Downside Risk and the Momentum Effect Andrew Ang Columbia University and NBER Joseph Chen University of Southern California Yuhang Xing Columbia University First Version: 31 Aug, 2001 This Version: 10 Dec, 2001 JEL Classification: C12, C15, C32, G12 Keywords: asymmetric risk, cross-sectional asset pricing, downside correlation, downside risk, momentum effect The authors would like to thank Brad Barber, Alon Brav, Geert Bekaert, John Cochrane, Randy Cohen, Kent Daniel, Bob Dittmar, Rob Engle, Cam Harvey, David Hirschleiffer, Qing Li, Terence Lim, Bob Stambaugh, Akhtar Siddique and Zhenyu Wang. We especially thank Bob Hodrick for detailed comments. We thank seminar participants at Columbia University, the Five Star Conference at NYU and USC for helpful comments. This paper is funded by a Q-Group research grant. Andrew Ang: [email protected]; Joe Chen: [email protected]; Yuhang Xing: [email protected].
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Downside Risk and the Momentum Effectpeople.stern.nyu.edu/rengle/corrfac.pdf · 2002-06-26 · Abstract Stocks with greater downside risk, which is measured by higher correlations
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little discernablepatternsin Section2.3. We provide aninterpretationof our resultsin Section
2.4.
2.1 Downside and Upside Correlations
In Table(1), weshow thatstockswith highdownsiderisk with themarkethavehigherexpected
returnsthan stockswith low downsiderisk. We measuredownsiderisk and upsiderisk by
downsideconditionalcorrelations,@BA , and upsideconditionalcorrelations,@ � , respectively.
Wedefinetheseconditionalcorrelationsas:
@ A �corr
���� �0� 25476 �DC 25476 �/E 25476 �@ � �
corr���� �0� 2�476 �>C 2�476 �GF 25476 � � (3)
4
where���� �
is theexcessstockreturn, 25476 � is theexcessmarket return,and 25476 � is themean
excessmarket return.
To ensurethat we do not capturethe endogenousinfluenceof contemporaneouslyhigh
returnson higher-ordermoments,we form portfoliossortedby pastreturncharacteristicsand
examineportfolio returnsover a future period. To sort stocksbasedon downsideandupside
correlationsat a point in time, we calculate@ A and @ � usingdaily continuouslycompounded
excessreturnsover thepreviousyear. We first rankstocksinto deciles,andthenwe calculate
theholdingperiodreturnover thenext monthof thevalue-weightedportfolio of stocksin each
decile. We rebalancetheseportfolios eachmonth. Appendix A provides further detailson
portfolio construction.
PanelsA andB of Table(1) list monthlysummarystatisticsof theportfoliossortedby @ Aand @ � , respectively. We first examinethe @ A portfolios in Panel A. The first column lists
the meanmonthly holding period returnsof eachdecile portfolio. Stockswith the highest
pastdownsidecorrelationshave the highestreturns. In contrast,stockswith the lowestpast
downsidecorrelationshave the lowestreturns.Going from portfolio 1, which is theportfolio
of lowest downsidecorrelations,to portfolio 10 which is the portfolio of highestdownside
Panel B of Table (1) shows the summarystatisticsof stockssortedby @ � . In contrast
to stockssortedby @ A , there is no discernablepatternbetweenmeanreturnsand upside
correlations.However, thepatternsin the H ’s,market capitalizationsandbook-to-market ratios
of stockssortedby @ � are similar to the patternsfound in @ A sorts. In particular, high @ �stocksalsotendto havehigherbetas,tendto belargestocks,andtendto begrowth stocks.The
last two columnslist thepost-formation@ � and H � statistics.Here,both @ � and H � increase
monotonicallyfrom decile1 to 10,but portfolio cutsby @ � do not giveany patternin expected
returns.
In summary, Table (1) shows that assetswith higher downsidecorrelationshave higher
returns.This resultis consistentwith modelsin which themarginal investoris morerisk-averse
on the downsidethanon the upside,anddemandhigherexpectedreturnsfor bearinghigher
downsiderisk.
2.2 Coskewness and Cokurtosis
Table(2) shows thatstockssortedby pastcoskewnessandpastcokurtosisdo not produceany
Table (1) shows that portfolios with higher downsidecorrelationhave higher H ’s and Table
(4) shows that market loadingsincreasewith downsiderisk. This raisesthe issuethat the
phenomenonof increasingreturnswith increasing@ A maybedueto arewardfor bearinghigher
exposureson H , ratherthan for greaterexposuresto downsiderisk. To investigatethis, we
performa sort on @ A , after controlling for H . Eachmonth,we placehalf of the stocksbased
on their H ’s into a low H groupandthe otherhalf into a high H group. Then,within eachHgroup,werankstocksbasedon their @ A into threegroups:a low @ A group,amedium@ A group
anda high @ A group,with thecutoffs at 33.3%and66.7%.This sortingprocedurecreatessix
portfoliosin total.
We calculatemonthly value-weightedportfolio returnsfor eachof these6 portfolios, and
reportthesummarystatisticsin thefirst panelof Table(5). Within thelow H group,theaverage
returnsincreasefromthelow @ A portfolio to thehigh @ A portfolio,with anannualizeddifference
of 2.40%(0.20%permonth).Moving acrossthelow H group,meanreturnsof the @ A portfolios
increase,while thebetaremainsflat ataroundH = 0.66.In thehigh H group,weobservethatthe
returnalsoincreaseswith @ A . Thedifferencein returnsof thehigh @ A andlow @ A portfolios,
within the high H group, is 3.24%per annum(0.27%per month), with a t-statisticof 1.98.
However, the H decreaseswith increasing@ A . Therefore,the higher returnsassociatedwith
higherdownsiderisk arenot rewardsfor bearinghighermarket risk, but arerewardsfor bearing
higherdownsiderisk.
In PanelB of Table(5), for each@BA group,we take the simpleaverageacrossthe two Hgroupsandcreatethreeportfolios,whichwecall the H -balanced@ A portfolios.Moving fromthe
H -balancedlow @ A portfolio to the H -balancedhigh @ A portfolio, meanreturnsmonotonically
increasewith @ A . This increaseis accompaniedby amonotonicdecreasein H from H = 0.94to
H = 0.87. HenceH is not contributing to thedownsiderisk effect, sincewithin eachH group,
hasa small averagereturnper month(0.10%)andis not statisticallysignificant. In contrast,
the WML factorhasthe highestaveragereturn,over 0.90%per month. However, unlike the
otherfactors,WML is constructedusingequal-weightedportfolios,ratherthanvalue-weighted
portfolios.
We list thecorrelationmatrix acrossthevariousfactorsin PanelB of Table(6). CMC has
a slightly negative correlationwith the market portfolio of –16%,a magnitudelessthan the
correlationof SMB with the market (32%) andlessin absolutevaluethanthe correlationof
HML with themarket(–40%).CMC is positivelycorrelatedwith WML (35%).Thecorrelation2 An alternative sortingprocedureis to performindependentsortson p and qsr , andtake the intersectionsto
bethe6 p /qsr portfolios. This procedureproducesa similar result,but givesanaveragemonthlyreturnof 0.22%
(t-stat= 1.88),which is significantat the 10% level. This procedureproducespoor dispersionon qkr becausepand qsr arehighly correlated,sothe independentsortplacesmorefirms in the low p /low qsr andthehigh p /highqkr portfolios,thanin thelow p /high qsr andthehigh p /low qkr portfolios.Oursortingprocedurefirst controlsforp andthensortson qsr , creatingmuchmorebalancedportfolioswith greaterdispersionover qkr .
12
matrix shows that SKS and CMC have a correlationof –3%, suggestingthat asymmetric
downsidecorrelationrisk hasa differenteffect thanskewnessrisk.
Table(6) showsthatCMC is highly negatively correlatedwith SMB (–64%).To allay fears
thatCMC is not merelyreflectingtheinverseof thesizeeffect,weexaminetheindividualfirm
jointly equalto zero,fails to rejectwith ap-valueof 0.49.
Downsiderisk portfolioswith low @ A have negative loadingson CMC. That is, thelow @ Aportfoliosarenegatively correlatedwith theCMC factor. SincetheCMC factorshortslow @ Astocks,many of the stocksin the low @ A portfolios have shortpositionsin the CMC factor.
Similarly, thehigh @ A portfolio hasapostitiveloadingonCMC because,by construction,CMC
goeslonghigh @ A stocks.
Whenweaugmenttheregressionin equation(9) with theFama-Frenchfactors,theintercept
coefficientsbo�
aresmaller. However, thefit of thedatais notmuchbetter, with theadjustedx 9 ’sthatarealmostidenticalto theoriginal modelof around90%. While theloadingsof SMB and
HML arestatisticallysignificant,theseloadingsstill go the wrong way, as they do in Table3 SMB is long morefirms thanit is shortsincethebreakpointsaredeterminedusingmarket capitalizationsof
to 4 =3. Figure(1) alsoshows the H ’s and @ A of themomentumportfolios. While theaverage
returnsincreasefrom decile1 to decile10, thepatternsof betaareU-shaped.In contrast,the
@ A of the decilesincreasegoing from the losersto the winners,exceptat the highestwinner
decile. Therefore,the momentumstrategiesgenerallyhave a positive relationwith downside
risk exposure.4 We now turn to formal estimationsof the relationbetweendownsiderisk and4 Ang andChen(2001)focuson correlationasymmetriesacrossdownsideandupsidemoves,ratherthanthe
level of downsideandupsidecorrelatio. They find that, relative to a normaldistribution, loserportfolios have
greatercorrelationasymmetrythan winner portfolios, even thoughpastwinner stockshave a higher level of
in correlationscanalsobeproducedby economieswith frictionsandhiddeninformation(Hong
and Stein, 2001) or with agentsfacing binding wealth constraints(Kyle and Xiong, 2001).
Similarly, thesestudiesdo not model large cross-sectionsof heterogeneousassets.Which of
theseexplanationsbestexplains the driving mechanismbehindcross-sectionalvariationsin
downsiderisk remainsto beexplored.
26
Appendix
A Data and Portfolio Construction
DataWe usedatafrom the Centerfor Researchin SecurityPrices(CRSP)to constructportfolios of stockssortedbyvarioushighermomentsof returns.We confineour attentionto ordinarycommonstockson NYSE, AMEX andNASDAQ, omittingADRs,REITs,closed-endfunds,foreignfirmsandothersecuritieswhichdonothaveaCRSPsharetypecodeof 10or11. Weusedaily returnsfrom CRSPfor theperiodcoveringJanuary1st,1964to December31st,1999,includingNASDAQ datawhich is only availablepost-1972.We usetheone-monthrisk-freeratefromCRSPandtake CRSP’s value-weightedreturnsof all stocksasthemarketportfolio. All our returnsareexpressedascontinuouslycompoundedreturns.
Higher Moment PortfoliosWe constructportfolios basedon correlationsbetweenassetÇ ’s excessreturn È�É and the market’s excessreturnÈ�Ê conditional on downside moves of the market (qkr ) and on upsidemoves of the market (qkË ). We alsoconstuctportfoliosbasedoncoskewness,cokurtosis,p , p conditionalondownsidemarketmovements(pÌr ), andpconditionalonupsidemarketmovements(pwË ). At thebeginningof eachmonth,wecalculateeachstock’smomentmeasuresusingthe pastyear’s daily log returnsfrom the CRSPdaily file. For the momentswhich conditionondownsideor upsidemovements,we defineanobservationat time Í to bea downside(upside)market movementif theexcessmarket returnat Í is lessthanor equalto (greaterthanor equalto) theaverageexcessmarket returnduringthepastoneyearperiodin consideration.Werequireastockto haveat least220observationsto beincludedin thecalculation.Thesemomentmeasuresarethenusedto sortthestocksinto decilesandavalue-weightedreturnis calculatedfor all stocksin eachdecile.Theportfoliosarerebalancedmonthly.
SMB, HML, SKS and WML Factor ConstructionTheFamaandFrench(1993)factors,SMB andHML, arefrom thedatalibrary at KennethFrench’swebsiteathttp://web.mit.edu/kfrench/www/datalibrary.html.
Following Carhart(1997),we constructWML (calledPR1YRin his paper)astheequally-weightedaverageof firms with thehighest30 percenteleven-monthreturnslaggedonemonthminustheequally-weightedaverageof firms with the lowest30 percenteleven-monthreturnslaggedonemonth. In constructingWML, all stocksinNYSE,AMEX andNASDAQ areusedandportfoliosarerebalancedmonthly.
Theconstructionof CMC is detailedin Section3.2.
Momentum PortfoliosTo constructthe momentumportfolios of JegadeeshandTitman (1993),we sort stocksinto portfolios basedontheir returnsover thepast6 months.We considerholdingperiodof 3, 6, 9 and12 months.This procedureyields4 strategiesand40 portfoliosin total. We illustratetheconstructionof theportfolioswith theexampleof the’6-6’strategies. To constructthe ’6-6’ deciles,we sortour stocksbaseduponthepastsix-monthsreturnsof all stocksin NYSEandAMEX. Eachmonth,anequal-weightedportfolio is formedbasedonsix-monthsreturnsendingonemonthprior. Similarly, equal-weightedportfoliosareformedbasedon pastreturnsthatendedonemonthsprior,threemonthsprior, andsoonupto six monthsprior. Wethentakethesimpleaverageof six suchportfolios.Hence,ourfirst momentumportfolio consistsof Î>Ï^Ð of thereturnsof theworstperformersonemonthago,plus ÎDÏ�Ð of thereturnsof theworstperformerstwo monthsago,etc.
27
Liquidity Factor and Liquidity BetasWe follow PastorandStambaugh(2001)to constructanaggregateliquidity measure,Ñ . Stockreturnandvolumedataareobtainedfrom CRSP. NASDAQ stocksareexcludedin theconstructionof theaggregateliquidity measure.Theliquidity estimate,Ò É�Ó Ô , for anindividualstockÇ in monthÍ is theordinaryleastsquares(OLS) estimateof Ò É�Ó Ôin thefollowing regression:
È�ÕÉ�Ó Ö Ëk× Ó Ô�Ø¹Ù ÉªÓ ÔsÚ7Û�É�Ó Ô�ÜÈ�É�Ó Ö�Ó Ô�Ú�ÒoÉªÓ Ô Ý)ǪÞDß È�ÕÉªÓ Ö�Ó Ô à ÉªÓ Ö�Ó ÔkÚ7á�É�Ó Ö Ëk× Ó Ô âäã Ø Î>â¢å�å¢å!â�æ å (A-1)
In equation(A-1), ÜÈ ÉªÓ Ö�Ó Ô is theraw returnon stockÇ on day ã of monthÍ , È ÕÉ�Ó Ö�Ó Ô Ø È É�Ó Ö�Ó Ôèç È ÊzÓ Ö�Ó Ô is thestockreturnin excessof themarket return,and à É�Ó Ö�Ó Ô is thedollar volumefor stock Ç on day ã of month Í . Themarket returnon dayon day ã of monthÍ , È ÊzÓ Ö�Ó Ô , is takenasthereturnon theCRSPvalue-weightedmarketportfolio. A stock’sliquidity estimate,Ò É�Ó Ô , is computedin a givenmonthonly if thereareat least15 consecutiveobservations,andifthestockhasa month-endsharepricesof greaterthan$5 andlessthan$1000.
Theaggregateliquidity measure,Ñ , is computedbasedontheliquidity estimates,ÒoÉªÓ Ô , of individualfirmslistedon NYSE andAMEX from August1962to December1992.Only theindividual liquidity estimatesthatmeettheabove criteria is used.To constructthe innovationsin aggregateliquidity, we follow PastorandStambaughandfirst form thescaledmonthlydifference:
é�êÒ Ô Ø ë Ôë ×ÎìíÉ�î ×ï Ò ÉªÓ Ôðç Ò É�Ó Ô rk×0ñ â (A-2)
whereì
is thenumberof availablestocksat monthÍ , ë Ô is thetotaldollar valueof theincludedstocksat theendof month Í ç Î , andë × is thetotal dollar valueof thestocksat theendof July 1962.Theinnovationsin liquidityarecomputedastheresidualsin thefollowing regression:é�êÒoÔ Ø¹ò Ú¤ó éôêÒoÔ rk× Ú¤õ ï ë Ô Ï ë × ñ êÒoÔ rk× Ú÷ö�Ô å (A-3)
Finally, theaggregateliquidity measure,ÑèÔ , is takento bethefitted residuals,ÑðÔ Ø êö�Ô .To calculatethe liquidity betasfor individual stocks,at the endof eachmonthbetween1968and1999,we
identify stockslistedon NYSE,AMEX andNASDAQ with at leastfive yearsof monthlyreturns.For eachstock,we estimatea liquidity beta,pðøÉ , by runningthefollowing regressionusingthemostrecentfive yearsof monthlydata:
È É�Ó Ô Ø psùÉ Ú÷p øÉ Ñ Ô Ú÷pðúÉüûþý�ÿ Ô Ú÷p��É � û�� Ô Ú p��É�� û Ñ Ô Ú¤á ÉªÓ Ô â (A-4)
whereÈgÉªÓ Ô denotesassetÇ ’s excessreturnand ÑèÔ is theinnovationin aggregateliquidity.
Macroeconomic VariablesWe usethe following macroeconomicvariablesfrom FederalReserve Bank of St. Louis: the growth ratein theindex of leadingeconomicindicators(LEI), thegrowthratein theindex of HelpWantedAdvertisingin Newspapers(HELP),thegrowth rateof total industrialproduction(IP), theConsumerPriceIndex inflationrate(CPI),thelevelof theFedfundsrate(FED),andthetermspreadbetweenthe10-yearT-bondsandthe3-monthsT-bills (TERM).All growth rates(including inflation) arecomputedasthe differencein logs of the index at times Í and Í ç Î ,whereÍ is monthly.
B Time-Aggregation of Coskewness and CokurtosisSincewe computeall of themonthlyhighermomentsmeasuresusingdaily data,theproblemof time aggregationmayexist for someof thehighermoments.Assumingthatreturnsaredrawn from infinitely divisibledistributions,centralmomentsat first andsecondorder can scale. That is, an annualestimateof the mean � and volatility� can be estimatedfrom meansand volatilities estimatedfrom daily data � Ö and � Ö , by the time aggregatedrelations� Ø ���� Ö and � Ø � �� � Ö . Hence,daily measuresfor secondordermomentsareequivalentto theircorrespondingmonthly measures.We now prove that daily coskewnessandcokurtosisdefinedin equations(5)and(6) areequivalentto monthlycoskewnessandcokurtosis.
With theassumptionof infinitely divisibledistributions,cumulantsscalebut notcentralmoments(“cumulantscumulate”).Thecentralmoment,��� , of � is definedas:
� � Ø �r �ï � ç � ×!ñ � ã�� for È Ø Dâ��>â��>â¢åaå åaâ (B-1)
28
integratingover thedistribution of returns� , and � × Ø�� ï � ñ . Theproductcumulants,��� , arethecoefficientsintheexpansion:
� ï Í ñ Ø��! #"���î ù � �
ï ÇªÍ ñ �È%$ â (B-2)
where� ï Í ñ is themomentgeneratingfunctionof a univariatenormaldistribution. Thebivariatecentralmoment,� �'& , is definedas
Thelastequationfollows from thefact that � ×)× Ø � × × ØC� ï á?DGÓ E*á?F�Ó E ñ Ø � , where á�É�Ó Ô Ø ÈgÉªÓ Ô çHG É ç p(É ûþý�ÿ Ô , istheresidualfrom theregressionof thefirm Ç ’sexcessstockreturnÈ�É onthecontemporaneousmarketexcessreturn,and á�Ê�Ó Ô is theresidualfrom theregressionof themarketexcessreturnon a constant.
29
C Computing Hansen-Jagannathan (1997) Distances and P-values
ï ì ç ý ç Î ñ�M - × statistics.Theweightsaretheì ç ý ç Î
non-zeroeigenvaluesof:
N Ø � 87OQP 87SRO T ç P 87O æ O ï æVUO P O æ O ñ rk× æVUO P87(RO rk× P 87O � 87(RO â
where� 87O and P
87O aretheupper-triangularCholesky decompositionsof� O and P O respectively, and æ O Ø�WYX�ZW1[ .
JagannathanandWang(1996)show thatN
hasexactlyì ç ý ç Î positive eigenvaluesÙ × â¢å¢å�å!â Ù í r]\zrs× . The
asymptoticdistributionof theHJdistancemetricis:
ÿ^I ï �_K ñ -a`í rb\zrk×c Ù c M - ×
asÿ `ed. We simulatetheHJstatistic100,000timesto computetheasymtoticp-valueof theHJdistance.
To calculatea small samplep-valuefor the HJ distance,we assumethat the linear factormodelholdsandsimulatea datageneratingprocess(DGP) with 432 observations,the samelengthasin our samples.The DGPtakestheform:
È ÉªÓ Ô Ø ÈfÔ rk× Ú÷p�UÉ � Ô Ú¤á ÉaÔ â (C-1)
whereÈ É�Ó Ô is thereturnonthe Ç -th portfolio, È fÔ is therisk-freerate,p É is an ûgI Î vectorof factorloadings,and � Ôis the ûhI Î vectorof factors.We assumethattherisk-freerateandthefactorsfollow a first-orderVAR process.Let i Ô Ø ï ÈfÔ â+�(Ô ñ U , andi Ô follows:
process.In eachsimulation,wegenerate432observationsof factorsandtherisk-freeratefrom theVAR systeminequation(C-2). For theportfolio returns,we usethesampleregressioncoefficient of eachportfolio returnon thefactors,
êp É , asour factorloadings.Weassumetheerrortermsof thebaseassets,á Ô , follow IID multivariatenormaldistributionswith meanzeroandcovariancematrix,
êlk� ç êp U êlkm êp , whereêlk� is thecovariancematrix of theassets
andêl m is thecovariancematrixof thefactors.For eachmodel, we simulate5000 time-seriesas describedabove and computethe HJ distancefor each
simulationrun. Wethencountthepercentageof theseHJdistancesthatarelargerthantheactualHJdistancefromrealdataanddenotethis ratio empiricalp-value. For eachsimulationrun, we alsocomputethe theoreticp-valuewhich is calculatedfrom theasymptoticdistribution.
30
References[1] Ahn, D., J.Conrad,andR. Dittmar, 2001,“Risk AdjustmentandTradingStrategies,” forthcomingReview of
Financial Studies.
[2] Ang, A., andJ.Chen,2001,“AsymmetricCorrelationsof EquityReturns,” forthcomingJournal of FinancialEconomics.
[3] Ang, A., andM. Piazzesi,2001,“A No-ArbitrageVectorAutoregressionof TermStructureDynamicswithMacroeconomicandLatentVariables,” working paper, ColumbiaUniversity.
[4] Bansal,R., D. A. Hsieh,andS. Viswanathan,1993,“A New Approachto InternationalArbitragePricing,”Journal of Finance, 48,1719-1747.
[5] Barberis,N., A. Shleifer, and R. Vishny, 1998, “A Model of InvestorSentiment,” Journal of FinancialEconomics, 49,3, 307-343.
[7] Bawa, V. S., andE. B. Lindenberg, 1977,“Capital Market Equilibrium in a Mean-Lower Partial MomentFramework,” Journal of Financial Economics, 5, 189-200.
[8] Bekaert,G., R. J. Hodrick, andD. A. Marshall,1997,“The Implicationsof First-OrderRisk AversionforAssetMarketRisk Premiums,” Journal of Monetary Economics, 40,3-39.
[9] Carhart,M. M. 1997,“On Persistencein Mutual FundPerformance,” Journal of Finance, 52,1, 57-82.
[16] Daniel,K., D. Hirshleifer, andA. Subrahmanyam,1998,“InvestorPsychologyandSecurityMarket Under-andOverreactions,” Journal of Finance, 53,1839-1886.
[17] DeBondt,WernerF. M. and Thaler, R. H., 1987, “Further Evidenceon InvestorOverreactionand StockMarketSeasonality,” Journal of Finance, 42,557-581.
[18] Dittmar, R.,2001,“NonlinearPricingKernels,KurtosisPreference,andEvidencefrom theCross-SectionofEquityReturns,” forthcomingJournal of Finance.
[22] Gervais,S.,R. Kaniel,andD. Mingelgrin,2001,“The High VolumeReturnPremium,” forthcomingJournalof Finance.
[23] Ghysels,E.,1998,“On StableFactorStructurein thePricingof Risk: Do Time-VaryingBetasHelpor Hurt?”Journal of Finance, 53,457-482.
31
[24] Gibbons, M. R., S. A. Ross, and J. Shanken, 1989, “A Test of the Efficiency of a Given Portfolio,”Econometrica, 57,5, 1121-1152.
[25] Grundy, B. D., andS. J. Martin, 2001,“Understandingthe Natureof Risksandthe Sourcesof RewardstoMomentumInvesting,” Review of Financial Studies, 14,1, 29-78.
[34] Jagannathan,R., andZ. Wang,1996,“The ConditionalCAPM andtheCross-Sectionof ExpectedReturns,”Journal of Finance, 51,3-53.
[35] Jegadeesh,N., andS.Titman,1993,“Returnsto BuyingWinnersandSellingLosers:Implicationsfor StockMarketEfficiency,” Journal of Finance, 48,65-91.
[36] Jegadeesh,N., andS. Titman, 2001,“Profitability of MomentumStrategies: An Evaluationof AlternativeExplanations,” Journal of Finance, 56,2, 699-720.
[37] Jones,C., 2001, “A Century of Stock Market Liquidity and Trading Costs,” working paperColumbiaUniversity.
[38] Kahneman,D., and A. Tversky, 1979, “Prospect Theory: An Analysis of Decision Under Risk,”Econometrica, 47,263-291.
[40] Kyle, A. W., andW. Xiong, 2001,“Contagionasa WealthEffect of FinancialIntermediaries,” forthcomingJournal of Finance.
[41] Lamont, O., C. Polk, and J. Saa-Requejo,2001, “Financial Constraintsand Stock Returns,” Review ofFinancial Studies, 14,2, 529-554.
[42] Markowitz, H., 1959,Portfolio Selection. New Haven,YaleUniversityPress.
[43] Newey, W. K., and K. D. West, 1987, “A Simple Positive Semi-Definite, Heteroskedasticity andAutocorrelationConsistentCovarianceMatrix,” Econometrica, 55,703-8.
[44] Pastor, L., and R. F. Stambaugh,2001, “Liquidity Risk and ExpectedStock Returns,” working paper,Wharton.
The table lists the summarystatisticsof the value-weightedqsr and qkË portfolios at a monthly frequency,where qsr and qkË aredefinedin equation(3). For eachmonth,we calculateqsr (q�Ë ) of all stocksbasedon daily continuouslycompoundedreturnsover the pastyear. We rank the stocksinto deciles(1–10),andcalculatethe value-weightedsimplepercentagereturnover the next month. We rebalancethe portfolios ata monthly frequency. Meansandstandarddeviationsare in percentagetermsper month. Std denotesthestandarddeviation (volatility), Auto denotesthefirst autocorrelation,and p is thepost-formationbetaof theportfolio with respectto themarketportfolio. At thebeginningof eachmonthÍ , wecomputeeachportfolio’ssimpleaveragelog market capitalizationin millions (size)andvalue-weightedbook-to-market ratio (B/M).The columnslabeled q r (q Ë ) and p r (p Ë ) show the post-formationdownside(upside)correlationsanddownside(upside)betasof theportfolios.High–Low is themeanreturndifferencebetweenportfolio 10 andportfolio 1 and t-statgives the t-statisticfor this difference. T-statisticsare computedusingNewey-West(1987)heteroskedastic-robuststandarderrorswith 3 lags. T-statisticsthataresignificantat the5% level aredenotedby *. Thesampleperiodis from January1964to December1999.
The table lists the summarystatisticsfor the value-weightedcoskewnessand cokurtosisportfolios at amonthly frequency. For eachmonth,we calculatecoskewnessandcokurtosisof all stocksbasedon dailycontinuouslycompoundedreturnsoverthepastyear. Werankthestocksinto deciles(1–10),andcalculatethevalue-weightedsimplepercentagereturnover thenext month. We rebalancetheportfoliosmonthly. Meansandstandarddeviationsarein percentagetermsper month. Std denotesthe standarddeviation (volatility),Auto denotesthe first autocorrelation,and p is the post-formationbetaof the portfolio with respectto themarket portfolio. Coskew denotesthepost-formationcoskewnessof theportfolio asdefinedin equation(5);cokurt denotesthe post-formationcokurtosisof the portfolio asdefinedin equation(6). High–Low is themeanreturndifferencebetweenportfolio 10 andportfolio 1 and t-stat is the t-statisticfor this difference.T-statisticsarecomputedusingNewey-West(1987)heteroskedastic-robuststandarderrorswith 3 lags. Thesampleperiodis from January1964to December1999.
34
Table3: PortfoliosSortedon PastH , H A andH �
Panel A: Portfolios Sorted on Past pPortfolio Mean Std Auto p High–Low t-stat1 Low p 0.90 3.72 0.13 0.42 0.23 0.702 0.93 3.19 0.20 0.493 1.01 3.33 0.18 0.594 0.95 3.62 0.14 0.705 1.13 3.78 0.08 0.766 1.02 3.84 0.06 0.797 1.00 4.37 0.07 0.938 0.97 4.87 0.07 1.049 1.07 5.80 0.08 1.2310 High p 1.13 7.63 0.05 1.57
PanelA of this tableshowsthetime-seriesregressionof excessreturnÈ É onfactorsûþý�ÿ ,� û�� and � û Ñ .
Theten qsr portfoliosof Table(1) areusedin theregression.Í ï ñ is thet-statisticof theregressioncoefficientcomputedusingNewey-West(1987)heteroskedastic-robuststandarderrorswith 3 lags. The regressionq -is adjustedfor the numberof degreesof freedom. s q
�is the � -statisticof Gibbons,RossandShanken
(1989),testingthehypothesisthat theregressioninterceptarejointly zero. r ï s q � ñ is the r -valueof s q�
.The sampleperiodis from January1964to December1999. PanelB reportsthe ò ’s andt-statisticsin thetime seriesregressionin two subsamples.Column“10–1” is thedifferenceof the ò ’s for the10thdecileandthefirst decile.
Std= 5.82 Std= 5.28 Std= 4.89 t-stat=1.98p = 1.23 p = 1.16 p = 1.09q r = 0.89 q r = 0.94 q r = 0.96
Panel B: p -Balanced qsr Portfolios
Low qkr Medium qsr High qkr High qsr - Low qkrp -balanced Mean=0.86 Mean=0.98 Mean=1.10 Mean= 0.23Std=4.60 Std=4.29 Std=3.88 t-stat= 2.35p =0.94 p =0.93 p =0.87qkr =0.88 qkr =0.92 qkr =0.96
Summarystatisticsfor the portfoliosusedto constructdownsiderisk factor t û t at a monthly frequency.Eachmonth,we rankstocksbasedon their p , calculatedfrom thepreviousyearusingdaily data,into a lowp group anda high p group,eachgroupconsistingof onehalf of all firms. Then, within eachp group,we rank stocksbasedon their qkr , which is alsocalculatedusingdaily dataover the pastyear, into threegroups:a low qsr group,a mediumqkr groupanda high qkr group,with cutoff pointsat 33.3%and66.7%.We computethe monthly value-weightedsimplereturnsfor eachportfolio. The p -balancedgroupsaretheequal-weightedaverageof the portfolios acrossthe two p groups. T-statisticsarecomputedusingNewey-West(1987)heteroskedastic-robuststandarderrorswith 3 lags. Thesampleperiodis from January1964toDecember1999.
This table shows the summarystatisticsof the factors. MKT is the CRSPvalue-weightedreturnsof allstocks.SMB andHML arethesizeandthebook-to-marketfactors(constructedby FamaandFrench(1993)),WML is thereturnon thezero-coststrategy of goinglong pastwinnersandshortingpastlosers(constructedfollowing Carhart(1997)),andSKSis thereturnongoinglongstockswith themostnegativepastcoskewnessand shortingstockswith the most positive pastcoskewness(constructedfollowing Harvey and Siddique(2000)). CMC is thereturnon a portfolio going long stockswith thehighestpastdownsidecorrelationandshortingstockswith thelowestpastdownsidecorrelation.Thetwo columnsshow themeansandthestandarddeviationsof thefactors,expressedasmonthlypercetages.Skew andKurt aretheskewnessandkurtosisoftheportfolio returns.Auto refersto first-orderautocorrelation.Factorswith statisticallysignificantmeansatthe5% (1%) level aredenotedwith * (**), usingheteroskedastic-robustNewey-West(1987)standarderrorswith 3 lags.Thesampleperiodis from January1964to December1999.
This tablelists the optimalGMM estimationresultsof the modelsusing40 momentumportfolios with therisk-freerate.Coefficient ( v ) refersto thefactorcoefficientsin thepricingkernelandPremia( u ) refersto thefactorpremia( u ) in monthlypercentageterms.P-valuesof JandHJ testsareprovidedin [], with p-valuesoflessthan5% (1%) denotedby * (**). TheJ-testis Hansen’s (1982)M - teststatisticson theover-identifyingrestrictionsof themodel. HJ denotesthe Hansen-Jagannathan(1997)distancemeasurewhich is definedinequation(24). Asymptoticandsmall-samplep-valuesof theHJ testareboth0.00for all models.Statisticsthataresignificantat 5% (1%) level aredenotedby * (**). In all models,Wald testsof joint significanceofall premiumsarestatisticallysignificantwith p-valuesof lessthan0.01. Thesampleperiodis from January1964to December1999.
Panel A: Mispricing across Average Liquidity Beta Portfolios
ò ó Ý p Í ï ò ñ Í ï ó ñ Í ï Ý ñ Í ï p ñ q -1 Low p ø -0.17 1.05 0.42 0.19 -1.93 40.39 9.88 3.30 0.912 -0.09 0.91 0.13 0.23 -1.36 49.44 4.30 5.76 0.943 -0.06 0.88 0.09 0.29 -1.15 49.36 3.54 8.19 0.934 -0.03 0.95 0.16 0.21 -0.49 36.88 4.44 4.80 0.935 High p ø -0.08 1.01 0.42 0.11 -0.92 44.22 12.77 2.88 0.91
ò @ - ò × = 0.10t-stat=0.71
Panel B: Mispricing across Average Downside Correlation Portfolios
ò ó Ý p Í ï ò ñ Í ï ó ñ Í ï Ý ñ Í ï p ñ q -1 Low q r -0.18 0.81 0.54 0.45 -2.38 30.89 12.64 12.69 0.862 -0.17 0.91 0.44 0.37 -2.14 36.51 11.32 7.29 0.913 -0.12 0.98 0.26 0.23 -1.67 46.77 8.00 4.90 0.934 -0.05 1.02 0.11 0.10 -0.76 60.21 3.93 2.68 0.965 High q r 0.08 1.07 -0.13 -0.13 1.80 90.83 -7.70 -4.70 0.98
ò @ - ò × = 0.26t-stat=2.62
This table shows the time-seriesregressionof excessreturn È�É on factors ûþý ÿ ,� û�� and � û Ñ . In
eachmonth,we sort all NYSE, AMEX andNASDAQ stocksinto 25 portfolios. We first sort stocksintoquintilesby p ø andsort stocksinto quintilesby qsr , where p ø is computedusingequation(25) usingtheprevious5 yearsof monthlydata.Theintersectionof thesequintilesforms25 portfolioson p ø andqsr . Theaverageliquidity betaportfolios in PanelA arethe liquidity betaquintilesaveragedover the qsr quintiles.Theaverageqsr portfoliosin PanelB arethe qkr quintilesaveragedovertheliquidity betaquintiles.Thetablereportsthe coefficientsfrom a time-seriesregressionof the portfolio returnsonto the Fama-French(1993)factors:È ÉaÔ Øþò É Ú¤ó É ûþý�ÿ Ô Ú¤Ý É � û�� Ô ÚHp É � û Ñ Ô Ú7á ÉaÔ . Í ï ñ is thet-statisticof theregressioncoefficientcomputedusingNewey-West(1987)heteroskedastic-robuststandarderrorswith 3 lags.Theregressionq - isadjustedfor thenumberof degreesof freedom.January1968to December1999. ò @ - ò × is thedifferenceinthealphasò betweenthe5th quintileandthefirst quintile.
42
Table11: MacroeconomicVariablesandv 2 v
Panel A: t û t�Ô Ø¹ò Ú 2 É î × ó#É û N t qxw Ô r É(Ú 2 É î × õ#Éyt û t�Ô r É(Ú7á�Ôû N t qxw Ô rs× û N t qzw Ô r - û N t qxw Ô r 2 JointSig
LEI coef –0.27 0.22 0.06 0.09t-stat –2.27n 1.24 0.60
HELP coef –0.00 –0.04 0.05 0.24t-stat –0.12 –1.26 1.97
IP coef –0.13 0.18 –0.02 0.16t-stat –1.39 1.43 –0.22
TERM coef –0.02 0.02 –0.01 0.03nt-stat –2.21n 1.42 –0.82
This tableshows the resultsof the regressionsbetweenCMC andthe macroeconomicvariables. PanelAlists the resultsfrom the regressionsof t û t on lagged t û t andlaggedmacroeconomicvariables,butreportsonly the coefficientson laggedmacrovariables. PanelB lists the resultsfrom the regressionsofmacrovariableson laggedCMC andlaggedmacroeconomicvariables,but reportsonly the coefficientsonlaggedCMC. LEI is thegrowth rateof theindex of leadingeconomicindicators,HELP is thegrowth rateintheindex of HelpWantedAdvertisingin Newspapers,IP is thegrowth rateof industrialproduction,CPI is thegrowth rateof ConsumerPriceIndex, FEDis thefederaldiscountrateandTERM is theyield spreadbetween10 yearbondand3 monthT-bill. All growth rate(including inflation) arecomputedasthe differencesinlogsof the index at time Í andtime Í ç Î , whereÍ is in months.FED is thefederalfundsrateandTERMis the yield spreadbetweenthe 10 yeargovernmentbondyield andthe 3-monthT-bill yield. All variablesareexpressedaspercentages.T-statisticsarecomputedusingNewey-Westheteroskedastic-robuststandarderrorswith 3 lags,andarelistedbelow eachestimate.JointSig in PanelA denotesto thep-valueof thejointsignificancetestonthecoefficientson laggedmacrovariables.JointSig in PanelB denotesthep-valueof thejoint significanceteston thecoefficientsof laggedCMC. T-statisticsthataresignificantat the5% (1%) levelaredenotedwith * (**). P-valuesof lessthan5% (1%) aredenotedwith * (**). Thesampleperiodis fromJanuary1964to December1999.
43
Figure1: AverageReturn,H , @ A of MomentumPortfolios
2 4 6 8 100
0.5
1
1.5
2J=6 K=3
Decile
Mean β ρ−
2 4 6 8 100
0.5
1
1.5
2J=6 K=6
Decile
Mean β ρ−
2 4 6 8 100
0.5
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2J=6 K=9
Decile
Mean β ρ−
2 4 6 8 100
0.5
1
1.5
2J=6 K=12
Decile
Mean β ρ−
Theseplots show the averagemonthly percentagereturns,p and qsr of the JegadeeshandTitman (1993)momentumportfolios. K refersto formationperiodand ý refersto holding periods. For eachmonth,wesortall NYSE andAMEX stocksinto decileportfoliosbasedon their returnsover thepast K =6 months.Weconsiderholding periodsover the next 3, 6, 9 and12 months. This procedureyields 4 strategiesand40portfoliosin total. Thesampleperiodis from January1964to December1999.
44
Figure2: Loadingsof MomentumPortfoliosonFactors
2 4 6 8 10−1
−0.5
0
0.5
1
1.5J=6 K=3
Load
ings
on
Fact
ors
Decile
MKTSMBHMLCMC
2 4 6 8 10−1
−0.5
0
0.5
1
1.5J=6 K=6
Load
ings
on
Fact
ors
Decile
MKTSMBHMLCMC
2 4 6 8 10−1
−0.5
0
0.5
1
1.5J=6 K=9
Load
ings
on
Fact
ors
Decile
MKTSMBHMLCMC
2 4 6 8 10−1
−0.5
0
0.5
1
1.5J=6 K=12
Load
ings
on
Fact
ors
Decile
MKTSMBHMLCMC
Theseplots show the loadingsof the JegadeeshandTitman (1993)momentumportfolios on MKT, SMB,HML and CMC. Factor loadingsare estimatedin the first stepof the Fama-MacBeth(1973) procedure(equation(11)). { refersto formationperiodand | refersto holdingperiods.For eachmonth,we sortallNYSEandAMEX stocksinto decileportfoliosbasedontheir returnsoverthepast{ =6 months.Weconsiderholdingperiodsover thenext 3, 6, 9 and12 months.This procedureyields4 strategiesand40 portfolios intotal. MKT, SMB andHML areFamaandFrench(1993)’sthreefactorsandCMC is thedownsidecorrelationrisk factor. Thesampleperiodis from January1964to December1999.
Theseplots show the pricing errorsof variousmodelsconsideredin Section4.2. Eachstar in the graphrepresentsoneof the 40 momentumportfolios with {^}�~ or the risk-freeasset.The first ten portfolioscorrespondto the |�}�� monthholdingperiod,thesecondtento the |�}�~ monthholdingperiod,thethirdten to the |�}�� monthholdingperiod,andfinally the fourth ten to the |�}���� holdingperiod. The41stassetis therisk-freeasset.Thegraphsshow theaveragepricingerrorswith asterixes,with two standarderrorbandsin solid lines. Theunitson the � -axisarein percentageterms.Pricingerrorsareestimatedfollowingcomputationof theHansen-Jagannathan(1997)distance.