instrumentsinaportfolio.Forexample,foraportfoliocomposedofNinstruments - PDF document

instrumentsinaportfolio.Forexample,foraportfoliocomposedofNinstrumentswithweightswi;t�1,(PNi=1wi;t�1=1;wi;t�10)wehaveRpt=NXi=1wi;t�1Rit,(percentreturn)rpt=ln(NXi=1wi:t�1erit);(continuouslycompounded)OftenrptisapproximatedbyPNi=1wi;t�1rit.WhendividendsarepaidoutwehaveRt=(Pt�Pt�1)=Pt�1+Dt=Pt�1t
ln(Pt)+Dt=Pt�1whereDtisthedividendpaidoutduringtheholdingperiod.2.2Multi-periodreturnsSingle-periodpricechanges(returns)canbeusedtocomputemulti-periodpricechangesorreturns.DenotethereturnoverthemostrecenthperiodsbyRt(h)then(abstractingfromdividends)Rt(h)=Pt�Pt�h Pt�hor1+Rt(h)=Pt=Pt�handrt(h)=ln(Pt=Pt�h)=rt+rt�1+:::+rt�h+1wherert�i,i=0;1;2;:::;h�1arethesingle-periodreturns.Forexample,weeklyreturnsisde…nedbyrt(5)=rt+rt�1+:::+rt�4.Similarly,sincethereare25businessdaysinonemonth,thenthe1-monthreturncanbecomputedasthesumofthelast251-dayreturns,orrt(25).2.3OverlappingreturnsNotethatmulti-periodreturnshaveoverlappingdailyobservations.Inthecaseofweeklyreturns,rt(5)andrt�1(5)havethefourdailyreturns,rt�1+rt�2+rt�3+rt�43 incommon.Asaresultthemulti-periodreturnswillbeseriallycorrelatedeveniftheunderlyingdailyreturnsarenotseriallycorrelated.Onewayofavoidingtheoverlapproblemwouldbetosamplethemulti-periodreturnshperiodsapart.Butthisislikelytobeine¢cientasitdoesnotmakeuseofallavailableobservations.Amoreappropriatestrategywouldbetousetheoverlappingreturnsbutallowforthefactthatthiswillinduceserialcorrelations.ForfurtherdetailsseePesaran,Pick,andTimmerman(2010).3StatisticalmodelsofreturnsAsimplemodelofreturns(orlogpricechanges)isgivenbyrt+1=
ln(Pt+1)=pt+1�pt=t+t"t+1;t=1;2;:::;T(1)wheretand2taretheconditionalmeanandtheconditionalvarianceofreturns(withrespecttotheinformationset tavailableattimet)and"t+1representstheunpredictablecomponentofreturn.Twopopulardistributionsfor"t+1are"t+1j tsIIDZ"t+1j ts r v�2 v!IIDTvwhereZsN(0;1)standsforastandardnormaldistribution,andTvstandsforStudent’stwithvdegreesoffreedom.Unlikethenormaldistributionthathasmomentsofallorders,Tvonlyhasmomentsoforderv�1andsmaller.FortheStudent’sttohaveavariance,forexample,weneedv�2.Sincert+1=ln(1+Rt+1),whereRt+1=(Pt+1�Pt)=Pt,itthenfollowsthatunder"t+1j tsIIDZ,thepricelevel,Pt+1conditionalon twillbelognormallydistributed.Notethat t=(Pt;Pt�1;::::)and t=(rt;rt�1;::::)conveythesameinformationandareequivalent.Hence,Pt+1=Ptexp(rt+1),andwehave1E(Pt+1j t)=PtE(exp(rt+1)j t)=Ptexp(t+1 22t) 1Usingpropertiesofthemomentgeneratingfunctionofnormalvariates,ifxvN(x;2x)then,E[exp(x)]=exp(x+:52x):4 Similarly,Var(Pt+1j t)=P2texp(2t+2t)exp(2t)�1Inpractice,itismuchmoreconvenienttoworkwithlogreturns,rt+1,ratherthanassetprices.TheprobabilitydensityfunctionsofZandTvaregivenbyf(Z)=(2)�1=2exp�Z2 2;�1Z1(2)andf(Tv)=1 p vB(v=2;1=2)1+T2v v�(v+1)=2(3)where�1Tv1,andB(v=2;1=2)isthebetafunctionde…nedbyB(;)=�()�() �(+);�()=Z10u�1e�uduItiseasilyseenthatE(Tv)=0,andVar(Tv)=v v�2Alargepartof…nancialeconometricsisconcernedwithalternativewaysofmod-ellingtheconditionalmean(meanreturns),t,theconditionalvariance(assetre-turnvolatility),t,andthecumulativeprobabilitydistributionoftheerrors,"t+1.Anumberofissuesneedtobeaddressedinordertochooseanadequatemodel.Inparticular:-Isthedistributionofreturnsnormal?-Isthedistributionofreturnsconstantovertime?-Arereturnsstatisticallyindependentovertime?-Aresquaresorabsolutevaluesofreturnsindependentlydistributedovertime?-Whatarethecrosscorrelationofreturnsondi¤erentinstruments?Theabovemodellingissuescanbereadilyextendedtothecasewhereweareconcernedwithavectorofassetreturns,rt=(r1t;r2t;:::rmt)0.Inthiscasewealsoneedtomodelthepair-wiseconditionalcorrelationsofassetreturns,namelyCorr(rit;rjtj t)=Cov(rit;rjtj t) p Var(ritj t)Var(rjtj t)5 Typicallytheconditionalvariancesandcorrelationsaremodelledusingexponen-tialsmoothingproceduresorthemultivariategeneralizedautoregressiveconditionalheteroskedasticmodelsdevelopedintheeconometricliterature.3.1Percentiles,criticalvaluesandValueatRiskSupposearandomvariabler(saydailyreturnsonaninstrument)hastheprobabilitydensityfunctionf(r).Thenthepthpercentileofthedistributionofr,denotedbyCp,isde…nedasthatvalueofreturnsuchthatppercentofthereturnsfallbelowit.Mathematicallywehavep=Pr(rCp)=ZCp�1f(r)drIntheliteratureonriskmanagementCpisusedtocompute‘ValueatRisk’orVaRforshort.Forp=1%,Cpassociatedwiththeone-sidedcriticalvalueofthenormaldistributionisgivenby�2:33,whereisthestandarddeviationofreturns.InhypothesistestingCpisknownasthecriticalvalueofthetestassociatedwitha(one-sided)testofsizep.Inthecaseoftwo-sidedtestsofsizep,theassociatedcriticalvalueiscomputedasCp=2.3.2MeasuresofdeparturefromnormalityThenormalprobabilitydensityfunctionforrt+1conditionalontheinformationattimet, t,isgivenbyf(rt+1)=(22t)�1=2exp�1 22t(rt+1�t)2witht=E(rt+1j t)and2t=E[(rt+1�t)2j t]beingtheconditionalmeanandvariance.Ifthereturnprocessisstationary,unconditionallywealsohave=E(rt+1),and2=E[(rt+1�t)2].Skewnessandtail-fatnessmeasuresarede…nedbySkewness=p b1=m3=m3=22Kurtosis=b2=m4=m22wheremj=PTt=1(rt�r)j T;j=2;3;46 Table1:DescriptivestatisticsfordailyreturnsonSP500,FTSE100,GermanDAX,andNikkei225 VariablesSPFTSEDAXNK Maximum14.1110.0512.8320.70Minimum-9.88-9.24-8.89-13.07Mean(r)-0.01-0.01-0.01-0.01S.D.(^)1.391.331.651.68Skewness(p b1)0.350.060.240.16Kurtosis(b2)14.39.78.517.8JBstatistic13453.64713.13199.223000.8 Foranormaldistributionp b1t0,andb2t3.Inparticular^=r=TXt=1rt=T;^=s PTt=1(rt�r)2 T�1TheJarque-Bera’steststatisticfordeparturefromnormalityisgivenby,see(JarqueandBera1980)JB=T1 6b1+1 24(b2�3)2 Underthejointnullhypothesisthatb1=0andb2=3,theJBstatisticisasymp-toticallydistributed(asT!1)asachi-squaredwith2degreesoffreedom,22:Therefore,avalueofJBinexcessof5:99willbestatisticallysigni…cantatthe95percentcon…dencelevel,andthenullhypothesisofnormalitywillberejected.4Empiricalevidence:statisticalpropertiesofre-turnsTable1givesanumberofstatisticsfordailyreturns(100)onfourmainequityindexfutures,namelyS&P500(SP),FTSE100(FTSE),GermanDAX(DAX),andNikkei225(NK),overtheperiod3-Jan-00to31-Aug-09(foratotalof2,519observations).Thekurtosiscoe¢cientsareparticularlylargeforallthefourequityfuturesandexceedthebenchmarkvalueof3forthenormaldistribution.Thereissomeevi-denceofpositiveskewness,butitisofsecondorderimportanceascomparedtothe7 magnitudeofexcesskurtosiscoe¢cientgivenby,b2�3.Thelargevaluesofexcesskurtosisisre‡ectedinthehugevaluesoftheJBstatisticsreportedin1.Alsoundertheassumptionthatreturnsarenormallydistributed,wewouldhaveexpectedthemaximumandminimumofdailyreturnstofall(with99%con…dence)intheregionof2.33S.D.,whichis3:24forSP500,ascomparedtotheobservedvaluesof�9:88and14:11.SeealsoFigure1. Figure1:HistogramandNormalcurvefordailyreturnsonSP500(overtheperiod3-Jan-00to31-Aug-09)Thedeparturefromnormalityisparticularlypronouncedoverthepastdecadewheremarketshavebeensubjecttotwoimportantepisodesof…nancialcrises:thecollapseofmarketsin2000afterthedot-combubbleandthestockmarketcrashof2008afterthe2007creditcrunch.(seeFigure2).However,theevidenceofdeparturefromnormalitycanbeseenindailyreturnsevenbefore2000.Forexample,overtheperiod3-Jan-94to31-Dec-99(1565dailyobservations)kurtosiscoe¢cientofreturnsonSP500was9.5whichisstillwellabovethebenchmarkvalueof3.Therecent…nancialcrisishasaccentuatedthesituationbutcannotbeviewedasthecauseoftheobservedexcesskurtosisofequityreturns.Similarresultsarealsoobtainedifweconsiderweeklyreturns.Thekurtosiscoe¢cientsestimatedusingweeklyreturnsovertheperiodJanuary2000totheendofAugust2009(504weeks)were12.4,15.07,8.9and15.2forSP500,FTSE,DAX,andNikkei,respectively.Thesearesomewhatlowerthantheestimatesobtained8 Table2:DescriptivestatisticsfordailyreturnsonBritishpound,euro,Japaneseyen,Swissfranc,Canadiandollar,andAustraliandollar VariablesJPYEUGBPCHFCADAD Maximum4.533.173.414.585.256.21Minimum-3.93-3.01-5.04-3.03-3.71-9.50Mean(r)-.006.016.007.012.013.022S.D.(^)-.65.65.60.70.59.90Skewness(p b1)-.28.01-.35.12.09-.76Kurtosis(b2)5.994.57.24.99.113.8 Table3:DescriptivestatisticsfordailyreturnsonUST-Note10Y,EuropeEuroBund10Y,JapanGovernmentBond10Y,and,UKLongGilts8.75-13Y VariablesBUBEBGBJ Maximum3.631.482.431.53Minimum-2.40-1.54-1.85-1.41Mean(r).0.01.01.01S.D.(^).43.32.35.24Skewness(p b1)-.004-.18.02-.18Kurtosis(b2)6.674.496.026.38 10 4.1OtherstylizedfactsaboutassetreturnsAssetreturnsaretypicallyuncorrelatedovertime,aredi¢culttopredictandaswehaveseentendtohavedistributionsthatarefat-tailed.Incontrasttheabsoluteorsquaresofassetreturns(thatmeasurerisk),namelyjrtjorr2t,areseriallycorrelatedandtendtobepredictable.Itisinterestingtonotethatrtcanbewrittenasrt=sign(rt)jrtjwheresign(rt)=+1ifrt�0andsign(rt)=�1ifrt0.Sincejrtjispredictable,itis,therefore,thenon-predictabilityofsign(rt),orthedirectionofthemarket,whichliesbehindthedi¢cultyofpredictingreturns.Theextenttowhichreturnsarepredictabledependsontheforecasthorizon,thedegreeofmarketvolatility,andthestateofthebusinesscycle.Predictabilitytendstoriseduringcrisisperiods.Similarconsiderationsalsoapplytothedegreeoffat-tailednessoftheunderlyingdistributionandthecrosscorrelationsofassetreturns.Thereturndistributionsbecomelessfat-tailedasthehorizonisincreased,andcrosscorrelationsofassetreturnsbecomemorepredictablewiththehorizon.Crosscorrelationofreturnsalsotendtoincreasewithmarketvolatility.Theanalysisoftimevariationsinthecrosscorrelationofassetreturnsisbeyondthescopeofthispaper.Buttheinterestedreadermightwishtoconsult(PesaranandPesaran2010)wheremultivariateconditionalvolatilitymodelsare…ttedtoweeklyreturnsonequities,bondsandcurrencies.Inthecaseofdailyreturns,equityreturnstendtobenegativelyseriallycorre-lated.Duringnormaltimestheyaresmallandonlymarginallysigni…cantstatisti-cally,butbecomerelativelylargeandattainahighlevelofstatisticalsigni…canceduringcrisisperiods.Thesepropertiesareillustratedinthefollowingempiricalap-plication.The…rstandsecondorderserialcorrelationcoe¢cientsofdailyreturnsonSP500overtheperiod3-Jan-00to31-Aug-07are�0:015(0:0224)and�0:0458(0:0224),respectively,butincreaseto�0:068(0:0199)and�0:092(0:0200)oncethesampleisextendedtotheendofAugust2009whichcoversthe2008global…nancialcrisis.2Similarpatternsarealsoobservedforotherequityindices.Forcurrenciestheevi-denceismoremixed.Inthecaseofmajorcurrenciessuchaseuroandyenthereislittleevidenceofserialcorrelationinreturnsandthisoutcomedoesnotseemmucha¤ectedbywhetheroneconsidersnormalorcrisisperiods.Forothercurrenciesthereissomeevidenceofnegativeserialcorrelation,particularlyattimesofcrisis.Forex-ample,overtheperiod3-Jan-00to31-Aug-09the…rstorderserialcorrelationofdaily 2The…guresinbracketsarestandarderrors.11 returnsonAustraliandollaramountsto�0:056(0:0199),butbecomesstatisticallyinsigni…cantifweexcludethecrisisperiod.Thereisalsoverylittleevidenceofserialcorrelationindailyreturnsonthefourmajorgovernmentbondsthatwehavebeenconsidering.Thisoutcomedoesnotdependonwhetherthecrisisperiodisincludedinthesample.Irrespectiveofwhethertheunderlyingreturnsareseriallycorrelated,theirabsolutevalues(ortheirsquares)arehighlyseriallycorrelated,oftenovermanyperiods.Forexample,overthe3-Jan-00to31-Aug-09periodthe…rstandsecondorderserialcorrelationcoe¢cientsofabsolutereturnonSP500are0:2644(0:0199);0:3644(0:0204);foreurotheyare0:0483(0:0199)and0:1125(0:0200),andforUS10Ybondtheyare0:0991(0:0199)and0:1317(0:0201).Theserialcorrelationinabsolutereturnstendtodecayveryslowlyandcontinuetobestatisticallysigni…canteventafter120tradingdays.SeeFigure3. Figure3:AutocorrelationfunctionoftheabsolutevaluesofSP500(overtheperiod3-Jan-00to31-Aug-09)Itisalsointerestingtonotethatthereislittlecorrelationbetweenrtandjrtj.BasedonthefullsampleendinginAugust2009,thiscorrelationis�:0003forSP500,0:025foreuro,and0:009fortheUS10Ybond.12 (4)shouldbestatisticallysigni…cant.Somewritershaveevengonesofarastoequatestockmarkete¢ciencywiththenon-predictabilityproperty.Butthislineofargumentisnotsatisfactoryanddoesnothelpinfurtheringourunderstandingofhowmarketsoperate.Theconceptofmarkete¢ciencyneedstobede…nedseparatelyfrompredictability.Infact,itiseasilyseenthatstockmarketreturnswillbenon-predictableonlyifmarkete¢ciencyiscombinedwithriskneutrality.6.1RiskneutralinvestorsSupposethereexistsariskfreeassetsuchasagovernmentbondwithaknownpayout.Insuchacaseaninvestorwithaninitialcapitalof$At,isfacedwithtwooptions:-Option1:holdtherisk-freeassetandreceive$(1+rft)Atattheendofthenextperiod.-Option2:switchtostocksbypurchasingAt=Ptshares,holdthemforoneperiodandexpecttoreceive$(At=Pt)(Pt+1+Dt+1)attheendofperiodt+1.Arisk-neutralinvestorwillbeindi¤erentbetweenthecertaintyof$(1+rft)At;andthehis/herexpectationsoftheuncertainpayoutofoption2.Namely,forsuchariskneutralinvestor(1+rft)At=E[(At=Pt)(Pt+1+Dt+1)j t](7)where tistheinvestor’sinformationattheendofperiodt.Thisrelationshipiscalledthe‘arbitragecondition’.Using(5)wenowhavePt+1+Dt+1=Pt(1+Rt+1)andtheabovearbitrageconditioncanbesimpli…edtoE[(1+Rt+1)j t]=(1+rft)15 orERt+1�rftj t=0(8)Thisresultestablishesthatiftheinvestorformshis/herexpectationsoffuturestock(index)returnstakingaccountofallmarketinformatione¢ciently,thentheexcessreturn,Rt+1�rft,shouldnotbepredictableusinganyofthemarketinformationthatareavailableattheendofperiodt.Noticethatrftisknownattimetandisthereforeincludedin t.Hence,underthejointhypothesisofmarkete¢ciencyandriskneutralitywemustalsohaveE(Rt+1j t)=rft.Theabovesetupcanalsobeusedtoderiveconditionsunderwhichassetpricescanbecharacterisedasarandomwalkmodel.Suppose,theriskfreerate,rft;inadditiontobeingknownattimet,isalsoconstantovertimeandgivenbyrf.Thenusing(7)wecanalsowritePt=1 1+rE[(Pt+1+Dt+1)j t]orPt=1 1+rf[E(Pt+1j t)+E(Dt+1j t)]Undertherationalexpectationshypothesisandassumingthatthe‘transversalitycondition’limj!11 1+rfjE(Pt+jj t)=0holdswehavethefamiliarresultPt=1Xj=11 1+rfjE(Dt+jj t)(9)thatequatesthelevelofstockpricetothepresentdiscountedstreamofthedividendsexpectedtooccurtotheassetoverthein…nitefuture.Thetransversalityconditionrulesoutrationalspeculativebubblesandissatis…ediftheassetpricesarenotexpectedtorisefasterthantheexponentialdecayratedeterminedbythediscountfactor,01=(1+rf)1.ItisnoweasilyseenthatifDtfollowsarandomwalksowillPt.Forexample,supposeDt=Dt�1+"t(10)where"tisawhitenoiseprocess.ThenE(Dt+jj t)=Dt16 andPt=Dt rf(11)Therefore,wealsohavePt=Pt�1+ut(12)whereut="t=rf.Therandomwalkpropertyholdsevenifrf=0,sinceinsuchacaseitwouldbereasonabletoexpectnodividendsarealsopaidout,namelyDt=0.InthiscasethearbitrageconditionbecomesE(Pt+1j t)=Pt(13)whichissatis…edbytherandomwalkmodelbutisinfactmoregeneralthantherandomwalkmodel.Anassetpricethatsatis…es(13)isamartingaleprocess.Ran-domwalkprocesseswithzerodriftaremartingaleprocessesbutnotallmartingaleprocessesarerandomwalks.Forexample,thepriceprocessPt+1=Pt+(
Pt+1)2�E(
Pt+1)2j t +"twhere"tisawhitenoiseprocessisamartingaleprocesswithrespecttotheinfor-mationset t,butitisclearlynotarandomwalkprocess,unless=0.Othermodi…cationsoftherandomwalktheoryisobtainedifitisassumedthatdividendsfollowageometricrandomwalkwhichisamorerealisticthanthelineardividendmodelassumedin(10).InthiscaseDt+1=Dtexp(d+dt+1)(14)wheredanddaremeanandstandarddeviationofthegrowthrateofthedividends.Ifitisfurtherassumedthatt+1j tisN(0;1),wehaveE(Dt+jj t)=Dtexpjd+1 2j2dUsingthisresultin(9)nowyields[assumingthat(1+rf)�1exp�d+1 22d
1]Pt=Dt (15)where=(1+rf)exp�d�1 22d�117 presentvaluemodelpriceswillhavefat-tailedinnovationsonlyifthedividendsthatdriveassetpricesarealsofat-tailed.Butunderthegeometricrandomwalkmodelfordividends(14),E(Dt+jj t)neednotexistifthedividendinnovations,t,arefat-tailed.OneimportantexampleariseswhenthastheStudenttdistributionasde…nedby(3).ForthederivationofthepresentvalueexpressioninthiscaseweneedE(exp(dt+j)),whichisthemomentgeneratingfunctionoft+jevaluatedatd.ButtheStudenttdistributiondoesnothaveamomentgeneratingfunction,andhencethepresentvalueformulacannotbecomputedwheninnovationstothedividendsaretdistributed.6.2RiskaverseinvestorsInadditiontotheabovedocumentedempiricalshortcomings,itisalsoimportanttonotethatriskneutralityisabehavioralassumptionandneednotholdevenifallmarketinformationisprocessede¢cientlybyallthemarketparticipants.Amorereasonablewaytoproceedistoallowsomeoralloftheinvestorstoberiskaverse.Inthismoregeneralcasethecertainpayout,(1+rft)At,andtheexpectationsoftheuncertainpayout,E[(At=Pt)(Pt+1+Dt+1)j t],willnotbethesameanddi¤erbya(possibly)time-varyingriskpremiumwhichcouldalsovarywiththeleveloftheinitialcapital,At.Morespeci…cally,wehaveE[(At=Pt)(Pt+1+Dt+1)j t]=(1+rft)At+tAtwheretisthepremiumper$ofinvestedcapitalrequired(expected)bytheinvestor.ItisnoweasilyseenthatERt+1�rftj t=tanditisnolongernecessarytruethatundermarkete¢ciencyexcessreturnsarenon-predictable.Theextenttowhichexcessreturnscanbepredictedwilldependontheexistenceofahistoricallystablerelationshipbetweentheriskpremium,t;andthemacroandbusinesscycleindicatorssuchaschangesininterestrates,dividendsandvariousbusinesscycleindicators.Inthecontextoftheconsumptioncapitalassetpricingmodeltisdeterminedbytheexantecorrelationofexcessreturnsandchangesinthemarginalutilityofconsumption.Inthecaseofarepresentativeconsumerwiththesingleperiodutil-ityfunction,u(ct),the…rst-orderinter-temporaloptimizationcondition(theEulerequation)isgivenbyERt+1�rftu0(ct+1) u0(ct)j t=0(17)19 di¢culttoreconcilethehighvolatilityofexcessreturnswiththelowvolatilityofconsumptiongrowththatareobservedhistorically.7Returnpredictabilityandalternativeversionsofthee¢cientmarkethypothesisInhis1970review,Famadistinguishesbetweenthreedi¤erentformsoftheEMH:1.Theweakformassertsthatallpriceinformationisfullyre‡ectedinassetprices,inthesensethatcurrentpricechangescannotbepredictedfrompastprices.ThisweakformwasalsointroducedinanunpublishedpaperbyRoberts(1967).2.Thesemi-strongformthatrequiresassetpricechangestofullyre‡ectallpub-liclyavailableinformationandnotonlypastprices.3.Thestrongformthatpostulatesthatpricesfullyre‡ectinformationevenifsomeinvestororgroupofinvestorshavemonopolisticaccesstosomeinforma-tion.FamaregardedthestrongformversionoftheEMHasabenchmarkagainstwhichtheotherformsofmarkete¢cienciesaretobejudged.Withrespecttotheweakformversionheconcludesthatthetestresultsstronglysupportthehypothesis,andconsideredthevariousdeparturesdocumentedaseconomicallyunimportant.Hereachedasimilarconclusionwithrespecttothesemi-strongversionofthehypothesis;althoughashenoted,theempiricalevidenceavailableatthetimewasratherlimitedandfarlesscomprehensiveascomparedtotheevidenceontheweakversion.ThethreeformsoftheEMHpresentdi¤erentdegreeswherebypublicandprivateinformationarerevealedintransactionprices.Itisdi¢culttoreconcileallthethreeversionstothemainstreamassetpricingtheory,andasweshallseebelowacloserconnectionisneededbetweenmarkete¢ciencyandthespeci…cationofthemodeleconomythatunderliesit.7.1DynamicstochasticequilibriumformulationsandthejointhypothesisproblemEvidenceonthesemi-strongformoftheEMHwasrevisitedbyFamainasecondreviewoftheE¢cientCapitalMarketspublishedin1991.Bythenitwasclearthatthedistinctionbetweentheweakandthesemi-strongformsoftheEMHwas21 established.AsFamarecognizedaweakerandeconomicallymoresensibleversionofthee¢-ciencyhypothesiswouldbeneeded,namely“pricesre‡ectinformationtothepointwherethemarginalbene…tsofactingoninformation(thepro…tstobemade)donotexceedthemarginalcosts."Thisinturnmakesthetaskoftestingthemarkete¢ciencyevenmorecomplicatedandwouldrequireequilibriumassetpricingmodelsthatallowedforinformationandtradingcostsinmarketswithmanydi¤erenttradersandwithnon-convergentbeliefs.Inviewofthesedi¢cultiessomeadvocatesoftheEMHhaveoptedforatrade-basednotion,andde…nemarketsase¢cientifitwouldnotbepossiblefortheinvestors“...toearnabove-averagereturnswithoutacceptingabove-averagerisks.”Malkiel(2003)(seep.60).Thisnotioncantakeaccountofinformationandtransac-tioncostsanddoesnotinvolvetestingjointhypotheses.Butthisisfarremovedfromthebasicideaofmarketsase¢cientallocatorsofcapitalinvestmentacrosscountries,industriesand…rms.Beatingthemarketasatestofmarkete¢ciencyalsoposesnewchallenges.Whilstitiscertainlypossibletoconstructtradingstrategies(inclusiveoftransactioncosts)withSharperatiosthatexceedthoseofthemarketportfoliosexpost,suchevidenceareunlikelytobeconvincingtotheadvocatesoftheEMH.Itcouldbearguedthattheyarecarriedoutwiththebene…tofhindsight,andareunlikelytoberepeatedinrealtime.Inthisconnectionsthefollowingconsiderationswouldneedtobeborninmind:(a)Datamining/datasnooping(PesaranandTimmermann(2005)).(b)Structuralchangeandmodelinstability(choiceofobservationwindow).(c)Thepositiverelationshipthatseemtoexistbetweentransactioncostsandpredictability.(d)Marketvolatilityandlearning.(e)The‘Beatthemarket’testisnotthathelpfuleitherinsheddinglightonthenatureandtheextentofmarketine¢ciencies.Amorestructuralapproachwouldbedesirable.8TheoreticalfoundationsoftheEMHAtthecoreoftheEMHliesthefollowingthreebasicpremises:23 1.Investorrationality:Itisassumedthatinvestorsarerational,inthesensethattheycorrectlyupdatetheirbeliefswhennewinformationisavailable.2.Arbitrage:Individualinvestmentdecisionssatisfythearbitragecondition,andtradedecisionsaremadeguidedbythecalculusofthesubjectiveexpectedutilitytheoryalaSavage.3.Collectiverationality:Di¤erencesinbeliefsacrossinvestorscanceloutinthemarket.Toillustratehowthesepremisesinteract,supposethatatthestartofperiod(day,week,month)tthereareNttraders(investors)thatareinvolvedinactofarbitragebetweenastockandasafe(risk-free)asset.Denotetheone-periodholdingreturnsonthesetwoassetsbyRt+1andrft,respectively.Followingasimilarlineofargumentasinsection6.2,thearbitrageconditionfortraderiisgivenby^EiRt+1�rftj it=it+itwhere^EiRt+1�rftj itishis/hersubjectiveexpectationsoftheexcessreturn,Rt+1�rfttakenwithrespecttotheinformationset it= it[twheretisthecomponentoftheinformationwhichispubliclyavailable,it�0representstrader’sriskpremium,andit�0isher/hisinformationandtradingcostsperunitoffundsinvested.Intheabsenceofinformationandtradingcosts,itcanbecharacterizedintermsofthetrader’sutilityfunction,ui(cit),wherectishis/herrealconsumptionexpendituresduringtheperiodttot+1,andisgivenbyit=^EiRt+1�rftj it=�^Covi(mi;t+1;Rt+1j it) ^Ei(mi;t+1j it)where^Covi(:j it)isthesubjectivecovarianceoperatorconditiononthetrader’sinformationset, it,mi;t+1=iu0i(ci;t+1)=u0i(cit),whichisknownasthe‘stochasticdiscountfactor’,u0i(:)isthe…rstderivativeoftheutilityfunction,andiishis/herdiscountfactor.Theexpectedreturnscoulddi¤eracrosstradersduetothedi¤erencesinthetheirperceivedconditionalprobabilitydistributionfunctionofRt+1�rft,thedi¤er-encesintheirinformationsets, it,thedi¤erencesintheirriskpreferences,and/orendowments.Undertherationalexpectationshypothesis^EiRt+1�rftj it=ERt+1�rftj it24 placereasonablyfast,therewillstillbeperiodsofturmoilwheremarketparticipantswillbesearchinginthedark,tryingandexperimentingwithdi¤erentmodelsofRt+1�rftoftenwithmarkeddeparturesfromthecommonrationaloutcomes,givenbyERt+1�rftjt.HerdingandcorrelatedbehaviouracrosssomeofthetraderscouldalsoleadtofurtherdeparturesfromtheequilibriumREsolution.InfacttheobjectiveprobabilitydistributionofRt+1�rftmightitselfbea¤ectedbymarkettransactionsbasedonsubjectiveestimates^EiRt+1�rftj it.Marketine¢cienciesprovidefurthersourcesofstockmarketpredictabilitybyintroducingawedgebetweena‘correct’exantemeasureERt+1�rftjt,anditsaverageestimatebymarketparticipants,whichwewriteasNtXi=1wit^EiRt+1�rftj itwherewitisthemarketshareoftheithtrader.Letwt=NtXi=1wit^EiRt+1�rftj it�ERt+1�rftjtandnotethatitcanalsobewrittenas(sinceNtXi=1wit=1)wt=NtXi=1witit(20)whereit=^EiRt+1�rftj it�ERt+1�rftjt(21)itmeasuresthedegreetowhichindividualexpectationsdi¤ersfromthecorrect(butunobservable)expectations,ERt+1�rftjt.Anon-zeroitcouldarisefromindividualirrationality,butnotnecessarilyso.Rationalindividualsfacedwithanuncertainenvironment,costlyinformationandlimitationsoncomputingpowercouldrationallyarriveattheirexpectationsoffuturepricechangesthatwithhindsightdi¤erfromthecorrectones.5Anon-zeroitcouldalsoariseduetodisparityof 5ThisisinlinewiththepremiseoftherecentpaperbyAngeletos,Lorenzoni,andPavan(2010)whomaintaintheaxiomofrationality,butallowfordispersedinformationandthepossibilityofinformationspilloversinthe…nancialmarketstoexplainmarketine¢ciencies.26 informationacrosstraders(includinginformationasymmetries),andheterogeneouspriorsduetomodeluncertaintyorirrationality.Nevertheless,despitesuchindividualdeviations,wtwhichmeasurestheextentofmarketorcollectiveine¢ciency,couldbequitenegligible.WhenNtissu¢cientlylarge,individual‘irrationality’cancanceloutatthelevelofthemarket,solongasit,i=1;2;:::;Ntarenotcrosssectionallystronglydependent,andnosingletraderdominatesthemarket,inthesensethatwit=O(N�1t)atanytime.6Undertheseconditionsateachpointintime,t,theaverageexpectedexcessreturnsacrosstheindividualtradersconvergesinquadraticmeanstotheexpectedexcessreturnofarepresentativetrader,namelywehaveNtXi=1wit^EiRt+1�rftj itq:m:!ERt+1�rftjt,asNt!1Insuchperiodstherepresentativeagentparadigmwouldbeapplicableandpre-dictabilityofexcessreturnwillbegovernedsolelybychangesinbusinesscyclecon-ditionsandotherpubliclyavailableinformation.7However,inperiodswheretraders’individualexpectationsbecomestronglycor-related(sayastheresultofherdingorcommonover-reactionstodistressingnews)wtneednotbenegligibleeveninthickmarketswithmanytraders;andmarketine¢cienciesandpro…tableopportunitiescouldprevail.Marketscouldalsodisplayine¢ciencieswithoutexploitablepro…tableopportunitiesifwtisnon-zerobutthereisnostablepredictablerelationshipbetweenwtandbusinesscycleorothervariablesthatareobservedpublicly.Theevolutionandcompositionofwtcanalsohelpinsheddinglightonpossiblebubblesorcrashesdevelopinginassetmarkets.Bubblestendtodevelopintheafter-mathoftechnologicalinnovationsthatarecommonlyacknowledgedtobeimportant,butwithuncertainoutcomes.Theemergingcommonbeliefsaboutthepotentialad-vantagesofthenewtechnologyandthedi¢cultiesindividualagentsfaceinlearninghowtorespondtothenewinvestmentopportunitiescanfurtherincreasethegapbetweenaveragemarketexpectationsofexcessreturnsandtheassociatedobjectiverationalexpectationsoutcome.Similarcircumstancescanalsoprevailduringacrashphaseofthebubblewhentraderstendtomoveintandemtryingtoreducetheirriskexposuresallatthesametime.Therefore,onewouldexpectthatduringbubblesand 6Conceptsofweakandstrongcrosssectiondependencearede…nedanddiscussedinChudik,Pesaran,andTosetti(2010).7Theheterogeneityofexpectationsacrosstraderscanalsohelpinexplaininglargetradingvolumeobservedinthe…nancialmarkets;afeaturewhichhasproveddi¢culttoexplaininrepresentativeagentassetpricingmodels.ButseeScheinkmanandXiong(2003)whorelatetheoccurrenceofbubblesandcrashestochangesintradingvolume.27 2.Doyouexpectthemarketpricenextperiodto(a)stayaboutthelevelitiscurrently,(b)fall,or(c)rise?Incaseswherethemarketisequilibratingwewouldexpectacloseassociationbetweentheproportionofrespondentswhoselect1aand2a,1band2b,and1cand2c.Butinperiodsofbubbles(crashes)onewouldexpectalargeproportionofrespondentswhoselect1b(1c)toalsoselect2c(2b).Insituationswheretheequilibratingprocessiswellestablishedandcommonlyunderstood,thesecondquestionisredundant.Forexample,ifanindividualstatesthattheroomtemperatureistoohigh,itwillbeunderstoodthathe/shewouldpreferlessheating.Thesameisnotapplicableto…nancialmarketsandhenceresponsestobothquestionsareneededforabetterunderstandingoftheoperationsofthemarketsandtheirevolutionovertime.9Exploitingpro…tableopportunitiesinpracticeIn…nancialmarketstheEMHisrespectedbutnotworshipped.Itisrecognizedthatmarketsarelikelytobee¢cientmostofthetimebutnotallthetime.Ine¢cienciescouldariseparticularlyduringperiodsofimportantinstitutionalandtechnologicalchanges.Itisnotpossibletoknowwhenandwheremarketine¢cienciesariseinadvance-butitisbelievedthattheywillarisefromtimetotime.Markettraderslovevolatilityasitsignalsnewsandchangewithpro…tpossibilitiestoexploit.Iden-ti…cationofexploitablepredictabilitytendtobefullydiversi…edacrossmarketsforbonds,equitiesandforeignexchange.Misalignmentsacrossmarketsfordi¤erentassetsandindi¤erentcountriesoftenpresentthemostimportantopportunities.Examplesincludestatisticalarbitrageandglobalmacroarbitragetradingrules.Predictabilityandmarketliquidityareoftencloselycorrelated;lessliquidmar-ketsarelikelytobemorepredictable.Marketpredictabilityandliquidityneedtobejointlyconsideredindevelopingpro…tabletradingstrategies.ReturnforecastingmodelsusedinpracticetendtoberecursiveandadaptivealongthelinesdevelopedinPesaranandTimmermann(1995)andrecentlyreviewedinPesaranandTim-mermann(2005).Recursivemodelling(RM)approachisalsoinlinewiththemorerecentdevelopmentsinbehavioural…nance.TheRMapproachaimsatminimizingthee¤ectofhindsightanddatasnooping(aproblemthata­ictsallexpostreturnregressions),andexplicitlydesignedtotakeaccountofthepotentialinstabilityofthereturnregressionsovertime.Forexample,PesaranandTimmermann(1995)…ndthattheswitchingtradingrulemanagestobeatthemarketonlyduringperiodsofhighvolatilitywherelearningmightbeincompleteandmarketsine¢cient.29 Thesecondrelatestotheaveragedeviationsofindividualtraders’sexpectationsfromthe"correct"unknownexpectations,asmeasuredbywtandde…nedby(20)).Asnotedearlierthiscomponentcouldvaryconsiderablyovertimeandneednotbere-latedtobusinesscyclefactors.Ittendstobelargeduringperiodsof…nancialcrisiswhencorrelationofmis-pricingacrosstradersrise,andtendtobenegligibleduringperiodsofmarketcalmwhencorrelationsarelow.Overthepastthreedecadesmuchoftheresearchin…nanceandmacroeconomicshasfocussedonmodellingoft,andbycomparisonlittleattentionhasbeenpaidtowt.Thisisclearlyanimportantareaforfutureresearch.Ourdiscussionsalsopointtoanumberofrelatedareasforfurtherresearch.Thereareclearly-Limitstorationalexpectations(foranearlytreatmentseePesaran(1987)),alsoseetherecentpaperonSurveyExpectationsbyPesaranandWeale(2006).-Limitstoarbitrageduetoliquidityrequirementsandinstitutionalconstraints.-Herdingandcorrelatedbehaviourwithnoisetradersenteringmarketsduringbullperiodsanddesertingduringbearperiods.Behavioral…nance,complexitytheoryandtheAdaptiveMarketsHypothesisre-centlyadvocatedbyLo(2004)alltry,inonewayoranother,toaddresstheabovesourcesofthedeparturesfromtheEMH.Someoftherecentdevelopmentsinbehav-ioural…nancearereviewedinBaberisandThaler(2003).FarmerandLo(1999)focusontherecentresearchthatviewsthe…nancialmar-ketsfromabiologicalperspectiveand,speci…cally,withinanevolutionaryframeworkinwhichmarkets,instruments,institutions,andinvestorsinteractandevolvedynam-icallyaccordingtothe‘law’ofeconomicselection.Underthisview,…nancialagentscompeteandadapt,buttheydonotnecessarilydosoinanoptimalfashion.Specialcareshouldalsobeexercisedinevaluationofreturnpredictabilityandtradingrules.Tominimizethee¤ectsofhindsightinsuchanalysisrecursivemod-ellingtechniquesdiscussedinPesaranandTimmermann(1995),PesaranandTim-mermann(2000)andPesaranandTimmermann(2005)seemmuchmoreappropriatethanthereturnregressionsona…xedsetofregressors/factorsthatareestimatedexpostonhistoricaldata.ReferencesAngeletos,G.,G.Lorenzoni,andA.Pavan(2010).Beautycontestsandirrationalexuberance:Aneoclassicalapproach.WorkingPaper15883NationalBureauOfEconomicResearch.31 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