Persistence and Volatnity of Hedge Fund Returns : ARMA-GARCH Modeling 利用統計を見る

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Persistence and Volatnity of Hedge Fund

Returns : ARMA-GARCH Modeling

著者

Munechika Midori

journal or

publication title

経済論集

volume

40 number

2 page range

201-225

year

2015-03

URL

http://id.nii.ac.jp/1060/00006950/

Creative Commons : 表示 - 非営利 - 改変禁止 http://creativecommons.org/licenses/by-nc-nd/3.0/deed.ja

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東洋大学「経済論集」40巻2号2015年3月

PerSiStenCeandVOlatilityOfHedgeFundRetumS:

ARMA-GARCHModeling

MidoriMunechika

1．IntroduCtion

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thesametime,theintemationalfinancialcommunityhasexpressedseriousconcemaboutwhethertheyhave playedacrucialroleintriggeringfinancialcrises.Theyhavealsobeenattractingtheattentionofinstitutional investorssuchaspensionfimdssincethelTbubbleburstin2003.Oneofthemainreasonsfbrsuchinterest stemsfromthepeculiarperfbnnancecharacteristicsofthehedgefimdsector.Hedgefimdmanagersemploy 廿equentlydynamictradingstrategiesinvoIvingshortsales,leverageandderivatives,andthus,theytendto generateretumslessuncorrelatedtothoseofmarketbenchmarkreturns. Hedgefilndsarenowm"ormarketparticipantsandtheyarenolongerpreceivedasmavericksinglobal financialmarkets.Theirdynamic,multi-facetedinvestmentstrategieshavenowpenetratedpublicallytraded ETFs.Investablehedgefilndindicesarereallyregardedasthedisguiseoffilndsofhedgefimds(Jaeger [20081).Forexample,investablehedgefimdindicestrackingtheperfbnnancesoftheirstrategiesareusedas

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retumsmeansreplicatingtheirreturnsourcesandcorrespondingriskexposuresbasedontheirstrategies. The20086nancialcrisishassignificantlydecreasedtheretumsofmosthedgefimdstrategies.Manymarket participantsinthehedgefimdindustryrealizedthereisnosafeplacefbrinvestorstoavoidsystematicrisk, andquestionedwhetherdiversificationacrosshedgefilndsasanalternativeinvestmentisreallyasbeneficial astheyintended.Therefbre,investorswhoaimtoputmoneyintoinvestablehedgefimdindicesmust understandtheirreturnsourcestoachievereplication. Univariatetime-seriesdataofhedgefimdretumsthemselvesexhibitpeculiarcharacteristicsofnon-nonnaldistributionsuchasheavy-tailedandskeweddistribution,andvolatilityclustering.Vblatilityisone ofthemostimportantconceptsoffinance.Itisoffenregardedasameasureoffinancialrisk,calculatedby

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thevarianceorstandarddeviationofanasset'sremm.Itiswellknownthattherearesomeperiodsofhigh volatilityandotherperiodsoflowvolatilityofassetreturnsinfinancialmarkets.Volatilityclusteringimplies thatvolatilityshockstodaywillinnuencetheexpectationofvolatilitymanyperiodsinthefilture.This phenomenonrequiresanalyststodescribereturnsandvolatilitythatarenonlinear. Volatilityisnotdirectlyobservableinthefinancialmarket,suchasinstockprices.Itisdescribedasa parameterofthestochasticprocessesthatisappliedtomodelvariationsinfinancialassetprices.Itisonly quantifiableinthecontextofamodel,andthus,theresultsoftheestimatescanbequitedifferentdepending onthemodelandonthemarketconditions.Manystudieshavearguedthatnonlinearprocessesmodelthe volatilitybehaviorofhedgefilndstrategiesbetter(Fiiss,R.,D.G.KaiserandZ.Adams[2007],Blazsek,S. andA.Downarowicz[2011],DelBrio,EB.,A.Mora-ValenciaandJ.Perote[2014],Tbulon,F.,K.Guesmi andS.Jebri[2014]).Inthecontextofportfbliodiversification,includinghedgefilnds,precisevolatility modelingofhedgefimdretumsmayhelpinstitutionalinvestorstoevaluatethefiltureriskofhedgefilnd portlblioandareusefilltodetenninemarkettimingandcontroltherisklimit. Thepurposeofthispaperistoexaminetheconditionalvolatilitycharacteristicsofdailymanagement hedgefUndindexretumsandconstructanARMA-GARCHtypemodeling・Thispaperwilllimititselfto theunivariatetime-seriesanalysisofhedgefilndreturnsalthoughtheissuesstudiedherewillbesimilar inmultivariateanalysis.Ifbcusontheconstructionofnonlineartime-seriesmodelsthatcanbeusefillfbr describingpersistenceandvolatilityofhedgefilndindexreturns.Thispaperisorganizedasfbllows.Section 2describesfburmainhedgefimdstrategiesandsummarizestheempiricalpropertiesoftheirreturnseries usedinthisstudy.Section3reviewsARMAmodelingandpresentstheestimationresultsanddiagnostic checking・InSection4,GARCHmodelingisintroducedanddiscussestheresults.Someconcludingremarks areofferedinthe6nalsection.

2．HedgeFundStrategiesandDataDescription

Inthispaper,fburprincipalhedgefimdstrategiesindices(EquityHedge,EventDriven,Macro/CTA,and RelativeValueArbitrageintheHFRXGlobalHedgeFundlndex)areinvestigated.Dataaredailyandspan theperiodMarch31,2003toAugustll,2014.ThedataofhedgefimdindicesisobtainedfiomtheHedge FundResearchlnc.(hereafierHFR).TheHFRXGIobalHedgeFundlndexisdesignedtoberepresentative

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l)HFRXHedgeFundIndicesaretheglobalindustrystandardfbrperfbnnancemeasurementacrossallaspectsofthe hedgefimdindustry.Constituentsofallindicesareselected廿omaneligiblepoolofthemorethan6,800hmdsthat reportoftheHFRDatabase.MoredetailedstrategydescriptionscanbeseeninHedgeFundResearch{2014],HFRX He唯e”"‘"""ces:D醜"edFor〃"/α/cMe/ﾙo伽/ogy<www.hedgefimdresearch.com>.

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Figurel:FourHedgeFundIndexReturnsfromAprill,2003toAuguStll,2014 (a)EquityHedge:mdexvalue (b)EventDriven:mdexvalue 1,500 1,400 1,300 1,200 1,100 1,000 900 1,800 1,600 1,400 1,200 1,000 800 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 1 3 1 4 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 1 3 1 4 EquityHedge:return EventDrien：return

３２１０１２３４

■ ■

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。。 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 1 3 1 4 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 1 3 1 4 (c)Macro/CTA:mdexvalue (d)RelativeValueArbitrage:mdexvalue 1,600 1,500 1,400 1,300 1,200 1,100 1，000 900 1,300 1,200 1,100 1,000 900 800 700

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0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 1 3 1 4 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 1 3 1 4 Macro/CTA:return RelativeValueArbitrage:return

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口一 4 2 0 -2 4 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 1 3 1 4 0 3 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1 1 1 2 1 3 1 4 −204−

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PersistenceandVolatilityofHedgeFundReturns:ARMA-GARCHModeling Ⅱ間blel:SummaryStatisticsofHedgeFundIndexReturns Aprill,2003toAugustll,2014 Jarque -Bera Mean STDSkewnessKurtosis DailyReturn No.Obs HFRXGlobalHedgeFundlndex EquityHedge EventDriven Macro/CTA RelativeValueArbitrage 4162.9r* 17919.96*** 7300.02… 180971.40*** 0.0052 0.0171 0.0039 0.0065 0.4066 0.2959 0.4081 0.2712 -0.8442 -1.1558 -l.0193 -l.7268 8.6599 15.0343 10.5510 41.7891

糾糾糾糾

８８８８２２２２

Source:Author'scalculations,basedondata廿omHedgeFundResearch. Notes:TheJarque-BeranonnalitytestisasymptoticallydistributedasacentralX]with2degreesoffi･eedomunder thenullhypothesis,withlO%,5%andl%criticalvalues.*,**,***denotesignificanceatthelO%,5%,andl%levels, respectively. Second,allhedgefmdremmdistributionsarenegativelyskewed.Negativeskewnessmeansthatthelefttail isparticularlyextreme.ltindicatesthatlargenegativereturnsaremoreprobablethanlargepositiveones. Negativeskewnessandleptokurtosisareunattractivefeaturesfbrrisk-averseinvestors. Thestatisticalpropertiesofnon-normallydistributedhedgefimdindexretumsposedifficultproblemsfbr measuringrisk・Thestandarddeviationsimplyaveragedailyvolatilities,ofienusedasariskmeasurement. However,itcanonlybeappropriatefbrariskiftheobservedreturnsarenonnallydistributed・Traditional riskmanagementbasedonthemean-varianceapproachonlytakestwoparameters-meanremmandretum

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Iftheretumsarenonnallydistributed,the6rsttwomomentsofthedistributionsareenoughtocharacterize theirrisk-returnprofile.However,inthecaseofnon-normallydistributedretums,skewnessandkurtosis mightplayasignificantroleonriskperceptionfbrinvestors.AsisevidencedbytheirsignificantJB-test statistics,itseemsappropriatetoconcludethatallhedgefimdindexretumsarenotnonnallydistributed. 3．ARMAModeling:LinearStructureinUnivariateTimeSeries Theunivariatetime-seriesofourinterestisthehedgeiimdindexvaluePtattime/.Anytime-seriesdata, Ptsuchasfinancialassetpricescanbethoughtofasrandomvariableshavingbeengeneratedbyastochastic process.Aconcretesetofdata,Pt,Pt+1,Pt+2,･･･canberegardedasaparticularrealizationoftheunderlying stochasticprocess(i.e.thevaluesofarandomvariables). Intimeseriesregression,theideathathistoricalrelationships(i.e.thefiltureislikethepast)canbe generalizedtothefiltureisfbrmalizedbytheconceptofstationarity.ThepercCptionthatthefilturewillbe likethepastisanimportantassumptionintimeseriesregression,somuchsothatitisgivenitsownname,

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arestationary・Toconfinnthisfbrfburhedgefimdindexreturns,theunitroottestsareusedindetecting whetherthereturnsseriesarestationaryornonstationary.Accordingtotheunitroottests(theaugumented Dicky-FullertestandthePhillip-Perrontest)fbrthenullhypothesisthattheserieshasaunitroot(i.e・itis nonstationary),allindexreturnscan呵ectthenullhypothesisfbrsignificanceat99%confidencelevels,

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Withtime-seriesdata,itislikelythattheobservationswillbecorrelatedovertimebecausetheobservation attime/istheconsequencesofeconomicactionsordecisionstakenattime/,butalsoattimer+1、/+2,andso on.AsshowninFigure2,theseeffectsdonotoccurinstantaneouslybutarespreadoverfilturetimeperiods. Apopularmethodofmodelingstationarytimeseriesistheautoregressivemovingaverage(ARMA) methodwhichassemblestwoseparatetools(ARtennsandMAtenns)fbrmodelingtheserialcorrelation inthelaggeddependentvariableandinthedisturbance・Itcanbesayingthatthedependentvariable7fin oneperiodwilldependonwhatitwasinthepastperiods,7t_1,7f_2,…,whichisthepersistenceofhedge filndperfbnnanceovervarioustimeintervals・Anotherwayofmodelingthecontinuingimpactofchange overseveralperiodsisviatheerrorterm,whichrepresentsthecompositionofallfactors(apart廿omthe independentvariables)thatinHuencethebehaviorofthedependentvariable.Thebehaviorofthesefactorsin thecurrenttimeperiodmightbequitesimilartotheirbehaviorintheprevioustimeperiodandsuggeststhe possibilityofsomecorrelationbetweenerrorsclosetogetherintime. Inthissection,thesetwowaysinwhichdynamicscanenterregressionrelationship-laggedvaluesofthe dependentvariable(ARtenns),andlaggedvaluesoftheerrortenn(MAterms)areconsidered. First,considertheunconditionalmomentsofthereturnprocess.Themeanllisdefinedas l ' = E [ 7 t ] ( l ) whereE[･]denotestheexpectationoperatorandtheexpectedvalueoftheretum(i.e.theexpectedremrn)EM. Thevarianceof7tisameasureofdispersioninthepossiblevaluesfbr7t,denotedasvar(7t),isdefinedas 〃αγ[7t]=EI7t-u]2=02 （2） whereitssquarerootoisthestandarddeviationof7t,whichiscalledvolatilityandameasureofrisk. Ingeneral,thereturnonanyasset7tcanbedividedintotwoparts:theexpectedpartsoftheremrnEMand theunexpectedpartofthereturnEt. 7t=EM+Et （3） 7t=仙+Et （4） 2)ThedistributiontheorysupportingtheDickey-Fullertestassumesthatthedisturbancetennsareuncorrelatedand homogeneous.TheaugumentedDickey-FullertestallowsthedismrbancetermsarecorrelatedbUtstillassumetobe homogeneous.Moreover,thePhillip-Perrontestallowsthedismrbancetennstobecorrelatedandheterogeneously distributed.SeeEnders[1995],p.239. −206−

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PersistenceandVolatilityofHedgeFundRemms:ARMA-GARCHModeling Figure2:TheDistributedLagEffect Economicaction attimer

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Eftct attimet+Z Effbct attimet+k Effbct attimeォ Effect attimet糸2

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、ノー﹁ Source:Author'scompilationbasedonGri価ths,HillandLim[20081,p.227 whereEt,isknownasthedismrbance,orerrortenn. Theerrortennisarandomvariablethathastheprobabilisticpropertieswithzeromean,constantvariance

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between-ltol.Thissimplemodeliscalledanautoregressivemodeloforderp. TheMApartofthemodelreferstothestrucmreoftheerrortenn.Thefirst-ordermovingaveragemodel, MA(1),is 7t=ll+Et+91Et-1 （10） whereO,scalestheinHuenceofthewhitenoiseprocess. TheMA(9)processcanbewrittenas 好＝u＋Et＋e1Et-1+82Et-2+…+eqEt-q (ll) Equation(11)statesthatamovingaveragemodelissimplyalinearconditionofwhitenoiseprocess.Inother

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words,7tdependsonthecurrentandpreviousvaluesofawhitenoiseerrortenn.Moreconcisely,

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acombinationofcurrentandpreviousvaluesofawhitenoiseerrortenns.Namely;theautoregressiveand movingaveragespecificationscanbecombinedtofbnnanARMAtl,9)model. Modelidentification ThestrategyofanappropriateARMAmodelselectionissystematic,i.e.theso-calledBox-Jenkins approach.Thisapproachtakesthreesteps:identification,estimationanddiagnosticchecking. ThefirststepofbuildinganARMAmodelistoidentifytheorderofthemodelrequiredtocapturethe featuresofdatageneratingprocess.ItistodetenninetheappropriateARandMAorderspand9.Acentral concernofthisapproachistospecifyfbrthepredictablepartasaconstant"andmeasuretheerrortennEt, whichisthedifferenceoftheseries廿omitsmean好一βasshowninequations(4). Identificationofthestructureinthedataiscarriedoutbylookingattheautocorrelationandpartial autocorrelationcoefficientsafierplottingthedataovertime.Autocorrelationisthecorrelationofaseries withitsownlaggedvalues・Whentheobservationsindifferenttimeperiodsarecorrelated,itissaidthat autocorrelationexists.Thecoefficientofcorrelationbetweentheobservationsattwoadjacentperiodsis calledtheautocorrelationcoefficient.Thble2displaystheautocorrelationfimction(ACF)ofthehedgefUnd indexretums.Theestimatedautocorrelationcoefficientsfbrlaglto20togetherwiththeI_jung-Box(LB) statisticswithfive,tenandtwentyautocorrelationsarereported・Atfirstglance,theACFofthereturnseries showthatthereisaslightlyautoregressivestructureinthedata・Inparticular,RelativeValueArbitrageshows highlysignificantautocorrelationsoveralllags.Thus,itseemsthateitheranARoramixedARMAprocess mightbeappropriatefbrmodelingthesedata.Infact,itisnoteasytopreciselydetenninetheappropriatelag ordergiventheseestimatesatthisstage. Itispossibletotestthejointhypothesisthatallofthefirst"I(=maximumlaglength)autocorrelation

coefficientsaresimultaneouslyOointly)equaltozero("0:P,=P,=…=Pm=0).Q-statisticsisthe

LjungandBoxstatisticofACF(LB-Q),representedinthebottompartofThble2.Theremmsoffburindices exceptingfbrRelativeValueArbitragedonotshowhighautocorrelationcoeffIcients,butsomeofthemare stillhighlysigni6cantat95%confidencelevel・SincethefirstACFcoefficientsofallremmsseriesarehighly significant,theLiung-Boxjointteststatistic呵ectsthenullhypothesisofnoautocorrelationatthel%level.

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-208-A C F Lag(1) Lag(2) Lag(3) Lag(4) Lag(5) Lag(6) Lag(7) Lag(8) Lag(9) Lag(10) Lag(ll) Lag(12) Lag(13) Lag(14) Lag(15) Lag(16) Lag(17) Lag(18) Lag(19) Lag(20) LB-Q(5) LB-Q(10) LB-Q(20) PersistenceandVolatilityofHedgeFundRetums:ARMA-GARCHModeling Table2:Autocorrelations

E_quityHedge EventDriven Macro/CTA

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Iable3:ARMAModeling

Indices _EquityHedge EventDriven Macro/CTA RelativeValue

Arbitrage

Model AR(1) ARMA(l,2) AR(1) ARMA(1,2)

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PersistenceandVolatilityofHedgeFundRemms:ARMA-GARCHModeling Figure5:TimeVaryingVolatility

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(approximatelyO.99).Thisimpliesthatshockstotheconditionalvariancewillbehighlypersistentanda largepositiveandalargenegativeretumwillleadfilturefbrecastsofthevariancetobehighfbrasubsequent period.Avolatilityofhalflife(i.e.thehalflifeperiod:HLP)takes22.757daysfbrtheEquityHedgeand 29.921daysfbrtheEventDriven,wherebytheHLPof69.668andl47.l31daysfbrMacro/CTAandRelative