=0or1,whichwouldmeanthatsolutionwithanon-binarycomponentcannotbeoptimal.If1=2,thevalueof( 4.33 )doesnotchangebyselectionxk0t0=0or1,i.e.,thereexistsanoptimal0solutionof( 4.31 ). Thecasewithmorethanonefractionalcomponentintheoptimalsolutioncanbeconsideredsimilarly,withtheonlydifferencethatthevariationoftheobjectivefunctionj(j)withrespecttosomevariablewithnon-binaryoptimalvaluemayhavetobeconsideredinasmalle-vicinityofanoptimalpoint. 4.3.6 Proposition 4.3.6 impliesthatiftheoptimalsolutionof( 4.18 )isunique,thenitis0.

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4.3.6 Thebinarystructureofoptimaltradingstrategyinthefrictionlessmarketcanbeviewedasacriterionforselectingthegroupdecisionrule( 4.13 ).Functionf()notallowingforanoptimal0solutionofproblem( 4.14 )intheabsenceofmarketimperfections,cannotbeconsideredasacandidateforthegroupdecisionruleofatradingstrategy.Proposition 4.3.6 actuallyascertainsthatthelawn-mowerprinciple( 4.15 ),asthedecisionruleinproblem( 4.18 ),holdsthisproperty. ( 1998 )haveshownthatunderlineartemporarymarketimpacttheoptimaltradingstrategyexhibitslittledifferencewiththeso-callednaivestrategy,whichconsistsindividingthepositioninequalportionstobeexecutedateachtimemomentt(providedthattimemomentsareequallyspaced): Dependingontheassetpricedynamics,theoptimaltradingstrategymaydeviatefromthenaiverule( 4.34 ),butnevertheless,thetransactionsizeDxcanbeassumednon-zeroateachtimestep.Inthepresentwork,weareprimarilyinterestedinshorttimeframesforliquidation,thereforeintheoptimalclosingproblem( 4.18 )wemayimposeconstraintDxjte;j=1;:::;J;t=1;:::;T; 4.15 )becomesequiva-lenttothesimpledecisionrule( 4.11 ): whichsimpliessignicantlytheoptimizationproblem( 4.18 ).Moreover,thesameargumentisvalidwhenbothtemporaryandpermanentformsofmarketimpactarepresent. Ontheotherhand,Proposition 4.3.6 suggeststhatwithdiminishingofthemarketimpact,d(Dx)!Dx,theoptimaltradingstrategywouldtendtoconcentratethetransactionsatsomespe-cicpointoftime,approaching0strategy.Thus,theabovesimplicationshouldbeusedwithcautionwhenthemarketimpactisnotsignicant.

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4.15 )andpayofffunctionpt()thatincorporatespermanentmarketimpactreadsasmaxX1 4.36 )isnotseparableasinthecaseoftemporaryimpact( 4.8 ),inthesensethatitcannotbepresentedasasumoffunctionsofsingleargument:ifi(xi);xi2R.Thisandthegeneralnon-concavityoftheobjective( 4.36 )wouldnotallowustoconstructanequivalentformulationofproblem( 4.36 )withaconvexfeasibleregion,asitwasdoneinSection 4.3.2 Therefore,toconsidertheproblemofoptimalpositionclosinginthepresenceofpermanentmarketimpactweperformpathpartitioningwithtrivialgroupdecisionrulexjt=xk(j;t)t;k(j;t)p(S);j=1;:::;J;t=1;:::;T; 4.3.3 ,thelawn-mowerprinciple( 4.15 )isequivalenttothesimplerulexjt=xk(j;t)tprovidedthatattimetandpathjthetradeDxjtisnon-zero.Previousstudies( BertsimasandLo ( 1998 ), AlmgrenandChriss ( 2000 ))demonstratedthatunderlinearper-manentmarketimpact,theoptimaltradingstrategyusuallyconsistsofnon-zerotrades:Dxt>0,t=1;:::;T,whichisalsoconsistentwithourndings(seeSection 4.6.3 ).Thiscanbeviewedasajusticationforreplacingformulation( 4.36 )with( 4.37 ).

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Anothermotivationforreplacingthelawn-mowerprinciple( 4.15 )bysimpledecisionrule( 4.11 )isthatthesimplerulexjt=xk(j;t)tdramaticallyreducesthedimensionalityoftheproblem,whichisespeciallyimportantinviewofthegenerallynonlinearnon-convexobjectiveof( 4.37 ). Whenthetemporaryandpermanentmarketimpactfunctionsqt(Dx)andqt(Dx)arelinear,qt(Dx)=gDx;qt(Dx)=gDx; 4.37 )reducestoaquadraticprogrammingproblemwiththefollowingobjective: whichisalwaysconcaveforanypositiveSjt,g,andg.Thiscasecorrespondstotheframeworkofpermanentmarketimpactdiscussedin BertsimasandLo ( 1998 )and AlmgrenandChriss ( 2000 ).Moreinterestingandpracticallyplausiblerepresentationforthepermanentmarketimpactfunctionqt()is( 4.7 ).Ifthefunctionqt()satises( 4.7 ),thepermanentchangesofmarketpricescanbeonlycausedbylargeenoughtrades. RockafellarandUryasev ( 2000 2002 )).CVaRisadownsideriskmeasure,whichquantiestheriskintermsofpercentilesoflossdistri-bution.Probably,themostoutstandingfeatureofConditionalValue-at-Riskisitsconvexitywithrespecttodecisionvariables(providedthatthelossfunctionisalsoconvex),whichdramaticallysimpliestheproblemofestimatingandmanagingrisksusingtechniquesofmathematicalpro-gramming,andalsomakestheseproceduresmuchmorerobustcomparedtootherriskmeasures,e.g.,Value-at-Risk(see RockafellarandUryasev ( 2000 2002 )). Indevelopingriskconstraintsfortheoptimalliquidationproblemwefollow RockafellarandUryasev ( 2000 2002 ).GivenalossfunctionL(x;h),wherehisastochasticvectorstandingformarketuncertainties,andxisthedecisionvector,ConditionalValue-at-Riskfa(x)withcondencelevelacanbedescribedapproximatelyastheconditionalexpectationoflossesexceedingthea-quantileofthelossdistribution.IfthelossfunctionL(x;h)hasacontinuousdistribution,this

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denitionisexact:fa(x)=(1a)1ZL(x;h)za(x)L(x;h)dF(h): RockafellarandUryasev ( 2002 ),intheoptimizationproblemwithmultipleCVaRconstraintsmaxx2Xg(x)s.t.fat(x)wt;i=1;:::;T; 1at1 4.39 )isconvexprovidedthatfunctionL(x;h)isconvexinx. Recallthatinthepresenceofpermanentmarketimpactthegeneratedpayoffisanonlinearandgenerallynon-convexfunctionofpositionvaluesxjt.Thus,imposingriskconstraintsbasedon

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thelossfunction( 4.40 )ontothefeasiblesetofoptimalliquidationproblem( 4.18 )willmakeitcomputationallyintractable. Ontheotherhand,ifonlytemporarymarketimpactisconsidered,andthetemporaryimpactfunctiondt()satisestoconditions(i)(v),thelossfunction( 4.40 )becomesconvex: ensuringconvexityofCVaRconstraints( 4.39 ).Toincreasecomputationalefciency,weapproxi-matetheconvexsetdeterminedbyconstraints( 4.39 ),whereL()isdenedasabove,byasetoflinearconstraintsusingthepiecewise-linearapproximation( 4.26 )offunctiondt().TheoptimalpositionliquidationproblemwithCVaRconstraintsthenreadsasmaxX1 1at1 4.43 )makesurethatateachtimettheaverageof(1at)100%worstlosses,comparedtotheinitialwealthlevelS0,doesnotexceedwt. Nextweshowthatourpreviousargumentsonoptimalliquidationwiththelawn-mowerdecisionruleremainvalidifCVaRconstraintsareincorporatedinthecorrespondingoptimizationproblems.

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4.3.4 and 4.3.5 holdwithCVaRconstraintsxjt2Fa;wt;t=1;:::;T;j=1;:::;J; 4.18 ),( 4.21 )and( 4.27 ),wheresetFa;wtisdenedasin( 4.43 ). 4.3.3 and 4.3.4 withthecorrespondingcorollariesremainvalid. 4.3.6 4.18 ),( 4.21 ),and( 4.27 )canbefractional.Thiscorrespondstothegeneralprincipleofdiversicationofrisk. Weconsideredatestproblemofoptimalliquidationofalongpositioninastockinthepresenceofdifferenttypesofmarketimpact.Thetimehorizonforclosingofthepositionspanned5businessdays,withonetradingopportunityperday(T=5).Becauseoftheshorttimeframe,wedidnotconsiderdiscounting. Thesample-pathscenariosetScontained5,000paths(J=5;000),andwasconstructedinthescopeoftheabove-mentionedtradingstrategy.Thecollectionofsamplepathswasgeneratedbycuttingfromthehistoricaltrajectoryofastock5-daywindowsthatfollowedtheopeningsignalgen-eratedbythetradingalgorithm.Ateachtimet=1;:::;5,asamplepath(S0;Sj1;:::;Sj5)containedthedailyclosingpriceofastockandthedailytradingvolumeinthatstock:Sjt=(Sjt;Vjt).Thepricesofthesamplepathsinthecollectionwerenormalizedsothattheinitialpricewasequalto1.Althoughtheobtainedsamplepathsactuallybelongtodifferentstocks,theiruseasascenariosetforasinglesecurityisquitereasonable,sincethetradingalgorithmgeneratestheopeningsignalinsimilarsituations,andthesetofstocksusedinthecasestudypossessedsimilarproperties(varianceandtradingvolume).

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Thepathpartitioningp(S)wasperformedasfollows:ateachtimemomenttthepathsweresortedinascendingorderwithrespecttocurrentpriceSjtandthenpartitionedintoKgroupsofequalsize:8k:Gkt=J=K;8k1>k2;j12Gk1t;j22Gk2t:Sj1tSj2t;t=1;:::;T; Wepresenttheoptimaltradingstrategiesintableauform(seeTables 4 4 )byreportingtheoptimalvaluesofvariablesxktinthecorrespondingoptimizationproblems.Wehaveconsideredthesimpledecisionrule( 4.11 )andthelawn-mowerprinciple( 4.15 ).RecallthataccordingtoProposition 4.3.2 ,thesetofvariablesxktdenesuniquelythetradingstrategyineithercase( 4.15 )or( 4.11 )ofthegroupdecisionrule.Allthetablesreporttheoptimalvaluesofvariablesxktfor10-grouppartitionasdescribedabove.TheresultspresentedinTables 4 4 havetobeinterpretedasfollows:ifattimetthepriceoftheinstrumentathandfallsintothepricerangeofgroupk,executethetransactioninaccordancetothespeciedgroupdecisionrule.Forinstance,inthecaseofthelawn-mowerdecisionrule( 4.15 ),reducethepositionvaluetothelevelofvariablexktifthepositionexceededxkt,ordonothingotherwise. 4.27 )withzeromarketimpact(dt(y)=y)hasa0solution,whichbecomesfrac-tionalifriskconstraints(inourcase,basedontheConditionalValue-at-Riskmeasure)areimposed.Itimpliesthattherisk-neutraltradingalgorithmselectsthemostfavorablemomenttoselltheposi-tioninoneshotsoastogainthehighestexpectedprot.Risk-aversetradingstrategyrecommendsclosingthepositionbypartsbecausewaitingforsuchafavorablemomentisrisky.Thiscorrespondstothegeneralprincipleofdiversicationofrisk. Table 4 displaystherisk-neutral(i.e.,withoutriskconstraints)andCVaR-constrainedopti-malliquidationstrategiesdenedassolutionsofproblems( 4.27 )withdt(y)=yandthecorrespond-ingproblemwithCVaRconstraints( 4.43 ).ParametersofCVaRconstraintsin( 4.42 )werechosen

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asat=0:9andwt=0:1,meaningthatineachtimettheaveragelossin10%ofworstcasesshouldnotexceed10%oftheoriginaldollarvalueoftheposition. Table4: Optimaltradingstrategyinfrictionlessmarket. CVaR-constrained(at=0.9,wt=0.1) 4.27 )withzero-impactfunctiondt(y)=yisquitesensitivetothechoiceofpartitionp(S).Consider,forexample,anextremecasewhenthenumberofgroupsateachtisequaltothenumberofpaths:Kt=J,t=1;:::;T.Inthiscase,theoptimalclosingproblemreducestoform( 4.8 ),withfullyanticipativeoptimalsolution( 4.9 ). 4.24 )decreasesaswell.Figure 4 presentstheperformanceofthetradingstrategydependingonthenumberKofgroupsinpartitionp(S)(theexpectedrateofreturnisequaltotheoptimalobjectivevaluedividedbytheinitialwealthS0). Figure4: Performanceofoptimaltradingstrategyinfrictionlessmarket. 4.9 )deliversmaximalobjectivevalueof( 4.24 )withdt(y)=yoverallpossiblepartitionsp(S).

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Notethattheexpectedrateofreturnofzero-impacttradingstrategyvariesconsiderably,from1:0294for10groupsto1:03625incaseof500groups.Sincetheobjectivevalueofthefullyanticipativesolution( 4.9 )is1:05685,wecanconcludethatsolutionwithasmanyas500groupshasanacceptabledegreeofnon-anticipativity. Atthesametime,theoptimalobjectivevalueofproblemwithCVaRconstraintsisalmostinsensitivetothenumberofgroupsinpartition.Thesameistrueforoptimalliquidationproblemswithnon-zeromarketimpactdiscussedinthefollowingsubsections. 4.18 )withatreepartitioningincom-parisontothesolutionoftheoptimalclosingproblemobtainedbythemethodsofdynamicpro-grammingandscenariotrees.ThedynamicprogrammingapproachinapplicationtotheoptimalclosingprobleminfrictionlessmarketwasusedbyButenko,Golodnikov,andUryasev(2003).Theauthorsdevelopedatechniquefortransformingacollectionofhistoricalpathsintoascenariotreewithaprescribednumberofbranchespernode.Thetradingstrategy,obtainedasthesolutionofcorrespondingdynamicprogrammingproblem,wastestedonthehistoricaldatausedinthiscasestudy.Wecomparedtheirresultsforthe4-branchscenariotree,solutiontowhichwaspresentedamongothers,withthesolutionofproblem( 4.18 )underthetreepartition,constructedbysplittingeachgroupinto4groupsofequalsizeateachtimet.AccordingtoProposition 4.3.1 ,treeparti-tionreducesanydecisionrule( 4.13 )tothesimplerule( 4.11 ).Ontheproblemwith5120paths,thealgorithm( 4.18 )slightlyoutperformedthedynamicprogrammingapproach,producingexpectedreturnof1.0241versus1.02062obtainedbyButenko,Golodnikov,andUryasev(2003).Thesim-ilarityofsolutionsobtainedbydifferenttechniquesvalidatesthecorrectnessofourapproachandthatpresentedinButenko,Golodnikov,andUryasev(2003). 4.18 )and( 4.27 )tohavetheform Thisfunction,evidently,satisesassumptions(i)(iv).Inrepresentation( 4.44 ),c=1correspondstothemostseveremarketimpact;whentheparametercincreases,themarketimpactfunction

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approachestheno-impactcasedt(y)=y.Whenb=2,equality( 4.44 )describesthelineartempo-rarymarketimpact,andinthiscasetheconvexprogrammingproblem( 4.27 )reducestoquadraticoptimizationwithadiagonalmatrix. Ingeneral,theoptimalsolutionofconvexprogrammingprogram( 4.27 )canbeapproximatedbyreplacingtheimpactfunctiondt()intheobjectivebyapiecewiselinearfunctionlt()( 4.26 )andsolvingthecorrespondinglinearprogrammingproblem( 4.28 ).Fortherepresentation( 4.44 ),coefcientsant,bntoffunctionlt()thatapproximatesdt()denedby( 4.44 ),canbechosenas bcNb1;bnt=nb(n1)bnb1 Thisapproximationisexactatpointsyn=n=N;n=1;:::;N. 4 displaystheoptimalvaluesofthedecisionvariablesxkt,obtainedasthesolutionofcorrespondingquadraticprogrammingproblem( 4.27 )withfunctiondt()denedaboveforb=2.Theproblemwassolvedforcasesc=1(severemarketimpact)andc=10(lightimpact). Table4: Optimaltradingstrategyunderlineartemporarymarketimpact(b=2). wherepositionissoldinequalportionsatallavailabletimemoments.ThisisconsistentwiththeresultsbyBertsimasandLo(1998)forlinear-percentagepriceimpact.Astheeffectsofmarketimpactdiminish(c!,d(y)!y),thesolutionstartstoexhibitdeviationsfromthenaivestrategy.

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4.2.3 ,themarketimpactfunctiond()dependsonthemarketconditionsatthemomentoftransaction.Toaccommodatethisinourmodel,weconsiderfunction( 4.44 )withcoefcientcthatdependsonthecurrentvolumeVtoftradesinthemarket.Inparticular,weassumedthefollowingexpressionforc:c=1+e(M(VjtC)); Tosolvetheoptimalclosingproblemwithtemporarymarketimpactdependingonmarketconditions,weemployedLPmodel( 4.28 )withpiecewise-linearpriceimpactfunctionconsistingof100linearsegmentsdeterminedby( 4.45 ).Theoptimalsolutionshowssignicantdifferencescomparedtothenaivestrategy( 4.46 )(seeTable 4 ). Table4: 4.37 )wasconsid-eredintwosettings:inassumptionoflinearityofthemarketimpactfunctionsqt(Dx)andqt(Dx),andundermorerealisticassumption( 4.7 ).

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partofmarketimpactin( 4.37 )isequalto( 4.44 )withb=2,i.e.,qt(y)=qt(y)=1 2cy: 4.46 )thanthecorrespondingsolutionoftheproblemwithtemporaryimpact,seeTable 4 : Table4: Optimaltradingstrategyunderlinearpermanentmarketimpact(b=2). 2cy;qt(y)=max0;1 2c(yl):(4.47) Inthiscase,however,theoptimalliquidationproblem( 4.37 )losestheconcavityoftheobjectivefunction.WeusedtheMINOSpackagetosolvetheresultingnon-convexprogrammingproblem,andTable 4 showsthesolutionreportedbyMINOS(l=0:2). Observethatthesolutiondepartssignicantlyfromthenaivestrategyeveninthecaseoflightmarketimpact(c=10).Thissuggeststhatthemarketimpactlaginthefunctionqt()( 4.47 )mayhavethemostpronouncedeffectonthepropertiesofoptimaltradingstrategiescomparedtoothermodelsofmarketimpact,bothtemporaryandpermanent,consideredinthischapter.

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Table4: Optimaltradingstrategyunderpiecewise-linearpermanentmarketimpact. 100.8990.7980.6320.374090.8990.7980.6290.371080.8990.7980.6430.367070.8990.7990.6360.368060.8990.7990.6430.373050.8990.7980.6300.359040.8990.7990.6350.367030.90.7980.6350.366020.8990.7990.6450.373010.90.7980.6330.3630 100.90.80.6430.454090.90.7750.5440.393080.90.7750.6750.382070.90.80.6750.364060.90.7750.6750.403050.90.7750.5680.249040.90.80.6750.356030.90.7990.6750.337020.90.80.70.442010.90.80.6320.3010 3cy2;qt(y)=max0;1 3csign(yl)(yl)2 Table4: Optimaltradingstrategyundernonlinearpermanentmarketimpact. 4 showslesspronounceddifferenceswiththenaivestrategyduetotheconvexityofthenonlinearimpactfunctions,whichimpliessmallerexecutioncostscomparedtothelinearnon-convexpermanentmarketimpact.

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Toavoidanticipativityofthesolutions,causedbyspecicpropertiesofthesample-pathsce-nariomodel,weintroducedpathpartitioningwithanewlawn-mowerdecisionrule.Sincethelawn-mowerprincipleleadstonon-convexityoftheoptimalclosingproblems,wedevelopedaconvexrelaxationfortheseproblems,whichwasfurtherapproximatedbylinearprogramming.Themodelofoptimalpositionliquidation,implementedasalinearprogrammingproblem,canbesolvedveryefcientlyandrobustlyinverylarge-scaleinstanceswithlargenumberofscenarios(sample-paths).Propertiesoftheoptimaltradingstrategiesbasedonthelawn-mowerprinciplehavebeeninvestigated. Inafrictionlessmarket,theoptimaltradingstrategybasedonthelawn-mowerconstraint,has0structure,i.e.,itliquidatestheentirepositioninonetransaction.IftheriskofnanciallossesduringtradesiscontrolledbyCVaRconstraints,thetradingstrategybecomesfractional.Whenstrongtemporaryorpermanentmarketimpactsarepresent,theoptimaltradingstrategyapproachestheso-callednaivestrategy,whichconsistsinsellingequalportionsofthepositionateachtimestep.Underweakermarketimpacts,theoptimaltradingstrategystartstodeviatefromthenaivestrategy.Accordingtoournumericalexperiments,themostsignicantdeviationsfromthenaivetradingstrategyareobservedinthecasewhentemporarymarketimpactdependsonthe

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currentmarketconditions,orwhenpermanentmarketimpacthasalag(i.e.,whensmallenoughtradesdonotcausepermanentchangesinpriceofthesecurity).

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Thischapterdevelopsageneralapproachtoriskmanagementinmilitaryapplicationsinvolvinguncertaintiesininformationanddistributions.Theriskofloss,damage,orfailureismeasuredbytheConditionalValue-at-Riskmeasure.ThegreatestadvantageofusingCVaRasariskmeasureinmilitaryapplicationsisthatCVaRisadownsidepercentileriskmeasure.Atthesametime,itisconvexasafunctionofdecisionvariables,andthereforecanbeefcientlycontrolled/optimizedusingconvexor(underquitegeneralassumptions)linearprogramming.ThegeneralmethodologywastestedontwoWeapon-TargetAssignment(WTA)problems.ItisassumedthatthedistributionsofrandomvariablesintheWTAformulationsarenotknownwithcertainty.Inthemathematicalprogrammingproblemformulation,thetotalcostofthemission(includingweaponattrition)isminimized,whilesatisfyingoperationalconstraintsandensuringdestructionofalltargetswithhighprobabilities.WeconductcasestudiesthatshowsignicantqualitativeandquantitativedifferencesinsolutionsofdeterministicWTAandstochasticWTAproblems. 114

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awiderangeofpossiblescenarios.Inthisregard,riskmanagementinmilitaryapplicationsissim-ilartopracticesinothereldssuchasnance,nuclearsafety,etc.,wheredecisionstargetedonlyatachievingthemaximalexpectedperformancemayleadtoanexcessesiveriskexposure.How-ever,incontrasttootherapplications,distributionsofthestochasticrisk-inducingfactorsareoftenunknownoruncertaininmilitaryproblems.Uncertaintyindistributionsofriskparametersmaybecausedbyalackofdata,unreliabilityofdata,orthespecicnatureofariskfactor(e.g.,indifferentcircumstancesariskfactormayexhibitdifferentstochasticbehavior).Therefore,decisionmakinginmilitaryapplicationsmustaccountforuncertaintiesindistributionsofstochasticparametersandberobustwithrespecttotheseuncertainties. Inthischapter,weproposeageneralmethodologyformanagingriskinmilitaryapplicationsinvolvingvariousriskfactorsaswellasuncertaintiesindistributions.TheapproachistestedwithseveralstochasticversionsoftheWeapon-TargetAssignmentproblem. Thechapterisorganizedasfollows.Section 5.2 presentskeytheoreticalresultsonriskman-agementusingConditionalValue-at-Risk(CVaR)riskmeasure,anddescribesthegeneralapproachtocontrollingriskwhendistributionsofriskfactorsareuncertain.Section 5.3 developsvariousformulationsofthestochasticWeapon-TargetAssignment(WTA)problemwithCVaRconstraints.Resultsofnumericalexperimentsforone-stageandtwo-stagestochasticWTAproblemsarepre-sentedinSection 5.4 .Section 5.5 summarizestheobtainedresultsandoutlinesthedirectionsoffutureresearch.

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5 ).Then,theproblemofpreventinghighlossesisaproblemofcontrollingandshapingthelossdistributionand,morespecically,itsrighttail,wherethehighlossesreside.Toestimateandquantifythelossesinthetailofthelossdistribution,ariskmeasurehastobespeci-ed.Inparticular,ariskmeasureintroducestheorderingrelationshipsforrisks,sothatoneisabletodiscriminatelessriskydecisionsfromtheriskierones. 5 displayssomeofthesemeasures;Value-at-Riskwithcondencelevela(a-VaR),whichisthea-percentileoflossdistribution,MaximumLoss(.0-percentileoflossdistribution),anda-CVaR,whichmaybethoughtofastheexpectationoflossesexceedinga-VaR. Figure5: Lossfunctiondistributionanddifferentriskmeasures. WebuildourapproachforriskmanagementinmilitaryapplicationsontheCVaRmethodology,whichisarelativelynewdevelopment( RockafellarandUryasev ( 2000 ), RockafellarandUryasev ( 1997 1999 )haveintroducedaconceptofideal,orcoherent,riskmeasure.Acoherentriskmeasure,whichsatisestoasetofaxiomsdevelopedinthispaper,isexpectedtoproduceproperandconsistentestimatesofrisk.

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( 2002 )).ThissectionpresentsthegeneralframeworkofriskmanagementusingConditionalValue-at-Risk,andextendsittothecasewhenthedistributionsofstochasticparametersarenotcertain. 5 ). Figure5: AvisualizationofVaRandCVaRconcepts. Theseintuitivedenitionsarecorrectif,forexample,allscenariosareequallyprobable,and(1a)100isanintegernumber.Theformaldenitionsofa-VaRanda-CVaRthatapplytoanylossdistributionandvalueofcondencelevelaaremorecomplex( RockafellarandUryasev 2002 ).HerewementiononlythemostimportantpropertiesofCVaRandtheirpracticalimplications. TheConditionalValue-at-RiskfunctionCVaRa[L(x;x)]hasthefollowingproperties( Rock-afellarandUryasev ( 2000 ), RockafellarandUryasev ( 2002 ), AcerbiandTasche ( 2002 )): Artzneretal. ( 1999 )

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Fromtheviewpointofmanagingandcontrollingofrisk,themostimportantpropertyofCVaR,whichdistinguishesitfromallotherpercentileriskmeasures,istheconvexitywithrespecttode-cisionvariables,whichpermitstheuseofconvexprogrammingforminimizingCVaR.IfthelossfunctionL(x;x)canbeapproximatedbyapiecewiselinearfunction,theprocedureofcontrollingoroptimizationofCVaRisreducedtosolvingaLinearProgramming(LP)problem. ThetechniquesforoptimizingCVaRwhenthelossdistributionisdiscreteareofspecialim-portanceformilitaryapplications,aswillbedemonstratedinSection 5.3 AssumethatthereareSpossiblerealizations(scenarios)x1;:::;xSofvectorxwithprobabilitiesps(naturally,Ss=1ps=1),thenintheoptimizationproblemwithmultipleCVaRconstraints RockafellarandUryasev 2002 ) Intheriskmanagementmethodologydiscussedabovethedistributionofstochasticparameterxisconsideredtobeknown.Thenextsubsectionextendsthepresentedapproachtothecase,whenthedistributionofstochasticparametersinthemodelisnotcertain.

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subjecttoAxb;CVaRa[L(x;x)jQn]C;n=1;:::;N; 5.2 )weassumethattheperformancefunctionFisconcaveinx,andthelossfunctionLisconvexinx.Theseassumptionsarenotrestrictive;onthecontrary,theyindicatethatgivenmorethanonedecisionwithequalperformanceonefavorssaferdecisionsovertheriskierones. Model( 5.2 )explainshowtohandletheriskofgeneratinganincorrectdecisioninanuncertainenvironment.Inmilitaryapplications,differenttypesofrisksandlossesmaybeexplicitlyinvolved,forexample,alongwithlossfunctionL(x;x)onemayconsideralossfunctionR(x;x)fortheriskoffalsetargetattack.Controlforthistypeofriskcanalsobeincludedinthemodelbyasimilarset

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ofCVaRconstraints: Manne ( 1958 ), denBroegeretal. ( 1959 ), Murphey ( 1999 ))leadstoanonlinearprogrammingprob-lemwithlinearconstraints(NLP),andisthesubjectofthenextchapter.Inthischapterweadoptanothersetup,wherethetotalcostofthemission(includingbattledamageorloss)isminimized,whilesatisfyingconstraintsonmissionaccomplishment(i.e.,destructionofalltargetswithsomeprescribedprobabilities).Weassumethatdifferentweaponshavedifferentcostsandefciencies,and,ingeneral,eachmayhaveamultishotcapacitysothatitmayattackmorethanonetarget.

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Inthedeterministicsetupoftheproblemweincludealsotheconstraintthatprescribeshowmanytargetsasingleweaponcanattack. ThedeterministicWTAproblemis subjecttoKk=1xikmi;i=1;:::;I; Theobjectivefunctioninthisproblemequalstothetotalcostofthemission.Therstconstraint,( 5.3b ),statesthatthemunitionscapacityofweaponicannotbeexceeded.Thesecondandthethirdconstraints( 5.3c )and( 5.3d )areresponsiblefornotallowingweaponitoattackmorethantitargets,wheretiK.Thelastconstraint( 5.3e )ensuresthatafterallweaponsareassigned,targetkisdestroyedwithprobabilitynotlessthandk.Notethatthisnonlinearconstraintcanbelinearized:

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InthiswaythedeterministicWTAproblem( 5.3a )canbeformulatedasalinearintegerprogram-ming(IP)problem. 5.3a )( 5.3e )arenotdeterministic,butstochasticvalues.Forexample,theprobabilitiespikofdestroyingtargetkmaydependuponbattlesituation,weatherconditions,andsoon,andconsequently,maybetreatedasbeinguncertain.Sim-ilarly,thecostofringcik,whichincludesbattleloss/damage,mayalsobeastochasticparameter.ThenumberoftargetsKmaybeuncertainaswell. First,weconsideraone-stageStochasticWeapon-TargetAssignment(SWTA)problem,wheretheuncertaintyisintroducedintothemodelbyassumingthatprobabilitiespikarestochasticanddependentonsomerandomparameterx:pik=pik(x): Wenowreplacethelastconstraintin( 5.3a )byaCVaRconstraint,wherethelossfunctiontakesapositivevalueiftheprobabilityofdestroyingtargetkislessthandk: andtakesanegativevalueotherwise.RecallthataCVaRconstraintwithcondencelevelaboundsthe(weighted)averageof(1a)100%highestlosses.Inourcase,allowingsmallpositivevaluesoflossfunction( 5.5 )forsomescenariosimpliesthatforthesescenariostargetkisdestroyedwithprobabilityslightlylessthandk,whichmaystillbeacceptablefromapracticalpointofview.

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Exceptfortheconstraintonthetargetdestructionprobability,theone-stageStochasticWTAproblemisidenticaltoitsdeterministicpredecessor: subjecttoKk=1xikmi;i=1;:::;I;xikmivik;i=1;:::;I;k=1;:::;K;Kk=1vikti;i=1;:::;I;CVaRa[Lk(x;x)]Ck;k=1;:::;K: 5.1 ),fortheadoptedscenariomodelwithprobabilitiespik,theCVaRconstraintforthek-thtargetCVaRa[Lk(x;x)]Ck

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5.7 )isariskconstraintthatincorporatesmultipleprobabilitymeasures. Thisproblemismorerealisticsinceitmodelstheeffectoftargetdiscoveryasbeingdynamic;thatis,notalltargetsareknownatanysingleinstanceoftime.Toaddressthistypeofuncertainty,weneedtomodifyournotation. ConsiderIweaponsaredeployedinsomeboundedregionofinterestandintervaloftimeTwiththegoalofndingtargetsandthen,oncefound,attackingthosetargets.IfwedelayallassignmentsofweaponshotsuntiltotargetsuntilthenaltimeT,thenwehaveadeterministic,staticWTAproblemasin( 5.3a )( 5.3e ).If,ontheotherhand,weassumethatweaponshaveatleast2opportunitiestoshootduringtheintervalT,thentheWTAproblemisdynamic.InthelatercasewehavetheopportunitytoavoidexpendingallourshotsattargetsdiscoveredearlyinTbyexplicitlymodelingthenumberofundiscoveredtargetsintheobjectivefunction. AssumethatKnowrepresentsthenumberofcategoriesoftargets(thetargetsmaybecatego-rized,forexample,bytheirimportance,vulnerability,etc). Wewillassumetheproblemhas2stages.Thatis,atanygivenpointintime,wemayalwayspartitionalltargetsintothosethusfardeterminedandthosethatweconjecturetoexistbuthave

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notyetfound.Ourconjecturemaybebasedonevidenceobtainedbypriorreconnaissanceoftheregionofinterest.Atsomearbitrarytime0
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subjecttoKk=1nkxikmi;i=1;:::;I; 5.9a )( 5.10e ).Asbefore,weminimizethetotalcostofthemission.Therstconstraint( 5.9b )isthemunitionscapacityconstraint.Thesecondconstraint,( 5.9c ),allowsarst-stagetargetincategoryktosurvivewith(small)errore1k,andthethirdcon-straint( 5.9d )boundsthesumoferrorse1kbysome(small)constantC. Intherecoursefunction( 5.10a )therstconstraint( 5.10b )requirestheweaponitonotexceeditsmunitionscapacitywhiledestroyingtherst-andsecond-stagetargets.Thepossibleinfeasibilityofthemunitionscapacityconstraintcanberelaxedusingauxiliaryvariablesdithatentertheobjec-tivefunctionwithcostcoefcientM1.Thesecondandthirdconstraints( 5.10c )( 5.10d )formaCVaRconstraintthatcontrolsthefailureofdestroyingsecond-stagetargetswiththeprescribedprobabilitiesdk.Similarlytothedeterministicconstraintin( 5.9a ),CVaRoffailuretodestroya

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second-stagetargetincategorykisboundedby(small)errorvariablee2k.Thetotalsumoferrorse1kande2katbothstagesisboundedbysmallconstantC,whichmakespossibleatradeoffbetweenthedegreeofmissionaccomplishmentattherstandsecondstages. Theextensiveformofthetwo-stageSWTAproblem( 5.9a )( 5.10a )is subjecttoKk=1(nkxik+hksyik(s))mi+di;8i;s;Ii=1ln(1pik)xikln(1dk)e1k;8k;Ii=1ln(1pik)yik(s)ln(1dk)zkwks;8k;s;zk+(1ak)1S1Ss=1wkse2k;8k;Kk=1(e1k+e2k)C;xik;yik(s);di2Z+;wks;e1k;e2k2R+zk2R;M1:

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Accordingtotheaforementioned,weusedsimulateddataforprobabilitiespiksandcostscik.Itwasassumedthatprobabilitiespiks=pisareuniformlydistributedrandomvariables,andtheFig-ure 5.4.1 displaystherelationbetweenthecostofmissileofweaponianditsefciency(i.e.,prob-abilitytodestroyatarget): Figure5: Dependencebetweenthecostandefciencyfordifferenttypesofweaponsinone-stageSWTAproblem( 5.8 )deterministicWTAproblem( 5.3a ). Onthisgraph,diamondsrepresenttheaverageprobabilityofdestroyingatargetbyringoneshotfromweaponi,andthehorizontalsegmentsrepresentthesupportforrandomvariablepik(x)=pi(x).Theaverageprobabilities 5.3a ). Theefciencyandcostofweapons1to5increasewiththeindexofweapon,i.e.,Weapon1istheleastefcientandcheapest,whereasWeapon5isthemostprecise,butalsomostexpensiveone. Tables 5 and 5 representtheoptimalsolutions(variablesxik)ofthedeterministicandone-stagestochasticWTAproblems. OnecanobservethedifferenceinthesolutionsproducedbydeterministicandstochasticWTAproblems:thedeterministicsolutiondoesnotusethemostexpensiveandmostpreciseWeapon5,whereasthestochasticsolutionofproblem( 5.8 )withCVaRconstraintusesthisweapon.ItmeansthattheCVaR-constrainedsolutionofproblem( 5.8 )representsamoreexpensivebutsaferdecision.

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Table5: OptimalsolutionofthedeterministicWTAproblem( 5.3a ) Weapon1021014 Weapon2012003 Weapon3100113 Weapon4100113 Weapon5000000 Optimalsolutionoftheone-stagestochasticWTAproblem( 5.6 ),( 5.8 ) Weapon1011013 Weapon2001113 Weapon3200103 Weapon4011013 Weapon5110103 Wehavealsoperformedtestingofthedeterministicsolutionunderdifferentscenarios.Thedeterministicsolutionfailedtodestroymorethanonetargetunder13of20scenarios. Thisexamplehighlightstheimportanceofusingriskmanagementproceduresinmilitarydecision-makingapplicationsinvolvinguncertainties.

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Figure5: Dependencebetweenthecostandefciencyfordifferenttypesofweaponsintwo-stageSWTAproblem( 5.11 ). Fortheprobabilitiespikinthetwo-stageproblem,weusedtherstfouraverageprobabilitiesfromthedeterministicWTAproblem,andtheefciency-costdependenceisshowninFigure 5.4.2 Tables 5 to 5 illustratetheoptimalsolutionoftheproblem( 5.11 ).Table 5 containstherst-stagedecisionvariablesxik,andTables 5 and 5 displaythesecond-stagevariablesyik(s)forscenarioss=1ands=2,justforillustrativepurposes. Similarlytotheanalysisoftheone-stagestochasticWTAproblem,wecomparedthescenario-basedsolutionofproblem( 5.11 )withthesolutionofthedeterministictwo-stageproblem,wherethenumberofsecond-stagetargetsineachcategoryistakenastheaverageover15scenarios.Thecomparisonshowsthatthesolutionbasedontheexpectedinformationleadstosignicantmunitionsshortagesin5of15(i.e.,33%)scenarios,andconsequentlytofailingthemissionatthesecondstage.Recallfromtheanalysisoftheone-stageSWTAproblemthatthesolutionbasedontheexpectedinformationalsoexhibitedpoorrobustnesswithrespecttodifferentscenarios.Indeed,solutionsthatuseonlytheexpectedinformation,aresupposedtoperformwellonaverage,orinthelongrun.However,inmilitaryapplicationsthereisnolongrun,andthereforesuchsolutionsmaynotberobustwithrespecttomanypossiblescenarios. Table5: First-stageoptimalsolutionofthetwo-stagestochasticWTAproblem #ofdetectedtargets352 Weapon1000 Weapon2000 Weapon3111 Weapon4111

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Table5: First-stageoptimalsolutionofthetwo-stagestochasticWTAproblem( 5.11 )fortherstscenario K3 #ofundetectedtargets14 2 Weapon100 2 Weapon2 0 0 1 Weapon3 1 1 0 Weapon4 1 1 0 Second-stageoptimalsolutionofthetwo-stagestochasticWTAproblem( 5.11 )forthesecondscenario K3 #ofundetectedtargets35 3 Weapon120 2 Weapon2 1 0 1 Weapon3 0 1 0 Weapon4 0 1 0 RockafellarandUryasev ( 2002 ).Althoughthepresentedapproachhasbeenusedtosolveone-stageandtwo-stagestochas-ticWeapon-TargetAssignmentproblems,itisquitegeneralandcanbeappliedtowideclassofproblemswithrisksanduncertaintiesindistributions.AmongthedirectionsoffutureresearchweemphasizeconsiderationofastochasticWTAprobleminNLPformulation,wherethedamagetothetargetsismaximizedwhileconstrainingtheriskoffalsetargetattack.

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Inthischapterweconsideramorerealisticformulationsofthetwo-stageStochasticWeapon-TargetAssignmentproblem,wherethecumulativedamagetothetargetsismaximized.Thisprob-lemsetup,however,leadstoMixed-IntegerProgrammingproblemswithnonlinearobjectives.Byusingarelaxationtechniquethatpreservesintegralityoftheoptimalsolutions,wedevelopLPfor-mulationsforthedeterministicandtwo-stagestochasticWTAproblems.Similarlytotheapproachoftheprecedingchapter,theriskofincorrectsecond-stagedecisionsduetoerrorsinspecieddistri-butionsofthesecond-stagetargetsiscontrolledusingtheConditionalValue-at-Riskriskmeasure.AnLPformulationforthetwo-stageSWTAproblemwithuncertaintiesindistributionshasbeendeveloped,whichproducesintegeroptimalsolutionsfortherst-stagedecisionvariables,andalsoyieldsatightlowerboundforthecorrespondingMIPproblem. Krokhmaletal. 2003 )toastochasticversionoftheWeapon-TargetAssignment( Manne 1958 ; denBroegeretal. 1959 ; Murphey 1999 )problem.Theapproachsuggestedin( Krokhmaletal. 2003 )isbuiltontherecentlydevelopedtechnique( RockafellarandUryasev 2000 2002 )forriskmanagementus-ingtheConditionalValue-at-Risk(CVaR)riskmeasure.Thegeneralframeworkwasdevelopedforspecicmilitaryapplicationssuchassurveillance,planning,andscheduling,whichrequirerobustdecisionmakinginadynamic,distributed,anduncertainenvironment.Thefocusofthesuggestedapproachhasbeenonthedevelopmentofrobustandefcientproceduresfordecision-makinginstochasticframeworkwithmultipleriskfactorsanduncertaintiesinthedistributionsofstochasticparameters. InaprecedingchaptertheauthorstestedthedevelopedmethodologyofriskmanagementusingConditionalValue-at-Riskonone-stageandtwo-stagestochasticversionsofaWeapon-TargetAs-signment(WTA)problem.IntheWTAproblemformulationdevelopedin Krokhmaletal. ( 2003 ), 132

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anoptimaldecisionminimizedthetotalcostofthemission,includingbattledamage,whileensur-ingthatalltargetsaredestroyedwiththeprescribedprobabilitylevel.Insuchasetup,theWTAproblemcouldeasilybeformulatedasalinearprogramming(LP)problem,orintegerprogramming(IP)problemwithlinearobjectiveandconstraints. Inthischapter,weconsidertheWTAprobleminamorerealisticformulation,wherethecumu-lativedamagetothetargetsismaximized.Thoughthissetuphassomeadvantagesoverthepreviousformulation(forexample,itallowsforprioritizingthetargetsbyimportanceandachievingade-sirabletradeoffbetweenassigningmoreweaponstohigh-prioritytargetsandfewerweaponstolow-priorityones),itleadstoanintegerprogrammingproblemwithnonlinearobjectiveandlinearconstraints. Inthischapter,thenonlinearintegerprogrammingproblemwillbetransformedintoa(convex)linearprogrammingproblem,andthecorrespondingLPrelaxationsfordeterministicandtwo-stagestochasticWTAproblemswillbedeveloped.Further,aformulationofatwo-stagestochasticWTAproblemwithuncertaintiesinthedistributionsofsecond-stagescenarioparameterswillbepre-sented.WeemploytheConditionalValue-at-Riskriskmeasure( RockafellarandUryasev 2000 2002 )inordertoconstraintheriskofgeneratinganincorrectdecision. Thechapterisorganizedasfollows.ThenextsectionintroducesagenericnonlinearmodelfortheWTAproblem,anddemonstrateshowanLPrelaxationcanbeconstructedfortheoriginalIPproblemwithanonlinearobjective.Section 6.3 presentsafastalgorithmbasedonanLPrelaxationforthetwo-stagestochasticWTAproblem.Section 6.4 considersthetwo-stageSWTAproblemwithuncertaintiesindistributions.AcasestudyfortheproblemispresentedinSection 6.5 Krokhmaletal. ( 2003 ))wehaveconsideredaformulationfortheWeapon-TargetAssignmentproblemwherethetotalcostofthemissionisminimizedwhilesatisfyingsomeprobabilisticconstraintsonthetargetdestruction.Anadvantageofthisformulationisthelinearityofthemathematicalprogrammingproblemsitreducesto.Now,following Krokhmaletal. ( 2003 ),weconsideranothersetupfortheWTAproblem,wherethetotaldamagetothetargetsisminimizedwithconstraintsonweaponsavailability.Thoughthisformulationresultsinaninteger

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programmingproblemwithnonlinearobjectiveandlinearconstraints,wewilldemonstratehowalinearrelaxationofthisproblemcanbedeveloped. First,consideradeterministicformulationoftheWTAproblem.LetNdenotethenumberoftargetstobedestroyed,andMbethetotalnumberofweaponsavailable.Assumethattheweapons(aircraft,missiles,etc.)areidenticalintheircapabilitiesofdestroyingtargets.Leteachtargethaveavalue(priority)Vj,j=1;:::;N.DenetheprobabilitypjofdestroyingthetargetjbyasingleweaponasBernoullitrialwithindependentoutcomes:P[targetjisdestroyedbyasingleweapon]=pjP[targetjisnotdestroyedbyasingleweapon]=qj=1pj 6.1a )representstheweightedcumulativeprobabilityofsurvivalofthesetoftargets.Constraint( 6.1b )isthemunitionscapacityconstraint,wheretheequalitysignmeansthatallmunitionshavetobeutilizedduringthemission. denBroegeretal. ( 1959 )showedthatthisproblemcouldbesolvedusingagreedyalgorithminO(N+MlogN)time.However,wewishtoextendthemodelandunfortunately,thisstrategywillnolongerhold.Hence,analternativestrategyforquicksolvingofthisproblemisrequired. Theoptimizationproblem( 6.1a )hasaspecialstructure:theobjectiveof( 6.1a )isalinearcombinationofunivariatenonlinearfunctions,wherethej-thfunctionhasargumentxj.Takingthisintoaccount,wereplaceeverynonlinearsummandqxjjintheobjectiveofproblem( 6.1a )bya

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piecewiselinearfunctionjj(xj)suchthat8xj2Z+:jj(xj)=qxjj;j=1;:::;N; 6.1a )and( 6.2 )havethesameoptimalsolutions. Nowconsideralinearrelaxationof( 6.2a )obtainedbyrelaxingtheintegralityofthedecisionvariablesxjandrepresentingthepiecewiseconvexfunctionsjj(xj)bythemaximumofMlinearfunctionsjj(xj)=maxxjlj;0(xj);:::;lj;M1(xj); wherezjareauxiliaryvariables. 6.3 )hasanoptimalsolution,whichisintegerinvariablesx1;:::;xN.

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Therefore,toshowthatproblem( 6.3a )hasanoptimalsolutionwithinteger-valuedvariablesx1;:::;xN,wehavetodemonstratethatallverticesofthefeasibleregion( 6.3b )( 6.3d )haveintegercoordinatesxi,i=1;:::;N. Withoutlossofgenerality,weassumethatthefeasiblesetof( 6.3a )isbounded(wemayimposeconstraintsziCforsomelargeC>0withoutaffectingthesetofoptimalsolutions).Then,considerafeasiblepointy=(x1;:::;xN;z1;:::;zN)2R2NthathasKnon-integercompo-nentsxi1;:::;xiK,where2KN:Thispointmaysatisfyatmost2NK+1differentequalitiesthatdenetheboundaryofthefeasibleset( 6.3b )( 6.3d ).Indeed,eachofNKinteger-valuedcomponentsxj0=m02f1;:::;M1gandthecorrespondingzj0maysatisfy2equalities( 6.3b )zj0=qm0j0[(1qj0)(m0xj0)+1]andzj0=qm01j0[(1qj0)(m01xj0)+1]; 6.3d )and1equalityzj0=1fromset( 6.3b );thecasem0=Mistreatedsimilarly).Eachnon-integerxikandthecorrespondingzikmaysatisfyatmost1equality( 6.3b )zik=qmik[(1qik)(mxik)+1]: 6.3c )Nj=1xj=M: 6.3b )( 6.3d ),andthereforecannotbeanextremepointofthefeasibleregion.

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6.1 )coincideswiththesetofintegeroptimalvaluesofvariablesx1;:::;xNinproblem( 6.3 ).Optimalvaluesofobjectivefunctionsof( 6.1 )and( 6.3 )coincideaswell. (i) ThesetSZN+offeasiblevaluesx1;:::;xNofproblem( 6.1a )isasubsetofthefeasibleregionof( 6.3a ). (ii) Byconstructionofproblem( 6.3a ),objectivefunctions( 6.1a )and( 6.3a )takeidenticalvaluesonS. (iii) Proposition 6.2.1 impliesthatobjectivefunctionof( 6.3a )achievesglobalminimumonS. From(i)(iii)followsthestatementoftheProposition 6.2.2 6.1 )or( 6.3 )arenotknownwithcertainty.Inthissection,weconsidertheuncertainparameteristhenumberoftargetstobedestroyed. Withoutlostofgenerality,assumethatthereareKcategoriesoftargets.Thetargetsarecat-egorizedbytheirsurvivabilityandimportance,sothatallthetargetswithincategorykhavethesameprobabilityofsurvivalqkandpriorityVk.Assumethattherearenkdetectedtargetsandxkundetectedtargetsineachcategoryk=1;:::;K,wherefxkjk=1;:::;Kgarerandomnumbers.Theundetectedtargetsareexpectedtoappearatsometimeinthefuture.Thus,wehavetwoclearlyidentiedstagesinourproblem:intherststageonehastodestroythealreadydetectedtargets,andinthesecondstageonemustdestroythetargetsthatmightbedetectedbeyondthecurrenttimehorizon,butbeforesomeendtimeT.Consequently,onehastomakeanassignmentofweaponsintherststagethatallowsenoughremainingweaponstoattackthepossiblesecond-stagetargets.Thistypeofproblemiswellknownastwo-stagerecourseproblem. Accordingtothestochasticprogrammingapproach,theuncertainnumberoftargetsatthesecondstageismodeledbythesetofscenariosf(xs1;:::;xsK)js=1;:::;Sg,wherexskisthenumberofthesecond-stagetargetsincategorykunderscenarios.Letxkibeequaltothenumberofweaponsassignedtoarst-stagetargetiincategoryk,andykibethenumberofweaponsassignedtoa

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second-stagetargetiincategoryk,thentherecourseformofthetwo-stageStochasticWTA(SWTA)problemis 6.4b )protectsagainstweapondepletionattherststage,whereasequality( 6.5b )ensuresfullweaponutilizationatthesecondstage. Thetwo-stageSWTAproblem( 6.4 )( 6.5 )canbelinearizedinthesamewayasdescribedintheprecedingsection.Afterthelinearization,theextensiveformofthetwo-stageSWTAproblemreadsasmin(Kk=1nki=1Vkzki+1

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6.6 )ofthetwo-stageSWTAproblemhasanoptimalsolu-tion,whichisintegerinvariablesxkiandyski. 6.2.1 6.6 )wasfoundtounreasonablyfavorassignmentstotargetswithlargenumbersinacategory.Analternativeobjective,whichscalesassignmentsbynk(xskforthesecondstage)tendstoprovidemorerealisticsolution: Insomesituations,however,thenumberofsecond-stagetargetsmaystillremainuncertainuntilcompletionofthesecondstage.Asanexample,consideracombatreconnaissancemissionwhereacombatunit(e.g.,aUAV)hasrsttoliquidateallknown(previouslydetected,orrst-stage)targets,andthenperformareasearchinordertondanddestroyalltargetsthathavenotbeendetectedyetorsurvivedtherst-stageattack(thesecond-stagetargets).Supposethatatanymomentofthissearchthereisanon-zeroprobabilityofdetectinganewtarget,hencethetotalnumberofsecond-stagetargetsremainsunknownuntilthemissionisnished.Therefore,ratherthanassumingacertainnumberofsecond-stagetargets,itismoreappropriatetodealwithaprobabilitydistributionforthenumberoftargets.Thisdistributionmaydependonthebattlesituation,weatherconditionsetc.andconsequentlymaynotbeknowninadvance(beforethebeginningofthemission).However,weassumethatuponcompletionoftherststageofthemission,thebattleunitisabletodeterminethetruedistributionofthesecond-stagetargets(forexample,byanalyzingthevolumeofjamming,etc.)

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Inaccordancetothedescribedsetupweproposeatwo-stagestochasticWTAproblem,whereasecond-stagescenariosspeciesnotthenumberoftargetsinacategory,butaprobabilitydistri-butionofthenumberofsecond-stagetargets.Therst-andsecond-stagedecisionvariablesxkiandyskdeterminethenumberofshotstoberedatatargetincategorykunderscenarios.Notethatasthenumberofthesecond-stagetargetsincategorykunderscenariosisunknown,variablesyskdonotcontainsubscripti.Thus,thesecond-stagedecisionprescribesthenumberofweaponstobeusedforeachtargetdetectedincategoryk,giventherealizationofscenarios. Considerasetofscenarioss=1;:::;SthatspeciesthefamilyofdistributionsQskforrandomvariablesxskrepresentingthenumberofthesecond-stagetargetsincategories1;:::;K:PQsk[xsk=i]=qski;iqski=1: Havinganuncertainnumberoftargetsatthesecondstage,wehavetotakeintoaccounttheriskofmunitionsdepletion,and,consequently,failuretodestroyallthedetectedtargets.Onewaytohedgeagainstshortageofmunitionsistoperformaworst-caseanalysis,e.g.,torequirethat However,constraintoftype( 6.8 )maybetooconservativeandrestricting,especiallywhenImaxisalargenumberandtheprobabilityP[xsk=Imax]isrelativelysmall.Indeed,theeventofencounteringthelargestpossiblenumberoftargetsineverycategoryatthesecondstageshouldhaveverylowprobability. ReplacingImaxin( 6.8 )withtheexpectednumberofthesecond-stagetargetsE[xsk]maybealsoinappropriate,especiallyfordistributionsQskwithheavytails. Tocircumventthepossibilityofrunningoutofammoatthesecondstage,weproposetouseamunitionsconstraintwheretheaveragemunitionsutilizationin,say,10%ofworstcases(i.e.,whentoomanysecond-stagetargetsaredetected)doesnotexceedthemunitionslimitM.Thistype

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ofconstraintcanbeformulatedusingtheConditionalValue-at-Riskoperator: whereaisthecondencelevel.Inequality( 6.9 )constrainsthe(weighted)averageofmunitionsutilizationin(1a)100%ofworstcases. TocalculatetheConditionalValue-at-Riskofthefunctionf(xxxs;ys)=Kk=1nki=1xk+xskysk!; Notethatscenariosetf(x1j;x2j;:::;xKj)jj=1;:::;Jgisthesameforallscenarios1;:::;S.Ascenariosassignsprobabilitypsjtoeachvector(x1j;x2j;:::;xKj)fromthiscollection.Naturally,Jj=1psj=1;s=1;:::;S: 6.10 ).Expression( 6.10 )forprobabilitiespsjmaystillbeusedifweassumethatscenariosdenesajointprobabilitydistributionfornumberoftargetsoverallcategories.

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Thus,thetwo-stagestochasticWTAproblemwithuncertaintiesindistributionsreadsasmin(Kk=1nki=1Vk(qk)xki+EQEx[Q(x;xxx)]) 6.11 )( 6.12 )isasfollows: Inaccordancetothegeneralprincipleofdiversicationofrisk,theCVaRconstraint( 6.9 ),repre-sentedbyinequalities( 6.13e )( 6.13f ),doesnotallowforaninteger-valuedoptimalsolutionforthe

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second-stagedecisionvariables,whichinturnprecludesintegralityoftherst-stagedecision.How-ever,itispossibletoachieveaninteger-valuedsolutionattherststagebyintroducinganintegervariableM1inconstraints( 6.13c ),( 6.13e ),or,equivalently,solvingM+1problems( 6.13a ),whereM12f0;:::;Mgisaparameter. Iftheprimaryinterestistherst-stageweapon-targetassignment,andtheintegralityofthesecond-stagedecisionisnotcritical,thelinearprogrammingformulation( 6.13a )yieldsafastal-gorithmforthetwo-stagestochasticWTAproblemswithuncertainties.Thecorrespondingoptimalsolutionmayberegardedastheonethatallowsfordestructionofthedetectedtargetswhilepre-servingsufcientresourcesfordestroyingthepossiblefuturetargets. Integer-valuedoptimalsolutionof( 6.11 )( 6.12 )isachievedbydeclaringthevariablesyskasinteger.Inthiscase,solutionoftheLPproblem( 6.13a )representsalowerboundfortheoptimalsolutionoftheMIPproblem( 6.11 )( 6.12 ). 6.6 )andthetwo-stagestochasticWTAproblemwithuncertaintiesindistributions( 6.13a ).However,reportinganddiscussingtheoptimalsolutionsoftheconsideredproblemswouldbearatherunwieldytaskeveninsmallinstances,sinceformulations( 6.6 )and( 6.13a )containindividualdecisionvariablesforeachrst-stageand,incase( 6.6 ),second-stagetarget.Therefore,tomakethepresentationcompact,weconsiderandreportsolutionsofthefollowingMIPanalogsofproblems( 6.6 )and( 6.13a )correspondingly:

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and Intheformulation( 6.14 )and( 6.15 )variablesxkandyskrepresentthenumberofweaponsusedforeveryrst-stageandsecond-stagetargetinacategoryk,respectively.Infact,thelinearprogrammingformulations( 6.6 )and( 6.13 )arethelinearizationsoftheMIPformulations( 6.14 )and( 6.15 )correspondingly.SolutionoftheLPproblem( 6.13 )wasusedasthelowerboundfortheMIPproblem( 6.15 ). ThedevelopedmodelsforthestochasticWeapon-TargetAssignmentproblemswereimple-mentedinC++andweusedCPLEX7.0solvertondsolutionstothecorrespondingMIPandLPproblems.Thesetupofthetestproblemwasasfollows: 6 6 wereusedasthescenarioinformationfortheclassicaltwo-stagestochasticWTAproblem( 6.14 )and( 6.6 ).

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Table6: Theexpectedvaluesforthenumberofthesecond-stagetargetsintwocategoriesforscenarioss=1;2;3. CategoryII134 6 comparestheoptimalsolutionsoftwo-stagestochasticWTAproblemindifferentformulations.Thevaluesincolumnscorrespondingtoa=0:01;0:50;0:99presenttheoptimalsolutionofthestochasticWTAproblem( 6.15 )withtheindicatedcondencelevelaintheCVaRconstraint.Columnwitha=0:00displaystheoptimalsolutionoftheclassicaltwo-stageSWTAproblem( 6.14 ),wherethescenariovaluesforthenumberthesecond-stagetargetsweretakenfromTable 6 6.15 ),wheretheCVaRconstraintisreplacedbytheworst-caseconstraint( 6.5a ). Table6: SolutionoftheMIPproblemandproblem( 6.15a )fordifferentvaluesofcondencelevela.Thecasea=1:00correspondstothecasewhenCVaRconstraintsin( 6.15a )isreplacedwith( 6.8 ).Columna=0:00presentsthesolutionofproblem( 6.6 ). IIIIIIIIIIIIIII Firststage 2424262424 Secondstages=23323221212 6 showsthattheclassictwo-stagestochasticWTAproblemthatusesonlytheexpectedvaluesofthesecond-stagetargetdistributionsyieldsthesolution,whichisquantitativelyverysim-ilartothesolutionofthetwo-stagestochasticWTAproblemwithuncertaintiesindistributions( 6.15 )forlowcondencelevelainCVaRconstraint.Thesesolutionsallocateamplemunitionsfordestructionofeachoftheencounteredsecond-stagetargets,whichmayleadtoseveralmu-nitionsshortageatworst-casescenarios.Ascondencelevelaincreases,and,correspondingly,worst-casescenariosgainmoreweightintheoptimizationproblem( 6.15 ),thenumberofweaponstobereservedforeachofthesecond-stagetargetsdecreases(suchsolutionsaremorerobustwith

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Figure6: MIPcurverepresenttheoptimalobjectivevaluesoftheMIPproblem( 6.15a );Relax-LPcurvecorrespondtothesolutionof( 6.13 )withallcontinuousvariables;Relax-MIPpointsshowthesolutionoftheproblem( 6.13 )wherethesecond-stagedecisionvariablesyskareinteger. respecttoencounteringmore-than-expectedsecond-stagetargets).Finally,athighcondencelev-els(a=0:99),problem( 6.15 )producesveryconservativesolutionthatcoincideswiththesolutionobtainedbyreplacingtheCVaRconstraintbytheworst-caseconstraint( 6.8 ). Figure 6 displaysthedegradationoftheoptimalobjectivevalueofthestochasticWTAproblemwithincreasingofthecondencelevela.Onthisgure,theMIPcurvecorrespondstotheoptimalobjectivevalueofproblem( 6.15 ),Relax-LPcurveshowstheoptimalobjectiveoftheLPproblem( 6.13 ),andRelax-MIPcurvedisplaystheobjectiveoftheproblem( 6.13 )withintegervariablesysk.Evidently,solutionoftheLPproblem( 6.13 )presentsagoodlowerboundforboththesolutionsofproblem( 6.15 )andMIPproblem( 6.14 ).

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unknown.Tocontroltherisksofgeneratinganincorrectdecisionduetotheuncertaintiesindis-tributions,wewiththeuncertaintiesindistributions,weappliedtherisk-managementtechniquesbasedontheConditionalValue-at-Risk(CVaR)riskmeasure.Forthetwo-stageSWTAproblemwithuncertaintiesindistributionswedevelopedanLPformulationthatyieldstightlowerboundforthecorrespondingMIPproblem,andhasintegeroptimalsolutionfortherst-stagevariables.

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Inourdissertation,westudiedriskmanagementtechniquesincontextofstochasticprogram-mingwithapplicationsindifferenteldsofdecisionmaking.Itwasdemonstratedthatriskman-agementtoolsandproceduresdevelopedinlastseveralyearshavegeneralnatureandcanbesuc-cessfullyappliednotonlytoproblemsofnancialengineering,butalsoinmilitarydecisionmakingproblems,whichinvolveastochasticenvironmentcompletelydifferentthanthatofnancialmar-kets.Wealsohavedevelopedanewsample-pathapproachtomulti-stagedecision-makingprob-lems,andtesteditwiththeoptimalpositionliquidationproblem. InChapter2,weextendedtheapproachforportfoliooptimization( RockafellarandUryasev 2000 ),whichsimultaneouslycalculatesVaRandoptimizesCVaR.Werstshowed(Theorem 2.3.1 )thatforrisk-returnoptimizationproblemswithconvexconstraints,onecanusedifferentoptimiza-tionformulations.ThisistrueinparticularfortheconsideredCVaRoptimizationproblem.Wethenshowed(Theorems 2.4.1 and 2.4.2 )thattheapproachby RockafellarandUryasev ( 2000 )canbeextendedtothereformulatedproblemswithCVaRconstraintsandtheweightedreturn-CVaRperformancefunction.TheoptimizationwithmultipleCVaRconstrainsfordifferenttimeframesandatdifferentcondencelevelsallowsforshapingdistributionsaccordingtothedecisionmaker'spreferences.WedevelopedamodelforoptimizingportfolioreturnswithCVaRconstraintsusinghistoricalscenariosandconductedacasestudyonoptimizingportfolioofS&P100stocks.Thecasestudyshowedthattheoptimizationalgorithm,whichisbasedonlinearprogrammingtechniques,isverystableandefcient,andtheapproachcanhandlelargenumberofinstrumentsandscenarios. InChapter3,wetestedtheperformanceofaportfoliorebalancingalgorithmwithdifferenttypesofriskconstraintsinanapplicationformanagingaportfolioofhedgefunds.Astheriskmeasureintheportfoliooptimizationproblem,weusedConditionalValue-at-Risk,Conditional 148

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Drawdown-at-Risk,Mean-AbsoluteDeviation,andMaximumLoss.Wecombinedtheseriskcon-straintswiththemarket-neutrality(zero-beta)constraintmakingtheoptimalportfoliouncorrelatedwiththemarket. Thenumericalexperimentsconsistedofin-sampleandout-of-sampletesting.Theout-of-samplepartofexperimentswasperformedintwosetups,whichdifferedinconstructingthescenariosetfortheoptimizationalgorithm. Theresultsobtainedaredataset-specicandwecannotmakedirectrecommendationsonport-folioallocationsbasedontheseresults.However,welearnedseverallessonsfromthiscasestudy.Imposingriskconstraintsmaysignicantlydegradein-sampleexpectedreturnswhileimprovingriskcharacteristicsoftheportfolio.In-sampleexperimentsshowedthatfortightrisktolerancelev-els,allriskconstraintsproducerelativelysimilarportfoliocongurations.Imposingriskconstraintsmayimprovetheout-of-sampleperformanceoftheportfolio-rebalancingalgorithmsinthesenseofrisk-returntradeoff.Especiallypromisingresultscanbeobtainedbycombiningseveraltypesofriskconstraints.Inparticular,wecombinedthemarket-neutrality(zero-beta)constraintwithCVaRorCDaRconstraints.Wefoundthattighteningofriskconstraintsgreatlyimprovesportfoliodynamicperformanceinout-of-sampletests,increasingtheoverallportfolioreturnanddecreasingbothlossesanddrawdowns.Inaddition,imposingthemarket-neutralityconstraintaddstothesta-bilityofportfolio'sreturn,andreducesportfoliodrawdowns.BothCDaRandCVaRriskmeasuresdemonstratedasolidperformanceinout-of-sampletests. Chapter4presentedanapproachforoptimaltransactionexecutionbasedonmathematicalprogrammingandsample-pathscenariomodel.Themaindistinguishingfeaturesofourmethodcomparedtootherapproachesareasfollows:(1)Thetradingstrategyistrulydynamic,i.e.,ital-lowsforadequateresponsetocurrentmarketconditionsateachtimestep;(2)Theapproachadmitsincorporationofvarioustypesofconstraintsinthetradingstrategy,suchaslegal,institutionalcon-straints,orthoseexpressinginvestor'spreferences.Inparticular,weconsideredrisk-aversetradingstrategies,whichcontrolriskofnanciallossesduringtransactionsusingtheConditionalValue-at-Riskmeasure;(3)Thesuggestedapproachcanaccommodatedifferentmodelsoftemporaryandpermanentmarketimpact,transactioncosts,etc.;(4)Akeyfeatureofourapproachisthesample-pathscenariomodel,whichdoesnotimposeanyrestrictionsonthepriceprocessofthesecurity.Historicaldataorsimulationcanbeusedtocreateascenariosetforourmethod.

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Toavoidanticipativityofthesolutions,causedbyspecicpropertiesofthesample-pathsce-nariomodel,weintroducedpathpartitioningwithanewlawn-mowerdecisionrule.Sincethelawn-mowerprincipleleadstonon-convexityoftheoptimalclosingproblems,wedevelopedaconvexrelaxationfortheseproblems,whichwasfurtherapproximatedbylinearprogramming.Themodelofoptimalpositionliquidation,implementedasalinearprogrammingproblem,canbesolvedveryefcientlyandrobustlyinverylarge-scaleinstanceswithlargenumberofscenarios(sample-paths).Propertiesoftheoptimaltradingstrategiesbasedonthelawn-mowerprinciplehavebeeninvestigated. Inafrictionlessmarket,theoptimaltradingstrategybasedonthelawn-mowerconstraint,has0structure,i.e.,itliquidatestheentirepositioninonetransaction.IftheriskofnanciallossesduringtradesiscontrolledbyCVaRconstraints,thetradingstrategybecomesfractional.Whenstrongtemporaryorpermanentmarketimpactsarepresent,theoptimaltradingstrategyapproachestheso-callednaivestrategy,whichconsistsinsellingequalportionsofthepositionateachtimestep.Underweakermarketimpacts,theoptimaltradingstrategystartstodeviatefromthenaivestrategy.Accordingtoournumericalexperiments,themostsignicantdeviationsfromthenaivetradingstrategyareobservedinthecasewhentemporarymarketimpactdependsonthecurrentmarketconditions,orwhenpermanentmarketimpacthasalag(i.e.,whensmallenoughtradesdonotcausepermanentchangesinpriceofthesecurity). Inthesecondpartofthedissertation,wedevelopedanapproachfordecisionmakingformili-taryapplicationswithuncertaintiesindistributionsinstochasticparameters.ThisapproachisbasedonthemethodologyofriskmanagementwithConditionalValue-at-Riskriskmeasuredevelopedby RockafellarandUryasev ( 2002 ).Althoughthepresentedapproachhasbeenusedtosolveone-stageandtwo-stagestochasticWeapon-TargetAssignmentproblems,itisquitegeneralandcanbeappliedtowideclassofproblemswithrisksanduncertaintiesindistributions.InChapters5and6wehaveconsideredseveralformulationsforthetwo-stagestochasticWeapon-TargetAssignmentproblem,wherethecumulativedamagetothetargetsismaximized.Intheoriginalformulation,theWTAproblemispresentedasanintegerprogrammingproblemwithanonlinearobjective.Wehavedevelopedalinearrelaxationforboththedeterministicandtraditionaltwo-stagestochasticformu-lationsoftheWTAproblem.Inthescopeofthetwo-stagestochasticWTAproblemweconsideredthesetupwhereprobabilitydistributionsforthenumberofthesecond-stagetargetsareunknown.To

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controltherisksofgeneratinganincorrectdecisionduetotheuncertaintiesindistributions,wewiththeuncertaintiesindistributions,weappliedtherisk-managementtechniquesbasedontheCondi-tionalValue-at-Risk(CVaR)riskmeasure.Forthetwo-stageSWTAproblemwithuncertaintiesindistributionswedevelopedanLPformulationthatyieldstightlowerboundforthecorrespondingMIPproblem,andhasintegeroptimalsolutionfortherst-stagevariables. IntheAppendix,wepresentedtheresultsofourongoingresearcheffortsintheareaofpricingofpath-dependentderivativesecuritiesusingtechniquesofmathematicalprogramming.

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Atthesametime,thewriterisobligatedtoselltheassettotheholderofoptionforthepriceXiftheholderchoosestobuytheunderlyingasset(i.e.,exercisethecontract.)Inthiscase,moneychangehandsattheexpirationdateT. ThecontractwhichgivestheholdertherighttoselltheunderlyingassettowriteratspecieddateTforpredeterminedpriceX,iscalledEuropeanputoption. Generally,derivativecontractsthatallowexerciseonlyatsomeprescribeddateTinthefuture,aresaidtobeEuropean-stylederivativesecurities.AnAmerican-stylederivativesecurity,onthecontrary,allowstheholderofcontracttoexerciseitonorpriortotheexpirationdateT. Forexample,anAmericancalloptiongivesitsholderaright,butnotanobligation,tobuytheunderlyingassetatthespeciedpriceXatanytimetT.Thecounter-party(thewriter)isobligatedtoselltheassetwhenevertheoptionholderchoosestoexercisethecontract.Analogously,anAmericanputoptiongivestheholdertherighttoselltheunderlyingassetatstrikepriceTnotonlyontheexpirydate,butalsobeforethatdate. ThedescribedaboveEuropeanandAmericanoptionsareoftencalledplainvanillaoptions,asopposedtoexoticoptions:Asianoptions,Russianoptions,barrieroptions,digitaloptionsetc.Formoredetaileddiscussionofdifferenttypesofoptionstradedinthemarket,see,forexample, CoxandRubinstein ( 1985 ), Wilmottetal. ( 1998 ),andothers. 152

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BlackandScholes ( 1973 ).Intheirseminalpapertheauthorshavesuggestedaverygeneralframeworkthatwasproventobeeffectiveinvaluatingvirtuallyanytypeofoptioncontract. Here,wepresentasketchBlack-ScholestheoryinitspartthatpertainstoderivingvalueofAmerican-styleoptions.ThepriceSoftheunderlyingassetisassumedtosatisfyastochasticdifferentialequation S=mdt+sdW;(A.1) wheremistheexpectedvalueoftherateoftheinstrument,sisthestandarddeviationoftherateofreturn,anddWisaWienerprocesswithmeanzeroandvariancedt:dWN(0;p BlackandScholes 1973 )isapplicablewhen,forexample,theparametersin( A.1 )aretime-dependent,orthepriceprocessS(t)followsotherstochasticmodels,e.g.theconstantelasticityofvariance(CEV)model,etc.(fordetails,see,forinstance, Wilmottetal. ( 1998 )). In BlackandScholes ( 1973 )itisshownthatinaperfectmarketeconomy,inabsenceofar-bitrageandtransactioncosts,anoptioncontractcanbereplicatedbycontinuouslytradinginthemarketaportfolioconsistingoftheunderlyingassetandabondwithrisk-freerater.Then,thepriceofbothEuropeanandAmericanoptionshavetosatisfythesamepartialdifferentialequation whereweusedPforthepriceofaputoption(acalloptionsatisesthesameequation).Themostremarkablefeatureoftheequation( A.2 )isthatthepriceofanoptiondoesnotdependonthedriftm.Thisconstitutesafundamentalinsightthatthoughinvestorsmaynotagreeontheexpectedrateofgrowthoftheasset'svalue,itwouldnotaffectthemarketpriceofanoptionderivativecontractonthatasset.

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Thedifferencebetweendifferentstylesandtypesofoptionsmanifestsitselfinboundarycon-ditionstobesatisedbythesolutionof( A.2 ).Inparticular,foraEuropeanputoptiontheboundaryconditionstobesatisedbytheoptionpriceP,are HereXisthestrikepriceandTistheexpirydateofthecontract.Theboundary-valueproblem( A.2 )( A.3 )hasanexplicitexactsolution.Indeed,bychangeofvariables 2x(1+g 2)2tu(x;t);g=r s2=2;(A.4) theboundary-valueproblem( A.2 )( A.3 )reducesto 2(g1)xe1 2(g+1)xo(A.6) Thesolutionoftheaboveboundary-valueproblemisasfollows whered1=ln(S=X)+r+1 2s2(Tt) 2s2(Tt) 2x2dx: Free-boundaryandcomplementarityproblemsforAmericanput.

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option.Clearly,sincethesetofexercisemomentsofanAmericanoptionincludesthatofaEuro-peanoption,thepriceofanAmericanoptionmustbegreaterorequaltothepriceofaEuropeanoption.Moreprecisely,itcanbedemonstratedthatanearlyexerciseisneveroptimalforanAmeri-cancalloptiononanon-dividendpayingstock(see,forexample, CoxandRubinstein ( 1985 )),andthereforethepricesofAmericanandEuropeanonastockthatdoesnotpaydividendsareequal.ForAmericanputoptions,however,thisisnotthecase. Moreover,uptothisdateanexactexplicitsolutionforAmericanputoptiononanon-dividendpayingstockisunknown.Therefore,moreattentionispaidtonumericalandapproximatemeth-odsofAmericanputpricing.Forexample, GeskeandJohnson ( 1984 )developedananalyticalexpressionforAmericanputoptionwithdiscreteexercisedates(theso-calledbermudanoption).Inthischapterwepresentdifferentapproachesthatuseoptimization,and,inparticular,linearpro-grammingtovalueanAmericanputoptiononannon-dividendpayingstock.First,wepresentatechniquethatdeliversanumericalsolutiontotheboundary-valueproblemcorrespondingtotheAmericanputoption. First,weassumetheexistenceoftheoptimalexerciseregionSsuchthatifS2Sforagivent,itisoptimaltoexercisetheoption,and,conversely,itisoptimaltokeepoptionwhenS=2S.Indeed,fromtheelementarypropertiesofAmericanoptionsitfollows(see CoxandRubinstein ( 1985 ))thatthereexistsanoptimalexerciseboundaryS(t)suchthatassoonasStS(t),thentheoptionmustbeexercised. Itturnsoutthattheboundary-valueproblemforAmericanputoptionbelongstotheclasssoffree-boundaryproblems,whereboundaryconditionsforapartialdifferentialequationareformu-latedonaboundary,partofwhichisnotknowninadvance. WenowdescribebrieyhowtheproblemofAmericanputoptioncanbereducedtoafree-boundaryproblemfortheBlack-Scholesequation( A.2 ).Fordetails,see,forexample, Wilmottetal. ( 1998 ).First,fromtheabsenceofarbitrageitfollowsthatthevalueofAmericanputisalwaysgreaterthanthepayoff: Otherwise,onecouldmakeaninniteprotbybuyingoptionsonthespotandimmediatelyexercis-ingthem.Inthesameway,thearbitrageargumentcanbeusedtoshowthatintheregionofoptimal

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exercise0St
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where 2)2tmaxn0;eg1 2xeg+1 2xo:(A.10d) Theboundaryconditionsatinnityforfunctionureadas Also,thecondition( A.8 )reducesto Finally,theproblemforAmericanputoptioncanbereducedtolinearcomplementarityproblemofthefollowingform: subjecttoinitialandboundaryconditionsujt=0=h(x;0); A.11 ).First,wereducetherangeofchangeofvariablexfrom
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byforwardandbackwardnitedifferences:utxut1x where0l1andd=Dt (Dx)2: 2,werepresenttherstinequalityof( A.11b )bythesystemoflinearalgebraicinequal-itiesd 2ukn+1+(1+d)uknd 2ukn1d 2uk1n+1+(1d)uk1n+d 2uk1n1;k=1;:::;K;n=N;:::;N; whereukn=u(xn;tk).Inthiswaythelinearcomplementarityproblem( A.11 )reducestoconsecutivesolvingof whereuk=ukN;ukN+1;:::;ukN1;ukNT,andM=0BBBBBBBBBB@1+dd 20:::0d 21+dd 2:::00d 21+d:::0...00:::d 21+d1CCCCCCCCCCA

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vectorpkhasthecomponentspkN=d 2hkN1+hk1N1+d 2uk1N+1+(1d)uk1Npkn=d 2uk1n+1+(1d)uk1n+d 2uk1n1;n=N+1;:::;N1;pkN=(1d)uk1N+d 2uk1N1; A.15 )canbereducedtothecanonicalformoflinearcomplementarityproblems:(Mw+q)w=0;Mw+q0;w0; wherew=ukhk;q=Mhkpk: A.2 ),weusetheapproachdevelopedby Mangasarian ( 1976 1978 ).HehasshownthatifthematrixMsatisescertainconditions,thesolutiontoalinearcomplementarityproblemoftheform( A.2 )canbeobtainedbysolvingthelinearprogrammincTws.t.Mw+q0; A.15 )and( A.2 )satisestoconditionMii>j6=ijMijj;8i; Mangasarian 1978 ),thevectorccanbeselectedasc=MTe,whereeisavectorof1's.Thus,wehavereducedtheproblemofpricingAmericanputoptiontoconsecutivesolving

PAGE 171

ofseriesofLPsmin(MTe)Tws.t.Mw+q0; A .Thefollowingparameterswereused:X=10:0,r=0:10,s=0:4,T=0:5(i.e.,anAmericanputwithstrikepriceof$10andexpirationinsixmonths).TocalculatethepriceoftheputoptionfordifferentvaluesoftheinitialstockpriceS0usingtheproceduredescribedabove,weperformeddiscretizationwithN=1000,K=10,andxmin=4andxmax=0:4771,whichcorrespondstoSmin=0:001,Smax=30:00. TableA: PricingofAmericanputusinglinearprogramming 160.04580.0460-0.60 140.13160.1320-0.27 120.36200.3622-0.06 100.90450.9211-1.84 82.09452.0951-0.03 6440 4660 2880 A .ThecolumnS0containsthevaluesoftheinitialstockpriceatt=0;valuesinPcolumnrepresenttheputpricesobtainedusingthedescribedprocedure;PexactistheexactpriceofAmericanputwithdiscreteexercisedatesobtainedbythetechniqueof GeskeandJohnson ( 1984 ). ThetabledemonstratesthatthepresentedprocedureofAmericanputvaluationusingLPallowstoobtainveryquicklyawholerangeofoptionpricescorrespondingtodifferentinitialpricesoftheunderlyingassetwithagoodaccuracy.TheloosenedaccuracyofourmethodinthecasewhentheinitialstockpriceS0isequaltotheexercisepriceXcanbeexplainedasfollows.Observethat

PAGE 172

transformingtheproblem( A.15 )to( A.2 )is,infact,equivalenttosetting whereg(x;t)= t2 DempsterandHutton ( 1999 )containsideassimilartothosepresentedabove.How-ever,thepresentedapproachwasdevelopedbytheauthorindependentlyoftheindicatedwork. A.1 ),etc. Formthisviewpoint,pricingmethodsbasedonsimulationtechniques,areofspecialimpor-tance.OneoftherstattemptsofderivingthepriceforAmericanputoptionsusingsimulationsbelongsto Tilley ( 1993 ).Later,simulationapproachestopricingofderivativeswereexploredinthepapersby BarraquandandMartineau ( 1995 ), Carriere ( 1996 ), Boyleetal. ( 1997 ), BroadieandGlasserman ( 1997 ),andothers.

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Theapproachpresentedbelowconstitutesasubjectoftheauthor'songoingresearchefforts.Wedevelopageneraltechniquethatusessimulationandstochasticprogrammingtopricepath-dependentderivativesecurities.Asavalidationoftheapproach,weconsiderpricingofAmericanputoption. Generally,theapproachtobepresentedisbasedonthesample-pathframeworkpresentedintheChapter3.Recallthatwehavementionedthatproblemofoptimalpositionliquidationiscloselyrelatedtopricingofderivativesecurities.Inthissectionwemakesomefurtherremarksalongthisanalogy. Weusetheabovestochasticdifferentialequationfordiscretesamplingofsamplepath(Dt=1=T):Sjt=S0exp(rs2 Obtainedinthiswaysamplepaths(S0;Sj1;:::;SjT)representpossiblerealizationsofstochasticpro-cessStgovernedby( A.20 ). CoxandRubinstein 1985 ).FromthisfollowsthemonotonicitypropertyoftheoptimalexerciseregionforAmericanput:8tTthereexistssuchStXsuchthatitisoptimaltoexercisetheoptionassoonasStSt,anditisoptimaltokeeptheoptionopenotherwise.

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Utilizingthisproperty,wearriveatthefollowinglinearprogrammingproblemforAmericanputpricing:max1 A.22 )equalstoaveragepayoffgeneratedalongthesamplepathsbyexer-cisingportionsoftheoption(yjtisaportionofoptiontobeexercisedattimetonasamplepathj.)Therstandthirdconstraintsof( A.22 )statethattheoptionhastobeexercisedonorbeforetimeT.Thoughthatmayseemtoconictwiththeverydenitionoftheoptioncontractbyeliminatingtheoptionalityinthedecisionsoftheoptionholder,itdoesnotimpacttheobjectiveof( A.22 ),becauseexercisingforSt>Xdoesnotgeneratecashow.Therefore,allexercisesimposedbytheseconstraints,whichhappenintheregionofSt>X,canbeignored. Thesecondconstraintimposesthemonotonicitystructureontheoptimalexerciseregion,i.e., In( A.22 ),jtisapermutationjt:f1;:::;Jg7!f1;:::;JgsuchthatSjt(1)tSjt(2)t:::Sjt(J)t: A.22 )hasa0-1optimalsolution. 6.2.1 A.3.1 hastwo-foldconsequences.First,itdemonstratesthattheoptimalso-lutionof( A.22 )doesindeedrepresentabinarydecision(exercisenotexercise)oftheoption-holder.Also,itfollowsfromtheproposition A.3.1 thatafastpolynomialalgorithmcanbedevel-opedforpricingoftheAmericanput.

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However,theformulation( A.22 ),thoughbeingverysimpleandperformingwellinpractice,mayleadtoincorrectresultswhensomeimpropersamplepaths(i.e.,samplepathsthatdonotfollowthegeometricBrownianmotionlaw( A.20 ))areincludedinthescenarioset. Thefollowingformulationoflinearprogrammingmodeldoesxthisdeciency:max1 A.24 )ensuresthatmonotonicityoftheoptimalexerciseregionholdsattimetnotforanypathsj1,j2,butforonlythosepaths,onwhichtheoptionhasnotbeenexercised.TheconstantMin( A.24 )ischosesoastoensure0-1structureoftheoptimalsolutionof( A.24 ) A.21 )withdiscretizationT=5(5exerciseopportunitiesareallowedbeforetheexpiry). TableA: PricingofAmericanputbysimulationandstochasticprogramming 160.04970.04608.12 140.13460.13201.94 120.35840.3622-1.06 100.91450.9211-0.72 82.02072.0951-3.55 63.92264-1.93 45.91456-1.43 27.90738-1.16 A .HereS0istheinitialpriceoftheunderlyingstock,Pistheoptimalvalueoftheobjectivefunction,andPexactistheexactvalueoftheoptioncalculatedusingtheprocedureof GeskeandJohnson ( 1984 ). Theadvantagesofthesuggestedmethodcomparedtothoseknownintheliteratureareasfollows:(i)theapproachallowsonetopricethederivativesecurities,whoseunderlyingassetis

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governedbylawsotherthanthestandardgeometricBrownianmotion( A.20 );(ii)scalability:thesizeofoptimizationproblemgrowslinearlywithrespecttobothnumberofpathsandnumberofpossibleexercisedates.

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IwasborninKiev,Ukraine.Sincechildhood,Ihadaninterestinnaturalsciencesandmath-ematics.In1991,IgraduatedwithaSilverMedalfromKievhighschool#179withanadvancedprograminmathematics.ThesameyearIwasadmittedtotheDepartmentofMathematicsandMechanicsofKievNationalTarasShevchenkoUniversity,oneofthepremierschoolsoftheformerSovietUnion.DuringtheperiodofstudyinKiev,Iacquiredasolidmathematicalbackgroundandhavedevelopedatasteforboththeoreticalandappliedresearch.IhavewonawardsandscholarshipsfromtheNationalAcademyofSciencesofUkraineandInternationalSorosScience&EducationProgram(ISSEP).InJune1996,IreceivedtheM.S.degreeinmechanicsandappliedmathematicswithhighesthonors.ThetopicofmyMastersthesiswasMotionofaRigidTorusinaViscousStokesFluid.InNovember1996,IwasadmittedtothePh.D.programinMechanicsofSolidsatKievUniver-sity.Myresearchactivitieswereconcentratedondevelopinganalyticalsolutionsofboundary-valueproblemsofelasticityandhydromechanicsforbodiesofcomplexgeometry.In1997,IreceivedaDiplomaoftheNationalAcademyofSciencesofUkraineinacompetitionofyoungscientistsofUkraineforthebestresearchproject.InSeptember1999,IsuccessfullydefendedmyPh.D.dissertationonTheSecondFundamentalBoundary-ValueProblemofElasticityforaTorus.In1999IjoinedtheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridatopursueadoctoraldegreeinnancialengineering,whereIcouldapplymymathematicalskills.UnderthesupervisionofProf.UryasevIhavebeenworkingonapplicationofoptimizationtechniquestoriskmanagementintheareasofnancialandmilitarydecision-makingproblems,whichconstitutesthetopicofmydissertation. 173


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RISK MANAGEMENT TECHNIQUES FOR DECISION MAKING
IN HIGHLY UNCERTAIN ENVIRONMENTS

















By

PAVLO A. KROKHMAL

















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2003




































Copyright 2003

by

Pavlo A. Krokhmal
















ACKNOWLEDGMENTS

I thank my supervisor, Prof Stanislav Uryasev, and the members of my Ph.D. committee for

their help and guidance. I am also grateful to my wife and my parents for their support.
















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS .................................. iii

LIST OF TABLES ........ ......... .......... .......... vii

LIST OF FIGURES ........ ......... .......... .......... .. viii

ABSTRACT. .......................................... x

CHAPTER

1 IN TRODU CTION . . . . . . . . . 1

2 PORTFOLIO OPTIMIZATION WITH CONDITIONAL VALUE-AT-RISK OBJEC-
TIVE AND CONSTRAINTS ................... ......... 5

2.1 Introduction . . . . . . . . . 5
2.2 Conditional Value-at-Risk ................... ........ 9
2.3 Efficient Frontier: Different Formulations ........ ............ 12
2.4 Equivalent Formulations with Auxiliary Variables ...... ........ 15
2.5 Discretization and Linearization ................... ..... 18
2.6 One Period Portfolio Optimization Model with Transaction Costs ........ 19
2.7 Case Study: Portfolio of S&P100 Stocks ........ ......... ... 23
2.7.1 Efficient Frontier and Portfolio Configuration ............. ..24
2.7.2 Comparison with Mean-Variance Portfolio Optimization . ... 25
2.8 Concluding Remarks .... .. ......... ........... .. .. 30

3 COMPARATIVE ANALYSIS OF LINEAR PORTFOLIO REBALANCING STRATE-
GIES: AN APPLICATION TO HEDGE FUNDS .................. 32

3.1 Introduction ................... . . .... 32
3.2 Linear Portfolio Rebalancing Algorithms ................ 35
3.2.1 Conditional Value-at-Risk .................. .... .. 37
3.2.2 Conditional Drawdown-at-Risk ........... ...... 41
3.2.3 Mean-Absolute Deviation .................. ...... .. 44
3.2.4 M aximum Loss .................. ......... .. 45
3.2.5 M arket-Neutrality .................. .......... .. 45
3.2.6 Problem Formulation .................. ........ .. 46
3.2.7 Conditional Value-at-Risk Constraint .................. 47
3.2.8 Conditional Drawdown-at-Risk Constraint ............... .. 48
3.2.9 MAD Constraint .................. .......... .. 48
3.2.10 MaxLoss Constraint .................. ......... .. 48
3.3 Case Study: Portfolio of Hedge Funds .................. ..... 49
3.3.1 In-Sample Results. .................. ......... .. 50










3.3.2 Out-of-Sample Calculations .................. .. 54
3.4 Conclusions .................. ................. .. 68

4 OPTIMAL POSITION LIQUIDATION .................. .... .. 76

4.1 Introduction ............... . . ... 76
4.2 General Definitions and Problem Statement .................. 79
4.2.1 Representation of Uncertainties by a Set of Paths ........... ..79
4.2.2 A Generic Problem Formulation .................. .. 81
4.2.3 Modeling the Market Impact .................. .. .82
4.3 Optimal Position Liquidation under Temporary Market Impact . ... 85
4.3.1 Paths Grouping and "Lawn-Mower Strategy" ............. 85
4.3.2 Approximation by Convex and Linear Programming . ... 90
4.3.3 Properties of Solutions of the Non-Convex and Convex Optimal Liqui-
dation Problems ........... . . ...... 97
4.4 Optimal Liquidation with Permanent Market Impact .............. ..100
4.5 Risk Constraints. ................ .............. 101
4.6 Case study: Optimal Closing of Long Position in a Stock . ..... 104
4.6.1 Optimal Closing in Frictionless Market ................ .. 105
4.6.2 Optimal Closing under Temporary Market Impact . ..... 107
4.6.3 Optimal Closing under Permanent Market Impact . ..... 109
4.7 Conclusions .................. ................. .. 112

5 ROBUST DECISION MAKING: ADDRESSING UNCERTAINTIES IN DISTRIBU-
TIONS ................ ... .............. ... .. 114

5.1 Introduction .................. ................. .. 114
5.2 The General Approach ... . . . . 115
5.2.1 Risk Management Using Conditional Value-at-Risk . . .. 117
5.2.2 Risk Management Using CVaR in the Presence of Uncertainties in Dis-
tributions ... . . ..... .. ........... 119
5.3 Example: Stochastic Weapon-Target Assignment Problem . . .. 120
5.3.1 Deterministic WTA Problem . . . ... ..120
5.3.2 One-Stage Stochastic WTA Problem with CVaR Constraints . 122
5.3.3 Two-Stage Stochastic WTA Problem with CVaR constraints . 124
5.4 Numerical results ............ . . . . ... 127
5.4.1 Single-stage deterministic and stochastic WTA problems . ... 127
5.4.2 Two-Stage Stochastic WTA Problem .................. 129
5.5 Conclusions .................. ................. .. 131

6 USE OF CONDITIONAL VALUE-AT-RISK IN STOCHASTIC PROGRAMS WITH
POORLY DEFINED DISTRIBUTIONS .................. ...... 132

6.1 Introduction ...... ...... ....... ... ........ 132
6.2 Deterministic Weapon-Target Assignment Problem .............. ..133
6.3 Two-Stage Stochastic WTA Problem . . . ..... 137
6.4 Two-Stage WTA Problem with Uncertainties in Specified Distributions . 139
6.5 Case Study .................. ................. .. 143
6.6 Conclusions .................. ................. .. 146










7 CONCLUSIONS ................... .............. 148

A PPEN D IX . . . . . . . . . . 152

A ON THE PRICING OF AMERICAN-STYLE DERIVATIVE SECURITIES USING
LINEAR PROGRAMMING ........ ..................... 152

A. 1 Black-Scholes Approach to Pricing of Options .......... ........ 152
A.2 Solution of the Free-Boundary Problem for American Put Using Linear Pro-
gramming. .............. ... ..... ............. 154
A.3 Pricing of Path-Dependent Options Using Simulation and Stochastic Programming 161

REFEREN CES ........ .......... .......... ... .... . .. 166

BIOGRAPHICAL SKETCH .................. ............... .. 173
















LIST OF TABLES
Table page

2-1 Portfolio configuration: assets' weights (%) in the optimal portfolio depending on
the risk level .............. ................ .. 26

3-1 Instrument weights in the optimal portfolio with different risk constraints . 53

3-2 Weights of residual instruments in the optimal portfolio with different risk constraints 53

3-3 Summary statistics for the "real" out-of-sample tests ................ 70

3-4 Summary statistics for the "mixed" out-of-sample tests ............... ..73

4-1 Optimal trading strategy in frictionless market. .............. . 106

4-2 Optimal trading strategy under linear temporary market impact (/ = 2). . 108

4-3 Optimal trading strategy under linear temporary market impact that depends on price dynamics. 109

4-4 Optimal trading strategy under linear permanent market impact (/ = 2). . 110

4-5 Optimal trading strategy under piecewise-linear permanent market impact. ..... .111

4-6 Optimal trading strategy under nonlinear permanent market impact . .... 111

5-1 Optimal solution of the deterministic WTA problem (5.3a) ............. ..129

5-2 Optimal solution of the one-stage stochastic WTA problem (5.6), (5.8) ...... ..129

5-3 First-stage optimal solution of the two-stage stochastic WTA problem . ... 130

5-4 First-stage optimal solution of the two-stage stochastic WTA problem (5.11) for the
first scenario ..... ......... .... ...... ............ 131

5-5 Second-stage optimal solution of the two-stage stochastic WTA problem (5.11) for
the second scenario .. .......... ....... . . ... 131

6-1 The expected values for the number of the second-stage targets in two categories for
scenarios s = 1, 2, 3. .................. ............. .145

6-2 Solution of the MIP problem and problem (6.15a) for different values of confidence
level a . . . . . .. . .. . 145

A-1 Pricing of American put using linear programming ................. .. 160

A-2 Pricing of American put by simulation and stochastic programming . ... 164
















LIST OF FIGURES


Efficient frontier (optimization with CVaR constraints) . . .

Efficient frontier of optimal portfolio with CVaR constraints in presence of transac-
tion costs . . . . . . . . . .. .


page

25


27


2-3 Efficient frontiers of CVaR- and MV-optimal portfolios (a =
sc ale) . . . . . . .

2-4 Efficient frontiers of CVaR- and MV-optimal portfolios (a -
sc ale) . . . . . . .

2-5 Efficient frontiers of CVaR- and MV-optimal portfolios (a =
sc ale) . . . . . . .

2-6 Efficient frontiers of CVaR- and MV-optimal portfolios (a =
sc ale) . . . . . . .

3-1 Loss distribution, VaR, CVaR, and Maximum Loss. ......

3-2 Portfolio value and drawdown ................

3-3 Efficient frontiers for portfolio with various risk constraints .


S0.95, Return/CVaR


S0.99, Return/CVaR


0.95, Retum/StDev


0.99, Retum/StDev


-4 Efficient frontier for market-neutral portfolio with various risk constraints .....

-5 Historical performance and rate of return dynamics for residual assets .......

-6 Historical trajectories of optimal portfolio with CVaR constraints .. .......

-7 Historical trajectories of optimal portfolio with CDaR constraints .. .......

-8 Historical trajectories of optimal portfolio with MAD constraints. ..........

-9 Historical trajectories of optimal portfolio with MaxLoss constraints .. ......

-10 Historical trajectories of market-neutral optimal portfolio with CVaR constraints .

-11 Historical trajectories of market-neutral optimal portfolio with CDaR constraints .

-12 Historical trajectories of market-neutral optimal portfolio with MAD constraints .

-13 Historical trajectories of market-neutral optimal portfolio with MaxLoss constraints

-14 Optimal portfolios vs. benchmarks .. ......................

-15 Market-neutral optimal portfolios vs. benchmarks .. ...............

-16 Portfolio with CVaR constraints (mixed test) .. ..................


Figure

2-1

2-2










3-17 Portfolio with CDaR constraints (mixed test) .................. .. 65

3-18 Portfolio with MAD constraints (mixed test) .................. .. 65

3-19 Portfolio with MaxLoss constraints (mixed test) .................. .. 65

3-20 Market-neutral portfolio with CVaR constraints (mixed test) . ..... 66

3-21 Market-neutral portfolio with CDaR constraints (mixed test) . 66

3-22 Market-neutral portfolio with MAD constraints (mixed test) ............ ..67

3-23 Market-neutral portfolio with MaxLoss constraints (mixed test) . . 67

4-1 Sample paths and and scenario trees .............. .80

4-2 Impact function ft(.). .................. .............. 83

4-3 Path grouping .................. .................. .. 86

4-4 Tree-like grouping. .................. ............... .. 87

4-5 The "lawn-mower" principle. .................. ........ 88

4-6 Illustration to proof of Proposition 4.3.2. .................. .... 93

4-7 Illustration to proof of Proposition 4.3.5 .................. .. 96

4-8 Performance of optimal trading strategy in frictionless market. . 106

5-1 Loss function distribution and different risk measures. .. . . ...... 116

5-2 A visualization of VaR and CVaR concepts. .................. .. 117

5-3 Dependence between the cost and efficiency for different types of weapons in one-
stage SWTA problem (5.8) deterministic WTA problem (5.3a). . ... 128

5-4 Dependence between the cost and efficiency for different types of weapons in two-
stage SW TA problem (5.11). .................. ....... 130

6-1 Solution of the MIP problem .................. .......... .. 146
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

RISK MANAGEMENT TECHNIQUES FOR DECISION MAKING
IN HIGHLY UNCERTAIN ENVIRONMENTS

By

Pavlo A. Krokhmal

August 2003

Chair: Stanislav Uryasev
Major Department: Industrial and Systems Engineering

The dissertation studies modern risk management techniques for decision making in highly

uncertain environments. The traditional framework of decision making under uncertainties relies

on stochastic programming or simulation approaches to surpass simpler quasi-deterministic tech-

niques, where the uncertainty is modeled by relevant statistics of stochastic parameters. In many

applications, however, the mentioned methodologies in their conventional form fail to generate ef-

ficient and robust decisions. Mathematical models for such class of applications, therefore, are

referred to as highly uncertain environments, with the defining features such as: large number of

mutually correlated stochastic factors with dynamically changing or uncertain distributions, multi-

ple types of risk exposure, out-of-sample application of the solution, etc. Robust decision making

in such environments requires an explicit control of the risk induced by uncertainties.

The first part of the dissertation considers risk management approaches for financial applica-

tions. We present a general framework of risk/reward optimization that establishes the equivalence

of different formulations of optimization problems w ilh i risk and reward functions. As an application

of the general result, we consider optimization of portfolio of stocks with Conditional Value-at-Risk

objective and constraints, and compare this approach with the classical Markowitz Mean-Variance

methodology. An extensive study of out-of-sample performance of trading algorithms based on

different risk measures is performed on the example of managing of a hedge fund portfolio. A

chapter dedicated to multi-stage decision-making problems presents a new sample-path approach










for multi-stage stochastic programming problems and applies it to the problem of optimal trans-

action implementation. The generality of the developed model allows for using it in pricing of

complex derivative securities, such as exotic options.

The second part of the dissertation considers risk management techniques for military decision-

making problems. The main challenges of military applications attribute to various types of risk

exposure, uncertain probability measures of risk-inducing factors, and inapplicability of the "long

run" convention. Different formulations for stochastic Weapon-Target Assignment problem with

uncertainties in distributions are considered, and relaxation and linearization techniques for the

resulting nonlinear mixed-integer programming problems are suggested.
















CHAPTER 1
INTRODUCTION

The dissertation is devoted to study of risk management techniques for decision making in

highly uncertain environments. The traditional framework of decision making under the presence

of uncertainties relies on stochastic programming (Birge and Louveaux, 1997; Pr6kopa, 1995) or

simulation (Ripley, 1987) approaches to surpass simpler quasi-deterministic techniques, where the

uncertainty is modeled by relevant statistics of the stochastic parameters, such as expectation or

variance.

The study of decision-making mathematical programming models that involve uncertain pa-

rameters was originated by Dantzig and Madansky (1961), and received major development in

works of R. J.-B. Wets and colleagues (Rockafellar and Wets, 1976a,b; Walkup and Wets, 1967,

1969; Wets, 1966a,b, 1974). In contrast to the quasi-deterministic "averaging" approaches in de-

cision making under uncertainties, stochastic programming introduces the concept of stages in the

decision making, where some decisions have to be made without prior knowledge of the realization

of the random parameters in the system, and followed by corrective actions, taken after the uncer-

tainties have been realized. For a comprehensive presentation of stochastic programming theory

and applications, see Birge and Louveaux (1997) or Pr6kopa (1995).

Use of stochastic programming solutions in uncertain environments ensures increased robust-

ness and effectiveness in comparison to deterministically obtained solutions. However, there ex-

ist applications, where the stochastic programming methodology in its conventional form fails to

generate efficient and robust decisions. Mathematical models for such class of applications usu-

ally feature a large number of mutually correlated stochastic factors with dynamically changing

or uncertain distributions, multiple types of risk exposure, application of the obtained solution in

situations that cannot be described by the scenario set of the original problem (so-called out-of-

sample applications), etc. We refer to mathematical models that exhibit such or similar properties

as highly uncertain environments. Robust decision making in such environments requires, besides

maximizing the expected performance, an explicit control on the risk induced by uncertainties.










The area of stochastic analysis and mathematical programming, known as risk management,

focuses on the effects of extremely adverse outcomes or events on the process of decision making.

The first outstanding contribution to this field is due to Markowitz (1952, 1991), who identified the

riskiness of a portfolio of financial instruments with the volatility of assets' returns, and proposed

the famous Mean-Variance portfolio optimization model for constructing a portfolio with lowest

risk for a given level of expected return. The transparency and efficiency of Markowqitz's approach

made it a very popular tool, which is widely used in the finance industry even nowadays. Since

Markowitz's seminal work, a considerable progress has been achieved in the area of risk analysis

and management, which evolved into a sophisticated discipline combining rigorous and elegant the-

oretical results with practical effectiveness. It was recognized that neither the Mean-Variance model,

which utilizes variance to measure the uncertainty in the system, nor the Value-at-Risk concept (the

upper a-quantile of distribution, see Jorion (1996, 1997)), which became the official standard in the

finance industry, can provide an adequate figure of risk exposure. The theory of coherent risk mea-

sures, developed by Artzner et al. (1997, 1999, 2001), Delbaen (2000), established a set of axioms

to be satisfied by a risk measure (usually, a certain statistic of the distribution) in order to yield cor-

rect estimation of risk under different conditions. At the same time, a series of developments of new

risk measures such as Conditional Value-at-Risk, Expected Shortfall (Acerbi, 2002; Acerbi et al.,

2001; Acerbi and Tasche, 2002; Rockafellar and Uryasev, 2000, 2002) demonstrated the theoretical

and practical importance and efficiency of the concept of coherent risk measures.

The first part of the dissertation (Chapters 2 to 4) considers risk management approaches in

the scope of financial applications. In Chapter 2, we present a general framework of risk/reward op-

timization that establishes the equivalence of different formulations of optimization problems with

risk and reward functions. As an application of the general result, we consider optimization of port-

folio of stocks with Conditional Value-at-Risk objective and constraints, and compare this approach

with the classical Markowitz Mean-Variance methodology. Chapter 3 contains an extensive study

of in-sample and out-of-sample performance of trading algorithms based on different risk measures

(Conditional Value-at-risk, Conditional Drawdown-at-Risk, Mean-Absolute Deviation, and Maxi-

mum Loss) on the example of managing a portfolio of hedge funds. Based on the results of this and

other case studies, we make recommendations on how to improve the "real" (i.e., out-of-sample)

performance of portfolio rebalancing strategies.










Chapter 4 introduces a new sample-path approach to the problem of optimal transaction im-

plementation. Based on the ideas of multi-stage stochastic programming, and using a sample-path

collection instead of the traditional scenario tree, this approach produces an optimal trading strat-

egy that admits a differentiate response to realized market conditions at each time step. In contrast

to other existing approaches to optimal execution, our approach admits a seamless incorporation

of different types of constraints, e.g. regulatory or risk constraints, into the trading strategy. The

generality of the developed approach allows to use it for pricing of complex derivative securities,

such as exotic options.

Although the field of risk management is rooted in finance and financial applications, the suc-

cesses of risk management in theoretical and, especially, applied contexts have drawn attention to

these techniques in other fields where uncertainties impact the decision making process. One of

the areas where systematic risk management approach will contribute significantly to robustness of

decisions and policies is military applications, which typically involve decision making in dynamic,

distributed, and highly uncertain environments. The main challenges of military applications at-

tribute to various types of risk exposure, uncertain probability measures of risk-inducing factors,

and inapplicability of the "long run" convention.

Indeed, in contrast to the problems of finance, where the only type of risk is the risk of financial

loss, military incorporate different types of risk, e.g., risk of not killing a target, risk of false target

attack, risk of being destroyed by enemy, etc. In addition, distributions of risk factors in military

problems are rarely known with certainty. Also, a typical financial model is expected perform well

in repetitive applications, i.e., it is required to be effective on average, or in a long run. Obviously,

there is no "long run" in military applications, therefore the generated decision must be both safe

and effective here and now.

In the second part of the dissertation (Chapters 5 to 6), we consider risk management tech-

niques for military decision-making problems. A general approach for military decision-making

problems with uncertainties in distributions is developed in Chapter 5, and, as an example, we

consider a stochastic version of the Weapon-Target Assignment problem. We demonstrate that

employing risk management techniques based on the Conditional Value-at-Risk measure leads to

solutions that perform robustly under a wide range of scenarios.







4

In Chapter 6, we consider a nonlinear integer programming model for the Stochastic Weapon-

Target Assignment problem, and develop a linear programming relaxation, which allows for effi-

cient incorporation of risk constraints in the problem.

Finally, Appendix presents the results of our ongoing research efforts in the area of pricing

of path-dependent derivative securities using the techniques of mathematical programming. We

discuss a linear programming-based algorithm for solving free-boundary problem arising in pricing

of American put option within the classical Black-Scholes framework, as well as a new approach

for pricing of path-dependent derivative securities using simulation and stochastic programming.
















CHAPTER 2
PORTFOLIO OPTIMIZATION WITH CONDITIONAL VALUE-AT-RISK OBJECTIVE AND
CONSTRAINTS

In this chapter, we extend the recently suggested approach (Rockafellar and Uryasev, 2000)

for optimization of Conditional Value-at-Risk (CVaR) to the case of optimization problems CVaR

constraints. We derive a general equivalence result for different types of optimization problems with

broad class of risk and reward functions. In particular, the approach can be used for maximizing

the expected returns under CVaR constraints. As a further extension, multiple CVaR constraints

with various confidence levels can be used to shape the profit/loss distribution. A case study for the

portfolio of S&P 100 stocks is performed to demonstrate how the new optimization techniques can

be implemented. The approach is compared with the classic Markowitz Mean-Variance model for

portfolio optimization.

2.1 Introduction

Portfolio optimization has come a long way from Markowitz (1952) seminal work which in-

troduces return/variance risk management framework. Developments in portfolio optimization are

stimulated by two basic requirements:

Adequate modeling of utility functions, risks, and constraints

Efficiency, i.e., ability to handle large numbers of instruments and scenarios.

Current regulations for finance businesses formulate some of the risk management require-

ments in terms of percentiles of loss distributions. An upper percentile of the loss distribution

is called Value-at-Risk (VaR).1 For instance, 95%-VaR is an upper estimate of losses which is

exceeded with 5% probability. The popularity of VaR is mostly related to a simple and easy to

understand representation of high losses. VaR can be quite efficiently estimated and managed when



1 By definition, VaR is the percentile of the loss distribution, i.e., with a specified confidence
level a, the a-VaR of a portfolio is the lowest amount such that, with probability a, the loss is
less or equal to 5. Regulations require that VaR should be a fraction of the available capital.










underlying risk factors are normally (log-normally) distributed. For comprehensive introduction

to risk management using VaR, we refer the reader to Jorion (1997). However, for non-normal

distributions, VaR may have undesirable properties (Artzner et al., 1997, 1999) such as lack of sub-

additivity, i.e., VaR of a portfolio with two instruments may be greater than the sum of individual

VaRs of these two instruments.2 Also, VaR is difficult to control/optimize for discrete distributions,

when it is calculated using scenarios. In this case, VaR is non-convex (see definition of convexity

in (Rockafellar, 1970) and non-smooth as a function of positions, and has multiple local extrema.

An extensive description of various methodologies for the modeling of VaR can be seen, along with

related resources, at URL http://www.gloriamundi.org/. Mostly, approaches to calculating VaR rely

on linear approximation of the portfolio risks and assume a joint normal (or log-normal) distribu-

tion of the underlying market parameters (Duffie and Pan (1997), Jorion (1996), Pritsker (1997),

RiskMetricsTM (1996), Simons (1996), Stublo Beder (1995), Staumbaugh (1996)). Also, historical

or Monte Carlo simulation-based tools are used when the portfolio contains nonlinear instruments

such as options (Jorion (1996), Mausser and Rosen (1991), Pritsker (1997), RiskMetricsTM (1996),

Stublo Beder (1995), Staumbaugh (1996)). Discussions of optimization problems involving VaR

can be found in Litterman (1997a,b), Kast et al. (1998), Lucas and Klaassen (1998).

Although risk management with percentile functions is a very important topic and in spite of

significant research efforts (Andersen and Sornette (2001), Basak and Shapiro (2001), Emmer et al.

(2000), Gaivoronski and Pflug (2000), Gourieroux et al. (2000), Grootweld and Hallerbach (2000),

Kast et al. (1998), Puelz (1999), Tasche (1999)), efficient algorithms for optimization of percentiles

for reasonable dimensions (over one hundred instruments and one thousand scenarios) are still not

available. On the other hand, the existing efficient optimization techniques for portfolio allocation3



2 When returns of instruments are normally distributed, VaR is sub-additive, i.e., diversification
of the portfolio reduces VaR. For non-normal distributions, e.g., for discrete distribution, diversifi-
cation of the portfolio may increase VaR.

3 High efficiency of these tools can be attributed to using linear programming (LP) techniques.
LP optimization algorithms are implemented in number of commercial packages, and allow for
solving of very large problems with millions of variables and scenarios. Sensitivities to parameters
are calculated automatically using dual variables. Integer constraints can also be relatively well
treated in linear problems (compared to quadratic or other nonlinear problems). However, recently
developed interior point algorithms work equally well both for portfolios with linear and quadratic










do not allow for direct controlling4 of percentiles of distributions (in this regard, we can mention the

mean absolute deviation approach (Konno and Yamazaki, 1991), the regret optimization approach

(Dembo and Rosen, 1999), and the minimax approach (Young, 1998). This fact stimulated our

development of the new optimization algorithms presented in this chapter.

This chapter suggests to use, as a supplement (or alternative) to VaR, another percentile risk

measure which is called Conditional Value-at-Risk. The CVaR risk measure is closely related to

VaR. For continuous distributions, CVaR is defined as the conditional expected loss under the con-

dition that it exceeds VaR, see Rockafellar and Uryasev (2000). For continuous distributions, this

risk measure also is known as Mean Excess Loss, Mean Shortfall, or Tail Value-at-Risk. However,

for general distributions, including discrete distributions, CVaR is defined as the weighted average

of VaR and losses strictly exceeding VaR (Rockafellar and Uryasev, 2002). Recently, Acerbi et al.

(2001), Acerbi and Tasche (2002) redefined expected shortfall similarly to CVaR.

For general distributions, CVaR, which is a quite similar to VaR measure of risk has more

attractive properties than VaR. CVaR is sub-additive and convex (Rockafellar and Uryasev, 2000).

Moreover, CVaR is a coherent measure of risk in the sense ofArtzner et al. (1997, 1999). Coherency

of CVaR was first proved by Pflug (2000); see also Rockafellar and Uryasev (2002), Acerbi et al.

(2001), Acerbi and Tasche (2002). Although CVaR has not become a standard in the finance indus-

try, CVaR is gaining in the insurance industry (Embrechts et al., 1997). Similar to CVaR measures

have been introduced earlier in stochastic programming literature, but not in financial mathemat-

ics context. The conditional expectation constraints and integrated chance constraints described in

Prekopa (1995) may serve the same purpose as CVaR.

Numerical experiments indicate that usually the minimization of CVaR also leads to near op-

timal solutions in VaR terms because VaR never exceeds CVaR (Rockafellar and Uryasev, 2000).

Therefore, portfolios with low CVaR must have low VaR as well. Moreover, when the return-loss



performance functions, see for instance Duarte (1999). The reader interested in various applica-
tions of optimization techniques in the finance area can find relevant papers in Ziemba and Mulvey
(1998).

4 It is impossible to impose constraints in VaR terms for general distributions without deterio-
rating the efficiency of these algorithms.










distribution is normal, these two measures are equivalent (Rockafellar and Uryasev, 2000), i.e.,

they provide the same optimal portfolio. However for very skewed distributions, CVaR and VaR

risk optimal portfolios may be quite different. Moreover, minimizing of VaR may stretch the tail

exceeding VaR because VaR does not control losses exceeding VaR, see Larsen et al. (2002). Also,

Gaivoronski and Pflug (2000) have found that in some cases optimization of VaR and CVaR may

lead to quite different portfolios.

Rockafellar and Uryasev (2000) demonstrated that linear programming techniques can be used

for optimization of the Conditional Value-at-Risk (CVaR) risk measure. A simple description of the

approach for minimizing CVaR and optimization problems with CVaR constraints can be found in

Uryasev (2000). Several case studies showed that risk optimization with the CVaR performance

function and constraints can be done for large portfolios and a large number of scenarios with

relatively small computational resources. A case study on the hedging of a portfolio of options using

the CVaR minimization technique is included in (Rockafellar and Uryasev, 2000). This problem

was first studied in the paper by Mausser and Rosen (1991) with the minimum expected regret

approach. Also, the CVaR minimization approach was applied to credit risk management of a

portfolio of bonds, see Andersson et al. (2001).

This chapter extends the CVaR minimization approach (Rockafellar and Uryasev, 2000) to

other classes of problems with CVaR functions. We show that this approach can be used also for

maximizing reward functions (e.g., expected returns) under CVaR constraints, as opposed to mini-

mizing CVaR. Moreover, it is possible to impose many CVaR constraints with different confidence

levels and shape the loss distribution according to the preferences of the decision maker. These

preferences are specified directly in percentile terms, compared to the traditional approach, which

specifies risk preferences in terms of utility functions. For instance, we may require that the mean

values of the worst 1%, 5% and 10% losses are limited by some values. This approach provides a

new efficient and flexible risk management tool.

The next section briefly describes the CVaR minimization approach from (Rockafellar and

Uryasev, 2000) to lay the foundation for the further extensions. In Section 3, we formulated a

general theorem on various equivalent representations of efficient frontiers with concave reward

and convex risk functions. This equivalence is well known for mean-variance, see for instance,

Steinbach (1999), and for mean-regret, (Dembo and Rosen, 1999), performance functions. We have










shown that it holds for any concave reward and convex risk function, in particular for the CVaR risk

function considered in this chapter. Using auxiliary variables, we formulated a theorem on reducing

the problem with CVaR constraints to a much simpler convex problem. A similar result is also

formulated for the case when both the reward and CVaR are included in the performance function.

As it was earlier identified in (Rockafellar and Uryasev, 2000), the optimization automatically sets

the auxiliary variable to VaR, which significantly simplifies the problem solution. Further, when the

distribution is given by a fixed number of scenarios and the loss function is linear, we showed how

the CVaR function can be replaced by a linear function and an additional set of linear constraints. In

section 2.7, we developed a one-period model for optimizing a portfolio of stocks using historical

scenario generation. A case study on the optimization of S&P100 portfolio of stocks with CVaR

constraints is presented in section 2.7. We compared the return-CVaR and return-variance efficient

frontiers of the portfolios.

2.2 Conditional Value-at-Risk

The approach developed in (Rockafellar and Uryasev, 2000) provides the foundation for the

analysis conducted in this chapter. First, following (Rockafellar and Uryasev, 2000), we formally

define CVaR and present several theoretical results which are needed for understanding this chapter.

Let f(x, y) be the loss associated with the decision vector5 x, to be chosen from a certain subset X

of Rn, and the random vector y in R". The vector x can be interpreted as a portfolio, with X as the

set of available portfolios (subject to various constraints), but other interpretations could be made

as well. The vector y stands for the uncertainties, e.g., market prices, that can affect the loss. Of

course the loss might be negative and thus, in effect, constitute a gain.

For each x, the loss f(x, y) is a random variable having a distribution in R induced by that

of y. The underlying probability distribution of y in R"' will be assumed for convenience to have

density, which we denote by p(y). This assumption is not critical for the considered approach. The

paper by Rockafellar and Uryasev (2002) defines CVaR for general distributions; however, here, for

simplicity, we assume that the distribution has density. The probability of f(x, y) not exceeding a


5 We use boldface font for vectors to distinguish them from scalars.









threshold 5 is then given by

y(x,/) ( p(y)dy. (2.1)

As a function of 5 for fixed x, Y(x, 5) is the cumulative distribution function for the loss

associated with x. It completely determines the behavior of this random variable and is fundamental

in defining VaR and CVaR.

The function Y(x, 5) is nondecreasing with respect to (w.r.t.) 5 and we assume that Y(x, 5) is

everywhere continuous w.r.t. 5. This assumption, like the previous one about density in y, is made

for simplicity. In some common situations, the required continuity follows from properties of the

loss f(x, y) and the density p(y); see Uryasev (1995).

The a-VaR and a-CVaR values for the loss random variable associated with x and any speci-

fied probability level a in (0, 1) will be denoted by 5,(x) and 0a(x). In our setting they are given

by

5C(x) =min{ ER: Y(x,) > a} (2.2)

and

a (x) (1- a)1() f(x,y)p(y)dy. (2.3)

In the first formula, 5 (x) comes out as the left endpoint of the nonempty interval6 consisting

of the values 5 such that actually Y(x, 5) = a. In the second formula, the probability that f(x, y) >

5a (x) is therefore equal to 1 a. Thus, 0, (x) comes out as the conditional expectation of the loss
associated with x relative to that loss being 5 (x) or greater.

The key to the approach is a characterization of 0a (x) and C (x) in terms of the function F,

on X x R that we now define by


Fa(x, )- = +(1 a) [f(x,y) ]p(y)dy, (2.4)
JyeRn

where [t]+ = max{t, 0}. The crucial features of Fa, under the assumptions made above, are as

follows (Rockafellar and Uryasev, 2000).



6 This follows from Y(x, 5) being continuous and nondecreasing w.r.t. 5. The interval might
contain more than a single point if Y has "flat spots."










Theorem 2.2.1 As a function of Fa(x, 5) is convex and continuously differentiable. The a-CVaR

of the loss associated with any x E X can be determined from the formula


(x) = minFa(x, ). (2.5)

In this formula, the set consisting of the values of for which the minimum is attained, namely


A(x) = argminF (x, ), (2.6)
cR

is a nonempty, closed, bounded interval (perhaps reducing to a single point), and the a-VaR of the

loss is given by

5a(x) = left endpoint ofA,(x). (2.7)

In particular, one always has


C~(x) E argminF,(x, 5) and (a(x) Fa(x, 5(x)). (2.8)
cR

For background on convexity, which is a key property in optimization that in particular eliminates

the possibility of a local minimum being different from a global minimum, see, for instance, Rock-

afellar (1970). Other important advantages of viewing VaR and CVaR through the formulas in

Theorem 1 are captured in the next theorem, also proved in (Rockafellar and Uryasev, 2000)

Theorem 2.2.2 Minimizing the a-CVaR of the loss associated with x over all x E X is equivalent

to minimizing Fa(x, 5) over all (x, 5) E X x R, in the sense that


min 0(x) min F,(x, ), (2.9)
xX (x,)cXxR

where moreover a pair (x*, 5*) achieves the right hand side minimum if and only if x* achieves

the left hand side minimum and 5* E A,(x*). In particular, therefore, in circumstances where the

interval Aa(x*) reduces to a single point (as is typical), the minimization ofF(x, 5) over (x, 5) E

X x Rproduces a pair (x*, 5*), not necessarily unique, such that x* minimizes the a-CVaR and 5"

gives the corresponding a-VaR.

Furthermore, Fa(x, ) is convex w.rt. (x, ), and 0a(x) is convex w.rt. x, when f(x,y) is

convex with respect to x, in which case, if the constraints are such that X is a convex set, thy joint

minimization is an instance of convex programming.










According to Theorem 2.2.2, it is not necessary, for the purpose of determining a vector x that

yields the minimum a-CVaR, to work directly with the function 0a(x), which may be hard to do

because of the nature of its definition in terms of the a-VaR value 50(x) and the often trouble-

some mathematical properties of that value. Instead, one can operate on the far simpler expression

Fa(x, 5) with its convexity in the variable 5 and even, very commonly, with respect to (x, 5).

2.3 Efficient Frontier: Different Formulations

The paper by Rockafellar and Uryasev (2000) considered minimizing CVaR, while requiring a

minimum expected return. By considering different expected returns, we can generate an efficient

frontier. Alternatively, we also can maximize returns while not allowing large risks. We, therefore,

can swap the CVaR function and the expected return in the problem formulation (compared to Rock-

afellar and Uryasev (2000), thus minimizing the negative expected return with a CVaR constraint.

By considering different levels of risks, we can generate the efficient frontier.

We will show in a general setting that there are three equivalent formulations of the optimiza-

tion problem. They are equivalent in the sense that they produce the same efficient frontier. The

following theorem is valid for general functions satisfying conditions of the theorem.

Theorem 2.3.1 Let us consider the functions 0 (x) and R(x) dependent on the decision vector x,

and the hollowing three problems:


(P1) min 0(x) R(x), x X, 9 > 0,
x


(P2) min (x), R(x)>p, xeX,
x


(P5) min -R(x), O(x) < ), xeX.
x

Suppose that constraints R(x) > p, O(x) < o have internalpoints.7 Varying the parameters pl1,

p, and o, traces the efficient frontiers for the problems (PO)-(P3), accordingly. If (x) is convex,

S(x) is concave and the set X is convex, then the three problems, (Pl)-(PO), generate the same

efficient frontier


7 This condition can be replaced by some other regularity conditions used in duality theorems.










The proof of Theorem 2.3.1 is based on the Kuhn-Tucker necessary and sufficient conditions

stated in the following theorem.

Theorem (Kuhn-Tucker, Theorem 2.5 (Pshenichnyi, 1971)). Consider the problem


min lo (x),


S(x) < 0 i = -m,...,-1,

y, (x) =0 i = 1,...,n,

x EX.

Let y, (x) be functionals on a linear space, E, such that y, (x) are convex for i < 0 and linear for

i > 1 and X is some given convex subset ofE. Then in order that s8 (x) achieves its minimum point

at x* E E it is necessary that there exists constants A,, i = -m,...,n, such that

1n 1n
i (x*) < i l ,(X)
I=-m i=-m

for all x e X. Moreover, ;> 0 for each i < 0, and Ai, y(xo) = 0 for each i 0. If o > 0, then the

conditions are also sufficient.

Let us write down the necessary and sufficient Kuhn-Tacker conditions for problems (P1), (P2),

and (P3). After some equivalent transformations these conditions can be stated as follows8 :

K-T conditions for (P1):


(KT1) 0(x*) (x*) p < (x) R(x), pl > 0, x EX .


K-T conditions for (P2):


S(x*)+ (p -R(x*)) < A(x)+ (p -R(x)),


(p-R(x)) 0, 02 > 0, o > 0, x EX.


8 Kuhn-Tacker conditions for (P1) are, actually, a definition of the minimum point.












(KT2) p(x*) 2R(x*) < O(x) 2R(x),

P2(P-R(x*)) 6, P2> 0, x X.

K-T conditions for (P3):


S03(-R(x*)) + 3(0(X*) ) _< 03(-R(x)) + 3 (0(X) ()),

;~( (x*)- )- 0, 1 >0, ;3 >0, X EX.




(KT3) -R(x*) +p30(x*) < -R(x) +p3(x),


p3(0(x*) ) 0, o > 0, x X.

Following Steinbach (1999), we call pi in (KT2) the optimal reward multiplier, and p3 in (KT3)

the risk multiplier. Further, using conditions (KT1) and (KT7), we show that a solution of problem

(P1) is also a solution of(P2) and vice versa, a solution of problem (P2) is also a solution of (P 1).

Lemma 2.3.1 Ifa point x* is a solution of (PI), then the point x* is a solution of(P2) with param-

eter p R(x*). Also, stated in the other direction, ifx* is a solution of(P2) and P2 is the optimal

reward multiplier in (KT2), then x* in a solution of(PI) with Pl = P2.

Proof of Lemma 2.3.1. Let us prove the first statement of Lemma A4. If x* is a solution of (P4),

then it satisfies condition (KT1). Evidently, this solution x* satisfies (KT2) with p R(x*) and

P2 1.
Now, let us prove the second statement of Lemma 2.3.1. Suppose that x* is a solution of (P2)

and (KT2) is satisfied. Then, (KT1) is satisfied with parameter Pi = P2 and x* is a solution of (P 1).

Lemma 2.3.1 iy proved. U

Further, using conditions (KT1) and (KT3), we show that s solution of problems (P1) is also a

solution of (P3) and vice versa, a solution of problems (P3) is also a solution of (P1).










Lemma 2.3.2 If a point x* is a solution of (PI), then the point x* is a solution of (P3) with the

parameter o = (x). Also, stated in other direction, ifx* is a solution of(P3) and 3 is a positive

risk multiplier in (KT3), then x* is a solution of(PI) with pl = 8/t3.

Proof of Lemma 2.3.2. Let us prove the first statement of Lemma 2.3.2. If x* is a solution of (P 1),

then it satisfies the condition (KT1). If Mi > 0, then this solution x* satisfies (KT3) with p3 1/p9

and o = (x).

Now, let us prove the second statement of Lemma A2. Suppose that x* is a solution of (P3)

and (KT6) is satisfied with 3 > 0. Then, (KT1) is satisfied with parameter l = 1/p 3 and x* is a

solution of (Pl). Lemma 2.3.2 is proved. U

Proof of Theorem 2.3.1. Lemma 2.3.1 implies that the efficient frontiers of problems (P1) and

(P2) coincide. Similar, Lemma 2.3.2 implies that the efficient frontiers of problems (P1) and (P3)

coincide. Consequently, efficient frontiers of problems (P 1), (P2), and (P3) coincide. U

The equivalence between problems (Pl)-(P3) is well known for mean-variance Steinbach

(1999) and mean-regret Dembo and Rosen (1999) efficient frontiers. Be have shown that it holds

for any concave reward and convex risk functions with convex constraints.

Further, we consider that the loss function f(x, y) is linear w. r. t. x, therefore Theorem 2

implies that the CVaR risk function 0a(x) is convex w.r.t. x. Also, we suppose that the reward

function, R(x) is linear and this constraints are linear. The conditions of Theorem 3 are satisfied

for the CVaR risk function 0 (x) and the reward function R(x) Therefore, maximizing the reward

under a CVaR constraint, generates the same efficient frontier as the minimization of CVaR under a

constraint on the reward.

2.4 Equivalent Formulations with Auxiliary Variables

Theorem 2.3.1 implies that we can use problem formulations (P ), (P2), and (P3) for generat-

ing the efficient frontier with the CVaR risk function (x) and the reward function R(x). Theorem

2.2.2 shows that the function Fa(x, 5) can be used instead of 0,(x) to solve problem (P2). Further,

we demonstrate that, similarly, the function Fa(x, 5) can be used instead of 0a(x) in problems (P1)

and (P8).









Theorem 2.4.1 The two minimization problems below


(P3) min -R(x), 0a(x) < o, xEX
xeX

and


(P3') min -R(x), Fa(x, ) < o), x EX
(,x)eXxR

are equivalent in the sense that their objectives achieve the same minimum values. Moreover if the

CVaR constraint in (P4) is active, a pair (x*, *) achieves the minimum of(P5') if and only if x*

achieves the minimum of(P4) and *5 E A (x*). In particular when the interval A(x*) reduces to

a single point, the minimization of -R(x) over (x, 5) E X x Rproduces a pair (x*, 5*) such that*

maximizes the return and 5* gives the corresponding a-VaR.

Proof of Theorem 2.4.1. The necessary and sufficient conditions for the problem (P4') are stated

as follows


(KT3') -R(x*) + 3Fa(x*, 5*) < -R(x) + 3Fa(x, C),


p3(Fa(x*,*)- o) 9, p3>0, xEX.

First, suppose that x* is a solution of (P4) and 5* E A (x*). Let us show that (x*, 5*) is a

solution of (P4'). Using necessary and sufficient conditions (KT3) and Theorem 1 we have

-R(x*) +3Fa(x*, *) = -R(x*) +p30a(X*)


< -R(x)+p30a(x) -R(x)+p3minFa(x, )

< -R(x) +3Fa(x, ),

and

P3(Fa(x*, *) 0) = 3(0a(x*) )) 0, p3 >0, xEX.

Thus, (KT3') conditions are satisfied and (x*, 5*) is a solution of(P4').

Now, let us suppose that (x*, 5*) achieves the minimum of (P4') and p3 > 0. For fixed x*, the

point 5* minimizes the function -R(x*) + P3Fa(x*, 5), and, consequently, the function Fa(x*, ).









Then, Theorem 2.2.1 implies that 5* E A(x*). Further, since (x*, 5*) is a solution of(P4'), condi-

tions (KT3') and Theorem 2.2.1 imply that


-R(x*) + 30a(x*) -

< -R(x) + p3Fa(x, a(x


R (x*) + p3Fa (x*, 5*)

)) -R(x) + p30 a(x)


p5(0,a(x*) W) = 3(Fa(x*, 5*) W) = 0, P3 > 0, x X.

We proved that conditions (KT3) are satisfied, i.e., x* is a solution of (P4).

Theorem 2.4.2 The two minimization problems below


min 0c(x) oR(x), pl > 0, xEX
xeX


mm
(x,n)ex;


E(X(x, ) PR(x), Pi > 0, x EX


are equivalent in the sense that their objectives achieve the same minimum values. Moreover, a

pair (x*, *) achieves the minimum of (P7) if and only if x* achieves the minimum of(P5) and

5* E Ac(x*). In particular when the interval Aa(x*) reduces to a single point, the minimization

ofFas(x, 5) piR(x) over (x, 5) E X x R produces a pair (x*, 5*) such that x* minimizes 0a(x) -

piR(x) and *5 gives the corresponding a-VaR.

Proof of Theorem 2.4.2. Let x* is a solution of (P5), i.e.,


0ac(x*) pR(x*) < Oa(x) plR(x), pi > 0, X EX .

and 5* e A(x*). Using Theorem 1 we have

Fa(x*, *) iR(x*) = Oa(x*)- piR(x*)


< Oa(x) iR(x) minFa(x, ) -p 1R(x)

< Fa(x, ) -p R(x), x E X,


i.e, (x*, 5*) is a solution of problem (P5').


(P5)


(P5')









Now, let us consider that (x*, 5*) is a solution of problem (P5'). For the fixed point x*, the
point 5* minimizes the functions Fa(x*, 5) lR(x*) and, consequently, the point 5* minimizes
the function Fa(x*, 5). Then, Theorem 2.2.1 implies that 5* E Aa(x*). Further, since (x*, *) is a

solution of (P5'), Theorem 2.2.1 implies


0a(x*) p R(x*) Fa (x*, n*)- jlR(x*)


This proves the statement of Theorem 2.4.2. U

2.5 Discretization and Linearization

The equivalent problem formulations presented in Theorems 5, 4 and 5 can be combined with

ideas for approximating the integral in Fa(x, 5), see (2.4). This offers a rich range of possibilities.
The integral in Fa(x, 5) can be approximated in various ways. For example, this can be done

by sampling the probability distribution of y according to its density p(y). If the sampling generates

a collection of vectors yi, y2, ,y, then the corresponding approximation to

Fa (x, ) 1+ (9 a)- [f(x,y)- ] p(y)dy

is
J
F(x,) = +(1-a) Xl y [f(x,yy)-] (2.10)
j=1
where rJ are probabilities of scenarios y,. If the loss function f(x, y) is linear w.r.t. x, then the

function Fa(x, 5) is convex and piecewise linear.

The function Fa(x, 5) in optimization problems in Theorems 2.2.2, 2.4.1, and 2.4.2 can be

approximated by the function Fa(x, 5). Further, by using dummy variables zj, j= 1,...,J, the

function Fa(x, 5) can be replaced by the linear function + (1 a)- 1J1 r zj and the set of

linear constraints


zJ > f(x,y,) z> 0, j 1,...,J, e R.

For instance, by using Theorem 2.4.1 we can replace the constraint


a(x) < 0)










in optimization problem (P4) by the constraint


Fa(x, ) < 0.

Further, the above constraint can be approximated by


F(x, )< ,), (2.11)

and reduced to the following system of linear constraints
J
S+(1-a) 1 ,tiz <_0, (2.12)
j=1

zi > f(x,y,) z > 0, j 1,...,J, CR. (2.13)

Similarly, approximations by linear functions can be done in the optimization problems in Theorems

1 and 5.

2.6 One Period Portfolio Optimization Model with Transaction Costs

Loss and reward functions. Let us consider a portfolio of n, (i = 7,...,n) different financial

instruments in the market, among which there is one risk-free instrument (cash, or bank account

etc). Let xo = (x,x, ...,x )T be the positions, i.e., number of shares, of each instrument in the

initial portfolio, and let x = (xi,x2, ...,X)T be the positions in the optimal portfolio that we intend

to find using the algorithm. The initial prices for the instruments are given by q = (l, q2, ..., q)T.

The inner product qTx0 is thus the initial portfolio value. The scenario-dependent prices for each

instrument at the end of the period are given by y = (yl,y2,...,yn)T. The loss function over the

period is

f(x,y;x, q) -yTx+q xo. (2.14)

The reward function R(x) is the expected value of the portfolio at the end of the period,


R(x) E[yx]= Y Ey,]x, (2.15)
i=1

Evidently, defined in this way, the reward function R(x) and the loss function f(x, y) are related as


R(x) -E[f(x,y)] + qTx


The reward function R(x) is linear (and therefore concave) in x.










CVaR constraint. Current regulations impose capital requirements on investment compa-

nies, proportional to the VaR of a portfolio. These requirements can be enforced by constraining

portfolio CVaR at different confidence levels, since CVaR > VaR. The upper bound on CVaR can be

chosen as the maximum VaR. According to this, we find it meaningful to present the risk constraint

in the form

0a(x) < qTx0, (2.16)

where the risk function 0a(x) is defined as the a-CVaR for the loss function given by (2.14), and

o is a percentage of the initial portfolio value qTx0, allowed for risk exposure. The loss function

given by (2.14) is linear (and therefore convex) in x, therefore, the a-CVaR function 0a(x) is also

convex in x. The set of linear constraints corresponding to (2.16), is

J n
c+(1- a) 9 ICjz, j o< ,q,xo, (2.17)
j=1 /=1

n
zj > (-yx +q,x) z > 0, j 1,...,. (2.18)
i=1

Transaction costs. We assume a linear transaction cost, proportional to the total dollar value

of the bought/sold assets. For a treatment of non-convex transaction costs, see Konno and Wi-

jayanayake (1999). With every instrument, we associate a transaction cost c,. When buying or

selling instrument i, one pays c, times the amount of transaction. For cash we set cash = 0. That is,

one only pays for buying and selling the instrument, and not for moving the cash in and out of the

account.

According to that, we consider a balance constraint that maintains the total value of the port-

folio including transaction costs

n n n
qx4 = c,q, |x x, + q,x, .
1=4 1=1 1=1










This equality can be reformulated using the following set of linear constraints9

n n n
Zqx2 = c,q, (u +u, ) + qx, ,


x, -x5 =,+-u i 1, ...,n,

u>O, u > O, i= 1 n.


Value constraint. We do not allow for an instrument i to constitute more than a given per-

cent, v,, of the total portfolio value
n
qx, < v, xkqk .
k=1
This constraint makes sense only when short positions are not allowed.

Change in individual positions (liquidity constraints) and bounds on positions. We con-

sider that position changes can be bounded. This bound could be, for example, a fixed number or

be proportional to the initial position in the instrument


O I --I I -- I


These constraints may reflect limited liquidity of instruments in the portfolio (large transactions

may significantly affect the price q,).

We, also, consider that the positions themselves can be bounded


x, x, < x,, i 1,...,n. (2.19)


The optimization problem. Below we present the problem formulation, which optimizes

the reward function subject to constraints described in this section:

n
min Ey,]x,, (2.20)
x, 7=1

subject to




9 The nonlinear constraint u, u, = 0 can be omitted since simultaneous buying and selling of
the same instrument, i, can never be optimal.











J n
+(1 -a) ) jz, = 1 k 1
n
zj > C(-y yxl + qxo) 5, z, > 0, j= 1, ., J,(2.22)
I=1
n
qx, < v i 1,...,n, (2.23)
k=1
n n n
qx c,q, (u+ +u-)+ )~ qx,, (2.24)
=1 =1 =1

x, -x u -u, i 1,...,n, (2.25)

0

x, < x < i 1,...,n. (2.27)

By solving this problem, we get the optimal vector x*, the corresponding VaR, which equals1 5*,

and the maximum expected return, which equals E[y]x*/(q'x0). The efficient return-CVaR frontier

is obtained by taking different risk tolerance levels w.

Scenario generation. With our approach, the integral in the CVaR function is approximated

by the weighed sum over all scenarios. This approach can be used with different schemes for

generating scenarios. For example, one can assume a joint distribution for the price-return process

for all instruments and generate scenarios in a Monte Carlo simulation. Also, the approach allows

for using historical data without assuming a particular distribution. In our case study, we used

historical returns over a certain time period for the scenario generation, with length At of the period

equal to the portfolio optimization period. For instance, when minimizing over a one day period, we

take the ratio of the closing prices of two consecutive days, pr and pt +l. Similarly, for a two week

period, we consider historical returns ptJ+l1/pb. In such a fashion, we represent the scenario set

for random variable y,, which is the end-of-period price of instrument i, with the set of J historical

returns multiplied by the current price q,,

ty,+A = t,
yjJ qlpl /p j 1,.-,J,


10 If there are many optimal solutions, VaR equals the lowest optimal value 5*.










where ti,..., t are closing times for J consecutive business days. Further, in the numerical simu-

lations, we consider a two week period, At = 10. The expected end-of-period price of instrument i

is
J J

J=1 7=1
where we assumed that all scenarios yj are equally probable, i.e., j = 1/J.

2.7 Case Study: Portfolio of S&P100 Stocks

We now proceed with a case study and construct the efficient frontier of a portfolio consisting

of stocks in the S&P100 index. We maximized the portfolio value subject to various constraints on

CVaR. The algorithm was implemented in C++ and we used the CPLEX 6.0 Callable Library to

solve the LP problem.

This case study is designed as a demonstration of the methodology, rather than a practical

recommendation for investments. We have used historical data for scenario generation (10-day his-

torical returns). While there is some estimation error in the risk measure, this error is much greater

for expected returns. The historical returns over a 10-day period provide very little information

on the actual "to-be-realized out-of-sample" returns; i.e., historical returns have little "forecast-

ing power." These issues are discussed in many academic studies, including (Jorion, 1996, 2000;

Michaud, 1989). The primary purpose of the presented case study is the demonstration of the novel

CVaR risk management methodology and the possibility to apply it to portfolio optimization. This

technology can be combined with more adequate scenario generation procedures utilizing expert

opinions and advanced statistical forecasting techniques, such as neural networks. The suggested

model is designed as one stage of the multistage investment model to be used in a realistic in-

vestment environment. However, discussing this multistage investment model and the scenario

generation procedures used for this model is beyond the scope of this chapter.

The set of instruments to invest in was set to the stocks in the S&P 100 as of the first of Septem-

ber 1999. Due to insufficient data, six of the stocks were excluded.11 The optimization was run

for two-week period, ten business days. For scenario generation, we used closing prices for five




11 Citigroup Inc., Hartford Financial Svc.Gp., Lucent Technologies, Mallinckrodt Inc., Raytheon
Co., U.S. Bancorp.










hundreds of the overlapping two-week periods (July 1, 1997 July 8, 1999). In effect, this was an

in-sample optimization using 500 overlapping returns measured over 10 business days.

The initial portfolio contained only cash, and the algorithm should determine an optimal in-

vestment decision subject to risk constraints. The limits on the positions were set to x, = 0 and

x, = respectively, i.e., short positions were not allowed. The limits on the changes in the indi-

vidual positions, u- and u+, were both set to infinity. The limit on how large a part of the total

portfolio value one single asset can constitute, v,, was set to 20% for all i. The return on cash was

set to 0.16% over two weeks. We made calculations with various values of the parameter a in CVaR

constraint.12

2.7.1 Efficient Frontier and Portfolio Configuration

Figure 2-1 shows the efficient frontier of the portfolio with the CVaR constraint. The values on

the Risk scale represent the tolerance level o), i.e., the percentage of the initial portfolio value which

is allowed for risk exposure. For example, setting Risk = 10% (o = 0.10) and a = 0.95 implies that

the average loss in 5% worst cases must not exceed 10% of the initial portfolio value. Naturally,

higher risk tolerance levels o) in CVaR constraint (2.21) allow for achieving higher expected returns.

It is also apparent from Fig. 2-1 that for every value of risk confidence level a there exists some

value o), after which the CVaR constraint becomes inactive (i.e., not binding). A higher expected

return cannot be attained without loosening other constraints in problem (2.20)-(2.27), or without

adding new instruments to the optimization set. In this numerical example, the maximum rate of

return that can be achieved for the given set of instruments and constraints equals 2.96% over two

weeks. However, very small values of risk tolerance o) cause the optimization problem (2.20)-

(2.27) to be infeasible; in other words, there is no such combination of assets that would satisfy

CVaR constraints (2.21)-(2.22) and the constraints on positions (2.23)-(2.27) simultaneously.

Table 2-1 presents the portfolio configuration for different risk levels (a = 0.90). Recall that

we imposed the constraint on the percentage v of the total portfolio value that one stock can consti-

tute (2.23). We set v = 0.2, i.e., a single asset cannot constitute more than 20% of the total portfolio

value. Table 1 shows that for higher levels of allowed risk, the algorithm reduces the number of the


12 (0 was set as some percentage of the initial portfolio value.











Efficient Frontier of Portfolio with CVaR constraints

3.5
3-


S2 Ua=0.90
.----a0.95
0 1.5 ---a=0.99


0.5 -
0
0 5 10 15 20 25
Risk, %


Figure 2-1: Efficient frontier (optimization with CVaR constraints). Rate of Return is the expected
rate of return of the optimal portfolio during a 2 week period. The Risk scale displays the risk
tolerance level wo in the CVaR risk constraint as the percentage of the initial portfolio value.


instruments in the portfolio in order to achieve a higher return (due to the imposed constraints, the

minimal number of instruments in the portfolio, including risk-free cash, equals five). This confirms

the well-known fact that "diversifying" the portfolio reduces the risk. Relaxing the constraint on

risk allows the algorithm to choose only the most profitable stocks. As we tighten the risk tolerance

level, the number of instruments in the portfolio increases, and for more "conservative" investing

(2% risk), we obtain a portfolio with more than 15 assets, including the risk-free asset (cash). The

instruments not shown in the table have zero portfolio weights for all risk levels.

Transaction costs need to be taken into account when employing an active trading strategy.

Transaction costs account for a fee paid to the broker/market, bid-ask spreads, and poor liquidity.

To examine the impact of the transaction costs, we calculated the efficient frontier with the following

transaction costs, c = 0%, 0.25%, and 1%. Figure 2-2 shows that the transaction costs nonlinearly

lower the expected return. Since transaction costs are incorporated into the optimization problem,

they also affect the choice of stocks.

2.7.2 Comparison with Mean-Variance Portfolio Optimization

In this section, we illustrate the relation of the developed approach to the standard Markowitz

mean-variance (MV) framework. It was shown in (Rockafellar and Uryasev, 2000) that for normally

distributed loss functions these two methodologies are equivalent in the sense that they generate









26


Table 2-1: Portfolio configuration: assets' weights (%) in the optimal portfolio depending on the
risk level (the instruments not included in the table have zero portfolio weights).


Risk o, %
Exp.Ret,%
St.Dev.
CVaR
Cash
AA
AIT
BEL
CGP
CSC
CSCO
ETR
GD
IBM
LTD
MOB
MSFT
SO
T
TAN
TXN
UCM
UIS
WMT


2 3 4 5 6 7 8 9 10


1.508
0.0220
0.02
7.7
1.1
7.2
2.0
0.2
1.0
1.0
5.0
10.0
13.7
3.6
4.2
0
3.7
10.7
8.4
0.4
20.0
0.2
0


1.962
0.0290
0.03
0
0
11.3
0.8
0
0
0.4
0
9.9
13.7
3.3
0
0
0
20.0
9.5
1.7
20.0
6.3
3.1


2.195
0.0333
0.04
0
0
14.4
0
0
0
9.4
0
3.9
7.9
0
0
0
0
20.0
12.4
0.4
20.0
11.6
0


2.384
0.0385
0.05
0
0
20.0
0
0
0
13.3
0
0
1.7
0
0
0
0
20.0
14.6
0
13.8
16.7
0


2.565
0.0439
0.06
0
0
19.1
0
0
0
14.5
0
0
0
0
0
0
0
20.0
20.0
0
6.4
20.0
0


2.719
0.0486
0.07
0
0
13.1
0
0
0
20.0
0
0
1.2
0
0
0
0
20.0
20.0
1.4
0
20.0
4.3


2.838
0.0532
0.08
0
0
0
0
0
0
20.0
0
0
1.4
0
0
0
0
20.0
20.0
9.3
0
20.0
9.2


2.915
0.0586
0.09
0
0
0
0
0
0
20.0
0
0
0
0
0
0
0
10.5
20.0
20.0
0
20.0
9.5


2.956
0.0637
0.10
0
0
0
0
0
0
20.0
0
0
0
0
0
13.8
0
0
20.0
20.0
0
20.0
6.2


the same efficient frontier. However, in the case of non-normal, and especially non-symmetric

distributions, CVaR and MV portfolio optimization approaches may reveal significant differences.

Indeed, the CVaR optimization technique aims at reshaping one tail of the loss distribution, which

corresponds to high losses, and does not account for the opposite tail representing high profits. On

the contrary, the Markowitz approach defines the risk as the variance of the loss distribution, and

since the variance incorporates information from both tails, it is affected by high gains as well as by

high losses.

Here, we used historical returns as a scenario input to the model, without making any assump-

tions about the distribution of the scenario variables.We compared the CVaR methodology with the

MV approach by running the optimization algorithms on the same set of instruments and scenarios.

The MV optimization problem was formulated as follows (Markowitz, 1991):


n n
min Y Y, gkXXk,
x 1k=-1


subject to


n
x, =1,
i=1


(2.28)


(2.29)












Efficient Frontier of Portfolio with CVaR Constraints
and Transaction Costs

3.5
3
2.5 -
"- c = 0%
,0 2
.2 2 -------- C00
-c = 0.25%
1.5
0 / --- 1%
a 1
0.5
0
0 5 10 15 20 25
Risk, %


Figure 2-2: Efficient frontier of optimal portfolio with CVaR constraints in presence of transaction
costs c = 0%, 0.25%, and 1%. Rate of Return is the expected rate of return of the optimal portfolio
during a 2 week period. The Risk scale displays the risk tolerance level (o in the CVaR risk constraint
(a = 0.90) as the percentage of the initial portfolio value.





E[r, x, rp, (2.30)
i=1



0< x, < v, i 1,...,n, (2.31)


where x, are portfolio weights, unlike problem (2.20)-(2.27), where x, are numbers of shares of

corresponding instruments. r, is the rate of return of instrument i, and ,k is the covariance between

returns of instruments i and k: Gk = cov(r, rk). The first constraint (2.29) is the budget constraint;

(2.30) requires portfolio's expected return to be equal to a prescribed value rp; finally, (2.31) im-

poses bounds on portfolio weights, where v, are the same as in (2.23). The set of constraints

(2.29)-(2.31) is identical to (2.23)-(2.27), except for transaction cost constraints. The expectations

and covariances in (2.28), (2.30) are computed using the 10-day historical returns, which were used

for scenario generation in the CVaR optimization model:


-1 J -1


Figure 2-3 displays the CVaR-efficient portfolios in Return/CVaR scales for the risk confi-

dence level a = 0.95 (continuous line). Also, for each return it displays the CVaR of the MV











optimal portfolio (dashed line). Note, that for a given return, the MV optimal portfolio has a higher

CVaR risk level than the efficient Return/CVaR portfolio. Figure 2-4 displays similar graphs for

a = 0.99. The discrepancy between CVaR and MV solutions is higher for the higher confidence

level.

0.95-CVaR- and MV-Efficient Portfolios

3.5
3
2.5

S- CVaR portfolio
S- MV portfolio
o 1 /

0.5
0
0 0.05 0.1 0.15
CVaR


Figure 2-3: Efficient frontiers of CVaR- and MV-optimal portfolios. The CVaR-optimal portfolio
was obtained by maximizing expected returns subject to the constraint on portfolio's CVaR with
95%-confidence level (a = 0.95). The horizontal and vertical scales respectively display CVaR
and expected rate of return of a portfolio over a two week period.


Figure 2-5 displays the efficient frontier for Return/MV efficient portfolios (continuous line).

Also, for each return it displays the standard deviation of the CVaR optimal portfolio with confi-

dence level a = 0.95 (dashed line). As expected, for a given return, the CVaR optimal portfolio has

a higher standard deviation than the efficient Return/MV portfolio. Similar graphs are displayed in

Figure 2-6 for a = 0.99. The discrepancy between CVaR and MV solutions is higher for the higher

confidence level, similar to Figures 2-3, 2-4.

However, the difference between the MV and CVaR approaches is not very significant. Rela-

tively close graphs of CVaR- and MV-optimal portfolios indicate that a CVaR optimal portfolio is

"near optimal" in MV-sense, and vice versa, a MV-optimal portfolio is "near optimal" in CVaR-

sense. This agreement between the two solutions should not, however, be misleading in deciding

that the discussed portfolio management methodologies "are the same". The obtained results are

dataset-specific, and the closeness of solutions of CVaR and MV optimization problems is caused by

apparently "close-to-normal" distributions of the historical returns used in our case study. Including
















0.99-CVaR- and MV-Efficient Portfolios

3.5

3



2

1.5 ,

1

0.5


0 0.05 0.1 0.15 0.2 0.25
CVaR



S.... 2-4: ) :: .... ..:...:. of CVaR- and .. .... portfolios. i .. '' optim al :.... .. ,
was obtained by maximizing expected returns subject to :1.. constraint on portfolio's : with
Confidence level (a 0.99). i .. horizontal and vertical scales .. :.. ..... *..
and expected rate of return of a :.. -..: .. ... over a two week :. -1


0.95-CVaR- and MV-Efficient Portfolios


2.5

2

S1.5


-a


0 0.01 0.02 0.03 0.04 0.05
Standard deviation


CVaR.
MV portfolio


0.06 0.07


Figure 2-5: ii:.: ::: :..:::: : of CVaR- and M" i:H .. :: : .. The CVaR-optimal port-
S was obtained by maximizing .1 returns ::.i to :i: constraint on po:::. :. 's C' :'
with .. .:... .. level (a = 0.95). The horizontal and vertical scales :. i. 1 <, ;. .: the
standard deviation and .1 rate of return of a i.... '. :.. over a two week i..











0.99-CVaR- and MV-Efficient Portfolios

3.5

3
2.5 -

2
S- CVaR portfolio
S.- MV portfolio

( 1

0.5

0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Standard deviation


Figure 2-6: Efficient frontiers of CVaR- and MV-optimal portfolios. The CVaR-optimal port-
folio was obtained by maximizing expected returns subject to the constraint on portfolio's CVaR
with 99%-confidence level (a = 0.99). The horizontal and vertical scales respectively display the
standard deviation and expected rate of return of a portfolio over a two week period.


options in the portfolio or credit risk with skewed return distributions may lead to quite different

optimal solutions of the efficient MV and CVaR portfolios (Mausser and Rosen, 1999; Larsen et al.,

2002).

2.8 Concluding Remarks

The chapter extended the approach for portfolio optimization (Rockafellar and Uryasev, 2000),

which simultaneously calculates VaR and optimizes CVaR. We first showed (Theorem 2.3.1) that

for risk-return optimization problems with convex constraints, one can use different optimization

formulations. This is true in particular for the considered CVaR optimization problem. We then

showed (Theorems 2.4.1 and 2.4.2) that the approach by Rockafellar and Uryasev (2000) can be

extended to the reformulated problems with CVaR constraints and the weighted retur-CVaR per-

formance function. The optimization with multiple CVaR constrains for different time frames and

at different confidence levels allows for shaping distributions according to the decision maker's

preferences. We developed a model for optimizing portfolio returns with CVaR constraints using

historical scenarios and conducted a case study on optimizing portfolio of S&P100 stocks. The case

study showed that the optimization algorithm, which is based on linear programming techniques,

is very stable and efficient. The approach can handle large number of instruments and scenarios.







31

CVaR risk management constraints (reduced to linear constraints) can be used in various applica-

tions to bound percentiles of loss distributions.
















CHAPTER 3
COMPARATIVE ANALYSIS OF LINEAR PORTFOLIO REBALANCING STRATEGIES: AN
APPLICATION TO HEDGE FUNDS

In the previous chapter, we considered a case study of portfolio optimization of S&P 100

stocks under Conditional Value-at-Risk constraints. In this chapter, we perform further numeri-

cal analysis of the performance of portfolio optimization techniques based on CVaR risk measure,

as well as several other risk measures (Conditional Drawdown-at-Risk, Mean-Absolute Deviation,

and Maximum Loss). In particular, we perform in-sample and out-of-sample runs for a portfolio of

hedge funds (fund of funds). The common property of the considered risk management techniques

is that they admit formulation of a portfolio optimization model as a linear programming (LP) prob-

lem. The possibility to formulate and solve portfolio optimization problem as a linear programming

problem leads to efficient and robust portfolio allocation algorithms, which can successfully handle

optimization problems with thousands of instruments and scenarios.

We use in-sample and out-of-sample tests, which simulate a "real-life" portfolio behavior, to

investigate the performance of various risk constraints in the portfolio management algorithm. Our

numerical experiments show that imposing risk constraints may improve the "real" performance of

a portfolio rebalancing strategy in out-of-sample runs. It is also beneficial to combine several types

of risk constraints that control different sources of risk.

3.1 Introduction

This chapter applies risk management methodologies to the optimization of a portfolio of

hedge funds (fund of funds). We compare risk management techniques based on two recently

developed risk measures, Conditional Value-at-Risk and Conditional Drawdown-at-Risk with more

established Mean-Absolute Deviation, Maximum Loss, and Market-Neutrality approaches. These

risk management techniques utilize stochastic programming approaches and allow for construction

of linear portfolio rebalancing strategies, and, as a result, have proven their high efficiency in vari-

ous portfolio management applications (Andersson et al. (2001), Chekhlov et al. (2000), Krokhmal

et al. (2002), Rockafellar and Uryasev (2000, 2002)). The choice of hedge funds, as a subject for










the portfolio optimization strategy, is stimulated by a strong interest to this class of assets by both

practitioners and scholars, as well as by challenges related to relatively small datasets available for

hedge funds.

Recent studies1 of the hedge funds industry are mostly concentrated on the classification

of hedge funds and the relevant investigation of their activity. However, this chapter is focused

on possible realization of investment opportunities existing in this market from the viewpoint of

portfolio rebalancing strategies (for an extensive discussion of stochastic programming approaches

to hedge fund management, see Ziemba (2003)).

Hedge funds are investment pools employing sophisticated trading and arbitrage techniques

including leverage and short selling, wide usage of derivative securities etc. Generally, hedge funds

restrict share ownership to high net worth individuals and institutions, and are not allowed to offer

their securities to the general public. Many hedge funds are limited to 99 investors. This private

nature of hedge funds has resulted in few regulations and disclosure requirements, compared for ex-

ample, with mutual funds (however, stricter regulations exist for hedge funds trading futures). Also,

the hedge funds may take advantage of specialized, risk-seeking investment and trading strategies,

which other investment vehicles are not allowed to use.

The first official2 hedge fund was established in the United States by A. W. Jones in 1949,

and its activity was characterized by the use of short selling and leverage, which were separately

considered risky trading techniques, but in combination could limit market risk. The term "hedge

fund" attributes to the structure of Jones fund's portfolio, which was split between long positions in

stocks that would gain in value if market went up, and short positions in stocks that would protect

against market drop. Also, Jones has introduced another two initiatives, which became a common

practice in hedge fund industry, and with more or less variations survived to this day: he made the

manager's incentive fee a function of fund's profits, and kept his own capital in the fund, in this way

making the incentives of fund's clients and of his own coherent.



1 See, for example, papers by Ackermann et al. (1999), Amin and Kat (2001), Brown and Goet-
zmann (2000), Fung and Hsieh (1997, 2001, 2000), and Lhabitant (2001)
2 Ziemba (2003) traces early unofficial hedge funds, such as Keynes Chest Fund etc., that existed
in the 1920's to 1940's.










Nowadays, hedge funds become a rapidly growing part of the financial industry. According to

Van Hedge Fund Advisors, the number of hedge funds at the end of 1998 was 5830, they managed

311 billion USD in capital, with between $800 billion and $1 trillion in total assets. Nearly 80% of

hedge funds have market capitalization less than 100 million, and around 50% are smaller than $25

million, which indicates high number of new entries. More than 90% of hedge funds are located in

the U.S.

Hedge funds are subject to far fewer regulations than other pooled investment vehicles, espe-

cially to regulations designed to protect investors. This applies to such regulations as regulations

on liquidity, requirements that fund's shares must be redeemable an any time, protecting conflicts

of interests, assuring fairness of pricing of fund shares, disclosure requirements, limiting usage of

leverage, short selling etc. This is a consequence of the fact that hedge funds' investors qualify

as sophisticated high-income individuals and institutions, which can stand for themselves. Hedge

funds offer their securities as private placements, on individual basis, rather than through public

advertisement, which allows them to avoid disclosing publicly their financial performance or asset

positions. However, hedge funds must provide to investors some information about their activity,

and of course, they are subject to statutes governing fraud and other criminal activities.

As market's subjects, hedge funds do subordinate to regulations protecting the market integrity

that detect attempts of manipulating or dominating in markets by individual participants. For exam-

ple, in the United States hedge funds and other investors active on currency futures markets, must

regularly report large positions in certain currencies. Also, many option exchanges have developed

Large Option Position Reporting System to track changes in large positions and identify outsized

short uncovered positions.

In this chapter, we consider problem of managing fund of funds, i.e., constructing optimal

portfolios from sets of hedge funds, subject to various risk constraints, which control different types

of risks. However, the practical use of the strategies is limited by restrictive assumptions3 imposed

in this case study:



3 These assumptions can be relaxed and incorporated in the model as linear constraints. Here
we focus on comparison of risk constraints and have not included other constraints.










Liquidity considerations are not taken into account

No transaction costs

Considered funds may be closed for new investors

Credit and other risks which directly are not reflected in the historical return data are not taken

into account

Survivorship bias is not considered.

The obtained results cannot be treated as direct recommendations for investing in hedge funds mar-

ket, but rather as a description of the risk management methodologies and portfolio optimization

techniques in a realistic environment. For an overview of the potential problems related to the data

analysis and portfolio optimization of hedge funds, see Lo (2001).

Section 3.2 presents an overview of linear portfolio optimization algorithms and the related

risk measures, which are explored in this chapter. Section 3.3 contains description of our case

study, results of in-sample and out-of-sample experiments and their detailed discussion. Section 3.4

presents the concluding remarks.

3.2 Linear Portfolio Rebalancing Algorithms

Formal portfolio management methodologies assume some measure of risk that impacts allo-

cation of instruments in the portfolio. The classical Markowitz theory, for example, identifies risk

with the volatility (standard deviation) of a portfolio. In this study we investigate a portfolio opti-

mization problem with several different constraints on risk: Conditional Value-at-Risk (Rockafellar

and Uryasev, 2000, 2002), Conditional Drawdown-at-Risk (Chekhlov et al., 2000), Mean-Absolute

Deviation (Konno and Yamazaki, 1991; Konno and Shirakawa, 1994; Konno and Wijayanayake,

1999), Maximum Loss (Young, 1998) and the market-neutrality ("beta" of the portfolio equals

zero).4 CVaR and CDaR risk measures represent relatively new developments in the risk manage-

ment field. Application of these risk measures to portfolio allocation problems relies on the scenario

representation of uncertainties and stochastic programming approaches.



4 There are different interpretations for the term "market-neutral" (see, for instance, BARRA
RogersCasey 2002). In this chapter market neutrality means zero beta.










A linear portfolio rebalancing ,:ig /, t ,i is a trading (investment) strategy with mathematical

model that can be formulated as a linear programming (LP) problem. The focus on LP techniques in

application to portfolio rebalancing and trading problems is explained by exceptional effectiveness

and robustness of LP algorithms, which become especially important in finance applications. Recent

developments (see, for example, Andersson et al. (2001), Carifio and Ziemba (1998), Carifio et al.

(1998), Chekhlov et al. (2000), Consigli and Dempster (1997, 1998), Dembo and King (1992),

Duarte (1999), Krokhmal et al. (2002), Rockafellar and Uryasev (2000, 2002), Turner et al. (1994),

Zenios (1999), Ziemba and Mulvey (1998), Young (1998)) show that LP-based algorithms can

successfully handle portfolio allocation problems with thousands and even millions of decision

variables and scenarios, which makes those algorithms attractive to institutional investors.

In the cited papers, along with Conditional Value-at-Risk and Conditional Drawdown-at-Risk,

other, much earlier established measures of risk, such as Maximum Loss, Mean-Absolute Deviation,

Low Partial Moment with power one and Expected Regret5 have been employed in the framework

of linear portfolio rebalancing algorithms (see, for example, Ziemba and Vickson (1975)). Some

of these risk measures are quite closely related to CVaR concept.6 We restricted ourselves to

considering CVaR- and CDaR-based risk management techniques.

However, the class of linear trading or portfolio optimization techniques is far from encom-

passing the entire universe of portfolio management techniques. For example, the famous portfolio



5 Low partial moment with power one is defined as the expectation of losses exceeding some
fixed threshold, see Harlow (1991). Expected regret (see, for example, Dembo and King (1992)) is
a concept similar to the lower partial moment. However, the expected regret may be calculated with
respect to a random benchmark, while the low partial moment is calculated with respect to a fixed
threshold.
6 Maximum Loss is a limiting case of CVaR risk measure (see below). Also, Testuri and Uryasev
(2000) showed that the CVaR constraint and the low partial moment constraint with power one are
equivalent in the sense that the efficient frontier for portfolio with CVaR constraint can be generated
by the low partial moment approach. Therefore, the risk management with CVaR and with low
partial moment leads to similar results. However, the CVaR approach allows for direct controlling
of percentiles, while the low partial moment penalizes losses exceeding some fixed thresholds.









optimization model by Markowitz (1952, 1991), which utilizes the mean-variance approach, be-

longs to the class of quadratic programming (QP) problems; the well-known constant-proportion

rule leads to nonconvex multiextremum problems, etc.

3.2.1 Conditional Value-at-Risk

The Conditional Value-at-Risk (CVaR) measure (Rockafellar and Uryasev, 2000, 2002) devel-

ops and enhances the ideas of risk management, which have been put in the framework of Value-at-

Risk (VaR) (see, for example, Duffie and Pan (1997), Jorion (1997), Pritsker (1997), Staumbaugh

(1996)). Incorporating such merits as easy-to-understand concept, simple and convenient represen-

tation of risks (one number), applicability to a wide range of instruments, VaR has evolved into a

current industry standard for estimating risks of financial losses. Basically, VaR answers the ques-

tion "what is the maximum loss, which is expected to be exceeded, say, only in 5% of the cases

within the given time horizon?" For example, if daily VaR for the portfolio of some fund XYZ is

equal to 10 millions USD at the confidence level 0.95, it means that there is only a 5% chance of

losses exceeding 10 millions during a trading day.

The formal definition of VaR is as follows. Consider a loss function f(x, y), where x is a

decision vector (e.g., portfolio positions), and y is a stochastic vector standing for market uncertain-

ties (in this chapter, y is the vector of returns of instruments in the portfolio). Let Y(x, 4) be the

cumulative distribution function of f(x, y),


S(x, ) P [f(x, y) <].

Then, the Value-at-Risk function (x) with the confidence level a is the a-quantile of f(x, y) (see

Figure 3-1):

a(x) =mmin (x, ) > a}.
eR
Using VaR as a risk measure in portfolio optimization is, however, a very difficult problem, if the

return distributions of a portfolio's instruments are not normal or log-normal. The optimization

difficulties with VaR are caused by its non-convex and non-subadditive nature (Artzner et al., 1997,

1999; Mausser and Rosen, 1999). Non-convexity of VaR means that as a function of portfolio

positions, it has multiple local extrema, which precludes using efficient optimization techniques.









The difficulties with controlling and optimizing VaR in non-normal portfolios have forced the
search for similar percentile risk measures, which would also quantify downside risks and at the
same time could be efficiently controlled and optimized. From this viewpoint, CVaR is a perfect

candidate for conducting a "VaR"-style portfolio management.
For continuous distributions, CVaR is defined as an average (expectation) of high losses re-
siding in the a-tail of the loss distribution, or, equivalently, as a conditional expectation of losses

exceeding the a-VaR level (Fig. 3-1). From this follows that CVaR incorporates information on
VaR and on the losses exceeding VaR.
For general (non-continuous) distributions, Rockafellar and Uryasev (2002) defined a-CVaR

function a (x) as the a-tail expectation of a random variable z,


0a(x) Ea tail [z],

where the a-tail cumulative distribution functions ofz has the form


oY(x, ) P [z = 0, C < a(x),
S [T(x,C)-a]/[l-a], _>Ca(x).

Also, Acerbi et al. (2001), Acerbi and Tasche (2002) redefined the expected shortfall similar to the
CVaR definition presented above.

Along with a CVaR function 0a(x), the following functions called "upper" and "lower"
CVaR (a-CVaR+ and a-CVaR ), are considered:

a+(x) E [f(x,y) f(x, y) > Ca (x)],

(x) = E [f(x,y) I f(x, y) > Ca(x)].

The CVaR functions satisfy the following inequality:


X(x) 0a(x) _< + (x).

Rockafellar and Uryasev (2002) showed that a-CVaR can be presented as a convex combination of

a-VaR and a-CVaR+,


00,(X) ;LO,(X) 0, (X) + I I ;LO,(X) I OU,(X),















c

o" i Maximum
h | VaR loss
--



Portfolio loss

Figure 3-1: Loss distribution, VaR, CVaR, and Maximum Loss.


where

aa(x) [T(x, Ca(x)) a]/[ a], O< Aa(x)< 1.

For a discrete loss distribution, where the stochastic parameter y may take values yi, y2,..., J with

probabilities 0,, j = 1,...,J, the a-VaR and a-CVaR functions respectively are7


Ca(x) = f(x, y,),


a(x = [(IX-a) f(x,y j)+ 0 f(x,y) ,
a (x) 1-a X tJIykAY+}
L\=1 I I=Ja 1+
where ja satisfies
Ja-1 Ja
X QJ J=1 -=1
For values of confidence level a close to 1, Conditional Value-at-Risk coincides with the Maximum

Loss (see Figure 3-1).

While inheriting some of the nice properties of VaR, such as measuring downside risks and

representing them by a single number, applicability to instruments with non-normal distributions

etc., CVaR has substantial advantages over VaR from the risk management standpoint. First of all,

CVaR is a convex function8 of portfolio positions. Hence, it has a convex set of minimum points




7 This proposition has been derived in assumption that, without loss of the generality, scenarios
Y1, Y2, ...,YJ satisfy inequalities f(x, yi) < ... < f(x, y).

8 For a background on convex functions and sets see Rockafellar (1970).











on a convex set, which greatly simplifies control and optimization of CVaR. Calculation of CVaR,

as well as its optimization, can be performed by means of a convex programming shortcut (Rock-

afellar and Uryasev, 2000, 2002), where the optimal value of CVaR is calculated simultaneously

with the corresponding VaR; for linear or piecewise-linear loss functions these procedures can be

reduced to linear programming problems. Also, unlike a-VaR, a-CVaR is continuous with respect

to confidence level a. A comprehensive description of the CVaR risk measure and CVaR-related

optimization methodologies can be found in Rockafellar and Uryasev (2000, 2002). Also, Rock-

afellar and Uryasev (2000) showed that for normal loss distributions, the CVaR methodology is

equivalent to the standard Mean-Variance approach. Similar result also was independently proved

for elliptic distributions by Embrechts et al. (1997).




0007
0 006
0005
0004
0 003
0002 -
0001

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35
time (business days)
-*-Portfolio value-- DrawDown function


Figure 3-2: Portfolio value and drawdown.


According to Rockafellar and Uryasev (2000, 2002), the optimization problem with multiple

CVaR constraints
min g(x)
xCX

subject to 0a,(x) < i ,...,I,

is equivalent to the following problem:


min g(x)
xGX, kcR, Vk
1 J
subjectto +-1 0,max{0, f(x, yj)- } < k, k 1,...,K,
1 ak1










provided that the objective function g(x) and the loss function f(x, y) are convex in x E X. When

the objective and loss functions are linear in x and constraints x EX are given by linear inequalities,

the last optimization problem can be reduced to LP, see Rockafellar and Uryasev (2002, 2000).

Except for the fact that CVaR can be easily controlled and optimized, CVaR is a more adequate

measure of risk as compared to VaR because it accounts for losses beyond the VaR level. The

fundamental difference between VaR and CVaR as risk measures are: VaR is the "optimistic" low

bound of the losses in the tail, while CVaR gives the value of the expected losses in the tail. In risk

management, we may prefer to be neutral or conservative rather than optimistic. Moreover, CVaR

satisfies several nice mathematical properties and is coherent in the sense of Artzner et al. (1999,

1997).

3.2.2 Conditional Drawdown-at-Risk

Conditional Drawdown-at-Risk (CDaR) is a portfolio performance measure (Chekhlov et al.,

2000) closely related to CVaR. By definition, a portfolio's drawdown on a sample-path is the drop

of the uncompounded9 portfolio value as compared to the maximal value attained in the previous

moments on the sample-path. Suppose, for instance, that we start observing a portfolio in Jan-

uary 2001, and record its uncompounded value every month.10 If the initial portfolio value was

$100,000,000 and in February it reached $130,000,000, then, the portfolio drawdown as of February

2001 is $0. If, in March 2001, the portfolio value drops to $90,000,000, then the current drawdown

equals $40,000,000 (in absolute terms), or 30.77%. Mathematically, the drawdown function for a

portfolio is

f(x, t) max {v,(x)} vt(x), (3.1)




9 Drawdowns are calculated with uncompounded portfolio returns. This is related to the fact that
risk measures based on drawdowns of uncompounded portfolios have nice mathematical properties.
In particular, these measures are convex in portfolio positions. Suppose that at the initial moment
t = 0 the portfolio value equals v and portfolio returns in the moments t =1,..., T equal rl,..., rT.
By definition, the uncompounded portfolio value v, at time moment z equals v, = vtl rt. We
assume that the initial portfolio value v = 1.

10 Usually, portfolio value is observed much more frequently. However, for the hedge funds
considered in this chapter, data are available on monthly basis.










where x is the vector of portfolio positions, and vt(x) is the uncompounded portfolio value at time

t. We assume that the initial portfolio value is equal to 1; therefore, the drawdown is the uncom-

pounded portfolio return starting from the previous maximum point. Figure 3-2 illustrates the

relation between the portfolio value and the drawdown.

The drawdown quantifies the financial losses in a conservative way: it calculates losses for the

most "unfavorable" investment moment in the past as compared to the current (discrete) moment.

This approach reflects quite well the preferences of investors who define their allowed losses in

percentages of their initial investments (e.g., an investor may consider it unacceptable to lose more

than 10% of his investment). While an investor may accept small drawdowns in his account, he

would definitely start worrying about his capital in the case of a large drawdown. Such drawdown

may indicate that something is wrong with that fund, and maybe it is time to move the money to

a more successful investment pool. The mutual and hedge fund concerns are focused on keeping

existing accounts and attracting new ones; therefore, they should ensure that clients' accounts do

not have large drawdowns.

One can conclude that drawdown accounts not only for the amount of losses over some period,

but also for the sequence of these losses. This highlights the unique feature of the drawdown

concept: it is a loss measure "with memory" taking into account the time sequence of losses.

For a specified sample-path, the drawdown function is defined for each time moment. How-

ever, in order to evaluate performance of a portfolio on the whole sample-path, we would like to

have a function, which aggregates all drawdown information over a given time period into one

measure. As this function one can pick, for example, the Maximum Drawdown,


MaxDD= max {f(x,t)},
0
or the Average Drawdown,
T
AverDD J(x, t)dt.
0
However, both these functions may inadequately measure losses. The Maximum Drawdown is

based on one "worst case" event in the sample-path. This event may represent some very specific

circumstances, which may not appear in the future. The risk management decisions based only on

this event may be too conservative.









On the other hand, the Average Drawdown takes into account all drawdowns in the sample-
path. However, small drawdowns are acceptable (e.g., 1-2% drawdowns) and averaging may mask
large drawdowns.
Chekhlov et al. (2000) suggested a new drawdown measure, Conditional Drawdown-at-Risk,
that combines both the drawdown concept and the CVaR approach. For instance, 0.95-CDaR can
be thought of as an average of 5% of the highest drawdowns. Formally, a-CDaR is a-CVaR with

drawdown loss function (x, t) given by (3.1). Namely, assume that possible realizations of the
random vectors describing uncertainties in the loss function is represented by a sample-path (time-
dependent scenario), which may be obtained from historical or simulated data. In this chapter, it

is assumed that we know one sample-path of returns of instruments included in the portfolio. Let
r,j be the rate of return of i-th instrument in j-th trading period (that corresponds toj-th month in
the case study, see below), j = 1,...,J. Suppose that the initial portfolio value equals 1. Let x,,
i = l,...,n be weights of instruments in the portfolio. The uncompounded portfolio value at time j
equals

vj(x) 1 +[ rzs x,.
,=1 s=l
The drawdown function f(x, r ) at the time j is defined as the drop in the portfolio value compared

to the maximum value achieved before the time moment j,


f(x,j) max rIsxl, rls, x,.
1k Ij _=1 s=l 1=1 s=l

Then, the Conditional Drawdown-at-Risk function Aa(x) is defined as follows. If the parameter a
and number of scenarios J are such that their product (1 a) J is an integer number, then Aa(x) is
defined as

Aa(x)= ra + (- max 0, max [ (kris, x,
(1a)J 1 k

,=1 \s=l 1

where rja = 7a(x) is the threshold that is exceeded by (1 a)J drawdowns. In this case the

drawdown functions Aa (x) is the average of the worst case (1 a) J drawdowns observed in the
considered sample-path. If (1 a)J is not integer, then the CDaR function, Aa(x), is the solution










of


Aa(x) = mmn r+ +
TI I a \
xJmaxn nkn
x max 0, max rn s x( rs( x, r
J=1 1
The CDaR risk measure holds nice properties of CVaR such as convexity with respect to portfolio

positions. Also CDaR can be efficiently treated with linear optimization algorithms (Chekhlov et al.,

2000).

3.2.3 Mean-Absolute Deviation

The Mean-Absolute Deviation (MAD) risk measure was introduced by Konno and Yamazaki

(1991) as an alternative to the classical Mean-Variance measure of a portfolio's volatility,


MAD E[rp(x) -E[rp(x)]|],


where rp(x) = rlx + r2x2 +.. + rnn is the portfolio's rate of return, with rl, ...,rn being the random

rates of return for instruments in the portfolio. Since MAD is a piecewise linear convex function

of portfolio positions, it allows for fast efficient portfolio optimization procedures by means of

linear programming, in contrast to Mean-Variance approach, which leads to quadratic optimization

problems. Konno and Shirakawa (1994) showed that MAD-optimal portfolios exhibit properties,

similar to those of Markowitz MV-optimal portfolios, and that one can use MAD as a risk measure in

deriving CAPM-type relationships. Later, it was also proved (Ogryczak and Ruszczynski, 1999) that

portfolios on the MAD efficient frontier correspond to efficient portfolios in terms of the second-

order stochastic dominance.

Formally, the Mean-Absolute Deviation (MAD) g(x) of portfolio's rate of return equals

n n
(x) = E rx, E [ x, ,
1=1 i=1
see Konno and Yamazaki (1991). We suppose that j = 1,...,J scenarios of returns with probabilities

0, are available. Let us denote by r,j the return of i-th asset in the scenario j. The portfolio's MAD
can be written as











J n k n
;(x) = O Irx, Ok r, k
=l1 z=1 k=l z=1

3.2.4 Maximum Loss

The Maximum Loss (MaxLoss) of a portfolio in a specified time period is defined as the

maximal value over all random loss outcomes (Fig. 3-1), see for instance, Young (1998). When the

distribution of losses is continuous, this risk measure may be unbounded, unless the distribution is

"truncated". For example, for normal distribution, the maximum loss is infinitely large. However,

for discrete loss distributions, especially for those based on small historical datasets, the MaxLoss

is a reasonable measure of risk. We also would like to point out that the Maximum Loss admits an

alternative definition as a special case of a-CVaR with a close to 1.

Let us suppose that = 1,...,J scenarios of returns are available (r,j denotes return of i-th asset

in the scenario j). The Maximum Loss (MaxLoss) function has the form (see for instance, Young

(1998))



(x) max X rI, .
1
It is worth noting that a-CVaR function 0, (x) coincides with MaxLoss for values a close to 1. Sup-

pose that scenarios j 1,...,J have equal probabilities l/J. When the confidence level & satisfies
J-1
a > the MaxLoss equals to a-CVaR function MT(x) = a(x).

3.2.5 Market-Neutrality

It is generally acknowledged that the market itself constitutes a risk factor. If the instruments

in the portfolio are positively correlated with the market, then the portfolio would follow not only

market growth, but also market drops. Naturally, portfolio managers are willing to avoid situations

of the second type, by constructing portfolios, which are uncorrelated with market, or market-

neutral. To be market-uncorrelated, the portfolio must have zero beta,


p=_ XPI,x,= 0,
i=-










where xi,...,xn denote the proportions in which the total portfolio capital is distributed among n

assets, and 0, are betas of individual assets,

Cov (r, rM)
Var (rm)'

where rM stands for market rate of return. Instruments' betas, j,, can be estimated, for example,

using historical data:


0-- (r (;-, -r
S ( riw)2) (r1,1 r1) (rI r-M),


where J is the number of historical observations, and r denotes the sample average, r = 1 rj.

As a proxy for market returns rM, historical returns of the S&P500 index can be used.

In our case study, we investigate the effect of constructing a market-neutral (zero-beta) portfo-

lio, by including a market-neutrality constraint in the portfolio optimization problem. We compare

the performance of the optimal portfolios obtained with and without market-neutrality constraint.

3.2.6 Problem Formulation

This section presents the "generic" problem formulation, which was used to construct an opti-

mal portfolio. We suppose that some historical sample-path of returns ofn instruments is available.

Based on this sample-path, we calculate the expected return of the portfolio and the various risk

measures for that portfolio. We maximize the expected return of the portfolio subject to different

operating, trading, and risk constraints,

n
max E [ rix (3.2)
x =1

subject to

0 n
Ix, < 1, (3.4)
i=1
(

i=1
where


x, is the portfolio position (weight) of asset i










r, is the (random) rate of return of asset i

3, is market beta of instrument i.

The objective function (3.2) represents the expected return of the portfolio. The first constraint

(3.3) of the optimization problem imposes limitations on the amount of funds invested in a single

instrument (we do not allow short positions). The second constraint (3.4) is the budget constraint.

Constraints (3.5) control risks of financial losses. The key constraint in the presented approach is the

risk constraint (3.5). Function (Risk(X1, ...,n) represents either a CVaR or a CDaR risk measure,

and risk tolerance level o is the fraction of the portfolio value that is allowed for risk exposure.

Constraint (3.6), with /, representing market's beta for instrument i, forces the portfolio to be

market-neutral in the "zero-beta" sense, i.e., the portfolio correlation with the market is bounded.

The coefficient k in (3.6) is a small number that sets the portfolio's beta close to zero. To investigate

the effects of imposing a "zero-beta" requirement on the portfolio-rebalancing algorithm, we solved

the optimization problem with and without this constraint. Constraint (3.6) significantly improves

the out-of-sample performance of the algorithm.

The risk measures considered in this chapter allow for formulating the risk constraint (3.5)

in terms of linear inequalities, which makes the optimization problem (3.2)-(3.6) linear, given the

linearity of objective function and other constraints. Below we present the explicit form of the risk

constraint (3.5) for CVaR and CDaR risk measures.

3.2.7 Conditional Value-at-Risk Constraint

The loss function incorporated into CVaR constraint, is the negative portfolio's return,


f(x,y) r,x,, (3.7)
i=1

where the vector of instruments' returns y = r = (ri,..., r) is random. The risk constraint (3.5),

0a(x) < o), where CVaR risk function replaces the function ORisk(x), is


c+ -max 0, rx,- < 0), (3.8)
0(1 a) j=l =1









where r,, is return of i-th instrument in scenario j, j = 1,...,J. Since the loss function (3.7) is linear,

the risk constraint (3.8) can be equivalently represented by the linear inequalities,

1 1 J
+ j j < ),


-_r'jx> -C < w, j=l 1,...,J, (3.9)
S=1
CeR, w, > 0, j 1,..., J.

This representation allows for reducing the optimization problem (3.2)-(3.6) with the CVaR con-

straint to a linear programming problem.

3.2.8 Conditional Drawdown-at-Risk Constraint

The CDaR risk constraint Aa (x) < o has the form

1 1 J n ( k \ X 1 n 1 X I
1r + max 0, max r x, rl r, x, 1-a J i Fl 1 1 \S=l I= \S=1

and it can be reduced to a set of linear constraints similarly to the CVaR constraint.

3.2.9 MAD Constraint

Given equal scenario probabilities, the MAD constraint g(x) < o has the form
J n J n

J =1 1Xr1 -k=l=l <

This constraint admits representation by linear inequalities,

1J
S(u+ +u) < ,
j=l
= X 1 i
n 1 J n
rZX' rZ x I u j 1,... ,J,

u > 0, j 1,...,J.


3.2.10 MaxLoss Constraint

The MaxLoss constraint m(x) < 0) in (4.5) can be written as

lmaxj I <0).
max -rux<}
1









Similar to other considered risk constraints, it can be replaced by a system of linear inequalities

W < 0),

rx, < w, j 1,...,J.
z=1




3.3 Case Study: Portfolio of Hedge Funds

The case study investigates investment opportunities and tests portfolio management strategies

for a portfolio of hedge funds. Hedge funds are subject to less regulations as compared with mutual

or pension funds. Hence, very little information on hedge funds' activities is publicly available (for

example, many funds report their share prices only monthly). On the other hand, fewer regulations

and weaker government control provide more room for aggressive, risk-seeking trading and invest-

ment strategies. As a consequence, the revenues in this industry are on average much higher than

elsewhere, but the risk exposure is also higher (for example, the typical "life" of a hedge fund is

about five years, and very few of them perform well in long run). Data availability and sizes of

datasets impose challenging requirements on portfolio rebalancing algorithms. Also, the specific

nature of hedge fund securities imposes some limitations on using them in trading or rebalancing

algorithms. For example, hedge funds are far from being perfectly liquid: hedge funds may not be

publicly traded or may be closed to new investors. From this point of view, our results contain a

rather schematic representation of investment opportunities existing in the hedge fund market and

do not give direct recommendations on investing in that market. The goal of this study is to com-

pare the recently developed risk management approaches and to demonstrate their high numerical

efficiency in a realistic setting.

The dataset for conducting the numerical experiments was provided to the authors by the Foun-

dation for Managed Derivatives Research. It contained a monthly data for more than 5000 hedge

funds, from which we selected those with significantly long history and some minimum level of

capitalization. To pass the selection, a hedge fund should have 66 months of historical data from

December 1995 to May 2001, and its capitalization should be at least 5 million U.S. dollars at the

beginning of this period. The total number of funds, which satisfied these criteria and accordingly

constituted the investment pool for our algorithm, was 301. In this dataset, the field with the names











of hedge funds was unavailable; therefore, we identified the hedge funds with numbers, i.e., HF 1,

HF 2, and so on. The historical returns from the dataset were used to generate scenarios for al-

gorithm (3.2)-(3.6). Each scenario is a vector of monthly returns for all securities involved in the

optimization, and all scenarios are assigned equal probabilities.

We performed separate runs of the optimization problem (3.2)-(3.5), with and without con-

straint (3.6) with CVaR and CDaR risk measure in constraint (3.5), varying such parameters as

confidence levels, risk tolerance levels etc.

The case study consisted from two sets of numerical experiments. The first set of in-sample

experiments included the calculation of efficient frontiers and the analysis of the optimal portfolio

structure for each of the risk measures. The second set of experiments, out-of-sample testing, was

designed to demonstrate the performance of our approach in a simulated historical environment.

3.3.1 In-Sample Results

Efficient frontier. For constructing the efficient frontier for the optimal portfolio with differ-

ent risk constraints, we solved the optimization problem (3.2)-(3.5) with different risk tolerance lev-

els o in constraint (3.5), varied from o = 0.005 to o = 0.25. The parameter a in CVaR and CDaR

risk constraints was set to a = 0.90. The efficient frontier is presented in Figure 3-3, where the

portfolio rate of return means expected monthly rate of return. In these runs, the market-neutrality

constraint (3.6) is inactive.


Efficient Frontier


6 CVaR MAD
50

I 40 U-o CVaR
CDaR Maxloss CDaR
) 30
S/ MAD
o 20 MaxLoss
10
0
0 .
05 2 35 5 65 8 95 11 125 14 155 17 185 20 215 23 245
Risk tolerance, %


Figure 3-3: Efficient frontiers for portfolio with various risk constraints (k = 0.01). The market-
neutrality constraint is inactive.











Figure 3-3 shows that three CVaR-related risk measures (CVaR, CDaR and MaxLoss) pro-

duce relatively similar efficient frontiers. However, the MAD risk measure produces a distinctively

different efficient frontier.

For optimal portfolios, in the sense of problem (3.2)-(3.5), there exists an upper bound (equal

to 48.13%) for the portfolio's rate of return. Optimal portfolios with CVaR, MAD and MaxLoss

constraints reach this bound at different risk tolerance levels, but the CDaR-constrained portfolio

does not achieve the maximal expected return within the given range of w0 values. CDaR is a

relatively conservative constraint imposing requirements not only on the magnitude of loses, but

also on the time sequence of losses (small consecutive losses may lead to large drawdown, without

significant increasing of CVaR, MaxLoss, and MAD).

Figure 3-4 presents efficient frontiers of optimal portfolio (3.2)-(3.5) with the active market-

neutrality constraint (3.6), where coefficient k is equal to 0.01. As one should expect, imposing

the extra constraint (3.6) causes a decrease in in-sample optimal expected return. For example, the

"saturation" level of the portfolio's expected return is now 41.94%, and all portfolios reach that level

at much lower values of risk tolerance w. However, the market-neutrality constraint almost does not

affect the curves of efficient portfolios in the leftmost points of efficient frontiers, which correspond

to the lowest values of risk tolerance w. This means that by tightening the risk constraint (3.5) one

can obtain a nearly market-neutral portfolio without imposing the market-neutrality constraint (3.6).


Efficient Frontier with Zero-beta Constraint

50
CVaR


S40 CVaR
MaxLoss CDaR
| 35 MAD CDaR MAD
2- MAD
o 30 MaxLoss
25
a.
05 15 25 35 45 55 65 75 85 95 105 115 125
Risk tolerance, %


Figure 3-4: Efficient frontier for market-neutral portfolio with various risk constraints (k = 0.01).










Quite high rates of return for the efficient portfolios can be explained by the fact that 301

funds, selected to form the optimal portfolios, constitute about 6% of the initial hedge fund pool,

and already are "the best of the best" in our data sample.

Optimal portfolio configuration. Let us discuss now the structure of the optimal portfolio

with various risk constraints. For this purpose, we selected the corresponding efficient frontiers for

four optimal portfolios with expected return of 35% and constraints on CVaR, CDaR, MaxLoss,

and MAD (market-neutrality is inactive). For three risk measures (CVaR, CDaR and MaxLoss), the

optimal portfolios with expected return of 35% are located in the vicinity of the leftmost points on

corresponding efficient frontiers (see Fig. 3-3). Table 3-1 presents portfolio weights for the four

optimal portfolios. It shows how a particular risk measure selects instruments given the specified

expected return. The left column of Table 3-1 contains the set of the assets, which are chosen by

the algorithm (3.2)-(3.5) under different risk constraints. Note that among the 301 available instru-

ments, only a few of them are used in constructing the optimal portfolio. Moreover, a closer look

at Table 3-1 shows that nearly two thirds of the portfolio value for all risk measures is formed by

three hedge funds HF 209, HF 219 and HF 231 (the corresponding lines are typeset in boldface).

These three hedge funds have stable performance, and each risk measure includes them in the op-

timal portfolio. Similarly, lines typeset in slanted font, indicate instruments that are included in the

portfolio with smaller weights, but still are approximately evenly distributed among the portfolios.

Thus, the instruments HF 93, 100, 209, 219, 231, 258, and 259 constitute the "core" of the optimal

portfolio under all risk constraints. The last row in Table 3-1 lists the total weight of these instru-

ments in corresponding optimal portfolio. The remaining assets (without highlighting in the table)

are "residual" instruments, which are specific to each risk measure. They may help us to spot differ-

ences in instrument selection of each risk constraint. Table 3-5 displays the residual weights of HF

49, 84, 106, 124, 126, 169, 196, and 298. The "residual" weights are calculated as the instrument's

weight with respect to the residual part of the portfolio. For example, in the optimal portfolio with

the CDaR constraint, the hedge fund HF 49 represents 11.02% of the total portfolio value, and at

the same time it represents 49.06% of the residual (1.00 0.775) 100% portfolio value. In other

words, it occupies almost half of the portfolio assets, not captured by hedge funds in the grayed

cells. Also, note that neither of the residual instruments is simultaneously present in all portfolios.
















Table 3-1: Instrument- *. 1 '. in the ... .1 .. :'W..::.. with different risk constraints

CDaR CVaR MAD MaxLoss
HF 49 0.110216 0.043866 0 0.191952
HF 84 0.041898 0 0 0.078352
HF 93 0.081394 0.08754 0., 0.062609
HF 100 0.06629 0.073659 0.023899 0.068993
HF 106 0 0 0 0.00191
HF 124 0 0 0.020289 0
HF 126 0 0.008673 0.027908 0
HF 169 0.054298 0.010144 0 0
HF 196 0 0.015627 0.084791 0
HF 209 I : ::: :.- :.;.:. 0.260824 0.226262
HF 219 0.137165 0.259254 0.169239 0.111746
HF 231 0.183033 i: :' .:. 0.137068 0.177008
HF 258 0.034083 0.014156 0.097597 0.012606
HF 259 0.058684 0.089403 0.133104 0.068562
HF 298 0.018257 0.0038 0 0
0.775331 0.917889 0.867012 0.727785











Table 3-2: Weights of residual instruments in the ( : :.. :1 portfolio with .:.- :..: risk constraints

CDaR CVaR MAD MaxLoss
HF 49 0.490571 0.534229 0 0.705151
HF 84 0.186487 0 0 0.287833
HF 93 XXXXXXXX XXXXXXXX XXXXXXXX XXXXXXXX
HF 100 XXXXXXXX XXXXXXXX XXXXXXXX XXXXXXXX
HF 106 0 0 0 0.007016
HF 124 0 0 0.152561 0
HF 126 0 0.105624 0.209855 0
HF 169 0.24168 0.123544 0 0
HF 196 0 0.19031 0.637582 0
HF 209 XXXXXXXX XXXXXXXX XXXXXXXX XXXXXXXX
HF 219 XXXXXXXX XXXXXXXX XXXXXXXX XXXXXXXX
HF 231 XXXXXXXX XXXXXXXX XXXXXXXX XXXXXXXX
HF 258 XXXXXXXX XXXXXXXX XXXXXXXX XXXXXXXX
HF 259 XXXXXXXX XXXXXXXX XXXXXXXX XXXXXXXX
HF 298 0.081261 0.046285 0 0










Figure 3-5 contains graphs of the historical return and price dynamics for the residual hedge

funds. We included these graphs to illustrate differences in risk constraints and to make some

speculations on this subject.

For example, instrument HF 84 is selected by CDaR and MaxLoss risk measures, but is ex-

cluded by CVaR and MAD. Note that graph of returns for the instrument HF 84 shows no negative

monthly returns exceeding 10%, which is probably acceptable for the MaxLoss risk measure. Also,

the price graph for the instrument HF 84 shows that it exhibits few drawdowns; therefore, CDaR

picked this instrument. However, MAD probably excluded instrument HF8 because it had a high

monthly return of 30% (recall that MAD does not discriminate between high positive returns and

high negative returns). It is not clear from the graphs why CVaR rejected the HF 84 instruments.

Probably, other instruments had better CVaR-return characteristics from the view point of the over-

all portfolio performance.

The hedge fund HF 124 has not been chosen by any risk measures, with the exception of

MAD. Besides rather average performance, it suffers long-lasting drawdowns (CDaR does not like

this), has multiple negative return peaks of -10% magnitude (CVaR does not favor that), and its

worst negative return is almost -20% (MaxLoss must protect from such performance drops). The

question why this instrument was not rejected by MAD cannot be clearly answered in this case. Do

not forget that such a decision is a solution of an optimization problem, and different instruments

with properly adjusted weights may compensate each other's shortcomings. This may also be an

excuse for MAD not picking the HF 49 fund, whose merits are confirmed by high residual weights

of this fund in CDaR, CVaR, and MaxLoss portfolios.

Fund HF 126 has the highest expected return among residual instruments, but it also suffers the

most severe drawdowns and has the highest negative return (exceeding -20%) that's, probably,

why algorithms with CDaR and MaxLoss measures rejected this instrument.

3.3.2 Out-of-Sample Calculations

The out-of-sample testing of the portfolio optimization algorithm (3.2)-(4.6) sheds light on

the "actual" performance of the developed approaches. In other words, the question is how well do

the algorithms with different risk measures utilize the scenario information based on past history in

producing a successful portfolio management strategy? An answer can be obtained, for instance,

by interpreting the results of the preceding section as follows: suppose we were back in May 2001,




























-0 -

-02

-03



HF 84

04

03

02

0 1



-0 1

-02

-03



HF 124

04

03

02

01
-01 ... ... -V----------



-0 2

-03



HF 126

04

03



-0 1




-0 2

-03



HF 169

04

03

02




-0 1

-02

-03


40000

30000 -

20000

10000

0



HF 84

60000

50000

40000

30000

20000

10000

0



HF 124

60000

50000

40000

30000

20000

10000

0



HF 126

120000

100000

80000 I-

60000

40000

20000






HF 169

60000 -

50000 -

40000 -

30000 -

20000 -- ---

10000 -


Figure 3-5: Historical performance in percent of initial value (on the right), and the rate of return

dynamics in percentage terms (on the left) for some of the "residual" assets in optimal portfolios.


HF 49










and we would like to invest a certain amount of money in a portfolio of hedge funds to deliver the

highest reward under a specified risk level. Then, according to in-sample results, the best portfolio

would be the one on the efficient frontier of a particular rebalancing strategy. In fact, such a portfolio

offers the best return-to-risk ratio provided that the historical distribution of returns will repeat in

the future.

To get an idea about the "actual" performance of the optimization approach, we used some part

of the data for scenario generation, and the rest for evaluating the performance of the strategy. This

technique is referred to as out-of-sample testing. In our case study, we perform the out-of-sample

testing in two setups: 1) "Real" out-of-sample testing, and 2) "Mixed" out-of-sample testing. Each

one is designed to reveal specific properties of risk constraints pertaining to the performance of the

portfolio-rebalancing algorithm in out-of-sample runs.

"Real" out-of-sample testing. First, we present the results of a "plain" out-of-sample test,

where the older data is considered as the 'in-sample' data for the algorithm, and the newer data

are treated as 'to-be-realized' future. First, we took the 11 monthly returns within the time period

from December 1995 to November 1996 as the initial historical data for constructing the first port-

folio to invest in, and observed the portfolio's "realized" value by observing the historical prices

for December 1996. Then, we added one more month, December 1996, to the data which were

used for scenario generation (12 months of historical data in total) to generate an optimal portfolio

and to allocate to investments in January, 1997, and so on. Note that we did not implement the

"moving window" method for out-of-sample testing, where the same number of scenarios (i.e., the

most recent historical points) is used for solving the portfolio-rebalancing problem. Instead, we

accumulated the historical data for portfolio optimization.

First, we performed out-of-sample runs for each risk measure in constraint (3.5) for different

values of risk tolerance level o (market-neutrality constraint, (3.6), is inactive). Figures 3-6 to 3-9

illustrate historical trajectories of the optimal portfolio under different risk constraints (the portfolio

values are given in % relatively to the initial portfolio value). Risk tolerance level o was set to

0.005, 0.01, 0.03, 0.05, 0.10, 0.12, 0.15, 0.17 and 0.20, but for better reading of figures, we report

only results with o = 0.005, 0.01, 0.05, 0.10, and 0.15. The parameter a in CDaR and CVaR

constraints (in the last case a stands for risk confidence level) was set to a = 0.90.

















Real Out-of-Sample: Portfolio with CVaR Constraints


600


500

a 400

S300



0
i200

100


--a 0.005
--o 0.01
a 0.05
--o 0.10
--o 0.15


Figure 3-6: Historical trajectories of optimal portfolio with CVaR constraints.




Real Out-of-Sample: Portfolio with CDaR Constraints


600

500

d 400

S 300


200


100


-- 0.005
-- 0.01
a 0.05
--o 0.10
- o= 015


Figure 3-7: Historical trajectories of optimal portfolio with CDaR constraints.




Real Out-of-Sample: Portfolio with MAD Constraints


350U

300

250



0
= 150-

o 100
a- 5
50^


-- 0.005
-- 0.01
) 0.05
-- 0.10
-- 0.15


Figure 3-8: Historical trajectories of optimal portfolio with MAD constraints.


S= 0.05 o = 0.01 = 0.005
\\ /


z 0.10






0= 0.15












Real Out-of-Sample: Portfolio with MaxLoss
Constraints

600

o = 0.01 o o=0.15
o 400- --j 0.005
SW0= -0.005 ---= 0.01
200------ -- o2=0.10
S 00 = 0.15

0.10.10
W 0.10
0 .
2 00 ^ .





Figure 3-9: Historical trajectories of optimal portfolio with MaxLoss constraints.


Figures 3-6 to 3-9 shows that risk constraint (3.5) has significant impact on the algorithm's

out-of-sample performance. Earlier, we had also observed that this constraint has significant impact

on the in-sample performance. It is well known that constraining risk in the in-sample optimiza-

tion decreases the optimal value of the objective function, and the results reported in the preceding

subsection comply with this fact. The risk constraints force the algorithm to favor less profitable

but safer decisions over more profitable but "dangerous" ones. From a mathematical viewpoint, im-

posing extra constraints always reduces the feasibility set, and consequently leads to lower optimal

objective values. However, the situation changes dramatically for an out-of-sample application of

the optimization algorithm. The numerical experiments show that constraining risks improves the

overall performance of the portfolio rebalancing strategy in out-of-sample runs; tighter in-sample

risk constraint may lead to both lower risks and higher out-of-sample returns. For all considered

risk measures, loosening the risk tolerance (i.e., increasing o) values) results in increased volatility

of out-of-sample portfolio returns and, after exceeding some threshold value, in degradation of the

algorithm's performance, especially during the last 13 months (March 2000 May 2001). For all

risk functions in constraint (3.5), the most attractive portfolio trajectories are obtained for risk tol-

erance level ) = 0.005, which means that these portfolios have high returns (high final portfolio

value), low volatility, and low drawdowns. Increasing a to 0.01 leads to a slight increase of the final

portfolio value, but it also increases portfolio volatility and drawdowns, especially for the second

quarter of 2001. For larger values of o) the portfolio returns deteriorate, and for all risk measures, a










portfolio curve with o = 0.10 shows quite poor performance. Further increasing the risk tolerance

to o = 0.15, in some cases allows for achieving higher returns at the end of 2000, but after this

high peak the portfolio suffers severe drawdowns (see figures for CDaR, MAD, and MaxLoss risk

measures).

The next series of Figures 3-10 to 3-13 illustrates effects of imposing market-neutrality con-

straint (3.6) in addition to risk constraint (3.5). Recall that primary purpose of constraint (3.6) is

making the portfolio uncorrelated with market. The main idea of composing a market-neutral port-

folio is protecting it from market drawdowns. Figures 3-10 to 3-13 compare trajectories of market-

neutral and without risk-neutrality optimal portfolios. Additional constraining resulted in most

cases in further improvement of the portfolio's out-of-sample performance, especially for CVaR

and CDaR-constrained portfolios. To clarify how the risk-neutrality condition (3.6) influences the

portfolio's performance, we displayed only figures for lowest and highest values of the risk toler-

ance parameter, namely for o = 0.005 and o = 0.20. Coefficient k in (3.6) was set to k = 0.01,

and instruments' betas 3, were calculated by correlating with S&P 500 index, which is traditionally

considered as a market benchmark. For portfolios with tight risk constraints (0 = 0.005) imposing

market-neutrality constraint (3.6) straightened their trajectories (reduced volatility and drawdowns),

which made the historic curves almost monotone curves with positive slope. On top of that, CVaR

and CDaR portfolios with market-neutrality constraint had a higher final portfolio value, compared

to those without market-neutrality. Also, for portfolios with loose risk constraints (0 = 0.20) impos-

ing market-neutrality constraint had a positive effect on the form of their trajectories, dramatically

reducing volatility and drawdowns.

Finally, Figures 3-14 and 3-15 demonstrate the performance of the optimal portfolios versus

two benchmarks: 1) S&P500 index; 2) "Best20", representing the portfolio distributed equally

among the "best" 20 hedge funds. These 20 hedge funds include funds with the highest expected

monthly returns calculated with past historical information. Similarly to the optimal portfolios

(3.2)-(3.6), the "Best20" portfolio was monthly rebalanced (without risk constraints).

According to Figures 3-14 to 3-15, the market-neutral optimal portfolios as well as portfolios

without the market-neutrality constraint outperform the index under risk constraints of all types,

which provides an evidence of high efficiency of the risk-constrained portfolio management algo-

rithm (3.2)-(3.6). Also, we would like to emphasize the behavior of market-neutral portfolios in























Real Out-of-sample: Portfolios with CVaR constraints


S- 0.005, p = 0

,0.005 /q -

S. 0.20, P 0




j=0.20



eX X e^ ^ c t< eie ee e1


--o 0.005
--o0.20
-- 0.005, = 0
= 0.20, = 0


Figure 3-10: Historical trajectories of optimal portfolio with CVaR constraints. Lines with /

correspond to portfolios with market-neutral constraint.

















Real Out-of-Sample: Portfolios with CDaR Constraints


o 0.005, P 0

S= .005





0.20, P 0 -

S= 0.20



'5 o ,S


-- 0.005
--o 0.20
--0 0.005, [ 0
S-0.20, [ 0


Figure 3-11: Historical trajectories of optimal portfolio with CDaR constraints. Lines with /

correspond to portfolios with market-neutral constraint.


400

350

300

S250

> 200
0
S150

O 100

50
0


400

350

300

S250

> 200

o 150
S100

50

0






















Real Out-of-Sample: Portfolios with MAD Constrains


-oj 0.005
-o 0.20
-ow 0.005, P0
w0) 20, [3 0


Figure 3-12: Historical trajectories of optimal portfolio with MAD constraints. Lines with = 0

correspond to portfolios with market-neutral constraint.

















Real Out-of-Sample: Portfolios with MaxLoss
Constraints


/ 0.20, P 0
0.005




S0.005, P 0




^ ^ 1 ,0^


-- =0.005
-- 0.20
-- = 0.005, = 0
= 0.20, = 0


Figure 3-13: Historical trajectories of optimal portfolio with MaxLoss constraints. Lines with

/ = 0 correspond to portfolios with market-neutral constraint.


600

500

W 400

300
0
S200
0
i-


500
450
S400
D 350
300
250
0
S200
150
0
iO 100
50
0


W = 0.20


~~~~~~~~~










"down" market conditions. Two marks on Fig. 3-15 indicate the points when all four portfolios

gained positive returns, while the market was falling down. Also, all risk-neutral portfolios seem

to w ilhsltnd the down market in 2000, when the market experienced significant drawdown. This

demonstrates the efficiency and appropriateness of the application of market-neutrality constraint

(3.6) to portfolio optimization with risk constraints (3.2)-(3.5).

The "Best20" benchmark evidently lacks the solid performance of its competitors. It not only

significantly underperforms all the portfolios constructed with algorithm (3.2)-(3.6), but also un-

derperforms the market half of the time. Unlike portfolios (3.2)-(3.6), the "Best20" portfolio pro-

nouncedly follows the market drop in the second half of 2000, and moreover, it suffers much more

severe drawdowns than the market does. This indicates that the risk constraints in the algorithm

(3.2)-(3.6) play an important role in selecting the funds.

An interesting point to discuss is the behavior of algorithm (3.2)-(3.6) under the MAD risk

constraint. Figures 3-8 and 3-12 show that a tight MAD constraint makes the portfolio curve al-

most a straight line, and imposing of market-neutrality constraint (3.6) does not add much to the

algorithm's performance, and even slightly lowers the portfolio's return. At the same time, portfo-

lios with CVaR-type risk constraints (CVaR, CDaR and MaxLoss) do not exhibit such remarkably

stable performance, and take advantage of constraint (3.6). Note that CVaR, CDaR and MaxLoss

are downside risk measures, whereas the MAD constraint suppresses both high losses and high

returns. The market-neutrality constraint (3.6) by itself also puts symmetric restrictions on the port-

folio's volatility; that's why it affects MAD and CVaR-related constraint differently. However, here,

we should point out that we just "scratched the surface" regarding the combination of various risk

constraints. We have imposed CVaR and CDaR constraints only with one confidence level. We

can impose combinations of constraints with various confidence levels including constraining per-

centiles of high returns and as well as percentiles of high losses. These issues are beyond the scope

of our study.

Summarizing, we emphasize the general inference about the role of risk constraints in the

out-of-sample and in-sample application of an optimization algorithm, which can be drawn from

our experiments: risk constraints decrease the in-sample returns, while out-of-sample performance

may be improved by adding risk constraints, and moreover, stronger risk constraints usually ensure

better out-of-sample performance.



















Portfolios vs. Benchmarks


-CVaR
CDaR
-. MAD
- MaxLoss
- S&P500
- Best20


Figure 3-14: Performance of the optimal portfolios with various risk constraints versus
index and benchmark portfolio combined from 20 best hedge funds. Risk tolerance level (o
parameter a= 0.90. Market-neutrality constraint is inactive.


S&P500
=0.005,


Market-Neutral Portfolios vs. Bencmarks


--CVaR
CDaR
-. MAD
--MaxLoss
- S&P500
- Best20


Figure 3-15: Performance of market-neutral optimal portfolio with various risk constraints versus
S&P500 index and benchmark portfolio combined from 20 best hedge funds. Risk tolerance level
is o = 0.005, parameter is a= 0.90.











"Mixed" out-of-sample test. The second series of out-of-sample tests for the portfolio opti-

mization algorithm (3.2)-(3.6) uses an alternative setup for splitting the data in in-sample and out-

of-sample portions. Instead of utilizing only past information for generating scenarios for portfolio

optimization, as it was done before, now we let the algorithm use both past and future information

for constructing scenarios. The design of this experiment is as follows. The portfolio rebalancing

procedure was performed every five months, and the scenario model utilized all the historical data

except for the 5-month period directly following the rebalancing date. The procedure was started

on December 1995. The information for scenario generation was collected from May 1996 to May

2001. The portfolio was optimized using these scenarios and invested on December 1995. After

5 months the money gained by the portfolio was reinvested and, at this time, the scenario model

was built on information contained in the entire time interval 12/1995-05/2001 except window

05/1996-09/1996 and so on.

Figures 3-16 to 3-19 display dynamics of the optimal portfolio under various risk constraints

and with different risk tolerance levels. To avoid overloading the presentation with details, we

report results only for wo = 0.01, 0.05 and 0.10. As earlier, we set parameter a in CVaR and CDaR

constraints to a = 0.90.


Mixed Out-of-Sample: Portfolio with CVaR Constraints

300

250
2000.01
2000
1 ) \ --=-0.01
> 150 0.05
.2
0 -- o 0.1
100
50-
0 0.10





Figure 3-16: "Mixed" out-of-sample trajectories of optimal portfolio with CVaR constraints


The general picture of these results is consistent with conclusions derived from "real" out-of-

sample tests: tightening of risk constrains improve performance of the rebalance algorithms. Lower

overall performance of the portfolio optimization strategy under all risk constraints comparing to
















Mixed Out-of-Sample: Portfolio with CDaR Constraints

350

300
o =0.01
250

"5 200 0.01
o j 0.05
1500.10
o 0.10o -0.05
S100---

iL 5 0 ----------------------------
5-
0








Figure 3-17: "Mixed" out-of-sample trajectories of optimal portfolio with CDaR constraints




Mixed Out-of-Sample: Portfolio with MAD Constraints

350

300
o 0.01
250
Z \o3 0.05
200 --j =0.01
>= 0.05
150 oj 005
-<0 --o=0.10
o 100
a-


0







Figure 3-18: "Mixed" out-of-sample trajectories of optimal portfolio with MAD constraints




Mixed Out-of-Sample: Portfolio with MaxLoss
Constraints

300
W = 0.01
250

S 200
--o3=0.01
150 =0.05

W --00=0.10
100
0 W =0.10
i 50 -5o
0
50








Figure 3-19: "Mixed" out-of-sample trajectories of optimal portfolio with MaxLoss constraints












that in "real" out-of-sample testing is explained by longer rebalancing period. It is well known that

more frequent rebalancing may give higher returns (at least, in the absence of transaction costs).

The next four Figures 3-20 to 3-23, demonstrate the influence of the market-neutrality con-

straint on the performance of the portfolio.


Mixed Out-of-Sample:
Zero-beta Portfolio with CVaR Constraints

300
o 0.01
250
S 200
S--o0.01
> 150 0 005

S 100 005 0. 010


0
0 d0.10








Figure 3-20: "Mixed" out-of-sample trajectories of market-neutrality optimal portfolio with CVaR
constraints




Mixed Out-of-Sample:
Zero-beta Portfolio with CDaR Constraints

350
o 0.01
300 0
250
S200 00----01
> To = 0.05 to =0.05

100 =0.10 --o010
100--
i.
50-
50







Figure 3-21: "Mixed" out-of-sample trajectories of market-neutrality optimal portfolio with CDaR
constraints


Imposing of market-neutrality constraint (3.6) in problem (3.2)-(3.5) for the "mixed" out-of-

sample testing has a similar impact as in "real" out-of-sample testing.

We finalize the out-of-sample testing of by presenting summary statistics of both "real" and

"mixed" out-of-sample tests in Tables 3-3 and 3-4.





















Mixed Out-of-Sample:
Zero-beta Portfolio with MAD Constraints


-- 0.01
S= 0.05
--o =0.10


Figure 3-22: "Mixed" out-of-sample trajectories of market-neutrality optimal portfolio with MAD
constraints
















Mixed Out-of-Sample: Zero-beta Portfolio with
MaxLoss Constraints


150


. 1 7N I 1- 0 0 1
o- 005
I -- 0.10


= 0.10


o100o -/


vo -' j ,0 c < 0 0' c 0 ',, c 6' S'


Figure 3-23: "Mixed" out-of-sample trajectories of
market-neutrality optimal portfolio with MaxLoss constraints


S= 0.01

o 0.05




o 0.10










Summarizing, we emphasize the general inference about the role of risk constraints in the

out-of-sample and in-sample application of an optimization algorithm, which can be drawn from

our experiments: risk constraints decrease the in-sample returns, while out-of-sample performance

may be improved by adding risk constraints, and moreover, stronger risk constraints usually ensure

better out-of-sample performance.

3.4 Conclusions

We tested the performance of a portfolio allocation algorithm with different types of risk

constraints in an application for managing a portfolio of hedge funds. As the risk measure in

the portfolio optimization problem, we used Conditional Value-at-Risk, Conditional Drawdown-at-

Risk, Mean-Absolute Deviation, and Maximum Loss. We combined these risk constraints with the

market-neutrality (zero-beta) constraint making the optimal portfolio uncorrelated with the market.

The numerical experiments consist of in-sample and out-of-sample testing. We generated ef-

ficient frontiers and compared algorithms with various constraints. The out-of-sample part of ex-

periments was performed in two setups, which differed in constructing the scenario set for the

optimization algorithm.

The results obtained are dataset-specific and we cannot make direct recommendations on port-

folio allocations based on these results. However, we learned several lessons from this case study.

Imposing risk constraints may significantly degrade in-sample expected returns while improving

risk characteristics of the portfolio. In-sample experiments showed that for tight risk tolerance lev-

els, all risk constraints produce relatively similar portfolio configurations. Imposing risk constraints

may improve the out-of-sample performance of the portfolio-rebalancing algorithms in the sense

of risk-return tradeoff. Especially promising results can be obtained by combining several types

of risk constraints. In particular, we combined the market-neutrality (zero-beta) constraint with

CVaR or CDaR constraints. We found that tightening of risk constraints greatly improves portfolio

dynamic performance in out-of-sample tests, increasing the overall portfolio return and decreasing

both losses and drawdowns. In addition, imposing the market-neutrality constraint adds to the sta-

bility of portfolio's return, and reduces portfolio drawdowns. Both CDaR and CVaR risk measures

demonstrated a solid performance in out-of-sample tests.







69

We thank the Foundation for Managed Derivatives Research for providing the dataset for con-

ducting numerical experiments and partial financial support of this case study.













Table 3-3: :: .......:.' statistics for the "real" -- -.- .:' '= tests.


Type of

constraint



CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD


Standard CVaR

Deviation


Risk

tolerance



0.005

0.005

0.005

0.005

0.01

0.01

0.01

0.01

0.03

0.03

0.03

0.03

0.05

0.05

0.05

0.05

0.07

0.07

0.07

0.07

0.1

0.1

0.1

0.1

0.12

0.12

0.12

0.12

0.15

0.15

0.15


Zero- Aver.

beta Rate of

Return

0.258

0.265

0.233

0.273

0.245

0.269

0.234

0.278

0.32

0.3

0.234

0.266

0.27

0.327

0.299

0.289

0.248

0.262

0.201

0.284

0.282

0.239

0.264

0.238

0.237

0.234

0.251

0.237

0.163

0.222

0.231


Drawdown Sharpe

Ratio


0.709

0.7078

0.239

0.703

0.803

0.804

0.38

0.742

1.045

1.073

0.752

0.939

1.256

1.303

1.221

1.150

1.41

1.473

1.489

1.345

1.45

1.591

1.62

1.487

1.53

1.606

1.624

1.547

1.48

1.618

1.607


0.818

0.834

0.11

0

1.035

1.011

0.38

0

1.32

1.37

0.794

09

1.712

1.638

1.529

0

1.983

2.067

2.234

0

1.98

2.401

2.46

0

2.311

2.43

2.467

0

2.407

2.468

2.458


0.178

0.183

0.019

0.154

0.2887

0.2541

0.0526

0.154

0.338

0.393

0.243

0.339

0.465

0.468

0.365

0.460

0.562

0.569

0.597

0.553

0.5

0.62

0.651

0.605

0.6

0.634

0.662

0.61

0.594

0.662

0.658


0.321

0.332

0.848

0.346

0.267

0.297

0.539

0.33

0.278

0.25

0.271

0.2519

0.193

0.228

0.22

0.224

0.155

0.158

0.115

0.189

0.174

0.131

0.144

0.14

0.135

0.127

0.136

0.134

0.09

0.119

0.125


Yearly

Aver.

Return

0.231

0.239

0.218

0.245

0.216

0.241

0.211

0.249

0.284

0.272

0.215

0.236

0.252

0.306

0.273

0.267

0.213

0.241

0.175

0.265

0.255

0.211

0.232

0.216

0.205

0.201

-0.097

0.211

0.145

0.1803

0.178













Type of Risk Zero- Aver. Standard CVaR Drawdown Sharpe Yearly

constraint tolerance beta Rate of Deviation Ratio Aver.

Return Return

MaxLoss 0.15 0.234 1.61 0 0.634 0.127 0.201

CDaR 0.17 0.195 1.52 2.399 0.607 0.108 0.162

CVaR 0.17 0.222 1.618 2.468 0.662 0.119 0.1805

MAD 0.17 0.247 1.612 2.458 0.658 0.135 0.208

MaxLoss 0.17 0.231 1.61 0 0.64 0.125 0.196

CDaR 0.2 0.233 1.505 2.397 0.499 0.135 0.219

CVaR 0.2 0.222 1.618 2.468 0.6628 0.119 0.18

MAD 0.2 0.233 1.639 2.562 0.685 0.124 0.193

MaxLoss 0.2 0.227 1.611 0 0.648 0.122 0.189

CDaR 0.005 + 0.251 0.529 0.642 0.089 0.42 0.225

CVaR 0.005 + 0.267 0.543 0.642 0.086 0.437 0.241

MAD 0.005 + 0.22 0.183 0.056 0.011 1.041 0.2078

MaxLoss 0.005 + 0.249 0.537 0 0.092 0.408 0.227

CDaR 0.01 + 0.255 0.581 0.735 0.117 0.387 0.231

CVaR 0.01 + 0.268 0.604 0.712 0.122 0.393 0.242

MAD 0.01 + 0.231 0.301 0.216 0.0295 0.668 0.211

MaxLoss 0.01 + 0.254 0.5870 0 0.108 0.382 0.231

CDaR 0.03 + 0.258 0.757 0.991 0.194 0.302 0.233

CVaR 0.03 + 0.276 0.774 0.923 0.193 0.318 0.248

MAD 0.03 + 0.238 0.611 0.657 0.104 0.341 0.211

MaxLoss 0.03 + 0.253 0.707 0 0.165 0.315 0.233

CDaR 0.05 + 0.266 0.754 0.891 0.184 0.312 0.237

CVaR 0.05 + 0.265 0.841 1.122 0.231 0.279 0.237

MAD 0.05 + 0.261 0.829 1.159 0.241 0.27 0.25

MaxLoss 0.05 + 0.256 0.765 0 0.173 0.295 0.238

CDaR 0.07 + 0.263 0.802 1.08 0.227 0.29 0.244

CVaR 0.07 + 0.252 0.875 1.22 0.232 0.254 0.218

MAD 0.07 + 0.247 0.859 1.272 0.271 0.252 0.232

MaxLoss 0.07 + 0.284 1.345 0 0.553 0.189 0.265

CDaR 0.10 + 0.223 0.848 1.194 0.276 0.228 0.211

CVaR 0.10 + 0.248 0.882 1.268 0.244 0.246 0.214

MAD 0.10 + 0.237 0.861 1.268 0.252 0.241 0.209

MaxLoss 0.10 + 0.258 0.854 0 0.239 0.266 0.230













Type of


Risk


constraint tolerance beta


CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss


0.12

0.12

0.12

0.12

0.15

0.15

0.15

0.15

0.17

0.17

0.17

0.17

0.20

0.20

0.20

0.20


Zero- Aver. Standard CVaR


Rate of

Return

0.244

0.247

0.257

0.252

0.22

0.247

0.251

0.247

0.243

0.247

0.251

0.247

0.216

0.247

0.251

0.247


Deviation


0.851

0.882

0.883

0.865

0.858

0.882

0.88

0.882

0.843

0.882

0.88

0.882

0.864

0.882

0.88

0.882


1.215

1.27

1.268

0

1.141

1.268

1.268

0

1.108

1.268

1.268

0

1.248

1.27

1.268

0


Drawdown Sharpe Yearly

Ratio Aver.

Return


0.276

0.244

0.245

0.244

0.246

0.244

0.245

0.244

0.223

0.244

0.245

0.244

0.278

0.244

0.2458

0.244


0.252

0.246

0.257

0.257

0.221

0.246

0.251

0.246

0.252

0.246

0.251

0.246

0.215

0.246

0.251

0.246


0.224

0.214

0.233

0.223

0.186

0.214

0.222

0.214

0.228

0.214

0.222

0.214

0.186

0.214

0.221

0.214













Table 3-4: Summary statistics for the "mixed" .:: .- : .... tests.


Type of

constraint



CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD


Standard CVaR

Deviation


Risk

tolerance



0.005

0.005

0.005

0.005

0.01

0.01

0.01

0.01

0.03

0.03

0.03

0.03

0.05

0.05

0.05

0.05

0.07

0.07

0.07

0.07

0.10

0.10

0.10

0.10

0.12

0.12

0.12

0.12

0.15

0.15

0.15


Zero- Aver.

beta Rate of

Return

0.1790

0.1564

0.2184

0.177

0.19

0.143

0.204

0.165

0.170

0.112

0.174

0.154

0.154

0.125

0.11

0.143

0.114

0.074

0.040

0.114

0.132

0.001

-0.027

0.071

0.134

-0.012

-0.014

0.0576

0.124

-0.026

-0.016


Drawdown Sharpe

Ratio


0.4246

0.4175

0.0949

0.396

0.437

0.493

0.182

0.45

0.518

0.67

0.486

0.628

0.592

0.807

0.8556

0.757

0.681

0.861

0.993

0.811

0.741

1.07

1.157

0.943

0.774

1.140

1.16

1.039

0.773

1.153

1.185


0.5024

0.6162

-0.026

0.852

0.508

0.788

0.16

1.01

0.649

1.19

0.738

1.44

0.840

1.452

1.5936

1.886

1.042

1.689

2.05

2.34

1.19

2.32

2.346

2.92

1.27

2.4256

2.346

3.106

1.36

2.336

2.346


0.112

0.1698

0.005

0.129

0.120

0.218

0.032

0.17

0.171

0.349

0.230

0.275

0.217

0.366

0.4086

0.305

0.279

0.432

0.550

0.361

0.331

0.647

0.69

0.487

0.354

0.681

0.667

0.545

0.352

0.691

0.674


0.3504

0.3027

1.984

0.370

0.366

0.229

0.96

0.30

0.271

0.123

0.296

0.198

0.21

0.117

0.0936

0.149

0.123

0.0510

0.010

0.104

0.137

-0.027

-0.049

0.043

0.134

-0.037

-0.038

0.0265

0.122

-0.048

-0.039


Yearly

Aver.

Return

0.165

0.142

0.212

0.166

0.175

0.133

0.19

0.155

0.161

0.104

0.161

0.145

0.15

0.126

0.105

0.141

0.117

0.076

0.025

0.115

0.132

-0.0006

-0.0448

0.067

0.136

-0.0184

-0.033

0.049

0.124

-0.044

-0.0349









74



Type of Risk Zero- Aver. Standard CVaR Drawdown Sharpe Yearly

constraint tolerance beta Rate of Deviation Ratio Aver.

Return Return

MaxLoss 0.15 0.0108 1.152 3.75 0.644 -0.017 -0.0039

CDaR 0.17 0.142 0.84 1.35 0.429 0.134 0.139

CVaR 0.17 -0.0151 1.164 2.356 0.67 -0.039 -0.034

MAD 0.17 -0.017 1.167 2.35 0.67 -0.040 -0.036

MaxLoss 0.17 -0.008 1.19 4.0429 0.682 -0.032 -0.024

CDaR 0.20 0.142 0.84 1.354 0.429 0.134 0.14

CVaR 0.20 -0.0151 1.164 2.356 0.67 -0.039 -0.034

MAD 0.20 -0.0169 1.167 2.356 0.67 -0.04 -0.036

MaxLoss 0.20 -0.008 1.189 4.042 0.682 -0.032 -0.024

CDaR 0.005 + 0.190 0.374 0.416 0.089 0.428 0.176

CVaR 0.005 + 0.18 0.401 0.467 0.128 0.374 0.165

MAD 0.005 + 0.208 0.097 -0.006 0.0031 1.841 0.202

MaxLoss 0.005 + 0.203 0.384 0.815 0.111 0.452 0.19

CDaR 0.01 + 0.207 0.419 0.435 0.078 0.422 0.19

CVaR 0.01 + 0.173 0.453 0.61 0.154 0.315 0.157

MAD 0.01 + 0.227 0.173 0.088 0.022 1.144 0.218

MaxLoss 0.01 + 0.195 0.441 1.11 0.146 0.374 0.181

CDaR 0.03 + 0.210 0.478 0.587 0.109 0.378 0.197

CVaR 0.03 + 0.153 0.629 1.059 0.290 0.196 0.142

MAD 0.03 + 0.166 0.456 0.593 0.179 0.298 0.149

MaxLoss 0.03 + 0.235 0.563 1.175 0.175 0.364 0.223

CDaR 0.05 + 0.205 0.533 0.672 0.136 0.328 0.193

CVaR 0.05 + 0.131 0.701 1.208 0.378 0.143 0.12

MAD 0.05 + 0.103 0.706 1.222 0.368 0.1038 0.088

MaxLoss 0.05 + 0.208 0.639 1.383 0.212 0.278 0.203

CDaR 0.07 + 0.191 0.592 0.747 0.114 0.271 0.182

CVaR 0.07 + 0.118 0.716 1.23 0.375 0.122 0.102

MAD 0.07 + 0.105 0.716 1.23 0.391 0.105 0.090

MaxLoss 0.07 + 0.16 0.677 1.865 0.282 0.192 0.155

CDaR 0.10 + 0.071 0.943 2.92 0.487 0.043 0.067

CVaR 0.10 + 0.1098 0.722 1.23 0.389 0.110 0.094

MAD 0.10 + 0.104 0.721 1.23 0.388 0.102 0.088

MaxLoss 0.10 + 0.142 0.701 1.749 0.324 0.16 0.133













Type of


Risk


constraint tolerance beta


CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss

CDaR

CVaR

MAD

MaxLoss


0.12

0.12

0.12

0.12

0.15

0.15

0.15

0.15

0.17

0.17

0.17

0.17

0.20

0.20

0.20

0.20


Zero- Aver. Standard CVaR


Rate of

Return

0.161

0.109

0.104

0.128

0.142

0.109

0.105

0.110

0.142

0.109

0.105

0.11

0.11

0.109

0.105

0.109


Deviation


0.627

0.723

0.722

0.709

0.635

0.723

0.716

0.721

0.635

0.723

0.716

0.721

0.647

0.723

0.721

0.723


0.818

1.23

1.23

1.66

0.877

1.23

1.23

1.798

0.877

1.23

1.23

1.798

0.993

1.23

1.23

1.8


Drawdown Sharpe Yearly

Ratio Aver.

Return


0.209

0.392

0.388

0.353

0.241

0.391

0.391

0.388

0.241

0.39

0.391

0.388

0.285

0.391

0.39

0.391


0.209

0.109

0.102

0.139

0.176

0.109

0.104

0.112

0.177

0.11

0.104

0.112

0.123

0.109

0.104

0.11


0.148

0.093

0.089

0.117

0.128

0.093

0.09

0.0952

0.128

0.094

0.09

0.0950

0.094

0.093

0.09

0.093
















CHAPTER 4
OPTIMAL POSITION LIQUIDATION

We consider the problem of optimal position liquidation with the aim of maximizing the ex-

pected cash flow stream from the transaction in the presence of a temporary or permanent market

impact. We use a stochastic programming approach to derive trading strategies that differentiate

decisions with respect to different realizations of market conditions. The scenario set consists of

a collection of sample paths representing possible future realizations of state variable processes

(price of the security, trading volume etc.) At each time moment the set of paths is partitioned into

several groups according to specified criteria, and each group is controlled by its own decision vari-

able(s), which allows for adequate representation of uncertainties in market conditions and circum-

vents anticipativity in the solutions. In contrast to traditional dynamic programming approaches,

the stochastic programming formulation admits incorporation of different types of constraints in

the trading strategy, e.g. risk constraints, regulatory constraints, various decision-making policies

etc. We consider the lawn-mower principle, which increases stability of the solution with respect

to paths partitioning and saturation of the scenario pattern, but leads to non-convex optimization

problems. It is shown that in the case of temporary market impact the optimal liquidation strategy

with the lawn-mower principle can be approximated by a solution of convex or linear programming

problems. Implemented as a linear programming problem, our approach is capable of handling

large-scale instances and produces robust optimal solutions. A risk-averse trading strategy was con-

structed by incorporating risk constraints in the stochastic programming problem. We controlled

the risk, associated with trading, using the Conditional Value-at-Risk measure. Numerical results

and optimal trading patterns for different forms of market impact are presented.

4.1 Introduction

This chapter presents numerical techniques for optimal transaction implementation, e.g., de-

termining the best way to sell (buy) a specified number of shares on the market during a prescribed

period of time.










The main challenge of constructing optimal trading strategies consists in preventing or mini-

mizing losses caused by the so-called market impact, or price slippage, that affect payoffs during

market transactions. The market impact phenomenon manifests itself by adverse price movements

during trades and is caused by disturbances of the market equilibrium. As a rather simplistic expla-

nation, consider an investor executing a market order to sell a block of shares. If at the moment of

execution no one is willing to buy this block for the current market price, the seller will be forced

to offer a lower price per share in order to accomplish the transaction, and consequently suffer a

loss due to the market impact. In reality the function of adjusting the price during transactions is

performed by market intermediaries.

Traditionally, two types of market impact are considered: temporary and permanent. Perma-

nent market impact denotes the changes in prices that are caused by the investor's trades and persist

during the entire period of his/her trading activity. If the deviations in prices caused by investor's

transaction are unobservable by the time of his/her next trade, the market impact is said to be tem-

porary. Naturally, the magnitude of market impact and, consequently, the loss due to adverse price

movements, depend on the size of trades, as well as on the time windows between transactions. For

a detailed discussion of issues related to market impact, see, among others, Chan and Lakonishok

(1995), Kein and Madhavan (1995), Kraus and Stoll (1972). Some of the latest developments on the

optimal transaction implementation and optimal trading policies are presented in papers by Bertsi-

mas and Lo (1998), Bertsimas et al. (1999), Rickard and Torre (1999), Almgren and Chriss (2000),

and Almgren (2001).

Bertsimas and Lo (1998) employed a dynamic programming approach for devising optimal

trading strategies that minimize the expected cost of trading a block of S shares within a fixed num-

ber of periods T. They derived analytical expressions for best-execution strategies in the standard

framework of discrete random walk models, under assumption that market impact is linear in the

number of shares traded. To gain an insight on how information component can influence the op-

timal strategy, authors introduce a serially correlated "information" variable in the price process of

the security. Extension of this methodology to the case of multiple assets and optimal execution for

portfolios is presented in Bertsimas et al. (1999).

Almgren and Chriss (2000) constructed risk-averse trading strategies using the classical Markowitz

mean-variance methodology. With permanent and temporary market impact functions being also










linear in the number of shares traded, the risk of the trading strategy is associated with the volatility

of the market value of the position. A continuous-time approximation of this approach, as well as

nonlinear and stochastic temporary market impact functions, was considered by Almgren (2001).

An application of fuzzy set theory to optimal transaction execution, which helps to mimic "non-

rational" human behaviour of traders, is discussed in paper by Rickard and Torre (1999).

However, a common drawback of the described methodologies can be seen in the inability of a

trading strategy to respond dynamically to realized or changed market conditions. Instead of having

a prescribed sequence of trades based on the parameters of security's price process, we would like

to develop a trading strategy that differentiates decisions with respect to actual realization of market

conditions at each moment of transaction. Also, the trading strategy should be able to incorporate

different types of constraints that may reflect investor's preferences, including risk preferences,

institutional regulations, etc.

In this chapter, we consider the optimal liquidation problem in the scope of maximizing the

expected cash flow from selling a block of shares in the market. The problem is formulated and

solved in the stochastic programming framework, which allows for creating multi-stage decision-

making algorithms with appropriate response to different realizations of uncertainties at each time

moment. The key feature of our approach is a sample-path scenario model that represents the

uncertain price process of the security by a set of its possible future trajectories (sample paths). The

approach admits a seamless incorporation of various types of constraints in the definition of trading

strategy, and is applicable under different forms of market impact.

The sample-path scenario models is a relatively new technique in the area of multi-stage deci-

sion making problems, where the dominant scenario models are the classical multinomial trees or

lattices. Sample-path based simulation models have been recently used for pricing of American-

style options (see Tilley (1993), Boyle et al. (1997), Broadie and Glasserman (1997), Carriere

(1996), Barraquand and Martineau (1995), and others), where the optimal decision policy contains

single binary decision (i.e., exercise-or-not-exercise the security contract). In this sense, the prob-

lems of optimal transaction execution are more complex, since the optimal strategy is a sequence of

non-binary decisions.

The chapter is organized as follows. The next section introduces the general formulation of

the optimal liquidation problem, definitions of market impact, etc. Sections 4.3 and 4.4 deal with










optimal position liquidation under temporary and permanent market impacts, correspondingly. In

Section 4.5, we discuss construction of risk-averse trading strategies by incorporating Conditional

Value-at-Risk constraints in the optimal liquidation problem. Section 4.6 presents a case study and

numerical results, and Section 4.7 concludes the chapter.

4.2 General Definitions and Problem Statement

In this section we introduce formal mathematical definitions and develop several formulations

for the optimal closing problem. The formal setup for the optimal liquidation problem is as follows.

Suppose that at time t = 0 there is an open position in some financial instrument (stock, bond, option

etc.), which has to be liquidated (closed) within the predefined time interval 0 < t < T. Assume

also that trades are only allowed at the specified time moments t = 1,2,..., T (integer indexing is

used for briefness of notation; by writing t = 0, 1,2,... we understand t = to, ti, t2,..., with time

moments to, ti, t2,..., not necessarily equally spaced). Information on future market conditions at

t =1,2,..., T is represented by a set of J sample paths (Fig. 4-1a). The objective is to generate an

optimal trading policy which maximizes the expected cash flow stream incurred from liquidating

the position. Throughout the chapter, we implicitly assume that the position to be liquidated is a

long position.

4.2.1 Representation of Uncertainties by a Set of Paths

Traditional approaches to solving multistage decision-making problems with uncertainties are

represented by the techniques of stochastic and dynamic programming, where the evolution of

stochastic parameters is modeled by tree or lattice structures (Fig. 4-1b). These classical techniques

have proved to be effective tools in dealing with multistage problems, especially in analytical frame-

work. However, in many real-life financial applications that require solving large-scale optimization

problems, use of the tree- or lattice- based scenario models may lead to considerable computational

difficulties, known as the "curse of dimensions". Therefore, many financial institutions in their re-

search and investment practice adopt scenario models different from the classical multinomial trees

and lattices. One of the most popular alternative approaches is representing the uncertain future

as a collection of sample paths, each being a possible future trajectory of a financial instrument or

group of instruments (Fig. 4-1a). This type of scenario model is supposed to overcome the "curse










of dimensions" that often defeats large-scale instances of optimization models based on scenario

trees or lattices.

Price S I Price

j=J 2






0 T 0 T

a. Set of sample paths b. Scenario trees

Figure 4-1: Sample paths and and scenario trees.


A significant deficiency of multinomial trees or lattices is a poor balance between randomness

and the tree/lattice size. A small number of branches per node in a tree/lattice clearly makes a rough

approximation of the uncertain future, whereas increasing of this parameter results in exponential

growth of the overall number of scenarios in the tree.

In contrast to this, a sample-path scenario model represents the uncertain value of stochastic

parameter at each time by a (large) number of sample points (nodes) belonging to different sample

paths of the scenario set. Increasing the number of nodes at each time step for better accuracy

results in a linear increase of the number of sample paths in the set. Similarly, increasing the

number of time periods in the model leads to a linear increase of the number of nodes, as opposed

to exponential growth in the number of nodes in scenario trees.

Besides superior scalability, sample-path concept allows for effortless incorporation of histor-

ical data into the scenario model, which is an important feature from a practical point of view.

There has been an increasing interest to use of sample paths in describing uncertain market

environment in problems of finance and financial engineering during recent years. For the most

part, this approach was employed in the area of pricing of derivatives (Titley (1993), Boyle et al.

(1997), Broadie and Glasserman (1997), Carriere (1996), Barraquand and Martineau (1995) etc.).

Recently, a sample paths framework was applied in solving dynamic asset and liability management

problems (Hibiki (1999, 2001)). In this chapter, we consider the concept of sample-path scenario









sets and corresponding optimization techniques in application to a problem of optimal transaction

execution in presence of market imperfections and friction.

4.2.2 A Generic Problem Formulation

Let e be a collection of sample paths


S (SoS{,S2,...,Si) 1,...,J},


where each term S' stands for, generally, a vector of relevant market parameters, such as the mid-

price of the security St, bid-ask spread sJ, volume V/, etc. at time t according to sample path

j:

St (S,s, v/,...).

SJ may also include other information observable at time t, e.g., major financial indices, interest
rates etc. In the simplest case, SJ may only contain price Sj of the security.

We define a trading strategy, corresponding to the sample-path collection e, as a set


5 -{(,O [,...,T) l 1- o> >I-...>- T 0, j= 1,...,J (4.1)

where / is the normalized value of the position at time t on path j. The instant proceeds incurred

by transaction at time t are determined by the payofffunction pt (), whose general form is


pt( ;S'1C < t), j 1,...,J, t= l ,...,T.

Let the payoff function pt(.) incorporate discounting and transaction costs, market slippage etc. The

explicit form of the function pt () and its impact on properties of the optimal liquidation problem is

discussed below; now we stress that payoff at time t cannot depend on the values of S' and for

> t.

The objective of our trading strategy is to maximize the expected cash flow stream incurred

from selling the asset:
T
max EC t( ;S S|T t) (4.2)
7 t=1

Problem (4.2) is a stochastic programming formulation of the optimal closing problem based on the

sample-path scenario model. Here E,,3 is the expectation operator defined on e, and E denotes the

set of all possible trading strategies (4.1). It will be seen later that the generic formulation (4.2) is









far from being perfect; before diving into further details, we have to discuss the form and properties

of the payoff function pt() in (4.2), which has a major impact on the properties of the problem of

optimal closing in general.

4.2.3 Modeling the Market Impact

Generally, the payoff on path j at time t is a function of the preceding series S{ and position

values :

Pt(.) Pt(o0,45,...,5/ ; So,S{,...,Sj).

Dependence of the payoff function pt(-) on the preceding decisions 10, ~,.., Jt 1 constitutes the

generally nonlinear effect of permanent market impact, when the trading activity of the market

player contributes to changes in prices that do not vanish during the trading period. When the

price at current moment t is unaffected by our preceding trades, but does depend on the size of

current transaction, it is said that only temporary market impact is present. First, we consider the

temporary market impact as it allows for computationally more tractable formulations of the optimal

liquidation problem.

Temporary market impact. Let the current payoff pt () depend only on the size Ait of the

current transaction

pt() =pt(A/; So,SJ,...,SJ), A/ = i i

This form of the payoff function is appropriate when the price changes caused by the transaction

at time t are negligibly small by the next time moment, t + 1. Assume further that function Pt ()

admits the representation

pt(A^; So, SJ,..., SJ) = StJ (A/; S I Z < t), (4.3)

where the price Sj is always positive: St > 0. The term 8(A/ ; SC| z < t) in (4.3) captures the

effects of market friction. It may depend parametrically on information observable at path j up

to time t, e.g., volume V/, prices So,...,SJ, etc.1 To lighten the notation, we suppress the term


1 The current volume of trades Vt may be defined as the number of trades between t 1 and t.









S I < t in 8(.) and introduce subscript t:

_(A
As a function of the trade size A( = y, t(y) satisfies

(i) t : [0, 1] [0, 1], t(0) = 0.
In the perfect frictionless market, obviously, 8(y) =y, y E [0, 1]. We assume that in the presence of

temporary market impact, proportional transaction costs2 etc., t (-) further satisfies

(ii) 8t(y)< y, y [0, 1];
(iii) V 0 < Y1 < y2 < 1 : 8t(y1) < 8t(2);
(iv) 8t(y) is concave on [0,1].

Condition (ii) states that payoff in the market with friction is always less than that in the frictionless

market. Condition (iii) and (iv) ensure that larger trades lead to higher revenues, however, the

marginal revenues become smaller with increasing of trade size (Fig. 4-2).







0

Figure 4-2: Impact function t(t).

The importance of the concavity requirement (iv) will be clear later when we present refined

formulations for the optimization problem (4.2). Observe also that (iv) implies that 5t() is contin-

uous on [0,1]: 5t E C([0, 1]).
Summarizing the aforesaid, in the case of temporary market impact we write the payoff func-

tion pt(.) in the objective of (4.2) in the form


pt (;S^ S| l< t) = St t (A/) (4.4)



2 Fixed transaction costs and fees are usually much smaller than the losses caused by market
impact, and can be neglected.










where St = const > 0, and 5(-) satisfies conditions (i)-(iv).

Permanent market impact. A more realistic model of market response to the trading ac-

tivity involves the permanent market impact as the dependence of current market prices on the

preceding transactions of the market player. To incorporate the permanent market impact in the

model, we assume the following price dynamics for S/

t 1
StJ S -StO(A/)- (, S 6 (A ), (4.5)
c=1

where S, = 1,...,t is the "undisturbed" price trajectory that would realize in the absence of

our trades, and S7J is the actual price at moment t and path j received by investor for liquidating

portion At/ of the position. Functions Ot (A ) and Ot (A ) are equal to the percentage drop in market

price due to the temporary and permanent market impact, correspondingly. Similarly to the above

notation 4.(), functions Or(0) and r(.) may contain, in general, values So,...,Sf as parameters:

6t(A/J) =A 0(A/; S|" T < t), t (A'/) A 0(A'/; Sl J < t).


Using (4.5), the total payoff attained over a path (So, Sj,..., SJ) is

T T t-

t=l t=l

SJ A/ A/ O(A/) Ot(A/) A I (4.6)
t=1 =t+l 1

The first term in braces in (4.6) attributes to the profit of selling the portion A(/ of the position

in perfect frictionless market. The second and third summands in (4.6) represent the losses due to

effects of temporary and permanent market impacts, correspondingly.

For consistency with the above discussion of temporary market impact, we assume that Ot ()

is such that the function

y-y Ot(y)

satisfies conditions (i)-(iv). This allows us to think of Ot(') as a non-negative non-decreasing func-

tion on [0, 1] (it may be convex or concave).










Similarly, the permanent market impact function Ot (.) is assumed to be non-negative and non-

decreasing. We suppose that function Or(-) satisfies


Ot(A) 0, 0 < A < At for some At E (0, 1).

This condition reflects the presumption that small enough trades Ait should not cause permanent

changes in market prices given the time windows tk tk1 between transactions. A plausible form

of Ot(-) is as follows:

0t(A) max{0, KcOt(A A)}, (4.7)

where K = const e (0, 1].

4.3 Optimal Position Liquidation under Temporary Market Impact

In this section we present a sample-path approach to generation of optimal liquidation policies

when only temporary market impact is observable. The framework includes the special case of zero

market impact, which constitutes an important extension of the considered technique to pricing

of derivative securities. First, we present the path grouping method that eliminates anticipativity

in solutions of the optimization problem based on sample-path approach (4.2). Then, the "lawn-

mower" decision-making rule, which results in trading strategies of specialized form, is introduced.

The properties of "lawn-mower"-compliant trading strategies are discussed.

4.3.1 Paths Grouping and "Lawn-Mower Strategy"

In Section 4.2.2, we introduced a generic formulation (4.2) for the optimal closing problem,

which is rewritten here taking into account the form of the payoff function (4.4) under temporary

market impact, and replacing the expectation operator by an average over the set of paths:

1 J T
max St 5tt(A /). (4.8)
j= t=1

It turns out, however, that the trading strategy based on the optimal solution of (4.8) is contingent

on perfect knowledge of the future and therefore inappropriate. As an example, consider a simple

case when Sj8t(At/) = S!Al/, i.e., the payoff equals to the dollar profit of selling a portion A(/ of

position for price Sj in frictionless market. Then, it is easy to see that the optimal solution of (4.8)










is given by


5I0,


for t
for t>tj,


where t = argmax{St}, j 1,...,J.
t=1,...,T


(4.9)


The optimal strategy (4.9) liquidates the whole position once the maximal price over the entire

trajectory (So, S, ...,S ) is achieved. In practice this would mean that the trader perfectly forecasts

the future by identifying some path j with the actual price trajectory (for example, by observing

price S1 at time t = 1).

From the formal point of view, the anticipativity of the optimal solution of (4.2) is attributable

to the possibility of making separate decisions regarding the position value / at each t and j.


Price Gk Si

j 2

So



J J
0 T


Figure 4-3: Path grouping


Paths grouping. A remedy for the anticipativity in formulation (4.2) is the so-called path

grouping (see similar approaches in Titley (1993) and Hibiki (1999, 2001)). The set of paths 6 is

partitioned into Kt groups G/t at each time t (Fig. 4-3)


SKt=
O r(e ) -a V to m1,.,: al,...,Jiin at Gt in a gin grk


One possible approach is to make identical decision at each path in a given group:


(4.10)


(4.11)


Here x('t) is the new decision variable, equal to the value of position variables f/ in the group Gt,

and k(j, t) is the function that returns the index k of set Gt, which contains path j at moment t. We


tJ k(,t) VjE k 1,...,T. T










denote this as

k(j, t) 7r(e).

Hibiki (1999, 2001) employed the "tree-like" grouping, satisfying the following property: for all

t = 2,...,T, andfor any k {1,...,Kt} there exists k* E 1,...,Kt- such that


G 1 c G 1. (4.12)

Kt Kt 1
In other words, time t partition U Gt is obtained from time t 1 partition U Gt 1 by splitting
k=1 k=1
groups Gtk1 into subgroups, similarly to splitting the scenario tree into branches at each time step

(Fig. 4-4). In this setting, the decision rule (4.11) is analogous to what is long known in stochastic

programming as non-anticipativity constraints (see, for example, Birge and Louveaux (1997)).









Figure 4-4: Tree-like grouping.


We generalize the path grouping method, compared to Titley (1993) and Hibiki (1999, 2001)

in two respects. First, we relax the "tree" grouping condition (4.12), thus allowing for "path inter-

mixing" (i.e., paths from different groups can merge into the same group at the next step)


3 t,k, j,j2 : l,j2 E Gk, Gk(Ji,'t1) Gk('2,t 1)

In this way we avoid having smaller and smaller number of paths per group at each next time step,

which is inevitable for the "tree" grouping.

Second, we consider general decision rules in the following form:


Vje G: ~' f(xEk(',t) -1 .***., ). (4.13)


Dependence of all the position variables / within a group Gt on the common decision variable

xk(,t) ensures the non-anticipativity of the solution. Below we prove that with the "tree" grouping

(4.12) the decision policy (4.13) is identical to the simple rule (4.11).










Proposition 4.3.1 Ifpartition 7r() satisfies the "tree" grouping condition (4.12), then the gener-

alized group decision rule (4.13) is equivalent to (4.11).

Proof. Consider time t = 1. Then Vj E G : j f(x' o'l,) (x('), 1) = x'). At the next

step, t 2, we have Vj E Gc f(x ('2),, ) k(J,2), k(J), 1). Since () is constant

Vj E Gk, then f(xk('2 k(' ,1) = X(,2). Proceeding to t = T, in finite number of steps we obtain

that Vt, VjeC: EGk kt(jt)

The general formulation of the optimal liquidation strategy under temporary market impact is

T
max E,,e I S5t(A, )
t=l
s.t. VjG>: 5/ f(xt), ), k(j, t) (4(.). (4.14)

"Lawn-mower" strategy. In what follows, we consider a special case of the group decision

rule (4.13) that is pertinent to the construction of an optimal liquidating strategy with a temporary

or zero market impact, namely


Vj E Gk: / min~ (t) }, O < (t) < 1. (4.15)


The intuition behind relation (4.15) is as follows. First of all, the decision variable xt(J't) represents

an upper bound for the position values tj in a group Gt:


/ < xk(j't), k(t, j) (o). (4.16)

Moreover, observe that position value _-1 is allowed to change (i.e., decrease) at time t only by

being "trimmed" to the level x$-(',t). If the position value i! 1 is not high enough to be "trimmed"

by x-(Jt), it remains unchanged at time t. This resembles lawn mowing, with position variables 5/

being the "grass" and decision variables x(J't) being the "blades" (Fig. 4-5).


Figure 4-5: The "lawn-mower" principle.


X ~k(j y I
tk(t,N










It follows from (4.15) and the trading strategy definition (4.1), which requires zero position at

the final time moment T, that variables x must satisfy

k(' =T 0, j= 1,...,J. (4.17)


By incorporating the "lawn-mower" principle with path grouping 7r(e) into the formulation

(4.8), we obtain the optimal closing problem in the form

1J T
max I SJt (A1) (4.18)
S j=lt=l

s.t. = min{j 1, Xk(J't)}, t= 1,...,T,j = 1,...,J,
S< (t) < 1, k(j, t) c ( ), t 1,...,T, j 1,...,J,
it t -1T _



By solving the problem (4.18) we find the optimal allocation of thresholds xt(J't) determining the

following optimal trading strategy: if scenario j belongs to the group Gt, then sell the portion of

position in excess ofxt; if current position value is less than xk, do i, 'i,,g

The next proposition shows that the optimal closing problem (4.18) is well-posed, i.e., for any

allocation of thresholds xt the corresponding trading strategy is unique.

Proposition 4.3.2 Under the "lawn-mower" decision rule (4.15) each variable / is a single-

valued function of variables x-(, t). Inverse statement does not hold.

Proof. Let us show that the "lawn-mower" principle (4.15) implies that the position variable / can

be expressed as


/ = min{x j ) xk('2), ...,x k(t)}, j 1,...,J, t 1,...,T. (4.19)

Obviously, the assertion holds for t = 1: j= min{ ,xx } = x k(1) min{x (1')}. Assume that

(4.19) holds for some t = z, then


1= min{J, x~ j } min min{x1),...x(')} x'+1

=min{xk(j x kj,x + 1)


which proves (4.19) by induction.