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首页 The Factor Tau in the Black-Litterman Model

The Factor Tau in the Black-Litterman Model.pdf

The Factor Tau in the Black-Lit…

mike柏林
2013-12-06 0人阅读 举报 0 0 暂无简介

简介:本文档为《The Factor Tau in the Black-Litterman Modelpdf》,可适用于经济金融领域

Electroniccopyavailableat:http:ssrncomabstract=TheFactorTauintheBlackLittermanModelJayWalters,CFAjwaltersblacklittermanorgThisVersion:OctoberInitialVersion:August,LastversionavailableatwwwssrncomAbstractThispaperconsidersthefactortau(τ)intheBlackLittermanmodelτisoneofthemoreconfusingaspectsofthemodel,asauthorsprovidecontradictoryinformationregardingitsuseandcalibrationWewillconsidertheoriginofthemixedestimationmodelusedintheBlackLittermanModelsothatwecandeveloparicherunderstandingofτanditsplaceinthemodelWewilldiscussthevariousmodelswhichusethenameBlackLittermanandhowtheydoordonotuseτFinally,wewillshowseveralwaystocalibrateτwhenusingthecanonicalBlackLittermanmodelJELClassification:C,GTheauthorgratefullyacknowledgeshelpfulfeedbackfromAttilioMeucciAnyerrorsoromissionsaresolelytheresponsibilityoftheauthor©Copyright–JayWaltersElectroniccopyavailableat:http:ssrncomabstract=IntroductionTheBlackLittermanModelisanassetallocationmodelIthasthreemainfeatures,theuseofaninformativepriorderivedfromtheICAPMequilibrium,amixingmodelthatallowstheinvestortospecifyviewsonanylinearcombinationoftheassets,andaportfoliochoicefeaturetoidentitytheoptimalportfoliofortheinvestorbasedontheirviewsTheequilibriumpriorestimateisgeneratedusingareverseoptimizationprocedurefromthemarketportfoliousingtheMarkowitzMeanVariancemodelTheinvestorknowsthecovariancematrixwhichallinvestorsareusing,theweightsofthemarketportfolioandtheriskaversionoftheaggregatemarketportfolioTheinvestorisuncertaininthisestimateanditisthusspecifiedasadistributionThemixedestimationmodelwasoriginallydevelopedbyTheil()ItisequivalenttotheBayesianproblemofanunknownmeanandknownvariance,DeGroot()Themixingmodelallowsabsoluteandrelativeviews,andviewsmaybeonanycombinationoftheassetsJustastheinvestorisuncertainintheirestimateoftheprior,theyarealsouncertainintheirestimatesofthereturnstotheviewsThispaperwillfocusontheroleofthefactortau(τ)τisusedtoscaletheinvestorsuncertaintyintheirpriorestimateofthereturnsThereareseveraldifferentapproachestocalibratingit,orevenincludingitdescribedintheliteratureJusttoillustratethedifferenceofopinion,wewilllookatcommentsfromthreeauthorsHeandLitterman()statetheysetτ=SatchellandScowcroft()statemanypeopleuseavalueofτaroundMeucci()proposesaformulationoftheBlackLittermanmodelwithoutτWewillusetheconceptofReferenceModelstoexplainthedifferencesbetweenthevariousauthorsWewillstartbypresentingtheCanonicalReferenceModelasderivedfromTheil'smixedestimationapproachinBlackandLitterman(),andfurtherexplainedinHeandLitterman()ThemaindifferencebetweenthecanonicalreferencemodelandotherreferencemodelsisuncertaintyNextwewillpresenttheAlternativeReferenceModelasproposedbyMeucci()whichestimatesthereturnswithoutτAthirdmodel,theHybridReferenceModelwasshowninSatchellandScowcroft(),butwewillshowthatanyexpressionofthehybridmodelcanbeduplicatedintheAlternativeReferenceModel,sowedonotneedtodiscusstheHybridReferenceModelfurtherFinally,wewillproviderecommendationsonhowaninvestorcantocalibrateτTheBlackLittermanModelThissectionwillprovideanoverviewoftheBlackLittermanModelThereadercanconsultWalters()formoredetailsWecanviewtheprocessofusingtheBlackLittermanModelashavingthreedistinctstepsThefirststepisthecalculationoftheinformativepriorestimateofreturnsThepriorisderivedfromtheICAPMequilibriumportfoliousingthefollowingformulawhichistheclosedformsolutiontounconstrainedMeanVarianceoptimization©Copyright–JayWalters()=wFormula()istherelationshipwhichwecallreverseoptimizationGiventheriskaversionofthemarket,δ,thecovarianceofreturns,Σ,andtheequilibriumweights,w,wecanbackouttheexpectedequilibriumexcessreturnsΠwillbethemeanofthedistributionofourpriorestimateaboutthetrueknownmeanreturn,orthepointestimateofthemean,dependingonthereferencemodelusedThesecondstepisthespecificationoftheinvestor'sviewsHeretheinvestorformulatesestimatedreturnsanduncertaintiesforoneormoreviewportfolios,orlinearcombinationsoftheassetsThethirdstepisthemixedestimationprocessusedtoblendthepriorestimatesofreturnswiththeviewstocreatetheposteriorestimatesofthereturnsalongwithestimatesoftheuncertaintyoftheestimatesWecanarriveatthesameformulationfortheblendingprocessusingthestandardcaseofanunknownmeanandaknownvariancefromBayesiantheory,DeGroot()WestartassumingthattheexpectedreturnsarenormallydistributedThegoaloftheBlackLittermanmodelistoestimatetheparametersofthisdistribution()E(r)∼N(μ,Σr)MeanreturnΣrCovariancematrixforthedistributionofreturnsaboutthemeanTherearetwomodelsfortheblendingprocessWecalltheseReferenceModelsEachReferenceModelcontainsdifferentassumptionsaboutwhatparameterstouseinordertomodelformula()TheCanonicalReferenceModelThissectionreviewstheCanonicalReferenceModelasdefinedinthepapers,BlackandLitterman(),BlackandLitterman()andHeandLitterman()FirstweintroducethesimplelinearmodelfromTheil()fortheestimatedreturn()μ=πϵHereμistheunknownpriorreturns,πtheequilibriumreturns,andεtheresidualThecoreoftheOriginalReferenceModelistheconceptthattheinvestorestimatesadistributionratherthanapointestimateofμbecausetheyareuncertainBlackandLitterman()state,“Themeanisanunobservablerandomvariable”ItisfurtherassumedtobenormallydistributedabouttheequilibriumreturnsWeassumeεisnormallydistributedwithmeanandvarianceΣμThus,wecanstatethatΣμisthevarianceoftheinvestor'sestimateaboutthemeanreturnμTheil()usesthephrase“samplingvariance”forΣμStandarderroristhesquarerootofsamplingvarianceThisleadstothefollowingexpressionforthedistributionoftheestimatedreturnaboutthetrueunknownmeanreturn©Copyright–JayWalters()μ∼N(π,Σμ)Formula()showsthedistributionoftheunknownmeanreturnaboutthepriorestimateInthecanonicalBlackLittermanreferencemodel,Σμisassumedtobeproportionaltothecovarianceofthereturnsaboutthemean,Σ,wheretheconstantofproportionalityistheparameterτIfweweretobootstrapanormaldistributionusingresamplingwithreplacement,calculatingmsamplemeanseachusingndraws,thecentrallimittheorytellsusthatasmapproachesinfinitythatΣμapproachesΣnThus,thesamplingvarianceis()Σμ=ΣnWeassertforsimplicitythatΣandΣμareindependentanduncorrelated,thenΣr,thevarianceofthedistributionofreturnsabouttheestimatedmean,π,isgivenbyformula()()Σr=ΣμΣWecancheckthereferencemodelattheboundaryconditionstoensurethatitiscorrectIntheabsenceofestimationerror,egε≡,thenπ=μandΣr=ΣAsourestimategetsworse,egΣμincreases,thenΣrincreasesaswellThisbehaviorisconsistentwithourearlierassertionthatourposteriorestimateofthemeanismoreprecisethaneithertheviewsorthepriorInadditionitisalsoconsistentwiththeideathatestimatesofthevarianceofadistributionofafinancialtimeseriesaboutanestimatedmean,canatbestapproachalowerlimitwhichisthevarianceofthedistributionaboutthepopulationmeanItcannotgobelowthatvalueGivenourpreviousassumptionthatΣμisproportionaltoΣwithconstantofproportionalityτ,thenthefollowingformulaholds()Σμ=τΣIfwecombineformulas()and(),wecanrelateτandnΣμ=τΣ=Σn()=nIfweusedastatisticalprocesstoformulateourpriorestimate,thenwewouldhaveaclearmethodforcalibratingitbasedonthisrelationshipHerewecanusethenfromthecomputationofthecovariancematrixorwecandirectlyassertouruncertaintyinthepriorestimateNowwecanintroduceourexpressionfortheCanonicalReferenceModelfortheestimateddistributionofexpectedreturns()E(r)∼N(μ,Σ),μ∼N(π,Σμ)Formula()representsthecompleteCanonicalReferenceModelwhichcorrespondstoourgoalasdefinedinformula()Thisreferencemodelmatchesupwithformulas,andinHeandLitterman()©Copyright–JayWaltersWewillnotshowthederivationhereasitcanbefoundinseveralofthereferences,butthestandardexpressionfortheBlackLittermanposteriorestimatedmeanandsamplingvarianceis:()Π̂∼N((τΣ)−ΠPTΩ−Q(τΣ)−PTΩ−P−,(τΣ)−PTΩ−P−)PosteriorestimateofthemeanreturnsΠPriorestimateofthemeanreturnsΣKnowncovariancematrixofreturndistributionabouttheunknownmeanPViewselectionmatrixCovarianceoftheestimatedviewmeanreturnsabouttheactualviewmeanreturnsQEstimatedmeanreturnsfortheviewsΩisatermsimilartoτΣ,representingtheuncertaintyoftheestimatedreturnsoftheviewsInthisreferencemodel,ΩisnotthevarianceofthedistributionofreturnsoftheviewsThediscussionofformula()iseasierintermsoftheinverseofthecovariancematrix,atermknownasprecisionintheBayesianliteratureWecansummarizetheposteriorestimatedmeaninformula()astheprecisionweightedaverageofthepriorestimateandtheviewestimatesTheposteriorprecisionisthesumofthepriorandviewprecisionsBoththeseformulationsmatchourintuitionasweexpecttheprecisionofourposteriorestimatetobemorethantheprecisionofeitherthepriorortheviewsSecond,themixedestimationprocessshouldmakeuseoftheprecisionoftheestimatesintheweightingofthemixing,eganimpreciseestimateshouldhavelessimpactontheposteriorthanapreciseestimateWithasmallmodificationtothecovariancetermwecanrewrite()using()tobeanexpressionfortheBlackLittermanposteriorestimateofthemeanandcovarianceofreturnsaroundthemean()E(r)∼N((τΣ)−ΠPTΩ−Q(τΣ)−PTΩ−P−,(Σ(τΣ)−PTΩ−P−))Theupdatedsamplingvarianceofthemeanestimatewillbelowerthaneitherthepriororconditionalsamplingvarianceofthemeanestimate,indicatingthattheadditionofmoreinformationwillreducetheuncertaintyoftheposteriorestimatesInBayesianterms,theposteriorestimateismoreprecisethaneitherthepriorortheviewestimatesThevarianceofthereturnsaboutthemeanfromformula()willneverbelessthantheknownvarianceofreturnsaboutthemean,ΣThismatchesourintuitionabouthowthevarianceofreturnscanchangeAddingmoreinformationshouldreducetheuncertainty(increasetheprecision)oftheestimates,butcannotreducethecovariancebeyondthatlimitGiventhatthereissomeuncertaintyinthevarianceofthereturnsaboutthemean,thenformula()providesabetterestimatorofthevarianceofreturnsaboutourestimatedmeanthantheknownvarianceaboutthemeanfromtheequilibriumTheHybridReferenceModelTheHybridReferenceModelpresentedbySatchellandScowcroft()setsτtoandthenappliestheusualBlackLittermanformulasInthiscasethepriorandviewestimatesarepoint©Copyright–JayWaltersestimates,theyarenotdistributionsAssuch,thebasisofthemodelisnolongeraBayesianormixedestimationmodel,itisnowjustashrinkagemodelwheretheweightsareproportionaltothecovarianceoftheassetsTheweightonthepriorwillbeΣandtheparametersτorΩcanbeadjustedtoregulatetheamountofshrinkagerealizedGiventhatτorΩcancontroltheshrinkage,wearriveattheAlternativeReferenceModelwheretheshrinkageiscontrolledbyΩandwenolongerneedtheparameterτTheAlternativeReferenceModelTheAlternativeReferenceModelisalsocommonlyusedintheliterature,thoughusuallynotexplicitlyWhileithasbeenusedbyotherauthors,Meucci()explicitlydescribeditsfeaturesHehaslabeleditBeyondBlackLitterman,butotherauthorstreatitasavariantoftheBlackLittermanmodelWewillfurtherassertthatanyauthorwhosuggestsτwithascaleof,ordoesnotuseanupdatedposteriorvarianceisusingthisreferencemodelimplicitlyTheAlternativeReferenceModeldoesnotconsideruncertaintyinanyoftheparameters,itisratherashrinkagemodelfortheexpectedreturnsliketheHybridReferencemodelRememberfromformula()theCanonicalReferenceModelincludesanerrorterminthepriorcovariance,τΣ,whichismissingfromthismodelTheposteriorcovariancealsodoesnotgetupdated(norincludetheremainingerrorterm)TheAlternativeReferenceModelstartsfromthepositionthattheinvestorhassomeviewsontheexpectedreturnsIntheBlackLittermanModel,theseviewscanbeonrelativeorabsolutereturns,andcanbepartialanddonotneedtocoverallassetsWherenoviewisspecifiedforanasset,ortheviewsarerelative,somereturntouseasastartingpointisrequiredInaddition,theinvestorbelievesthatdirectlyusingthereturnsimpliedbytheviewswillleadtoextremeportfoliosTheblendingmodelinsideBlackLittermanallowstheuseoftheICAPMequilibriumreturns(Π)bothasastartingpointandashrinkagetarget,whichresultsinlessextremeportfoliosandmorestableoptimizationresultsGiventhattheinvestorhasnouncertaintyabouttheirestimates,theyknowthecovariancematrixoftheexcessreturnsisΣTheinvestordoesnotneedtoexpressviewsonthecovariancematrix,andthereisnoshrinkageofthecovariancematrixThemixedestimationmodelattheheartoftheBlackLittermanModelallowsforeasyblendingofthepriorandviewsWewillshowitcanalsobeviewedasashrinkagemodelThelinearformofashrinkagemodelforreturnsis:()=−VHere,δisascalarshrinkageparameterandVistheinvestorsviewsonestimatedreturnsIntheBlackLittermanmodel,wemodeltheinvestorsviewsas,akxvector(Q)oftheexpectedreturntoeachviewandakxnmatrix(P)ofviewportfoliosindicatingtheweightoftheassetsineachoftheviewportfoliosWewillreplaceVfrom()withP−QWealsowanttheshrinkagefactortobeamatrixNotethatifPisnotsquare,thenwewillusethepseudoinversePTP−PT©Copyright–JayWaltersratherthanascalarallowingadifferentamountofshrinkagetobeappliedtoeachviewThisgivesusanupdatedmodel()=I−P−QOnlytheshrinkagefactor,Δ,stillneedstobemappedbacktotheBlackLittermanmodelIfweconsiderformula()inrelationtoformula()fromtheOriginalReferenceModelwecanarriveatonepossiblewaytocomputeΔwhichisconsistentwiththeusualformofBayesianshrinkageandprovidesindependentcontrolovertheshrinkageofeachview()=−−PT−PandI−=PT−P−PT−PHere,Ωisadiagonalmatrixcontaininganonnegativemeasure(ωi)foreachviewwhichcorrespondstouncertaintyintheviewByvaryingtheuncertainty,ωi,from(,∞)wecanvarythevalueofδiovertheinterval(,)NotethatωiisinverselyproportionaltoδiThisissufficienttoparametrizeourmixingmodelOnedrawbackisthatspecifyingΩisnotanintuitivewaytoquantifythedesiredamountofshrinkageWecansubstitute()into()andarriveattheAlternativeReferenceModel()Er~N−PT−Q−PT−P−,Aswehaveseenτdoesnotenterthemodel,thereisnoneedforanotherfreeparameterIdzorek()takesadifferentapproachtoexpresstheshrinkagefactorδspecifyingaconfidenceineachviewasapercentage,whichmapsontotheterm(δi)Usingthisconfidencelevelforeachview,hismethodcomputesthevaluesofωiSpecifyingtheconfidenceinthismannerisamoreintuitivewaytospecifytheshrinkageChoosingaReferenceModelNowthatthemainreferencemodelshavebeendescribedwecanprovidesomeguidanceonusingonemodelortheotherThemostcommonreferencemodelintheliteratureistheCanonicalReferenceModelMeucciandafewotherauthorsworkwiththeAlternativeReferenceModelTheHybridReferenceModelisusedlessfrequentlysincetheAlternativeReferenceModelhasbeendevelopedTheprimaryreasontousetheCanonicalReferenceModelwouldbetopickuptheadditionalinformationfromthemodelviatheupdatedposteriorcovariancematrixThisadditionalinformationfromthemodelcomesatthecostofneedingtodeterminetheadditionalfactorτInvestorswillingtoacceptthesimplerAlternativeReferenceModelcanavoidtheneedtoconsiderτWorkedExamplewithBothReferenceModelsWhenweusetheCanonicalReferenceModel,theposteriorcovariancemaybesmallerthanthepriorcovarianceiftheviewsimprovetheprecisionoftheestimateAbsoluteviewsgenerallymakelargerimprovementsintheprecisionoftheestimateRelativeviewsmakeweakorno©Copyright–JayWaltersimprovementsintheprecisionoftheestimateIntuitivelythismakessenseasrelativeviewsdonotprovideanimprovedestimateofthemean,justextrainformationontherelationshipbetweentheestimatesWecanmeasuretheprecisionoftheestimatesbysummingtheunconstrainedweightsWecancomputeaneffectiveposteriormeasureofuncertaintyprecisionasshownbelow()=−∑nwi∑nwiThevalueηinformula()canbecomparedtoτandcanbeusedtomeasuretheuncertaintyinthepriororposteriorestimatesofthemeanWhenviewingthepriorestimates,η=τWecancompareηbetweenthepriorandtheposteriortodeterminetherelativeimprovementintheprecisionoftheestimatesOneoftheinterestinglyartifactsoftheBlackLittermanmodelisthatwhiletheestimatedreturnofanassetwithoutviewscanchange,theunconstrainedweightoftheassetintheportfoliodoesnotchangeatallWecanprovethispointbyexaminingtheformulainHeandLitterman()ΛisakxmatrixwithonerowperviewThePTΛtermwillalwaysbezeroforanassetinnoviewsUndertheCanonicalReferencemodel,ourinvestorwithlessthanconfidenceintheirpriorestimatesisnotinvested,butisonlyinvestedinthefraction(τ)Thisisbecauseofformula()whichshowsthepriordispersionofrealizedreturnsabouttheestimatedmeanis(τ)ΣμTheassetweightsinanunconstrainedportfoliobasedonlyonthepriorwillbeweq(τ)Becausetheposteriorprecisionoftheestimatedmeanwillbeequalorhigher,theinvestorwillinvestanequalorlargerfractionoftheirwealthintheportfolioandtheassetallocationwillexperiencesomechangesolelybecauseofthischangeOfcourse

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The Factor Tau in the Black-Litterman Model

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