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PAGE 1 OPTIMIZATIONWITHGENERALIZEDDEVIATIONMEASURESINRISKMANAGEMENTByKONSTANTINP.KALINCHENKOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012 PAGE 2 2012KonstantinP.Kalinchenko 2 PAGE 3 IdedicatethisthesistomyparentsPavelandOlga,andmybrotherAlexander,whosupportedmeinallmyendeavours. 3 PAGE 4 ACKNOWLEDGMENTS IamverythankfultomyadvisorProf.StanUryasevandtoProf.R.TyrrellRockafellarfortheirsupportduringmydoctoratestudyingattheUniversityofFlorida.WiththeirresearchguidanceandinvaluablehelpIwasabletogrowonbothprofessionalandpersonallevels.Iwouldliketoexpressmygratitudetoothermembersofmydoctoratecommittee,Prof.PanosPardalos,Prof.VladimirBoginskiandProf.LiqingYanfortheircontributiontomyresearch.Also,IamparticularlythankfultoProf.MichaelZabarankin(StevensInstituteofTechnology),Prof.MarkJ.Flannery(UniversityofFlorida,WarringtonCollegeofBusinessAdministration)andProf.OlegBondarenko(UniversityofIllinois)forvaluablefeedbacks.IwouldalsoliketoexpressmygreatestappreciationtomycolleaguesfromtheRiskManagementandFinancialEngineeringlabandCenterforAppliedOptimization.Intensivediscussions,exchangeofideasandjointresearchwithmyfellowgraduatestudentsfromthesetwolabshelpedmesignicantlytoachievemygoals.Also,Iwouldliketothankmyfamilyandfriends,whosupportedandencouragedmeinallofmybeginnings. 4 PAGE 5 TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 11 2GENERALIZEDMEASURESOFDEVIATION,RISKANDERROR ....... 14 2.1ClassicalRiskandDeviationMeasures ................... 14 2.2GeneralizedRiskandDeviationMeasures .................. 19 2.3ConditionalValue-at-Risk ........................... 22 2.4ApplicationtoGeneralizedLinearRegressions ............... 23 2.4.1MeasuresofError ........................... 23 2.4.2GeneralizedLinearRegressions ................... 25 2.4.3DistributionofResidual ......................... 27 3ROBUSTCONNECTIVITYISSUESINDYNAMICSENSORNETWORKSFORAREASURVEILLANCEUNDERUNCERTAINTY .............. 33 3.1Multi-SensorSchedulingProblems:GeneralDeterministicSetup ..... 36 3.1.1FormulationwithBinaryVariables ................... 36 3.1.2CardinalityFormulation ......................... 38 3.2QuantitativeRiskMeasuresinUncertainEnvironments:ConditionalValue-at-Risk ........................ 39 3.3OptimizingtheConnectivityofDynamicSensorNetworksUnderUncertainty ............................ 41 3.3.1EnsuringShortTransmissionPathsvia2-clubFormulation ..... 44 3.3.2EnsuringBackupConnectionsviak-plexFormulation ........ 46 3.4ComputationalExperiments .......................... 47 4CALIBRATINGRISKPREFERENCESWITHGENERALIZEDCAPMBASEDONMIXEDCVARDEVIATION ....................... 52 4.1DescriptionoftheApproach .......................... 55 4.1.1GeneralizedCAPMBackground .................... 55 4.1.2PricingFormulasinGCAPM ...................... 59 4.1.3MixedCVaRDeviationandBetas ................... 60 4.1.4RiskPreferencesofaRepresentativeInvestor ............ 63 4.2CaseStudyDataandAlgorithm ........................ 66 5 PAGE 6 4.3CaseStudyComputationalResults ...................... 70 5CONCLUSIONS ................................... 75 5.1DissertationContribution ........................... 75 5.2FutureWork ................................... 76 APPENDIX:PROOFS ................................... 77 REFERENCES ....................................... 81 BIOGRAPHICALSKETCH ................................ 85 6 PAGE 7 LISTOFTABLES Table page 3-1CPLEXResults:Problemwith2-clubConstraints ................. 48 3-2CPLEXResults:Problemwithk-plexConstraints ................. 49 3-3CPLEXandPSGResults:StochasticSetup .................... 50 4-1CaseStudyDataforSelectedDates ........................ 68 4-2CaseStudyCommonData ............................. 68 4-3DeviationMeasureCalibrationResults ....................... 69 7 PAGE 8 LISTOFFIGURES Figure page 2-1RelationsBetweenMeasuresofError,DeviationMeasures,RiskMeasuresandStatistics ............................ 24 2-2ProbabilityDensityFunctionsforDExp(,1) .................... 31 2-3ProbabilityDensityFunctionsforDExp(,2) .................... 32 3-1GraphicalrepresentationofVaRandCVaR. .................... 40 4-1CVaR-typeRiskIdentierforaGivenOutcomeVariableX ............ 62 4-2CalculatedPricesandMarketPricesintheScaleofImpliedVolatilitiesPart1outof3 .................................... 71 4-3CalculatedPricesandMarketPricesintheScaleofImpliedVolatilitiesPart2outof3 .................................... 72 4-4CalculatedPricesandMarketPricesintheScaleofImpliedVolatilitiesPart3outof3 .................................... 73 4-5S&P500ValueandRiskAversityDynamics .................... 74 8 PAGE 9 AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyOPTIMIZATIONWITHGENERALIZEDDEVIATIONMEASURESINRISKMANAGEMENTByKonstantinP.KalinchenkoMay2012Chair:StanUryasevMajor:IndustrialandSystemsEngineering Ourworkprovidesanoverviewoftheso-calledgeneralizeddeviationmeasuresandgeneralizedriskmeasures,anddevelopsstochasticoptimizationapproachesutilizingthem.Thesemeasuresaredesignedtoquantifyriskwhenimplieddistributionsareknown.Weprovideusefulexamplesofdeviationandriskmeasures,whichcanbeefcientlyappliedinsituations,whentheclassicalmeasureseitherdonotproperlyaccountforrisk,ordonotsatisfypropertiesdesiredforefcientapplicationinstochasticoptimization.Wediscusstheimportanceofconsideringalternativeriskanddeviationmeasuresintheclassicalmodels,suchasthecapitalassetpricingmodelandquantileregression.Weapplystochasticoptimizationandriskmanagementtechniquesbasedontheconditionalvalue-at-risk(CVaR)tosolveadynamicsensorschedulingproblemwithrobustnessconstraintsonawirelessconnectivitynetwork.WealsodevelopanefcientapplicationofthegeneralizedCapitalAssetPricingModelbasedonmixedCVaRdeviationtoestimatingriskpreferencesofinvestorsusingS&P500stockindexoptionprices. Intherstpartweprovideanoverviewofthemainclassesofgeneralizeddeviationmeasuresandcorrespondingriskmeasures,andcomparethemtotheclassicalriskanddeviationmeasures,suchasmaximumrisk,value-at-riskandstandarddeviation.Inaddition,weprovidearelationbetweendeviationmeasuresandmeasuresoferror,whichareusedinregressionmodels.Insomeapplications,suchassimulation,a 9 PAGE 10 distributionoftheresidualtermhastobespecied.Weapplytheentropymaximizationprincipletoidentifytheappropriatedistributionforthequantileregression(factor)model. Inthesecondpartweconsiderseveralclassesofproblemsthatdealwithoptimizingtheperformanceofdynamicsensornetworksusedforareasurveillance,inparticular,inthepresenceofuncertainty.Theoverallefciencyofasensornetworkisaddressedfromtheaspectsofminimizingtheoverallinformationlosses,aswellasensuringthatallnodesinanetworkformarobustconnectivitypatternateverytimemoment,whichwouldenablethesensorstocommunicateandexchangeinformationinuncertainandadverseenvironments.TheconsideredproblemsaresolvedusingmathematicalprogrammingtechniquesthatincorporateCVaR,whichallowsonetominimizeorboundthelossesassociatedwithpotentialrisks.Theissueofrobustconnectivityisaddressedbyimposingexplicitrestrictionsontheshortestpathlengthbetweenallpairsofsensorsandonthenumberofconnectionsforeachsensor(i.e.,nodedegrees)inanetwork.Specicformulationsoflinear0-1optimizationproblemsandthecorrespondingcomputationalresultsarepresented. InthethirdpartweapplythegeneralizedCapitalAssetPricingModelbasedonmixedCVaRdeviationtocalibrateriskpreferencesofinvestors.WeintroducethenewgeneralizedbetatocapturetailperformanceofS&P500returns.CalibrationisdonebyextractinginformationaboutriskpreferencesfromoptionpricesonS&P500.Actualmarketoptionpricesarematchedwiththeestimatedpricesfromthepricingequationbasedonthegeneralizedbeta.Theseresultscanbeusedforvariouspurposes.Inparticular,thestructureoftheestimateddeviationmeasureconveysinformationabouttheleveloffearamonginvestors.Highleveloffearreectsatendencyofmarketparticipantstohedgetheirinvestmentsandsignalsinvestors'anticipationofpoormarkettrend.Thisinformationcanbeusedinriskmanagementandforoptimalcapitalallocation. 10 PAGE 11 CHAPTER1INTRODUCTION Themainobjectiveofourstudyistodevelopmodelsutilizinggeneralizedmeasuresofdeviation,riskanderrorinstochasticoptimizationandriskmanagementapplications.Uncertaintyisgenerallymodeledusingrandomvariables,anddifferentmodelsutilizevariousfunctionals(ormeasures)onthespaceofrandomvariablestoproperlyaccountforrisk.Dependingonaparticularapplication,thefunctionalmusthavecertainpropertiestoquantifyacertainaspectofuncertainty.Althoughmanydifferentfunctionalsmaysatisfytheseproperties,mostmodelsutilizejustseveralclassicfunctionals,suchasstandarddeviationorquantile. Optimalityofanyparticularchoiceofmeasureaccountingforuncertaintycanoftenbeargued.Thisledtomultiplestudiesintroducingnewmeasuresanddevelopingalternativemodelsutilizingthem.Theconceptsofgeneralizedmeasuresofdeviation,riskanderrorweredevelopedtowraptheseandothermeasuresinseveralclasses,whereeachclasssatisescertainproperties(axioms)requiredinaparticularapplication.Ifacertainmodelbasedonsomeriskordeviationfunctionalisadjustedtocertainassumptions,itsapplicationcanoftenbegeneralizedbysubstitutingthefunctionalwithageneralizedmeasureofriskordeviation.Differentinstancesofthegeneralizedmodelcanthusbecompared.Moreover,ifthefunctionalhasaparameter,itcanalsobeoptimized. Inourwork,weutilizethreeclassesoffunctionals:generalizedmeasuresofrisk,generalizeddeviationmeasures,andgeneralizedmeasuresoferror.Generalizedmeasuresofriskweredesignedtoquantifypotentiallosses.Generalizeddeviationmeasuresaccountonlyforvariabilityoflosses.Generalizedmeasuresoferrorcanbeviewedastoolstoestimatesignicanceofaresidualterminapproximation,oritsdeviationfrom0. 11 PAGE 12 Itisimportanttomentionthatthereisaone-to-onecorrespondencebetweenmeasuresofriskandmeasuresofdeviation,andeverymeasureoferrorhasaparticularmeasureofdeviationcorrespondingtoit.Thisfeaturelinkstogethercertainverydifferentmodels,andprovidesasolutiontoproperlychoosingfunctionalinonemodeldependingonthefunctionalusedinanothermodel,whenthetwomodelsareappliedtothesameproblem. Althoughthetheoreticalfoundationsofthedeviationmeasureshavealreadybeendeveloped,theirpracticalapplicationshavenotyetbecomepopular.Inourwork,wedemonstrateseveralapplicationsofthegeneralizedmeasuresofdeviation,riskanderrorinstochasticoptimization.Inparticular,weconsiderinstancesbasedonconditionalvalue-at-risk(CVaR)andmixedCVaR.CVaRisariskmeasure,whichgainedsubstantialattentioninacademicpublicationsduetoseveralreasons.First,CVaRhasanintuitivedenitionasexpectedlossescorrespondingtothe1)]TJ /F4 11.955 Tf 12.56 0 Td[(tailofdistribution.Second,CVaRisacoherentmeasureofrisk,andisthereforeapplicableinoptimization.Third,theproblemofoptimizingCVaRhasalinearprogrammingformulation.MixedCVaRisaconvexcombinationofseveralCVaRtermswithdifferentvalues.Byvaryingthenumberoftermsandthevaluesofcoefcientsand,onecanpreciselyspecifysignicanceofdifferentpartsofadistributionaccordingtohisperceptionofrisk. InChapter2weappliedentropymaximizationmethodologytospecifyingdistributionoftheresidualtermingeneralizedlinearregression.Generalizedlinearregressionisdenedasastochasticoptimizationproblemofminimizingageneralizedmeasureoferroroftheresidual.Itisimportanttomentionthatthegeneralizedlinearregressionhasanalternativeformulationutilizingdeviationmeasureandso-calledstatistic,bothcorrespondingtothesamemeasureoferror.Twoinstancesofthegeneralizedlinearregressionarewellknown:classicallinearregression,basedonmeansquarederror,andquantileregression,basedonKoenker-Bassetterror.Incertainapplications, 12 PAGE 13 suchassimulation,thedistributionoftheresidualterminthelinearregressionhastobespecied.AcommonwaytospecifythedistributionunderlimitedinformationisbymaximizingShannonentropy.Thisapproachisjustiedbythecommonviewthatentropyisameasureofuncertainty.Inthecaseoftheclassicallinearregression,whenexpectationandvarianceoftheresidualtermareknown,thedistributionwithmaximumentropyisnormal.Followingthesameintuition,inquantileregressionweestimatedthedistributionbymaximizingentropysubjecttoconstraintsonquantileandCVaRdeviation. InChapter3weappliedCVaR-basedoptimizationtoasensorschedulingproblem.Suchproblemsarecommoninapplicationswhereinformationlossesoccurduetoinabilitytocollectinformationfromallsourcessimultaneously.Informationlossesassociatedwithnotobservingacertainsiteatsomemomentintimearemodeledasapenalty,whichconsistsoftwocomponents:axedpenaltyandapenaltyproportionaltothetimethesitewasnotobserved.InthissetupCVaRisappliedtominimizetheaverageofthe1)]TJ /F4 11.955 Tf 12.32 0 Td[(greatestpenalties.Themodelalsoincludestwotypesofwirelessconnectivityrobustnessconstraints:2-clubandk-plex. WediscussanexampleoftheproblemrequiringoptimizationofadeviationmeasureinChapter4.WeconsideredthegeneralizedCapitalAssetPricingModelbasedonmixedCVaRdeviationtoestimateriskpreferencesofinvestors.Theproblemofriskpreferencesestimationwasactivelydiscussedinmanystudies.OnemotivationforthesediscussionsisthecriticismoftheclassicalCAPM,whichisbasedontheassumptionthatinvestors'perceptionofriskcanberepresentedbystandarddeviation.Thiscriticismalignswithourmotivationforapplyingtheconceptofgeneralizedmeasures.ContrarytotheclassicalCAPMandsomerecentmodications,generalizedCAPMconsidersaclassofmixedCVaRdeviationswithcoefcientsspecifyingparameterization. 13 PAGE 14 CHAPTER2GENERALIZEDMEASURESOFDEVIATION,RISKANDERROR Inthischapterweprovideanoverviewofgeneralizeddeviationmeasures1andrelatedquantitativemeasures,e.g.riskmeasures,measuresoferror,statisticsandentropy.MostofthesemeasureshavebeenintroducedinarecentlineofresearchbyR.T.Rockafellar,S.Uryasev,M.ZabarankinandS.Sarykalin.Wedemonstratethatsomesubclassesofthesemeasureshaveproperties,whichallowthemtobemoreefcientlyappliedinriskmanagementapplications. Anapplicationofthedeviationmeasuresinregressionmodelsisproposedinthischapter.Inparticular,weidentiedthedistributionwhichisthemostapplicabletomodeltheresidualinthequantileregression. Thefollowingsectionprovidesanoverviewofthepopularmeasuresusedinriskmanagement.Section3introducesso-calledgeneralizeddeviationmeasuresandgeneralizedriskmeasures,andprovidesmathematicalrelationsbetweenthem.SomeimportantsubclassesofthesemeasuresarealsointroducedinSection3.Anoverviewofconditionalvalue-at-riskdeviationandrelatedmeasuresisprovidedinsection4.Section5containsanoverviewofso-calledmeasuresoferrorandtheirrelationtodeviationmeasures.Thesamesectionintroducesapplicationsofthedeviationmeasuresandthemeasuresoferrorinregressionmodels. 2.1ClassicalRiskandDeviationMeasures Sinceriskmanagementbecameastandardpracticeforalmostallinstitutionsandcommercialcompanies,anumberofquantitativemeasureshavebeendevelopedto 1Weusethewordgeneralizedtoaccentuatethatthesemeasuresbelongtoaclassgeneralizingthestandarddeviation.Inthecontextofthisandthefollowingchaptersthetermsdeviationmeasureandgeneralizeddeviationmeasurehavethesamemeaning. 14 PAGE 15 provideanaturalwaytoestimaterisk.Themostpopularandcommonlyusedmeasureshavebeenmaximumrisk,value-at-riskandstandarddeviation. Maximumrisk(maxrisk)providesaquantitativeevaluationoflossesintheworstpossiblescenario.Theexpressionbelowprovidesaformaldenitionofmaxrisk:maxrisk(X)=)]TJ /F2 11.955 Tf 11.29 0 Td[(inf(X) Itistheleastconvenientmeasureduetoitsover-conservatism:formostcommondistributionsitprovidesameaninglessvalueofinnity.Evenwhenthismeasureisapplicable(e.g.whenasetofpossibleoutcomesisnite),thismeasurelacksrobustness.Forexample,ifthesetofpossibleoutcomesisextendedbyaddingonemoreoutcomecorrespondingtolosses,whicharesubstantiallygreaterthanthepreviousvalueofmaxrisk,thenthevalueofmaxriskchangesbythesameamountregardlessoftheprobabilityofthisoutcome.Itisimportanttomention,however,thatthismeasurecanbeefcientlyusedinsomeoptimizationapplications.Inparticular,anyconstraintonmaxriskisequivalenttoasetofsimilarconstraintsforeachpossibleoutcome.Thiswasdemonstrated,forexample,in Boykoetal. ( 2011 ). Value-at-riskhasbeenapopularmeasureinthelast20yearsduetoitsnaturalinterpretationasanamountofreservesrequiredtopreventdefaultwithagivenprobability.Belowistheformaldenitionfrom Artzneretal. ( 1999 ): Denition.Given2(0,1),andareferenceinstrumentr,thevalue-at-riskVaRatlevelofthenalnetworthXwithdistributionP,isdenedbythefollowingrelation:VaR(X)=)]TJ /F2 11.955 Tf 11.29 0 Td[(inffxjP[Xxr]>gBaselCommitteeonBankingSupervision(BCBS)issuesso-calledBaselAccords,whicharerecommendationsonbankinglawsandregulations.Accordingtotheserecommendations,value-at-riskisthepreferredapproachtomarketriskmeasurement(forexample, Bas ( 2004 )).Inparticular,theserecommendationsspecifyminimum 15 PAGE 16 capitalrequirements,estimatedwithvalue-at-risk.Inmanycountries(includingtheUSA)regulatorsenforcenancialcompaniestocomplywithsomeoralloftheserecommendations.Thisledtovalue-at-riskbecomingoneofthemostcommonlyusedriskmeasures. Despiteitspopularity,value-at-riskhasaseriousdrawback.TheproblemisthatthefunctionalVaRdoesnothaveaconvexityproperty.Inriskmanagementconvexityisoftenanecessaryrequirement.Forexample,inportfoliomanagement,convexityofariskmeasurejustiesdiversicationofinvestments.Also,thelackofconvexitymakesthevalue-at-riskmeasureinefcientinoptimization,whereconvexityoftheoptimizedfunctionorconstraintsisalwaysadesiredproperty. Standarddeviation()isafunctiondenedonthespaceofrandomvariablesasasquarerootofvariance.Insomeapplications,atermvolatilityisusedinsteadofstandarddeviation.Contrarytovalue-at-risk,standarddeviationsatisestheconvexityproperty.Thismeasureisveryuseful,because,inparticular,volatilityinmanymodelsisaparameter.Forexample,inportfoliomanagementastockpricerandomprocessStisdescribedbythefollowingstochasticdifferentialequation:dSt=tStdt+tStdWt withtdenotingstockvolatility. Volatilityandvariancecanbeviewedasthemostpopularmeasuresinportfoliooptimizationandriskmanagement.Forexample,theCapitalAssetPricingModel(CAPM, Sharpe ( 1964 ), Lintner ( 1965 ), Mossin ( 1966 ), Treynor ( 1961 ), Treynor ( 1999 ))andtheArbitragePricingTheory(APT, Ross ( 1976 ))arefactormodelsfocusingonexplainingvariabilityinstockreturns.CAPMassumptionsimply,inparticular,thatallinvestorsinthemarketareoptimizingtheirinvestmentportfoliosconsideringvarianceasameasureofrisk.Basedonvariance,CAPMintroducestheso-calledBeta,aquantity 16 PAGE 17 determiningexposureofstock(orportfolio)returnstofuturemarkettrend:i=iM 2M whereiMdenotescovariancebetweenstockireturnsandmarketreturns,andMdenotesthemarketvolatility.IfallCAPMassumptionshold,thetotalvarianceofastockreturncanbeseparatedintosystematicandnonsystematic(idiosyncratic)components,wheresystematicpartofthevariancecorrespondstomarketreturns:2i=2i2M+2i,n IntheexpressionaboveidenotestheBetaofthestocki,andi,nisthenonsystematic(stock-specic)partofvolatility.Thisexpressionprovidesatoolforriskmanagement.Forexample,ifamanagerislookingforaportfoliowithweights(wi)withnoexposuretomarkettrend,hehastoconsideronlyportfolioswithtotalBetaequal0:NXi=1wii=0 Theconvexitypropertyofvolatilityguaranteesthatthenonsystematiccomponentoftheportfoliovariancecanbereducedbydiversication. ItisalsoimportanttomentionthatforanormalrandomvariableX,apair(E[X],(X))providescompleteinformationaboutthedistributionofX.Moreover,assumethatthedistributionofYhastobespecied,andtheonlyavailableinformationaboutarandomvariableYisthevaluesy=E[Y]andy=(Y).Followingthemaximumentropyprinciple,whichwasrstintroducedin Jaynes ( 1957 1968 ),itisnaturaltoassumetheleast-informativedistributionofYwithgivenmeanandvariance.Specically,considerthefollowingoptimizationproblem: maxEntr(f)s.t.Z1tf(t)dt= 17 PAGE 18 Z1t2f(t)dt)]TJ /F4 11.955 Tf 11.95 0 Td[(2=2fisaPDF whereEntr(f)denotestheShannonentropy( Shannon ( 1948 )):Entr(f)=)]TJ /F12 11.955 Tf 11.29 16.27 Td[(Z1f(t)lnf(t)dt whichisacommonmeasureofuncertainty.Theoptimalsolutionf(t)=1 p 2e)]TJ /F15 5.978 Tf 7.78 3.86 Td[((t)]TJ /F18 5.978 Tf 5.76 0 Td[()2 22istheprobabilitydensityfunctionofanormaldistributionwithmeanandvariance2(theproofcanbefoundin CozzolinoandZahner ( 1973 )).Therefore,iftheonlyavailableinformationaboutadistributionismeanandvariance,incanbenaturaltoassumeanormaldistribution.Itisveryconvenient,becauseinmanymodels,suchasfactormodels,uncertaintyismodeledbynormaldistribution. Thestandarddeviation,however,hasdisadvantages.Inparticular,itdoesn'tsatisfythemonotonicitypropertyX PAGE 19 canbefoundin BradleyandTaqqu ( 2003 )).Theworst-casep%outcomesparticularlyinterestinvestors,becausetheyaretheonesmostlikelytocauseadefault. Theinefcienciesofthestandardmeasures,mentionedabove,canbesummarizedasfollows: 1. Theclassicalvalue-at-riskmeasurelackstheconvexityproperty,thereforeitissometimesdifculttoimplementVaRinstochasticoptimization. 2. Themaximumriskmeasurecanbeeasilyimplementedinstochasticoptimization,butitsover-conservatismoftenmakesitmeaningless. 3. Thestandarddeviationandthevariancedonotsatisfythemonotonicityproperty,whichwouldbenaturalforariskmeasure. 4. Asadeviationmeasure,thestandarddeviationdoesnotdistinguishbetweenthepositiveoutcomes(gains)andnegativeoutcomes(losses),andmeasuresoverallvariability,whileriskmanagerisprimarilyconcernedaboutthepartofvariabilityassociatedwiththemostundesirablescenarios. 2.2GeneralizedRiskandDeviationMeasures AnewsystematizationofmeasuresevaluatingprobabilitydistributionswasintroducedbyRockafellar,Sarykalin,UryasevandZabarankin( Rockafellaretal. ( 2006a ), Sarykalin ( 2008 )).Theyintroduceanumberofaxiomsdeningtwoseparateclasses:deviationmeasuresandriskmeasures.Theyalsoprovideaone-to-onecorrespondencebetweentheseclassesandspecialsubclasses. Considerthefollowingsetofaxioms: (D1)D(X+C)=D(X)forallXandconstantsC, (D2)D(0)=0andD(X)=D(X)forallXandall>0, (D3)D(X+Y)D(X)+D(Y)forallXandY, (D4)D(X)0forallXwithD>0fornonconstantX, (D5)fXjD(X)CgisclosedforeveryconstantC, (D6)D(X)EX)]TJ /F2 11.955 Tf 11.96 0 Td[(infXforallX. Theaxiom(D2)denespositivehomogeneity,axioms(D2)and(D3)togetherdeneconvexity.Accordingtothedenitionin Rockafellaretal. ( 2006a ),afunctional 19 PAGE 20 D:L2![0,1]isadeviationmeasureifitsatisesaxioms(D1)-(D4).Theaxiom(D5)denesclosedness.Inthisdocumentwewillonlyconsidercloseddeviationmeasures.Theproperty(D6)deneslowerrangedominance. Undertheseaxioms,D(X)dependsonlyonX)]TJ /F3 11.955 Tf 12.27 0 Td[(EX(fromthecaseof(D1)whereC=)]TJ /F3 11.955 Tf 9.29 0 Td[(EX),anditvanishesonlyifX)]TJ /F3 11.955 Tf 10.59 0 Td[(EX=0(asseenfrom(D4)withX)]TJ /F3 11.955 Tf 10.59 0 Td[(EXinplaceofX.ThiscapturestheideathatDmeasuresthedegreeofuncertaintyinX.Proposition4in Rockafellaretal. ( 2006a )provestheconvexitypropertyoftheclassofdeviationmeasuresandthesubclassoflowerrangedominateddeviationmeasures.Itcanbeseenthatthestandarddeviationtsintotheclassofdeviationmeasures,butitdoesn'tsatisfythelowerrangedominanceaxiom(D6). AlthoughthedeviationmeasuresasmeasuresofuncertaintyprovidesomeinformationabouttheriskinessassociatedwiththeoutcomeofX,theyarenotriskmeasuresinthesenseproposedin Artzneretal. ( 1999 ).Consider,forexample,asituationinthenancialmarketwithanarbitrageopportunitywithanetpayoffX.Bydenitionofthearbitrage,X0almostsurelyandP(X>0)>0.Arbitrageisgenerallyviewedasaprotablerisklessopportunity,therefore,forariskmeasureRthevalueR(X)shouldnotbegreaterthan0.IfXisrandom,adeviationmeasurewillalwaysbegreaterthan0. Rockafellaretal. ( 2006a )introducestheclassofcoherentriskmeasures,whichextendstheriskmeasuresdenedin Artzneretal. ( 1999 ).Considerthefollowingaxioms: (R1)R(X+C)=R(X))]TJ /F3 11.955 Tf 11.96 0 Td[(CforallXandconstantsC, (R2)R(0)=0,andR(X)=R(X)forallXandconstants>0, (R3)R(X+Y)R(X)+R(Y)forallXandY, (R4)R(X)R(Y)forallXY, (R5)R(X)>E[)]TJ /F3 11.955 Tf 9.29 0 Td[(X]forallnon-constantX. 20 PAGE 21 Theaxiom(R2)denespositivehomogeneity,(R3)denessubadditivity.Theaxioms(R2)and(R3)combinedimplythatRisaconvexfunctional.(R4)iscalledmonotonicity. Accordingtothedenitionsin Rockafellaretal. ( 2006a ),afunctionalR:L2!(,1]is a) acoherentriskmeasureifitsatisesaxioms(R1)-(R4), b) strictlyexpectationboundedriskmeasureifitsatises(R1)-(R3)and(R5),and c) coherent,strictlyexpectationboundedriskmeasureifitsatisesallaxioms(R1)-(R5). ItcanbenoticedthatVaRdoesn'ttinanyofthesecategoriesbecauseitdoesn'thavethesubadditivityproperty(R3). Thesamepaperdenesrelationsbetweendeviationmeasuresandriskmeasures: D(X)=R(X)]TJ /F3 11.955 Tf 11.96 0 Td[(EX) (2) R(X)=E[)]TJ /F3 11.955 Tf 9.29 0 Td[(X]+D(X) (2) Inparticular,equations( 2 )and( 2 )provideone-to-onecorrespondencebetweenstrictlyexpectationboundedriskmeasures(satisfying(R1)-(R3)and(R5))anddeviationmeasures(satisfyingaxioms(D1)-(D4)),andone-to-onecorrespondencebetweencoherent,strictlyexpectationboundedriskmeasures(satisfying(R1)-(R5))andlowerrangedominateddeviationmeasures(satisfying(D1)-(D4)and(D6)).Itcanbeshowndirectlyorusingtheserelationsthatthesetsofcoherentriskmeasures,strictlyexpectationboundedriskmeasures,andcoherent,strictlyexpectationboundedriskmeasuresareallconvex. Accordingtotheserelations,thestandarddeviationcorrespondstothestrictlyexpectationboundedriskmeasureR(x)=E[)]TJ /F3 11.955 Tf 9.3 0 Td[(X]+(X),whichisnotacoherentriskmeasure. 21 PAGE 22 2.3ConditionalValue-at-Risk Theclassicalriskmeasuresdiscussedinthebeginningofthechapterhaveanumberofimperfections.Thisledtothedevelopmentofnewkindsofriskmeasures.Oneofthemostnoticeableriskmeasuresistheconditionalvalue-at-risk(CVaR),alsoknownasexpectedshortfall,ortail-VaR.Wedenethismeasureaccordingto Pug ( 2000 ):CVaR(X)=minCC+(1)]TJ /F4 11.955 Tf 11.96 0 Td[())]TJ /F11 7.97 Tf 6.59 0 Td[(1E[X+C])]TJ /F12 11.955 Tf 7.09 11.48 Td[( where[y])]TJ /F1 11.955 Tf 10.41 1.79 Td[(equals)]TJ /F3 11.955 Tf 9.3 0 Td[(yfory<0and0otherwise.Theparametermusthaveavalueintheinterval(0,1).ItisimportanttonoticethattheoptimalCequalsvalue-at-risk:argminCC+(1)]TJ /F4 11.955 Tf 11.95 0 Td[())]TJ /F11 7.97 Tf 6.58 0 Td[(1E[X+C])]TJ /F12 11.955 Tf 7.09 11.48 Td[(=VaR(X) whereVaRdenotesthevalue-at-risk:VaR(X)=)]TJ /F2 11.955 Tf 11.29 0 Td[(supfzjFX(z)<1)]TJ /F4 11.955 Tf 11.95 0 Td[(g Anequivalentdenitioncanbefoundin Acerbi ( 2002 ):CVaR(X)=(1)]TJ /F4 11.955 Tf 11.96 0 Td[())]TJ /F11 7.97 Tf 6.59 0 Td[(1Z1VaR(X)d IfCVaRiscontinuousat)]TJ /F2 11.955 Tf 9.3 0 Td[(VaR,thenitcanbeexpressedviathefollowingrelation: CVaR(X)=)]TJ /F3 11.955 Tf 9.3 0 Td[(E[XjX)]TJ /F2 11.955 Tf 21.91 0 Td[(VaR(X)](2) Equation( 2 )showsthatconditionalvalue-at-riskequalstheweightedaverageoflossesexceedingvalue-at-risk.Therefore,conditionalvalue-at-riskestimateshowseverecanbepotentiallossesassociatedwithatailofdistribution.Thiscanbeviewedasanadvantageovertheclassicalvalue-at-risk,whichisnotsensitivetochangesinthetailofdistribution.Also,conditionalvalue-at-riskisacoherent,strictlyexpectationboundedriskmeasure.ThisfeatureallowsconsideringCVaRinavarietyofstochasticoptimizationapplications.Inparticular,asitwillbediscussedinChapter3,conditional 22 PAGE 23 value-at-riskcanbeusedinlinearprogramming.Thelowerrangedominateddeviationmeasure,denedvia( 2 )forR=CVaRiscalledaconditionalvalue-at-riskdeviation(CVaRdeviation),denotedasCVaR.Duetoconvexityofcoherent,strictlyexpectationboundedriskmeasures,aconvexcombinationofseveralCVaRswithdifferentcondencelevelsisalsoacoherentstrictlyexpectationboundedriskmeasure.ConvexcombinationofseveralCVaRsiscalledamixedconditionalvalue-at-risk(mixedCVaR).Thelowerrangedominateddeviationmeasure,denedvia( 2 )forR=mixedCVaR,iscalledamixedconditionalvalue-at-riskdeviation(mixedCVaRdeviation).InChapter4wedemonstrateapplicationofmixedCVaRdeviationintheframeworkofthegeneralizedcapitalassetpricingmodelforestimatingriskpreferencesofinvestors. 2.4ApplicationtoGeneralizedLinearRegressions Inthissectionweprovideanoverviewoftheso-calledmeasuresoferror,includingtheirrelationtogeneralizeddeviationmeasuresandapplicationtogeneralizedlinearregressions. 2.4.1MeasuresofError ConsiderafunctionalE:L2![0,1]andthesetofaxioms: (E1)E(0)=0butE(X)>0whenX6=0;also,E(C)<1forconstantsC, (E2)E(X)=E(X)forconstants>0, (E3)E(X+Y)E(X)+E(Y)forallX,Y, (E4)fX2L2()jEcgisclosedforallC<1, (E5)infX:EX=CE(X)>0forconstantsC6=0. Accordingtothedenitionin Rockafellaretal. ( 2008 ),thefunctionalEisameasureoferrorifitsatisesaxioms(E1)-(E4).Theproperty(E5)iscallednondegeneracy. ConsiderafunctionalD:L2![0,1]denedaccordingtothefollowingrelation: D(X)=infCE(X)]TJ /F3 11.955 Tf 11.96 0 Td[(C)(2) 23 PAGE 24 Figure2-1. RelationsBetweenMeasuresofError,DeviationMeasures,RiskMeasuresandStatistics whereEisanondegeneratemeasureoferror.Then,accordingtoTheorem2.1in Rockafellaretal. ( 2008 ),Disadeviationmeasure.Wewillfurtherusethetermprojecteddeviationmeasuretospecifythatthisdeviationmeasureisobtainedaccordingtoequation( 2 ).ThesetS(X)=argminCE(X)]TJ /F3 11.955 Tf 11.95 0 Td[(C)iscalledstatistic.InmanycasesS(X)isreducedtoasinglevalue. Considerameansquareerror:EMS(X)=E[X2].Itisawell-knownfactthatSMS(X)=EX.Accordingto( 2 ),thecorrespondingdeviationmeasureisvariance:DMS=2. Figure 2-1 illustratesrelationsbetweenmeasuresoferror,statistics,deviationmeasuresandriskmeasures. AnotherimportantexampleistheKoenker-Bassetterror,introducedin KoenkerandBassett ( 1978 ):EKB=EX++ 1)]TJ /F4 11.955 Tf 11.95 0 Td[(X)]TJ /F12 11.955 Tf 7.09 18.65 Td[( whereX+=maxf0,XgandX)]TJ /F2 11.955 Tf 10.78 1.8 Td[(=maxf0,)]TJ /F3 11.955 Tf 9.3 0 Td[(Xg.Itwasshownin Rockafellaretal. ( 2008 )thatEKBisanondegeneratemeasureoferror,whichcorrespondstotheconditional 24 PAGE 25 value-at-riskdeviation:DKB(X)=infCEKB(X)]TJ /F3 11.955 Tf 11.96 0 Td[(C)=CVaR(X) Moreover,thecorrespondingSKB(X)equalsvalue-at-risk:SKB(X)=argminCEKB(X)]TJ /F3 11.955 Tf 11.96 0 Td[(C)=VaR(X) 2.4.2GeneralizedLinearRegressions Generalizedlinearregressionwasdenedin Rockafellaretal. ( 2008 )asthefollowingproblem: minE(Z(c0,c1,...,cn)) (2) s.t.Z(c0,c1,...,cn)=Y)]TJ /F2 11.955 Tf 11.95 0 Td[((c0+c1X1+...+cnXn) (2) IntheaboveformulationtherandomvariablesX1,...,Xnarefactorsandc0,c1,...,cnareregressioncoefcients.IfE(Y)<1,thenthesetofoptimalregressioncoefcients(c0,c1,...,cn)alwaysexists. Theorem3.2inthesamepaperprovesequivalenceoftheproblem( 2 )( 2 )tothefollowingproblemofminimizingtheprojecteddeviationmeasure: minD(Z(c0,c1,...,cn)) (2) s.t.02S(Z(c0,c1,...,cn)) (2) Z(c0,c1,...,cn)=Y)]TJ /F2 11.955 Tf 11.95 0 Td[((c0+c1X1+...+cnXn) (2) Fortheoptimalregressioncoefcientsc0,c1,...,cnwecanwrite: Y=c0+c1X1+...+cnXn+"(2) where"istheerrorterm,equaltotheresidualfortheoptimalregressioncoefcients:"=Z(c0,c1,...,cn) 25 PAGE 26 TheTheorem3.2andequation( 2 )implythatfortheoptimalsetofcoefcientsstatisticofYequalsstatisticoftheoptimalcombinationoffactors:S(Y)=S(c0+c1X1+...+cnXn) TheyalsoimplythattheoptimalsetofcoefcientsminimizesthedeviationmeasureoftheresidualZ(c0,c1,...,cn): D(")=minc0,c1,...,cnD(Z(c0,c1,...,cn))(2) Itisimportanttonoticethatthevalueofc0hasnoeffectontherighthandsideofequation( 2 ). Belowweconsidertwoimportantexamples,whichwillbeusedinfurtheranalysis. ConsiderthemeansquarederrorEMS.Forthismeasureoferrortheproblem( 2 )( 2 )isequivalenttotheclassicallinearregression.SMS()=E[]impliesthattheexpectationoftheoptimalcombinationoffactorsequalsexpectationofY:EY=E[c0+c1X1+...+cnXn] Equation( 2 )impliesthattheerror"canbeinterpretedasarandomvariable,minimizingresidualvariance:2(")=minc1,...,cn2(Y)]TJ /F3 11.955 Tf 11.95 0 Td[(c1X1)]TJ /F2 11.955 Tf 11.96 0 Td[(...)]TJ /F3 11.955 Tf 11.96 0 Td[(cnXn) Anotherexampleisthequantileregression(forexample, Koenker ( 2005 )).Itcanbeformulatedaccordingto( 2 )( 2 )withE=EKB,or,equivalently,accordingto( 2 )( 2 )withageneralizeddeviationmeasureD=CVaRandstatisticS=VaR.Followingthesamelogicthatweusedfortheclassicallinearregression,SKB()=VaR()impliesthefollowing:VaR(Y)=VaR(c0+c1X1+...+cnXn) 26 PAGE 27 Equation( 2 )forquantileregressionimpliesthefollowinginterpretationoftheoptimalresidual":CVaR(")=minc1,...,cnCVaR(Y)]TJ /F3 11.955 Tf 11.96 0 Td[(c1X1)]TJ /F2 11.955 Tf 11.95 0 Td[(...)]TJ /F3 11.955 Tf 11.95 0 Td[(cnXn) 2.4.3DistributionofResidual Insomeapplicationsitisconvenienttospecifythedistributionoftheresidualerror".Thechoiceofthedistributionshouldbeonlybasedontheavailableinformation.Forthegeneralizedlinearregression,onlythestatisticanddeviationmeasureoftheerrortermareknown.Giventhisinformation,itisnaturaltopickthedistributionwhichhasthegreatestuncertainty. WeconsidertheShannonentropyEntr(f),whichiscommonlyusedasameasureofuncertainty( Shannon ( 1948 )):Entr(f)=)]TJ /F12 11.955 Tf 11.29 16.28 Td[(Z1f(t)lnf(t)dt wherefistheprobabilitydensityfunction. Forconvenience,considerthefollowingnotation.ForafunctionalF:L2![,1],deneF:ffjfisaPDFg![,1] accordingtothefollowing:F(f)=F(X)forXsuchthatfisaPDFofX WeapplythisnotationforF()=E[],F()=(),F()=CVaR(),F()=VaR(). Forageneralizedlinearregression,denedin( 2 )( 2 ),wechoosethedistributionf"bysolvingthefollowingentropymaximizationproblem: maxEntr(f) (2) s.t.S(f)=0 (2) 27 PAGE 28 D(f)=D(") (2) fisaPDF (2) whereS(f)andD(f)arethestatisticandthedeviationmeasureofarandomvariablewiththeprobabilitydensityfunctionf. Fortheclassicallinearregressiontheoptimizationproblem( 2 )( 2 )istheentropymaximizationproblemwithconstraintsonexpectationandexpectedvalue.Thesolutiontothisproblemisthenormaldistribution. Forthequantileregressiontheoptimizationproblem( 2 )( 2 )isequivalenttothefollowing: maxEntr(f) (2) s.t.VaR(f)=0 (2) CVaR(f)=CVaR(") (2) fisaPDF (2) Wederivethesolutiontothisproblemfromthesolutiontoasimilarproblem: maxEntr(g) (2) s.t.E(g)=0 (2) CVaR(g)=v (2) gisaPDF (2) Thesolutionto( 2 )( 2 )forv=1canbefoundin Grechuketal. ( 2009 ): g",1(t)=8>><>>:(1)]TJ /F4 11.955 Tf 11.96 0 Td[()exp)]TJ /F11 7.97 Tf 6.68 -4.98 Td[(1)]TJ /F9 7.97 Tf 6.59 0 Td[( )]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t)]TJ /F11 7.97 Tf 13.15 4.7 Td[(2)]TJ /F11 7.97 Tf 6.58 0 Td[(1 1)]TJ /F9 7.97 Tf 6.59 0 Td[(t2)]TJ /F11 7.97 Tf 6.58 0 Td[(1 1)]TJ /F9 7.97 Tf 6.58 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[()exp)]TJ /F5 11.955 Tf 5.48 -9.68 Td[()]TJ /F12 11.955 Tf 11.29 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t)]TJ /F11 7.97 Tf 13.15 4.7 Td[(2)]TJ /F11 7.97 Tf 6.59 0 Td[(1 1)]TJ /F9 7.97 Tf 6.59 0 Td[(t2)]TJ /F11 7.97 Tf 6.58 0 Td[(1 1)]TJ /F9 7.97 Tf 6.58 0 Td[( 28 PAGE 29 Derivationofthesolutionforothervaluesofvisidenticaltothecasev=1.Thefollowingfunctionistheoptimalprobabilitydensityfunction: g",v(t)=8>><>>:1)]TJ /F9 7.97 Tf 6.59 0 Td[( vexp)]TJ /F11 7.97 Tf 6.68 -4.97 Td[(1)]TJ /F9 7.97 Tf 6.59 0 Td[( v)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t)]TJ /F11 7.97 Tf 13.15 4.71 Td[(2)]TJ /F11 7.97 Tf 6.59 0 Td[(1 1)]TJ /F9 7.97 Tf 6.58 0 Td[(t2)]TJ /F11 7.97 Tf 6.59 0 Td[(1 1)]TJ /F9 7.97 Tf 6.58 0 Td[(1)]TJ /F9 7.97 Tf 6.59 0 Td[( vexp)]TJ /F5 11.955 Tf 5.48 -9.68 Td[()]TJ /F11 7.97 Tf 10.68 4.71 Td[(1 v)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t)]TJ /F11 7.97 Tf 13.15 4.71 Td[(2)]TJ /F11 7.97 Tf 6.59 0 Td[(1 1)]TJ /F9 7.97 Tf 6.59 0 Td[(t2)]TJ /F11 7.97 Tf 6.59 0 Td[(1 1)]TJ /F9 7.97 Tf 6.58 0 Td[((2) Note,that VaR(g",v)=)]TJ /F2 11.955 Tf 10.5 8.09 Td[(2)]TJ /F2 11.955 Tf 11.96 0 Td[(1 1)]TJ /F4 11.955 Tf 11.96 0 Td[((2) whatfollowsfrom( 2 )andthefollowingequality:Z2)]TJ /F15 5.978 Tf 5.76 0 Td[(1 1)]TJ /F18 5.978 Tf 5.76 0 Td[(1)]TJ /F4 11.955 Tf 11.95 0 Td[( vexp1)]TJ /F4 11.955 Tf 11.95 0 Td[( vt)]TJ /F2 11.955 Tf 13.15 8.08 Td[(2)]TJ /F2 11.955 Tf 11.95 0 Td[(1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(dt=1)]TJ /F4 11.955 Tf 11.96 0 Td[( Theorem.Theoptimalprobabilitydensityfunctionfintheproblem( 2 )( 2 )isthefollowingfunctionf",v: f",v(t)=8>><>>:1)]TJ /F9 7.97 Tf 6.59 0 Td[( vexp)]TJ /F11 7.97 Tf 6.68 -4.98 Td[(1)]TJ /F9 7.97 Tf 6.59 0 Td[( vtt01)]TJ /F9 7.97 Tf 6.59 0 Td[( vexp)]TJ /F5 11.955 Tf 5.48 -9.68 Td[()]TJ /F11 7.97 Tf 10.68 4.7 Td[(1 vtt0(2) wherev=CVaR("). ProofofTheorem.First,considerthefollowingentropyproperty:Entr(f)=Entr(g)forg(t)=f(t)]TJ /F3 11.955 Tf 11.96 0 Td[(c) wherecisanyconstant.Thispropertyguaranteesthattheproblem( 2 )( 2 )isequivalenttotheproblem: maxEntr(g) (2) s.t.g(t)=f(t+E(f))8t (2) VaR(f)=0 (2) CVaR(f)=v (2) f,garePDFs (2) 29 PAGE 30 Theaxiom(D1)allowsustosubstituteconstraint( 2 )withconstraintCVaR(g)=d.Alsonote:( 2 )guaranteesthatE(g)=0fortheoptimalg,and( 2 )togetherwith( 2 )guaranteethatf(t)=g(t)]TJ /F2 11.955 Tf 12.11 0 Td[(VaR(g)).Therefore,thisproblemisalsoequivalenttothefollowing: maxEntr(g) (2) s.t.g(t)=f(t+E(f))8t (2) f(t)=g(t)]TJ /F2 11.955 Tf 11.95 0 Td[(VaR(g))8t (2) VaR(f)=0 (2) CVaR(g)=v (2) E(g)=0 (2) f,garePDFs (2) Notethatinthisproblemtheconstraints( 2 ),( 2 )and( 2 )areredundant:foranyginthefeasibleset,iffisdenedaccordingto( 2 ),thensuchfalreadysatises( 2 ),and( 2 )isenforcedby( 2 ).Therefore,thisproblemisequivalenttotheproblem( 2 )( 2 ).g",din( 2 )istheoptimalPDFg,andtheoptimalfisobtaineddirectlyfrom( 2 )and( 2 ). Thisdistributionhasparametersandv.Wewillfurthersaythatarandomvariable"isdistributedaccordingtoDExp(,v),ifitsprobabilitydensityfunctionf"isexpressedby( 2 ). Figures 2-2 and 2-3 depicttheprobabilitydensityfunctionsofdistributionsDExp(,v)fordifferentvaluesandv.Eachdistributioncanbeviewedasatwo-sidedexponentialdistribution. 30 PAGE 31 Figure2-2. ProbabilityDensityFunctionsforDExp(,1) 31 PAGE 32 Figure2-3. ProbabilityDensityFunctionsforDExp(,2) 32 PAGE 33 CHAPTER3ROBUSTCONNECTIVITYISSUESINDYNAMICSENSORNETWORKSFORAREASURVEILLANCEUNDERUNCERTAINTY Inthischapter1,weaddressseveralproblemsandchallengesarisinginthetaskofutilizingdynamicsensornetworksforareasurveillance.Thistaskneedstobeefcientlyperformedindifferentapplications,wherevarioustypesofinformationneedtobecollectedfrommultiplelocations.Inadditiontoobtainingpotentiallyvaluableinformation(thatcanoftenbetime-sensitive),onealsoneedstoensurethattheinformationcanbeefcientlytransmittedbetweenthenodesinawirelesscommunication/sensornetwork.Inthesimpleststaticcase,thelocationofsensors(i.e.,nodesinasensornetwork)isxed,andthelinks(edgesinasensornetwork)aredeterminedbythedistancebetweensensornodes,thatis,twonodeswouldbeconnectediftheyarelocatedwithintheirwirelesstransmissionrange.However,inmanypracticalsituations,thesensorsareinstalledonmovingvehicles(forinstance,unmannedairvehicles(UAVs))thatcandynamicallymovewithinaspeciedareaofsurveillance.Clearly,inthiscasethelocationofnodesandedgesinanetworkandtheoverallnetworktopologycanchangesignicantlyovertime.Thetaskofcrucialimportanceinthesesettingsistodevelopoptimalstrategiesforthesedynamicsensornetworkstooperateefcientlyintermsofbothcollectingvaluableinformationandensuringrobustwirelessconnectivitybetweensensornodes. Intermsofcollectinginformationfromdifferentlocations(sites),oneneedstodealwiththechallengethatthenumberofsitesthatneedtobevisitedtogatherpotentiallyvaluableinformationisusuallymuchlargerthanthenumberofsensors.Undertheseconditions,oneneedstodevelopefcientschedulesforallthemovingsensorssuchthattheamountofvaluableinformationcollectedbythesensorsismaximized.Arelevant 1ThischapterisbasedonthejointpublicationwithA.Veremyev,V.Boginski,D.E.JeffcoatandS.Uryasev( Kalinchenkoetal. ( 2011 )) 33 PAGE 34 approachthatwaspreviouslyusedbytheco-authorstoaddressthischallengedealtwithformulatingthisproblemintermsofminimizingtheinformationlossesduetothefactthatsomelocationsarenotundersurveillanceatcertaintimemoments.Inthesesettings,theinformationlossescanbequantiedasbothxedandvariablelosses,wherexedlosseswouldoccurwhenagivensiteissimplynotundersurveillanceatsometimemoment,whilevariablelosseswouldincreasewithtimedependingonhowlongasitehasnotbeenvisitedbyasensor.Takingintoaccountvariablelossesofinformationisoftencriticalinthecasesofdealingwithstrategicallyimportantsitesthatneedtobemonitoredascloselyaspossible.Inaddition,theparametersthatquantifyxedandvariableinformationlossesareusuallyuncertain,therefore,theuncertaintyandriskshouldbeexplicitlyaddressedinthecorrespondingoptimizationproblems. Theotherimportantchallengethatwillbeaddressedinthischapterisensuringrobustconnectivitypatternsindynamicsensornetworks.Theserobustnesspropertiesareespeciallyimportantinuncertainandadverseenvironmentsinmilitarysettings,whereuncertainfailuresofnetworkcomponents(nodesand/oredges)canoccur. Theconsideredrobustconnectivitycharacteristicswilldealwithdifferentparametersofthenetwork.First,thenodeswithinanetworkshouldbeconnectedbypathsthatarenotexcessivelylong,thatis,thenumberofintermediarynodesandedgesintheinformationtransmissionpathshouldbesmallenough.Second,eachnodeshouldbeconnectedtoasignicantnumberofothernodesinanetwork,whichwouldprovidethepossibilityofmultiple(backup)transmissionpathsinthenetwork,sinceotherwisethenetworktopologywouldbevulnerabletopossiblenetworkcomponentfailures. Clearly,theaforementionedrobustconnectivitypropertiesaresatisediftherearedirectlinksbetweenallpairsofnodes,thatis,ifthenetworkformsaclique.Cliquesareveryrobustnetworkstructures,duetothefactthattheycansustainmultiplenetworkcomponentfailures.Notethatanysubgraphofacliqueisalsoaclique,whichimpliesthatthisstructurewouldmaintainrobustconnectivitypatternsevenifmultiplenodesin 34 PAGE 35 thenetworkaredisabled.However,thepracticaldrawbacksofcliquesincludethefactthatthesestructuresareoftenoverlyrestrictiveandexpensivetoconstruct. Toprovideatradeoffbetweenrobustnessandpracticalfeasibility,certainothernetworkstructuresthatrelaxthedenitionofacliquecanbeutilized.Thefollowingdenitionsaddresstheserelaxationsfromdifferentperspectives.GivenagraphG(V,E)withasetofvertices(nodes)VandasetofedgesE,ak-cliqueCisasetofverticesinwhichanytwoverticesaredistanceatmostkfromeachotherinG Luce ( 1950 ).LetdG(i,j)bethelengthofashortestpathbetweenverticesiandjinGandd(G)=maxi,j2VdG(i,j)bethediameterofG. Thus,iftwoverticesu,v2Vbelongtoak-cliqueC,thendG(u,v)k,howeverthisdoesnotimplythatdG(C)(u,v)k(thatis,othernodesintheshortestpathbetweenuandvarenotrequiredtobelongtothek-clique).ThismotivatedMokken( Mokken ( 1979 ))tointroducetheconceptofak-club.Ak-clubisasubsetofverticesDVsuchthatthediameterofinducedsubgraphG(D)isatmostk(thatis,thereexistsapathoflengthatmostkconnectinganypairofnodeswithinak-club,whereallthenodesinthispathalsobelongtothisk-club).Also,~VVissaidtobeak-plexifthedegreeofeveryvertexintheinducedsubgraphG(~V)isatleastj~Vj)]TJ /F3 11.955 Tf 18.02 0 Td[(k( SeidmanandFoster ( 1978 )).Acomprehensivestudyofthemaximumk-plexproblemispresentedinarecentworkby Balasundarametal. ( 2010 ). Inthischapter,weutilizetheseconceptstodeveloprigorousmathematicalprogrammingformulationstomodelrobustconnectivitystructuresindynamicsensornetworks.Moreover,theseformulationswillalsotakeintoaccountvariousuncertainparametersbyintroducingquantitativeriskmeasuresthatminimizeorrestrictinformationlosses.Overall,wewilldevelopoptimalschedulesforsensormovementsthatwilltakeintoaccountboththeuncertainlossesofinformationandtherobustconnectivitybetweenthenodesthatwouldallowonetoefcientlyexchangethecollectedinformation. 35 PAGE 36 3.1Multi-SensorSchedulingProblems:GeneralDeterministicSetup Thissectionintroducesapreliminarymathematicalframeworkfordynamicmulti-sensorschedulingproblems.Thesimplestdeterministicone-sensorversionofthisproblemwasintroducedin YavuzandJeffcoat ( 2007 ).Theone-sensorschedulingproblemwasthenextendedandgeneralizedtomorerealisticcasesofmulti-sensorschedulingproblems,includingthesetupsinuncertainenvironmentsin Boykoetal. ( 2011 ).Inthesubsequentsectionsofthischapter,thissetupwillbefurtherextendedtoincorporaterobustconnectivityissuesintotheconsidereddynamicsensornetworkmodels. Tofacilitatefurtherdiscussion,werstintroducethefollowingmathematicalnotationsthatwillbeusedthroughoutthischapter.Assumethattherearemsensorsthatcanmovewithinaspeciedareaofsurveillance,andtherearensitesthatneedtobeobservedateverydiscretetimemomentt=1,...,T.Onecaninitiallyassumethatasensorcanobserveonlyonesiteatonepointoftimeandcanimmediatelyswitchtoanothersiteatthenexttimemoment.Sincemisusuallysignicantlysmallerthann,therewillbebreachesinsurveillancethatcancauselossesofpotentiallyvaluableinformation. Apossibleobjectivethatarisesinpracticalsituationsistobuildastrategythatoptimizesapotentiallossthatisassociatedwithnotobservingcertainsitesatsometimemoments. 3.1.1FormulationwithBinaryVariables Onecanintroducebinarydecisionvariables xi,t=8><>:1,ifi-thsiteisobservedattimet0,otherwise(3) andintegervariablesyi,tthatdenotethelasttimesiteiwasvisitedasoftheendoftimet,i=1,...,n,t=1,...,T,m PAGE 37 Onecanthenassociateaxedpenaltyaiwitheachsiteiandavariablepenaltybiofinformationloss.Ifasensorisawayfromsiteiattimepointt,thexedpenaltyaiisincurred.Moreover,thevariablepenaltybiisproportionaltothetimeintervalwhenthesiteisnotobserved.Weassumethatthevariablepenaltyratecanbedynamic;therefore,thevaluesofbimaybedifferentateachtimeinterval.Thusthelossattimetassociatedwithsiteiis ai(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)(3) Intheconsideredsetup,wewanttominimizethemaximumpenaltyoveralltimepointstandsitesi maxi,tfai(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t)g(3) Furthermore,xi,tandyi,tarerelatedviathefollowingsetofconstraints.Nomorethanmsensorsareusedateachtimepoint;therefore nXi=1xi,tm,8t=1,...,T(3) Timeyi,tisequaltothetimewhenthesiteiwaslastvisitedbyasensorbytimet.Thisconditionissetbythefollowingconstraints: 0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F11 7.97 Tf 6.59 0 Td[(1txi,t,8i=1,...,n,8t=1,...,T (3) txi,tyi,tt,8i=1,...,n,8t=1,...,T (3) Further,usinganextravariableCandstandardlinearizationtechniques,wecanformulatethemulti-sensorschedulingoptimizationprobleminthedeterministicsetupasthefollowingmixedintegerlinearprogram: minC (3) s.t.Cai(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t),8i=1,...,n,8t=1,...,T (3) nXi=1xi,tm,8t=1,...,T (3) 37 PAGE 38 0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F11 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,T (3) txi,tyi,tt,8i=1,...,n,8t=1,...,T (3) yi,0=0,8i=1,...,n (3) xi,t2f0,1g,8i=1,...,n,8t=1,...,T (3) yi,t2IR,8i=1,...,n,8t=0,...,T (3) Weallowedrelaxation( 3 )ofvariablesyi,ttothespaceofrealnumbers,becausetheconstraints( 3 )and( 3 )enforcethefeasiblevaluesofvariablesyi,ttobeinteger. 3.1.2CardinalityFormulation Lemma.Constraint( 3 )isequivalenttothefollowingcombinationoftwoconstraints: 0xi,t18i=1,...,n,8t=1,...,T (3) card(xt)nXi=1xi,t8t=1,...,T (3) wherext=(x1,t,...,xn,t)T,andcard(xt)denotesthecardinalityfunctionforthevectorxt.Bydenition,card(xt)equalsthenumberofnon-zeroelementsintheinputvectorxt. Proof.Assumethematrix(xi,t)satisesconstraint( 3 ).Obviously,itthensatises( 3 ).Atthesametime,foreveryt,sumofallelementsisequaltothenumberofvalues1init.Andthesearetheonlynon-zeroelementsinit.Therefore,constraint( 3 )isalsosatised. Nowassumethematrix(xi,t)doesnotsatisfyconstraint( 3 ).Thusthereisapair(i,t),forwhichxi,t=and6=0and6=1.If<0or>1,thenconstraint( 3 )isviolated.Thus,forallpairs(i,t),0xi,t1,and0<<1.Therefore,forallpairs(i,t),card(xi,t)xi,t,andcard()>.Takingintoaccountthatcard(xt)=Picard(xi,t)weconcludethat( 3 )isviolated. 38 PAGE 39 Nowwecanwritealternative,cardinalityformulationforthegeneraldeterministicsensor-schedulingproblem. minC (3) s.t.Cai(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t),8i=1,...,n,8t=1,...,T (3) nXi=1xi,tm,8t=1,...,T (3) 0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F11 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,T (3) txi,tyi,tt,8i=1,...,n,8t=1,...,T (3) yi,0=0,8i=1,...,n (3) 0xi,t1,8i=1,...,n,8t=1,...,T (3) card(xt)nXi=1xi,t,8t=1,...,T (3) yi,t2IR,8i=1,...,n,8t=0,...,T (3) Althoughthetwoformulationsareequivalent,someoptimizationsolvers,suchasPortfolioSafeguard(thatwillbementionedlaterinthischapter),canprovideanear-optimalsolutionfasteriftheformulationwithcardinalityconstraintsisusedinsteadoftheonewithbooleanvariables,whichmaybeimportantintime-criticalsystemsinmilitarysettings. 3.2QuantitativeRiskMeasuresinUncertainEnvironments: ConditionalValue-at-Risk Tofacilitatefurtherdiscussionontheformulationsoftheaforementionedproblemsunderuncertainty,inthissectionwebrieyreviewbasicdenitionsandfactsrelatedtotheConditionalValue-at-Riskconcept. ConditionalValue-at-Risk(CVaR) RockafellarandUryasev ( 2000 2002 ); Sarykalinetal. ( 2008 )isaquantitativeriskmeasurethatwillbeusedinthemodelsdevelopedinthenextsection,whichwilltakeintoaccountthepresenceofuncertainparameters.CVaRiscloselyrelatedtoawell-knownquantitativeriskmeasurereferredtoas 39 PAGE 40 Value-at-Risk(VaR).Bydenition,withrespecttoaspeciedprobabilitylevel(inmanyapplicationsthevalueofissetratherhigh,e.g.95%),the-VaRisthelowestamountsuchthatwithprobability,thelosswillnotexceed,whereasforcontinuousdistributionsthe-CVaRistheconditionalexpectationoflossesabovethatamount.Asitcanbeseen,CVaRisamoreconservativeriskmeasurethanVaR,whichmeansthatminimizingorrestrictingCVaRinoptimizationproblemsprovidesmorerobustsolutionswithrespecttotheriskofhighlosses(Figure 3-1 ). Figure3-1. GraphicalrepresentationofVaRandCVaR. Formally,-CVaRforcontinuousdistributionscanbeexpressedas CVaR(x)=(1)]TJ /F4 11.955 Tf 11.95 0 Td[())]TJ /F11 7.97 Tf 6.58 0 Td[(1ZL(x,w)(x)L(x,w)p(w)dw(3) whereL(x,w)istherandomloss(penalty)variabledrivenbydecisionvectorxandhavingadistributioninIRinducedbythatofthevectorofuncertainparametersw. 40 PAGE 41 CVaRisdenedinasimilarwayfordiscreteormixeddistributions.ThereadercanndtheformaldenitionofCVaRforgeneralcasein RockafellarandUryasev ( 2002 ); Sarykalinetal. ( 2008 ). Ithasbeenshownin RockafellarandUryasev ( 2000 )thatminimizing( 3 )isequivalenttominimizingthefunction F(x,)=+(1)]TJ /F4 11.955 Tf 11.96 0 Td[())]TJ /F11 7.97 Tf 6.59 0 Td[(1Zw2IRd[L(x,w))]TJ /F4 11.955 Tf 11.95 0 Td[(]+p(w)dw(3) overwand,where[t]+=twhent>0but[t]+=0whent0,andoptimalvalueofthevariablecorrespondstotheVaRvalue,introducedabove. 3.3OptimizingtheConnectivityofDynamicSensorNetworks UnderUncertainty Thissectionextendstheprevioussensorsschedulingproblemtoastochasticenvironment.WeuseCVaRmeasuretomodelandoptimizevariousobjectivesassociatedwiththeriskoflossofinformation. Inthestochasticformulation,thepenaltiesaiandbi,tarerandom.WegenerateSdiscretescenarios,whichapproximateimpliedjointdistribution.Thus,everyscenarioconsistsoftwoarrays:one-dimensionalfaigsandtwo-dimensionalfbi,tgs. Now,considerthetermofthelossfunctioncorrespondingtothesitei,timet,andscenarios:Ls(x,y;i,t)=asi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t) Underuncertainty,itisoftenmoreimportanttomitigatethebiggestpossiblelosses,ratherthantheaveragedamage.Followingthisidea,wetake(1)]TJ /F4 11.955 Tf 12.17 0 Td[()biggestpenalties,andminimizeaveragepenaltyoveralli,tands.Thisobjectivefunctionisexactlytheconditionalvalue-at-risk. Wenowhavethefollowingclassofoptimizationproblems: minx,yCVaRfL(x,y;i,t)g(3) 41 PAGE 42 Thisclasshasoneextremecase:=1,whentheproblembecomesequivalenttominimizingmaximumpossiblepenaltyoverallscenarios,locationsandtimepoints: minx,ymaxi,t,s(asi(1)]TJ /F3 11.955 Tf 11.96 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t))(3) ThisproblemhasanequivalentLPformulation: minC (3) s.t.Casi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t) (3) 8i=1,...,n,8t=1,...,T,8s=1,...,S InordertoformulateageneralCVaRoptimizationprobleminLPtermswehavetointroduceadditionalvariablessi,t,s=1,...,S,i=1,...,n,t=1,...,T,and.WiththesevariablestheproblemofminimizingCVaRwillbereducedtothefollowing: minC (3) s.t.C+1 (1)]TJ /F4 11.955 Tf 11.96 0 Td[()nSTXs=1,...,Si=1,...,nt=1,...,Tsi,t (3) si,tasi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t))]TJ /F4 11.955 Tf 11.95 0 Td[( (3) 8i=1,...,n,t=1,...,T,s=1,...,Ssi,t0,8i=1,...,n,t=1,...,T,s=1,...,S (3) Wehavediscussedvariousobjectivefunctionswithobjective-specicconstraintsforsensorschedulingproblemsinthestochasticenvironment.Inadditiontothat,everysensorschedulingproblem,includingthoseinstochasticenvironment,musthaveconstraintslimitingnumberofsensors( 3 )anddeningvariablesofthelasttimeofobservation( 3 )( 3 ).Theseconstraintsarereferredtoasmandatoryconstraintsforeverysensor-schedulingproblem. FurtherwedeneawirelessconnectivitynetworkG(V,E)onthesetoflocationsV.Weinterpretitintermsofthe0-1adjacencymatrixE=feijgi,j=1,...,n,whereeacheijisa 42 PAGE 43 0-1indicatorofwirelesssignalreachabilitybetweennodesiandj,thatis,iflocationsiandjarewithindirecttransmissiondistancefromeachother,thentheyareconnectedbyanedge,andeij=1(eij=0otherwise).Wealsodeneasubnetwork~GofG(V,E)containingonlythosemnodes(locations)thataredirectlyobservedbysensorsataparticulartimemoment. Schedulingofobservationoftenrequiressensorstomaintainacertainlevelofwirelessconnectivityrobustness.Ifanenemysendsajammingsignalthatbreaksconnectivitybetweenapairofnodes,thenthesubnetwork~Geithershouldstayconnected,oratleastshouldmaintainunitywithprobabilitycloseto1.Further,wewillutilizeseveraltypesofnetworkstructuresthatcanbeappliedtoensurethatthenetworksatisescertainrobustnessconstraints. Themostrobustnetworkstructureisaclique,whichimpliesthateachpairofnodesisdirectlyconnectedbyanedge.Obviously,maintainingacliquestructureofthesubnetwork~Gateverymomentintimeisveryexpensiveintermsofpenalty,andcanbeevenimpossible,iftheoverallwirelessconnectivitynetworkisnotdenseenough.Hence,itisreasonabletoutilizeappropriatetypesofcliquerelaxationstoensurerobustnetworkconnectivityateverytimemoment. Oneoftheconsideredconceptsisak-plex.Bydenition,asmentionedabove,ak-plexisasubgraphinwhicheverynodeisconnectedtoatleastm)]TJ /F3 11.955 Tf 12.05 0 Td[(kothernodesinit(wheremisthenumberofnodesinthissubgraph).Thisnetworkcongurationensuresthateachnodeisconnectedtomultipleneighbors,whichmakesitmorechallengingforanadversarytodisconnectthenetworkandisolatethenodesbydestroying(jamming)theedges. Anotherconsideredclassofnetworkcongurationsisak-club.Recallthateverypairofnodesink-clubisconnectedinitthroughachainofnomorethankarcs(edges).Themotivationforstudyingthistypeofconstraintsisbasedonthefactthatiftwosensorsareconnectedthroughashorterpath,itlowerstheprobabilityoferrorsin 43 PAGE 44 informationtransmissionthroughintermediaries,sincethenumberofintermediariesissmaller.Laterinthepaper,wewillspecicallyuseastrongerrequirementonthelengthofthesepaths.Werequirethatanytwonodesareconnectedeitherdirectlybyanedge,orthroughatmostoneintermediarynode,whichisoftenadesiredrobustnessrequirementundertheconditionswhenthenumberofintermediaryinformationtransmissionsneedstobeminimizedduetoadversarialconditions.Clearly,a2-clubisastructurethatsatisesthisrequirement.Inthenextsubsection,weshowthatthisconditioncanbeincorporatedintheconsideredoptimizationmodels. 3.3.1EnsuringShortTransmissionPathsvia2-clubFormulation Thegeneralrequirementforasubnetwork~Gtorepresentak-clubcanbeformulatedasthefollowingsetofconstraints: eij+...+nXq=1eiqeqjxq,t+...+nXq=1nXl=1eiqeqleljxq,txl,t+...+nXq=1nXl=1nXp=1eiqeqlelpepjxq,txl,txp,t+...+nXi1=1nXi2=1nXik)]TJ /F15 5.978 Tf 5.76 0 Td[(2=1nXik)]TJ /F15 5.978 Tf 5.75 0 Td[(1=1eii1ei1i2...eik)]TJ /F15 5.978 Tf 5.76 0 Td[(2ik)]TJ /F15 5.978 Tf 5.76 0 Td[(1eik)]TJ /F15 5.978 Tf 5.76 0 Td[(1jxi1,t...xik)]TJ /F15 5.978 Tf 5.75 0 Td[(1,txi,t+xj,t)]TJ /F2 11.955 Tf 11.96 0 Td[(1(3) wherei=1,...,n)]TJ /F2 11.955 Tf 12.26 0 Td[(1,j=i+1,...,n,t=1,...,T.Foreveryktheseconstraintscanbelinearized,however,thesizeoftheproblemmaysubstantiallyincrease.Inthispaper,welimitourdiscussiononlyto2-clubconstraintsduetothepracticalreasonsmentionedearlierinthissectionandduetothefactthattheformulationforthecaseofk=2willnotaddtoomanynewentities(nomorethanO(n2))totheproblemformulation.Theyrequireeverypairofnodes(i,j)tobeconnecteddirectly,orthroughsomeothernodep. 44 PAGE 45 Suchtypeofcommunicationbetweensensors(i,j)hasaconciseformulation:eij+nXp=1eipepjxp,txi,t+xj,t)]TJ /F2 11.955 Tf 11.96 0 Td[(18i=1,...,n)]TJ /F2 11.955 Tf 11.96 0 Td[(1,8j=i+1,...,n8t=1,...,T Here,theleft-handsideisalwaysnonnegative.Theright-handsidebecomespositiveonlyifbothlocationsiandjareobservedbysensors,andthenitequals1.Accordingtothe2-clubdenition,thesesensorshavetobeconnected(andexchangeinformation)eitherdirectly,orthroughoneotherintermediarysensornode.Intherstcaseeijequals1.Inthesecondcase,thesumPnp=1eipepjxp,twillalsobepositive. Itisalsoimportanttonotethatthoseconstraints,forwhicheij=1,canbeomitted.Thus,a2-clubwirelessnetworkcongurationcanbeensuredbythefollowingsetofconstraints:Xp2(i)\(j)xp,txi,t+xj,t)]TJ /F2 11.955 Tf 11.95 0 Td[(18i=1,...,n)]TJ /F2 11.955 Tf 11.95 0 Td[(1,8j=i+1,...,n,j=2(i),8t=1,...,T where(i)and(j)arethesetsofneighborsofnodesiandj,respectively. Belowwepresentthecompletegeneralformulationforthedynamicsensorschedulingoptimizationprobleminastochasticenvironmentwith2-clubwirelessconnectivityconstraints. minCs.t.C+1 (1)]TJ /F4 11.955 Tf 11.96 0 Td[()nSTXs=1,...,Si=1,...,nt=1,...,Tsi,tsi,tasi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t))]TJ /F4 11.955 Tf 11.95 0 Td[(8i=1,...,n,t=1,...,T,s=1,...,Ssi,t0,8i=1,...,n,t=1,...,T,s=1,...,SnXi=1xi,tm,8t=1,...,T 45 PAGE 46 0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F11 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,Ttxi,tyi,tt,8i=1,...,n,8t=1,...,Tyi,0=0,8i=1,...,nXp2(i)\(j)xp,txi,t+xj,t)]TJ /F2 11.955 Tf 11.95 0 Td[(18i=1,...,n)]TJ /F2 11.955 Tf 11.95 0 Td[(1,8j=i+1,...,n,j=2(i),8t=1,...,Txi,t2f0,1g,8i=1,...,n,8t=1,...,Tyi,t2IR,8i=1,...,n,8t=0,...,T 3.3.2EnsuringBackupConnectionsviak-plexFormulation Constraintsthatrequireawirelessnetworktohavethek-plexstructure,canbedenedusingasymmetricadjacencymatrixE=feijgi,j=1,...,n,asdenedabove.Recallthatxt=(x1,t,...,xn,t)T.Considerthevectorzt=(z1,t,...,zn,t)=Ext.Theelementzi,tcanbeinterpretedasthenumberofsensorswhichhaveawirelessconnectionwithnodeiattimet.Thus,theconstraintExtxtor(E)]TJ /F3 11.955 Tf 11.96 0 Td[(I)xt0ensuresthateachsensornodehasatleastoneneighbor,i.e.,itisnotisolated.Ifwewanteachsensortohaveatleast(m)]TJ /F3 11.955 Tf 11.99 0 Td[(k)wirelessconnections(edges)withothersensors,thenweshouldmaketheconstraintsmorerestrictive:Ext(m)]TJ /F3 11.955 Tf 11.95 0 Td[(k)xt,or(E)]TJ /F2 11.955 Tf 11.95 -.16 Td[((m)]TJ /F3 11.955 Tf 11.95 0 Td[(k)I)xt08t=1,...,T Theserestrictionsbydenitionensurethatasubnetwork~Gisak-plex. Belowwepresentthecompletegeneralformulationforthedynamicsensorschedulingoptimizationprobleminastochasticenvironmentwithk-plexwirelessconnectivityconstraints. minCs.t.C+1 (1)]TJ /F4 11.955 Tf 11.96 0 Td[()nSTXs=1,...,Si=1,...,nt=1,...,Tsi,t 46 PAGE 47 si,tasi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(xi,t)+bsi,t(t)]TJ /F3 11.955 Tf 11.96 0 Td[(yi,t))]TJ /F4 11.955 Tf 11.95 0 Td[(8i=1,...,n,t=1,...,T,s=1,...,Ssi,t0,8i=1,...,n,t=1,...,T,s=1,...,SnXi=1xi,tm,8t=1,...,T0yi,t)]TJ /F3 11.955 Tf 11.95 0 Td[(yi,t)]TJ /F11 7.97 Tf 6.58 0 Td[(1txi,t,8i=1,...,n,8t=1,...,Ttxi,tyi,tt,8i=1,...,n,8t=1,...,Tyi,0=0,8i=1,...,n(E)]TJ /F2 11.955 Tf 11.95 -.17 Td[((m)]TJ /F3 11.955 Tf 11.95 0 Td[(k)I)xt0,8t=1,...,Txi,t2f0,1g,8i=1,...,n,8t=1,...,Tyi,t2IR,8i=1,...,n,8t=0,...,T 3.4ComputationalExperiments ComputationalexperimentsonsampleprobleminstanceshavebeenperformedonIntelXeonX53552.66GHzCPUwith16GBRAM,usingtwocommercialoptimizationsolvers:ILOGCPLEX11.2andAORDAPSG64bit(MATLAB64bitenvironment).Itshouldbenotedthatduetothenatureoftheconsideredclassofproblems,theyarecomputationallychallengingevenonrelativelysmallnetworks.Therefore,inmanypracticalsituations,ndingnear-optimalsolutionsinareasonabletimewouldbesufcient.ThePSGpackagewasusedinadditiontoCPLEXbecauseithasattractivefeaturesintermsofcodingtheoptimizationproblems,andthereforeitmaybemorepreferabletouseinpracticaltime-criticalsettings.Inparticular,inadditiontolinearandpolynomialfunctions,PSGsupportsanumberofdifferentclassesoffunctions,suchasCVaRandcardinalityfunctions.ForthepurposesofthecurrentcasestudywedenedinPSGtheobjectiveusingtheCVaRfunction,andwealsousedcardinalityfunctionforthecardinalityconstraintonxi,tinsteadofbooleanconstraint. 47 PAGE 48 Table3-1. CPLEXResults:Problemwith2-clubConstraints Case PSGCARPSGTANKCPLEX26secCPLEX1minnm valuetimevaluetimevaluetimevaluetime 104 97.9622.3100.6121.684.6226.184.6260.1105 79.2922.883.6925.674.3026.071.5760.0106 74.7924.271.1523.663.3226.163.3260.1107 64.8225.661.6826.557.5426.157.5460.0108 52.2726.751.8625.250.7626.150.2160.0114 106.1923.2105.1624.292.2126.192.2160.1115 86.6123.085.2122.475.5426.175.5460.1116 75.0823.976.2122.570.7926.166.3260.1117 68.4325.469.9324.058.3726.058.3760.1118 60.4424.461.0523.857.0126.157.0160.1124 122.3224.1124.0022.8105.4526.1105.4560.1125 91.2522.898.0222.682.0826.281.9660.1126 84.6922.881.7623.072.6526.072.6560.1127 73.4423.676.9522.164.1126.164.1160.1128 64.7825.661.9524.056.1026.156.1060.1134 126.3724.5119.7628.998.4626.098.4660.1135 94.4824.0104.7826.586.3126.088.2060.1136 82.4624.283.2927.276.6126.176.6160.1137 73.9725.574.5930.870.5326.167.9360.0138 71.5726.469.7933.959.9226.159.9260.1144 135.7525.6139.0627.3118.4126.0112.7460.1145 109.7527.1114.2727.295.0126.194.8760.1146 89.5824.393.8227.279.5426.179.5460.1147 80.7025.980.8923.370.5326.270.3760.1148 75.3126.076.8826.665.2626.161.6760.1154 155.6727.9145.0026.7127.0026.2126.8260.1155 113.1825.8115.6528.8104.0626.1102.3460.1156 95.5124.899.1128.190.8026.182.7460.1157 85.9625.086.4927.874.2226.174.2260.1158 77.8126.176.8334.468.0926.168.2260.0 avgavgavgavg 24.826.026.160.1 Forcomparisonpurposes,multipleexperimentshavebeenperformed.Allexperimentsweredividedintotwogroups:with2-clubconnectivityconstraintsonsubnetwork~G,andk-plexconstraintswithk=m 2.Ineachofthesegroups,numberoflocationsn=10,11,12,13,14,15andnumberofsensorsm=4,5,6,7,8.AllproblemshaveCVaR-typeobjectivewith=0.9,deterministicsetup(1scenario),20 48 PAGE 49 Table3-2. CPLEXResults:Problemwithk-plexConstraints Case PSGCARPSGTANKCPLEX27secCPLEX1minnm valuetimevaluetimevaluetimevaluetime 104 106.5225.1102.9422.785.3427.185.3460.1105 79.9823.783.4822.072.4027.172.4060.1106 74.2225.078.2526.063.2227.063.2260.1107 65.7127.062.2726.956.7327.056.7360.0108 55.7025.850.9124.350.2627.050.2660.0114 117.6123.7101.2826.887.5727.087.5760.1115 89.1624.792.7722.975.9727.175.9760.0116 77.2724.384.0826.268.2527.167.0360.0117 70.426.067.2423.558.6127.158.6160.0118 67.9825.964.8423.453.8027.153.1560.0124 134.6231.1128.6026.1109.8927.1102.5260.1125 100.8024.4103.7023.080.1127.180.1160.1126 83.2124.387.5625.570.5427.170.5460.1127 69.9125.874.9926.263.2627.163.2660.1128 70.0327.969.2822.756.7527.156.7360.1134 134.3932.2121.7228.5103.8727.1103.8760.1135 97.2425.4103.8923.784.6327.184.5760.1136 90.5125.789.4029.577.9427.177.9460.1137 78.1526.277.3428.768.5227.168.5260.1138 77.4926.772.6124.363.3627.159.6960.1144 134.0130.3140.1232.1119.1727.1112.1860.1145 114.7226.9113.6127.690.0027.189.3460.1146 97.8327.496.7129.178.3727.178.3760.1147 86.0926.487.0031.470.4627.170.2460.1148 77.7827.275.7526.062.4827.062.4860.1154 153.1834.4189.4132.3136.8927.1120.8160.1155 123.6128.6123.8427.0110.9427.198.1560.2156 97.5327.2101.7231.483.1127.082.5960.1157 93.2127.386.4830.174.4327.174.4360.0158 80.2928.975.7025.767.0827.167.3760.1 avgavgavgavg 26.926.527.160.1 timeintervals.Theedgedensityoftheconsideredoverallwirelessconnectivitynetworkwas=0.8(80%pairsofnodesareconnected). WehaverunPSGusingtwobuilt-insolvers:CARandTANK.Thesesolverstookonaverage26secondstodeliversolutionoverallcaseswith2-clubconstraints,and27secondsforthecaseswithk-plexconstraints.AfterthatwerunCPLEXoncases 49 PAGE 50 Table3-3. CPLEXandPSGResults:StochasticSetup CPLEXPSGCARPSGTANKtypeS valuetimegapvaluetimevaluetime k-plex10 85.14300.127.0%96.7825.396.8129.4k-plex20 89.36300.234.6%95.1035.396.1037.1k-plex50 92.27300.544.4%110.06154.797.40273.1k-plex100 93.81301.749.7%104.57300.6115.69300.62-club10 86.92300.130.5%100.13229.9100.13300.12-club20 84.92300.132.0%97.5579.497.55300.12-club50 89.75300.544.3%104.30300.1103.42300.12-club100 95.87301.850.7%116.63300.2116.23300.2 with2-clubconstraintswithtimelimit26seconds,oncaseswithk-plexconstraintswithtimelimit27seconds.Then,weadditionallyrunCPLEXonallcaseswithtimelimit1minute.ComputationalresultsarepresentedinTable 3-1 andTable 3-2 ,forthecaseswith2-clubconstraintsandk-plexconstraints,respectively. TheresultsshowthatonaveragethebestsolutionisproducedbyCPLEX1minuterun.Values,obtainedbyCPLEXrunswith26and27secondslimitsareby1.2%and2.2%greaterfor2-clubandfork-plexrespectively.Inmostcasessolutionsobtainedbytworunswereequal.Therefore,CPLEXobtainssolutionclosetooptimalinaboutlessthan30seconds.PSGTANKsolutionvalueisgreaterthanCPLEX1minutesolutionvalueby15.8%and22.4%for2-clubandfork-plexrespectively.PSGCARperformsslightlybetterthananothersolver,providingthesolutionvaluesgreaterthanCPLEX1minutesolutionvaluesby15.0%and22.0%. Inadditiontodeterministicsetup,wehaveruntheaforementionedoptimizationproblemsunderuncertaintyonseveralstochasticprobleminstanceswiththenumberofsensorsm=6,thenumberoflocationsn=12,theCVaR-typeobjectivewith=0.9,T=10timeintervals,fordifferentnumbersofscenarios:S=10,20,50,100.Asbefore,thewirelessconnectivitynetworkedgedensitywas=0.8.Thetimelimitwassetto5minutes.PSGsolversinmostcasesprovidedsolutionbeforethetimelimitwasreached 50 PAGE 51 (Table 3-3 ).However,thequalityofsolutionwasworsethenprovidedbyCPLEXby15%onaverage. 51 PAGE 52 CHAPTER4CALIBRATINGRISKPREFERENCESWITHGENERALIZEDCAPMBASEDONMIXEDCVARDEVIATION ThischapterisbasedonthejointpublicationwithS.UryasevandR.T.Rockafellar( Kalinchenkoetal. ( 2012 )) TheCapitalAssetPricingModel(CAPM, Sharpe ( 1964 ), Lintner ( 1965 ), Mossin ( 1966 ), Treynor ( 1961 ), Treynor ( 1999 ))afteritsfoundationinthe1960'sbecameoneofthemostpopularmethodologiesforestimationofreturnsofsecuritiesandexplanationoftheircombinedbehavior.Thismodelassumesthatallinvestorswanttominimizeriskoftheirinvestments,andallinvestorsmeasureriskbythestandarddeviationofreturn.ThemodelimpliesthatalloptimalportfoliosaremixturesoftheMarketFundandriskfreeinstrument.TheMarketFundiscommonlyapproximatedbysomestockmarketindex,suchasS&P500. AnimportantpracticalapplicationoftheCAPMmodelisthepossibilitytocalculatehedgedportfoliosuncorrelatedwiththemarket.Toreducetheriskofaportfolio,aninvestorcanincludeadditionalsecuritiesandhedgemarketrisk.TheriskoftheportfoliointermsofCAPMmodelismeasuredbybeta.Thevalueofbetaforeverysecurityorportfolioisproportionaltothecorrelationbetweenitsreturnandmarketreturn.Thisfollowsfromtheassumptionthatinvestorshaveriskattitudesexpressedwiththestandarddeviation(volatility).Thehedgingisdesignedtoreduceportfoliobetawiththeideatoprotecttheportfolioincaseofamarketdownturn.However,betaisjustascaledcorrelationwiththemarketandthereisnoguaranteethathedgeswillcoverlossesduringsharpdownturns,becausetheprotectionworksonlyonaverageforthefrequentlyobservedmarketmovements.Recentcreditcriseshaveshownthathedgeshavetendenciestoperformverypoorlywhentheyaremostneededinextrememarketconditions.Theclassicalhedgingproceduresbasedonstandardbetasetupadefencearoundthemeanofthelossdistribution,butfailinthetails.ThisdeciencyhasledtomultipleattemptstoimprovetheCAPM. 52 PAGE 53 OneapproachtoCAPMimprovementistoincludeadditionalfactorsinthemodel.Forexample, KrausandLitzenberger ( 1976 ), FriendandWestereld ( 1980 ),and Lim ( 1989 )providetestsforthethree-momentCAPM,includingco-skewnessterm.Thismodelaccountsfornon-symmetricaldistributionofreturns. FamaandFrench ( 1996 )addedtotheassetreturnlinearregressionmodeltwoadditionalterms:thedifferencebetweenthereturnonaportfolioofsmallstocksandthereturnonaportfoliooflargestocks,andthedifferencebetweenthereturnonaportfolioofhigh-book-to-marketstocksandthereturnonaportfoliooflow-book-to-marketstocks.Recently, BarberisandHuang ( 2008 )presentedCAPMextensionbasedonprospecttheory,whichallowstopricesecurity'sownskewness. Thesecondapproachistondalternativeriskmeasures,whichmaymorepreciselyrepresentriskpreferencesofinvestors.Forinstance, KonnoandYamazaki ( 1991 )appliedanL1riskmodel(basedonmeanabsolutedeviation)totheportfoliooptimizationproblemwithNIKKEI225stocks.TheirapproachledtolinearprogramminginsteadofquadraticprogrammingintheclassicalMarkowitz'smodel,butcomputationalresultsweren'tsignicantlybetter.Furtherresearchhasbeenfocusedonriskmeasuresmorecorrectlyaccountingforlosses.Forexample, Estrada ( 2004 )applieddownsidesemideviation-basedCAPMforestimatingreturnsofInternetcompanystocksduringtheInternetbubblecrisis.Downsidesemideviationcalculatesonlyforthelossesunderperformingthemeanofreturns.Nevertheless,semideviation,similarlytostandarddeviation,doesn'tpayspecialattentiontoextremelosses,associatedwithheavytails. SortinoandForsey ( 1996 )alsopointoutthatdownsidedeviationdoesnotprovidecompleteinformationneededtomanagerisk. AmuchmoreadvancedlineofresearchisconsideredinpapersofRockafellar,UryasevandZabarankin( Rockafellaretal. ( 2006a ), Rockafellaretal. ( 2006b ), Rockafellaretal. ( 2007 )).Theassumptionhereisthattherearedifferentgroupsofinvestorshavingdifferentriskpreferences.ThegeneralizedCapitalAssetPricing 53 PAGE 54 Model(GCAPM)proposesthatthereisacollectionofdeviationmeasures,representingriskpreferencesofthecorrespondinggroupsofinvestors.Thesedeviationmeasuressubstituteforthestandarddeviationoftheclassicaltheory.WiththegeneralizedpricingformulafollowingfromGCAPMonecanestimatethedeviationmeasureforaspecicgroupofinvestorsfrommarketprices.Thisisdonebyconsideringparametricclassesofdeviationmeasuresandcalibratingparametersofthesemeasure.TheGCAPMprovidesanalternativetotheclassicalCAPMmeasureofsystematicrisk,so-calledgeneralizedbeta.Similarlytoclassicalbeta,thegeneralizedbetacanbeusedinportfoliooptimizationforhedgingpurposes. Weconsidertheclassofso-calledmixedCVaRdeviations,havingseveralattractiveproperties.First,differenttermsinthemixedCVaRdeviationgivecredittodifferentpartsofthedistribution.Therefore,byvaryingparameters(coefcients),onecanapproximatevariousstructuresofriskpreferences.Inparticular,so-calledtail-betacanbebuiltwhichaccountsforheavytaillosses(e.g.,lossesinthetop5%ofthetaildistribution).Second,mixedCVaRdeviationisacoherentdeviationmeasure,anditthereforesatisesanumberofdesiredmathematicalproperties.Third,optimizationofproblemswithmixedCVaRdeviationcanbedoneveryefciently.Forinstance,fordiscretedistributions,theoptimizationproblemscanbereducedtolinearprogramming. Weconsiderasetupwithonegroupofinvestors(representativeinvestor).WeassumethattheseinvestorsestimateriskswiththemixedCVaRdeviationshavingxedquantilelevels:50,75,85,95and99percentofthelossdistribution.Bydenition,thismixedCVaRdeviationisaweightedcombinationofaveragelossesexceedingthesequantilelevels.TheweightsforCVaRswiththedifferentquantilelevelsdetermineaspecicinstanceoftheriskmeasure.Thegeneralizedpricingformulaandgeneralizedbetaforthisclassofdeviationmeasuresareusedinthisapproach.Withmarketoptionpricestheparametersofthedeviationmeasurearecalibrated,thusestimatingriskpreferencesofinvestors. 54 PAGE 55 Severalnumericalexperimentscalibratingriskpreferencesofinvestorsatdifferenttimemomentswereconducted.Wehavefoundthatthedeviationmeasure,representinginvestors'riskpreferences,hasthebiggestweightontheCVaR50%term,whichequalstheaveragelossbelowmedianreturn.Onaverage,about11%oftheweightisassignedtoCVaR85%,CVaR95%andCVaR99%evaluatingheavy-lossscenarios.Experimentsalsoshowedthatriskpreferencestendtochangeovertimereectinginvestors'opinionsaboutthestateofmarket. Wearenottherstwhoattemptedtoextractriskpreferencesfromoptionprices.Itisacommonknowledgethatoptionpricesconveyriskneutralprobabilitydistribution.Somestudies,suchas Ait-SahaliaandLo ( 2000 ), Jackwerth ( 2000 ), BlissandPanigirtzoglou ( 2002 ),containvariousapproachestoextractingriskpreferencesintheformofutilityfunctionbycomparingobjective(orstatistical)probabilitydensityfunctionwithriskneutralprobabilitydensityfunction,estimatedfromoptionprices.Inourworkriskpreferencesareexpressedintheformofdeviationmeasure,thusmakingitimpossibletocompareresultswithpreviousstudies.Webelieve,however,thatawiderangeofapplicabilityofgeneralizedCAPMframeworkmakeourresultsbeingusefulinagreatervarietyofapplicationsinpracticalnance. Theremainingpartofthischapterisstructuredasfollows.Section2recallsthenecessarybackground,describestheassumptionsofthemodel,providesthemaindenitionsandstatements,andpresentsthederivationofthegeneralizedpricingformula.Section3containsdescriptionofthecasestudy.Section4presentstheresultsofthecasestudy.Theconclusionsectionprovidesseveralideasforfurtherresearchthatcanbeperformedinthisarea. 4.1DescriptionoftheApproach 4.1.1GeneralizedCAPMBackground IntheclassicalMarkovitzportfoliotheory( Markowitz ( 1952 ))allinvestorsaremean-varianceoptimizers.Contrarytotheclassicalapproach,considernowagroupof 55 PAGE 56 investorswhoformtheirportfoliosbysolvingoptimizationproblemsofthefollowingtype:P()minx0r0+xTErr0+x0+xTe=1D)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(x0r0+xTr whereDissomemeasureofdeviation(notnecessarystandarddeviation),r0denotestherisk-freerateofreturn,risacolumnvectorof(uncertain)ratesofreturnonavailablesecurities,andeisacolumn-vectorofones.ProblemP()minimizesdeviationoftheportfolioreturnsubjecttoaconstraintonitsexpectedreturnandthebudgetconstraint.Differentinvestorswithintheconsideredgroupmaydemanddifferentexcessreturn.Unlikeclassicaltheory,insteadofvarianceorstandarddeviation,investorsmeasureriskwiththeirgeneralizeddeviationmeasureD.Accordingtothedenitionin Rockafellaretal. ( 2006a ),afunctionalD:L2![0,1]isadeviationmeasureifitsatisesthefollowingaxioms: (D1)D(X+C)=D(X)forallXandconstantsC. (D2)D(0)=0andD(X)=D(X)forallXandall>0. (D3)D(X+Y)D(X)+D(Y)forallXandY. (D4)D(X)0forallXwithD>0fornonconstantX. Similarlyto Rockafellaretal. ( 2006b ),wecaneliminatex0,whichisequalto1)]TJ /F3 11.955 Tf 11.95 0 Td[(xTe:P0()minxT(Er)]TJ /F10 7.97 Tf 6.58 0 Td[(r0e)D)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xTr Apair(x0,x)isanoptimalsolutiontoP()ifandonlyifxisanoptimalsolutiontoP0()andx0=1)]TJ /F3 11.955 Tf 11.95 0 Td[(xTe.Theorem1in Rockafellaretal. ( 2006c )showsthatanoptimalsolutiontoP0()exists,ifdeviationmeasureDsatisestheproperty(D5): (D5)fXjD(X)CgisclosedforeveryconstantC. AdeviationmeasureDsatisfyingthispropertyiscalledlowersemicontinuous.Forfurtherresultswewillalsorequireanadditionalproperty,calledlowerrangedominance: (D6)D(X)EX)]TJ /F2 11.955 Tf 11.96 0 Td[(infXforallX. 56 PAGE 57 Inthispaperweconsideronlylowersemicontinuous,lowerrange-dominateddeviationmeasures. Rockafellaretal. ( 2006b )showthatifagroupofinvestorssolvesproblemsP(),theoptimalinvestmentpolicyischaracterizedbytheGeneralizedOne-FundTheorem(Theorem2inthatpaper).Accordingtotheresult,theoptimalportfolioshavethefollowinggeneralstructure:x=x1,x0=1)]TJ /F2 11.955 Tf 11.95 0 Td[((x1)Te wherex0istheinvestmentinriskfreeinstrument,xisavectorofpositionsinriskyinstruments,and(x10,x1)isanoptimalsolutiontoP(),with=1.Portfolio(x10,x1)iscalledabasicfund.Itisimportanttonotethat,infullgenerality,(x1)TErcouldbepositive,negative,orequal0(thresholdcase),althoughformostsituationsthepositivecaseshouldprevail. Accordingtothesamepaper,aportfolioxDiscalledamasterfundofpositive(negative)typeif(xD)Te=1((xD)Te=)]TJ /F2 11.955 Tf 9.3 0 Td[(1),andxDisasolutiontoP0()forsome>0.Fromthedenitionfollowsthatmasterfundcontainsonlyriskysecurities,withnoinvestmentinriskfreesecurity.Withthisdenition,thegeneralizedOne-FundTheoremcanbereformulatedintermsofthemasterfund.Belowwepresentitsformulationasitwasgivenin Rockafellaretal. ( 2006b ). Theorem1(One-FundTheoreminMasterFundForm).Supposeamasterfundofpositive(negative)typeexists,furnishedbyanxD-portfoliothatyieldsanexpectedreturnr0+forsome>0.Then,forany>0,thesolutionfortheportfolioproblemP()isobtainedbyinvestingthepositiveamount=(negativeamount)]TJ /F2 11.955 Tf 9.3 0 Td[(=)inthemasterfund,andtheamount1)]TJ /F2 11.955 Tf 12.43 -.16 Td[((=)(amount1+(=)>1)intheriskfreeinstrument. 57 PAGE 58 FromTheorem1followsthatforeveryinvestorintheconsideredgroup,theoptimalportfoliocanbeexpressedasacombinationofinvestmentinthemasterfund,andinvestmentintheriskfreesecurity. Rockafellaretal. ( 2007 )extendstheframeworktothecasewithmultiplegroupsofinvestors.Everygroupofinvestorsi,wherei=1,...,I,solvestheproblemP()withtheirowndeviationmeasureDi.Itwasshownthatthereexistsamarketequilibrium,andoptimalpolicyforeverygroupofinvestorsisdenedbytheGeneralizedOne-FundTheorem.Inthisframeworkinvestorsfromdifferentgroupsmayhavedifferentmasterfunds.Fromnowonweassumethatageneralizeddeviationmeasurerepresentsriskpreferencesofagivengroupofinvestors. ConsideraparticulargroupofinvestorswithriskpreferencesdenedbyageneralizeddeviationD.Iftheirmasterfundisknown,thecorrespondingGeneralizedCAPMrelationscanbeformulated.Theexactrelationdependsonthetypeofthemasterfund.LetrMdenotetherateofreturnofthemasterfund.ThenrM=(xD)Tr=nXj=1xDjrj wheretherandomvariablesrjstandforratesofreturnonthesecuritiesintheconsideredeconomy,xDjarethecorrespondingweightsofthesesecuritiesinthemasterfund,andPnj=1xDj=1. Thegeneralizedbetaofasecurityj,replacingtheclassicalbeta,isdenedasfollows: j=cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDM) D(rM)(4) InthisformulaQDMdenotestheriskidentiercorrespondingtothemasterfund,takenfromtheriskenvelopecorrespondingtothedeviationmeasureD.Examplesofriskidentiersforspecicdeviationmeasureswillbepresentedinthenextsubsection. Rockafellaretal. ( 2006c )derivesoptimalityconditionsforproblemsofminimizingageneralizeddeviationofthereturnonaportfolio.Theoptimalityconditionsareapplied 58 PAGE 59 tocharacterizethreetypesofmasterfunds.Theorem5inthatpaper,presentedbelow,formulatestheoptimalityconditionsintheformofCAPM-likerelations. Theorem2.LetthedeviationDbeniteandcontinuous. Case1.AnxD-portfoliowithxD1+...+xDn=1isamasterfundofpositivetype,ifandonlyifErM>r0andErj)]TJ /F3 11.955 Tf 11.95 0 Td[(r0=j(ErM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0)forallj. Case2.AnxD-portfoliowithxD1+...+xDn=)]TJ /F2 11.955 Tf 9.3 0 Td[(1isamasterfundofnegativetype,ifandonlyifErM>)]TJ /F3 11.955 Tf 9.29 0 Td[(r0andErj)]TJ /F3 11.955 Tf 11.96 0 Td[(r0=j(ErM+r0)forallj. Case3.AnxD-portfoliowithxD1+...+xDn=0isamasterfundofthresholdtype,ifandonlyifErM>0andErj)]TJ /F3 11.955 Tf 11.95 0 Td[(r0=jErMforallj. FromnowonwecalltheconditionsspeciedintheTheorem2theGeneralizedCAPM(GCAPM)relations. 4.1.2PricingFormulasinGCAPM Letrj=j=j)]TJ /F2 11.955 Tf 12.03 0 Td[(1,wherejisthepayofforthefuturepriceofsecurityj,andjisthepriceofthissecuritytoday. Similarlytoclassicaltheory,pricingformulascanbederivedfromtheGeneralizedCAPMrelations,asitwasdonein Sarykalin ( 2008 ).ThefollowingLemmapresentsthesepricingformulasbothincertaintyequivalentform,andriskadjustedform. Lemma1.Case1.Ifthemasterfundisofpositivetype,thenj=Ej 1+r0+j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0=1 1+r0Ej+cov(j,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0Case2.Ifthemasterfundisofnegativetype,thenj=Ej 1+r0+j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM+r0=1 1+r0Ej+cov(j,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM+r0Case3.Ifthemasterfundisofthresholdtype,thenj=Ej 1+r0+jErDM=1 1+r0Ej+cov(j,QDM) D(rDM)ErDM 59 PAGE 60 SeeproofinAppendix. Rockafellaretal. ( 2007 )provedtheexistenceofequilibriumformultiplegroupsofinvestorsoptimizingtheirportfoliosaccordingtotheirindividualriskpreferences,andthereforethepricingformulasinLemma1holdtrueforallgroupsofinvestors. 4.1.3MixedCVaRDeviationandBetas ConditionalValue-at-Riskhasbeenstudiedbyvariousresearchers,sometimesunderdifferentnames(expectedshortfall,Tail-VaR).Wewillusenotationsfrom RockafellarandUryasev ( 2002 ).FormoredetailsonstochasticoptimizationwithCVaR-typefunctionssee Uryasev ( 2000 ), RockafellarandUryasev ( 2000 ), RockafellarandUryasev ( 2002 ), Krokhmaletal. ( 2002 ), Krokhmaletal. ( 2006 ), Sarykalinetal. ( 2008 ). SupposerandomvariableXdeterminessomenancialoutcome,futurewealthorreturnoninvestment.Bydenition,Value-at-Riskatlevelisthe-quantileofthedistributionof()]TJ /F3 11.955 Tf 9.3 0 Td[(X):VaR(X)=q()]TJ /F3 11.955 Tf 9.3 0 Td[(X)=)]TJ /F3 11.955 Tf 9.29 0 Td[(q1)]TJ /F9 7.97 Tf 6.59 0 Td[((X)=)]TJ /F2 11.955 Tf 11.29 0 Td[(inffzjFX(z)>1)]TJ /F4 11.955 Tf 11.95 0 Td[(g whereFXdenotestheprobabilitydistributionfunctionofrandomvariableX. ConditionalValue-at-RiskforcontinuousdistributionsequalstheexpectedlossexceedingVaR:CVaR(X)=)]TJ /F3 11.955 Tf 9.3 0 Td[(E[XjX)]TJ /F2 11.955 Tf 21.91 0 Td[(VaR(X)] ThisformulaunderliesthenameofCVaRasconditionalexpectation.Forthegeneralcasethedenitionismorecomplicated,andcanbefound,forexample,in RockafellarandUryasev ( 2000 ).ConditionalValue-at-Riskdeviationisdenedasfollows:CVaR(X)=CVaR(X)]TJ /F3 11.955 Tf 11.95 0 Td[(EX) AsfollowsfromTheorem1in Rockafellaretal. ( 2006a ),thereexistsaone-to-onecorrespondencebetweenlower-semicontinuous,lowerrange-dominateddeviation 60 PAGE 61 measuresDandconvexpositiveriskenvelopesQ:Q=nQQ0,EQ=1,EXQEX)-222(D(X)forallXo D(X)=EX)]TJ /F2 11.955 Tf 15.05 0 Td[(infQ2QEXQ(4) TherandomvariableQX2Q,forwhichD(X)=EX)]TJ /F3 11.955 Tf 12.06 0 Td[(EXQX,iscalledtheriskidentier,associatedbyDwithX. ForagivenXandCVaRdeviation,theriskidentiercanbeviewedasastepfunction,withajumpatthequantilepoint: QX(!)=1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(1InX(!)q1)]TJ /F9 7.97 Tf 6.59 0 Td[((X)o(4) where!denotesanelementaryeventontheprobabilityspace,and1Inconditionoisanindicatorfunction,denedonthesameprobabilityspace,whichequals1ifconditionistrue,and0otherwise.Figure 4-1 illustratesthestructureoftheCVaRriskidentier,correspondingtosomerandomoutcomeX.Forsimplicity,theprobabilityspace,assumedintheFigure 4-1 ,isthespaceofvaluesoftherandomvariableX. IfthegroupofinvestorsconstructsitsmasterfundbyminimizingCVaRdeviation,andallrjarecontinuouslydistributed,betaforsecurityjhasthefollowingexpression,derivedinRockafellaretal.(2006c): j=cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QM) CVaR(rM)=E[Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaR(rM)] E[ErM)]TJ /F3 11.955 Tf 11.96 0 Td[(rMjrM)]TJ /F2 11.955 Tf 21.91 0 Td[(VaR(rM)](4) Classicalbetaisascaledcovariancebetweenthesecurityandthemarket.Thenewbetafocusesoneventscorrespondingtobiglossesinthemasterfund.Forbig(>0.8),thisexpressioncanbecalledtail-beta. ThefollowingtwotheoremsleadtothedenitionofmixedCVaRdeviation,whichisusedforthepurposeofthispaper. 61 PAGE 62 Figure4-1. CVaR-typeRiskIdentierforaGivenOutcomeVariableX Theorem3.LetdeviationmeasureDlcorrespondtoriskenvelopeQlforl=1,...,L.IfdeviationmeasureDisaconvexcombinationofthedeviationmeasuresDl:D=LXl=1lDl,withl0,LXl=1l=1 thenDcorrespondstoriskenvelopeQ=PLl=1lQl. SeeproofinAppendix. Thefollowingtheorempresentsaformulaforthebetacorrespondingtoadeviationmeasurethatisaconvexcombinationofanitenumberofdeviationmeasures. Theorem4.IfthemasterfundM,correspondingtothedeviationmeasureD,isknown,andDisaconvexcombinationofanitenumberofdeviationmeasuresDl,l=1,...,L:D=1D1+...+LDL 62 PAGE 63 thenj=1cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QD1M)+...+Lcov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDLM) 1D1(rM)+...+LDL(rM) whereQDlMisariskidentierofmasterfundreturncorrespondingtodeviationmeasureDl. SeeproofinAppendix. Foragivensetofcondencelevels=(1,...,L)andcoefcients=(1,...,L)suchthatl0foralll=1,...,L,andPLl=1l=1,mixedCVaRdeviationCVaR;isdenedinthefollowingway: CVaR,(X)=1CVaR1(X)+...+LCVaRL(X)(4)Corollary1.IfD=CVaR,,where=(1,...,L)and=(1,...,L),anddistributionofrMiscontinuous,then j=1E[Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaR1(rM)]+...+LE[Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRL(rM)] 1CVaR1(rM)+...+LCVaRL(rM)(4)SeeproofinAppendix. 4.1.4RiskPreferencesofaRepresentativeInvestor Howcanriskpreferencesofinvestorsbeextractedfrommarketprices? AccordingtoGCAPM,riskpreferencesofagroupofinvestorsarerepresentedbyadeviationmeasure.Thisdeviationmeasuredeterminesthestructureofamasterfund.Foraknowndeviationmeasureandamasterfund,ariskidentierforthemasterfundcanbespecied.Ifajointdistributionofpayoffsforsecuritiesisalsoknown,thenonecancalculatethebetasforsecurities,andthencalculateGCAPMpricesforthesesecurities.Therefore,accordingtoGCAPM,thedeviationmeasureandthedistributionofpayoffdeterminethepriceforeachsecurity.Toestimatethedeviationmeasure,havingexpectedreturnsonsecuritiesandmarketprices,onecanndacandidatedeviationmeasureDforwhichtheGCAPMpricesareequaltothemarketprices. 63 PAGE 64 Inthisandfollowingsectionsthepaperconsidersasetupwithonegroupofinvestors.Inotherwords,allinvestorsevaluaterisksoftheirinvestmentsaccordingtothesamedeviationmeasure.Therefore,allfurtherresultscanbereferredtoasdescribingaso-calledrepresentativeinvestor.Frommarketequilibriumfollowsthatthemasterfundforarepresentativeinvestorisknown,and,therefore,canbeapproximatedwithamarketindex,suchasS&P500. Alternativelytostandarddeviation,whichmeasuresthemagnitudeofpossiblepricechangesinbothdirections,ConditionalValue-at-Riskdeviationmeasurestheaveragelossfortheworst-casescenarios.WeassumethatriskpreferencescanbeexpressedwithamixedCVaRdeviation,denedbyformula( 4 ),whichisaweightedcombinationofseveralCVaRdeviationswithappropriateweights,tocapturedifferentpartsofthetailofthedistribution. Amongthewholevarietyofsecuritiestradedinthemarket,inadditiontotheIndexfunditself,weconsiderS&P500putoptionswithonemonthtomaturity.Byconstruction,putoptions'pricesprovidemonetaryevaluationofthetailsofdistribution,sotheyareexpectedtobeperfectcandidatetocalibratecoefcientsinthemixedCVaRdeviation. Toestimatethecoefcients1,...,LwewilluseGCAPMformulas,presentedinTheorem2.LetPKdenotethemarketpriceofaputoptionwithstrikepriceKand1monthtomaturity,Kdenoteits(random)monthlyreturn,andrK=K PK)]TJ /F2 11.955 Tf 12.51 0 Td[(1denoteits(random)returninonemonth.LetrMbe(random)returnonthemasterfund,withitsdistributionatthismomentassumedtobeknown;r0isthereturnonariskfreesecurity.IfmarketpricesareexactlyequaltoGCAPMprices,andthedeviationmeasureisamixedCVaRdeviationwithxedcondencelevels1,...,L,thenthesetofcoefcients1,...,Lisasolutiontothefollowingsystemofequations: ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(r0=K()(ErM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0),K=K1,K2...,KJ)]TJ /F11 7.97 Tf 6.59 0 Td[(1,KJ(4) 64 PAGE 65 where K()=PLl=1lE[ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(rKjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRl(rM)] PLl=1lCVaRl(rM)(4) LXl=1l=1(4) and l0,l=1,...,L(4) Equations( 4 )areGCAPMformulasfromTheorem2,appliedtomarketpricesPKofputoptionswithstrikepricesK=K1,...,KJ,andrandompayoffsK.Systematicriskmeasure()isexpressedthroughthecoefcientslaccordingtoCorollary1. Bymultiplyingbothsidesofequation( 4 )byPLl=1lCVaRl(rM)andtakingintoaccount( 4 ),weget (ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)LXl=1lCVaRl(rM)=(ErM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)LXl=1lE[ErK)]TJ /F3 11.955 Tf 11.96 0 Td[(rKjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRl(rM)],K=K1,...,KJ or,equivalently, LXl=1)]TJ /F2 11.955 Tf 5.48 -9.68 Td[((ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)CVaRl(rM))]TJ /F2 11.955 Tf 11.95 0 Td[((ErM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)E[ErK)]TJ /F3 11.955 Tf 11.96 0 Td[(rKjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRl(rM)]l=0,K=K1,...,KJ(4) Ifthenumberofequations(optionswithdifferentstrikepricesK)isgreaterthanthenumberofvariables,thensystemofequations( 4 )maynothaveasolution.Forthisreasonwereplacetheequations( 4 )withalternativeexpressionswitherrortermseK: LXl=1)]TJ /F2 11.955 Tf 5.48 -9.68 Td[((ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)CVaRl(rM))]TJ /F2 11.955 Tf 11.95 0 Td[((ErM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)E[ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(rKjrM)]TJ /F2 11.955 Tf 21.91 0 Td[(VaRl(rM)]l=eK,K=K1,...,KJ(4) 65 PAGE 66 Weestimatethecoefcients1,...,Lastheoptimalpointtothefollowingoptimizationproblemminimizinganormofvector(eK1,...,eKJ): min1,...,Lk(eK1,...,eKJ)k(4) subjectto LXl=1)]TJ /F2 11.955 Tf 5.48 -9.69 Td[((ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)CVaRl(rM))]TJ /F2 11.955 Tf 11.95 0 Td[((ErM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0)E[ErK)]TJ /F3 11.955 Tf 11.95 0 Td[(rKjrM)]TJ /F2 11.955 Tf 21.91 0 Td[(VaRl(rM)]l=eK,K=K1,...,KJ(4) l0,l=1,...,L,LXl=1l=1(4) Intheaboveformulationkkissomenorm.Weconsidertwonorms: L1-norm: k(eK1,...,eKJ)k1=1 JJXj=1jeKjj(4) andL2-norm:k(eK1,...,eKJ)k2=vuut 1 JJXj=1e2Kj 4.2CaseStudyDataandAlgorithm Wedid153experimentsofestimatingriskpreferences,eachforaseparatedate(henceforth:dateofexperiment)startingwith1/22/1998.Dateswerechosenwithintervalsapproximately1monthinsuchawaythateachdateis1monthpriortoanextmonthoptionexpirationdate.However,wepresentdetailedanalysisfor12dateswithintervalsapproximately1=2yearstartingwith12/23/2004.ForeveryexperimentweusedasetofS&P500putoptionswithstrikepricesK1,...,KJ,whereKJisastrikepriceoftheat-the-moneyoption(optionwithstrikepriceclosesttotheIndexvalue).WedeneoptionmarketpricePKasanaverageofBIDandASKprices:PK=1 2Pask,K+Pbid,K 66 PAGE 67 WechoseK1asaminimumstrikeprice,forwhichthefollowingtwoconditionsaresatised:1)StartingwiththeoptionK1,pricesPKjarestrictlyincreasing,i.e.PKj+1>PKj;2)Openinterestforalloptionsintherangeisgreaterthan0. ForeveryexperimentwedesignedasetofscenariosofmonthlyIndexratesofreturninthefollowingway.ObservinghistoricalvaluesofS&P500overtheperiodfrom1/1/1994to10/1/2010,foreverytradingdaysfromhistoricalobservationswerecordedthevaluefr(s)I=Is+21 Is)]TJ /F2 11.955 Tf 11.95 0 Td[(1,whereIsistheIndexvalueondays. Wefurthercalculateimpliedvolatilityoftheat-the-moneyoption(theoptionwithstrikepriceKJ),andthevalueb=standarddeviation(fr(s)I) Next,everyscenarioreturnwasmodiedasfollows: r(s)I= bfr(s)I)]TJ /F3 11.955 Tf 11.96 0 Td[(EerI+r0+(4) wherethevalueforthemonthlyrisk-freerateofreturnr0wasselectedequalto0.01%,and>0issomeparameter.Thenewscenarioswillhavevolatilityequaltothevolatilityoftheat-the-moneyoptions,andexpectedreturnr0+.Informula( 4 )thevalueofwaschosensuchthatexpectedreturnsonoptionsarenegative.Weselected=1 3.Numericalexperimentsshowedthatresultsarenotverysensitivetotheselectionoftheparameter. Suppose,formodelingpurposes,thattheinvestors'preferencesaredescribedbyamixedCVaRdeviationwithcondencelevels50%,75%,85%,95%and99%: D()=LXl=1lCVaRl(4) whereL=5,1=99%,2=95%,3=85%,4=75%,5=50% 67 PAGE 68 Table4-1. CaseStudyDataforSelectedDates DateofexperimentIndexvalueLoweststrikepriceHigheststrikepriceNotationI0KminKmax 12/23/041210.13112012106/16/051210.961080121012/22/051268.12110012706/22/061245.601150124512/21/061418.30131014206/21/071522.191375152012/20/071460.12125514606/19/081342.831110134512/18/08885.286308856/18/09918.3773592012/17/091096.0890010956/17/101116.049401115 Table4-2. CaseStudyCommonData DecriptionNotationValue Risk-freemonthlyinterestrater00.4125%NumberoftermsinmixedCVaRdeviationL5Condencelevel1199%Condencelevel2295%Condencelevel3385%Condencelevel4475%Condencelevel5550%Numberofscenarios(days)S5443 and l0,5Xl=1l=1(4) TheinputdataforthecasestudyarelistedinTable 4-1 andTable 4-2 Multipletestsdemonstratedthattheresultsdonotdependsignicantlyonthechoiceofnormintheoptimizationproblem( 4 ).FurtherinthispaperwepresentresultsobtainedusingL1-norm. Thefollowingstepsdescribethealgorithm,whichwasusedtoestimateriskpreferencesfromtheoptionprices. 68 PAGE 69 Step1.Calculatescenariosindexedbys=1,...,Sforpayoffsandnetreturnsofputoptionsaccordingtotheformula:(s)K=max(0,K)]TJ /F3 11.955 Tf 11.96 0 Td[(I0(1+r(s)I)),r(s)K=(s)K PK)]TJ /F2 11.955 Tf 11.96 0 Td[(1 whereK=K1,...,KJarethestrikeprices,andI0istheIndexvalueattimeoftheexperiment. Step2.Calculatethefollowingvalues:E[ErK)]TJ /F3 11.955 Tf 11.96 0 Td[(rKjrI)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRl(rI)]forallK=K1,...,KJandl=1,...,L andCVaRl(rI)foralll=1,...,L Step3.Buildthedesignmatrixfortheconstrainedregression( 4 )-( 4 )withrM=rI. Table4-3. DeviationMeasureCalibrationResults DateofExperiment99%95%85%75%50% 12/23/20040.0000.0200.2350.0000.7456/16/20050.0360.0160.0000.0000.94812/22/20050.0580.0000.0000.0000.9426/22/20060.0710.0330.0000.0000.89512/21/20060.0810.0000.0000.0000.9196/21/20070.0550.0400.0000.0000.90512/20/20070.0000.0410.2750.0000.6846/19/20080.0000.0550.1810.0000.76512/18/20080.0000.0000.1150.0000.8856/18/20090.0150.0140.1680.0000.80312/17/20090.0490.0010.0830.0000.8686/17/20100.0410.0480.0230.0000.889mean(12dates)0.0340.0220.0900.0000.854standarddeviation(12dates)0.0300.0200.1020.0000.085mean(153dates)0.0290.0290.0520.0070.883standarddeviation(153dates)0.0280.0330.0770.0470.078 69 PAGE 70 Step4.Findasetofcoefcientslbysolvingconstrainedregression( 4 )-( 4 )withL1norm,givenbyequation( 4 ).VectorgivescoefcientsinmixedCVaRdeviation. 4.3CaseStudyComputationalResults ComputationswereperformedonthelaptopPCwithIntelCore2DuoCPUP88002.66GHz,4GBRAMandWindows7,64-bit.Algorithm,describedinprevioussection,wasprogrammedinMATLAB.Bothoptimizationproblems,theconstrainedregressionandCVaRportfoliooptimization,oneachiterationofthealgorithmweresolvedwithAORDAPortfolioSafeguarddecisionsupporttool( PSG ( 2009 )).Foronedatethecomputationaltimeisaround15seconds. ThesetofcoefcientsinthemixedCVaRdeviationforeverydateispresentedinTable 4-3 .ThistableshowsthatinallexperimentstheobtaineddeviationmeasurehasthebiggestweightonCVaR50%,andsmallerweightsoneitherCVaR85%,CVaR95%,orCVaR99%.Thiscanbeinterpretedasthatinvestorsareconcernedbothwiththemiddlepartofthelossdistribution,expressedwithCVaR50%,andextremelossesexpressedwithCVaR85%,CVaR95%,orCVaR99%. LetusdenotebyKtheGCAPMoptionprices,calculatedwithpricingformulasinLemma1,usingcalculatedmixedCVaRdeviationmeasureandthemasterfund.WemappedtheobtainedoptionpricesKandthemarketpricesPKintotheimpliedvolatilityscale.ThismappingisdenedbytheBlack-Scholesformulainimplicitform.ThegraphsofKandPKfor12datesinthescaleofmonthlyimpliedvolatilitiesarepresentedinFigures 4-2 4-3 and 4-4 .AllgraphsshowthattheGCAPMpricesareclosetomarketprices,exceptforthegraphsfor6/16/2005andfor12/22/2005. Figure 4-5 comparesdynamicsofthevalue=1)]TJ /F4 11.955 Tf 10.81 0 Td[(50%on153datesofexperimentwithS&P500dynamics.Highvaluesofindicategreaterinvestors'apprehensionaboutpotentialtaillossesandgreaterinclinationtohedgetheirinvestmentsinS&P500.Itcanbeseenthatriskpreferenceswererelativelystableuntil2008,whenthedistressed 70 PAGE 71 Figure4-2. CalculatedPricesandMarketPricesintheScaleofImpliedVolatilitiesPart1outof3 71 PAGE 72 Figure4-3. CalculatedPricesandMarketPricesintheScaleofImpliedVolatilitiesPart2outof3 72 PAGE 73 Figure4-4. CalculatedPricesandMarketPricesintheScaleofImpliedVolatilitiesPart3outof3 73 PAGE 74 Figure4-5. S&P500ValueandRiskAversityDynamics periodbegan.Itcanalsobeseenthatmarketparticipantsdidn'talwaysproperlyanticipatefuturemarkettrends.Inparticular,inDecember2008thevalueofwaslow(0.115),whichindicatedthatmarketwronglyanticipatedthatIndexreacheditsbottomandwillgoup.Nevertheless,2009startedwithfurtherdeclineintheIndex. 74 PAGE 75 CHAPTER5CONCLUSIONS 5.1DissertationContribution Inthisdissertationweprovidedanoverviewofthefollowingclasses:generalizeddeviationmeasures,riskmeasures,measuresoferror,andsomeoftheirsubclasses.Webrieydiscussedthemotivationforapplyingthesemeasuresinstochasticoptimizationapplications. Wereviewedgeneralizedlinearregressionmodels.Forthequantileregressionwedeterminedthedistributionfortheresidual.Thisresultcanbeusedinapplications,whichrequirethedistributionoftheerrorterminaquantilefactormodeltobespecied,suchassimulationprocedures.Interpretationofthisdistributionasatwo-sidedexponentialdistributionmakesitpossibletoestimatevariouspropertiesofthedistributionwithoutapplyingnumericalintegrationmethods.Therefore,implementationsofcomputationalmodelsbasedonthisdistributionareexpectedtobehighlyefcient. Wehavedenedandextendedaclassofdynamicsensorschedulingproblems,basedonconditionalvalue-at-risk,byintroducingexplicitrobustconnectivityrequirements,specically,k-clubandk-plexconstraints,takingintoaccountwirelessconnectivityrequirementsforsensorsateverytimemoment.Wehavealsopresentedcomputationalresultsformoderate-sizeinstancesinbothdeterministicandstochasticproblemsetups.SincethesizeofthestochasticversionoftheproblemisStimeslargerthanforthedeterministicversion(whereSisthenumberofimpliedpenaltyscenarios),solvingthesestochasticproblemsisclearlychallengingfromthecomputationalperspective. Theclassesofproblemsconsideredinthisresearchareprimarilymotivatedbymilitaryapplications;however,thedevelopedformulationsaregeneralenoughsothattheycanbeappliedinavarietyofsettings. Wehavedescribedanewtechniqueofexpressingriskpreferenceswithgeneralizeddeviationmeasures.Wehavepresentedamethodforextractingriskpreferences 75 PAGE 76 frommarketoptionpricesusingtheseformulas.Wehaveconductedacasestudyforextractingriskpreferencesofarepresentativeinvestorfromputoptionprices. Weextractedriskpreferencesfor153dateswith1monthintervals,andexpressedthemwithmixedCVaRdeviation.Resultsdemonstratethatinvestorsareconcernedbothwiththemiddlepartofthelossdistribution,expressedwithCVaR50%,andextremelossesexpressedwithCVaR85%,CVaR95%,orCVaR99%.Exactproportionsvary,reectinginvestorsanticipationofhighorlowreturns. Animportantapplicationofthetheoryisthatitprovidesanalternative,morebroadviewonsystematicrisk,comparedtotheclassicalCAPMbasedonstandarddeviation.SimilarlytotheclassicalCAPM,wecalculatednewbetasforsecurities,whichmeasuresystematicriskinadifferentway,capturingtailbehaviorofamasterfundreturn.Thesebetascanbeusedforhedgingagainsttaillosses,whichoccurindownmarket. Potentialapplicationsgobeyondidentifyingriskpreferencesofconsideredinvestors.Aninvestorcanexpressriskattitudesintheformofadeviationmeasure,andthenrecalculatebetasforsecuritiesusingthisdeviationmeasure.Withthesebetastheinvestorcanbuildaportfoliohedgedaccordingtohisriskpreferences. 5.2FutureWork Sensorsschedulingproblemformulationscanbefurtherextendedbyaddingmovementnetworkandcorrespondingconstraintsasintroducedin Boykoetal. ( 2011 ),thusmodelingthemapofpossiblesensormovements. Itwouldbeinterestingtoexamineinvestors'perceptionofriskfordifferenttimehorizonsbyestimatingriskpreferencesusingmarketpricesofputoptionswith2,3,andmoremonthstomaturity. 76 PAGE 77 APPENDIX:PROOFS Belowwepresentproofsofstatementsformulatedinthearticle.Forthereader'sconvenience,werepeatformulationsbeforeeveryproof.Lemma1.Case1.Ifthemasterfundisofpositivetype,thenj=Ej 1+r0+j)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ErDM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0=1 1+r0Ej+cov(j,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0Case2.Ifthemasterfundisofnegativetype,thenj=Ej 1+r0+j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM+r0=1 1+r0Ej+cov(j,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ErDM+r0Case3.Ifthemasterfundisofthresholdtype,thenj=Ej 1+r0+jErDM=1 1+r0Ej+cov(j,QDM) D(rDM)ErDMProofofLemma1.Proofsforallthreecasesaresimilar,sowepresenttheproofonlyforamasterfundofpositivetype.AccordingtotheGCAPMrelationspeciedinCase1,Erj)]TJ /F3 11.955 Tf 11.95 0 Td[(r0=j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0Sincerj=j=j)]TJ /F2 11.955 Tf 11.95 0 Td[(1,thenErj=Ej=j)]TJ /F2 11.955 Tf 11.95 0 Td[(1,fromwhichweget Ej j)]TJ /F2 11.955 Tf 11.96 0 Td[((1+r0)=j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0(A)ThisyieldstheGeneralizedCapitalAssetPricingFormulainthecertaintyequivalentform: j=Ej 1+r0+j)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0(A)Usingtheexpressionforbeta( 4 )wecanalsowrite j=Ej 1+r0+cov()]TJ /F10 7.97 Tf 6.59 0 Td[(rj,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0(A) 77 PAGE 78 Bymultiplyingbothsidesoftheequality( A )byj,weget Ej)]TJ /F4 11.955 Tf 11.95 0 Td[(j(r0+1)=jj)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ErDM)]TJ /F3 11.955 Tf 11.95 0 Td[(r0(A)Withexpressionforbeta( 4 )weget jj=jcov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDM) D(rDM)=cov()]TJ /F4 11.955 Tf 9.29 0 Td[(jrj,QDM) D(rDM)=cov()]TJ /F4 11.955 Tf 9.3 0 Td[(j(1+rj)+j,QDM) D(rDM)=cov()]TJ /F4 11.955 Tf 9.3 0 Td[(j(1+rj),QDM) D(rDM)+cov(j,QDM) D(rDM)(A)Herejisaconstant,consequentlythesecondterminthelastsumequals0.Therefore,jj=cov()]TJ /F4 11.955 Tf 9.3 0 Td[(j(1+rj),QDM) D(rDM)Sincej(1+rj)=j,thenjj=)]TJ /F2 11.955 Tf 10.49 8.09 Td[(cov(j,QDM) D(rDM)Substitutingexpressionforjjinto( A )gives:Ej)]TJ /F4 11.955 Tf 11.96 0 Td[(j(r0+1)=)]TJ /F2 11.955 Tf 10.49 8.08 Td[(cov(j,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0Thelastequationimpliestherisk-adjustedformofthepricingformula: j=1 1+r0Ej+cov(j,QDM) D(rDM))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(ErDM)]TJ /F3 11.955 Tf 11.96 0 Td[(r0(A)Theorem3.LetdeviationmeasureDlcorrespondtoriskenvelopeQlforl=1,...,L.IfdeviationmeasureDisaconvexcombinationofthedeviationmeasuresDl:D=LXl=1lDl,withl0,LXl=1l=1thenDcorrespondstoriskenvelopeQ=PLl=1lQl. 78 PAGE 79 ProofofTheorem3.Withformula( 4 )weget: D(X)=LXl=1lDl(X)=EX)]TJ /F10 7.97 Tf 18.18 14.94 Td[(LXl=1linfQ2QlEXQ==EX)]TJ /F2 11.955 Tf 48.68 0 Td[(inf(Q1,...,QL)2(Q1,...,QL)EX LXl=1lQl!=EX)]TJ /F2 11.955 Tf 30.58 0 Td[(infQ2PLl=1lQlEXQ(A)Theorem4.IfthemasterfundM,correspondingtothedeviationmeasureD,isknown,andDisaconvexcombinationofanitenumberofdeviationmeasuresDl,l=1,...,L:D=1D1+...+LDLthenj=1cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QD1M)+...+Lcov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDLM) 1D1(rM)+...+LDL(rM)whereQDlMisariskidentierofmasterfundreturn,correspondingtodeviationmeasureDl.ProofofTheorem4.FromTheorem3follows:j=cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDM) D=cov()]TJ /F3 11.955 Tf 9.29 0 Td[(rj,1QD1M+...+LQDLM) 1D1(rM)+...+LDL(rM)= =1cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QD1M)+...+Lcov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDLM) 1D1(rM)+...+LDL(rM)(A)Next, cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDlM)=E(Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rj)(QDlM)]TJ /F3 11.955 Tf 11.96 0 Td[(EQDlM)(A)Accordingtothedenitionofriskenvelope,EQDlM=1.Therefore,from( A )wehave:cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDlM)=E(Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rj)(QDlM)]TJ /F2 11.955 Tf 11.96 0 Td[(1)=E(Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rj)QDlM)]TJ /F3 11.955 Tf 11.96 0 Td[(E(Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rj)=E(Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rj)QDlM 79 PAGE 80 Corollary1.IfD=CVaR,,where=(1,...,L)and=(1,...,L),anddistributionofrMiscontinuous,then j=1E[Erj)]TJ /F3 11.955 Tf 11.95 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaR1(rM)]+...+LE[Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRL(rM)] 1CVaR1(rM)+...+LCVaRL(rM)(A)ProofofCorollary1.ForDl=CVaRl,accordingto( 4 ):QDlM(!)=1 1)]TJ /F4 11.955 Tf 11.96 0 Td[(l1InrM(!))]TJ /F2 11.955 Tf 21.92 0 Td[(VaRl(rM)oThen, cov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDlM)=E(Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rj)1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(l1InrM(!))]TJ /F2 11.955 Tf 21.91 0 Td[(VaRl(rM)o==E[Erj)]TJ /F3 11.955 Tf 11.96 0 Td[(rjjrM)]TJ /F2 11.955 Tf 21.92 0 Td[(VaRl(rM)](A)Substitutingexpressionforcov()]TJ /F3 11.955 Tf 9.3 0 Td[(rj,QDlM)andexpressionformixedCVaRdeviation( 4 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Yavuz,Mesut,andDavidE.Jeffcoat.2007.Singlesensorschedulingformulti-sitesurveillanceTechnicalreport,AirForceResearchLaboratory. 84 PAGE 85 BIOGRAPHICALSKETCH KonstantinKalinchenkowasbornin1985inProtvino,Russia.Hereceivedhis5-yearSpecialistdegreeinmathematicsfromMoscowStateUniversityUniversityin2007.KonstantinKalinchenkoworkedasaneconomistandsenioreconomistin2005-2008foraleadingRussiancommercialbank:SberabankRossii,wherehehadanopportunitytotakepartindevelopingcreditriskmeasurementmethodologyandcomputermethodsincontrolofoperationswithprivatesecurities.In2008,KonstantinKalinchenkojoinedthegraduateprograminindustrialandsystemsengineeringdepartmentattheUniversityofFlorida.HereceivedhisMasterofSciencedegreeinindustrialandsystemsengineeringfromtheUniversityofFloridainthespringof2011,andhereceivedhisPh.D.fromtheUniversityofFloridainthespringof2012.KonstantinKalinchenkoalsoworkedasaninternin2011foranassetmanagementcompanyStateStreetGlobalAdvisors(Boston,MA),whereheworkedondevelopingquantitativefactorsbasedonnewsparsingsolutionsforstockranking.KonstantinKalinchenkoistheauthorofseveralscienticpapers.HewasalsoaTAinEngineeringEconomyandFinancialOptimizationCaseStudiesclasses. 85 |